A Principled Framework to Assess the Information-Theoretic Fitness of Brain Functional Sub-Circuits

Abstract

In systems and network neuroscience, many common practices in brain connectomic analysis are often not properly scrutinized. One such practice is mapping a predetermined set of sub-circuits, like functional networks (FNs), onto subjects’ functional connectomes (FCs) without adequately assessing the information-theoretic appropriateness of the partition. Another practice that goes unchallenged is thresholding weighted FCs to remove spurious connections without justifying the chosen threshold. This paper leverages recent theoretical advances in Stochastic Block Models (SBMs) to formally define and quantify the information-theoretic fitness (e.g., prominence) of a predetermined set of FNs when mapped to individual FCs under different fMRI task conditions. Our framework allows for evaluating any combination of FC granularity, FN partition, and thresholding strategy, thereby optimizing these choices to preserve the important topological features of the human brain connectomes. By applying to the Human Connectome Project with Schaefer parcellations at multiple levels of granularity, the framework showed that the common thresholding value of 0.25 was indeed information-theoretically valid for group-average FCs, despite its previous lack of justification. Our results pave the way for the proper use of FNs and thresholding methods, and provide insights for future research in individualized parcellations.

1. Introduction

The success of large-scale brain connectomics—which subserves a myriad of neuroimaging research endeavors based on fMRI [1,2,3], MEG [4], and EEG [5]—hinges on choosing representations of functional connectivity that are as well-defined as possible. Functional connectomes (FCs) are often constructed by computing a statistical dependency measure, such as the Pearson correlation coefficient, across all specified pairs of the brain’s regions of interest (ROIs) using the aggregated voxel-level blood-oxygen-level-dependent (BOLD) signals. However, constructing FCs from BOLD signals with activation delays (due to inhibitory–excitatory dynamics possibly causing negative ROI correlations) can significantly impact estimates of population-level FCs [1] and the associated functional brain network topological features, such as nodes’ centrality [6], global network measures [2], and geometry–topology relation [7]. Recent efforts have focused on improving FC construction by taking into account neuronal signal activation delays [2] and negative correlations [3]. Nonetheless, much effort is still needed to quantify the efficacy of each FC construction framework, especially in terms of preserving the “true” features of the population FCs that shed light on fundamental principles of the brain.

Functional sub-circuits, e.g., functional networks (FNs) [8,9], and their modularity characteristics [10,11,12,13] are crucial to understanding such fundamental neural principles, including brain complexity [14], differential configurational properties [13], modular structures [10,15], and information processing [16,17]. Studies on the modular organizations of the human brain have also informed applied research on aging [18,19] and disorders including schizophrenia [20]. Moreover, research consistently shows that executive subsystems in the brain are reproducible across many individuals at rest, e.g., [8,21], indicating a widespread application of these FNs in various studies—from control groups [22] to pathological investigations [23,24] and predicting individual differences [25]. Even so, there have been few (if any) systematic studies addressing the validity of a common and rarely challenged practice in brain connectomics, which is applying one specific set of a priori FNs to multiple FCs. In other words, FC processing usually involves mapping an a priori fixed set of FNs onto the constructed FCs, across different subjects and fMRI task conditions, without examining whether those mappings are relevant information-theoretically fit to the constructed FCs, given the existence of human brain fingerprint [13,17,26,27,28,29,30,31,32,33].

Among the many decisions influencing whole-brain functional connectivity estimates, like FCs, and circuit-level representations, like FNs, the choice of brain parcellations (i.e., how nodes in functional brain networks are defined) is, undoubtedly, one of the most critical steps [34,35,36]. In fact, this choice determines the network topology used in downstream analyses. Recent studies have shown that different levels of parcellation granularity can affect the identification of subject-level FC fingerprints [30,37]. In an effort to register the raw neuroimaging data into a sequence of increasing granularity, Schaefer and colleagues have recently published a scheme of atlases that increase in network sizes. These parcellations refine the robust set of resting state networks initially identified by Yeo et al. [8], offering various granularity levels for in-depth analysis. Thanks to these advancements, the brain connectivity research community can now explore characteristics of sequential functional brain networks, especially those coupled with the corresponding a priori set of FNs.

Regardless of which parcellation scheme is employed during large-scale FC and FN analyses, another common practice in network neuroscience is thresholding (or, more generally, eliminating statistically spurious functional edges) based on some arbitrary rules or research hypotheses. Careful design of the thresholding process is central to ensuring scientific rigor not only in healthy control studies, but also in those studying disorders such as schizophrenia [38], unipolar depression, and bipolar disorder [39]. Unrigorous application of thresholding can therefore undermine the validity of such important studies by affecting downstream analyses, including parametric statistical tests [40] and network characterization [41]. To mitigate such issues, various thresholding strategies have been proposed to retain particular desired attributes of the original weighted networks. These strategies include proportional thresholding aimed at keeping the absolute number of edges across different subjects and tasks [41], modular similarity [39], and percolation aimed at preserving the topological features of the original weighted graph [42]. Spurious edge elimination also involves methods based on wavelets [38], mixture modeling [43], topological data analysis through persistent homology [44,45], branch-and-bound-based algorithms (to study cognitive activity [5]), and orthogonal minimal spanning trees for dynamical functional brain networks [46]. Furthermore, alternatives to thresholding treatment for FCs have also been proposed using hierarchical Bayesian mixture models [47]. However, this multitude of strategies further complicates the already complex decision-making process of brain data preprocessing and analysis. After all, how can one determine which combination of FC parcellation, FN partitioning, and edge pruning techniques is optimal for their dataset? To the best of our knowledge, no studies have offered a mathematically justified and robust process for choosing such combinations.

