Carina Curto

Neural circuits in the brain perform a variety of essential functions, including input classification, pattern completion, and the generation of rhythms and oscillations that support functions such as breathing and locomotion. There is also substantial evidence that the brain encodes memories and processes information via sequences of neural activity. Traditionally, rhythmic activity and pattern generation have been modeled using coupled oscillators, whereas input classification and pattern completion have been modeled using attractor neural networks. Here, we present a theoretical framework that demonstrates how attractor-based networks can also generate diverse rhythmic patterns, such as those of central pattern generator circuits (CPGs). Additionally, we propose a mechanism for transitioning between patterns. Specifically, we construct a network that can step through a sequence of five different quadruped gaits. It is composed of two dynamically distinct modules: a “counter” network, that can count the number of external inputs it receives via a sequence of fixed points; and a locomotion network, that encodes five different quadruped gaits as limit cycles. A sequence of locomotive gaits is obtained by connecting the counter network with the locomotion network. Specifically, we introduce a new architecture for layering networks that produces “fusion” attractors, binding pairs of attractors from individual layers. All of this is accomplished within a unified framework of attractor-based models using threshold-linear networks.

Threshold-linear networks consist of simple units interacting in the presence of a threshold nonlinearity. Competitive threshold-linear networks have long been known to exhibit multistability, where the activity of the network settles into one of potentially many steady states. In this work, we find conditions that guarantee the absence of steady states, while maintaining bounded activity. These conditions lead us to define a combinatorial family of competitive threshold-linear networks, parametrized by a simple directed graph. By exploring this family, we discover that threshold-linear networks are capable of displaying a surprisingly rich variety of nonlinear dynamics, including limit cycles, quasi-periodic attractors, and chaos. In particular, several types of nonlinear behaviors can co-exist in the same network. Our mathematical results also enable us to engineer networks with multiple dynamic patterns. Taken together, these theoretical and computational findings suggest that threshold-linear networks may be a valuable tool for understanding the relationship between network connectivity and emergent dynamics.

Combinatorial threshold-linear networks (CTLNs) are a special class of inhibition-dominated TLNs defined from directed graphs. Like more general TLNs, they display a wide variety of nonlinear dynamics including multistability, limit cycles, quasiperiodic attractors, and chaos. In prior work, we have developed a detailed mathematical theory relating stable and unstable fixed points of CTLNs to graph-theoretic properties of the underlying network. Here we find that a special type of fixed points, corresponding to core motifs, are predictive of both static and dynamic attractors. Moreover, the attractors can be found by choosing initial conditions that are small perturbations of these fixed points. This motivates us to hypothesize that dynamic attractors of a network correspond to unstable fixed points supported on core motifs. We tested this hypothesis on a large family of directed graphs of size n = 5, and found remarkable agreement. Furthermore, we discovered that core motifs with similar embeddings give rise to nearly identical attractors. This allowed us to classify attractors based on structurally-defined graph families. Our results suggest that graphical properties of the connectivity can be used to predict a network’s complex repertoire of nonlinear dynamics.

Nonlinear network dynamics are notoriously difficult to understand. Here we study a class of recurrent neural networks called combinatorial threshold-linear networks (CTLNs) whose dynamics are determined by the structure of a directed graph. They are a special case of TLNs, a popular framework for modeling neural activity in computational neuroscience. In prior work, CTLNs were found to be surprisingly tractable mathematically. For small networks, the fixed points of the network dynamics can often be completely determined via a series of graph rules that can be applied directly to the underlying graph. For larger networks, it remains a challenge to understand how the global structure of the network interacts with local properties. In this work, we propose a method of covering graphs of CTLNs with a set of smaller directional graphs that reflect the local flow of activity. While directional graphs may or may not have a feedforward architecture, their fixed point structure is indicative of feedforward dynamics. The combinatorial structure of the graph cover is captured by the nerve of the cover. The nerve is a smaller, simpler graph that is more amenable to graphical analysis. We present three nerve theorems that provide strong constraints on the fixed points of the underlying network from the structure of the nerve. We then illustrate the power of these theorems with some examples. Remarkably, we find that the nerve not only constrains the fixed points of CTLNs, but also gives insight into the transient and asymptotic dynamics. This is because the flow of activity in the network tends to follow the edges of the nerve.

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.