Sequential attractors in combinatorial threshold-linear networks

Abstract

Sequences of neural activity arise in many brain areas, including cortex, hippocampus, and central pattern generator circuits that underlie rhythmic behaviors like locomotion. While network architectures supporting sequence generation vary considerably, a common feature is an abundance of inhibition. In this work, we focus on architectures that support sequential activity in recurrently connected networks with inhibitiondominated dynamics. Specifically, we study emergent sequences in a special family of threshold-linear networks, called combinatorial threshold-linear networks (CTLNs), whose connectivity matrices are defined from directed graphs. Such networks naturally give rise to an abundance of sequences whose dynamics are tightly connected to the underlying graph. We find that architectures based on generalizations of cycle graphs produce limit cycle attractors that can be activated to generate transient or persistent (repeating) sequences. Each architecture type gives rise to an infinite family of graphs that can be built from arbitrary component subgraphs. Moreover, we prove a number of graph rules for the corresponding CTLNs in each family. The graph rules allow us to strongly constrain, and in some cases fully determine, the fixed points of the network in terms of the fixed points of the component subnetworks. Finally, we apply these results to identify all graphs up to size n = 5 that are parameter-independent core motifs. Core motifs are special graphs with a single fixed point that are in some sense irreducible, and typically support a single attractor. Additionally, we show how the structure of certain architectures gives insight into the sequential dynamics of the corresponding attractor. Conclusion 49

Accurate angular integration with only a handful of neurons

“…This analysis relied on characterizing the behavior of threshold-linear networks in terms of a separation between different linear dynamical regimes. This separation of linear dynamics has recently been used to infer the underlying connectivity of biological networks [55], and to design different connectivity motifs that use transitions between linear regimes to keep count, to coarsely represent different positions, or to generate distinct dynamical patterns [56, 57]. Here, we showed how the precise tuning of interactions within a single connectivity motif shapes the properties of these linear regimes, and how these properties in turn impact the accuracy of integration in the network.…”

Unitary Evolutions From Interacting Quantum Memories: Closed Quantum Systems Directing Themselves Using Their State Histories

“…It is worthwhile to highlight that stable and multistable attractors of QMM-UEs not only occur in the smallest quantum system with simplest interactions between only two arbitrary quantum memories, but also take place in the complete absence of any kind of stochasticity. Furthermore, straightforward generalization of our behavioral analyses in sections (4,5) to larger systems should reveal much richer kinds of multi-attractor and dynamic-attractor phases, such as sequential limit cycles and higher dimensional attractors in biologically-inspired neural networks [160,161]. QMM-UEs are natural to realize these behaviors in quantum neural computations.…”