This work tackles the complexity posed by that abundance of choices and lack of theoretical justification. Our objectives are two-fold: (i) formalizing and quantify the level of information prominence of a given fixed set of FNs across different subjects and tasks, and (ii) using the level of prominence as guidance to eliminate spurious functional edges in whole-brain FCs. To do so, we utilize Schaefer parcellations [34] with nine distinct granularity levels, ranging from 100 to 900 nodes in 100-node increments. We first present an overview of Schaefer parcellations, as well as a formalization of stochastic block models (SBMs) and its relevance to our quest in Section 2. We then propose an SBM reconstruction pipeline in Section 3. We wrap up with the results in Section 4 and a discussion in Section 5. Our framework can be generalized to any given pair of an FN partition and a parcellation (e.g., [48,49]).

2. A Principled Framework to Assess Information Theoretical Fitness of Brain Functional Sub-Circuits/Networks

2.1. Schaefer’s Atlas of the Cerebral Cortex

The brain parcellations used in this work are sequential, in the sense that their granularity increases, ranging from 100 nodes to 900 nodes with increments of 100 nodes, and are registered on the cortical surface of the brain. These sequential atlases are made possible thanks to the work of Schaefer and colleagues [34]. To ensure our framework is consistent with prior works in human brain connectomics [31,50], we added 14 sub-cortical regions, resulting in network sizes of 114, 214, …, 914 nodes for all fMRI conditions in the Human Connectome Project (HCP) repository, Released Q3 [51,52]. Schaefer parcellations are subdivisions of Yeo’s functional networks [8].

2.2. Stochastic Block Models (SBMs)

Stochastic Block Models (SBMs) have recently gained traction due to exciting developments in both theoretical and practical domains (see Preliminaries—Stochastic Block Models in the Supplementary Materials for further details on notations and a brief introduction). Theory-wise, phase transitions in the fundamental limits of community detection (or, more generally, mesoscopic structures) were discovered through the measure Signal-to-Noise Ratio (SNR) [53]. In the domain of brain connectivity, SBM has demonstrated its advantages in exploring and uncovering diverse types of brain functional sub-circuits (e.g., dis-assortative or core-periphery) beyond the traditional assortative mesoscopic structures [54,55]. Specifically, Sandon and Abbe, in [53], laid out a comprehensive treatment of criteria for mesoscopic structure recovery for any pair of a networked system and an a priori set of communities (or functional networks in brain connectomic domain). Specifically, the recovery requirements were classified under:

• Weak Recovery (also known as community detection);
• Almost Exact Recovery;
• Exact Recovery.

The recoverability of the ground-truth partition depends on the degree regime (indicated by the degree scaling factor $s_t$) in which the network resides. For instance, weak recovery only requires the necessary condition for a limiting graph ($n\to \infty$) to be in the constant degree regime, i.e., $O\left({\displaystyle \frac{1}{n}}\right)$. On the other hand, exact recovery requires the necessary condition (for the limiting graph) that the graph is asymptotically connected, i.e., in the degree regime of logarithmic $O\left({\displaystyle \frac{log\left(n\right)}{n}}\right)$. The sufficient condition for all the recovery criteria is stated in the respective theorems with different proposed measures with sharp phase transitions, as seen in [53]. If a measure (say, for weak or exact recovery) is below a certain algebraic threshold (stated in the respective theorems), recovery is not possible, although the necessary condition is satisfied. Further details on recovery theorems are located in the Supplementary Materials.

Here, we chose the weak-recovery requirements as guidance for whole-brain functional connectivity estimation for four reasons:

• Although Schaefer parcellations with an increasing number of nodes allow us to project some empirical insights onto their degree regime, a rigorous theoretical argument on the degree regime is not possible for any empirical graph sequence. Hence, the exact recovery of an a priori unique ground-truth partition is not relevant in the case of brain functional connectomes;
• Even in the empirical domain, we observe that both group-average and individual FCs become disconnected (i.e., the number of connected components is more than 1) after a relatively small threshold value in the interval $\tau \in [0.2,0.3]$. Theoretically, a graph sequence is required to be connected, asymptotically, to fulfill the requirements for exact recovery. On the other hand, weak-recovery (detection of mesoscopic structures) offers a more realistic and relaxed set of criteria for this particular application. This facilitates estimating a whole-brain FC that is most suitable for an a priori set of FNs without evaluating the number of connected components of the thresholded FC.
• Most (if not all) mesoscopic studies of brain functional sub-circuits, such as [54,55], are based on pre-defined hypotheses, e.g., that the brain functional sub-circuits involve a more diverse class of community than just assortative ones [54]. Such an assumption leads to the appropriate usage of different community detection algorithms, such as Weighted Stochastic Block Models in the case of [54,55]. As mentioned above, weak-recovery is equivalent to community detection in the theoretical SBM literature;
• No set of functional sub-circuits is universally agreed upon and uniquely identified as the ground-truth communities. Hence, all proposed brain functional sub-circuit parcellations, e.g., [8], are relative.

