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The first human electroencephalogram, recorded in 1924 by Hans Berger.

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• Article: What's Your Brain Doing?
• Author(s): Brett Szymik
• Publisher: Arizona State University School of Life Sciences Ask A Biologist
• Site name: ASU - Ask A Biologist
• Date published: May 9, 2011
• Date accessed: April 17, 2024
• Link: https://askabiologist.asu.edu/brain-regions

Brett Szymik. (2011, May 09). What's Your Brain Doing?. ASU - Ask A Biologist. Retrieved April 17, 2024 from https://askabiologist.asu.edu/brain-regions

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Brett Szymik. "What's Your Brain Doing?". ASU - Ask A Biologist. 09 May 2011. ASU - Ask A Biologist, Web. 17 Apr 2024. https://askabiologist.asu.edu/brain-regions

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Original research article, analyzing complex problem solving by dynamic brain networks.

• 1 Department of Computer Engineering, Middle East Technical University, Ankara, Turkey
• 2 Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN, United States

Complex problem solving is a high level cognitive task of the human brain, which has been studied over the last decade. Tower of London (TOL) is a game that has been widely used to study complex problem solving. In this paper, we aim to explore the underlying cognitive network structure among anatomical regions of complex problem solving and its subtasks, namely planning and execution . A new computational model for estimating a brain network at each time instant of fMRI recordings is proposed. The suggested method models the brain network as an Artificial Neural Network, where the weights correspond to the relationships among the brain anatomic regions. The first step of the model is preprocessing that manages to decrease the spatial redundancy while increasing the temporal resolution of the fMRI recordings. Then, dynamic brain networks are estimated using the preprocessed fMRI signal to train the Artificial Neural Network. The properties of the estimated brain networks are studied in order to identify regions of interest, such as hubs and subgroups of densely connected brain regions. The representation power of the suggested brain network is shown by decoding the planning and execution subtasks of complex problem solving. Our findings are consistent with the previous results of experimental psychology. Furthermore, it is observed that there are more hubs during the planning phase compared to the execution phase, and the clusters are more strongly connected during planning compared to execution.

1. Introduction

Complex problem solving is a very crucial ability of the human brain, which covers a large number of high-level cognitive processes, including strategy formation, coordination, sequencing of mental functions, and holding information online. These complex high-level cognitive processes make the inner workings of problem solving a challenging task.

The standard method for neuro-analysis of complex problem solving in the literature is to study the fMRI data recorded while the subjects play the Tower of London (TOL) game, designed by Shallice (1982) . TOL game consists of three bins having different capacities with colored balls placed in the bins; the aim is to rearrange the balls from their initial state to a predetermined goal state while moving one ball at a time and taking into consideration the limited capacity of each bin (as shown in Figure 1 ).

Figure 1 . Example tower of London (TOL) puzzle.

TOL game has been primarily employed to study the effect of various properties of complex problem solving performance in healthy subjects. The predictive power of working memory, inhibition, and fluid intelligence on TOL performance has been investigated with consideration of factors such as age, gender, exercise, etc. ( Unterrainer et al., 2004 , 2005 ; Zook et al., 2004 , 2006 ; Boghi et al., 2006 ; Albert and Steinberg, 2011 ; Chang et al., 2011 ; Desco et al., 2011 ; Kaller et al., 2012 ). Additionally, TOL has been used to investigate the effect of various clinical disorders on functions associated with the prefrontal cortex such as planning. For example, the task has been utilized in neuroimaging studies to identify executive dysfunction by examining differential cognitive activation patterns in people suffering from neurological disorders like epilepsy, seizures, depression, Parkinson's and schizophrenia ( Goethals et al., 2005 ; Rasser et al., 2005 ; Rektorova et al., 2008 ; MacAllister et al., 2012 ).

The classic work of Newell and Simon ( Newell et al., 1957 ; Simon and Newell, 1971 ) hypothesized three distinct phases of complex problem solving: construction of problem representation, elaboration to search for operators to solve the problem, and execution to implement the solution. Despite being a well respected theory, there is little to no evidence from cognitive neuroimaging that supports this hypothesis directly. Consequently, refinements of the theory, such as online planning in which elaboration and execution phases are interspersed, and mechanisms such as schema development that may suggest qualitative differences between good and poor problem-solvers, are understood even less. The primary reason for this state of affairs is that cognitive neuroimaging in general and fMRI analysis, in particular, tends to ignore the temporal aspect of how brain activation and network connectivity evolve during complex cognitive tasks. Further, much of the existing methods have tended to test theoretical cognitive models by searching for brain data that fit those models, rather than using the brain data themselves to inform us about cognition.

Numerous studies have proposed various computational models in order to build brain networks from fMRI measurements, both during cognitive tasks or during resting state. These studies represent a shift in the literature toward brain decoding algorithms that are based on the connectivity patterns in the brain motivated by the findings that these patterns provide more information about cognitive tasks than the isolated behavior of individual groups of voxels or anatomical regions ( Lindquist, 2008 ; Ekman et al., 2012 ; Shirer et al., 2012 ; Richiardi et al., 2013 ; Onal et al., 2017 ). Some of these studies focused on the pairwise relationships between voxels or brain regions. For example, Pearson correlation has been used in order to construct undirected functional connectivity graphs at different frequency resolutions in Richiardi et al. (2011) . Also, pairwise correlations and mutual information have been used in order to build functional brain networks in various studies aiming to investigate the network differences between patients with Schizophrenia or Alzheimer's disease and healthy subjects ( Lynall et al., 2010 ; Menon, 2011 ; Kurmukov et al., 2017 ). Others used partial correlation along with constrained linear regression to generate brain networks in Lee et al. (2011) .

In our previous studies, we take advantage of the locality property of the brain by constructing local mesh networks around each brain region. Then, we represent the entire brain network as an ensemble of local meshes. In these studies, we estimated the Blood-Oxygenation Level Dependent (BOLD) response of each brain region as a linear combination of the responses of its "closest" neighboring regions. Then, we solved the systems of linear equations using various regression techniques. Our team applied Levinson-Durbin recursion in order to estimate the edge weights of each local star mesh, where the nodes are the neighboring regions of the seed brain region ( Fırat et al., 2013 ; Alchihabi et al., 2018 ). We also used ridge regression to estimate edge weights while constructing the local mesh networks across windows of time series of fMRI recordings ( Onal et al., 2015 , 2017 ).

In this study, we present a novel approach for estimating dynamic brain networks, which represent the relationship among the brain anatomic regions at each time instant of the fMRI recordings. The approach models the relationship among the anatomical brain regions as an Artificial Neural Network (ANN), where the edge weights correspond to the arc weights of the brain network. The idea of modeling the brain network as an ANN is first introduced in our lab ( Kivilcim et al., 2018 ), where the model can be constructed to estimate both directed and undirected brain graphs. In this study, we further extend this idea to estimate dynamic brain networks. We also explore the validity and representation power of the suggested brain network by analyzing its statistical properties using the methods suggested in Bassett and Bullmore (2006) , Power et al. (2010) , Rubinov and Sporns (2010) , and Park and Friston (2013) .

Several network measures, such as measures of centrality, which identify potential hubs and measures of functional segregation, which detect densely interconnected clusters of nodes, provide means to analyze both individual components of brain network and the brain network as a whole. As a result, these network measures reveal and characterize various aspects of inter-dynamics of brain regions enabling us to analyze and compare different brain network snapshots. The properties of the dynamic brain networks are studied in order to identify the active anatomical regions during both planning and execution phases of complex problem solving. Potential hubs and clusters of densely connected brain regions are identified for both subtasks. Furthermore, the distinctions and similarities between planning and execution networks are highlighted. The results identify both active and inactive hub regions as well as clusters of densely connected anatomical regions during complex problem-solving. In addition, results show that there are more potential hubs during the planning phase compared to the execution phase. Also, the clusters of densely interconnected regions are significantly more strongly connected during planning compared to execution. Finally, we studied the decoding power of the suggested brain network model by using simple machine learning methods to classify two phases of complex problem solving, namely, planning and execution.

2. TOL Experiment Procedure

In this section, we introduce the details of the experiment as well as data collection and preprocessing methods.

2.1. Participants and Stimuli

18 college students aged between 19 and 38 participated in the experiment after signing informed, written consent documents approved by the Indiana University Institutional Review Board. The subjects solved a computerized version of TOL problem; two configurations were presented at the beginning of each puzzle: the initial state and the goal state. The subjects were asked to transform the initial state into the goal state using the minimum number of moves. However, the subjects were not informed of the minimum number of moves needed to solve a given puzzle nor of the existence of multiple solution paths.

2.2. Procedure

Each subject underwent a practice session before entering the scanning session to acquaint subjects with the TOL problem. The subjects were given the following instructions: "You will be asked to solve a series of puzzles. The goal of the puzzle is to make the 'start' or 'current' state match the 'goal' state (They were shown an example). Try to solve the problems in the minimum number of moves by planning ahead. Work as quickly and accurately as possible, but accuracy is more important than speed."

The scanning session consisted of 4 runs, each run included 18 timed puzzles, with a 5-s planning only time slot during which subjects were not allowed to move the balls. However, they were allowed to continue planning after the 5 s planning only time slot if they chose to do so. Following every puzzle, there was a 12-s rest period where subjects focused on a plus sign in the center of the screen. Each run was also followed by a 28-s fixation period.

The planning task is defined from the start of the puzzle until the subject's first move. The execution task is defined from the subject's first move until the end of the puzzle. During the experiments, both planning and execution times change across the subjects, runs and puzzles. The average planning time instances per puzzle is 5.91 and the average time instances for execution per puzzle is 5.63 over all puzzles and subjects. The average total planning time instances per run is 106.35 and the average total execution time instances per run is 101.28 over all subjects. Figure 2 shows the total number of planning and execution instances for each run of all subjects. Each mark on the horizontal axis refers to a given run of a specific subject. As it can be observed from Figure 2 , the data is quite balanced between the planning and execution phases (average of 101 planning and 106 execution across the subjects per each run). For this reason, we did not augment the classes or eliminate some samples in the dataset for balancing the classes. The details of the dataset are summarized in Table 1 .

Figure 2 . The number of planning and execution instances for each run of all subjects. Horizontal axis indicates the subject ID for each run (there are total of 18 subject x4 runs = 72 runs), whereas the vertical axis shows the number of time instances for planning (blue) and execution (orange) per run.

Table 1 . Summary of TOL dataset.

2.3. fMRI Data Acquisition and Preliminary Analysis

The fMRI images were collected using a 3 T Siemens TRIO scanner with an 8-channel radio frequency coil located in the Imaging Research Facility at Indiana University. The images were acquired in 18 5 mm thick oblique axial slices using the following set of parameters: TR = 1,000 ms, TE = 25 ms, flip angle = 60°, voxel size = 3.125 mm × 3.125 mm × 5 mm with a 1 mm gap.

The statistical parametric mapping toolbox was used to perform the preliminary data analysis that included: image correction for slice acquisition timing, resampling, spatial smoothing, motion correction and normalization to the Montreal Neurological Institute (MNI) EPI template. Further details concerning the procedure and data acquisition can be found in Newman et al. (2009) as we use the same data/participants in this study. It is also worth noting that we perform our analysis on all recorded puzzles, not only correctly solved ones, given that the aim of this study is to investigate the planning and execution networks in general. In future work, we aim to study the differences in the planning and execution networks between good problem-solvers and bad problem-solvers; in that case, we will make the distinction between correctly solved puzzles and unsolved puzzles. Furthermore, the entirety of our analysis is performed on the raw fMRI recordings; no first-level modeling or regressors are applied; rather, we use the recorded time series as our raw BOLD response.

In order to investigate the inter-subject variability, we estimate the mean values and variance of BOLD activation of each brain anatomic region across all subjects. Figure 3 clearly shows the relatively low variations of the BOLD activation around the mean values of brain anatomic regions across 18 subjects.

Figure 3 . Estimated mean values and variations of BOLD activation of each brain anatomic region, over 18 subjects. Horizontal axis shows the index of anatomic regions, whereas the vertical axis shows the mean values (blue bars) and variances (black bars) of measured BOLD activation.

3. Modeling Dynamic Brain Network as an Artificial Neural Network

Can we model the relationship among the anatomic regions as an Artificial Neural Network? If so, what is the validity and representation power of this network to analyze cognitive tasks such as complex problem solving? In this section, we suggest a computational model to represent the complex problem solving task as a dynamic brain network. In the next section, we shall explore the validity of this network and try to analyze the complex problem solving task of the human brain.

3.1. Preprocessing of the fMRI Recordings

In order to be able to estimate a dynamic brain network among the anatomic regions, we need to process the raw fMRI data for

• representation of anatomic regions,

• interpolation in time,

• injecting additive noise,

as explained below.

3.1.1. Representation of Anatomic Regions

Each anatomic region is represented by a time series using voxel selection and averaging methods. Voxel selection reduces the dimension of fMRI data (185,000 voxels per brain volume) and eliminates the irrelevant voxels that do not contribute to the underlying cognitive process. ANOVA method is used to choose the most discriminative voxels and to discard the remaining ones ( Cox and Savoy, 2003 ; Pereira et al., 2009 ; Afrasiyabi et al., 2016 ). The f -value score of each voxel v i is calculated from Equation (1):

where y label is the label indicating the subtask (Planning or Execution). MSB ( v i , y label ) is the mean square value between raw measured BOLD response of voxel i and the label vector y label , which is calculated by Equation (2):

SSB ( v i , y label ) is the sum of squares between y label and v i , df between is the number of groups minus one. MSW ( v i , y label ) is the mean square value within voxel i and the label vector y label and it is calculated by Equation (3):

where SSW ( v i , y label ) is the sum of squares within group and df within is the degree of freedom within (total number of elements in v i and y label minus the number of groups).

We order the voxels according to their f -value scores. Then, the distribution of f -value scores of all voxels is plotted in order to determine the number of voxels to retain. Voxel selection is applied to the voxels of all brain regions except the ones located in the cerebellum, which we exclude during network extraction.

Each anatomic region is represented by averaging the BOLD response of the selected voxels, which resides in that region defined by automated anatomical labeling (AAL) atlas ( Tzourio-Mazoyer et al., 2002 ) as shown in Equation (4):

where r j is the representative BOLD response of region j , v i is the raw measured BOLD response of voxel i and ζ[ j ] is the set of selected voxels located in region j . The representative BOLD responses, r j , enables us to investigate the role and contribution of each region to the planning and execution phases of the problem solving task.

3.1.2. Interpolation

It is well-known that despite its high spatial resolution, fMRI signal has very low temporal resolution compared to EEG signal. In this study, we interpolate the fMRI signal in order to compensate for this drawback and study the effect of interpolation for estimating the brain networks and on decoding the planning and execution phases of TOL game.

In the TOL study, subjects solved a puzzle in at most 15 s and the sampling rate, TR, is 1,000 ms. Interpolation is used to increase the temporal resolution by estimating z extra brain volumes between each two consecutive measured brain volumes. As a result, the total number of available brain volumes for each puzzle becomes n + z * ( n − 1), where n is the number of measured brain volumes of a given puzzle. We use the cubic spline interpolation function rather than linear interpolation methods in order to prevent edge effects and smoothing out the spikes between the measured brain volumes ( McKinley and Levine, 1998 ).

In order to analyze the effect of time interpolation and to estimate an acceptable number of inserted brain volumes z , we compare the Fourier Transform of the fMRI signal computed before and after interpolation so that the frequency content of the signal is not distorted by interpolation. The original single-sided amplitude of the signal and the one obtained after interpolation are compared in order to ensure that interpolation is preserving the smooth peaks of the data in the frequency domain ( Cochran et al., 1967 ; Frigo and Johnson, 1998 ).

3.1.3. Injecting Gaussian Noise

When modeling a deterministic signal by a probabilistic method, adding noise to the signal decreases the estimation error in most of the practical applications. The final phase of preprocessing is adding Gaussian noise to the interpolated time series of the BOLD response in each anatomical region. For this purpose, instead of just injecting white noise, a rather informed noise, colorful Gaussian noise , is added. In order to reflect the corresponding brain region's properties, for each sample, the additive noise sample is generated from a Gaussian distribution having mean and variance of that anatomical region. These newly generated samples not only act like a natural regularizer to improve the generalization performance of brain decoding but also help making the Artificial Neural Network more stable when estimating the edge weights of the brain networks ( Matsuoka, 1992 ; Reed et al., 1992 ).

Given a representative time series from a particular brain region, i represents the index of an anatomical region. The new samples are generated with vector addition of noise while preserving the signal-to-noise ratio (SNR) as in r ~ j = r j + τ j , where τ j is a noise vector sampled from N ( α n o i s e μ ( r j ) , β n o i s e σ 2 ( r j ) ) , α noise and β noise are the scaling factors which are set empirically, to optimize the decoding performance.

3.2. Building Dynamic Brain Networks With Artificial Neural Networks

The above preprocessing methods yield a relatively high temporal resolution and smooth time series for each anatomic region compared to the row fMRI recordings.

In this section, we use the output of the preseprocessing step to estimate the relationship among the time series of anatomic regions at each time instance to generate a dynamic brain network, where the arc weights vary with respect to time instances.

3.2.1. Partitioning the Time Series Into Fixed Size Internals and Defining the Brain Network for Each Window

As the first step, we partition each time series, which represents an anatomical region, into fixed-size windows. Each window, win ( t ), is centered at the measured brain volume at time instance, t . The size of each window is Win _ Size = z + 1 brain volumes, where z is the number of interpolated brain volumes in each window. Equation (5) shows the time instances included in each window.

We define a dynamic brain network, N ( t ) = ( V, W ( t )), for each time window win ( t ), where V is the set of nodes of the graph corresponding to the brain anatomical regions and W ( t ) = { w t,j,i |∀ i, j ∈ V } is the directed weighted edges between the nodes of the graph within time window win ( t ). The nodes of the graph represent the AAL-defined brain regions ( Tzourio-Mazoyer et al., 2002 ), except for the regions located in the cerebellum. The nodes are then pruned using voxel selection, as some anatomical regions contribute no voxels at all and get deleted from the set of nodes of the graph V .

Note that our aim is to label the BOLD responses measured at each brain volume as it belongs to one of the two phases of complex problem solving, namely, planning and execution . For this purpose, we represent each brain volume measured at a time instant t by a network, which shows the relationship among the anatomical regions. This dynamic network representation will allow us to investigate the network properties of planning and execution subtasks.

Note also that the nodes, V , of the network are fixed to the active anatomic regions, and our goal is only to estimate the weights of the edges, W ( t ), of the brain network, N ( t ), for each time instance, t For this purpose we adopt the method suggested by our team in Kivilcim et al. (2018) .

3.2.2. Forming Local Meshes

It is well-known that the human brain operates with two contradicting principles, namely locality and centrality . Our suggested network model incorporates these two principals by defining a set of spatially local meshes then ensembling the local meshes to form the brain network. This representation not only avoids to define fully connected brain networks by omitting the connectivity among irrelevant brain regions but also reduces the computational complexity.

In order to define local meshes, for each window win ( t ), we define the functional neighborhood matrix, Ω t , for each time instant t . The entries of Ω t are binary, either 1 or 0, indicating if there is a connection between two regions or not. The size of the matrix is M × M , where M is the number of brain anatomical regions. The functional neighborhood matrix contains no self-connections, thus, Ω t ( i, i ) = 0∀ i ∈ [1, M ]. Recall that the brain regions are pruned by voxel selection. Thus, the regions which do not contain any voxels have no in/out connections, and the corresponding entries in Ω t are all zero.

The connectivity of each region to the rest of the regions is determined by using Pearson correlation, as follows: first, for every region i , we measure the Pearson correlation between its BOLD response r i,t and the BOLD responses of all the other remaining regions as shown below:

where r i,t is the BOLD response of region i across time window win ( t ), cov ( r i,t , r j,t ) is the covariance between the corresponding BOLD responses of regions i and j . σ is the standard deviation of the BOLD response of a given region. Thus, the higher the Pearson correlation between two regions the closer they are to each other in the functional neighborhood system.

Then, we select p of the regions with the highest correlation scores with region i . Thus, a local mesh for each anatomic region i is formed by obtaining the neighborhood set η p [ i ], which contains the p closest brain regions to region i . The degree of neighborhood, p , is determined empirically as will be explained in the next section. Finally, we define the Ω t ( i, j ) as the connectivity between the regions i and j , using the constructed neighborhood sets as follows:

Note that each anatomical region is connected to its p closest functional neighbors. This approach forms a star mesh around each anatomical region.

The ensemble of all of the local meshes creates a brain network at each time instance. Note, also, that Pearson correlation values are not used as the weights between two regions. They are just used to identify the nodes of each local mesh formed around an anatomical region. The estimated brain network becomes sparser as p gets smaller. When p is set to the number of anatomic regions, M , the network becomes fully connected. This approach of defining the connectivity matrix makes the network representation sparse for small p -values and constructs a network that is connected in functionally closest regions, satisfying the locality property of the human brain.

3.2.3. Estimating the Edge Weights of the Brain Network

After having determined the edges of the brain graph using the functional neighborhood matrix Ω t , all that is left is to estimate the weights of these edges at each local mesh. At this point, we could use the Pearson correlation values as edge weights between two anatomic regions. However, Pearson values are restricted to measure the connectivity among the pairs only. A better approach is to consider the multiple relationships among an anatomic region and all of its neighbors in the local mesh. In order to estimate the edge weights in a mesh all at once, we represent the time series of each region i ( r i,t ) as a linear combination of its closest p -functional neighbors as shown in Equation (8):

In Equation (8), r ^ i , t is the representative time series of of region i within the time window win ( t ), w t,j,i is the estimated edge weight between node (region) i and node j at time instance t . η p [ i ] is the p -closest functional neighbors of region i .

Ertugrul et al. (2016) showed that representing the time series of an anatomic region as a linear combination of its closest neighbors provides better performance compared to using pairwise Pearson correlation in brain decoding. They estimated the arc-weights for each mesh formed around region i for each time window win ( t ) by minimizing the mean-squared error loss function using Ridge regression. In this approach, the mean-squared error loss function is minimized with respect to w t,j,i , for each mesh, independent of the other meshes, where the expectation is taken over the time-instances, in window win ( t ) as shown in Equation (9).

where λ is the L2 regularization parameter whose value is optimized using cross-validation. L2 regularization is used in order to improve the generalization of the constructed mesh networks. Note that the estimated arc-weights, w t,j,i ≠ w t,i,j . Therefore, the ensemble of meshes yields a directed brain network.

