Mesoscopic architecture enhances communication across the macaque connectome revealing structure-function correspondence in the brain

Anand Pathak, Shakti N. Menon, and Sitabhra Sinha

Phys. Rev. E 106, 054304 – Published 9 November 2022

Abstract

Analyzing the brain in terms of organizational structures at intermediate scales provides an approach to unravel the complexity arising from interactions between its large number of components. Focusing on a wiring diagram that spans the cortex, basal ganglia, and thalamus of the macaque brain, we identify robust modules in the network that provide a mesoscopic-level description of its topological architecture. Surprisingly, we find that the modular architecture facilitates rapid communication across the network, instead of localizing activity as is typically expected in networks having community organization. By considering processes of diffusive spreading and coordination, we demonstrate that the specific pattern of inter- and intramodular connectivity in the network allows propagation to be even faster than equivalent randomized networks with or without modular structure. This pattern of connectivity is seen at different scales and is conserved across principal cortical divisions, as well as subcortical structures. Furthermore, we find that the physical proximity between areas is insufficient to explain the modular organization, as similar mesoscopic structures can be obtained even after factoring out the effect of distance constraints on the connectivity. By supplementing the topological information about the macaque connectome with physical locations, volumes, and functions of the constituent areas and analyzing this augmented dataset, we reveal a counterintuitive role played by the modular architecture of the brain in promoting global coordination of its activity. It suggests a possible explanation for the ubiquity of modularity in brain networks.

Received 9 July 2021
Revised 6 June 2022
Accepted 13 September 2022

DOI:https://doi.org/10.1103/PhysRevE.106.054304

Physics Subject Headings (PhySH)

Biological neural networks Community structure Neuroscience, neural computation & artificial intelligence Scaling laws of complex systems

Brain network

Community detection algorithms Diffusion & random walks Simulated annealing

Nonlinear DynamicsInterdisciplinary PhysicsNetworksPhysics of Living Systems

Authors & Affiliations

Anand Pathak ^1,2, Shakti N. Menon ¹, and Sitabhra Sinha ^1,2

¹The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
²Homi Bhabha National Institute, Anushaktinagar, Mumbai 400 094, India

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Vol. 106, Iss. 5 — November 2022

