Effective Brain Connectivity for fNIRS With Fuzzy Cognitive Maps in Neuroergonomics

Effective connectivity (EC) among functional near-infrared spectroscopy (fNIRS) signals is a quantitative measure of the strength of influence between brain activity associated with different regions of the brain. Evidently, accurate deciphering of EC gives further insight into the understanding of the intricately complex nature of neuronal interactions in the human brain. This article presents a novel approach to estimate EC in the human brain signals using enhanced fuzzy cognitive maps (FCMs). The proposed method presents a regularized methodology of FCMs, called effective FCMs (E-FCMs), with improved accuracy for predicting EC between real and synthetic fNIRS signals. Essentially, the revisions made in the FCM methodology include a more powerful prediction formula for FCM combined with independent tuning of the transformation function parameter. A comparison of EC in fNIRS signals obtained from E-FCM with that obtained from standard FCM, general linear model (GLM) parameters that power dynamic causal modeling (DCM), and Granger causality (GC) manifests the greater prowess of the proposed E-FCM over the aforementioned methods. For real fNIRS data, an empirical investigation is also made to gain an insight into the role of oxyhemoglobin and deoxy-hemoglobin (oxy-Hb, deoxy-Hb) in representing the cognitive activity. We believe this article has profound implications for neuroergonomics research communities.


I. INTRODUCTION
A N EFFECTIVE connectivity (EC) amongst the brain signals represent influence between neural systems which can be both activity, and/or time dependent [1], [2]. Hence EC elucidates the relationships in a brain network thus formed during a cognitive task undertaken by a subject. An accurate deciphering of connections amongst the brain signals is a significant step toward a better understanding of EC linkages between different brain regions. A comprehensive picture of EC between brain regions, for a particular mental task, can lead to a thorough assimilation of the workings of the human brain [3]. Owing to the significance associated with the detection and subsequent analysis of the discerned influence between brain regions, an accurate interpretation of EC in fNIRS data is crucial.
Owing to the multitude of advantages of functional near-infrared spectroscopy (fNIRS) over functional magnetic resonance imaging (fMRI), such as the former being safe, cost efficient, and a portable technique with reasonable spatial and excellent temporal resolution, fNIRS is fast replacing fMRI for empirical studies in the field of neuroscience. This is true in particular for psychological studies where brain activity is inspected to understand human factors at work and normal life settings. This new paradigm of research has been termed as neuroergonomics [3].
The de facto standard for EC analysis in fMRI studies is dynamic causal modeling (DCM) [4]. However, DCMs cannot be used for discerning EC in fNIRS data since the temporal resolution of fNIRS is sufficient to be affected by inter-regional axonal conduction delay [5]. This is because extending DCMs to account for time delays will not only introduce parameterized complexity but also pose a problem of formulation. In this regard, an alternative approach that is used to estimate connections in fNIRS data is Granger causality (GC) [6]. The GC is a relatively fast, scalable technique that can be used to compute EC while taking into account time delays. However, the GC analysis cannot transcribe nonlinear cause effect relationships without compromising GC scalability owing to the linear vector auto regression in GC [6], [7].
The GC analysis is also predominantly dependent on the pertinent information presented beforehand. Essentially, it implies that the GC analysis may identify spurious connections, or even estimate wrong feedback relations [8] on account of missing relevant information. Furthermore, the limitations of GC analysis comprise its inability to account for connections that change direction over time [8]. Another consequential constraint that limits GC analysis applicability is its dependence on the temporal resolution in the data to be analyzed for accurate deciphering of EC. Hence, if the data has low temporal resolution than the minimum resolution to identify the time This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ lags between which an influence transpires, the GC analysis will not be able to decipher the underlying EC in the data [9].
In the realm of neuroimaging, and in particular for fNIRS data, the sensitivity of GC analysis to the lack of relevant information is critical. Essentially, it renders the inference made by GC rather obscure for identification of EC in fNIRS data. Moreover, the GC analysis is unable to transcribe nonlinear effective connections and is also unable to assimilate effective links changing direction over time. This has implications for GC application in fNIRS data since the underlying EC in fNIRS data are inherently nonlinear. Furthermore, the lack of flexibility of GC to cater for changing effective connections renders GC particularly inapplicable for an insightful EC analysis between fNIRS data. Owing to the aforementioned shortcomings of GC analysis for uncovering the EC in fNIRS data in particular, we propose the novel approach of using fuzzy cognitive maps (FCMs) to delineate EC in fNIRS data.
The major four contributions of the proposed methods are as follows.
1) An automatic data-driven method is proposed to compute EC from neural hemodynamic data. 2) A new FCM formulation, called effective FCMs (E-FCMs), for regularizing the network by weight reinforcement using error-driven learning.