In this study, we define an Artificial Neural Network to estimate the values of mesh arc-weights for all anatomical regions jointly in each time window, as proposed in Kivilcim et al. (2018) . In this method, we estimate the mesh arc-weights matrix W ( t ) = { w t,j,i | j,i ∈ V } using a feed-forward neural network. The architecture of this network consists of an input layer and an output layer, both containing M nodes corresponding to each anatomic region. The edges of the feed-forward neural network are constructed using the neighborhood matrix Ω t . There is an edge between node i of the output layer and node j from the input layer, if Ω t ( i,j ) = 1.

The loss function of the suggested Artificial Neural Network is given in Equation (10), where W is the weight matrix of the entire neural network that corresponds to directed edge weights of the brain graph and W i is the row of matrix W corresponding to region i :

We train the aforementioned Artificial Neural Network in order to obtain the weights of the brain network at each time instance t that minimize the loss function by applying a gradient descent optimization method as shown in Equation (11),

where w t , j , i ( κ ) is the weight of the edge from node j to node i at epoch (iteration) κ, α learning is the learning rate. The number of epochs and learning rate used to train the network are optimized empirically using cross-validation.

Finally, the weights of the above artificial neural network, computed for each win ( t ), correspond to the edge weights of the dynamic brain network, N ( t ) = ( V, W ( t )), at each time instant t . Thus, we refer to the brain networks using their window indices in order to obtain a set of dynamic brain networks T = { N (1), N (2), … N ( tot _ win )}, where N ( t ) is the brain network for time window win ( t ) and tot _ win is the total number of time windows.

3.3. Network Metrics for Analyzing Brain Networks

In this section, we introduce some measures which we will use to investigate the network properties of each phase of the complex problem solving task, namely, planning and execution, using the estimated dynamic brain functional networks. The connectivity patterns of anatomical regions are analyzed by the set of network measures given below. Two separate sets of measures are used, namely, measures of centrality and segregation. Since our estimated brain networks are directed, we distinguish the incoming and outgoing edges in the network while defining the measures.

Recall that the suggested brain network N ( t ) = ( V, W ( t )) consists of a set of nodes, V , each of which corresponding to one of the M anatomical regions. W ( t ) is the dynamic edge weight matrix with the entries, w i,j , representing the weight of the edge from node i to node j . For the sake of simplicity, we omit the time dependency parameter t , since we compute the network properties at each time instant.

3.3.1. Measures of Centrality

Measures of centrality aim to identify brain regions that play a central role in the flow of information in the brain network or nodes that can be identified as hubs. It is commonly measured using node degree, node strength and node betweenness centrality, which are defined below.

3.3.1.1. Node Degree

The degree of a node is the total number of its edges as shown in Equation (12), where degree i is the degree of node i , V is the set of all nodes in the graph and a i,j is the edge between node i and node j .

where a i,j takes value 0 if ( w i,j == 0) and takes value 1 otherwise.

In the case of a directed graph, we distinguish two different metrics: node in-degree d e g r e e i i n and node out-degree d e g r e e i o u t metrics which are shown in Equations (13) and (14), respectively, where a j,i = 1, if there is a directed edge from node j to node i .

Node degree is a measure of centrality of the given nodes, where it aims to quantify the hub brain regions interacting with a large number of brain regions. Thus, a node with high degree indicates its central role in the network.

3.3.1.2. Node Strength

Node strength is the sum of the weights of edges connected to a given node (Equation 15), where w i,j is the weight of the edge between node i and node j .

Similar to node degree, node strength, also, distinguishes two metrics in the case of directed graphs, namely, node in-strength s t r e n g t h i i n and out-strength s t r e n g t h i o u t shown in Equations (16) and (17), respectively, where w j,i is the weight of the edge from node j to node i .

Node strength is a node centrality measure that is similar to node degree, which is used in the case of weighted graphs. Nodes with large strength values are tightly connected to other nodes in the network forming hub nodes.

3.3.1.3. Node Betweenness Centrality

Betweenness centrality of node i is the fraction of the shortest paths in the network that pass through node i as shown in Equation (18)

where ρ j,k is the number of shortest paths between nodes j and k , ρ j , k i is the number of shortest paths between nodes j and k that pass through node i , nodes i , j and k are distinct nodes.

Before measuring the betweenness centrality of a node, we need to change our perspective from connection weight matrix to connection length matrix since betweenness centrality is a distance-based metric. In connection weights matrix, larger weights imply higher correlation and shorter distance while it is the opposite in the case of length matrix. Connection length matrix is obtained by inverting the weights of the connection weight matrix. Then, the algorithm suggested in Brandes (2001) is employed in order to calculate the node betweenness centrality for each anatomical region.

Nodes with high betweenness centrality are expected to participate in many of the shortest paths of the networks. Thus, taking a crucial role in the information flow of the network.

3.3.2. Measures of Segregation

Measures of segregation aim to quantify the existence of subgroups within brain networks, where the nodes are densely interconnected. These subgroups are commonly referred to as clusters or modules. The existence of such clusters in functional brain networks is a sign of interdependence among the nodes forming the cluster. Measures of segregation include clustering coefficient, transitivity and local efficiency. While global efficiency is a measure of functional integration representing how easy it is for information to flow in the network.

3.3.2.1. Clustering Coefficient

The clustering coefficient of a node i is the fraction of triangles around node i which is calculated by Equation (19) as proposed in Fagiolo (2007) . It is defined as the fraction of the neighbors of node i that are also neighbors of each other.

where d i i n is the in-degree of node i and d i o u t is the out-degree of node i . χ i is the weighted geometric mean of triangles around node i that is calculated by Equation (20). Recall that a j,i = 1, if there is a directed edge from node j to node i and a j,i = 0, otherwise.

The clustering coefficient of a node is the fraction of triangles around the node. It is defined as the fraction of the neighbors of the node that are also the neighbors of each other.

3.3.2.2. Transitivity

Transitivity of a node is similar to its clustering coefficient. However, transitivity is normalized over all nodes, while cluster coefficient for each node is normalized independently, which makes clustering coefficient biased toward nodes with low degree. Transitivity can be expressed as the ratio of triangles to triplets in the network. It is calculated by Equation (21), as suggested in Fagiolo (2007) :

where d j i n is the in-degree of node j and d j o u t is the out-degree of node j . χ i is the weighted geometric mean of triangles around node i that is calculated by Equation (20). Note that a h,j a j,h = 1, if there exits an edge in both directions.

3.3.2.3. Global and Local Efficiency

The global efficiency of a brain network is a measure of its functional integration. It measures the degree of communication among the anatomical regions. Thus, it is closely related to the small-world property of a network. Formally speaking, global efficiency is defined as the average of the inverse shortest path lengths between all pairs of nodes in the brain network. Equation (22) shows how to calculate the global efficiency of a brain network, where ϱ i , j w is the weighted shortest path length between two distinct nodes i and j ( Rubinov and Sporns, 2010 ).

On the other hand, the local efficiency of a network is defined as the global efficiency calculated over the neighborhood of a single node. The local efficiency is, thus, a measure of segregation rather than functional integration as it is closely related to clustering coefficient. While global efficiency is calculated for the entire network, local efficiency is calculated for each node in the network ( Rubinov and Sporns, 2010 ).

4. Experiments and Results

In this section, we explore the validity of the suggested dynamic brain network model and study the network properties of complex problem solving task on TOL dataset. First, we analyze the effect of the preprocessing step on the brain decoding performance of planning and execution phases of complex problem solving. Then, we investigate the validity of the dynamic functional brain network model proposed in this study. Finally, we analyze the network properties of the constructed functional brain networks for planning and execution subtasks.

4.1. Voxel Selection

First, we discarded all of the voxels located in the cerebellum anatomical regions. Then, we calculated the f -score for each one of the remaining voxels and order the obtained f -scores of the voxels. Following that, we plotted the ordered f -scores of the voxels in order to determine the appropriate number of voxels to retain. Figure 4 shows the ordered f -scores of the voxels averaged across all subjects. It can be observed from this figure that a relatively small number of voxels is crucial for discriminating the subtasks of problem solving while the remaining voxels do not have significant information concerning the subtasks of problem solving. Based on the f -score distribution shown in Figure 4 , we kept the 10,000 voxels with the highest f -scores observing the elbow point, whereas we discarded the remaining ones.

Figure 4 . Ordered f -scores of voxels for all subjects.

After selecting 10,000 voxels with the highest f -scores of each run, we calculated the number of selected voxels contained in each one of the 90 anatomical regions. We also calculated the percentage of selected voxels to the total number of voxels located in each anatomical region. These values of selected voxels can be considered as measures of participation of an anatomic region into the complex problem solving task. The Figure 5A shows the average number of voxels contributed by each region across all subjects with its corresponding standard deviation, Figure 5B shows the average percentage of voxels contributed by each region across all subjects with its corresponding standard deviation.

Figure 5 . Distribution of selected voxels across anatomical regions, measured by number of selected voxels (top) and percentage of selected voxels (bottom) from each anatomical region. A large value in both figures is an indication of relatively high activity in a particular anatomic region. (A) Average number of voxels selected from each anatomical region across all subjects. (B) Average percentage of voxels selected from each anatomical region across all subjects.

It is clear from these figures that a large number of regions contribute little to no voxels, such as the amygdala, caudate, heschl gyrus, hippocampus, pallidum, putamen, temporal pole, superior temporal cortex, thalamus and parahippocampus. A small number of regions contribute a significantly large number of voxels (over 300 voxels each) during complex problem solving, such as occipital, precentral, precuneus and parietal regions.

Furthermore, Figure 5B ensures that there is no bias against tiny anatomical regions with small number of voxels by normalizing the number of voxels selected from each region by its total number of voxels. Figure 5B clearly shows that in the left prefrontal and inferior occipital regions a significant percentage of voxels are active during complex problem solving. Both figures also show high standard deviations across subjects, which indicates high inter-subject variability.

4.2. Interpolation

After selecting the most discriminative voxels and averaging their BOLD responses with respect to their corresponding brain anatomical regions, we employed temporal interpolation to each representative time series to increase the temporal resolution of the TOL dataset. As a result, the total number of obtained brain volumes is equal to n + z * ( n − 1) where n is the number of measured brain volumes of a given puzzle and z is the number of estimated brain volumes plugged between each pair of measured brain volumes. The optimal value of z is equal to 8, which is determined empirically using cross-validation to maximize the brain decoding performance. Figure 6 shows the interpolated BOLD response of a randomly selected anatomical region from the given subjects, where the blue dots represent the measured BOLD response of the region and the orange dashes are the interpolated values. It is clear from Figure 6 that the interpolated points using cubic spline function do not introduce sharp edges, nor do they smooth out the spikes between measured brain volumes.

Figure 6 . Interpolated BOLD response of a randomly selected anatomic region.

Furthermore, Figure 7 shows the single-sided amplitude spectrum of a randomly selected anatomical region from a given subject before interpolation, after interpolation and finally, after adding Gaussian noise. The figure clearly demonstrates that both interpolation and injecting Gaussian noise preserve the envelope of the signal in the frequency domain.

Figure 7 . Single-Sided amplitude spectrum of a time series of an randomly selected anatomic region.

4.3. Gaussian Noise

In order to control the signal-to-noise ratio (SNR), we used cross-validation to choose the optimal pair of values for α noise and β noise , the ratios of mean and standard deviation of the added noise, respectively. As a result, the optimal values obtained are α noise = 0.025 and β noise = 0.075 from the following set of values α noise , β noise ∈ [0.025, 0.05, 0.075, 0.1].

4.4. Brain Decoding With Preprocessed fMRI Data

We use brain decoding in order to investigate the validity of our proposed preprocessing steps on the TOL dataset. We aim to distinguish the two phases of complex problem solving, namely: planning and execution. At first, we used ANOVA to select the 10,000 voxels with the highest f -scores. Then we averaged the selected voxels into their corresponding anatomical regions defined by Tzourio-Mazoyer et al. (2002) . Following that, we employed temporal interpolation to increase the temporal resolution of each puzzle by estimating z = 8 brain volumes between each pair of measured brain volumes. Finally, we added Gaussian noise in order to regularize the BOLD responses of each region to improve the generalization performance of the classifiers. We used k -fold Cross validation for each subject in all of the experiments introduced in this section, with k = 8. After we obtained the results, we averaged them across the different folds, then we calculated the average and standard deviation across all subjects. We used both supervised and unsupervised brain decoding methods. A linear support-vector machine (SVM) ( Fan et al., 2008 ) was used for supervised brain decoding while k -means clustering was used for unsupervised brain decoding. The input to the decoders is formed by concatenating the values of representative time series computed per each time instant, across the anatomic regions. Considering the fact that there is a total of 90 anatomic regions, the dimension of the input vectors is 90. If there are no selected voxels in an anatomic region after the voxel selection process, the corresponding entry of the input vector becomes 0.

Table 2 shows the effect of our preprocessing pipeline on the brain decoding of complex problem solving subtasks. The first row shows the performances of brain decoding on the raw dataset without any preprocessing, simply averaging all of the voxels into their corresponding anatomical regions. While the second row shows the results of applying voxel selection then averaging the selected voxels into their anatomical regions. The third row shows the results of brain decoding after applying temporal interpolation, while the forth row shows the results after injecting the data with Gaussian noise.

Table 2 . Decodinge performances of preprocessing pipeline after each step.

From the results of the preprocessing experiments, it is observed that voxel selection improves the brain decoding performance for both supervised and unsupervised methods from 60 to 74% and from 63 to 85%, respectively. This can be attributed to the fact that voxel selection retains only the most discriminative voxels and trashes the remaining non informative ones. In addition, voxel selection manages to sparsify the representation of the data since some brain regions contribute no voxels at all; thus have no contribution to brain decoding.

The table also shows that temporal interpolation further improves the supervised brain decoding performance from 74 to 81%; this significant increase is due to increasing the number of brain volumes, thus, increasing the number of training samples for the SVM classifier. However, temporal interpolation slightly reduces the performance of unsupervised methods from 85 to 84%. This result can be partially attributed to the fact that the additional brain volumes smooth the mixture distribution, thus reducing the distinction between the two phases of problem solving, planning and execution. This is due to the method used to label the estimated brain volumes, where each estimated brain volume is given the labels of its closest neighboring measured brain volume.

Finally, the addition of Gaussian noise slightly boosts the performance of both supervised and unsupervised methods from 81 to 82% and from 84 to 85%, respectively. The table also shows high standard deviation across subjects, which is consistent with voxel selection plots, revealing high inter-subject variability.

4.5. Brain Decoding With Dynamic Brain Networks

In this section, we compare our model for building dynamic functional brain networks with some of the popular network methods proposed in the literature in terms of their brain decoding performance. Brain decoding performance can be considered as a measure of validity of the proposed brain networks. High decoding performance indicates that the constructed brain network has a good representation power of the underlying cognitive subtasks, namely planning and execution.

For this purpose, we built the dynamic brain networks, as explained in the previous sections, after having successfully applied the preprocessing pipeline. It is important to remark that each time instance has either a planning label or an execution label. While constructing the brain networks, we define a feature vector for each time instance by using the interpolated time instances (4 extra instances at each side of a measured instance). Thus, for each measured time sample, we form 4+4+1= 9 brain volumes to estimate the brain network weights. These weights represent a network among 90 anatomic regions for each measured time instance.

The optimal values for learning rate α learning and the number of epochs were chosen empirically using cross-validation, obtaining the following values, respectively 1 * 10 −8 and 10. As for p , the number of neighbors used to represent each anatomical region; we chose p equal to the total number of regions, which is 90. In this way, a fully-connected brain network is obtained at each time window. However, the total number of nodes is less than 90, given that some regions have flat BOLD responses; therefore, they were pruned along with all their edges from the brain network.

We also constructed brain networks using Pearson correlation and ridge regression as proposed in Richiardi et al. (2011) , Onal et al. (2015) , Ertugrul et al. (2016) , and Onal et al. (2017) , respectively, in order to compare the performance of our methods with other works in the literature. In the case of Pearson correlation, the functional brain networks were constructed using Pearson correlation scores between each pair of brain regions ( Richiardi et al., 2011 ; Ertugrul et al., 2016 ). As for the case of ridge regression, the mesh arc-weight descriptors were estimated using ridge regression in order to represent each region as a linear combination of its neighbors ( Onal et al., 2015 , 2017 ).

Since our goal is to represent the fMRI data by an informative dynamic network structure, we used generic classification/clustering methods with relatively small learning capacity in order to highlight the representative power of the constructed brain networks. For this reason, we used simple classifiers/clustering methods, such as SVM and K-means. It would be possible to improve the brain decoding performances by using methods with higher learning capacity, such as Multi-Layer Perceptrons. In this case, the dynamic network representation of the Artificial Neural Networks is expected to obtain much better performances compared to the ones reported in the paper. However, the reported performances are sufficient to show that the decoding performance of the dynamic network, which is 90 x 90 = 8,100 edges of the brain network, is compatible with the fMRI data, although it is much more compact and more informative than the raw data (185,000 voxels per brain volume).

The main advantage of representing the fMRI data by dynamic brain networks is that they are neuroscientifically interpretable and much more comprehensive compared to the voxel based representations. The constructed dynamic brain networks allow us to investigate a large variety of network properties in order to identify regions of interest, such as hubs and subgroups of densely connected brain regions with the aim of deriving neuro-scientifically valid insights into the planning and execution phases of the complex problem solving task.

Table 3 shows the brain decoding results of the aforementioned brain network construction methods compared against the results of multi-voxel pattern analysis (MVPA), which feeds the preprocessed BOLD response representing a time instant of all brain regions into a classifier. The first row shows the brain decoding results of preprocessed fMRI data, where there is no network representation at all. This data is obtained by applying voxel selection, interpolation then noise addition to the raw fMRI data. The first row of Table 3 is the same as the last row of Table 2 .

Table 3 . Braine decoding performances of proposed dynamic brain network model compared to the state of the art models, namely, pearson correlation and ridge regression.

The second and third rows show the brain decoding performances of the networks extracted using Pearson correlation and Ridge regression methods, respectively. The Pearson Correlation data is generated using the preprocessed fMRI data, where Pearson correlation is used to generate the brain networks. The weights of the edges in the constructed brain networks are the Pearson correlation scores between the preprocessed BOLD responses of the corresponding anatomic regions (as shown in Equation 6). The Ridge Regression data is generated using the preprocessed fMRI data, where Ridge Regression is used to estimate the weights of the edges in the constructed brain networks. The weights of the edges in the constructed brain networks are estimated using Ridge regression by minimizing the cost function of Equation (9).

The last row shows the brain decoding performances of our proposed Dynamic Brain Network model.

The edge weights of the Dynamic Brain Network is computed by training an Artificial Neural Network with the preprocessed fMRI data. The nodes in the dynamic brain networks represent the brain anatomic region. The edge links of the brain networks are determined by using Equation (7). The edge weights are estimated using Artificial Neural Network as shown in Equations (10) and (11).

The inputs to SVM and K-Means in the case of Pearson Correlation, Ridge Regression and Artificial Neural Networks are the estimated weights of the brain networks. A feature vector of edge weights, with 90 x 90 = 8,100 dimension, is defined at each recorded time instance of fMRI data, as a single training/testing sample. Each feature vector has its corresponding class label, as Planning or Execution.

Table 3 clearly shows that both Pearson correlation and Ridge regression fail to construct valid brain networks that are good representatives of the underlying cognitive tasks. However, our model managed to get brain decoding results similar or slightly better than those obtained from MVPA both in the cases of supervised and unsupervised methods. This can be attributed to the challenging nature of the TOL dataset; Pearson correlation does not manage to capture the interdependencies between the anatomical regions over short time windows. While ridge regression fails to correctly estimate the mesh arc-weights as it estimates the arc-weights for each region independently of the other ones. Our proposed model, with a relatively small number of epochs, manages to obtain mesh arc-weight values that capture the activation patterns of anatomical regions and their relationships.

It is important to note that, it is possible to obtain higher brain decoding accuracy using voxel-level MVPA rather than anatomic-region-level MVPA, and by normalizing the BOLD response of individual voxels to having 0 mean and 1 standard deviation. However, we do not employ either of them in our analysis for the following reasons. Firstly, any analysis at the voxel-level comes with a very high computational cost, especially when attempting to build functional brain networks. Also, the analysis at voxel level does not allow us to perform the study roles and contributions of brain anatomic region level to the complex problem solving task, which is the main goal of this paper. Secondly, normalizing the BOLD responses of individual voxels prevents us from constructing informative, functional brain networks as this normalization distorts the information of relative activation patterns between the voxels and the brain anatomic region, which is essential in the process of building functional brain networks.

5. Brain Network Properties

In this section, we aim to analyze the network properties of the constructed functional brain networks. We investigate the network properties for each anatomical brain region during both planning and execution subtasks in order to understand which regions are most active and which regions work together during each one of the two subtasks of complex problem solving.

Given that the constructed brain functional networks are both weighted, directed, fully-connected and contain both negative and positive weights, we preprocessed the networks before measuring their network properties. Firstly, we got rid of all the negative weights by shifting all the mesh arc-weights values by a positive quantity equal to the absolute value of the largest negative arc-weight. We then normalized the mesh arc-weights to ensure that all of the weights are within the range of [0, 1]. Finally, we measured the network properties on the pruned brain graph, where the brain regions (nodes) contributing no voxels (have a flat BOLD response) and all of their corresponding arc-weights (edges) were deleted from the brain graph. Thus, the networks contained less than 90 regions with their corresponding edges. We used the brain connectivity toolbox to calculate the investigated network properties ( Rubinov and Sporns, 2010 ).