2.3. SBM Description, Inference and Extended Usage

2.3.1. Model Description

In this subsection, we define some of the key components of SBM. Other fundamental mathematical notations are referred to in Section titled Stochastic Block Model Preliminaries in the Supplementary Materials.

• $G=\left[a_{uv}\right]=\left\{\begin{array}{cc}FC,\hfill & \text{weighted-graphs},\hfill \\ M,\hfill & \text{binarized-graphs}\hfill \end{array}\right.$: network/graph (e.g., FCs in the context of this work);
• $V\left(G\right)=\left\{u\right\}$, and $E\left(G\right)=\left\{uv\mid u,v\in V\left(G\right)\right\}$ are sets of vertices and edges, respectively;
• The size and order of a network are denoted by $\left|V\right(G\left)\right|=n$ and $\left|E\right(G\left)\right|$, respectively;
• $\left\{G_t\right\},\forall t\in \mathbb{N}$ is the graph sequence. In the empirical domain, the number of graphs in the sequence is defined as $|\left\{G_t\right\}|=T$;
• k is the number of communities/clusters;
• $\sigma =\left[{\sigma }_u\right]\in {\left[k\right]}^n$ is the pre-defined, well-understood community assignment in vector form of length n. It is the mathematical map $\sigma =\left\{u\mapsto i,\forall u\in \left[n\right],i\in \left[k\right]\right\}$. In general, $\sigma$ is also referred to as a graph partition;
• $\mathsf{\Omega }=\left[\right|{\mathsf{\Omega }}_i\left|\right]$ is the vector containing the cardinality of community, where

  $$|{\mathsf{\Omega }}_i|=|\left\{u\mid {\sigma }_u=i\right\}|,\forall i\in \left[k\right],u\in \left[n\right].$$
• C is the statistical summary of edge properties within and between communities in matrix form. Mathematically,

  $$C=\left\{\begin{array}{c}{C}_{bin}\in {\mathbb{N}}_{+}^{k\times k},\hfill \\ C_{wei}\in {\mathbb{R}}^{k\times k},\hfill \end{array}\right.$$

  where $C_{bin}\in {\mathbb{N}}_{+}^{k\times k}$ denoted the simple edge count matrix within or between communities and $C_{wei}$ denoted the weighted edge sum (also within or between communities);
• $C_{max}\in {\mathbb{N}}_{+}^{k\times k}$ is the maximum number of edges within or between communities;
• $p=\left[p_i\right]$: the probability that a node u belongs to community $i\in \left[k\right]$; $P=\mathrm{diag}\left(p\right)$ is a $k\times k$ matrix filled with $p_i$ in the diagonal;
• $Q=\left[Q_{ij}\right]\in {\mathbb{R}}^{k\times k}$ is the expected node degree matrix, i.e., the expected number of connections a node in community i has with community j;
• $s_t$: scalable factor of degree regime in a graph sequence $G_t$ where $t\in T$;
• $W_{s_t}={\left[w_{ij}\right]}_{s_t}$ is the edge probability between two nodes in community i and j in terms of the scaling factor $s_t$ (it is worth noting that if $w_{ij}$ is the same for all $i,j\in \left[k\right]$, then SBM collapses to classical ER random graph model). We use W to denote the edge probability matrix with $s_t=1$;
• $PQ=nP\left[{\displaystyle \frac{W}{s_t}}\right]=nPW$ is the community profile matrix where i column is the expected number of edges that community i has with all communities. Note that for weak-recovery (detection), scaling factor $s_t=1$.

2.3.2. Inference and Extended Usage

The basis of SBM parameter inference is reverse engineering by the maximum likelihood principle. Specifically, since both G and $\sigma$ (subsequently, $k={max}_{u\in \left[n\right]}{\sigma }_u$) are priors, in expectation, we can infer $SBM(P,W)$ using the Bayesian approach as follows:

• $P={\displaystyle \frac{\mathsf{\Omega }}{n}}=\left[p_i\right]=\left[{\displaystyle \frac{|{\mathsf{\Omega }}_i|}{n}}\right]$.
• Infer $W_{bin}={\displaystyle \frac{C_{bin}}{C_{max}}}$.
• Compute $W_{wei}={\displaystyle \frac{C_{wei}}{C_{max}}}$.
• $Q=nW$ as $s_t=1$ for weak-recovery.
• Compute $PQ$ (Matrix Multiplication).

$C_{bin}$ is a simple edge count of $M_{\tau }^{GA}$ between or within blocks of communities, whereas $C^{wei}$ is the sum of weighted edges of $F{C}_{\tau }$ (also between or within communities). Specifically,

$$C_{bin}=\sum _{u,v\in \left[n\right]}{\mathbf{1}}_{{\sigma }_u={\sigma }_v},$$

$$C_{wei}=\sum _{u,v\in \left[n\right]}\left|\right|{\mathbf{w}}_{uv}\left|\right|,{\sigma }_u={\sigma }_v,$$

and

$$C_{max}=\mathsf{\Omega }{\mathsf{\Omega }}^T.$$

The inference of matrix P is based on the law of large numbers [53]. For $W_{bin}$, we perform entry-wise divisions of matrix $C_{bin}$ by matrix $C_{max}$, which infers the Bernoulli random variable parameter p representing the probability of successful edge formation between each pair of stochastically equivalent nodes within or between communities. In the case of $C_{wei}$, note that we use the term computing instead of inferring because we have extended the usage of $SNR$ to mesoscopic prominence measure. We use the absolute values $\parallel {\mathbf{w}}_{uv}\parallel$ only to consider the overall magnitude (and not the sign) of functional couplings within/between FNs.

