A morphospace of functional configuration to assess configural breadth based on brain functional networks

The quantification of human brain functional (re)configurations across varying cognitive demands remains an unresolved topic. We propose that such functional configurations may be categorized into three different types: (a) network configural breadth, (b) task-to task transitional reconfiguration, and (c) within-task reconfiguration. Such functional reconfigurations are rather subtle at the whole-brain level. Hence, we propose a mesoscopic framework focused on functional networks (FNs) or communities to quantify functional (re)configurations. To do so, we introduce a 2D network morphospace that relies on two novel mesoscopic metrics, trapping efficiency (TE) and exit entropy (EE), which capture topology and integration of information within and between a reference set of FNs. We use this framework to quantify the network configural breadth across different tasks. We show that the metrics defining this morphospace can differentiate FNs, cognitive tasks, and subjects. We also show that network configural breadth significantly predicts behavioral measures, such as episodic memory, verbal episodic memory, fluid intelligence, and general intelligence. In essence, we put forth a framework to explore the cognitive space in a comprehensive manner, for each individual separately, and at different levels of granularity. This tool that can also quantify the FN reconfigurations that result from the brain switching between mental states.

Each subplot represent rest and seven tasks in HCP dataset. The optimal reconstructed number of orthogonal components is indicated by a black dot.
We used the standard HCP functional pre-processing pipeline, which includes artifact removal, motion 73 correction and registration to standard space, as described in Glasser Figure S1.

MORPHOSPACE ANALYSIS
The concept of a morphospace can be used to analyze many other mathematical objects, including In this section, we analyze both measurements in depth. We first introduce the formulation of 97 each axis and then provide further characteristics and/or requirements/assumptions, if any. Any theory 98 that already introduced in the main text will not be re-introduced here.

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It is important to be aware that any generic network can be fragmented (disconnected). This is rarely the case for brain functional networks because of how we compute functional couplings, typically using Pearson Correlation Coefficient. Despite of that, a priori functional community induced from global thresholded adjacency structure is not guaranteed to be connected. As pointed out in Malliaros and This is the x-coordinate of u(C). Module trapping efficiency assesses the characteristic of a functional community based on how well it sustains its topology under rich repertoire of task-evoked conditions, relatively to its segregation/integration role, simultaneously. Recall that module Trapping efficiency is formalized as followed: As claimed in the main text, TE is finitely bounded. There are several ways to observe this; one approach involves applying hierarchical community detection algorithm Fortunato and Hric (2016) and look for the first time G split into more than one subgraphs. Thus, let i be indices representing communities belong to the first hierarchical layer, then where l represents the number of communities. Such value is well-defined and finite. An alternative way to see the trivial bound of the measures is as follows: Let us consider the entire network G, we have: because there is no exits if the configurations is the entire network; moreover, there is zero leakages.

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Hence, any cut into G would have to be strictly less than this upper bound. Note that ∞ 0 is undefined.   In the context of the data set at hand, we can, however provide a better bound then finiteness. We  Realistically, since larger communities carry more exits which is driven purely from a 111 topological viewpoint, L C is a logical choice to normalize the magnitude of τ .

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Additionally, L C is deemed to perform as ||τ || 2 -damping. Notice that there also exists functional 113 communities with low total exiting strength with large cardinality, theoretically. In such case, these 114 structures are rewarded from the standpoint of TE as it converges to TE(C ≡ G). This is the y-coordinate of u(C). Module exit entropy represents communicating preferences of C with respect to the rest of network G from information theoretical viewpoint. The magnitude of this measure assesses the specificity of integration of a given community in varied environment.

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For example, let us say that we have two communities with the same state set S trans = 1, 2, 3 and 122 S abs = a, b, c, d. In community 1, w ij = 0.01, ∀i ∈ S trans , j ∈ S abs ; and community 2, 123 w ij = 0.9, ∀i ∈ S trans , j ∈ S abs . Once we compute the entropy, for both cases, we see that they both have 124 no communication preference, hence numerator is one for both cases.

Normalization
This is the coordinate where normalization is possible. Note that since entropy is 126 normalized by its maximum value (i.e. log(|S trans |)), the number of exits |S trans | impacts is, 127 consequently, neutralized. Thus, one does not need to concern about the cardinality of a community with 128 respect to its number of exits as, in reality, a typically larger community usually carries more exits.

TE, EE behavior across thresholds 130
In this section, we explore some further characterization of these new metrics used in the morphospace.   corresponding changes. This algorithm preserves network basic topological characteristics such as size, density and degree sequence. As the desired number of changes increases, the difference between the original matrix, denoted as A orig , and the randomized counterpart, denoted as A rand , also increases.
The difference between two graphs can be quantified as follows: where n is graph's size and Diss ∈ [0, 1]. It is important to note that the difference between two graphs   convex hull is denoted as Vol(Conv(W )). In R d , the convex hull dimension can take on the values d(x i , x j ) denotes the pre-defined metric distance between two generic points.

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3. h = 2 and h ≥ 3 which constitutes the notion of area and volume, respectively.  Recall that, in the main text, we define the equivalent notion of configural breadth using functional reconfiguration and preconfiguration for a given FN. In the main text, we address that once the points are well-defined to represent tasks per each functional community, we need now the notion that highlights subject capacity to exploring this cognitive space. We provide a deeper analysis of the drive behind the usage of volume of the convex hull.
First of all, since we can only obtain finite number of tasks (hence, points in this space), we see that Analogously, once the points are well-defined in this space, in order to effectively measure the notion of functional preconfiguration, we need to highlight the functional readiness, from a cognition standpoint, to switch between resting configuration to a generic task. Here, we first provide the formula proposed in main-text for functional preconfiguration:

Model Description
We apply iteratively multi-linear correlation models (MLM) to correlate ) with various behavioral measures, i . We hypothesize that highly subject sensitive 211 predictor, as described in Fig. 7C (main text) should be prioritized in MLM model. Iteratively, we start 212 by using only 1 predictor (P F P ); in every subsequent step, we append one extra predictor to the existing 213 one(s), again, accordingly per panel C of Fig. 7 (main text). At the end of iterative process, we 214 consequently obtain 16 MLMs.  and not anything else, say a randomized vector.

Model Description
We further test the strength of our hypothesis by splitting available data into two subsets: test and validation set. Specifically, we first extract the optimal number of predictors by applying