In order to measure for centrality, the number of neighbors for each anatomical region (P) was chosen to be equal to 89, which is equal to the total number of neighbors for any given node as the total number of brain anatomical regions defined by the AAL atlas ( Tzourio-Mazoyer et al., 2002 ) after deleting the regions residing in the cerebellum equals 90. In addition, since we pruned the nodes that correspond to regions from which no voxels were selected, our constructed brain networks were weighted directed fully-connected networks. Therefore, the in-degree, out-degree and total degree of all nodes in the graph were equal to the total number of anatomical regions retained after voxel selection.

Therefore, we used node strength and node betweenness centrality to identify nodes with high centrality, which are potential hubs in the brain networks controlling the flow of information in the network. In our proposed model, the node in-strength of node i is the sum of the mesh arc-weight values, which is estimated using our proposed neural network method in order to minimize the reconstruction error of the BOLD response of anatomical region i using its neighbors. Thus, node in-strength is not used as part of our network properties analyses; we rather used node out-strength to measure the centrality of all anatomicalregions.

As for measures of segregation, quantifying the existence of subgroups within brain networks is based on densely interconnected nodes. These subgroups are commonly referred to as clusters or modules. The existence of such clusters in functional brain networks is a sign of interdependence among the nodes forming the cluster. Therefore, clustering coefficient, transitivity and local efficiency were measured in order to identify potential clusters with dense interconnections in the brain networks.

5.1. Planning and Execution Brain Networks

In this section, we discuss the network properties of the planning and execution networks. For each aforementioned network metric, we ranked the brain regions in descending order according to their score on that network measure for all subjects across all runs. Then, we retained the 10 anatomical regions with the highest scores. Following that, we measured the frequency of occurrence of each brain region among the top 10 regions across all runs in order to identify the shared regions and patterns across all subjects for both planning and execution subtasks. The results of the analysis are shown in tables: Table 4 shows the brain regions that have high scores for the reported network properties during planning subtask, and Table 5 shows the brain regions that have high scores during execution subtask. There are a number of processes taking place during planning and execution. Plan generation involves a series of recursive events including 1) problem encoding; 2) decision-making in order to decide which ball to move and where to move it; 3) mental imagery to imagine the ball moving, and 4) working memory to maintain the intermediate steps as well as the move number. During plan execution, there is 1) retrieval of the steps from memory; 2) confirming the correct steps are being performed, and 3) the motor execution of those steps. As the results demonstrate, the networks for planning and execution are overlapping. These results are similar to the activation results reported in Newman et al. (2009) in that the regions that were found to be active during the task are also regions that are most prominently found with the highest network measures. These regions include the right and left middle frontal gyrus, anterior cingulate cortex, precentral cortex, and superior parietal cortex.

Table 4 . Planning: Anatomical regions with the highest network measures across subjects, regions are painted if they overlap with execution.

Table 5 . Execution: Anatomical regions with the highest network measures across subjects, regions are painted if they overlap with planning.

Previous work has suggested that the regions found in the current study to show high network measures are directly related to the sub-tasks associated with TOL performance. For example, both the left and right prefrontal cortex have been found to be involved in the TOL task, with the two regions performing distinguishable functions. The right prefrontal cortex is involved in constructing the plan for solving the TOL problem while the left prefrontal cortex is involved in supervising the execution of that plan ( Newman et al., 2003 , 2009 ). The anterior cingulate has been linked to error detection and is particularly involved in the TOL when the number of moves is higher, or the problem difficulty is manipulated. The right superior parietal cortex and precentral cortex have been linked to visuo-spatial attention necessary for planning ( Newman et al., 2003 ), and the left parietal cortex has been linked to visuo-spatial working memory processing ( Newman et al., 2003 ). The overlap between the regions with the highest network measures and those that have been linked to the task is an important feature and is not due to the voxel selection process. Many regions that passed threshold were not in the top ranked list of network measures. For example, the basal ganglia, including the caudate has been found in previous studies to be involved in TOL performance ( Dagher et al., 1999 ; Rowe et al., 2001 ; Beauchamp et al., 2003 ; Van den Heuvel et al., 2003 ; Newman et al., 2009 ); however, the region appears to not be an important network hub. Figures 8A,B visualize the reported brain regions in Tables 4 , 5 , respectively, using Brain Net Viewer ( Xia et al., 2013 ). In Figures 8A,B , the color of the node (brain region) implies the following: red indicates that the region has high transitivity, clustering coefficient or local efficiency. Green indicates that the node has high node centrality measured by node out-strength and node betweenness. As for blue, it shows the nodes that have high node centrality and is part of a subgroup of densely interconnected regions.

Figure 8 . Regions with the highest network measures for planning brain network (A) and execution brain network (B) .

5.2. Differences Between Planning and Execution Networks

In this section, we explore the network differences between planning and execution by calculating the difference between the network property scores for planning and execution for each run. To achieve that, we took the difference between the network property scores for brain anatomical regions during planning and the network property scores for brain anatomical regions during execution for each run. Then, we counted the frequency of times a given anatomical region is more active during planning than execution and vice-versa in order to identify consistent patterns of the disagreements between planning brain networks and execution brain networks across all subjects. Results showed, generally, that the network measures were higher for planning than execution. This, too, mirrors the findings from Newman et al. (2009) in which planning resulted in greater activation than execution.

Node out-strength is a measure of how connected the node is to other nodes in the network. Figures 9A,B visualize the brain regions with higher node out-strength during planning and during execution, respectively. Planning showed greater out-strength than execution in the following regions: occipital regions (calcarine, cuneus), parietal regions (bilateral superior parietal cortex and precunues), the right superior frontal cortex, and inferior occipito-temporal regions (fusiform and lingual gyri). The left angular gyrus and bilateral medial superior frontal cortex showed greater out-strength for execution. As for node betweenness, Figures 10A,B visualize the brain regions with higher betweenness during planning and during execution, respectively. The following brain regions had higher node betweenness during planning than execution: occipital regions (calcarine, cuneus, right middle, right superior); inferior occipito-temporal (fusiform, lingual); parietal (bilateral superior parietal, left postcentral, precuneus). Bilateral medial superior frontal had higher node betweenness during execution than planning.

Figure 9 . Anatomical regions with higher node out-strength during planning (Top) and during execution (Bottom). (A) Anatomical regions with higher node out-strength during planning. (B) Anatomical regions with higher node out-strength during execution.

Figure 10 . Anatomical regions with higher node betweenness during planning (Top) and during execution (Bottom). (A) Anatomical regions with higher node betweenness during planning. (B) Anatomical regions with higher node betweenness during exection.

These results suggest that there is greater information flow during planning than execution. This matches our expectations. Planning is more computationally demanding than execution. Again, during planning, participants must explore the problem space, which requires generating and manipulating a mental representation of the problem. The regions that show greater information flow during planning are all regions involved in that generation and manipulation, particularly parietal, occipital and inferior occipito-temporal. On the other hand, execution requires recall of the plan generated and stored and therefore, greater information flow from frontal regions related to memory retrieval is observed.

Clustering coefficient, local efficiency and transitivity are measures of segregation that aim to identify sub-networks. Figure 11A visualizes the brain regions with higher local efficiency and higher clustering coefficient during planning phase compared to execution phase. While Figure 11B visualizes the brain regions with higher transitivity during planning than during execution phase. Each of these measures was larger for planning than execution, with no regions showing larger measures for execution.

Figure 11 . Anatomical regions with higher local efficiency and clustering coefficient (Top) and higher transitivity (Bottom) during planning. (A) Anatomical regions with higher local efficiency and clustering coefficient during planning. (B) Anatomical regions with higher transitivity during planning.

The regions that showed a higher clustering coefficient in planning included: the cuneus, left middle occipital cortex, and right precuneus. Local efficiency was higher in a similar set of regions (the cuneus, left middle occipital cortex, and right precuneus). The clustering coefficient and local efficiency identified a visual-spatial sub-network that is more strongly connected during planning. Transitivity identified an overlapping but more extensive set of regions that included: bilateral angular gyrus, calcarine sulcus, cuneus, bilateral middle frontal cortex, bilaterial superior frontal cortex, bilateral fusiform and lingual gyri, bilateral occipital cortex, bilatral superior parietal cortex, postcentral and precentral cortex, precuneus, supplementary motor area, right supramarginal gyrus, and right inferior and middle temporal cortex.

5.3. Global Efficiency

Since global efficiency is measured over the entire brain network, not for a given node in the network, we measured the global efficiency for all planning and execution networks within all runs across subjects. Then, global efficiency of planning is compared against that of execution. Results show that the majority of runs had higher global efficiency scores during planning than execution; 43 out of 72 runs had higher global efficiency during planning than execution. Furthermore, Table 6 shows the number of runs where global efficiency was higher during planning and during execution across all subjects for all 4 runs of each subject. The first column shows the number of subjects that had a higher global efficiency score during planning than during execution. The second column shows the number of subjects that had a higher global efficiency score during execution than during planning.

Table 6 . Global efficiency.

Although there was no significant difference in global efficiency between planning and execution, from the table, it is clear that the majority of subjects had a higher global efficiency for planning for the first runs. Some subjects switched from having higher global efficiency during planning to having higher global efficiency during execution. A potential explanation for this change across runs is switching from pre-planning to online planning or planning intermixed with execution. Although there is a dedicated planning phase in the current study, that does not mean that planning is not taking place during execution. In fact, it has been debated as to whether efficient pre-planning is possible in the TOL or whether TOL performance is controlled by online planning ( Kafer and Hunter, 1997 ; Phillips, 1999 , 2001 ; Unterrainer et al., 2004 ). According to Phillips (1999 , 2001) , pre-planning the entire sequence is not natural, but that people instead plan the beginning sequence of moves and then intersperse planning and execution. If this is the case, then it may be expected that some participants will switch to online planning. This intermixing of planning and execution is also likely to impact the performance of the machine learning algorithms to detect planning and execution phases.

The relationship between global efficiency and behavioral performance was examined. Global efficiency was found to be positively correlated with the mean number of extra moves (a measure of error) during problem-solving (for execution r = 0.73, p = 0.0006). Previous studies have shown a relationship between global efficiency and task performance ( Stanley et al., 2015 ).

This suggests that the variance in global efficiency is indicative of individual differences in neural processing and further suggests that the changes in global efficiency across runs are also likely indicative of changes in neural processing related to changing strategy. Further research using a larger sample is necessary to explore this hypothesis.

6. Conclusion

In this paper, we propose a new computational method to estimate dynamic functional brain networks from the fMRI signal recorded during a complex problem solving task. Our model recognizes the two phases of complex problem solving with more than 80% accuracy, indicating the representation power of the suggested dynamic brain network model. We study the properties of the constructed brain networks during planning and execution phases in order to identify essential anatomic regions in the brain networks related to problem solving. We investigate the potential hubs and densely connected clusters. Furthermore, we compare the network structure of the estimated dynamic brain networks for planning and execution tasks.

There are some limitations to the study. Although the primary aim of this study was to demonstrate the feasibility of the methods, the sample size is somewhat small, making the interpretation of the results difficult. Second, a goal of this method is to identify brain states that are interspersed with each other. In the current study, planning was expected to occur both prior to execution as well as during execution; therefore, planning states are interspersed within the execution phase. The temporal sampling rate of the fMRI data may be a limiting factor. Alternatively, the sluggish and blurred underlying hemodynamic response may be the factor preventing the ability to detect brain states. We plan to explore this factor in future work.

Data Availability Statement

The dataset used in this study and the code required to reproduce our results can be found at https://osf.io/krch2/?view_only=df8aaf1f4fa46129a69f89486f65a83 .

Ethics Statement

The studies involving human participants were reviewed and approved by Indiana University Institutional Review Board. The patients/participants provided their written informed consent to participate in this study. All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. All individual participants in the study signed an informed, written consent documents approved by the Indiana University Institutional Review Board.

Author Contributions

AA proposed and implemented the computational model with support from OE and BK. Performed the experiments with support from BK and OE. Prepared the visualizations with support from BK. Wrote the manuscript with support from OE, SN, and FY. BK proposed the idea for building brain networks using neural networks. SN provided the neuroscientific interpretations of the results of the proposed computational model, designed the Tower of London experiment procedure and collected the corresponding fMRI recordings. FY supervised the project, conceived the study and was in charge of the overall direction and planning. All authors provided critical feedback and helped shape the research, analysis and manuscript. All authors contributed to the article and approved the submitted version.

This study was funded by TUBITAK (Scientific and Technological Research Council of Turkey) under grant No: 116E091 as well as the Indiana METACyt Initiative of Indiana University, funded in part through a major grant from the Lilly Endowment, Inc. AA, OE, BK, and FY received funding from TUBITAK (Scientific and Technological Research Council of Turkey) under the grant no: 116E091. SN received research funding from the Indiana METACyt Initiative of Indiana University and a major grant from the Lilly Endowment, Inc. The authors declare that this study received funding from the Lilly Endowment Inc. The funder was not involved in the study design, collection, analysis, interpretation of data, the writing of this article or the decision to submit it for publication.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

We thank Gonul Gunal Degirmendereli for her contributions to this manuscript and her help with data analysis of the TOL experiment procedure.

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Keywords: fMRI, machine learning, brain networks, tower of London (TOL), complex problem solving

Citation: Alchihabi A, Ekmekci O, Kivilcim BB, Newman SD and Yarman Vural FT (2021) Analyzing Complex Problem Solving by Dynamic Brain Networks. Front. Neuroinform. 15:670052. doi: 10.3389/fninf.2021.670052

Received: 20 February 2021; Accepted: 10 November 2021; Published: 10 December 2021.

Reviewed by:

Copyright © 2021 Alchihabi, Ekmekci, Kivilcim, Newman and Yarman Vural. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Abdullah Alchihabi, abdullahalchihabi@cmail.carleton.ca

This article is part of the Research Topic

Machine Learning Methods for Human Brain Imaging

• People/Staff

How the brain solves problems

12 August 2021 - Delia du Toit

The connections among areas of our brain and how they interact is what counts when trying to find solutions to problems.

In trying to think of an introduction for this article it occurred to me that had I been inside an MRI, the screen would have showed several brain regions lighting up like Times Square as my mind was attempting to solve the problem. 

First, the prefrontal cortex, basal ganglia and thalamus would recognise that the blank page meant that there was a problem that needed to be solved. The thought that the editor might not favour this first-person account in a science article would send the limbic system, the primal part of the brain where emotions are processed, into overdrive. The amygdala, that little almond-shaped nugget at the base of the brain, would look like a Christmas tree as anxiety ticked up.

Finally, as words started filling the screen, the prefrontal cortex behind the forehead would flicker and flash. The hippocampus would access memories of previous similar articles, the information-gathering process and even school-level English classes decades ago, to help the process along. And all this activity would happen at once.

Holistic problem solving

Depending on the problem in front of you, the entire brain could be involved in trying to find a solution, says Professor Kate Cockcroft , Division Leader of cognitive neuroscience at the Neuroscience Research Laboratory (Wits NeuRL) in the School of Human and Community Development .

“You would use many different brain regions to solve a problem, especially a novel or difficult one. The idea of processes being localised in one or two parts of the brain has been replaced with newer evidence that it is the connections among brain areas and their interaction that is important in cognitive processes. Some areas may be more activated with certain problems – a visual problem would activate the visual cortices, for example. 

“All this activity takes place as electrochemical signals. The signals form within neurons, pass along the branch-like axons and jump from one neuron to the next across gaps called synapses, with the help of neurotransmitter chemicals. The pattern, size, shape and number of these signals, what they communicate with, and the region of the brain in which they happen, determine what they achieve.”

Although problem solving is a metacognitive - ‘thinking about thinking’ - process, that does not make it solely the domain of the highly evolved human prefrontal cortex, adds Dr Sahba Besharati, Division Leader of social-affective neuroscience at NeuRL.

“This is the most recently evolved part of the human brain, but problem solving does not happen in isolation – it’s immersed in a social context that influences how we interpret information. Your background, gender, religion or emotions, among other factors, all influence how you interpret a problem. This means that it would involve other brain areas like the limbic system, one of the oldest brain systems housed deep within the cortex,” says Besharati.

“Problem-solving abilities are not a human peculiarity. Some animals are even better than us at solving certain problems, but we all share basic problem-solving skills – if there’s danger, leave; if you’re hungry, find food.”

None of this would be possible without memory either, says Cockcroft. “Without it, we would forget what it is that we are trying to solve and we wouldn't be able to use past experiences to help us solve it.”

And memory is, again, linked to emotion. “We use this information to increase the likelihood of positive results when solving new problems,” she says.

Improving your skills

It has been proven time and again that just about any brain process can be improved – including problem-solving abilities. “Brain plasticity is a real thing – the brain can reorganise itself with targeted intervention,” says Besharati. “Rehabilitation from neurological injury is a dynamic process and an ever-improving science that has allowed us to understand how the brain can change and adapt in response to the environment. Studies have also shown that simple memorisation exercises can assist tremendously in retaining cognitive skills in old age.” 

Of course, all these processes depend on your brain recognising that there’s a problem to be dealt with in the first place – if you don’t realise you’re spending money foolishly, you can’t improve your finances. “Recognition of a problem can happen at both a conscious and unconscious level. Stroke patients who are not aware of their motor paralysis, for example, deludedly don’t believe that they are paralysed and will sometimes not engage in rehabilitation. But their delusions often spontaneously recover, suggesting recognition at an unconscious level and that, over time, the brain can restore function.” 

If all else fails, there might be some value to the adage ‘sleep on it’, says Cockcroft. “Sleep is believed to assist memory consolidation – changing memories from a fragile state in which they can easily be damaged to a permanent state. In doing so, they become stored in different brain regions and new neural connections are formed that may assist problem solving. On waking, you may have formed associations between information that you didn’t think of previously. This seems to be most effective within three hours of learning new information – perhaps we should institute compulsory naps for students after lectures!”

• Delia du Toit is a freelance writer.
• This article first appeared in  Curiosity ,  a research magazine produced by   Wits Communications  and the  Research Office .
• Read more in the 12 th issue, themed: #Solutions. We explore #WitsForGood solutions to the structural, political and socioeconomic challenges that persist in South Africa, and we are encouraged by astounding ‘moonshot moments’ where Witsies are advancing science, health, engineering, technology and innovation.

Common and specific neural correlates underlying insight and ordinary problem solving

• Original Research
• Published: 24 July 2020
• Volume 15 , pages 1374–1387, ( 2021 )

Cite this article

• Jiabao Lin 1 , 2 , 3 ,
• Xue Wen 4 ,
• Xuan Cui 2 , 3 ,
• Yanhui Xiang 5 ,
• Jiushu Xie 6 ,
• Yajue Chen 2 , 3 ,
• Ruiwang Huang 2 , 3 &
• Lei Mo   ORCID: orcid.org/0000-0001-8586-7508 2 , 3  

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Previous studies have investigated the cognitive and neural mechanisms underlying insight problem solving (INPS). However, it is still unclear which mechanisms are common to both INPS and ordinary problem solving (ORPS), and which are distinctly involved in only one of these processes. In this study, we selected two types of Chinese character chunk decompositions, ordinary Chinese character chunk decomposition (OCD) and creative Chinese character chunk decomposition (CCD), as representatives of ORPS and INPS, respectively. By using functional magnetic resonance imaging (fMRI) to record brain activations when subjects executed OCD or CCD operations, we found that both ORPS and INPS resulted in significant activations in the widespread frontoparietal cognitive control network, including the middle frontal gyrus, inferior frontal gyrus, and inferior parietal lobe. Furthermore, compared with ORPS, INPS led to greater activations in higher-level brain regions related to symbolic processing in the default mode network, including the anterior cingulate cortex, superior temporal gyrus, angular gyrus, and precuneus. Conversely, ORPS induced greater activations than INPS in more posterior brain regions related to visuospatial attention and visual perception, such as the inferior temporal gyrus, hippocampus, and middle occipital gyrus/superior parietal gyrus/fusiform gyrus. In addition, an ROI analysis corroborated the neural commonalities and differences between ORPS and INPS. These findings provide new evidence that ORPS and INPS rely on common as well as distinct cognitive processes and cortical mechanisms.

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Abbreviations

• Ordinary problem solving
• Insight problem solving

Ordinary Chinese character chunk decomposition

Creative Chinese character chunk decomposition

Ordinary chunk decomposition-low difficult level

Ordinary chunk decomposition-high difficult level

Creative chunk decomposition-low creative level

Creative chunk decomposition-high creative level

Middle frontal gyrus

Inferior frontal gyrus

Supramarginal gyrus

Medial prefrontal cortex

Inferior parietal lobe

Superior frontal gyrus

Angular gyrus

Anterior cingulate cortex

Superior temporal gyrus

Middle temporal gyrus

Hippocampus

Inferior temporal gyrus

Middle occipital gyrus

Middle cingulate cortex

Sensorimotor areas

Superior parietal lobe

Fusiform gyrus

Montreal Neurological Institute

Full width at half maximum

Analysis of variance

Response times

Region of interest

Correct rates

Default mode network

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Acknowledgments

This work was supported by funding from the Project of Guangzhou Philosophies and Social Sciences (Grant Number: 2020GZQN43), the National Social Science Foundation of China (Grant Number: 14ZDB159) and the Project of Key Institute of Humanities and Social Sciences, MOE (16JJD880025).

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Jiabao Lin, Xuan Cui, Yajue Chen, Ruiwang Huang & Lei Mo

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Lin, J., Wen, X., Cui, X. et al. Common and specific neural correlates underlying insight and ordinary problem solving. Brain Imaging and Behavior 15 , 1374–1387 (2021). https://doi.org/10.1007/s11682-020-00337-z

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The Anatomy of the Brain

The brain controls your thoughts, feelings, and physical movements

Associated Conditions

The brain is a unique organ that is responsible for many functions such as problem-solving, thinking, emotions, controlling physical movements, and mediating the perception and responses related to the five senses. The many nerve cells of the brain communicate with each other to control this activity.

Each area of the brain has one or more functions. The skull, which is composed of bone, protects the brain. A number of different health conditions can affect the brain, including headaches , seizures , strokes , multiple sclerosis , and more. These conditions can often be managed with medical or surgical care.

The brain is primarily composed of nerve cells, which are also called neurons. Blood vessels supply oxygen and nutrients to the neurons of the brain. Cerebrospinal fluid (CSF), a fluid that provides nourishment and immune protection to the brain, flows around the brain and within the ventricular system (spaces between the regions of the brain).