Technically, this inference is less challenging than traditional inference problems where $\sigma$ is also a latent variable in the model and graph ensemble G is the only observable ensemble available. Specifically,

$$(G,n,\sigma ,k)\sim SBM(P,W)$$

where G and $\sigma$ are priors.

2.4. Weak Recovery of Ground-Truth Partition

Definition 1.

Weak recovery of a ground-truth partition can be rigorously equivalent to the existence of an algorithm that infers a partition that agrees with the ground-truth one up to ${max}_i{p}_i+ϵ,\forall i\in \left[k\right]$. This level of accuracy is the minimal requirement for most community detection methods.

Theorem 1

(Sandon and Abbe [53]). Let $(G,\sigma )\sim SBM\left(n,p,{\displaystyle \frac{s_{t}Q}{n}}\right)$ for $p,Q$ arbitrary and $s_t=1$. If $SNR>1$, then weak recovery is efficiently solvable, where

$$SNR={\displaystyle \frac{{\lambda }_2^2}{{\lambda }_1}}$$

and ${\lambda }_i$ is the $i^{th}$ eigen value of the community profile matrix $PQ$.

Weak recovery of given ground-truth communities means that, through that algorithm, the recovered partition outperforms a random guess, i.e., ${max}_i{p}_i$, by a small factor $ϵ$. The criteria for weak recovery are driven by a hard threshold approach presented in the below theorem. Importantly, achieving weak recovery does not necessitate the graph being connected under an asymptotic regime. Loosely speaking, we only need every graph in the graph sequence to have a large connected component. In other words, we only need $\left\{G_{t\in T}\right\}$ to be in the constant degree regime, i.e., $s_t=1$. Consider a network of n nodes divided into two equal-sized ground-truth communities (i.e., ${\displaystyle \frac{n}{2}}$ nodes for each community). In a weak recovery scenario, it is feasible to accurately identify the community membership of each node with a probability marginally above $50\%$, say by an additional $5\%$. This implies that if an ensemble is generated under a constant degree regime, one can arbitrarily assign any community membership to isolated nodes, i.e., leaves; hence, exact recovery is impossible in this regime. On the other hand, for exact recovery, since W scales with n through the factor $s_t$, the community profile matrix M consequently grows with the factor $s_t$ as well.

2.5. Problem Formulation and the Fitness Assessment Framework

2.5.1. Problem Formulation

The current problem of interest is stated as follows: given an a priori set of functional sub-circuits, we would like to investigate and subsequently quantify the “fitness” of such pre-determined partition for a given brain network topology.

We formulate this problem as the “dual” of the primal community detection problems [56,57,58,59]. Specifically, in the primal problem, a partition (i.e., set of mesoscopic structures) is constructed for a given network topology. In the dual, we look to evaluate and subsequently quantify the fitness of a pre-determined (e.g., a priori) set of mesoscopic structures for the given network.

2.5.2. Fitness Assessment Framework

For a given pair of a complex network (e.g., functional connectome) and an a priori set of ground-truth communities (e.g., Yeo’s functional sub-circuits), we propose the following steps to access the information-theoretic fitness of ground-truth communities (Figure 1) as follows:

• Step 1: obtain an average representation (e.g., a group-average FC) from the collection of individual networks, binarize the group-average representation, apply Theorem 1 across the thresholding parameter space $\mathbb{S}$, and yield the masked, binarized group-average FC;
• Step 2 (vetting step): Compute the SNR for the masked group-average FC and investigate the SNR across all finite combinations in $\mathbb{S}$. Compute the the weak-recoverability sub-interval $I\subseteq \mathbb{S}$;
• Step 3: compute the a priori community prominence for each individual FC; note that this prominence can also be computed for the group-average FC.
• Step 4: For each individual FC, obtain the $\overrightarrow{\tau }$ maximizing the prominence computed in Step 3. Check if ${\tau }_{opt}$ belongs to the weak-recoverability sub-interval I. Note that, similarly to Step 3, this step can also be performed on the group-average FC.

3. Application: A Pipeline for Thresholding Functional Connectomes

The below pipeline (Figure 2) describes the process to compute the optimal threshold for a given fMRI condition, a Schaefer granularity, and a cohort in two particular cases:

• Individually driven threshold ${\tau }_{opt}^{\gamma }$;
• Constant (cohort-driven) threshold ${\tau }_{opt}^{GA}$ where $GA$ stands for group-average.

Here, we see that the parameter space $\mathbb{S}$ reduces to the line search of threshold value $\tau =[0,1]\in \mathbb{S}$.

The pipeline contains four distinct steps:

• Step 1: for each Schaefer granularity level and task, compute the binarized (masked) group-average FC (denoted as $M_{\tau }^{GA}$) using the entry-average of individual FCs (the number of individual FC is denoted as $\mathsf{\Gamma }$).
• Step 2 (vetting step):

  (a)

  For each threshold value $\overrightarrow{\tau }=\tau \in [0,1]$, infer the Stochastic Block Model (SBM) parameters to compute the Signal-to-noise ratio (SNR) of $M_{\tau }^{GA}$:

  $$SNR\left[M_{\tau }^{GA}\right]={\displaystyle \frac{{\lambda }_2^2}{{\lambda }_1}}\left\{{\left[PQ\right]}_{bin}^{GA}\right\}.$$

  (b)

  Repeat this computation for all threshold values, apply Theorem 1 to determine the weak-recoverability sub-interval $(a_w,b_w)⊊\tau =[0,1]$ for the group-average FC, i.e., $M_{\tau }^{GA}$.

