Colloquium: Multiscale modeling of brain network organization

A complete understanding of the brain requires an integrated description of the numerous scales and levels of neural organization. This means studying the interplay of genes and synapses, but also the relation between the structure and dynamics of the whole brain, which ultimately leads to different types of behavior, from perception to action, while asleep or awake. Yet multiscale brain modeling is challenging, in part because of the difficulty to simultaneously access information from multiple scales and levels. While some insight has been gained on the role of specific microcircuits on the generation of macroscale brain activity, a comprehensive characterization of how changes occurring at one scale or level can have an impact on other ones remains poorly understood. Recent efforts to address this gap include the development of new frameworks originating mostly from network science and complex systems theory. These theoretical contributions provide a powerful framework to analyze and model interconnected systems exhibiting interactions within and between different layers of information. Recent advances for the characterization of the multiscale brain organization in terms of structure-function, oscillation frequencies, and temporal evolution are presented. Efforts are reviewed on the multilayer network properties underlying the physics of higher-order organization of neuronal assemblies, as well as on the identification of multimodal network-based biomarkers of brain pathologies such as Alzheimer’s disease. This Colloquium concludes with a perspective discussion of how recent results from multilayer network theory, involving generative modeling, controllability, and machine learning, could be adopted to address new questions in modern physics and neuroscience.

Figure 1

Multiscale brain organization. The different organizational aspects of the brain system are represented over a multidimensional manifold. Three type of dimensions, or levels, are illustrated here, i.e., time, space, and topology. From the top to the bottom of the manifold, the scales of each organizational level go from micro to macro. From Thibault Rolland.

Figure 2

Bottom-up hierarchical modeling. (a) The so-called K-set hierarchy showing the model progression from cell level to entire brain. K0 is a noninteracting collection of neurons. KI corresponds to a cortical column with sufficient functional connection density. KII represents a collection of excitatory and inhibitory populations. KIII is formed by the interaction of several KII sets and simulates the known dynamics of sensory areas with $1/f$ spectra; see the inset in (b). KIV is formed by the interaction of three KIII sets that models the genesis of simple forms of intentional behaviors. Adapted from [146]. (b) Schematic view of major components involved in thalamocortical interactions. Different shading patterns code for different zones of the system, i.e., from micro [relay nuclei, thalamic reticular nuclei (TRN)] to macro scales (cortex). As indicated by the key, all connections shown are excitatory except for the connection from the reticular cell to the relay cell, which is inhibitory. Adapted from [215]. Inset from [207].

Figure 3

Main configurations of multilayer networks. (a) Full multilayer network. Both within- and between-layer connections are allowed with no specific restrictions. This configuration is typically adopted to model multifrequency brain networks; see Sec. 4a. (b) Multiplex network. Only interlayer connections between the replica nodes are allowed. There are no restrictions on connections within layers. This configuration is typically used to model multimodal brain networks; see Sec. 4a. (c) Temporal network. Interlayer connections are allowed only between adjacent layers. There are no restrictions on connections within layers. This configuration is typically adopted to model time-varying brain networks; see Sec. 4a. From Thibault Rolland.

Figure 4

Structural reducibility of multifrequency brain networks. (a) For each combination of layers a quality function measures the amount of new information added with respect to an equivalent single-layer model. (b) Median values of a quality function obtained from fMRI multifrequency brain networks in healthy subjects. Shaded areas indicate the standard deviation around each value. Adapted from [76].

Figure 5

Emergent properties in full multilayer brain networks. (a) Intralayer and interlayer edges in the multifrequency MEG network. 1-edge between regions at the same frequency. 2-edge of the same area between different frequency bands. 3-edge between different nodes at different frequency bands. (b) Algebraic connectivity ${\lambda }_2$ as a function of the total interlayer connectivity ($S_p$). The vertical solid line corresponds to the actual value of the interlayer connectivity, i.e., without modifying their weights. Adapted from [45].

Figure 6

Multiplex motif analysis of multimodal brain networks. (a) Structural-functional two-layer brain network. Interlayer links between replica nodes are omitted for visibility. Five nontrivial multiplex motifs of two nodes are possible based on the type of connectivity in the DTI structural layer [green (upper-layer) nodes] and in the fMRI functional layer [yellow (lower-layer) nodes]. The $Z$ scores show the motifs that are overrepresented and underrepresented compared to equivalent random networks. Adapted from [20]. (b) Patterns of multiplex triangles comprising directed structural tuples (solid connections) closed by a functional edge (dashed connections). The overall motif counts normalized by equivalent random multiplexes are illustrated as a function of the basal activation parameters $P$ and $Q$ of the Wilson-Cowan model. Adapted from [63].

Figure 7

Multiplex core-periphery structure of the human connectome. Scatterplot of multiplex coreness against single-layer corenesses obtained from structural (DTI) and functional (fMRI) layers. Labels indicate brain areas whose multiplex coreness cannot be predicted by looking at the coreness values in the respective structural and functional layers. Adapted from [19].

