Stable fixed points of combinatorial threshold-linear networks.

Combinatorial threshold-linear networks (CTLNs) are a special class of recurrent neural networks whose dynamics are tightly controlled by an underlying directed graph. Recurrent networks have long been used as models for associative memory and pattern completion, with stable fixed points playing the role of stored memory patterns in the network. In prior work, we showed that target-free cliques of the graph correspond to stable fixed points of the dynamics, and we conjectured that these are the only stable fixed points possible [1, 2]. In this paper, we prove that the conjecture holds in a variety of special cases, including for networks with very strong inhibition and graphs of size n≤4. We also provide further evi-dence for the conjecture by showing that sparse graphs and graphs that are nearly cliques can never support stable fixed points. Finally, we translate some results from extremal com-binatorics to obtain an upper bound on the number of stable fixed points of CTLNs in cases where the conjecture holds.

Fixed Points of Competitive Threshold-Linear Networks.

Threshold-linear networks (TLNs) are models of neural networks that consist of simple, perceptron-like neurons and exhibit nonlinear dynamics determined by the network's connectivity. The fixed points of a TLN, including both stable and unstable equilibria, play a critical role in shaping its emergent dynamics. In this work, we provide two novel characterizations for the set of fixed points of a competitive TLN: the first is in terms of a simple sign condition, while the second relies on the concept of domination. We apply these results to a special family of TLNs, called combinatorial threshold-linear networks (CTLNs), whose connectivity matrices are defined from directed graphs. This leads us to prove a series of graph rules that enable one to determine fixed points of a CTLN by analyzing the underlying graph. In addition, we study larger networks composed of smaller building block subnetworks and prove several theorems relating the fixed points of the full network to those of its components. Our results provide the foundation for a kind of graphical calculus to infer features of the dynamics from a network's connectivity.