3) An E-FCM fusion from individual E-FCMs obtained
from bi-dimensional hemodynamic data. 4) An empirical analysis of the individual contribution of oxy-Hb and deoxy-Hb in representing the underlying cognitive activity. In particular, this article presents an approach at fusion of bi-dimensional fNIRS data while retaining their individualism by utilizing individual E-FCM models representative of EC generated by both oxy-and deoxy-hemoglobin (oxy-Hb, deoxy-Hb). This approach has the potential for uncovering the EC in fNIRS data more accurately since it explores its dual dimensions both separately at first, and then by combination later.

II. HIGH-ORDER FUZZY COGNITIVE MAPS
An FCM is inherently a signed directed fuzzy graph which is capable of representing fuzzy connectivity between variables as fuzzy degrees of relationships in a complex system. The advantages of FCMs over GC analysis are numerous owing to their expressiveness and flexibility they inherit from neuro-fuzzy systems. FCMs are scalable, can handle nonlinear data, and combined with their prowess to accommodate changing effective connections along with the flexibility to modify the memory order of the model, they offer a powerful paradigm to analyze EC in fNIRS data. The extension of the generic FCM model to a higherorder memory implies that those complex systems, such as fNIRS data, which exhibit higher order effective connections among their concepts can also be modeled by FCM [10], [11].
FCMs offer a graphical representation of the dynamics of a given system which can be seen as a combination of fuzzy logic and neural networks [10]. FCMs are widely recognized for discerning the presence and strength of relationships between nodes in a given dynamic system from various domains, such as engineering, medicine, control, and political affairs [12]- [16].
The following is a formal mathematical description of a higher order FCM [15]: where C j (t) is the fuzzy value of the concept or node (or signal in fNIRS paradigm) j at time t based on the strength of the interaction, e ij in the range [−1, 1], of concept j with other concepts in the system and the past values of the concept j, C j (t − k). The total number of concepts in a given system is represented by N, and T denotes the total length of a signal. The number of past fuzzy values of a given concept to be considered by the FCM when computing the new fuzzy value for that concept is dictated by the order of the FCM, represented by k. The FCM also allows for tuning the strength of influence of the preceding fuzzy values on the current fuzzy value for a given concept by optimizing values of g j (k), and e ij using genetic algorithm (GA).
The new fuzzy value for a given concept is normalized using a transformation function with a specified normalized range. The most commonly used transformation function is the sigmoid function (2) that restricts the weighted sum to the usual range of [0, 1], [17], [18]. The transformation function f (x), used in this article (3), is inherently a sigmoid function with range of [−1, 1]. This normalization of fuzzy degree of relationship of a particular concept with respect to any other concept in the fuzzy graph facilitates the comparative analysis such that a value of 1 means fully interconnected, a value of −1 means fully interconnected in the opposite direction, a value of 0 means disconnected, and a value between 0 and 1 (or −1) means interconnected to a certain extent where C is a parameter used to define a particular shape of the sigmoid function. The most common value of C found in the literature is 5 [14], [19]. A quantitative assessment of the predicted states generated by the FCM can be done using the two standard FCM error functions Errors 1 and 2 outlined in (4) and (5) [14], respectively. The error is estimated by comparing the reconstructed signals from the resultant FCM model with the original signals-synthetic fNIRS signals for Experiment 1, and real fNIRS signals for Experiment 2 whereĈ is the predicted state of the concept of resultant FCM, and C is the rate of change in the state of a concept.

A. EC in fNIRS
One of the most common approaches for computing EC in fNIRS data is through GC analysis. The work by Holper et al. [20] focuses on computing EC via GC between brain signals recorded from two subjects undertaking the same task simultaneously. They conclude that GC values are greater for signal of the model to cause the imitator signal than when the direction is reversed, i.e., from the signal of the imitator to that of the model. This could be attributed to the limitation of the GC model to read the causality within the same set of signals when the direction and/or time lag during which causality transpires is changed. In the study by Im et al. [21], the fNIRS signal analysis is conducted using GC. Their conclusion is also limited to unidirected coupling between the somatosensory areas and the motor areas, owing to limitations imposed by unidirectional data analysis of GC.
A comparative study of the brain activity recorded during rest and movement using fNIRS is done by Bajaj et al. [22]. The estimation of EC in the recorded fNIRS data is done by GC. However, they credited the use of DCM over GC analysis when there is varying time lag in activation of causation between two signals. This is due to limited prowess of GC to analyze signals with time lag, or the order of causality, unknown and/or varying. An interesting study by Yuan [23] presents estimating EC by using GC mapping on regions of interest (ROIs) identified by independent component analysis (ICA). The improved results of the GC-ICA hinges on accurate over simplification of the brain regions identified by ICA along with the associated shortcomings of GC analysis for fNIRS data.
An attempt to estimate EC using DCM is done by Tak et al. [5]. Though their results are consistent with findings of the previous fMRI studies, successful application of DCM for EC analysis on fNIRS data is highly contingent on the undertaken task and hence the corresponding characteristic hemodynamic response (HR) function used in DCM analysis.

