Wild oscillations in a nonlinear neuron model with resets: (I) Bursting, spike adding and chaos

TL;DR

This paper analyzes how complex oscillations, including bursting and chaos, arise in a nonlinear neuron model with resets, using a mathematical approach to characterize the adaptation map and its bifurcation structure.

Contribution

It introduces a mathematical framework for understanding bursting and chaos in neuron models with resets, focusing on the adaptation map and its period-incrementing structure.

Findings

The adaptation map converges to a piecewise linear discontinuous map.

Bursting patterns correspond to periodic orbits of the adaptation map.

The period-incrementing structure persists with non-constant adaptation, with more complex transitions.

Abstract

In a series of two papers, we investigate the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons. This first paper presents mathematical analysis showing that the system can support bursts of any period as a function of model parameters. In continuous dynamical systems with resets, period-incrementing structures are complex to analyze. In the present context, we use the fact that bursting patterns correspond to periodic orbits of the adaptation map that governs the sequence of values of the adaptation variable at the resets. Using a slow-fast approach, we show that this map converges towards a piecewise linear discontinuous map whose orbits are exactly characterized. That map shows a period-incrementing structure with instantaneous transitions. We show that the period-incrementing structure persists for the full system with non-constant adaptation, but the transitions are more complex. We investigate the presence of chaos at the transitions.