Controlled Perturbation-Induced Switching in Pulse-Coupled Oscillator Networks

TL;DR

This paper investigates how symmetry and perturbations influence switching behaviors in pulse-coupled oscillator networks, revealing explicit transition rules and heteroclinic structures that enable complex neural computations.

Contribution

It derives explicit switching transition rules for small pulse-coupled oscillator networks, highlighting how symmetry affects system dynamics and heteroclinic connections.

Findings

Symmetry enforces or forbids certain switching transitions.

Explicit transition rules are derived for networks of five oscillators.

Pulse-coupled systems can generate complex, rule-based spatiotemporal patterns.

Abstract

Pulse-coupled systems such as spiking neural networks exhibit nontrivial invariant sets in the form of attracting yet unstable saddle periodic orbits where units are synchronized into groups. Heteroclinic connections between such orbits may in principle support switching processes in those networks and enable novel kinds of neural computations. For small networks of coupled oscillators we here investigate under which conditions and how system symmetry enforces or forbids certain switching transitions that may be induced by perturbations. For networks of five oscillators we derive explicit transition rules that for two cluster symmetries deviate from those known from oscillators coupled continuously in time. A third symmetry yields heteroclinic networks that consist of sets of all unstable attractors with that symmetry and the connections between them. Our results indicate that pulse-coupled systems can reliably generate well-defined sets of complex spatiotemporal patterns that conform to specific transition rules. We briefly discuss possible implications for computation with spiking neural systems.