Figure 8

Temporal network flexibility correlates with brain perfomance. (a) Overview of network switching (or flexibility) in a temporal network. The red and blue circles identify the nodes belonging to two different communities according to the multilayer network modularity metric. (b) Brain maps of switching rate and dynamic fMRI connectivity. Values were normalized into $z$ scores to ensure that the connectivity dynamics and switching values were equally scaled. Adapted from [196].

Figure 9

Stabilization of critical dynamics in multilayer glia-neuronal networks. (a) Left side: glia cells redistribute metabolic resources from the bloodstream to neural synapses. Right side: associated two-layer network model. Black arrows indicate neural synaptic interactions. The arrow thickness indicates the synaptic strength that evolves according to spiking-time-dependent plasticity (STDP). Red arrows that terminate on black arrows represent the resource supply to the corresponding synapse. (b) Stability analysis of the two-layer STDP model. The largest eigenvalue $\lambda$ of the neuronal network layer and the total resource $\mathcal{R}$ of all glia and synapses are illustrated as a function of time. The data plotted in black correspond to a “baseline” condition. For the data plotted in red (labeled “instability”), the initial evolution is the same as that for the baseline data until the diffusion of resources between the glial cells is turned off (vertical arrow). Adapted from [252].

Figure 10

Temporal network flexibility predicts future learning rate. Significant predictive Spearman correlations between flexibility in session 1 and learning in session 2 (black curve, $p\approx 0.001$) and between flexibility in session 2 and learning in session 3 [red (lighter gray) curve, $p\approx 0.009$]. Each point corresponds to a subject. Note that the relationships between learning and fMRI network flexibility in the same experimental sessions (1 and 2) were not significant; $p>0.13$ was obtained using permutation tests. Adapted from [16].

Figure 11

Multifrequency and temporal reorganization of brain networks in Alzheimer’s disease. (a) Multiplex brain networks are constructed by layering different frequency-specific networks, while temporal networks were constructed by concatenating time-specific networks within frequency bands. (b) Top: hub disruption of MEG multifrequency networks in patients with Alzheimer’s disease. Each point corresponds to a different brain area, and $k$ is the slope of the regressing line. Bottom: brain regions with significant between-group differences in the overlapping weighted degree. $\mathrm{PCUN}.\mathrm{R}$, right precuneus; $\mathrm{HIP}.\mathrm{L}$, left hippocampus; $\mathrm{IPL}.\mathrm{R}$, right inferior parietal but supramarginal and angular gyri; $\mathrm{SPG}.\mathrm{R}$, right superior parietal gyrus; $\mathrm{MOG}.\mathrm{L}$, left middle occipital gyrus; $\mathrm{SOG}.\mathrm{L}$, left superior occipital gyrus; $\mathrm{IOG}.\mathrm{L}$, left inferior occipital gyrus. Adapted from [267], and [50](c) Scatterplot showing the Mahalanobis distance of each subject from the AD or control group when the multiplex clustering coefficient (MCC) and the multiplex participation coefficient (MPC) extracted from time-varying networks (gray line indicates equal distance) are combined. Adapted from [50].

Figure 12

Multimodal brain networks revealing disrupted core-periphery structure in Alzheimer’s disease. (a) Multimodal brain networks (multiplex) constructed by layering DTI, fMRI, and several frequency-based MEG brain connectivities. Adapted from [118]. (b) Spearman correlation ($R=0.59$, $p=0.005$) between the coreness disruption index ($\kappa$) and the memory impairment of AD patients as measured using the free recall (FR) test. (c) Box plots showing the values of the coreness disruption index ($\kappa$) obtained by progressively removing the edges preferentially connected to the multiplex periphery of the healthy control (HC) group. The blue ($x$-axis HC) and red ($x$-axis AD) box plots illustrate, respectively, the $\kappa$ values for the HC and AD groups. (b),(c) Adapted from [119].

Figure 13

Temporal network flexibility as a clinical marker of shizophrenia genetic risk. (a) Significant increases in the mean dynamic reconfiguration of modular fMRI brain networks in unaffected first-grade relatives (REL) (gray bar) and patients with schizophrenia (SZ) (black bar) compared to matched healthy controls (HC) (white bar) [$F(2,196)=6.541$, $P=0.002$]. Bars indicate mean values, and whiskers represent standard error means (SEMs). (b) Significant increases in the mean dynamic reconfiguration of modular brain networks in healthy controls after application of dextrometorphan (DXM) [dark gray bars; repeated measures ANOVA placebo (PLA) vs DXM: $F(1,34)=5.291$, $P=0.028$] relative to PLA (light gray bars). Adapted from [39].

Figure 14

Illustration of the multi-node2vec algorithm. Beginning with a multilayer network (left), one first identifies a collection of multilayer neighborhoods (bag of nodes) via the variable neighborhood search procedure. Next, the optimization procedure calculates the maximum likelihood estimator $F$ through the use of the skip-gram neural network model (right) on the identified bag of nodes. Adapted from [263].