B. Learning Methods for FCMs
A hybrid technique for FCMs learning for predicting nonstationary time series is proposed by Yang and Liu [24]. They presented the use of wavelet transform with higher order FCMs for predicting nonstationary time series. The application of wavelet transform converts original time series into multivariate time series, which are then employed by higher order FCMs to model and predict the original nonstationary time series.
Toward automatic learning of large-scale FCMs, the work by Zou and Liu [25] presents the use of mutual informationbased two phase memtic algorithm (MA). Their work optimizes the edge weights of only those links identified in the first phase of MA, thereby reducing the search space for MA. The works by Wu and Liu [26], [27] offer two different approaches for large-scale and sparse FCMs learning, one using compressed sensing and the second using least absolute shrinkage and selection operator.
The use of multiagent genetic algorithm for large-scale FCMs learning is done by Liu et al. [28]. Their work demonstrates the strength of using multiagent genetic algorithm for reconstructing large-scale gene regulatory networks with 200 nodes.

C. Pruning Methods for FCMs
A critical problem in the simulation of computational networks, such as FCMs and artificial neural networks (ANNs), is overfitting due to a large number of learnable parameters introduced to accurately model a given system [29]. Among the prominent works done for regularization of neural networks to minimize overfitting, including dropout (DO) [30] and dropconnect (DC) [31]. In DO [30], nodes (along with any associated connections) from the hidden and/or output layer(s) are dropped out randomly during training. Whereas in DC [31], only some of the connections are randomly removed rendering DC to be a more liberal form of network regularization than DO. Both works demonstrate a higher network performance post implementation of network regularization.
One noteworthy solution for the pruning of FCM specifically is proposed by Averkin and Kaunov [18]. In their work, a regularization of FCMs for hybrid decision support system is presented. The work proposes pruning of weights as an approach to compensate for: 1) expert inaccessibility for modeling a system that vary a little from their experience and 2) multiple experts to collaborate on one system. The work is limited by the assumption that the system model does not change during the course of the analysis.
Another consideration in FCMs implementation is the normalization of weights using a transformation function to a certain specified range (usually [0, 1] for FCMs). The steepness or the gradient of the transformation function establishes the range of variability of the non-normalized weights that will have nonextreme (neither 0 nor 1) normalized equivalents. In essence, the parameter tuning defines the window of the nonnormalized weights that can contribute meaningfully to the model estimation by an FCM simulation.
In this regard, the work by Papageorgiou and Froelich [17] is a significant contribution toward the transformation function parameter optimization. The study investigates the influence of the transformation function parameter on the shape and properties of the transformation function. They conclude that optimization of the transformation function parameter is quintessential for the FCM model to fit a given data set.

