Overview of the present work:

We used the human connectome and a linear model of local node dynamics as the main network model. Moreover, we explored Macaque and mouse connectomes, as well as nonlinear models including a neural mass model of local dynamics. These two variables, network structure and local dynamics, define our ‘game constraints’, since they dictate how information flows in the network and how nodes respond to incoming signals. For every game, such as a linear model of dynamics on the human connectome, the approach of MSA (see Figure 2), uncovered the ‘optimal influence’ landscape of the network, that is, a game-theoretical equilibrium point where nodes cannot unilaterally increase their influence on each other any further. We compared the optimal influence profile of nodes at this point with various communication models and analyzed the most influential nodes in such a signal propagation regime. SPE: Shortest path efficiency, NE: Navigation efficiency, SI: Search information.

Visual summary of the MSA approach:

A. A simple toy network with two sources and one target, in which the sources have different amounts of influence on the target. B. For this toy network, the space of all possible ways in which the activity of the target node can be decomposed into contributions from source nodes has two dimensions. There exists a subset of potential solutions where the activity of the target node is correctly decomposed, but the decomposition is not optimal. There exists one equilibrium point where the decomposition is correct and optimal. C. For every game, that is, the simulation of a whole-brain computational model, MSA performs extensive multi-site lesioning analysis to uncover the influence of every node on every other node. For every target node, MSA lesions combinations of source nodes, tracks how the activity of the target node changes given each perturbation and, for all source nodes, computes the difference between two scenarios: one with the source node included in the lesion-set, and the other where the source node is not lesioned. The difference between these two cases defines the contribution of the source to that specific coalition/combination of sources. Averaged over all contributions, the time-varying contribution of each source is then inferred. Iterating over all nodes results in the ‘optimal influence’ landscape that can be decomposed into two components: Direct influences, where nodes influence their connected neighbors, and indirect influences, where nodes influence distant unconnected nodes.

Large-scale network organization of the optimal influence landscape:

Optimal influences were reorganized according to nine large-scale functional network modules and characterized by two metrics: 1. Average influence, which captures the normalized total influence of a given module within itself and to other modules. 2. Heterogeneity of influences, accounting for the normalized variation of influences within a functional module and to other modules. The egalitarian value represents a naive strategy for assigning contributions in which all players are assumed to contribute equally, i.e., by equal division of the total influence among all nine functional modules. The bars represent how much each module deviates positively or negatively from the egalitarian assumption.

Comparison of optimal influences with other putative communication measures:

Pairwise Pearson’s correlation between each communication measure and optimal influences, to characterize the alignment with existing communication models. Communicability (r=0.94) and Search Information (r=-0.83) had the best alignment in this univariate setting.

Strategies to alleviate the underestimated influence of long-range pathways:

A. Difference between adjacency matrices of tunable models and OI that indicates where these models underestimate (red) and overestimate (blue) the influence of nodes on each other. B. SAR obtains a correlation coefficient of r= 0.99 by increasing the depth by which signals can travel, however, the plot on the bottom shows that a wide range of values from 0.2 to 0.7 can be chosen with small degradation of fit. SAR: Spatial Auto-Regressive model, CO: Communicability, OI: Optimal Influence.

Uni-and-multivariate models of OI:

A. Predictive power of a linear regression model trained on walks up to 15 broadcasting steps. B. A regularized multivariate model trained on all steps shows a small set of steps to be sufficient to explain %96 of OI on average. Each dot represents one cross-validation trial (N=100). C. Sets of regularized multivariate models were trained on network features at each degree of spatial influence for SAR and LAM to capture the impact of this parameter on the performance of the statistical model. The dashed line on the left shows a scenario when the degree of spatial influence is zero, effectively rendering SAR an identity matrix. The second dashed line indicates where the model performed the best, explaining %99 OI. D. Bars represent the contribution of each feature to the model, averaged over degrees of influence. Error bars indicate the %95 confidence interval. OI: Optimal influence.