The brain and the CSF are protected by the meninges, composed of three layers of connective tissue: the pia, arachnoid, and dura layers. The skull surrounds the meninges.

The brain has many important regions, such as the cerebral cortex, brainstem, and cerebellum. The areas of the brain all interact with each other through hormones and nerve stimulation.

The regions of the brain include:

• Cerebral cortex : This is the largest portion of the brain. It includes two hemispheres (halves), which are connected to each other—physically and functionally—by the corpus callosum. The corpus callosum runs from the front of the cerebral cortex to the back of the cerebral cortex. The outer part of the cerebral cortex is often described as gray matter, and the deeper areas are often described as white matter due to their microscopic appearance.
• Lobes of the cerebral cortex : Each hemisphere of the cerebral cortex is composed of four lobes. The frontal lobes are the largest, and they are located at the front of the brain. The temporal lobes are located on the sides of the brain, near and above the ears. The parietal lobes are at the top middle section of the brain. And the occipital lobes, which are the smallest lobes, are located in the back of the cerebral cortex.
• Limbic system : The limbic system is located deep in the brain and is composed of several small structures, including the hippocampus, amygdala, thalamus, and hypothalamus .
• Internal capsule : This area is located deep in the brain and is considered white matter. The frontal regions of the cerebral cortex surround the left and right internal capsules. The internal capsule is located near the lateral ventricles.
• Thalamus : The left and right thalami are below the internal capsule, above the brainstem, and near the lateral ventricles.
• Hypothalamus and pituitary gland : The hypothalamus is a tiny region of the brain located directly above the pituitary gland. The pituitary gland is a structure that extends directly above the optic chiasm, where the optic nerves meet.
• Brainstem : The brainstem is the lowest region of the brain and is continuous with the spinal cord. It is composed of three sections: the midbrain, pons, and medulla. The cranial nerves emerge from the brainstem.
• Cerebellum : The cerebellum is located at the lower back of the brain, under the occipital lobe and behind the brainstem. It has two hemispheres (left and right) that are connected by a middle structure called the vermis.
• Blood vessels : The blood vessels that supply your brain include the anterior cerebral arteries , middle cerebral arteries , posterior cerebral arteries, basilar artery , and vertebral arteries . These blood vessels and the blood vessels that connect them to each other compose a collection of blood vessels described as the circle of Willis .
• Ventricular system : CSF flows in the right and left lateral ventricles, the third ventricle, the cerebral aqueduct, the fourth ventricle, and down into the central canal in the spinal cord.

The brain has a number of functions, including motor function (controlling the body’s movements), coordination, sensory functions (being aware of sensations), hormone control, regulation of the heart and lungs, emotions, memory, behavior, and creativity.

These functions often rely on and interact with each other. For example, you might experience an emotion based on something that you see and/or hear. Or you might try to solve a problem with the help of your memory. Messages travel very quickly between the different regions in the brain, which makes the interactions almost instantaneous.

Functions of the brain include:

• Motor function : Motor function is initiated in an area at the back of the frontal lobe called the motor homunculus. This region controls movement on the opposite side of the body by sending messages through the internal capsule to the brainstem, then to the spinal cord, and finally to a spinal nerve through a pathway described as the corticospinal tract.
• Coordination and balance : Your body maintains balance and coordination through a number of pathways in the cerebral cortex, cerebellum, and brainstem.
• Sensation : The brain receives sensory messages through a pathway that travels from the nerves in the skin and organs to the spine, then to the brainstem, up through the thalamus, and finally to an area of the parietal lobe called the sensory homunculus, which is directly behind the motor homunculus. Each hemisphere receives sensory input from the opposite side of the body. This pathway is called the spinothalamic tract.
• Vision : Your optic nerves in your eyes can detect whatever you see, sending messages through your optic tract (pathway) to your occipital lobes. The occipital lobes put those messages together so that you can perceive what you are seeing in the world around you.
• Taste and smell : Your olfactory nerve detects smell, while several of your cranial nerves work together to detect taste. These nerves send messages to your brain. The sensations of smell and taste often interact, as smell amplifies your experience of taste.
• Hearing : You can detect sounds when a series of vibrations in your ear stimulate your vestibulocochlear nerve. The message is sent to your brainstem and then to your temporal cortex so that you can make sense of the sounds that you hear.
• Language : Speaking and understanding language is a specialized brain function that involves several regions of your dominant hemisphere (the side of the brain opposite your dominant hand). The two major areas that control speech are Wernicke’s area , which controls the understanding of speech, and Broca’s area, which controls the fluency of your speech.
• Emotions and memory : Your amygdala and hippocampus play important roles in storing memory and associating certain memories with emotion.
• Hormones : Your hypothalamus, pituitary gland, and medulla all respond to the conditions of your body, such as your temperature, carbon dioxide level, and hormone levels, by releasing hormones and other chemicals that help regulate your body’s functions. Emotions such as fear can also have an influence on these functions.
• Behavior and judgment : The frontal lobes control reasoning, planning, and maintaining social interactions. This area of the brain is also involved in judgment and maintaining appropriate behavior.
• Analytical thinking : Mathematical problem solving is located in the dominant hemisphere. Often, this type of reasoning involves interaction with the decision-making regions of the frontal lobes.
• Creativity : There are many types of creativity, including the production of visual art, music, and creative writing. These skills can involve three-dimensional thinking, also described as visual-spatial skills. Creativity also involves analytical reasoning and usually requires a balance between traditional ways of thinking (which occurs in the frontal lobes) and "thinking outside the box."

There are many conditions that can affect the brain. You may experience self-limited issues, such as the pain of a headache, or more lasting effects of brain disease, such as paralysis due to a stroke. The diagnosis of brain illnesses may be complex and can involve a variety of medical examinations and tests, including a physical examination, imaging tests, neuropsychological testing, electroencephalography (EEG) , and/or lumbar puncture .

Common conditions that involve the brain include:

• Headaches : Head pain can occur due to chronic migraines or tension headaches. You can also have a headache when you feel sleepy, stressed, or due to an infection like meningitis (an infection of the meninges).
• Traumatic brain injury : An injury to the head can cause damage such as bleeding in the brain, a skull fracture, a bruise in the brain, or, in severe cases, death. These injuries may cause vision loss, paralysis, or severe cognitive (thinking) problems.
• Concussion : Head trauma can cause issues like loss of consciousness, memory impairment, and mood changes. These problems may develop even in the absence of bleeding or a skull fracture. Often, symptoms of a concussion resolve over time, but recurrent head trauma can cause serious and persistent problems with brain function, described as chronic traumatic encephalopathy (CTE).
• Transient ischemic attack (TIA) : A temporary interruption in the blood supply to the brain can cause the affected areas to temporarily lose function. This can happen due to a blood clot, usually coming from the heart or carotid arteries. If the interruption in blood flow resolves before permanent brain damage occurs, this is called a TIA . Generally, a TIA is considered a warning that a person is at risk of having a stroke, so a search for stroke causes is usually necessary—and stroke prevention often needs to be initiated.
• Stroke : A stroke is brain damage that occurs due to an interruption of blood flow to the brain. This can occur due to a blood clot (ischemic stroke) or a bleed in the brain (hemorrhagic stroke) . There are a number of causes of ischemic and hemorrhagic stroke, including heart disease, hypertension, and brain aneurysms.
• Brain aneurysm : An aneurysm is an outpouching of a blood vessel. A brain aneurysm can cause symptoms due to pressure on nearby structures. An aneurysm can also bleed or rupture, causing a hemorrhage in the brain. Sometimes an aneurysm can be surgically repaired before it ruptures, preventing serious consequences.
• Dementia : Degenerative disease of the regions in the brain that control memory and behavior can cause a loss of independence. This can occur in several conditions, such as Alzheimer’s disease , Lewy body dementia, Pick’s disease, and vascular dementia (caused by having many small strokes).
• Multiple sclerosis (MS) : This is a condition characterized by demyelination (loss of the protective fatty coating around nerves) in the brain and spine. MS can cause a variety of effects, such as vision loss, muscle weakness, and sensory changes. The disease course can be characterized by exacerbations and remissions, a progressive decline, or a combination of these processes.
• Parkinson’s disease : This condition is a progressive movement disorder that causes tremors of the body (especially the arms), stiffness of movements, and a slow, shuffling pattern of walking. There are treatments for this condition, but it is not curable.
• Epilepsy : Recurrent seizures can occur due to brain damage or congenital (from birth) epilepsy. These episodes may involve involuntary movements, diminished consciousness, or both. Seizures usually last for a few seconds at a time, but prolonged seizures (status epilepticus) can occur as well. Anti-epileptic medications can help prevent seizures, and some emergency anti-epileptic medications can be used to stop a seizure while it is happening.
• Meningitis or encephalitis : An infection or inflammation of the meninges (meningitis) or the brain (encephalitis) can cause symptoms such as fever, stiff neck, headache, or seizures. With treatment, meningitis usually improves without lasting effects, but encephalitis can cause brain damage, with long-term neurological impairment.
• Brain tumors : A primary brain tumor starts in the brain, and brain tumors from the body can metastasize (spread) to the brain as well. These tumors can cause symptoms that correlate to the affected area of the brain. Brain tumors also may cause swelling in the brain and hydrocephalus (a disruption of the CSF flow in the ventricular system). Treatments include surgery, chemotherapy, and radiation therapy.

If you have a condition that could be affecting your brain, there are a number of complex tests that your medical team may use to identify the problem. Most important, a physical exam and mental status examination can determine whether there is any impairment of brain function and pinpoint the deficits. For example, you may have weakness of one part of the body, vision loss, trouble walking, personality or memory changes, or a combination of these issues. Other signs, such as rash or fever, which are not part of the neurological physical examination, can also help identify systemic issues that could be causing your symptoms.

Diagnostic tests include brain imaging tests such as computerized tomography (CT), magnetic resonance imaging (MRI), or functional magnetic resonance imaging (fMRI). These tests can identify structural and functional abnormalities. And sometimes, tests such as CT angiography (CTA), MRI angiography (MRA), or interventional cerebral angiography are needed to visualize the blood vessels in the brain.

Another test, an evoked potential test, can be used to identify hearing or vision problems in some circumstances. And a lumbar puncture may be used to evaluate the CSF surrounding the brain. This test can detect evidence of infection, inflammation, or cancer. Rarely, a brain biopsy is used to sample a tiny area of the brain to assess the abnormalities.

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By Heidi Moawad, MD Dr. Moawad is a neurologist and expert in brain health. She regularly writes and edits health content for medical books and publications.

January 25, 2008

What Are We Thinking When We (Try to) Solve Problems?

New research indicates what happens in the brain when we're faced with a dilemma

By Nikhil Swaminathan

Aha! Eureka! Bingo! "By George, I think she's got it!" Everyone knows what it's like to finally figure out a seemingly impossible problem. But what on Earth is happening in the brain while we're driving toward mental pay dirt ? Researchers eager to find out have long been on the hunt, knowing that such information could one day provide priceless clues in uncovering and fixing faulty neural systems believed to be behind some mental illnesses and learning disabilities.

Researchers at Goldsmiths, University of London report in the journal PLoS ONE that they monitored action in the brains of 21 volunteers with electroencephalography (EEG) as they tackled verbal problems in an attempt to uncover what goes through the mind—literally—in order to observe what happens in the brain during an "aha!" moment of problem solving.

"This insight is at the core of human intelligence … this is a key cognitive function that the human can boast to have," says Joydeep Bhattacharya, an assistant professor in Goldsmiths's psychology department. "We're interested [in finding out] whether—there is a sudden change that takes place or something that changes gradually [that] we're not consciously aware of," he says. The researchers believed they could pin down brain signals that would enable them to predict whether a person could solve a particular problem or not.

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In many cases, the subjects hit a wall, or what researchers refer to as a "mental impasse." If the participants arrived at this point, they could press a button for a clue to help them untangle a problem. Bhattacharya says blocks correlated with strong gamma rhythms (a pattern of brain wave activity associated with selective attention) in the parietal cortex, a region in the upper rear of the brain that has been implicated in integrating information coming from the senses. The research team noticed an interesting phenomenon taking place in the brains of participants given hints: The clues were less likely to help if subjects had an especially high gamma rhythm pattern. The reason, Bhattacharya speculates, is that these participants were, in essence, locked into an inflexible way of thinking and less able to free their minds, and thereby unable to restructure the problem before them.

"If there's excessive attention, it somehow creates mental fixation," he notes. "Your brain is not in a receptive condition."

At the end of each trial, subjects reported whether or not they had a strong "Aha!" moment. Interestingly, researchers found that subjects who were aware that they had found a new way to tackle the problem (and so, had consciously restructured their thinking) were less likely to feel as if they'd had eureka moment compared to more clueless candidates.

"People experience the "Aha!" feeling when they are not consciously monitoring what they are thinking," Bhattacharya says, adding that the sentiment is more of an emotional experience he likens to relief. "If you're applying your conscious brain information processing ability, then you're alpha." (Alpha brain rhythms are associated with a relaxed and open mind; volunteers who unwittingly solved problems showed more robust alpha rhythms than those who knowingly adjusted their thinking to come up with the answer.)

He says the findings indicate that it's better to tackle problems with an open mind than by concentrating too hard on them. In the future, Bhattacharya says, his team will attempt to predict in real-time whether a stumped subject will be able to solve a vexing problem and, also, whether they can manipulate brain rhythms to aid in finding a solution.

The second probe into problem-solving focused on the anterior cingulate cortex (ACC), a region in the front of the brain tied to functions such as decision making, conflict monitoring and reward feedback. A team at the University of Lyon's Stem Cell and Brain Research Institute in Bron, France reports in Neuron that it verified that the ACC helps detect errors during problem solving (as previously discovered), but also that it does so by acting more as a general guide, monitoring and scoring the steps involved in problem solving, pointing out miscalculations as well as success.

The team discovered this by recording electrical activity in the brains of two male rhesus monkeys as they tried to determine which targets on a screen would result in a tasty drink of juice. "When you're trying to solve a problem, you need to search; when you discover the solution, you need to stop searching," says study co-author Emmanuel Procyk, coordinator of the Institute's Department of Integrative Neurobiology. "We need brain areas to do that."

He says that researchers observed increased neuronal activity in the animals' ACCs when they began searching. When the monkeys hit the jackpot, there was still heightened activity in the ACC (though only a selective population of nerve cells remained hopped up), indicating that the region is responsible for more than simply alerting the rest of the brain when errors are made. Once the monkeys got the hang of it—and routinely pressed the correct target—ACC activity slowed.

"What we think based on this experiment and other experiments," Procyk says, "is that this structure is very important in valuing things." It essentially scores each of the monkey's behaviors as successful or not successful. "It is an area," he adds, "that will help to decide when to shift from the functioning that goes on when [the brain is] learning to when the learning [is] done."

Procyk says that if this system is compromised, it could have implications for issues such as drug dependency. If the ACC is functioning abnormally, he says, it could overvalue drugs, leading to addiction. (Other studies have shown that an impaired cingulate cortex can result in maladaptive social behavior and disrupted cognitive abilities.)

Alas, the ultimate "Aha!" moment for researchers probing problem solving is likely is far off, but at least the latest research may help them avoid an impasse.

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• v.5; Jan-Dec 2021

Hot and cold executive functions in the brain: A prefrontal-cingular network

Mohammad ali salehinejad.

1 Department of Psychology and Neurosciences, Leibniz Research Centre for Working Environment and Human Factors, Dortmund, Germany

Elham Ghanavati

2 Department of Neuropsychology, Institute of Cognitive Neuroscience, Faculty of Psychology, Ruhr University Bochum, Bochum, Germany

Md Harun Ar Rashid

Michael a. nitsche.

3 Department of Neurology, University Medical Hospital Bergmannsheil, Bochum, Germany

Executive functions, or cognitive control, are higher-order cognitive functions needed for adaptive goal-directed behaviours and are significantly impaired in majority of neuropsychiatric disorders. Different models and approaches are proposed for describing how executive functions are functionally organised in the brain. One popular and recently proposed organising principle of executive functions is the distinction between hot (i.e. reward or affective-related) versus cold (i.e. purely cognitive) domains of executive functions. The prefrontal cortex is traditionally linked to executive functions, but on the other hand, anterior and posterior cingulate cortices are hugely involved in executive functions as well. In this review, we first define executive functions, their domains, and the appropriate methods for studying them. Second, we discuss how hot and cold executive functions are linked to different areas of the prefrontal cortex. Next, we discuss the association of hot versus cold executive functions with the cingulate cortex, focusing on the anterior and posterior compartments. Finally, we propose a functional model for hot and cold executive function organisation in the brain with a specific focus on the fronto-cingular network. We also discuss clinical implications of hot versus cold cognition in major neuropsychiatric disorders (depression, schizophrenia, anxiety disorders, substance use disorder, attention-deficit hyperactivity disorder, and autism) and attempt to characterise their profile according to the functional dominance or manifest of hot–cold cognition. Our model proposes that the lateral prefrontal cortex along with the dorsal anterior cingulate cortex are more relevant for cold executive functions, while the medial–orbital prefrontal cortex along with the ventral anterior cingulate cortex, and the posterior cingulate cortex are more closely involved in hot executive functions. This functional distinction, however, is not absolute and depends on several factors including task features, context, and the extent to which the measured function relies on cognition and emotion or both.

Introduction

Executive functions and their domains.

Executive functions (EFs), also called cognitive control, refer to a family to top-down cognitive processes required for goal-directed behaviours ( Diamond, 2013 ; Miller and Cohen, 2001 ). These higher-order cognitive functions involve active maintenance of goal presentations and the means to achieve these goals ( Miller and Cohen, 2001 ). In this process, different types of information processing, different sensory modalities (e.g. visual and auditory), and different systems responsible for response execution, memory updating and retrieval, and emotional evaluation are involved. Accordingly, a wide range of functions and brain regions are involved in EFs. These higher-order cognitive functions are also required for adapting and regulating behaviour, mental and physical health, and cognitive, social, and psychological development ( Diamond, 2013 ). Deficits of EFs or executive dysfunctions are commonly observed in patients with psychiatric and mental disorders ( Elliott, 2003 ; Reimann et al., 2020 ). It is important to consider EFs as a meta-cognitive, supervisory, or controlling system rather than being tied to particular cognition domains ( Ward, 2020 ). Nevertheless, EFs are commonly described in terms of specific types of information processing or cognitive functions.

Traditionally, the concept of EFs was closely related to the distinction between two types of information processing: automatic versus controlled processing ( Shiffrin and Schneider, 1977 ). In this framework, EFs refer to those behaviours and processes that require intentional, online exert of control. Another popular model of EFs is to categorise them into separate modular-type processes and specific cognitive functions. There is a general agreement about three core EFs: response inhibition (e.g. inhibitory control), working memory, and cognitive flexibility ( Miyake et al., 2000 ). Similar to this, early works attempted to describe EFs in terms of certain kinds of information processing associated with specific behavioural tasks. These processes can be summarised in (1) task-setting and problem-solving abilities, (2) response inhibition abilities, (3) task switching abilities, and (4) multitasking ( Ward, 2020 ). In addition to these well-established accounts of EFs, results of neuroimaging studies suggest several organising principles of EFs. One of these organising principles is related to hemispheric differences of the neural substrates of EFs, which considers dissociated functional roles of the left and right hemispheres. In one such model, left lateral prefrontal cortex (PFC) is considered specialised for task-setting functions and the right lateral PFC is specialised for monitoring performance ( Stuss and Alexander, 2000 ). Another proposed model organises the neural substrates of EFs anatomically from anterior to posterior parts of the brain. In one such model, a posterior to anterior gradient is considered for the lateral PFC with a differential functional specificity of the dorsal (linked to action planning) versus ventral (linked to language and objects) routes ( Badre and D’Esposito, 2009 ). The updated version of this theory emphasizes on separate brain networks that interact via local and global hierarchical structure ( Badre and Nee, 2018 ).

A recently emerging and perhaps the least controversial organising principle of EFs is to distinguish between EFs based on the extent they are related to emotion (e.g. hot EFs ) or purely cognitive aspects (e.g. cold EFs ) ( Ward, 2020 ). Hot EFs, involve processing of information related to reward, emotion, and motivation, while cold EFs involve purely cognitive information processing. Examples of hot and cold EFs are ‘monetary delay discounting’ and ‘working memory letter’ tasks, respectively. The hot versus cold principle has several advantages for organising EFs. First, in this model both cognition and emotion are considered. Second, it presents EFs in a spectrum-like model, indicating that all domains of EFs can be hot or cold depending on contextual information, and third, broader regions of the brain are considered for EFs. A network approach, however, is needed to more accurately depicts functional organization of hot vs cold domains of EF. The current knowledge of neural substrates of hot versus cold EFs distinct between lateral ( cold -related) and medial ( hot -related) regions of the PFC. This, however, is not limited to the PFC regions and other cortical and subcortical areas appear to be involved as well. Major domains, tasks, and neural substrates of hot versus cold EFs based on the currently available studies are summarized in Figure 1 .

Current knowledge about domains and behavioural tasks of executive functions (a), involved brain structures (b) and underlying assumptions/features (c) of hot versus cold executive functions.

SST: stop signal task; AX-CPT: AX Continuous Performance Task; ERT: emotional regulation task.

Although the hot–cold organising principle of EFs has been most often linked to the PFC (as shown in Figure 1 ), here we attempt to broaden this principle to cingulate areas as well, due to their significant involvement in executive and cognitive control functions. It is, however, notable that EFs and especially hot EFs are closely related to subcortical areas involved in emotional processing, including the amygdala, insula, striatum (including putamen, caudate, and nucleus accumbens) hippocampus, and brainstem ( Ardila, 2019 ; Heyder et al., 2004 ; Pessoa, 2009 ). As the scope of this review is focused on prefrontal and cingulate cortices, the specific contributions of these subcortical regions will not be covered in detail here, but will be mentioned where required. In the next section, we describe hot and cold EFs with a specific focus on majorly involved cortical regions, namely, the PFC and cingulate cortex. Before that, we briefly mention methods in cognitive neuroscience that provide important insights into the relevance of brain areas, and networks associated with EFs.