A. fNIRS Data
fNIRS is a popular neuroimaging technique used extensively for the study of brain activity. It is capable of recording brain activity in real time by detecting changes in metabolic profile of a given brain region under investigation. The changes in concentration of oxy-Hb, deoxy-Hb can be inferred from metabolic activity and oxygen consumption of the underlying neural tissues [32], [33]. The recorded HR can be used in the study of neuroergonomics and brain-computer interfaces [34]. Since fNIRS has a multitude of advantages over other neuroimaging techniques (e.g., fMRI), such as it is low-cost, offers better temporal resolution, has a higher sampling rate, and robustness to motion artifacts alongside safety, and portability, fNIRS is fast becoming a medium of choice for empirical investigations in cognitive neuroscience [35]- [37].
In this section, a description of synthetic [32] and real fNIRS [37] data used to evaluate the proposed E-FCM methodology for detecting EC is presented.
1) Experiment 1: In this article, synthetic fNIRS data is generated to evaluate the efficacy of the proposed E-FCM methodology for assessing EC in fNIRS signals. The method outlined by Scarpa et al. [32] is reproduced to generate a total of 16 synthetic fNIRS causal signals. The synthetic fNIRS signals are generated according to (6), with minor modifications to accommodate the needs of this article where u true (t) represents the true HR. The true HR, u true (t), is scaled by a constant k which can take the value of 0, 0.5, or 1. The channel location would determine the exact value of k for a given synthetic fNIRS signal. The processes of physiological noise, random noise, and noise due to possible motion artifacts are represented by φ sim (t), η(t), and r(t), respectively, in the overall simulated fNIRS signal, y sim (t). For more information on u true (t), and noise models in (6), the reader is referred to [32], [38], and [39]. Fig. 1 shows a representative synthetic fNIRS signal, y sim (t), corresponding to a given true HR stimulus, u true (t). The synthetic fNIRS signal is inherently noisier than true HR stimulus by contributions from physiological and random noise mainly. The incorporation of physiological noise also stretches the length of the HR stimulus in the synthetic fNIRS while the occurrence of the maximum displacement of the synthetic fNIRS is in sync with that of the true HR stimulus [39]. A total of 16 synthetic fNIRS signals are generated to remain consistent with the 16 real fNIRS signals available as outlined in Section IV-A2.   (7) a) Preprocessing of synthetic fNIRS data: Inherently, fNIRS signals are a measure of the metabolic changes in the blood owing to cognitive activity. In this regard, motion artifacts arising from possible head movement can cause fluctuations in fNIRS recordings which are not representative of the cognitive activity. Therefore, it is crucial to remove motion artifacts from fNIRS signals before analyzing it for more accurate assessment of cognitive activity in the brain.
In this article, the synthetic fNIRS signals are filtered using Kalman filter as outlined in [40]. The order of the state space model is one, and the autoregressive model parameters are estimated using the Yule-Walker equations.
The filtered synthetic fNIRS signals are also down sampled before introducing EC within the data. This is because the total length of the synthetic fNIRS signal (≈ 500 s) is significantly larger (≈ 10 times) than the length of the signal where the two true HR stimulus transpire (≈ 50 s) hence down sampling the signal will pronounce the connectivity within the data by omitting some nonsalient data points. In this article, the down sampling factor is experimentally chosen to be 10 such that the fNIRS data still retains its characteristics. b) Effective synthetic fNIRS connectivity data: The EC in the down sampled and filtered synthetic fNIRS data is incorporated according to Fig. 2 and (7) [41]. As can be seen in Fig. 2 and (7), signal 1 (X 1 t ) value at time t is independent of all other signals. However, signal 2 (X 2 t ) and signal 3 (X 3 t ) value is dependent on signal 1 (X 1 t−2 ) value with a time lag of 2, and signal 1 value with a time lag of 3 (X 1 t−3 ), respectively. Signal 4 (X 4 t ) values are influenced by signal 1 values (X 1 t−1 ) and (X 1 t−2 ), and signal 5 (X 5 t−1 ) values; whereas only signal 4 (X 4 t−1 ) influences on signal 5 (X 5 t ). The 16 synthetic fNIRS signals are enhanced with EC by forming three sets of five signals each having same dependencies as delineated in Fig. 2 and (7). The 16th signal (like the 1st, 6th, and 11th signal) has no causal dependencies on any other signal(s). A generic connection matrix, E, with elements e ij , corresponding to the causal network illustrated in Fig. 2, and (7) as shown in Table I. The resultant set of 16 synthetic fNIRS signals enhanced with EC is used by the E-FCM model to estimate the EC within the signals 2) Experiment 2: The real fNIRS data [37] used in this article comes from a neuroergonomics study that involved 27 right-handed male surgeons affiliated with National Health Service and Imperial College London. A local research ethics committee approval was obtained (project number: 05/Q0403/142). The participants performed a complex visualspatial task [namely, laparoscopic surgery (LS)] while their brain activity is recorded, as shown in Fig. 3. Specifically, among the 27 surgeons included nine novices (NVs), ten trainees (TNs), and eight expert (EX) consultants. The reader is referred to [37] for a more detailed description of the participants, the task, and the preprocessing of the fNIRS signals.
A total of 16 real fNIRS signals are used in this article. The fNIRS digitized probe positions were registered from a realworld coordinate to the Montreal Neurological Institute (MNI) space. The MNI coordinates were transformed to Talairach space [42], and look up in a brain atlas [43] to establish their relations with the ROI. A detailed explanation of the fNIRS probe positions transformation is provided in the study by Andreu-Perez et al. [37]. The measured channels locations are as shown in Fig. 4.
The reason this particular dataset is chosen to gauge the performance of the E-FCM method is because the participants can be categorized based on their expertise level for performing LS. A range of expertise within the participants implied a better differentiation can be inferred in the cognitive activity arising from the dexterity of the task.
The data set also entails separate records for oxy-Hb and deoxy-Hb levels in the investigated brain regions. Hence, in order to better assess the EC within the 16 real fNIRS signals grouped using the proposed E-FCM model, the oxy-Hb fNIRS signal and deoxy-Hb fNIRS signals are analyzed separately. This is important because no direct coupling of the two hemoglobin dimensions of fNIRS has been established in the literature yet [33], [44], [45], therefore, it would be counterintuitive to use total hemoglobin (THb) response instead of individual hemoglobin responses. However, the discrepancies between the oxy-Hb and deoxy-Hb signal may be meaningful to elucidate transient neural activity as revealed in [46]. A detailed discussion of how the dual dimensionality of fNIRS signals has been used in this article to establish EC in the brain regions is presented in Sections IV-A2a and IV-D.
a) Fusing hemodynamic varieties: In order to address the particular issue of which hemoglobin response is more representative of the metabolic activity in the underlying neural tissue, the individual effective links deduced by the E-FCM model for both hemoglobin dimensions of fNIRS are analyzed separately and used to compute an overall network that is a combination of the stand-alone networks of oxy-Hb and deoxy-Hb signals.
In particular, the resulting weights of the connection matrix from the E-FCM simulation of both oxy-Hb and deoxy-Hb for all NVs, TNs, and EXs are averaged based on their accuracy level as outlined in Section IV-D. This would facilitate minimization of any random errors that may arise due to slight environmental changes while recording of the data as well as any discrepancies within the subjects per category [such as intelligence quotient (IQ), degree of alertness, etc.]. The averaged weights of the oxy-Hb and deoxy-Hb are then analyzed to compute an overall E-FCM representing both dimensions of fNIRS. In essence, the combined E-FCM would be a representative of the THb response without undermining the individual characteristics of both oxy-Hb and deoxy-Hb.