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Images

Figure 1
Mesoscopic organization of the macaque brain. The network of brain areas, shown in (a) horizontal, (b) sagittal, and (c) coronal projections, clearly indicate that the nodes (filled circles) are organized into five modules, each characterized by dense intraconnectivity. The modular membership of each node is represented by its color (see color key to the right, containing the list of brain areas in each module), while node sizes provide a representation of the relative volumes of the corresponding brain areas (the spatial scale being indicated by the horizontal bar in each panel). The spatial positions of the nodes are specified by the three-dimensional stereotaxic coordinates of the corresponding areas (see Sec. 2). Links indicate the directed nerve tracts connecting pairs of brain areas, and are colored in accordance with their source nodes. For details of each of the brain areas see connectome_nodes.xls [36], and for the macroscopic properties of the network see Supplemental Material, Fig. S2 [36]. (d) Visual representation of the association between the network modules and cortical (in black), as well as, subcortical (in red) divisions of the brain, viz., FL: Frontal Lobe, PL: Parietal Lobe, TL: Temporal Lobe, OL: Occipital Lobe, Cing.: Cingulate gyrus, Ins.: Insula, BG: Basal Ganglia, Thal.: Thalamus, Hyp.: Hypothalamus, OFC: Olfactory complex, and MB: Mid-brain. For a detailed breakdown of the major subdivisions of the brain in terms of their module membership, see see Supplemental Material, Table S1, as well as, Figs. S3 and S4 [36]. All alluvial diagrams in this paper have been created using the online visualization tool RAW [62].
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Figure 2
Submodular organization of module 5 of the connectome. The network of brain areas, shown in (a) horizontal, (b) sagittal, and (c) coronal projections, indicate that the nodes in module 5 (highlighted), which is broadly associated with visual processing, are further organized into three submodules that are characterized by dense intraconnectivity. The submodule membership of each node in module 5 is represented by its color (see color key at the bottom) with the list of brain areas belonging to each of the three submodules shown in the table at the bottom. Submodule $5 A$ is seen to comprise primary visual and subcoritcal areas, while submodules $5 B$ and $5 C$ contain areas that belong to the ventral and dorsal visual pathways, respectively. The node sizes provide a representation of the relative volumes of the corresponding brain areas (the spatial scale being indicated by the horizontal bar in each panel). The spatial positions of the nodes are specified by the three-dimensional stereotaxic coordinates of the corresponding areas. Links indicate the directed nerve tracts connecting pairs of brain areas, and are colored in accordance with their source nodes.
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Figure 3
Robustness of the modular decomposition of the connectome. (a) Visual representation of the comparison between the modular decomposition of the macaque connectome obtained using spectral partitioning [left] with that obtained using the Infomap method [right]. The modules are represented as vertical bars, connected by bands which are colored according to the module obtained using the spectral method from which they originate [using the same color scheme as in Fig. 1]. (b) Modularity of the macaque connectome, shown as a function reconstructed from $10^{3}$ partitionings (circles) obtained through a simulated annealing method for determining communities [45]. The axes on the horizontal plane orthogonal to the vertical axis that corresponds to modularity $Q$ represent embedding dimensions that are themselves complex functions of the partition space, such that the scale of these axes are irrelevant. The distance between the partitionings (whose positions on the horizontal plane are obtained by Curvilinear Component Analysis) are indicative of the degree of dissimilarity between the corresponding modular partitions of the network. The partition obtained by the deterministic spectral method yielding a $Q$ value of $Q_{spectral} = 0.485$ (diamond), and which has been used for our analysis, is seen to occur in the high-modularity plateau comprising a large number of similar partitions, all having a high value of $Q$ . The 291 partitionings that occur above the translucent plane corresponding to $Q = 0.47$ , and whose $Q$ values differ by less than $3 %$ from $Q_{spectral} = 0.485$ , have been used to determine the robustness of the modular identities of the different nodes in the connectome. (c–e) The network of brain areas shown in (c) horizontal, (d) sagittal, and (e) coronal projections, indicating the areas (highlighted) whose modular memberships are invariant across the partitionings obtained by the spectral method (used in our analysis) as well as those obtained by simulated annealing, whose $Q$ differs by less than $3 %$ from $Q_{spectral} = 0.485$ . As in Figs. 1, the modular membership of each node is represented by its color (see color key at the bottom of each panel), the spatial positions of the nodes are specified by the three-dimensional stereotaxic coordinates of the corresponding areas, and node sizes provide a representation of the relative volumes of the corresponding brain areas (the spatial scale being indicated by the horizontal bar shown next to each projection). Within the 291 partitionings that have $Q > 0.47$ , around $70 %$ of the 266 brain areas always occur in the same module as that in the spectral modular decomposition, underlining the robustness of their modular identities (see Supplemental Material, Table S3 [36]). This is quantitatively established by the consensus matrix shown in panel (f), which indicates for each pair $i, j$ of brain areas the fraction of partitionings $P_{i j}$ in which they occur in the same module. The modules obtained using spectral partitioning, displayed in the order in which they are described in the text, are indicated using broken lines. Within each module, the areas are arranged in increasing order of versatility (see text). Module 5, in particular, is seen to be almost completely consistent across all 291 partitionings.
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Figure 4