Studying EFs: neuroimaging versus non-invasive brain stimulation methods

Recent advances in the cognitive neurosciences have provided us with novel, non-invasive methods for studying human cognition. Neuroimaging, particularly functional magnetic resonance imaging (fMRI), has become a dominant tool in cognitive neuroscience research and especially human cognition ( Dolan, 2008 ). The emergence of this method has revolutionized study of the living human brain and fMRI is the most widely used technique in cognitive neuroscience ( Newman, 2019 ). fMRI relies on blood oxygenation level–dependent (BOLD) contrast, which arises due to the magnetic susceptibility of deoxyhaemoglobin (deoxy-Hb). To put it briefly, an increase of neural activity leads to an increase of blood volume and thus the proportion of oxygenated haemoglobin (oxy-Hb) in the region, resulting in an increased BOLD signal. This BOLD signal is indicative of brain activity. When it is time-locked to an event/stimulus, it can be used to reveal neural correlates of cognition. A region with enhanced activity refers to a local increase of brain metabolism during performance of an experimental task compared to the baseline. With fMRI, we can investigate which brain regions are activated during cognitive task performance, including EFs. Most of our knowledge about the brain regions involved in EFs comes from neuroimaging studies ( Elliott, 2003 ; Yuan and Raz, 2014 ). However, they come with some limitations. Apart from a relatively poor temporal resolution, which is, however, not the case for electroencephalogram (EEG), the other well-known neuroimaging method, fMRI delivers correlational information about the involvement of brain areas and networks in human cognition. In other words, the evidence provided by brain imaging methods is purely correlative and does not allow us to infer causal relationships between brain and behaviour.

While such correlative information about brain–behaviour relations is valuable and informative, it does not allow to easily infer causality of brain–behaviour relationships. Here, tools that allow active manipulation of brain activity come into play. Non-invasive brain stimulation (NIBS) is a group of methods for modulating neural processes of the brain, enabling us to directly study how an experimentally altered neural activity affects behaviour ( Polania et al., 2018 ). Transcranial magnetic stimulation (TMS) and transcranial electrical stimulation (tES) are two commonly used and well-established NIBS techniques. TMS is based on principles of electromagnetism which ultimately leads to electrical stimulation of brain regions in a focal way, and transcranial direct current stimulation (tDCS), the most common used tES methods, uses a weak, painless electrical current applied to the scalp, thereby modulating brain excitability in a more non-focal way ( Nitsche and Paulus, 2000 ). Depending on a specific frequency (for TMS) and stimulation polarity/intensity/duration (for tDCS), different TMS and tDCS protocols can result in excitatory or inhibitory after-effects that might last for several minutes and in this case are linked to long-term potentiation or long-term depression ( Polania et al., 2018 ). Due to such effects on cortical excitability and neuroplasticity, which are physiological foundations of cognition, these techniques have great potential for experimental investigation of the physiological foundations behind human cognition.

As briefly mentioned, various cortical and subcortical regions are involved in EFs. While neuroimaging methods can show the functional and structural correlates of EFs in the brain, with NIBS (e.g. TMS and tDCS) we can further complement our knowledge of the brain regions/networks supporting EFs. In this review, we will mostly focus on the evidence coming from these methods (i.e. fMRI, TMS, and tDCS) in order to picture how hot versus cold EFs are organised in the brain. We focus mainly on studies conducted in healthy individuals in this review. However, due to high relevance of hot versus cold cognition in neuropsychiatric disorders, we discuss important clinical implications of this distinction at the end. A brief description of the research methods used in the studies of this review is summarised in Table 1 .

Characteristics of commonly applied neuroimaging and non-invasive brain stimulation methods for studying human cognition.

fMRI: functional magnetic resonance imaging; EEG: electroencephalogram; TMS: transcranial magnetic stimulation; tES: transcranial electrical stimulation.

Hot versus cold EFs in the PFC

In traditional and contemporary conceptualisations of EFs, there is a consensus that the frontal lobe and especially the PFC have a critical role ( Miller and Cohen, 2001 ). The PFC has extensive connections with almost all sensory systems, cortical regions, and subcortical structures involved in action, motor response, memory, emotion, and affect ( Miller and Cohen, 2001 ). Our focus here is on how PFC structures are related to hot versus cold EFs. Broadly speaking, the most basic anatomical division within the PFC defines three cortical areas: the lateral PFC, the medial PFC, and the orbital PFC. The lateral PFC lies anterior to the premotor areas and the frontal eye fields and is situated close to the surface of the skull. It includes the dorsolateral prefrontal cortex (DLPFC) (Brodmann’s areas 46 and 9) and the ventrolateral prefrontal cortex (VLPFC) (Brodmann’s areas 44, 45, and 47) ( Ward, 2020 ). The medial PFC lies between the two hemispheres and anterior to the corpus callosum and the anterior cingulate cortex (ACC) (Brodmann’s area 24 and adjacent regions). The orbitofrontal cortex (OFC) lies above the orbits of the eyes and the nasal cavity (Brodmann’s areas 11, 12, 13, and 14). It is of note that the OFC is functionally and anatomically related to the ventral part of the medial PFC and is sometimes referred to as the ventromedial prefrontal cortex (VMPFC) (Brodmann’s area 10, 14, 25, and 32 and parts of 11, 12, and 13) ( Öngür and Price, 2000 ), but these areas are not identical at finer anatomical divisions ( Ward, 2020 ).

The EF domains related to these areas can be classified in different ways. One popular classification is to functionally specify these areas based on the extent to which these are involved in hot (e.g. emotion and motivation-related) and/or cold EFs (e.g. purely cognitive). Hot EFs mainly involve the orbital and medial PFC, including the OFC and VMPFC, and cold EFs engage the lateral PFC, including the DLPFC and VLPFC ( Öngür and Price, 2000 ; Stuss, 2011 ; Ward, 2020 ). Functionally speaking, hot EFs are top-down cognitive processes that operate in contexts with significant emotional and motivational salience, gratification, rewards and/or punishment ( Zelazo and Carlson, 2012 ; Zelazo et al., 2005 ). Examples of hot EF are delay discounting, affective/risky decision-making, and interpersonal and social behaviour. Cold EFs are top-down cognitive processes that are logically based or mechanistic ( Chan et al., 2008 ) and operate in affectively neutral contexts ( Zelazo and Carlson, 2012 ). Examples of cold EFs include working memory, response inhibition, attentional control, and planning as far as these functions are not presented in an emotional context. In what follows, we provide evidence from neuroimaging (i.e. fMRI) and brain stimulation studies (i.e. TMS and tDCS) about the relation of hot versus cold EFs to different PFC areas.

Neuroimaging studies

Pfc and cold efs.

A large body of evidence from neuroimaging studies show that the lateral PFC, including DLPFC and VLPFC, are involved in cold EFs. Response inhibition, the ability to suppress unrelated or inappropriate stimuli/responses, is a core cold component of EFs. It is well-established that a specific region of the PFC, the right inferior frontal gyrus (r-IFG), is critical for inhibitory control ( Aron et al., 2004 , 2014 ; Hampshire et al., 2010 ). The r-IFG is moreover connected with the ACC, involved in error detection, and the lateral OFC when conveying information from non-reward systems ( Du et al., 2020 ). The left IFG is also involved in verbal fluency, another major cold EF domain ( Costafreda et al., 2006 ). The lateral PFC, including the DLPFC, is another well-documented region actively involved in working memory updating and tasks requiring executive control ( Lemire-Rodger et al., 2019 ; Wagner et al., 2001 ). The PFC, however, should be considered as a part of a larger brain network, the fronto–cingulo–parietal network, that supports cognitive control via interaction of different cortical (and also subcortical) structures.

A great example of a cold EF and its association with subregions of PFC is navigation behavior. Planning, decision-making, goal-coding, and adaptive behavior are those domains of EFs required for real-world navigation ( Patai and Spiers, 2021 ), all of which are functionally cold . At anatomical level, navigation behavior involves interaction of subregions of PFC (e.g., DLPFC, VLPFC), cingulate cortex (dorsal ACC) as well subcortical regions such as hippocampus ( Patai and Spiers, 2021 ). An fMRI meta-analytic study of 193 studies revealed a common pattern of activation in the lateral PFC, dorsal ACC, and parietal cortex across major cold EF domains (working memory, inhibition, flexibility, and planning) ( Niendam et al., 2012 ), indicating that cold EFs are supported by this cognitive control network with the DLPFC as a key region ( Figure 2 ). The connectivity between the lateral PFC and dorsal ACC indicates that these regions are rather involved in cold EFs which are discussed in more detail in the section dedicated to the cingulate cortex.

Lateral view of the prefrontal cortex (PFC) regions and association with hot and cold EFs. The lateral PFC includes dorsolateral prefrontal cortex (DLPFC) and ventrolateral prefrontal cortex (VLPFC) that are predominately involved in cold EFs (in blue). The medial PFC and orbitofrontal cortex (OFC) are predominantly involved in hot EFs. The hot PFC regions have extensive connections with several subcortical structures that process emotion and motivation will be discussed later (Figure 4).

Marked regions are close approximate to the intended regions. Also note that circuit nodes and connections are excluded in this and later figures for clarity.

PFC and hot EFs

A large and compelling body of evidence from neuroimaging studies shows that the medial and orbital PFC, specifically the VMPFC and OFC, are involved in cognitive functions related to reward, emotion, motivation, and social evaluation. During cognitive control of emotional stimuli, the medial PFC and OFC are usually activated ( Ochsner and Gross, 2005 ). These regions, however, interact with the lateral PFC (e.g. VLPFC) and ACC during effortful control mechanisms when it comes to emotional and motivational stimuli ( Ochsner and Gross, 2005 ; Pessoa, 2009 ). This indicates that hot EFs involve both brain regions involved in cold executive control (e.g. lateral PFC and ACC), and those involved in processing of emotion and motivation. One major hot EF is risky decision-making or decision-making under uncertainty. Neuroimaging studies have repeatedly shown that the VMPFC and OFC are involved in decision-making under uncertainty ( Bechara et al., 2000 ; Clark et al., 2008 ; Fellows and Farah, 2007 ; Lipsman et al., 2014 ; Windmann et al., 2006 ). Delay discounting or temporal discounting is another classic example of hot EFs. Here again, studies show a prominent involvement of the VMPFC and OFC ( Wang et al., 2016 ). What makes the medial–orbital PFC at least partially relevant for emotional and motivational processing is their connectivity with subcortical structures such as the limbic system, amygdala, and insula ( Matyi and Spielberg, 2020 ; Sharpe and Schoenbaum, 2016 ). These regions are also connected with the posterior cingulate cortex (PCC), the counterpart region in the cingulate cortex which is discussed in the next section. Indeed, proposed models for delay-discounting behaviour in humans based on neuroimaging data assume that a unitary system encompassing the medial PFC, including VMPFC, and PCC are involved in immediate and delayed reward evaluation ( Kable and Glimcher, 2007 ; Peters and Büchel, 2010 ). In this line, an fMRI study showed coactivation of the VMPFC and PCC during monetary reward encoding ( Lin et al., 2011 ). Another fMRI study showed that when people consider themselves to experience a positive future, greater activity was observed in the VMPFC and PCC, indicating the connecting of these regions as well ( Blair et al., 2013 ).

NIBS studies

Attentional control is a core component of cold EFs. NIBS studies have shown that both TMS and tDCS over the left, right, or bilateral DLPFC enhance selective attention ( Gladwin et al., 2012 ; Pecchinenda et al., 2015 ; Vanderhasselt et al., 2010 ). Other NIBS studies have moreover shown a performance-enhancing effect of increasing activity of the lateral PFC, including DLPFC and r-IFG, on attentional control and sustained attention ( Coffman et al., 2012 ; Hwang et al., 2010 ). Regarding inhibitory control, the DLPFC and the IFG, with a right hemispheric predominance, are involved in response inhibition by both tDCS and TMS studies (for a review, see Brevet-Aeby et al., 2016 ). Working memory is another major component of cold EFs which was widely studied by NIBS. Recent review and meta-analytic studies have confirmed an enhancing effect of increased activity of DLPFC on working memory task performance ( Bagherzadeh et al., 2016 ; Brunoni and Vanderhasselt, 2014 ; Hill et al., 2016 ; Mancuso et al., 2016 ). Another recent relevant meta-analysis investigated the effects of prefrontal tDCS here on executive function and found that anodal tDCS over the DLPFC increases performance of updating tasks and global EF performance under specific stimulation parameters ( Imburgio and Orr, 2018 ). A recent tDCS study that targeted the DLPFC, temporal cortex, and posterior parietal cortex showed that DLPFC activation contributes to EFs regardless of task modality (semantic, phonemic, and visuospatial) ( Ghanavati et al., 2019 ). Other studies show also enhanced problem-solving, and cognitive flexibility as a result of increased activity of lateral PFC regions via NIBS ( Cerruti and Schlaug, 2009 ; Lucchiari et al., 2018 ; Metuki et al., 2012 ; Nejati et al., 2018a ). These studies clearly show that the DLPFC and lateral PFC regions are involved in working memory and other cold EFs, although these structures are involved in specific aspects of emotional processing too ( Nejati et al., 2021 ; Lindquist et al., 2012 ).

Regarding hot EFs, numerous NIBS studies show involvement of medial and orbital PFC regions. The involvement of the medial–orbital PFC (e.g. VMPFC and OFC) in reward and emotion processing is well-documented by both tDCS ( Abend et al., 2019 ; Manuel et al., 2019 ) and TMS studies ( Konikkou et al., 2017 ). A recent tDCS–fMRI study showed a causal link between VMPFC activation and the experience and regulation of anger, a hot EF domain, in an anger-provoking game involving fair and unfair offers, supporting its role in anger regulation ( Gilam et al., 2018 ). Activity of the VMPFC in this study was coupled with both ACC and PCC activation, depending on the specific offer with more PCC activation during unpleasant offers. Social variables that include evaluation, interaction, theory of mind, and empathy are also considered hot EFs and NIBS studies have shown that activation of the VMPFC modulates such social variables ( Adenzato et al., 2017 ; Chib et al., 2013 ; Li et al., 2020 ; Salehinejad et al., 2020a ). A recent tDCS study specifically investigated the interaction of the DLPFC and OFC in hot versus cold EFs by applying excitability-enhancing anodal and excitability-reducing cathodal stimulation ( Nejati et al., 2018a ). Participants conducted response inhibition and problem-solving tasks as measures of cold EFs and risky decision-making and delay-discounting tasks as measures of hot EFs while receiving combined left DLPFC-right OFC stimulation. Increased activity of the left DLPFC concurrent with decreased activity of the OFC prominently improved cold EFs while hot EFs were enhanced under both protocols, those that activated the left DLPFC and the right OFC. The results of this study suggest that hot and cold EFs are placed on a spectrum, with lateral and medial–orbital contributions to cold and hot EFs, respectively, and that no EFs are purely cold or hot. This depends to the extent that each EF domain involves emotion/reward or cognition processing which determines engagement of relevant brain region. The brain regions should be predominantly, but not purely, considered cold and hot as well and this is determined by task feature too.

Hot versus cold EFs in the cingulate cortex

The major anatomical divisions in the cingulate cortex include the anterior, mid, and posterior cingulate cortices, named ACC, MCC and PCC, respectively ( Caruana et al., 2018 ; Vogt, 2005) although some classifications only include ACC and PCC. Here our focus is specifically on the ACC and PCC. Studying involvement of the cingulate cortex in EFs has been mostly limited to the anterior portion of the cingulate cortex or ACC (Brodmann’s area 24 and adjacent regions). The ACC is traditionally linked to the ability of error detection, a cognitive mechanism that monitors for errors and recalibrates task performance accordingly ( Carter et al., 1998 ). As mentioned earlier, one fundamental domain of EFs is inhibitory control or response inhibition ( Miyake et al., 2000 ) which is usually involved in situations with conflicting stimuli. The Stroop test and Go/No-Go tasks are well introduced behavioural examples with conflicting and competing stimuli. Conflict monitoring signals the need for increased cognitive control to resolve current conflicts, and here, the ACC can be linked to EFs ( Cole et al., 2009 ; Shenhav et al., 2013 ). The involvement of the ACC in error detection abilities, which requires attentional control, suggests its predominant role in cold EFs. However, there is compelling evidence for an involvement of the ACC in emotion and reward–related processes as well. In fact, the ACC can be subdivided into areas differentially related to cognitive versus emotional functions. The dorsal ACC is linked to cognitive, whereas the ventral ACC is linked to emotional processing ( Gasquoine, 2013 ; Lockwood and Wittmann, 2018 ). In addition to the ventral ACC, the PCC has been increasingly studied in recent years and linked to some domains of EFs related to hot cognition ( Platt and Plassmann, 2014 ). In what follows, we present evidence from fMRI and brain stimulation studies about how hot and cold EFs are linked to the cingulate cortex with a primary focus on the ACC and PCC.

ACC and cold/hot EFs

The relation of the ACC to cold EFs is based on its primary role in conflict detection during information processing, which signals the need to engage top-down attentional control and performance monitoring ( Petersen and Posner, 2012 ). In this line, an early fMRI study aimed to investigate which levels of processing are being monitored by the ACC during performance of a task with conflicting stimuli and responses. It was shown that the ACC has a highly specific contribution to EFs through detection of conflicts at response level which usually occurs late during information processing ( van Veen et al., 2001 ). This suggests that the executive control exerted by ACC is different from the contribution of the DLPFC. This was confirmed in another fMRI study that specifically compared the roles of the DLPFC versus ACC in attentional control. Attentional control is a clear example of cold EFs as it requires exerting control over the goal, monitoring of the goal, and the processes needed to achieve the goal. It is a fundamental component of executive control that comes into play in almost all EF domains and is traditionally linked to the DLPFC ( Miller and Cohen, 2001 ). In that fMRI study, it was, however, shown that not only the DLPFC takes a leading role in implementing top-down attentional control, but also that the ACC is involved in specific additional aspects of attentional control, such as response-related processes ( Milham et al., 2003 ). Regardless of the type of attentional control, the contribution of the ACC to this effortful process indicates that it is relevantly involved in cold EFs.

Involvement of the ACC in cold EFs is more precisely linked to the dorsal ACC. This region is a key hub in a network of brain regions involved in domain-general EFs in humans ( Petersen and Posner, 2012 ; Shenhav et al., 2013 ); however, this ‘domain-general’ region has also a domain-specific function related to stimuli valence. An fMRI study in healthy participants showed that while dorsal ACC activity is required for processing task-irrelevant information during the Stroop task performance, which is distracting due to its cognitive content, the ventral ACC is activated during presentation of task-irrelevant information which is distracting due to emotional content ( Mohanty et al., 2007 ). This study is a good example of how cognitive versus emotional information in the context of conflicting stimuli is processed by dorsal versus ventral ACC, respectively. In other words, both dorsal and ventral ACC seem to be involved in effortful control over stimuli, but these areas differ with respect to the kind of stimuli processed, that is, cognitive or emotional. Another fMRI study in healthy subjects, as well as in patients with ACC lesions, found that the dorsal ACC is actively involved in effortful cognitive and motor behaviour in healthy individuals, but that these activities were blunted in patients with focal lesions of the ACC ( Critchley et al., 2003 ). Together, these studies suggest that the dorsal ACC is involved in cold EFs. Another influential account, however, links dorsal ACC functions with motivation and reward–based decision-making ( Wallis and Kennerley, 2011 ). One of the most recent accounts for the role of the dorsal ACC integrates these two perspectives and suggests that that the dorsal ACC plays a central role in decisions about the allocation of cognitive control based on a cost/benefit analysis that identifies the highest expected value of control ( Shenhav et al., 2016 ). According to this theory, exerting effortful control ( cold EF) is based on the analysed value of control ( hot EFs), and the dorsal ACC has a central role in these processes.

In contrast to the controversy over the functions of the dorsal ACC, there is a general agreement that the ventral region of the ACC is rather linked to emotional, motivational, and social information processing ( Gasquoine, 2013 ; Lockwood and Wittmann, 2018 ). An fMRI study that measured performance monitoring with a valence-based task showed that the ventral ACC along with the PCC is specifically sensitive to the valence of feedback ( Mies et al., 2011 ), specifically the perigenual ACC and subgenual ACC. These are two subregions of the ventral ACC that are linked with hot EFs involving emotion, motivation, and social decision-making ( Lockwood and Wittmann, 2018 ). In accordance, a recent fMRI study showed connectivity between both perigenual and subgenual ACC and OFC/VMPFC, which are involved in emotional and motivational processing ( Du et al., 2020 ). This might partially explain why the ventral ACC is rather involved in hot EFs, as it is structurally connected and anatomically closer located to those regions of the PFC that are relevant for hot EFs ( Figure 3 ).

The cingulate cortex in the human brain and association with hot and cold EFs. The anterior cingulate cortex (ACC) includes dorsal ACC (dACC) that is predominately involved in cold (in blue) and ventral ACC, consisting of perigenual (pgACC) and subgenual (sgACC) that are predominately involved in hot EFs (in red), respectively. The posterior cingulate cortex (PCC) is predominantly involved in hot EFs (in red). Note that the anatomical borders of the cingulate cortex in this figure is based on the anatomical studies (see Caruana et al., 2018 and Vogt, 2005 for details). In some studies, the mid-cingulate cortex is part of the dACC.

Marked regions are close approximate to the intended regions. Also, note that most circuit nodes and connections (specially subcortical regions) are excluded for clarity.

