Nerve Theorems for Threshold-Linear Networks

Abstract: Combinatorial threshold-linear networks (CTLNs) are a mathematically tractable subfamily of threshold-linear networks whose dynamics are determined by the structure of a directed graph. Threshold-linear networks have been widely used to model neural networks with associative memory. ln spite of their simplicity, CTLNs exhibit the full range of nonlinear dynamics: multistability, limit cycles, quasiperiodic and chaotic attractors. One of the most remarkable features of CTLNs is the strong relationship between the attractors of the dynamics and the collection of fixed points. Critically, these fixed points can often be completely determined by the structure of the graph. When the fixed points are confined to a subset of the neurons, neural activity has been observed to converge to this subset. In this sense, attractors live where the fixed points live. This motivates the idea of directional graphs, networks that exhibit feedforward dynamics without requiring a feedforward architecture, as a proxy for the expected flow of activity. Specifically, any subnetwork that is a directional graph acts like a single directed edge in terms of the flow of activity through the larger network. We find that whenever a large network can be covered by directional graphs, then we can associate a simplified nerve graph where each directional subnetwork is replaced by a directed edge. Many properties of the larger network can then be understood in terms of those of the nerve graph, resulting in a dramatic dimensionality reduction. We will present a number of results relating the fixed points of the nerve graph to those of the larger network. We will also show examples of how to engineer complicated network architectures with desired dynamic properties and how to investigate the dynamics of complex networks using this powerful dimensionality reduction technique.

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