B. Effective Fuzzy Cognitive Maps
In this section, the proposed changes to the FCM are outlined. These revisions include a regularized FCM prediction formula motivated from DC for neural networks [31], optimization of the transformation function parameter value, C in (2), and the significance of employing a higher order FCM, k in (1), in particular, for detecting EC in the fNIRS signals.
The proposed regularization of FCM is achieved by straining the weights of the connection matrix, E, using a soft regularizer, Z. The soft regularizer Z is a generalization of DC mask since elements of Z ∈ R and can attain any value in the range of [0, 1] unlike a binary mask in DC whose elements can only be 0 or 1 [31]. The elements in E and Z are both optimized using GA; however, please note only elements of E can model the direction of the EC whereas Z can only optimize the strength of the EC, and hence cannot affect the direction of EC unlike E.
Equation (8) entails the formal mathematical definition of E-FCM where C j (t) is the fuzzy value of the concept or node (or signal in fNIRS paradigm) j at time t based on the strength and direction of the interaction, e ij z ij ( represent element wise multiplication) in the range [−1, 1], of concept j with other concepts in the system, and the past values of the concept j, C j (t − k). The total number of concepts in a given system is represented by N, and T denotes the total length of a signal. The number of past fuzzy values of a given concept to be considered by the E-FCM when computing the new fuzzy value for that concept is dictated by the order of the E-FCM, represented by k. The E-FCM also allows for tuning the strength of influence of the preceding fuzzy values on the current fuzzy value for a given concept by optimizing values of g j (k), z ij , and e ij using GA.
The effectiveness of the E-FCM arises from entailing greater degrees of freedom owing to the weight regularizer Z while finding the optimum EC direction, and strength without violating the set of constraints imposed (see tolerance criterion in Section IV-C) on it. In Section V-A and in particular Fig. 7 and Table III, a comparison of E-FCM results with the original FCM results is also made to further fortify the effectiveness of E-FCM.
In this article, both Error 1 (4) and Error 2 (5) are computed when analyzing the performance of the proposed E-FCM technique for evaluating the effective connections between synthetic signals and real fNIRS signals, separately.
1) Transformation Function Parameter Tuning: As mentioned earlier, transformation function is responsible for normalizing the fuzzy degrees of relationship (or weight values) in a specified range. The particular transformation function used in this article is the sigmoid function [see (3)]. Essentially, the gradient of the transformation function determines how fast the non-normalized fuzzy degrees of relationship are squeezed into the normalized range for the fuzzy degrees of relationship.  The value for C is assumed to be 5 for most practical applications [14], [19]. However, with C = 5, the gradient of the transformation function proved too steep to suit the needs for this article-the fuzzy degrees of relationship were being squashed to the normalized range too fast, and hence were not generating the required results.
In contrast, the empirically found value of C = 1, is more inclusive of the non-normalized fuzzy degrees of relationship to be translated to the nonextreme values in the normalized range for the fuzzy degrees of relationship by having a less steep gradient. This can also be seen in Fig. 5. The transformation function with C = 1 (red) is including values from approximately (−4, 4) to be converted to nonextreme normalized equivalents whereas with C = 5, values from the approximate range of (−1, 1) only are being translated to nonextreme counterparts in the normalized fuzzy degrees of relationship.
2) E-FCM Order, k: The E-FCM order dictates how many past fuzzy values, k, in a set of observations will be considered by the E-FCM when trying to discern a fuzzy effective connection between them. The implications of the choice of the order of the E-FCM are largely dependent on the particular application. In (8), the parameter k defines the order of the E-FCM. As is also evident in (8), the impact of the preceding state(s) on the current state can also be scaled by tuning the value of the parameter g.
The motivation for employing a higher order E-FCM for fNIRS signals lies in a more accurate deciphering of the fuzzy EC between the fNIRS signals. This is owing to complex, higher order fuzzy effective connections among the fNIRS signals which the first-order E-FCM dynamics cannot comprehend very well.