PCC and hot EFs

In comparison to the ACC, a relatively limited number of studies explored functional organisation of the PCC in EFs. Previous fMRI studies mainly investigated PCC activation during memory processes, specifically episodic memory. However, functional imaging studies consistently found that emotionally salient stimuli activate the PCC ( Maddock, 1999 ), and the involvement of PCC in episodic and autobiographical memory could be due to its role in the interaction between emotion and memory ( Maddock et al., 2001 ). The involvement of the PCC in emotion, and thus hot EFs, is anatomically related to its connectivity and coactivation with the amygdala, insula, and OFC ( Vogt et al., 2000 ). An fMRI study found that the PCC was activated bilaterally during both unpleasant and pleasant, as compared to neutral words in a memory task ( Maddock et al., 2003 ). Recent studies have provided more convincing evidence for the involvement of the PCC in emotional stimuli processing. Le et al. (2019) investigated neural substrates underlying behavioural avoidance in alcohol drinkers using a valence-based Go/No-Go task. Their major finding was increased activity in the PCC during motivated avoidance and incentivised inhibition of action which was correlated with sensitivity to punishment. In another recent TMS–fMRI study, 1 Hz rTMS was applied to the medial PFC of healthy participants who immediately thereafter underwent fMRI while performing an emotional self-referential task ( De Pisapia et al., 2019 ). Neuroimaging findings revealed that the PCC was the only region that was specifically activated by negative-valence stimuli and as a result of TMS (TMS–valence interaction). Another recent study showed elevated functional connectivity between the PCC and subgenual PFC (e.g. ventral–medial PFC) as a maker of rumination, in depressed individuals versus healthy controls ( Benschop et al., 2021 ). Rumination is a cognitive risk factor resulting from deficient cognitive control over negative emotions and a maladaptive self-referential processing and thus related to hot EFs. Overall, findings of these studies indicate relevant connectivity between the medial PFC and PCC, which will be discussed later in our prefrontal-cingular network model for hot versus cold EFs ( Figure 3 ).

In contrast to neuroimaging studies that provide correlates of brain–behaviour relations, NIBS methods allow to infer the causality of these associations. The feasibility of NIBS to modulate ACC and PCC physiology is, however, limited in part due to the anatomical depth of these regions. Regarding cold EFs, some NIBS studies investigated the impact of the ACC stimulation on EFs and support the contribution of this region to these EFs.

In a recent tDCS study, using a high-definition (HD) stimulation protocol (i.e. 4 × 1 electrode montage), anodal and cathodal tDCS were applied over the dorsal ACC during performance of a cognitive and emotional cognitive Stroop task ( To et al., 2018 ). Anodal stimulation over the dorsal ACC enhanced performance on the cognitive incongruent stimuli of the task, which requires effortful attentional control, while cathodal stimulation over the same region enhanced performance on the block including emotional incongruent stimuli. Furthermore, anodal stimulation significantly increased beta frequency band activity, which is associated with attentional control. A recent tDCS–fMRI study applied anodal tDCS over the ACC and measured behavioural performance in a colour-word Stroop task, and resting-state fMRI after stimulation ( Khan et al., 2020 ). While behavioural findings showed enhanced Stroop task performance as a result of improved cognitive control, neuroimaging findings showed a significant decrease of functional connectivity of the cognitive control network, including ACC, which is associated with less effortful information processing. In a TMS study that targeted the ACC during a counting Stroop task performance (a cold EF), excitatory 10 Hz rTMS ( Hayward et al., 2004 ) over both the dorsal and ventral ACC abolished Stroop interference. Together, NIBS studies show that stimulation of the dorsal ACC is associated with enhanced cold executive control.

Not many NIBS studies are conducted to modulate the activity of the PCC to explore its impact on cognitive functions. The TMS–fMRI study conducted by De Pisapia et al. (2019) , which found a TMS–valance interaction for the activation of the PCC after applying TMS over the medial PFC, is, however, a relavant example. In this study, the neural basis of emotional content in self-referential processing, a hot EF, was investigated by stimulating the medial PFC with 1 Hz TMS. Participants then conducted a valence-based self-referential task. Stimulating the medial PFC activated a network of regions including the PCC which was specifically sensitive to emotionally negative aspects of the stimuli. In another study, the right DLPFC was stimulated with inhibitory TMS, and delay-discounting task performance was monitored during positron emission tomography (PET) scan. The PCC, and especially the posterior parietal lobule, which is part of the PCC, were activated during this task ( Cho et al., 2012 ). Together, NIBS studies available so far show that activation of the PCC is observed during performing hot EF tasks ( Table 2 ).

Summary of hot versus cold executive functions in the studies of this review based on the applied tasks, techniques, and involved regions.

DLPFC: dorsolateral prefrontal cortex; VLPFC: ventrolateral prefrontal cortex; dACC: dorsal anterior cingulate cortex; sgACC: subgenual anterior cingulate cortex; vACC: ventral anterior cingulate cortex; rACC: rostral anterior cingulate cortex; OFC: orbitofrontal cortex; TBS: Theta Burst Transcranial Stimulation; HF-rTMS: high-frequency (excitatory) transcranial magnetic stimulation; LF-rTMS: low-frequency (inhibitory) transcranial magnetic stimulation; tDCS: transcranial direct current stimulation; PET: positron emission tomography; VMPFC: ventromedial prefrontal cortex; PASAT: Paced Auditory Serial Addition Test; AX-CPT: AX-Continuous Performance Test; WCST: Wisconsin Card Sorting Task; N/A: not applicable; r-IFG: right inferior frontal gyrus; l-IFG: left inferior frontal gyrus; ToM: theory of mind.

The studies in this table include selective neuroimaging/NIBS studies.

A prefrontal-cingular network model for hot versus cold EFs

So far, we discussed how hot and cold EFs can be linked to different regions within the PFC and the cingulate cortex. Anatomical and functional connectivity between the cingulate and prefrontal cortices can explain functional organisations of hot–cold EFs in this network. Two major functional connectivity branches are considered in this proposed model: (1) the connectivity between the lateral PFC (e.g. DLPFC) and ACC (specifically the dorsal ACC) and (2) the connectivity between the orbital–medial PFC (e.g. VMPFC and OFC), ventral ACC, and PCC ( Figure 4 ). In this section, we discuss a prefrontal-cingular network that may more comprehensively account for hot and cold EFs. As subcortical regions are highly involved in hot EFs, we also depicted major subcortical limbic structures involved in emotional and motivational processing which are connected to hot -related prefrontal-cingular structures.

The prefrontal-cingular network in the human brain and association with hot and cold EFs. The lateral PFC, including DLPFC and VLPFC, along with dorsal ACC are predominantly related to cold EFs and can be considered as the cold stream. The PCC, medial and orbital PFC (VMPFC and OFC), and ventral ACC constitute the hot stream and are predominantly related to hot EFs. The VLPFC is also connected to medial and orbital PFC. The hot EFs stream is closely connected with several limbic structures that are involved in emotional and motivational processing (red curve). The connectivity between the hippocampus and lateral prefrontal cortex subregions is also relevant for major cold EFs such as working memory and navigation behavior.

DLPFC: dorsolateral prefrontal cortex; VLPFC: ventrolateral prefrontal cortex; ACC: anterior cingulate cortex; dACC: dorsal anterior cingulate cortex; vACC: ventral anterior cingulate cortex; VMPFC: ventromedial prefrontal cortex; OFC: orbitofrontal cortex; PCC: posterior cingulate cortex; VA: ventral striatum; NA: nucleus accumbens; A: amygdala; H: hippocampus.

Marked regions are close approximate to the intended regions. Note that some circuit nodes and connections specially with subcortical areas are excluded for clarity and that some connections (shown by arrows) may be indirect.

The cingulate cortex in monkeys and humans has extensive connections with the PFC ( Pandya et al., 1981 ). The dorsal ACC (Brodmann’s area 32) projects mostly to the lateral PFC, including DLPFC, and the mid-OFC ( Pandya et al., 1981 ), and the ventral ACC (Brodmann’s area 25), which is part of the VMPFC, has connections with subcortical regions like amygdala and insula, and projects also to the VMPFC, OFC also lateral PFC. The PCC (Brodmann’s area 31), however, projects to the VMPFC, ventral ACC, and OFC ( Leech and Sharp, 2014 ). Although there is a relative overlap between the ACC and PCC connectivity with PFC regions at the anatomical level, anterior and posterior parts of the cingulate cortex show a more differentiated functional specificity. As discussed in the previous section, both neuroimaging and brain stimulation studies show that the dorsal anterior part of the cingulate cortex is mainly involved in cognitive control functions ( Critchley et al., 2003 ) such as response inhibition and conflict monitoring ( Botvinick et al., 2004 ). From a functional perspective, these EF domains are mostly relevant for cold EFs and similar to those domains the lateral PFC is involved in, although a functional difference is observed for the timing of attentional control exerted by the DLPFC and dorsal ACC ( Milham et al., 2003 ). The connectivity between the lateral PFC and ACC and their coactivation during cognitive control tasks supports this functional link ( Tik et al., 2017 ). The ventral ACC is mainly involved in exerting control over emotional stimuli which indicates that the ACC is involved in both cold and hot EFs ( Gasquoine, 2013 ; Lockwood and Wittmann, 2018 ). However, the cognitive functions specific to the posterior part of cingulate cortex (e.g. PCC) are shown to be involved in emotional processing, value-based decision-making, subjective valuation, and motivational states ( Platt and Plassmann, 2014 ; Vogt et al., 2000 ). In this connection, neuroimaging studies have shown coactivations between the PCC and the medial–orbital PFC (e.g. VMPFC and OFC) ( De Pisapia et al., 2019 ; Le et al., 2019 ; Lin et al., 2011 ; Vogt et al., 2000 ).

This relative functional specificity of the anterior versus posterior parts of the cingulate cortex seems similar to the ‘ hot – cold ’ organising principle of EFs in the PFC. In the PFC, the hot versus cold organisation is proposed based on functional differences between the lateral versus medial–orbital regions. Neuroimaging and brain stimulation studies have documented that cold EFs are rather supported by the lateral PFC, while hot EFs are related to the medial–orbital PFC ( Nejati et al., 2018a ; Ochsner and Gross, 2005 ; Öngür and Price, 2000 ; Peterson and Welsh, 2014 ; Ward, 2020 ). Integrating this functional differentiation in the PFC and cingulate cortex with respect to hot versus cold EF allows us to consider a broader network. According to the prefrontal-cingular network, the lateral PFC (e.g. DLPFC and VLPFC) is functionally more closely related to the dorsal ACC, while the medial–orbital PFC (e.g. VMPFC and OFC) is functionally and anatomically more closely related to the ventral ACC and PCC. This does not, however, exclude a contribution of the dorsal ACC, which is documented to be involved in emotion, affect, and pain, to hot EFs ( Shackman et al., 2011 ). Considering a purely segregationist model for the ACC seems to be no longer appropriate ( Shackman et al., 2011 ), however, our discussion here is limited to EFs and specifically hot versus cold domains, which seems to be functionally supported by different regions of the cingulate cortex.

It is important to consider the following additional aspects with regard to the proposed prefrontal-cingular network. First, the involvement of the lateral PFC and dorsal ACC in cold EFs, and the medial–orbital PFC, ventral ACC, and PCC in hot EFs does not imply that these regions are functionally limited to cold or hot cognition. They rather have a predominant functional specificity. As shown in previous studies, there is an interplay between these cold-related and hot-related regions ( Le et al., 2019 ; Nejati et al., 2018a ) and importantly these regions show coactivations depending on specific task features. Second, the hot versus cold EFs distinction should not likewise be considered as representing two separate and unrelated domains of EFs. Although some domains are most purely cognitive, such as inhibitory control or working memory, they might be enriched by emotional features depending on the task, stimuli, and the context used for measuring them. Finally, it should be taken into account that we narrowed our discussion to the prefrontal and cingulate cortices. The prefrontal-cingular network includes the cortical regions most closely involved in EFs based on previous studies. This is not meant to underestimate the engagement of other brain regions, especially subcortical limbic regions, the amygdala–hippocampal systems, and sensorimotor regions of the dorsal striatum (e.g. putamen and caudate nuclei) in EFs, which were, however, beyond the scope of this review.

Clinical implications of hot–cold EFs for neuropsychiatric disorders

The hot–cold distinction of cognition has important clinical implications for both characterising and applying appropriate treatment of neuropsychiatric disorders. In the majority of neuropsychiatric disorders, the core pathophysiology involves cortico-subcortical regions of the brain, and here the prefrontal-cingular network is highly involved ( Heilbronner and Haber, 2014 ; Miller and Cummings, 2017 ). This is in line with the network approach in cognitive neuroscience, which assumes that a dynamically changing pattern of activity over several brain regions is critical for cognitive processes ( Ward, 2020 ). Disturbances of these networks – structural and functional – are related to symptoms and pathophysiology of neuropsychiatric disorders. Accordingly, it is possible to identify symptom-relevant brain networks, and their disturbances, based on connectivity mapping of the human connectome which is one of the major approaches in current biological psychiatry. Abnormalities of the prefrontal-cingular and prefrontal-limbic networks are largely involved in the pathophysiology, symptom expression, and course of the major neuropsychiatric disorders, including but not limited to depression, schizophrenia, anxiety disorder, substance use, and impulse control disorders, as well as major neurodevelopmental disorders (attention-deficit hyperactivity disorder (ADHD) and autism). In what follows, we briefly discuss the respective pathophysiology of some of these disorders and outline whether their cognitive profiles (i.e. hot vs cold ) are fundamental (central) for or rather manifest (relevant expression) in the psychopathology of each disease. A summary of the specific hot versus cold profile of each disorder is shown in Table 3 .

Hot–cold cognitive profile in major neuropsychiatric disorders.

DLPFC: dorsolateral prefrontal cortex; ACC: anterior cingulate cortex; sgACC: subgenual anterior cingulate cortex; OFC: orbitofrontal cortex; PCC: posterior cingulate cortex.

Emotional dysregulation is the core phenotype in depression and in agreement with this, deficits of hot cognition are a common manifestation in depression. From a hot–cold perspective, however, dysfunctional cold EFs, especially cognitive control deficits, are central for the psychopathology of depression, in line with cognitive theories of depression ( Gotlib and Joormann, 2010 ). In other words, cold cognition turns hot in depression ( Roiser and Sahakian, 2013 ). Deficient cold EFs are observed in cognitive control, working memory, and attention ( Nikolin et al., 2021 ; Salehinejad et al., 2017 ), while hot EF deficits involve those EF tasks (mainly cold) that utilise emotionally valenced stimuli, and reward and punishment processing ( Roiser and Sahakian, 2013 ). In accordance, neuroimaging studies show abnormalities of frontal-limbic structures that account for both cold and hot cognitive deficits ( Keren et al., 2018 ; Rive et al., 2013 ). A central functional abnormality of the left and right PFC is proposed in depression with a hypoactivated left and hyperactivated right DLPFC, supported by the results of neuroimaging studies ( Grimm et al., 2008 ). Modulation of the activity of these regions is consequently one focus of NIBS treatment in depression ( Chen et al., 2020 ; Razza et al., 2020 ; Rostami et al., 2017 ). Treatment approaches in depression should consider fundamental cold executive dysfunctions as the primary target more than before in line with the hot – cold pathology of the disease explained above.

Schizophrenia

In schizophrenia, deficient cold cognition has been more extensively studied with respect to the disease pathophysiology and symptoms ( Sheffield et al., 2018 ). A deficient cold cognitive profile seems to be both fundamental to and manifest of the symptoms and underlying pathophysiology. This is also in agreement with the developmental aspect of schizophrenia, including onset in adolescents, where cold cognition deficits are central ( James et al., 2016 ). A well-documented deficient network in schizophrenia that is involved in cold cognition is the thalamocortical circuitry, especially the thalamus-PFC pathway ( Giraldo-Chica and Woodward, 2017 ) and prefrontal-hippocampal connectivity ( Meyer-Lindenberg et al., 2005 ). Deficient cognitive control, processing speed, memory (verbal, working), and reasoning are commonly reported cold EF deficits in schizophrenia ( Table 3 ). However, impaired hot EFs, including risky decision-making, theory of mind, and emotion recognition are also reported in schizophrenic patients, and associated with psychotic symptoms ( MacKenzie et al., 2017 ; Ruiz-Castañeda et al., 2020 ). Regarding treatment approaches, therapeutic targeting of cold EF deficits aligns, however, best with the fundamentally involved cold cognitive profile and pathophysiological characteristics of the disease.

Anxiety disorders

Anxiety disorders are traditionally linked to emotional and threat-related difficulties, and thus, hot cognition deficits (e.g. emotion regulation, threat perception, reward–punishment processing) are central for the pathology of these disorders. These emotional difficulties are, however, strongly linked to deficits in several cold EFs that are stable over time ( Nejati et al., 2018b ; Zainal and Newman, 2018 ). Two well-known theories in this respect are the ‘Attentional Control Theory’ ( Eysenck and Derakshan, 2011 ) and the ‘Cognitive Model’ of pathological worry ( Hirsch and Mathews, 2012 ). In these theories, impaired cold EFs (specifically inhibition and set-shifting abilities), on one hand, and threat-related perceptual and attentional bias, on the other hand, are proposed to be responsible for the overwhelming experience of worry and anxiety. This effect is, however, dependent on the extent to which respective EF tasks include threatening stimuli or significant cognitive load ( Leonard and Abramovitch, 2018 ). Neuroimaging and brain stimulation studies have shown functional and structural abnormalities of cortical regions related to both hot and cold EFs, with a specific focus on the prefrontal-amygdala network in anxiety disorders ( Ironside et al., 2019 ). Here, a hyperactivation of the medial PFC (VMPFC and OFC) and ventral ACC ( hot pathway), which is highly relevant for fear memory and extinction ( Marcovic et al., 2021 ), and a hypoactivation of the DLPFC and dorsal ACC ( cold pathway) are shown to be linked to hypersensitivity of the amygdala and other limbic structures ( Vicario et al., 2019 ).

Substance use disorders

A common and core feature of substance use disorder (SUD) is impaired control over craving, or impulsive behaviour. In accordance, here again deficits of both cold and hot EFs are central to the psychopathology of SUD. According to the neurocognitive model of addiction, hot and cold executive deficits play a fundamental and manifest role in different stages of addiction. In the first two stages (binge/intoxication and withdrawal/negative affect), a deficient reward system is central ( Koob and Volkow, 2016 ), which is related to a deficient hot cognition, and in the preoccupation/anticipation stage, where craving behaviour dominates, a deficient executive control system is relevantly involved ( Koob and Volkow, 2016 ), which is related to cold EFs. Therefore, both hot and cold EFs deficits are involved in the psychopathology of SUD, although hot manifestations of symptoms are predominant. The dual-process model of addiction similarly emphasises on both a cold -related ‘controlled’ system (related to the lateral PFC) and a hot -related ‘impulsive’ system (including mesolimbic and nigrostriatal pathways) ( Wise, 2009 ). Novel treatment approaches in SUD have shown the relevance of targeting the cold -related ‘controlled’ as well as the hot -related ‘impulsive’ system by modulating activity of brain structures including DLPFC and ACC which are connected to reward system ( Alizadehgoradel et al., 2020 ; Zhao et al., 2021 ).

ADHD is a major neurodevelopmental disorder, and executive dysfunctions are central for its psychopathology ( Willcutt et al., 2005 ). Results of functional and structural neuroimaging studies, and behavioural studies exploring EFs show predominantly cold EF deficits in the psychopathology, and pathophysiology of ADHD ( Antonini et al., 2015 ; Hobson et al., 2011 ; Molavi et al., 2020b ; Rubia, 2018 ). Regarding hot EFs, results are mixed, with some studies reporting deficits in affective/motivational EF tasks ( Nejati et al., 2020 ), while others report unimpaired hot EF functions ( Antonini et al., 2015 ). However, neuroimaging, brain stimulation, and behavioural studies have recently shown an impairment of several hot -related cognitive processes and an involvement of medial PFC regions in hot EF task performance ( Nejati et al., 2020 ; Rubia, 2018 ). It might be speculated that hot EF deficits in ADHD are caused by central cold executive deficits and do not exist independently ( Van Cauwenberge et al., 2015 ). In accordance, the pathology of the functional activity profile of the brain in ADHD is more closely aligned with predominantly cold EF deficits with a fundamental involvement of the frontoparietal network (including IFG, DLPFC, ACC, and temporoparietal regions), the basal ganglia, and the cerebellum ( Rubia, 2011 ). NIBS studies, in this line, have been mostly focused on improving cold EFs in ADHD ( Salehinejad et al., 2019 , 2020b ).

Autism spectrum disorder

Core symptoms in autism spectrum disorder (ASD) include deficits in reciprocity behaviours (required for successful social interaction), and repetitive behaviours. However, ASD is rather known as a disorder of social abilities, although this depends also on the phenotype of the disease. In contrast to this prevailing view, the majority of studies about EFs in ASD investigated cold EFs. Recently, however, hot EFs are studied more extensively in ASD. Briefly, these studies show that ASD involves deficits of both cold and hot EFs ( Kouklari et al., 2018 ; Zimmerman et al., 2016 ). However, deficits related to hot cognition (e.g. reciprocity abilities and theory of mind) seem to be predominant ( Kouklari et al., 2017 , 2018 ; Zimmerman et al., 2016 ), which aligns with the central role of the medial PFC and PCC in the pathophysiology of ASD ( Li et al., 2017 ; Patriquin et al., 2016 ). Cold EFs are also relevantly impaired, but these deficits might be secondary and largely restricted to those cold domains needed for the performance of hot EFs ( Zimmerman et al., 2016 ). This is in line with the development of EFs in ASD. Cold but not hot EFs improve significantly as a function of age ( Kouklari et al., 2018 ), suggesting that hot deficits are more fundamental for the psychopathology of ASD. Considering the heterogeneity of the disease, more detailed research is required to determine the hot – cold profile in ASD, and its subtypes.

Other relevant disorders

Hot–cold executive dysfunctions and respective pathophysiology in the prefrontal-cingular network are prominent in psychopathology other neuropsychiatric disorders as well including but not limited obsessive–compulsive disorders ( cold -deficits driven), borderline personality disorder ( hot -deficits driven), and impulse control disorders ( cold–hot deficits driven) ( Gruner and Pittenger, 2017 ; Schulze et al., 2019 ). In this line, recent NIBS studies have also shown than modulating activity of the prefrontal-cingular and relevant subcortical regions are promising for the treatment of these disorders ( Brevet-Aeby et al., 2016 ; Molavi et al., 2020a ; Rostami et al., 2020 ).