• Step 3: For a given individual FC and threshold value $\tau$, compute the associated thresholded FC, i.e., $F{C}_{\tau }^{\gamma }$, and then compute the Stochastic Block Model (SBM) parameters for $F{C}_{\tau }^{\gamma }$. Extend the usage of $SNR$ as a mesoscopic prominence measure:

  $$SNR\left[F{C}_{\tau }^{\gamma }\right]={\displaystyle \frac{{\lambda }_2^2}{{\lambda }_1}}\left\{{\left[PQ\right]}_{wei}^{\gamma }\right\}.$$
  Analogously, we can also compute the $SNR$ for the group-average FC ($F{C}^{GA}$) as follows:

  $$SNR\left[F{C}_{\tau }^{GA}\right]={\displaystyle \frac{{\lambda }_2^2}{{\lambda }_1}}\left\{{\left[PQ\right]}_{wei}^{GA}\right\}.$$
  Repeat Step 3 for all threshold values $\tau \in [0,1]$ and all individual FCs for a given fixed Schaefer parcellation and fMRI task pair.
• Step 4:

  (a)

  Obtain the threshold value that maximizes SNR of the thresholded FC and the corresponding optimally reconstructed whole-brain FC:

  $${\tau }_{opt}^{\gamma }=\underset{\tau }{\mathrm{argmax}}\left(SNR\left[F{C}_{\tau }^{\gamma }\right]\right).$$

  Note that if the group-average FC ($F{C}^{GA}$) is used in Step 3, then:

  $${\tau }_{opt}^{GA}=\underset{\tau }{\mathrm{argmax}}\left(SNR\left[F{C}_{\tau }^{GA}\right]\right).$$

  (b)

  Check if ${\tau }_{opt}$ is in the weak-recoverability sub-interval computed in Step 2:

  $${\tau }_{opt}\in (a_w,b_w),$$

  Note that one needs to check the optimal threshold against the weak-recovery sub-interval, regardless of whether it is an individualized threshold (${\tau }_{opt}^{\gamma }$) or a group-average one (${\tau }_{opt}^{GA}$).

4. Results

In this section, using weak recovery criteria, we investigate the level of information-theoretic prominence of an a priori set of FNs with respect to different FCs (both group-average and individual subject levels) across a range of threshold values. Additionally, we offer deeper insights into the use of SNR as a measure of the information-theoretic prominence of this predetermined set of FNs.

The dataset used in this paper contains 410 unrelated subjects from the Human Connectome Project repository, released Q3 [51,52]. This includes (test and retest) sessions for resting state and seven fMRI tasks: gambling (GAM), relational (REL), social (SOC), working memory (WM), language processing (LANG), emotion (EMOT), and motor (MOT). Whole-brain FCs estimated from this fMRI dataset include nine distinct Schaefer granularity levels that parcellate the cortical regions into $n=100$ to $n=900$ nodes, with a 100 nodes increment for each parcellation. The functional communities evaluated in this framework include seven cortical resting state FNs from [8]: visual (VIS), somatomotor (SM), dorsal attention (DA), ventral attention (VA), frontoparietal (FP), limbic (LIM), and default mode (DMN). Each Schaefer granularity has a corresponding Yeo’s FN parcellation. Additional details about the dataset are available in the Supplementary Materials.

4.1. Weak-Recoverability Sub-Interval $(a_w,b_w)$

Based on Panel (A) of Figure 3, we see that for most Schaefer granularity levels (except for $n=100$), the lower and upper bound of theoretically guaranteed sub-interval of weak-recovery stay fairly stable: $\tau \in [0.05,0.8]$. The lower bound $a_w$ stabilizes faster than the upper bound $b_w$ across Schaefer parcellations. Except for the low-resolution parcellation $n=100$, the weak-recovery valid range is relatively stable and parcellation-independent. This implies that the information-theoretic relevance of an a priori set of FNs is, to some extent, parcellation-free. In other words, for all investigated granularity levels, the thresholded graphs are in the weak-recoverability regime, except for the complete ($\tau \in [0,0.05)$) or empty ($\tau \in (0.8,1]$) graph extremes. Panel D of Figure 3 shows further details on the FC density. This is rather interesting because, at those two extremes, networks will contain either too much noise (complete graphs) or too little signal (empty graph) for any highly putative partitions to be information-theoretically relevant.

4.2. Resting State: Group-Average versus Individuals

Based on Panels (B) and (E) of Figure 3, it is evident that all SNR profiles (including the group average and individual levels) behave non-monotonically across the threshold range. There exists a threshold value such that SNR is maximized in the investigated range $\tau \in [0,1]$. In addition, all optimal threshold values, for both group-average and individual FCs, are within the weak-recoverability sub-interval $(a_w,b_w)$ for all investigated Schaefer granularity levels.

Secondly, we see that both group-average and individual SNR profiles scale with n. This is because the scaling factor $s_t$ for the Schaefer FC sequence is not constant. In other words, as the graph size gets larger, one can expect the community profile matrix $PQ$ with entries ${\left[PQ\right]}_{ij},\forall i,j\in \left[k\right]$ to represent the number of expected “friends” between FN i and j (e.g., between DMN and LIM) to become larger numerically. Further evidence on empirical exploration of the Schaefer graph sequence degree regime is located in the Supplementary Materials.

