Splay states in finite pulse-coupled networks of excitable neurons

TL;DR

This paper investigates the emergence and stability of splay states in finite networks of pulse-coupled excitable neurons, providing exact analytical methods and revealing their marginal stability and relation to phase oscillator models.

Contribution

It introduces an exact event-driven map for finite pulse-coupled neurons and derives analytical expressions for the stability of splay states, extending previous phase oscillator theories.

Findings

Splay states are marginally stable with N-2 neutral directions.

A family of periodic solutions surrounds the splay state.

In the limit of delta pulses, the number of neutral directions becomes N.

Abstract

The emergence and stability of splay states is studied in fully coupled finite networks of N excitable quadratic integrate-and-fire neurons, connected via synapses modeled as pulses of finite amplitude and duration. For such synapses, by introducing two distinct types of synaptic events (pulse emission and termination), we were able to write down an exact event-driven map for the system and to evaluate the splay state solutions. For M overlapping post synaptic potentials the linear stability analysis of the splay state should take in account, besides the actual values of the membrane potentials, also the firing times associated to the M previous pulse emissions. As a matter of fact, it was possible, by introducing M complementary variables, to rephrase the evolution of the network as an event-driven map and to derive an analytic expression for the Floquet spectrum. We find that, independently of M, the splay state is marginally stable with N-2 neutral directions. Furthermore, we have identified a family of periodic solutions surrounding the splay state and sharing the same neutral stability directions. In the limit of $\delta$-pulses, it is still possible to derive an event-driven formulation for the dynamics, however the number of neutrally stable directions, associated to the splay state, becomes N. Finally, we prove a link between the results for our system and a previous theory [Watanabe and Strogatz, Physica D, 74 (1994), pp. 197- 253] developed for networks of phase oscillators with sinusoidal coupling.