C. E-FCM Learning
The E-FCM learning can be achieved either manually using the information provided by experts or by employing an automated process that can use historical information to develop FCMs [15]. There is an increasing trend to use computerized techniques to uncover the fuzzy relations between concepts for an FCM simulation [47] since using an automated approach to learn the inherent model of a given system does not introduce a bias in the FCM simulation that may be incorporated into the results had the FCM evolution been governed by human knowledge.
In accordance with the greater advantages of automated FCM learning, in this article, E-FCM learning is achieved by utilizing the GA. GA is an optimization algorithm that is based on evolutionary ideas of natural selection and genetics and is capable of solving both constrained and unconstrained optimization problems. Owing to its robust and heuristic nature, GA can be applied to learn the inherent model of a given system using the historical data of the system [14].
The GA can also accommodate the learning of peculiar characteristics of a system by tuning its inherent parameters. In this article, the values of these parameters were empirically determined such that the resultant connectivity matrix, E, depicted the inherent structure of the 16 synthetic fNIRS signals as close as possible. The population size of the GA is set to 1000, the maximum number of generations is defined as 1000× the population. The maximum fitness is set to 0.99 and the tolerance criterion is set to 1e −1 . The genetic operator used is crossover and the selection is made by tournament. A detailed description of the GA parameters can be found in [14] and [47].
The flowchart in Fig. 6 outlines the steps for the generation of a resultant connection matrix, E Z, by GA using historical data to be incorporated in the E-FCM simulation. The E-FCM builds the next state vector, C(t+1), of a given system using the resultant connection matrix, E Z, and the historical data, C(t), till a chosen tolerance criterion is achieved.
The GA learning is driven by the fitness of the predicted stateĈ (t) with respect to the original state C(t). The fitness of the predicted stateĈ (t) is evaluated according to the fitness function, f f = 1 10(T − 1)N error (9) where T is the length of each of the total N signals and the error is computed using (5). The fitness of each predicted stateĈ (t) is gauged against an a priori termination criterion, and if achieved, the weight  Fig. 6]. In case the termination criterion is not met, the GA will try to look for new offsprings using selection, crossover, and mutation, to generate a new predicted stateĈ (t) which will then again be compared with the termination criterion, and the process will continue to repeat itself till the chosen termination criterion is satisfied.

D. E-FCM Fusion
The fNIRS channels locations, which are concepts/nodes for an E-FCM simulation, are identical for both oxy-Hb and deoxy-Hb, therefore, the only facet that needs conformation for E-FCM combination is the connections between the concepts. Within the connections realm, the details that need assimilation are the presence/absence, positive/negative relation, and the strength (weight) of the fuzzy degrees of relationship between the channels. In this article, the strength of the fuzzy degrees of relationship in combined E-FCM for the case where a connection of the same polarity (negative or positive) is present in both oxy-Hb and deoxy-Hb E-FCMs is computed by taking the average of their respective fuzzy degrees of relationship in individual E-FCMs. For the case where a connection is present only in one of the parent E-FCMs, oxy-Hb, or deoxy-Hb, it is included in the combined E-FCM with the same fuzzy degree of relationship as in the parent E-FCM. The last case where a connection is present in both E-FCMs albeit with different polarities, the overall E-FCM does not include it, the conflicting connection. A similar work for combining FCMs in group decision making is done by Hanafizadeh and  [12] where individual FCMs originating from different experts are consolidated to arrive at an FCM that represents the decisions of all the experts involved.
Owing to the absence of any established direct link between oxy-Hb and deoxy-Hb that confers the presence of brain activity [33], [44], [45], the approach of a merger E-FCM that reads from both oxy-Hb and deoxy-Hb E-FCM offers a greater accuracy into establishing the overall EC within different regions of the brain. Hence, connections with the same polarity in the parent E-FCMs are corroborated, and connections present in any one of the parent E-FCM are retained, to bespeak the significance of EC transpiring between the corresponding channels. However, the connections which have opposing effective connections in the parent E-FCMs are inconclusive, and consequently removed from the overall E-FCM.

A. EC Computation on Synthetic fNIRS Data
In order to assess the efficacy of the proposed E-FCM technique for evaluating the EC between 16 synthetic fNIRS signals, the reconstructed signals generated by the third-order   [48], [49], the reconstructed signals from bivariate linear autoregressive model [41], [50] that drives GC (GC-AR), and the original signals. A plot of the reconstructed signals from E-FCM (red) with reconstructed signals from standard FCM (green), GC-AR (cyan), and DCM-GLM parameters (black) are shown along with the original signals (blue) in Fig. 7. The multivariate GC (MVGC) toolbox [51] has been used to compute GC-AR parameters and DCM-GLM parameters are estimated using SPM-fNIRS toolbox [52]. A visual assessment of Fig. 7 indicates that the third-order E-FCM is able to discern well the characteristic peaks, i.e., the true HR stimulus, for most of the synthetic fNIRS signals with effective connections. In contrast, as can also be seen in Fig. 7, a total of four predicted signals from standard FCM technique (shown in green) had no HR stimulus [row (R) 1, column (C) 4; R2, C1; R3, C4; and R4, C2] at all. This is owing to overfitting in the absence of a weight regularizer Z in the standard FCM formulation (1) to find the optimum weights while satisfying the set of constraints imposed on it. Likewise, the reconstructed signals from the DCM-GLM parameters are unable to predict the second HR stimulus for almost all 16 signals.
The order, k, of the E-FCM is empirically chosen, third in this case, such that minimum possible values are attained by Error 1 (4) and Error 2 (5). The total error value, i.e., the sum of error for all 16 synthetic fNIRS signals, for Errors 1 and 2 for different orders of E-FCM are reported in Table II. As can be seen in Table II, the total error values for both Errors 1 and 2 are the least for E-FCM order k = 3. A quantitative assessment of the third-order E-FCM results is also done by computing values for Error 1 (4) and Error 2 (5), and comparing them with those obtained for standard FCM, GC-AR, and DCM-GLM results. The error values for all methods are tabulated in Table III The performance of the proposed third-order E-FCM is also gauged against the equivalent results obtained from GC. GC is a statistical tool and the de facto standard for evaluating future values of a time series variable (Y) from past values of both itself and another time series variable (X), if it is established that X Granger causes Y [41], [50]. The MVGC toolbox [51] has been used to compute the values of GC for the 16 synthetic fNIRS. Fig. 8 shows a comparison of established EC using GC [ Fig. 8(b)] and proposed the third-order E-FCM [ Fig. 8(c)] with the true effective connections [ Fig. 8(a)]. A filled black square connecting two signals indicates the presence of an effective connection between them, and a white square implies that there is no effective connection between the two connected signals, both measured at 95% confidence level. For example, in Fig. 8(a), signal 4 is influencing signal 5 (connected with a black square), and signal 5 is not influencing signal 3 (connected with a white square).
The GC results shown in Fig. 8(b) are not very representative of the true EC between the 16 synthetic fNIRS signals [ Fig. 8(a)]. The GC results depict many false positives and false negatives. An example of a false positive among GC results is signal 1 influencing signal 5 whereas signal 9 not influencing signal 10 is a false negative. The spurious results of GC indicate their inapplicability for the analysis of a network characterized by nonlinear effective connections such as fNIRS signals.  In contrast, the results from the proposed third-order E-FCM method are able to decode EC for 12 out of 16 signals [ Fig. 8(c)]. For the case of E-FCM, a filled black square represents the acceptance of the t-test null hypothesis, H 0 : the original and predicted signals connectivity values have the same mean at 95% confidence level.