Thirdly, for a fixed Schaefer granularity level, the group-average SNR peaks higher and earlier across the investigated threshold range than that of an individual subject. Interestingly, the topological property of connected components for both individual and group-average FCs, across all Schaefer parcellations, also exhibit a similar trend. Specifically, according to Figure S1 (Supplementary Materials), individual FCs become fragmented earlier, i.e., the number of connected components surpasses 1 faster, compared to the corresponding group-average FCs for a fixed granularity level. Topologically and numerically speaking, averaging FC entries across the subject domain damps down the individual fingerprints presented as high-magnitude Pearson correlation values in FCs. This results in magnitude-wise smaller functional connectivity entries, which are annihilated by smaller threshold value $\tau$. On the other hand, using the same analogy, one can see that it takes a higher threshold value for individual FC entries to be annihilated.

4.3. Individualized Optimal Thresholds

As one can observe from Figure 4, the individualized optimal threshold varies across different individuals, which demonstrates strong evidence of the existence of FN functional fingerprint across subjects. In addition, the average of these individualized thresholds, for a given parcellation granularity, is roughly equal to the group-average optimal threshold.

4.4. Group-Average: Resting State vs. fMRI Task Analysis

Next, we investigate the prominence of Yeo’s resting state networks with respect to different fMRI conditions, including seven tasks and the resting state, through SNR measures using group-average FCs across all Schaefer granularity levels and the entire threshold range. Using the resting state SNR profile as a baseline, we compare all task responses in these two scenarios:

• Constructing FCs with the maximum number of time points available for each fMRI condition;
• For all fMRI conditions, constructing FCs using 166 time points, which correspond to the number of time points associated with the EMOT task that is also the minimum across all conditions.

Firstly, in both scenarios, the maximum SNR values for all examined tasks surpass the hard threshold $SNR=1$ for weak recoverability (Figure 5). Moreover, the optimal threshold ${\tau }_{opt}$ consistently falls within the range $(a_w,b_w)$. Trivially, resting state SNR dominates all available tasks across all parcellation levels. This is expected because the selected set of FNs are Yeo’s resting state networks. Secondly, working memory (WM) fMRI responds fairly consistently across all granularity levels in both scenarios. From an information-theoretic perspective, EMOT is the most similar task to the resting state, with respect to Yeo’s resting state networks.

Thirdly, in the maximum-timepoint case, with the exception of $n=100$ parcellation, the SNR profiles for most tasks are roughly half the magnitude of the resting-state SNR. Furthermore, for all examined Schaefer parcellations, group-average task FCs appear to reach their SNR peak earlier than the resting-state counterpart. Further details are indicated in Figure 6, Panel A.

In the second scenario, when the minimum number of time points is used across all fMRI conditions, the gap in SNR magnitude between the resting state and each task condition is significantly narrowed, yet the SNR during rest still exceeds those during tasks. Further details are indicated in Figure 6, Panel B.

4.5. The SNR-Driven Inequality

It is important to check if SNRs are robust against randomness, i.e., whether they are a valid factor in deciding the threshold. To do so, we randomly shuffle Yeo’s resting state networks and recompute the SNR response. We repeat the random shuffling procedure 100 times, and record the results for all nine group-average FC induced by the nine Schaefer parcellations, each of which is under the $RES{T}_1$ condition with scanning pattern $LR$. Results for $RL$ pattern are available in the Supplementary Materials.

For every fixed Schaefer parcellation granularity, the null model SNR profiles are uniformly lower than those of all subjects across the entire thresholding range. Furthermore, the null model values do not exceed the hard threshold imposed by weak recovery criteria, i.e., $SNR=1$. This observation holds true across all investigated Schaefer parcellations, as seen in Panel F of Figure 3. Interestingly, the SNR gets uniformly smaller as Schaefer parcellation granularity increases, as seen also in Panel F.

Collectively, given the SNR results obtained at rest, under task conditions, and null models, we empirically form an inequality relation between resting state and task fMRI-induced FCs in terms of SNR response and the corresponding level of prominence of Yeo’s resting state networks across different fMRI conditions:

$$0<SN{R}_{null}<SN{R}_{task}<SN{R}_{rest}$$

(1)

This general order of SNR response is observed at the threshold $\tau$ that maximizes the objective function $SNR$ by the weak-recovery criteria. At such optimal threshold values, all SNR profiles for task fMRI are in the weak recoverability region while still smaller in magnitude than that at the resting state. Together, these inequalities constitute an empirical lower-bound and upper-bound for $SN{R}_{task}$, at least for all the tasks investigated in our study.

4.6. Maximum SNR and Threshold Relationship

As the granularity of Schaefer parcellation gets finer, the corresponding group-average SNR profiles become larger due to the natural scaling of the community profile matrix $PQ$. This observation applies to the majority of the threshold range. Moreover, per Figure 3, Panel F, we see that optimal thresholds, e.g., ${\tau }_{opt}$, tend to decrease as the granularity level increases, which suggests that larger Schaefer FCs do not need to be thresholded as much. Another interesting observation is that, with the exceptions of $n=\left\{100,200,900\right\}$, all other investigated granularity levels accept a very stable optimal threshold ${\tau }_{opt}^{GA}=0.25$. Being a computation pipeline that relies on discretized line search on threshold $\tau$ (of increments 0.05 for $\tau =[0,1]$), yielding this level of consistence of optimal value is unexpected.

