# Firing statistics in the bistable regime of neurons with homoclinic spike generation

## TL;DR

This paper analyzes the firing statistics of neurons exhibiting bistability due to homoclinic bifurcation, deriving interspike interval densities and highlighting features that distinguish HOM dynamics from other bifurcations.

## Contribution

It provides a novel derivation of interspike interval densities for neurons with homoclinic bifurcation-induced bistability, aiding in experimental identification.

## Key findings

- Interspike interval densities are unimodal and distinct from inverse Gaussian distributions.
- Transition between rest and spiking mainly occurs along the downstroke of the action potential.
- Deduced spike statistics can help identify HOM dynamics in experimental data.

## Abstract

Neuronal voltage dynamics of regularly firing neurons typically has one stable attractor: either a fixed point (like in the subthreshold regime) or a limit cycle that defines the tonic firing of action potentials (in the suprathreshold regime). In two of the three spike onset bifurcation sequences that are known to give rise to all-or-none type action potentials, however, the resting-state fixpoint and limit cycle spiking can coexist in an intermediate regime, resulting in bistable dynamics. Here, noise can induce switches between the attractors, i.e., between rest and spiking, and thus increase the variability of the spike train compared to neurons with only one stable attractor. Qualitative features of the resulting spike statistics depend on the spike onset bifurcations. This study focuses on the creation of the spiking limit cycle via the saddle-homoclinic orbit (HOM) bifurcation and derives interspike interval (ISI) densities for a conductance-based neuron model in the bistable regime. The ISI densities of bistable homoclinic neurons are found to be unimodal yet distinct from the inverse Gaussian distribution associated with the saddle-node-on-invariant-cycle (SNIC) bifurcation. It is demonstrated that for the HOM bifurcation the transition between rest and spiking is mainly determined along the downstroke of the action potential -- a dynamical feature that is not captured by the commonly used reset neuron models. The deduced spike statistics can help to identify HOM dynamics in experimental data.