This review was focused on how hot and cold EFs can be functionally organised in the prefrontal and cingulate cortices. Based on evidence from neuroimaging and NIBS studies, we propose a prefrontal-cingular network that can explain neuronal correlates of hot versus cold EFs more comprehensively and in line with the current network-driven approach. In this network, the lateral PFC and associated regions (e.g. DLPFC, VLPFC, and IFG) along with the ACC, specifically the dorsal ACC, are more closely involved in cold EFs (e.g. attentional control, inhibition, error detection, and working memory), whereas the medial and orbital PFC regions (e.g. VMPFC and OFC) and ventral ACC along with the PCC are more relevant for hot EFs that involve emotional, motivational, reward/punishment based, and social stimuli. The extent to which these regions are hot or cold EF-related does not exclude a role of these networks in the other EFs, but rather indicates a gradual dominance for the respective type of information processing. The hot–cold distinction in EFs, and broadly in cognition, provide a novel, network-based approach for studying underlying pathophysiology in major neuropsychiatric disorders that usually come with both cognitive and emotional disturbances. This can promote more effective therapeutic intervention congruent with cognitive profile of the diseases.

Author contributions: M.A.S. contributed to the conceptualisation, investigation, visualisation, validation, writing – original draft, and writing – review and editing. E.G. contributed to the investigation, visualisation, and writing – review and editing. M.H.A.R. contributed to the investigation. M.A.N. contributed to the supervision, and writing – review and editing.

Declaration of conflicting interests: The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: M.A.N. is a member of the Scientific Advisory Board of Neuroelectrics and NeuroDevic. All other authors declare no competing interests.

Funding: The publication of this article was funded by the Open Access Fund of the Leibniz Association.

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Strategies to Improve Your Problem-Solving Skills

Got problems? We all do.

They’re something we encounter daily, both at work and at home. Tackling problems and finding solutions are useful skills that are in high demand.

At a basic level, there are three steps to solving any problem:

• Define the problem
• Generate ideas for solutions
• Implement solutions

You might be tempted to think that the first step is unnecessary. After all, that’s why you’re here in the first place, to solve a problem. However, defining the problem is arguably the most important step in problem-solving.

Albert Einstein is famously quoted as saying, “If I had an hour to solve a problem I’d spend 55 minutes thinking about the problem and 5 minutes thinking about solutions.”

When you can spend more time defining the real problem, and not just a symptom, it will be easier to find a lasting solution.

How to better define the problem:

• Ask “why” questions
• Talk it through with others
• Write down the problem in words
• Use graphs or flow charts

Now that we’ve talked about the steps of solving a problem and how to better define it, let’s dig into some strategies to help your brain perform at its best for solving problems.

“Thanks to a process called neuroplasticity, your brain is continually reorganizing itself by forming new neural connections throughout your life, which gives you the power to make your brain better.” – Daniel G. Amen, MD

Neuroplasticity enables your brain to continue to learn and grow throughout your life. Like your muscles, your brain needs exercise to become stronger. Becoming a life-long learner will not only strengthen your brain, but also sharpen your memory, boost confidence, and bring new knowledge and skills into your life.

3 Ways to Improve Your Problem-Solving Skills

1.    Regularly Engage in Brain Boosting Activities

There are a number of easy and fun ways to strengthen your brain. Adding one or more of these activities into your daily routines can help boost your brain and result in better problem-solving abilities.

• Work on a jigsaw puzzle – Puzzles can be done on your own, or as a social activity. Putting together a puzzle requires concentration and spatial awareness, activating multiple parts of the brain and improving short-term memory.
• Play a musical instrument – Research has shown that learning to play an instrument can improve neuroplasticity and help improve your memory. Playing music engages multiple regions of the brain, providing numerous benefits. Maybe it’s been a while since you last played, or maybe you’ve never learned an instrument. Either way, it’s never too late to tap into your musical side and begin making music.
• Try a new hobby – Remember the “use it or lose it” concept when it comes to the brain. It’s recommended to never stop learning new things. Challenge yourself, no matter your age! Trying new hobbies is a great mental exercise to sharpen your brain. You also may find a new activity that brings more joy to your life.
• Meditate – The practice of meditation has been around for thousands of years as a tool for reducing stress, clearing your mind, promoting relaxation, and improving focus. Meditation is a powerful tool that can boost your brain anytime, anywhere.
• Play brain games – Chess, crossword puzzles, and sudoku all fall under this category. Brain games are an easy and fun way to improve concentration and strengthen memory. The best part is that they only take a few minutes to play and offer a nice break during the day.
• Read a book with a book club – Reading a book offers many benefits, including stimulating different areas of your brain to process and analyze information. When you participate in a reading group , your brain will need to remember information for later recall. This information recall is highly beneficial to protecting short-term memory. Book clubs also can provide a fun and supportive social network.

2.    Spend time NOT looking for the solution

This is counterintuitive, but it’s an important strategy to use when working on a problem. Allow yourself some downtime after defining the problem.

Let your subconscious do some work. Setting a task aside for a time can actually improve your efforts later. When you return to the problem at hand, you’ll likely have a fresh perspective.

What should you do while giving your brain a break from active problem-solving? Enjoy a hobby, get some rest, or move your body with a walk or other form of exercise.

3.    Practice healthy habits

You guessed it, those healthy habits that affect so many areas of your life are also tied to a healthy brain. Exercise, a healthy diet, and quality sleep can all help your brain function better and improve your problem-solving skills overall.

• Exercise – Moving your body increases blood flow to the brain, which can improve your ability to think critically, clearly, and creatively. Additionally, physical activity is a known way to reduce stress and anxiousness. Research has shown creativity and problem-solving to be negatively affected by stress. Using exercise to combat stress can improve your ability to find solutions with a clear mind. By exercising regularly, your overall physical, emotional, and brain health may be positively impacted.
• Healthy Diet – Dr. Daniel Amen teaches that one of the secrets to a healthy brain is to focus on detoxification in your diet. This includes avoiding alcohol, drinking plenty of water, and consuming detoxifying vegetables . Some good vegetables to incorporate into your diet would be lettuce, spinach, kale, broccoli, and asparagus. You also may try increasing your protein intake for a healthy brain, or try adding in turmeric , which can increase neuroplasticity.
• Quality Sleep – Finally, don’t forget about the impact quality sleep, or the lack of it, can have on your brain function and problem-solving abilities. Getting a good night’s rest gives your brain time to recharge and that necessary downtime of not actively thinking about the problems needing solving. While you sleep, your subconscious has a chance to do some work for you!

When you engage in brain-boosting activities, take some downtime, and practice healthy habits you’ll be better prepared for the problems in your days. And, next time you’re faced with the inevitable problems that come with life and work, you can address them with more clarity and confidence.

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• Published: 23 July 2021

Developmental brain dynamics of numerical and arithmetic abilities

• Stephan E. Vogel   ORCID: orcid.org/0000-0003-2630-4881 1 &
• Bert De Smedt 2  

npj Science of Learning volume  6 , Article number:  22 ( 2021 ) Cite this article

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• Cognitive neuroscience
• Human behaviour

The development of numerical and arithmetic abilities constitutes a crucial cornerstone in our modern and educated societies. Difficulties to acquire these central skills can lead to severe consequences for an individual’s well-being and nation’s economy. In the present review, we describe our current broad understanding of the functional and structural brain organization that supports the development of numbers and arithmetic. The existing evidence points towards a complex interaction among multiple domain-specific (e.g., representation of quantities and number symbols) and domain-general (e.g., working memory, visual–spatial abilities) cognitive processes, as well as a dynamic integration of several brain regions into functional networks that support these processes. These networks are mainly, but not exclusively, located in regions of the frontal and parietal cortex, and the functional and structural dynamics of these networks differ as a function of age and performance level. Distinctive brain activation patterns have also been shown for children with dyscalculia, a specific learning disability in the domain of mathematics. Although our knowledge about the developmental brain dynamics of number and arithmetic has greatly improved over the past years, many questions about the interaction and the causal involvement of the abovementioned functional brain networks remain. This review provides a broad and critical overview of the known developmental processes and what is yet to be discovered.

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Introduction

Understanding the development of numerical and arithmetic abilities is highly relevant for our modern and educated societies. Research has shown that these abilities are equally important for life success as literacy 1 and that deficits in these abilities can have severe effects on individuals’ well-being and nation’s economy 2 . Current estimates have shown that ~20% of the population in OECD countries have difficulties within mathematics, imposing great practical and occupation restrictions 3 . Around 5–7% of the population suffers from dyscalculia, a severe mathematical learning disorder 4 . In the last decades, cognitive neuroscientists have begun to investigate the brain mechanisms associated with the developmental dynamics of these foundational abilities. And although our current understanding is still limited, some key findings of the functional and structural organization of these networks are gradually emerging.

This review aims to reach a broad audience and only broadly summarizes our current knowledge about the functional and structural brain dynamics that are associated with the development of numerical (i.e., conceptual knowledge and representations about the structure and meaning of numbers) and arithmetic abilities (i.e., conceptual knowledge and representations about the manipulations of numbers, such as adding or subtracting, and their results). We start by discussing some general principles associated with the brain networks that support the development of these abilities. We next elaborate on the neurocognitive development of basic numerical and after that the neurocognitive development of arithmetic abilities. We then address how these abilities are impaired in the context of atypical development, i.e., in dyscalculia. We end this review by discussing the limitations of the existing body of evidence and by proposing some avenues to further investigate the developmental brain dynamics of these abilities. Throughout the review, we cautiously attempt to suggest some implications of these neuroscientific findings to everyday life, such as the classroom or clinical practice, acknowledging that such direct implications cannot be merely “dropped” in practice, yet require a collaborative effort between scientists and practitioners (for a discussion see ref. 5 ). Rather, our review aims to summarize what is scientifically know and what is not known, a base of reliable knowledge for teacher training and development that can aid practitioners to become critical consumers of so-called “brain-based” explanations.

Principles of the brain networks associated with numerical and arithmetic abilities

A core insight that has emerged from the research is that the development of numerical and arithmetic abilities cannot be restricted or reduced to a single cognitive mechanism or to a single brain region 6 . The development of these abilities is complex and multidimensional, and consequently its education or remediation cannot be reduced to one single factor or intervention. Number and arithmetic incorporate multiple cognitive abilities 7 , representational dimensions 8 , and brain regions 9 . The neurocognitive networks and their associated functions interact in various complex ways to enable efficient and flexible processing of the relevant numerical and arithmetic information. The efficient working of these brain regions is further modulated by genetic factors 10 , 11 , age 12 , 13 , 14 , 15 , ability level 15 , 16 , 17 , task constraints 18 , 19 , education 20 , and other environmental factors such socio-economic status 21 , 22 .

The involved brain dynamics are, therefore, best described as an interaction of multiple brain regions/networks that vary along a functional continuum with domain-specific functions on the one end, and domain-general functions on the other end (see Fig. 1 and Fig. 3 ). Domain-specific functions can be defined as mental operations that are largely restricted to a particular (academic) domain. In the case of arithmetic, this involves certain aspects of basic number processing that are less relevant in other learning domains (e.g., reading). Examples include the representation of numerical quantities 23 or symbolic knowledge about ordinal relationships 8 . Domain-general functions are less specific to a particular (academic) domain. They mainly reflect mental operations that are important for learning and information processing more generally 7 . Examples include cognitive functions such as working memory (i.e., the ability to temporally hold information in our mind) or visual–spatial reasoning (i.e., the ability to mentally manipulate and understand the spatial relation of objects), which are both relevant for learning to calculate or to read.

The approximate representation of numerosities, the verbal representation of numbers, and the visual representation of numbers build the core components of the triple-code model. Auxiliary functions such as verbal and spatial abilities are shown in white. The object-tracking system is part of rather domain-general visual–spatial abilities. Note that the classification of domain-specific and domain-general is rather continuous than categorical. The arrows suggest uni- or bidirectional influences.

Another important finding from brain imaging research is that cortical brain regions are not devoted to one specific task 24 . Brain regions rather contribute to a variety of domain-specific and domain-general functions. The formation of these brain regions into functional networks is highly dynamic (see also Fig. 3 ). These changes can be observed in real time (e.g., the functional synchronization of different neuronal population/regions/networks depending on task requirements) as well as over larger developmental timescales (e.g., the functional specialization of neuronal populations/regions/networks to process relevant stimuli dimensions more efficiently) 25 , 26 , 27 , 28 . We also know that environmental factors, such as learning and education, influence the formation of domain-general and domain-specific regions into functional brain networks in various ways. For example, cognitive tutoring or specific training reconfigures the functional connectivity of relevant brain regions in elementary school 20 , 29 , 30 . The educational transition from play-based learning in kindergarten to formal learning in grade one in primary school even changes the brain activity during domain-general processes, such as executive function 20 , indicating that even these domain-general processes and their brain networks are malleable via educational practice. In the next sections, we describe the brain functions and regions that support the development of these numerical and arithmetic abilities in more detail.

Numerical abilities

Tuning the mind for numerical quantities.

The ability to perceive and to nonverbally quantify the objects of a set (i.e., its numerosity) has been proposed as one important domain-specific precursor for the development of numerical and arithmetic abilities 31 , 32 . Two neurocognitive systems have been related to this development: the approximate number system and the object-tracking system 33 , 34 . Both systems show significant individual differences and developmental changes early in childhood. While it is undisputed that these systems provide a crucial basis for perceiving quantities, their causal relationship with the development of symbolic numerical and arithmetic abilities is debated (for reviews and detailed arguments see refs. 35 , 36 , 37 , 38 ).

The approximate number system

The approximate number system reflects an intuitive sense to estimate and to discriminate the number of objects of different sets (e.g., perceiving that a set of 16 dots differs from a set of 8 dots). In humans, this capacity increases continuously over developmental time. Six-month-old infants can reliably differentiate sets with a numerical ratio of 1:2, 9–12-month-old with a ratio of 2:3, 4–5-year-old’s with a ratio of 3:4 and adults with a ratio of 7:8 objects 31 , 39 , 40 , 41 , 42 , 43 . The approximate number system forms a central component of the triple-code model 32 and its associated brain correlates 44 . It provides the semantic basis for two other “codes”: the visual representation of numbers, which encodes number symbols (such as the Arabic numerals), and the verbal representation of numbers, which encodes number words. The developmental formation of these codes and their corresponding brain systems in the parietal, occipital, and temporoparietal cortex have been argued to build an important foundation for the continued development of arithmetic abilities.

Over the last two decades, an increasing number of neuroimaging studies have investigated the neural underpinnings of the approximate number system and its development in the human brain, using a variety of different methodological approaches 45 , 46 . Evidence from this research indicates an early functional specialization of the parietal cortex to represent numerical quantities 23 , 47 . Neuroimaging studies with newborns 40 , 6-month-old infants 45 , 46 as well as with young children 48 , 49 have found significant numerical ratio-dependent brain activations within regions of the parietal cortex, especially the right intraparietal sulcus (IPS) 40 , 45 , 46 , 48 , 49 . Neurocognitive models of number processing propose that this approximate activation arises from a neuronal representation that encodes numbers according to Weber’s law 50 , 51 . More specifically, numerical quantities reflect a linear representation in which each number is represented as a Gaussian distribution with a scalar variability—the width of the distribution (i.e., tuning curves) increases with numerical quantity (see Fig. 2 ). Single-cell recordings in non-human primates have demonstrated that the response profiles of neurons in homolog regions of the IPS (the ventral intraparietal area, VIP) and the prefrontal cortex (PFC) behave according to this prediction—showing a systematic activation decrease as the distance to the preferred numerical quantity increases 52 , 53 . This work provides evidence for the existence of single neurons that encode numerical quantities within the parietal cortex of non-human primates. In humans, the neuronal tuning curves of this particular brain area are similar and their width decreases with age 12 , 13 , possibly indicating a developmental tuning of the neuronal populations to represent numerical quantities with a higher precision 54 . As such, the existence of the approximate number system in specific brain regions of the parietal cortex, as proposed by the triple-code model, has received great support from the neuroscientific literature (the interested reader is referred to these detailed reviews 23 , 31 , 55 , 56 ). However, the empirical proof that the approximate number system serves as a unidirectional neurobiological foundation for the development of symbolic numerical and arithmetic abilities is rather weak 57 , 58 , 59 , 60 , 61 , 62 and even non-existent 35 , 63 , 64 . Consequently, (cognitive) interventions that merely focus on training this approximate number system with the aim to transferring improvements to arithmetic have not provided conclusive evidence (for a critical review see ref. 65 ).

Numbers are represented as Gaussian distributions with a scalar variability (i.e., Weber’s law). The increased variability with larger numbers reflects greater uncertainty to represent these numerical quantities.

The object-tracking system

The second neurocognitive system that is implicated in the development of early perceptual abilities is the object-tracking-system 33 , 34 . This system enables the precise and fast individuation of 1–4 objects within a visual scene (in the literature often termed subitizing). The primary function of the object-tracking system is to perceive object boundaries, to predict object movements, and to accurately retain a small number of objects in working memory. As such, the object-tracking system is considered to be more of a domain-general system of visual–spatial abilities 31 , 34 . The capacity of the object-tracking system to keep track of a small number of objects develops rapidly over the first month of infants. Whereas 6-month-olds display a capacity limit of one object, 12-month-olds already show adult-like object-tracking abilities (i.e., 3–4 objects) 66 . Because of its properties to perceive a limited number of individual objects with high precision, it is thought to function as an important primitive for the conceptual development of counting via subitizing (see section 2.2.3) 67 .

Neuroimaging studies with adults have identified several brain regions that correlate with the object-tracking system. These brain regions encompass the inferior parietal cortex, the posterior parietal cortex, the occipital cortex 68 , 69 , 70 , 71 , and the temporoparietal junction (TPJ) 72 , 73 . While early neural responses of the occipital cortex seem to be driven by basic visual parameters and object identification 74 , 75 , the activation of the parietal cortex has been linked to enumeration. The TPJ might play a particular role in this process as its brain activation has been linked to the ventral attention network, which directs attention to salient object properties of the visual scene (i.e., the number of elements).

The developmental brain mechanisms of the object-tracking system, as well as its separation (or overlap) from the approximate number system, are still unclear. The results of a study with 6-month-old infants suggested a separation of the two systems and an involvement of both systems early in development 46 . The overall question of whether the enumeration of objects is exclusively performed by the approximate number system (i.e., the single-system view), or whether small numerosities are encoded by the object-tracking-system and large numerosities by the approximate number system (i.e., the double system view) is not yet resolved and debated (the interested reader is referred to these detailed reviews 31 , 34 ).

Introducing cultural tools: the role of symbolic numerical knowledge

Sophisticated numerical abilities of children go well beyond the mere perception of quantities. One important stepping stone is the construction of symbolic numerical knowledge: children have to learn to represent numerical information via symbols, which can be number words or Arabic digits. One prominent account proposes that over developmental time (and education), number symbols, which are argued to be processed in the occipital cortex (see triple-code model), are mapped onto quantities represented in the IPS 31 , 76 . In other words, it is argued that due to a “symbolic-quantity-mapping” an associative link between cultural invented symbols and the neurobiological foundation of quantities is established: Children associate arbitrary visual symbols (e.g., the Arabic digit “5”; or the number word /five/ ) to corresponding quantities (e.g., “the five-ness”). They further learn via counting to use these symbols to determine the exact quantity of a set.

The symbolic representation of quantities

The neuroscientific evidence of how our human brain develops symbolic representations has substantially increased over the past years 77 , 78 . There is now good evidence to support the hypothesis that specific regions of the visual cortex are responsive to the visual properties of number symbols (the interested reader is referred to this meta-analysis 79 ). Several studies have revealed increases in brain activity in the right and/or left occipitotemporal cortex, close to the fusiform gyrus of the inferior occipitotemporal gyrus 79 , 80 , 81 . The locus of this left-hemispheric activity is close to the visual word form area, which is critical to recognize letters and to read words 82 . As such, it has been suggested that activity in the left and right occipitotemporal cortices reflect the increased efficiency to process visual number symbols such as Arabic digits 83 . Although this view is consistent with the triple-code model, the specifics for functional specialization in the left and/or the right hemispheres as well as their possible lateralization to process number symbols are still unresolved.

Findings from this work have also shown that similar and distinct brain regions as in perceiving the number of objects are activated during mere symbolic number processing, especially in the parietal and frontal cortex (the interested reader is referred to this meta-analysis 84 ). The overlapping brain activation is seen as one of the core findings to support the “symbolic-quantity mapping” account. It suggests that the human brain processes numerical quantities in the same brain region, independent of notation (i.e., symbolic or nonsymbolic) in which quantities are presented 85 , 86 . However, several findings, including that of distinct brain regions, have challenged this idea and questioned whether the observed overlap in brain activation provides conclusive evidence for such simple mapping (for a discussion, see ref. 38 ).

Specifically, neuroimaging studies were able to demonstrate that regions beyond the parietal cortex, especially the frontal cortex (the middle frontal gyrus, the inferior frontal gyrus, and the precentral gyrus), additionally engage in symbolic encoding 48 , 87 , 88 and that symbolic processing mechanisms mediate the relation between nonsymbolic encoding and math, even after controlling for multiple domain-general functions 38 . The activation of brain regions outside the parietal cortex is substantially greater in children compared to adults 87 . This additional brain activation might be explained by auxiliary functions that support symbolic computations (e.g., working memory; executive functioning), or by a direct encoding of symbolic-quantity information in these regions 47 , 89 . Although both alternatives are possible, the precise answer is not yet known. These more recent neuroimaging studies have also revealed that the neuronal responses to perceiving the number of objects and to process symbolic quantities are very heterogeneous with regard to their location, even within the IPS 90 , 91 , 92 , 93 . In other words, although similar brain regions may encode both nonsymbolic and symbolic representations, the precise neuronal encoding patterns differ within these regions. It is important to know that these studies were done in adults and that there are no studies available that have investigated developmental changes in relation to these different encoding patterns in children, an issue that warrants further investigation.