Classification of brain areas according to their intra- and intermodular connectivity. (a) Nodes of the macaque brain network [colored and scaled as per Figs. 1] are displayed in accordance with their within-module degree $z$ score ( $z$ ) and participation coefficient ( $Π^{c}$ ), which provide a measure of their intra- and intermodular connectivity respectively. This allows the brain areas to be categorized into one of seven possible categories (see Sec. 2), viz., R1: ultraperipheral, R2: peripheral, R3: satellite connector, R4: kinless, R5: provincial hub, R6: connector hub, and R7: global hub. Note that there are no areas in the macaque brain belonging to the categories R4 and R7. (b) The distribution of the areas of the entire macaque brain across the different categories R1–R7 is similar to the corresponding distributions observed in several anatomical divisions, viz., Thal: Thalamus, FL: Frontal Lobe, PL: Parietal Lobe, Cing: Cingulate Gyrus, Ins: Insula, TL: Temporal Lobe, OL: Occipital Lobe, Amyg: Amygdala and Str: Striatum. (c) The connectivity pattern between areas belonging to the different categories R1-R7 indicated by the $z$ scores for abundance of links between each pair of categories (the first symbol in Ri-Rj refers to the category of the source area and the second to that of the target), measured with respect to degree- and modularity-preserved randomized ensemble of networks (see Sec. 2). Large positive (or negative) $z$ scores, i.e., $z > 1$ (or $z < - 1$ ), imply that the corresponding connection types occur more (or less) often than expected from random networks that have degree sequence and community structure identical to the empirical network (for reference, dotted lines are used to indicate $| z | = 1$ ). (d) Sagittal projection of the network of brain areas [see Fig. 1] showing that connections between provincial hubs (highlighted nodes) are localized within each module (see Supplemental Material, Fig. S11 for an enlarged view [36]).
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Figure 5
The pattern of intra- and intermodular connections enhances the spread of information at local as well as global scales. (a) Temporal evolution of spreading processes, quantified in terms of distributions of first passage times ( $τ$ ) of random walkers starting from one node to reach another, contrasted between the empirical brain network (solid line, $τ_{emp}$ ) and randomized ensembles of networks, generated by preserving either the degrees alone (red, $τ_{D}$ ), or both the degree and the modular membership of each node (green, $τ_{D M}$ ). (b) The distribution of $τ$ differs significantly, depending on whether the target and source nodes belong to the same module (blue, $τ^{intra}$ ) or different modules (red, $τ^{inter}$ ). As in panel (a), spreading occurs significantly more rapidly in the empirical network (solid lines) compared to the networks belonging to the randomized ensemble (obtained by preserving degree and modular membership). (c) The empirical network also shows a reduction in the time ( $τ_{emp}^{ord}$ ) required for global coordination, viz., the maximal ordering achieved in a system of binary Ising spins for varying fluctuation levels (indicated by the parameter $T$ ), when compared with randomized network ensembles (degree-preserved: $τ_{D}^{ord}$ , degree- and module-preserved: $τ_{D M}^{ord}$ ). In panels (a)–(c), the dotted lines and the shaded regions around them represent the means and standard deviations calculated over the randomized ensembles. (d)–(i) To see how the different categories R1–R7 of brain areas allow spreading to occur faster in the empirical brain network than in equivalent randomized networks, we compare the case where the source node can belong to any category (d) with those where the source is either ultraperipheral R1 (e), peripheral R2 (f), satellite connector R3 (g), provincial hub R5 (h), or global hub R6 (i). The $z$ score indicates that there is a statistically significant shift in the empirical distribution towards lower values of $τ$ in all cases. However, while for R3 the increase in the rate of spreading is similar irrespective of whether the target is in the same module or in a different one, we observe that there is a relatively larger shift at lower values for $τ^{intra}$ as compared to $τ^{inter}$ for most of the other categories (in particular, R1 and R5). Indeed, the latter behavior dominates when we consider sources across all categories [see panel (c)].
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Figure 6
Physical distance between brain areas is seen to constrain their connectivity, but the modular organization of the network is independent of their three-dimensional spatial arrangement. (a, top) Probability distribution of the physical distances $d$ between all pairs of nodes (red) contrasted with that of connected pairs (blue). (a, bottom) The variation with physical distance $d$ of the connection probability $P (C | d)$ between a pair of nodes separated by $d$ (red) showing the best fit relation $P \sim 1 / d$ (solid line). (b) Joint representation of the space-independent modular organization of the network of brain areas showing the matrices indicating adjacency ${A_{i j}}$ (left surface), modularity ${B_{i j}}$ (normalized by total number of links $L$ , right surface), and physical distance ${d_{i j}}$ (top surface) between areas. Note that for matrix $A$ the background intensity of each block is proportional to the density of connections within that block, and for matrix $B$ only the values corresponding to linked pairs of nodes are shown. The nodes are grouped into partitions corresponding to the space-independent modules of the network with the boundaries indicated by solid lines. The relatively large positive values clustered along the diagonal blocks of $B$ indicate the occurrence of significantly higher density of connections within each module, compared to that expected from the degrees of, and the distance between, every pair of nodes. This characteristic signature of modularity is also visible in the adjacency matrix $A$ representing the connection topology, suggesting that the mesoscopic structure of the brain network is a consequence of factors beyond the constraints associated with physical distance. This is also apparent from the distance matrix which shows that the modules comprise many spatially proximal nodes even after discounting the effect of distance in identifying the modules. (c) Representation of the correspondence between the network modules determined using exclusively information about the connection topology (“Original”) and those obtained from space-independent partitioning of the network into communities. (d)–(f) The distributions of the degree of similarity between the original topological (orig.) and the space-independent (SI) modular partitionings of a network, as measured by normalized mutual information $I_{norm}$ for three types of random surrogate network ensembles. The corresponding values for the modularity $Q$ obtained using the two methods is shown in panels (g)–(i). The ensembles differ in the nature of dependence of connection probability $P$ between a pair of brain areas on the distance $d$ between them, viz., $P \sim d^{0}$ (d), (g), $P \sim 1 / d$ (e, h), and $P \sim exp (- d)$ (f), (i). As the macaque connectome exhibits a power-law dependence similar to that in (e) and (h), we have indicated in these panels the corresponding values for the empirical network (arrows).
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