B. EC Computation on Real fNIRS Data
The real fNIRS data [37] involving 27 surgeons (nine NVs, ten TNs, and eight EXs) performing an LS task is also used to assess the efficacy of the proposed method of a higher order E-FCM for delineating the underlying EC. In order to also better gauge the representation of oxy-Hb and deoxy-Hb in the resulting neuronal activity within the brain, the oxy-Hb and deoxy-Hb signals are simulated separately by the proposed E-FCM to compute their individual EC networks. The EC networks of the bi-dimensions of the fNIRS signals are then used to generate an overall E-FCM.
The average error values (4) along with corresponding standard deviations comparing the accuracy in estimating fuzzy effective connections between oxy-Hb and deoxy-Hb by the proposed third-order E-FCM are listed in Table V. Essentially, hemoglobin group E-FCM simulation for which the average error is greater, its weights are scaled down by the corresponding percentage hence taking forward more of the accurate EC estimates to contribute to the combined E-FCM. For example, for NVs, the oxy-Hb weights are scaled down by 3.1% before mapping into the overall E-FCM.
In order to assess the third-order E-FCM results for statistical significance, Table VI lists the number of the t-test H 0 being accepted for all subjects of category NVs, TNs, and EXs. The null hypothesis is defined as H 0 : the original and predicted signals connectivity values have the same mean at 95% confidence level. The results indicate that for most of the channels, across all subjects and expertise levels, the H 0 is accepted indicating the prowess of the proposed higher order E-FCM for delineating the EC in fNIRS data.
However, the results in Table VI indicate that underlying effective connections in oxy-Hb results are better understood by the proposed third-order E-FCM in comparison to the deoxy-Hb for less experienced subjects. However, the accuracy of the predicted signals, for both oxy-Hb and deoxy-Hb signals, can also be seen to decline as the expertise level increases. This could perhaps be owing to changes in the memory order of the EC of the underlying channels on account of increased experience but needs further investigation since, in this article, the order of the E-FCM model is not varied with the change in expertise level.
In order to facilitate the comparison of brain network evolution as an individual gains a certain degree of expertise in doing a certain task, LS in this case, Fig. 9 shows a plot of the E-FCMs generated from the oxy-Hb and deoxy-Hb fNIRS signals along with corresponding combined E-FCM for NVs, TNs, and EXs. Please note only the most significant connections are shown in Fig. 9 which have fuzzy degrees of relationship greater than 90th percentile. A noteworthy observation that can be made from Fig. 9 is that as the expertise level increases, the number of significant EC (strength of fuzzy degrees of relationship greater than 90th percentile) is more for deoxy-Hb E-FCMs as compared to oxy-Hb E-FCMs signifying that perhaps the deoxy-Hb signals hold more latent information with regards to channels underlying EC as compared to oxy-Hb signals if the brain networks have evolved owing to more experience. The quiescent EC structure within the deoxy-Hb signals for more experienced subjects could also explain why a greater number of t-test H 0 got rejected in Table VI against corresponding oxy-Hb channels.