Additional findings suggest a developmental specialization of several brain regions to process symbolic numerical information. This specialization is characterized by age and/or training/education-related increase in brain activation in the parietal cortex, the inferior frontal cortex, and the occipital cortex 14 , 30 , 94 , 95 , and an age and/or training/education-related decrease in brain activation in several regions of the prefrontal cortex, including the anterior cingulate gyrus 30 , 87 , 95 . This functional shift in brain activation has been related to the automatization of the parietal and occipital cortex, in particular the left IPS, to process symbolic numerical information as children get more experienced with this stimulus dimension 14 , 94 , 95 . The age and training/education-related decrease in frontal and cingulate brain regions is often associated with a reduced working memory load and less attentional effort in adults or more skilled participants compared to children 87 , 95 . These findings are in line with further evidence that has shown that the brain activation of the left parietal cortex is associated with behavioral skills that require the manipulation of numerals (e.g., arithmetic) 94 , 96 . The developmental specialization of the left IPS could therefore reflect a dynamic interaction of domain-general as well as domain-specific resources to efficiently act upon symbolic knowledge. Indeed, a recent study showed that functional connectivity patterns (i.e., the signal correlation between distant brain regions) between the right parietal and the left parietal cortex predicted individual scores in a standardized test of mathematical achievement 97 . Together, these findings indicate that the learning of symbolic representations is much more complex than simply mapping quantities onto symbols (see the following references for a detailed discussion 35 , 36 , 37 , 64 ). They suggest that the construction and learning of symbolic numerical information are related to the integration of multiple knowledge dimensions 98 , such as numerical order and counting, all of which should be fostered through (mathematics) education.

Knowledge of numerical order

Numerical order refers to our knowledge that an Arabic numeral or number word occupies a relative rank or position within a sequence 8 . Research that aims to better understand the functional development of numerical order processing has significantly increased over the past years. Results from this work have demonstrated that the understanding of numerical order (a) explains unique variance in children’s arithmetic abilities 15 , 99 , 100 , 101 , 102 , 103 , (b) mediates the well-established association between numerical quantity processing and arithmetic 15 , 100 , 104 , and (c) shows a specific developmental trajectory within the first years of formal education, becoming the best predictor and diagnostic marker of arithmetic performance at the end of primary school 99 .

Investigations into the developmental brain mechanisms of numerical order processing are extremely sparse. Only a handful of neuroimaging studies have directly explored the neural correlates of numerical order processing in children. The results of these studies have demonstrated an age-dependent increase in brain activation in the left IPS in response to numerical order processing 15 , 17 , 30 , 105 , 106 . In addition, a significant association between the neural responses of ordinal processing and arithmetic performance in regions of the semantic control network was found, especially in the right posterior middle temporal gyrus (pMTG) and at the right inferior frontal gyrus (IFG) 15 . The IFG is known for its functional relevance in visual working memory, which helps to monitor simple rules that are associated with the manipulation of numerical items 107 . These findings indicate again that domain-general and domain-specific resources work in concert to support multiple dimensions that are involved in the construction of symbolic representations in the brain.

Counting: the bridge to arithmetic?

Another knowledge dimension that is critical for the construction of symbolic numerical information is the ability to count. Counting allows children to determine the exact quantity of elements in a given set by using a symbolic representation, i.e., a number word and later an Arabic numeral. Counting is also related to the understanding of order, given that counting constitutes the precise matching of an ordered sequence of symbols with quantities. The development of counting begins at around 2–3 years of age when children start to learn number words and the associated stable order principle 108 , 109 , i.e., knowledge that a particular number word, such as /nine/ , comes after the number word /eight/ and before the number word /ten/ , which appears to increase with age and the understanding of this principle has been shown to be delayed in children with difficulties in learning arithmetic 110 . Over the ages of 4–5 children gradually develop an understanding that number words represent the number of objects and, therefore, their numerical quantities. This insight is associated with the knowledge that each counting word needs to be lined up with only one object in the set to be counted (i.e., one-to-one correspondence principle) and that the last word reflects the total number of objects (i.e., cardinal principle) 111 . The process of counting objects is often accompanied by the use of fingers, which might facilitate the spatial representation of numbers 13 , 112 , 113 . During this time children also conceptually grasp the idea that every number in the counting list has a predecessor ( n  − 1) and a successor ( n  + 1) 114 , 115 . Children who have mastered these general conceptual ideas are known as cardinal principle knowers and recent evidence indicates that symbolic numerical knowledge accelerates after children have mastered these cognitive steps 116 .

Unfortunately, we know hardly anything about the brain mechanisms that facilitate this development. One reason for this lack of knowledge is that neuroscientific data collection (e.g., fMRI) is particularly difficult in children of this age range, i.e., 3–5-year olds, as the acquisition of such data requires children to lay very still in the magnet for a given amount of time. The few studies that have investigated the brain mechanisms of counting, albeit in older populations, have indicated a link to the approximate number system (see section 2.1.1), and/or to the object-tracking system (see section 2.1.2). More specifically, effortful counting of larger sets appears to be associated with brain activity in fronto-parietal regions, while the enumeration of smaller sets is linked to temporal-parietal brain regions, especially to the TPJ 75 . The involvement of the TPJ in counting indicates that the precise individuation of smaller sets of elements via the object-tracking system (i.e., subitizing) may feed into the conceptual learning of counting 67 . This is further supported by the fact that children first understand the principles of counting in the small number range. Once children have grasped these principles for smaller numbers, they can apply this knowledge to an infinite number of elements 117 . However, no brain imaging study has yet systematically mapped the progressive development of counting: from understanding that counting words always occur in the same ordered sequence (the stable order principle) to knowing that the last word in the sequence represents the total number of objects in the set (cardinality principle). This clearly represents an area for future (brain imaging) research.

Arithmetic abilities

Arithmetic development as a strategy change.

Children’s counting is not only important for their acquisition of symbolic knowledge. It also provides a crucial foundation for their development of arithmetic. This development is characterized by a change in the distribution of strategies that children use to calculate, i.e., to determine the sum or difference of two (or more) quantities 118 . Initially, children use fingers or manipulatives to count the answer to a problem 119 , yet progressively, they execute strategies without external aids. The counting strategies become increasingly sophisticated and children move from counting all elements (sum strategy), to counting from the first (counting on), and subsequently from the larger (counting-on-larger) operand. Through practice, the strength of the problem–answer associations is increased, and eventually, this results in the storage and retrieval of arithmetic facts from long-term memory. These arithmetic facts further provide the basis for more complex procedural calculation strategies, such as decomposition strategies, in which a problem is decomposed into more simple problems, as is the case when children work with larger numbers (e.g., 8 + 6 is solved by doing 8 + 2 = 10 and 10 + 4 = 14, so 14). It is important to acknowledge that the abovementioned types of strategies all remain available over development, but that their distributions change 120 .

The arithmetic brain network

What are the brain regions that are correlated with the development of these strategies? The nascent body of brain imaging studies has revealed a widespread set of interconnected brain areas (see Fig. 3 ), reflecting the involvement of both domain-specific and domain-general processes 9 . As already discussed further above, a key region within this network is the IPS. It has been suggested that the brain activation of this region reflects quantity processing. Increases in activity in this region are typically observed during the execution of procedural strategies, such as counting and decomposition strategies, which are more often observed during the solution of larger problems and during subtraction. The retrieval of arithmetic facts from long-term memory has been associated with the more inferior part of the parietal cortex, which includes the angular gyrus (AG) and the supramarginal gyrus (SMG). Activity in these regions is typically correlated with multiplication, which is known to be dependent on fact retrieval. Although studies in adults have observed increases in the AG during the retrieval of arithmetic facts, this has not always been consistently observed in children. Here, activity in the hippocampus has been shown to correlate with arithmetic fact retrieval 121 , 122 . This is not unexpected in view of the fact that the hippocampus plays a role in memory encoding 123 . Therefore, it has been suggested that arithmetic fact retrieval is a graded phenomenon, in which the early and initial consolidation stages are more related to activity in the hippocampus and the later more automatized stages are related to activity in the AG.

The blue and orange color coding indicates the relative domain-specific and domain-general involvement of the particular brain regions. DLPFC dorsolateral prefrontal cortex, VLPFC ventrolateral prefrontal cortex, PSPL posterior superior parietal lobe, IPS intraparietal sulcus, SMG supramarginal gyrus, AG angular gyrus, FG fusiform gyrus, HC hippocampus (the dotted lines indicate the medial position of this brain region).

The arithmetic network also includes auxiliary areas, including the prefrontal cortex and the posterior superior parietal cortex (PSPL), related to working memory and the allocation of attentional resources. Activity in these areas typically increases with the difficulty of the task, and these increases are often observed during more complex problems or during the early stages of learning 9 . Arithmetic training studies in adults 124 and children 30 , 125 have shown that brain activity in these auxiliary areas decreases after training.

The brain activity during arithmetic undergoes important developmental changes, yet there are only but a handful of studies that have investigated these changes. One piece of evidence comes from studies that have compared children in different age groups 80 or that have correlated brain activity with age 126 , 127 . These studies have generally shown that with increasing age, brain activity decreases in prefrontal areas and increases in the parietal cortex and the occipitotemporal cortex. These changes seem to mirror the development that can be observed in children’s arithmetic strategy use, which is characterized by an increasing automatization and reliance on arithmetic fact retrieval. To the best of our knowledge, there is only one study that has investigated developmental changes at multiple time points in the same sample 122 . The results showed decreases in brain activity in the prefrontal cortex as well as increases in the hippocampus. An analysis of children’s arithmetic strategy use revealed that these changes were accompanied by an increase in fact retrieval and a decrease in counting strategies.

Children show large individual differences in their development of strategies 128 in the classroom. Are such differences also observed in the brain? A small number of studies have observed that individuals with lower mathematical skills show higher activity in the IPS during calculation 121 , 129 , 130 , 131 . The precise interpretation of this association remains, however, unclear 9 . It is possible that the increased IPS activity in children with lower mathematical skills reflects a protracted reliance on immature calculation strategies, such as counting. In contrast to their peers, poorer numerical processing skills prevent these children from developing a reliance on the arithmetic fact retrieval network (i.e., in the adjacent AG and SMG areas of the parietal cortex). This shift has to be differentiated from the developmental increase in brain activity of the IPS in response to numerical processing. While in the context of low mathematics achievement, the brain activity during arithmetic might relate to a delayed shift of strategies and the corresponding brain networks, the brain activity during numerical processing corresponds to an age-related deficiency to processes symbolic numerical quantities. As such, it is possible that individuals with lower mathematical skills show age-dependent reduced brain activation in the IPS in response to symbolic numerical quantity processing, as well as greater activation in the IPS in response to arithmetic problem-solving, due to their protracted reliance on symbolic-quantity processing instead of arithmetic fact retrieval.

Individual differences in arithmetic performance have also been correlated with structural characteristics of the arithmetic brain network, such as the anatomical structure or volume of brain areas (gray matter) as well as the connections between them (white matter), although the number of existing studies remains to be small. The evidence from this work suggests that the gray matter volume of the arithmetic network is positively correlated with higher arithmetic skills 132 , 133 . Two key findings have emerged from studies that have investigated white matter connections between distant brain regions and arithmetic performance 134 : (a) positive correlations between white matter tracts that connect the prefrontal cortex and the posterior parietal cortex (i.e., the superior longitudinal fasciculus and arcuate fasciculus) 135 , 136 , 137 , and (b) a positive association between the tract that connects the prefrontal cortex and the occipitotemporal cortex (i.e., the inferior longitudinal fasciculus) 138 , 139 . These structural brain imaging data indicate that larger gray matter volume and a better white matter organization of the connections between distant areas of the arithmetic network coincide with better arithmetic performance. In the absence of longitudinal data, we currently do not know whether these structural variations are the cause or consequence of arithmetic development. In other words, it is unclear whether these structural characteristics precede the individual differences in arithmetic development or whether they emerge as a result of these individual differences (and related expertise).

Atypical development: dyscalculia

Approximately 5–7% of children experience life-long and persisting difficulties in acquiring arithmetic skills and this condition is referred to as dyscalculia 4 . The difficulties are not merely explained by sensory problems, low intellectual ability, and mental or neurological conditions 4 . For decades, it has been emphasized that difficulties in arithmetic strategy use are the hallmark of dyscalculia 140 , 141 , 142 . In more recent years, it has been observed that children with dyscalculia also show consistent impairments in the processing of symbolic numbers 143 , 144 , for which reason it has been suggested that measures of symbolic number processing might be useful diagnostic markers for children at risk for dyscalculia 145 , 146 , 147 . These impairments seem to be persistent over time and they coincide with arithmetic fact retrieval deficits, although it remains to be unclear whether these difficulties in number processing are the cause or the consequence of poor arithmetic development or both.

The etiology of dyscalculia is still debated and it is likely that this etiology is heterogenous 6 . Evidence indicates that genetic, neurobiological, cognitive, and environmental factors might contribute to the atypical development of brain systems that impair the representation and/or acquisition of numerical abilities (for reviews, see refs. 140 , 148 , 149 ). Our neuroscientific evidence on the origins of dyscalculia is, however, limited because longitudinal investigations are still missing (for an exception, see ref. 106 ). The existing body of cross-sectional studies is descriptive in nature, revealing insights about the phenotype but not about the origin of a disorder. Stated differently, it is currently unclear whether the structural and functional brain abnormalities in dyscalculia are the cause or the consequence of the learning disorder. In other words, are the abnormalities present before the learning disorder manifests itself (= cause)? Or do these abnormalities emerge as a result of a poor learning process or experience with mathematics (= consequence)? This question can only be answered via longitudinal data, part of which is collected at an early age before the disorder emerges, which is currently lacking at the neurobiological level. Nevertheless, domain-specific, as well as domain-general neurocognitive deficits, have been hypothesized. The predominant domain-specific accounts propose either a neurocognitive deficit 148 to process and to represent numerical quantities or a deficit to access 150 numerical quantities via symbolic representations (e.g., Arabic numerals). The resulting consequence is similar in both accounts: individuals with dyscalculia have difficulties to develop an accurate representation of symbolic numbers that further impairs the acquisition of arithmetic abilities. The predominant domain-general account highlights the crucial role of working memory and executive functions. Especially, difficulties to inhibit irrelevant information in working memory are considered as the main cause of poor performances in arithmetic and mathematics 140 .

What do we know about the neurobiological correlates of dyscalculia? There are only a few studies that have investigated the neurocognitive correlates of basic numerical abilities in children with dyscalculia 78 , 151 , 152 . Findings from this research suggest significant differences, often deactivation, in the functional organization of the brain regions associated with nonsymbolic and symbolic number processing, in particular the IPS 153 , 154 , 155 . This indicates that children with dyscalculia engage similar brain networks compared with typically developing children, yet the efficiency of the involved brain regions to process numerical information seems to be altered. Although an easy explanation of these findings might be a deficient functioning of the IPS (i.e., the core deficit hypothesis) to process numerical quantities, the overall picture seems to be far more nuanced. Studies have also found greater activation during number processing in the left IPS, the frontal cortex, and in visual areas in children with dyscalculia, probably indicating that children with dyscalculia engage additional cognitive control resources to compensate for difficulties in quantity processing 17 , 106 , 153 , 156 . However, the exact reasons (cause or effect) for these different activation patterns are still unknown.

Only a handful of studies have investigated brain activity during arithmetic in children with dyscalculia 9 , 151 . These studies have observed brain activation differences in the abovementioned arithmetic network compared to typically developing children. Again, the results are difficult to interpret as both increases and decreases in brain activation have been observed. While one study showed increases in brain activity in parietal, prefrontal, and occipitotemporal regions in children with dyscalculia during addition and subtraction 157 , another study found decreases in brain activity during addition in the prefrontal cortex, right posterior parietal, and occipitotemporal areas 158 . The existing body of the data is simply too small to draw reliable conclusions on what these activation and deactivation differences mean. Although the possibility of biomarkers for clinical diagnosis of dyscalculia has been suggested 159 , this is currently not possible and we do not have enough studies available that allow us to use these brain imaging data in clinical practice.

A small number of studies have also examined the structural characteristics of the arithmetic brain in dyscalculia. This small number of morphological studies have revealed significantly less gray matter and white matter volume in the parietal cortex 160 , 161 , 162 , prefrontal cortex 161 , and hippocampal areas 162 in children with dyscalculia. One recent longitudinal study found persistent reduced gray and white matter volumes over a time span of 4 years in a widespread network of frontal, parietal, temporal, and occipital regions 163 . The reduced gray and white matter volume indicate morphological and compositional alterations of the cellular microstructure in these regions such as the degree of myelination (i.e., insulation of neuronal axons that is important for efficient information processing) or differences in the shape and size of dendritic spines 164 . Independent of the neurobiological nature of these differences, changes in cellular structure might have a significant impact on the information processing of brain regions and thus might transfer to functional connectivity measures, although we are currently far from a precise understanding of how abnormalities in brain structure relate to brain function and vice versa. Indeed, a handful of studies revealed differences in the structural 162 , 165 and functional 157 , 159 connections between the prefrontal and parietal areas that are involved in arithmetic. The findings from this work suggest deficient structural connections as well as a functional hyperconnectivity between relevant brain regions in children with dyscalculia. These findings indicate an atypical structural and functional network formation 165 , which possibly results in less efficient integration of information processing (which would be in line with the access deficit hypothesis). Whether structural and functional measures can be related to the same or different microstructural changes, needs to be further determined. Nevertheless, recent evidence also indicates that the observed hyperconnectivity is malleable and that it can be altered via specific numerical and arithmetic interventions 29 , 166 . This is a promising avenue for future interventions within this domain. Although our knowledge about the atypical development of these networks has significantly increased, the current body of data is too preliminary to draw reliable conclusions and future research is needed.

In this article, we have provided a succinct overview of our current understanding of how the human brain supports the development of basic numerical and arithmetic abilities. The overarching finding is that the development of these abilities constitutes a dynamic interaction or co-development of multiple domain-general and domain-specific cognitive dimensions: different cognitive functions and brain regions/networks interact depending on what dimension has to be processed and how this dimension is qualitatively (e.g., calculation vs. fact retrieval) and quantitatively (e.g., efficient access to semantic information) processed at a given point in time. The quality and quantity of domain-specific functions might moderate the involvement of domain-general function (e.g., greater engagement of working memory during calculation), and the quality and quantity of domain-general functions might in turn moderate domain-specific functions (e.g., forming associations, for example between symbolic numbers and the quantities they represent, to efficiently process semantic information). These interactions occur in real time (e.g., the functional synchronization of different neuronal populations/regions during a specific task) as well as over larger developmental timescales (e.g., the functional specialization of neuronal populations/regions/networks to process relevant stimuli dimensions more efficiently). In view of these multiple dimensions, it is imperative to study large-scale connectivity patterns between different brain regions that support these dimensions rather than their mere localization 27 . The findings from a handful of studies that have started to investigate these connectivity patterns suggest a tight coupling between the frontal and parietal areas, which appears to increase with age 80 . These findings support the notion that domain-general and domain-specific functions become integrated over developmental time to support numerical and arithmetic abilities.

Although our understanding of the neurocognitive mechanisms has substantially increased over the past years, numerous challenges and questions remain. A central limitation to our current understanding is that much of our knowledge about brain mechanisms is still related to adult studies or to cross-sectional comparisons between two age groups (often children versus adults). To understand the gradual shift of cognitive functions and their associated changes of neural networks over time, longitudinal studies and cross-sectional studies that target-specific functions at specific age periods are needed 167 .

Another methodological problem is that much of the basic numerical development happens at a time period during which brain activity is difficult to investigate with the existing imaging tools, due to technical limitations (e.g., the request to lie very still in the MRI scanner for a period of time). For example, the formation of symbolic knowledge starts in preschoolers well before children are introduced to the school system. To investigate the transition from nonsymbolic to symbolic knowledge, technological and methodological advances are needed to increase the reliability and the validity of neuroimaging data at this point in development 168 .

A critical aspect that requires further investigation is the extent to which environmental factors, such as home environment and schooling, impact on the development of numerical and arithmetic abilities. Indeed, the acquisition of numerical knowledge and arithmetic does not occur in isolation, but happens within in a wider educational context 62 . Behavioral data clearly indicate that these environmental factors predict the development of numerical and arithmetic abilities 169 , yet we do not understand how these environmental factors predict and change the abovementioned brain networks that are relevant for numerical and arithmetic abilities.

Another crucial agenda for subsequent research lies in separating causes from consequences, a problem that is particularly prominent in the understanding of dyscalculia, as well as many other neurodevelopmental disorders. As documented above, there are correlations between the functional and structural properties of the arithmetic network and individual differences in performance. These correlations are almost exclusively based on cross-sectional data, which cannot separate causes from consequences. This is further complicated by the observation that studies have typically included children of very broad age ranges during which massive maturational changes in brain development occur 170 , and these maturational changes may confound the observed association between arithmetic and brain measures. In the absence of longitudinal data, we do not know whether poor arithmetic performance occurs as a result of impairments in the arithmetic brain network, or whether poor arithmetic performance leads to deviances in this network. One way to answer this question would be to study children at genetic risk for developing difficulties in their mathematics learning. Such an approach has been successful in understanding the causes and consequences of dyslexia, showing that some brain abnormalities already exist before learning to read while others occur as a result of poor reading 171 . This work has also pointed to compensatory brain mechanisms in children who are genetically at risk for dyslexia but do not develop this learning disorder. It remains to be determined whether similar processes and compensatory mechanisms can be observed in children (at risk for) dyscalculia. The discovery of such compensatory mechanisms holds great promise for designing remedial interventions in dyscalculia that might exploit these compensatory processes.

To conclude, there is an emergent understanding of the neurocognitive mechanisms associated with the early development of basic numerical and arithmetic abilities. As can be seen from this review, the available body of brain imaging evidence leaves many questions unanswered, and much more studies are needed to better understand the dynamic integration of various neurocognitive functions to establish numerical and arithmetic knowledge across development.

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during this study.

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Acknowledgements

This work has been partially supported by research grants of Fund for Scientific Research Flanders (G.0638.17 and G.0707.20) awarded to B.D.S. We further acknowledge the financial support of S.E.V. by the University of Graz.

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Vogel, S.E., De Smedt, B. Developmental brain dynamics of numerical and arithmetic abilities. npj Sci. Learn. 6 , 22 (2021). https://doi.org/10.1038/s41539-021-00099-3

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DOI : https://doi.org/10.1038/s41539-021-00099-3

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