Relating network connectivity to dynamics: opportunities and challenges for theoretical neuroscience

Abstract

We review recent work relating network connectivity to the dynamics of neural activity. While concepts stemming from network science provide a valuable starting point, the interpretation of graph-theoretic structures and measures can be highly dependent on the dynamics associated to the network. Properties that are quite meaningful for linear dynamics, such as random walk and network flow models, may be of limited relevance in the neuroscience setting. Theoretical and computational neuroscience are playing a vital role in understanding the relationship between network connectivity and the nonlinear dynamics associated to neural networks.

Figure 1:. Graph-theoretic concepts for directed and undirected networks.

(A) A directed network is one in which each edge of the graph has a direction i → j. Bidirectional edges, such as 5 ↔ 6, reflect the presence of both the i → j and j → i edges. (B) The undirected graph corresponding to the network in A. (C) Motifs are induced subgraphs, obtained by selecting a subset of nodes and keeping all edges between them. The graph in A has a variety of motifs, depicted here with matching vertex labels. (D) A geometric graph consists of vertices embedded in a metric space, with (typically undirected) edges between nodes obeying rules based on the distance between them. (E) A small world network has an underlying geometric organization, but also randomly-selected long-range connections. (F) An Erdös-Renyi random graph assigns undirected edges with probability p, independently for each pair of vertices. (G) A hierarchical, or modular, network consists of local modules with long-range connections between them.

Figure 2:

Graph structures, dynamic models, and dynamic properties of interest. Many of the graph structures we look for in neural networks are motivated by their relevance in very simple dynamic models [10, 6]. These models are often linear, and may be poor predictors of nonlinear behavior that is more typical of neural activity.

Figure 3:

Overrepresented motifs and robust motifs. (A) Two motifs that were overrepresented in several distinct connectome studies [76, 63, 82]. (B) Generalized motifs obtained by doubling one of the nodes in the top graph of A. These have also been found to be overrepresented in the C. elegans connectome [41]. (C) Directed cliques have an ordering of the nodes for which i → j if i < j. Note that bidirectional edges are also allowed. (D) Robust motifs of TLNs.

Figure 4:

Motif embedding matters. (A) A simple 3-cycle motif produces a sequential limit cycle attractor in an inhibition-dominated TLN. (B) The attractor associated to a 3-cycle may or may not survive as an attractor of a larger network. In the 5-neuron network (left), there are two 3-cycles but only one of them, 235, has an associated limit cycle (right). (C-E) Three additional networks have identical connectivity statistics as the graph in B. However, they all exhibit qualitatively distinct dynamics. The network in C has two limit cycles, corresponding to the 3-cycles 125 and 253, but no attractor for 145. In contrast, the network in D has four chaotic attractors, while the one in E has three fixed point attractors, one for each 2-clique [56].