VI. NEUROPSYCHOLOGICAL DISCUSSION
In this article, the real fNIRS data [37] used in Experiment 2 entailed subjects which differed in their level of expertise for performing a predefined LS task. The fNIRS data recorded, while the subjects performed LS task, is consequently used to assess the efficacy of the novel, high-order and regularized E-FCM method presented. In this section, the results obtained for EC between ROIs from the proposed thirdorder E-FCM model are gauged against similar works in the literature.
The results from the third-order E-FCM model indicate network connections change from random activations to evenly spread out along with more positive influences as the expertise level increase as shown in Fig. 9. This is in agreement with the current findings [35]- [37] that as an individual progress in learning, their brain networks evolve and optimize their connections. Another perspective for underpinning the EC analysis in between ROIs is done by collapsing the weights according to prefrontal cortex (PFC) and motor cortex (MC) as shown in Fig. 10. This is done by first averaging the weights for all subject's oxy-Hb and deoxy-Hb of a given expertise level, and consequently finding the mean of the average values for the individual hemoglobin's ROIs weights. The resultant averaged values are then scaled to unit length for each connection, i.e., PFC to PFC, PFC to MC, MC to PFC, and MC to MC across the three expertise level. A similar trend can also be seen in Fig. 10, with NVs relying more on inter-PFCs and almost nondependent on MCs connections as compared to TNs and EXs. TNs and EXs rely more on inter-MC connections and less on inter-PFC with progression toward a balanced corroboration between ROIs as more experience is gained hence spreading the cognitive load in contrast to NVs [35], [37]. This progression trend with increased experience is also intercepted well by third-order E-FCM as can be seen in Fig. 9 with more strong positive cause-effect relations between PFC and MC for TNs and EXs as compared to NVs.
The shift in brain activations from PFC to MC on account of increased experience is also in line with the findings from the work of Sanes [53] which observed a decrease in EC in frontal pathway encompassing the regions of inferior frontal gyrus and precentral gyrus brain as a particular visual-motor coupled task is learned. Another distinct work by Kawai et al. [54] focusing on the particular role that MC takes while a certain task is learned concludes that MC is critical for learning of a task but not essential for execution of a previously learned motor task. They conclude that MC engages subcortical motor cortex circuits upon learning of a task and assigns them the execution of the learned task. This frees the MC for any new learning activity and is in coherence with the findings of this article that brain networks evolve to a more balanced configuration, on acquisition of a certain level of expertise, without draining any particular segment of the brain.
The work by Chersi et al. [55] also report a similar trend of high dependency on PFC when the subject is inexperienced with the task undertaken and with more experience internal brain areas, such as basal ganglia (BG), are more activated. The task carried out by the animal in the study [55] was to switch on flashing lights by pressing corresponding buttons. Once the animal learned the task, the button and flashing light sequence is changed so as to coerce it to suppress its learned behaviors. The study concluded that PFC regions are more involved during the learning of a task whereas BG takes over when a goal directed behavior is required.

VII. CONCLUSION
This article has demonstrated a novel approach to identify EC connections between fNIRS data using enhanced higher order E-FCM model. The significance of accurately deciphering EC from fNIRS data is paramount for the advancement of research in the working and evolution of the human brain. In this regard, the combination of a higher order E-FCM offers a novel, more robust, and fully automated solution for uncovering the EC networks between fNIRS data. The strength of the proposed model was shown by its superior accuracy in identifying the effective connections for both synthetic and real fNIRS data at 95% confidence level.
The improvements in the FCM framework, along with tuning of the transformation function parameter, renders the proposed higher order E-FCM more adaptable to the requirements of a given complex network. An attempt is also made to study the individual significance of the oxy-Hb and deoxy-Hb signals while uncovering the underlying EC within the fNIRS signals. The EC networks determined from individual oxy-Hb and deoxy-Hb E-FCMs are used in conjunction in a novel approach to arrive at a combined E-FCM representing the characteristics from both dimensions of fNIRS signals. The preliminary results are in line with the current literature that the dual dimension of fNIRS data are independent; however, deoxy-Hb signals might be able to identify more effective relations based on greater number of significant EC connections in Fig. 9 as compared to oxy-Hb signals. Likewise, in Table VI, the number of signals for which the t-test H 0 is accepted is greater for oxy-Hb signals with lesser experience (NVs: H 0 = 144, TNs: H 0 = 140, and EXs: H 0 = 102) than that for deoxy-Hb signals with greater experience (NVs: H 0 = 131, TNs: H 0 = 141, and EXs: H 0 = 111) implying the presence of a more latent EC information in deoxy-Hb signals, which becomes more pronounced with an increase in experience, that was not detected by the proposed fixed higher order E-FCM model. The novel finding of deoxy-Hb signals becoming more representative of the underlying EC as an individual gains experience in a certain motor task may help to understand the evolution of brain networks on acquisition of experience.
The future work aims at exploring the effect of modifying the order of the E-FCM model as the expertise level increases. This can be of significance because as the brain networks evolve the memory order of the underlying EC in the brain network may also increase or decrease. This needs to be further investigated but an intuitive hypothesis can be that optimization of brain networks would have a bearing on the memory order of EC within the ROIs.