4.7. Validation: Highly Putative Partition Back-Test

The theory of weak recovery and its extended usage proposed here allowed us to argue for the relevance of using SNR as a measure that, via thresholding as a specific use case, guides the estimation of well-defined functional connectivity given the mapping of an a priori set of FNs. In this section, our goals are two-fold:

• Validate our framework by solving the “forward” problem (e.g., community detection) and compare the detected partition with the ground truth;
• Benchmark different community detection algorithms to solidify the above validation.

Specifically, we juxtapose SNR as a guiding measure against objective-function community detection methods. One such method is Newman’s Q score Maximization [60,61,62]. Here, instead of using the Q score (modularity score) as an objective function for community detection, we use it as a guiding measure to investigate its behavior across the tested threshold range $[0,1]$. In broad strokes, the Q score measures the statistical difference between a network and its corresponding null model with similar topological properties such as the degree sequence. It can be computed as follows:

$$Q=\sum _{u,v}(A_{uv}-\alpha P_{uv})\delta ({\sigma }_u,{\sigma }_v).$$

where $\delta (·,·).$ and $\alpha$ are the Kronecker delta and tuning parameter (defaulted at $\alpha =1$), respectively. In network neuroscience, the majority of studies examining mesoscopic structures of brain functions heavily leverage the maximization of Q score, which unravels predominantly assortative communities, i.e., mesoscopic structures with denser internal edge density than the external one [15,63]. SBM inference methods like Weighted SBM Inference, in principle, uncover a more diverse type of communities beyond assortative ones, such as dis-assortative and core-periphery communities [54]. Because of such distinct differences in principle between the two types of approaches, the Q score would provide a good benchmark for comparing the robustness of SNR against various community detection approaches. Note that for Weighted SBM inference, we assume a Poisson distribution for the weighted graph [64]. Although other model assumptions are possible, our goal in this paper is not to select the most fitting model assumption, but rather to investigate the differences in the communities detected using two theoretically different approaches. In other words, we are not looking to see if the Q score or $SNR$ picks up the exact threshold where the inferred partition is information-theoretically aligned with Yeo’s FNs; rather, we are interested in seeing whether each of those two measures captures the threshold interval where the two partitions agree to a relatively high degree. To measure the information-theoretic agreement between the inferred and ground-truth partitions, we use adjusted mutual information (AMI), which is a measure adjusted to chance. Further details on the inference method and AMI are described in the Supplementary Materials.

Here, our evaluation criteria are as follows: there would only be a threshold sub-interval that are better aligned with the highly putative (ground-truth) community assignment. In the forward direction, we apply three different community detection algorithms across the investigated threshold range $[0,1)$ and measure the corresponding AMI between the inferred and ground-truth partitions. A robust guiding measure, since “blinded” from the ground-truth partition, should have a similar behavior, compared to the AMI curve.

Firstly, per the right panels in Figure 7, we see that both community detection methods, namely Newman’s Q score Maximization and Weighted SBM Inference, yield very similar trends. Specifically, both AMI profiles go up and down crossing the threshold range. Further, AMI gets smaller as n gets larger, which is expected for graphs with increasing numbers of nodes. Interestingly, the threshold values that maximize the AMI for Newman’s Q score Maximization tend to shift left as n increases. We see this particular behavior with $SNR$ in the earlier result section (Panel F of Figure 3). Secondly, the Q score keeps a fairly steady rise in magnitude across the threshold range. Further, it does not appear that the Q score is parcellation dependent; this is expected, because the measure is normalized by $2 \mathrm{m}$. Moreover, the Q score peaks and plateaus at a very high threshold range $\tau \in [0.6,0.8]$. In that range, the thresholded FC is highly fragmented (Figure S1) with extremely low edge density Figure 7, and ceases to retain interesting topological insights for further analysis. Lastly, we see that $SNR$ driven curves, with an a priori set of FNs, behave very similar to AMI profiles of both objective-function approaches. On the other hand, the Q score keeps rising across the threshold range and starts plateauing towards the end, which indicates failure at picking a threshold useful for an a priori partition such as Yeo’s FNs. Collectively, our results show, once again, that $SNR$ computation on weighted, thresholded FCs provides excellent guidance for reconstructing a graph with the most information-theoretic relevance to a particular fixed set of FNs.

5. Discussion

In recent years, the network neuroscience field has been striving forward with many exciting discoveries that are becoming more and more relevant to clinical applications and personal medicine. In network neuroscience, this urges the need to improve a popular proxy of brain function, namely functional connectivity. Having the most proper, state-of-the-art mathematical representation of functional brain circuits allows for more accurate and confident positioning of research endeavors. In this work, we put forth a simple framework that allows improving the mathematical representation of brain functions given the use of an a priori set of functional networks. This framework also doubles as a clear evaluation tool for any specific combination of FC parcellation, FN partition, and edge pruning techniques applied to a large-scale brain dataset, thereby streamlining the complex yet crucial studies in network neuroscience.