Correlations among linear, nonlinear, and neural mass models of local dynamics:

A. OI matrices for a linear and a nonlinear model show a near-perfect correlation. Moreover, a Hopf model also has a strong correlation with the linear model, indicating that a linear model of local dynamics represents the communication in brain networks under OI as well as more complex node models B. Scatter plot between two settings of the same linear model, one fitted to resting-state fMRI and the other using a random value for the global coupling parameter. The finding provides evidence for the greater role of network’s structure compared to nodal dynamics in shaping brain-wide optimal influence.

Hubs, rich-club nodes, and their influence:

A. Scatter plot of the influences of nodes on their connected neighbors (direct influence in Figure 2) versus the strength of the underlying connections. Despite the overall strong correlation between the total influence and the strength of structural connections, residuals (depicted by warm colors) reveal several weak connections with more influence than expected by their connection weights. B. The position of weak connections with strong influence in the adjacency matrix. C. The position of these connections in the structural brain network shows that they are mainly long-range connections among the hub regions of the cortex. D. shows those hub regions. E. The relationship between the influence of a region is and its centrality, by four centrality measures (see Material and Methods, Graph-theoretical Measures). F. The most indirectly influential nodes (indicated by red points in E.) that can propagate information to distant and unconnected regions. DMPFC: Dorsomedial prefrontal cortex, PCC: Posterior cingulate cortex, ACC: Anterior cingulate cortex, SFC: Superior frontal cortex, SMA: Supplementary motor area, VLPFC: Ventrolateral prefrontal cortex.

Relationship between centrality of individual connections and their influence:

A. Strength of individual connections versus their centrality in terms of shortest-path edge betweenness centrality. The lower panel shows the same relationship, but for the amount of influence a node has over its connected neighbor compared to the centrality of their connections. B. Depicts the same information for the random walk (diffusive) model of information flow. In both A and B, this relationship is compared against a null model in which the structural connectivity is shuffled while preserving the strength of each node, i.e., the weighted sum of its connections.

How influence decouples from the underlying structural connections:

The spread of points along the second principal component, i.e., orthogonal to the regression line, across different values of spatial influence. A very small degree of spatial influence represents a case in which the signal can only reach the neighboring nodes, while higher values indicate influences that extend over longer pathways and reverberate in the network, effectively decoupled from the underlying structural weight.

Communication Models and Measures

Graph-theoretical Measures

Fitting curve of the scaled communicability (left) and the linear attenuation model (right) to the optimal influence matrix of the human connectome.

Fitting curves of the spatial autoregressive model (top), scaled communicability (middle) and the linear attenuation model (bottom) to the optimal influence matrix of macaque (left) and mouse (right) connectomes.

Temporal characteristics of optimal influence, given the Hopf model. The part above shows how the amplitude of influence between two regions relates to the structural weight and the topological distance between them. The heatmaps in the middle represent the brain-wide incoming and outgoing influence to and from the posterior cingulate cortex (PCC). Together with the column-averaged time course of these influences, the middle plot suggests that PCC receives a diverse range of incoming influences but broadcasts the same message to the rest of the network. This is also apparent from the zoomed-in view of the heatmap at the bottom.

The relationship between the log-normalized structural weights and the fiber length, which had a correlation of -0.6, suggesting that the longer the fiber, the weaker its strength is.

Further analysis of the influence of weak connections in null, synthetic, and supplementary networks. A. The relationship between structural weights and the influence of two connected regions in the mouse and macaque connectome supports the results from the human connectome. There exists many weak connections that exert more influence than expected from their connection weights (the ‘bump’) B. Communication models capture the bump to different extent. C. The bump is also apparent in the strength-preserved null model, although localized in weaker connections. D. Two synthetic networks in which both the weights and the topology is randomized did not replicate the bump. E. Topology-preserved weight-shuffled networks replicated the bump, however, as with C. they are localized to weak connections, suggesting a complementary role for both weight distribution and topological features.

The relationship between influential nodes (in red) and their controllability measures.

Comparing the optimal influence matrices from 1,000 and 10,000 lesion samples per source node.

The amount of which univariate and multivariate models can predict optimal influence from summing individual walks, discounted exponentially (as with communicability).

Comparing optimal influence with functional connectivity.