Thresholding, which is an edge pruning technique used in post-FC processing, is seldom challenged as a standard practice that eliminates, albeit arbitrarily, statistically spurious edges. Since an increasing body of clinical research now involves FC thresholding in the data construction pipeline, careful scrutiny of thresholding is therefore imperative. To the best of our knowledge, our framework represents the first attempt to address the impact of thresholding on the fitness of functional networks. Other thresholding methods aim to optimize topological properties (such as network density [41,65,66,67]), thereby stabilizing graph metrics across datasets. On the other hand, our approach tackles a fundamentally different challenge, namely stabilizing the information-theoretic metrics of functional subcircuit assignment across parcellations. We believe that this challenge is at least as relevant, if not more so, to the current state of fMRI-based functional brain research, as most studies continue to rely on Yeo’s canonical functional subcircuit assignments without re-examining their fitness to the specific parcellation granularity.

By addressing this gap, we conclude that there is no single constant threshold value that is optimal across different parcellation granularity levels, such as the Schaefer ones. In particular, from coarser to finer Schaefer granularities, the optimal threshold value decreases. This result is partially observable in the behavior of matrix W across all studied Schaefer parcellations. According to Figure 7, we see that for a fixed threshold value, as Schaefer granularity increases, Yeo’s functional networks behave more in an assortative manner, i.e., denser internal edge density and sparser external one. We see that through a brighter diagonal and a darker off-diagonal regime of matrix W, across Schaefer parcellations with fixed threshold value $\tau$. Information-theoretically, it means that a larger graph (in size) tends to contain more relevant information to unravel the ground-truth partition (in our study, seven Yeo’s resting state networks); hence, we do not need to threshold the FCs as deeply as the lower granularity parcellations, such as $n=100$. This result also suggests that FC size is proportional to the level of prominence, or fitness, of the a priori set of FNs. Nonetheless, the exact limit of this behavior when granularity tends toward infinity is unknown, e.g., whether the optimal thresholding value will reach a plateau even if the granularity increases. Despite this uncertainty, we showed that for the majority of granularity levels between 100 and 900, the commonly used threshold of $\tau =0.25$ is a valid choice for group-average topology.

Moreover, when using SNR as a goodness-of-fit measure while fitting an a priori set of FNs onto the FC, while no significant differences are observed between resting state and task conditions for the low-resolution parcellation (n = 100), distinct differences emerge at higher resolutions. There are two ways of interpreting this result: (i) a priori FNs exhibit a poorer fit during rest compared to task states; (ii) there is an intrinsic shift of functional network dynamics at the individual level between the resting state and the task condition. Furthermore, there is also strong evidence suggesting a wide variance in the individualized thresholds across all Schaefer parcellation granularity levels. In the same vein, our results also support the concept of individualized parcellation suggested by the work of Salehi and colleagues [36]. While intuitive and insightful, individualized parcellation across subjects and tasks remains computationally expensive. To that end, our work offers a well-defined tool to examine the level of relevance a particular set of functional networks exhibits when mapped onto individual FCs under different conditions. In simpler terms, it allows us to, for the first time, quantify the individual differences (through information-theoretic gap) when the same atlas is mapped across cohort and/or task domains. This paves the way for alternative frameworks that build upon our work, potentially leading to task-dependent or subject-dependent parcellation methods beyond that proposed in [36].

Our work also extends the usage of the weak-recovery theorem by leveraging SNR as a goodness-of-fit measure. Specifically, our results suggest that for the majority of threshold values, the masked binarized FCs are in the regime of week recovery. However, an open question remains: when parcellated by the Schaefer atlas for a fixed individual and an fMRI condition, is the sequence of FCs in the exact recovery regime? Future studies are needed to address this information-theoretic gap between weak and exact recoverability requirements that is reflected by two measures: SNR for weak recovery and Chernoff–Hellinger distance for exact recovery. Although exact recovery is a stronger requirement, if the Schaefer graph sequence falls within the exact recovery degree regime, the mutual information between the inferred partition (through network inference and objective-based community detection methods) and the ground-truth one (e.g., Yeo’s parcellations) will be theoretically higher.

While our current work addresses the validity of multiple levels of analysis, such as parcellations and subcircuits, future extensions of this study should focus on examining the external validity and generalizability of the framework and its results. In particular, the study would benefit from replication on datasets drawn from healthy populations across different age groups, extreme populations such as those with neuropathological conditions, and data recorded under varying experimental conditions and task designs. Such extensions would introduce a vast range of network complexity and heterogeneous distributions of topological properties, thereby providing the ultimate tests for the power of our framework. Additionally, several technical aspects requiring further investigation include the limitations of fMRI voxel resolution, both spatial and temporal, as well as the corresponding Schaefer parcellations, particularly their impact on SNR when fitting Yeo’s functional networks. Specifically, further analysis is needed to explore the effects of voxel sizes (e.g., 2 mm isotropic for the HCP dataset [51]) and repetition times (e.g., 720 ms for the HCP dataset [51]).

Nevertheless, our findings highlight two crucial points that have substantial implications for brain connectomics research. First, because of the existence of individual brain fingerprints [30,31], we need to pay extra attention when applying a common, fixed atlas to individual FCs. Secondly, we show that thresholding FC matrices is not only an intuitive step during FC post-processing (e.g., to eliminate statistically spurious edges), but also a necessary one if we would like to use such FCs, coupled with an a priori set of FNs, to support any research endeavor in brain connectomics. These results suggest a promising new direction: individualized and task-dependent parcellation methods as an alternative to fixed atlases like that of Yeo. This opens up new directions for precision medicine, particularly targeting individual-specific neurological and neuropsychiatric disorders.