Widespread neuronal chaos induced by slow oscillating currents

TL;DR

This study explores how slow oscillating currents induce widespread chaos in a neuron model, revealing complex bifurcation structures and routes to chaos through advanced bifurcation and reduction techniques.

Contribution

It uncovers the detailed bifurcation structure and chaos mechanisms in a neuron model with fast-slow dynamics, including novel symbolic analysis methods.

Findings

Multiple homoclinic connections underpin chaos.

Bifurcation analysis reveals key codimension-2 points.

A one-dimensional map captures complex dynamics.

Abstract

This paper investigates the origin and onset of chaos in a mathematical model of an individual neuron, arising from the intricate interaction between 3D fast and 2D slow dynamics governing its intrinsic currents. Central to the chaotic dynamics are multiple homoclinic connections and bifurcations of saddle equilibria and periodic orbits. This neural model reveals a rich array of codimension-2 bifurcations, including Shilnikov-Hopf, Belyakov, Bautin, and Bogdanov-Takens points, which play a pivotal role in organizing the complex bifurcation structure of the parameter space. We explore various routes to chaos occurring at the intersections of quiescent, tonic-spiking, and bursting activity regimes within this space, and provide a thorough bifurcation analysis. Despite a high dimensionality of the model, its fast-slow dynamics allow a reduction to a one-dimensional return map, accurately capturing and explaining the complex dynamics of the neural model. Our approach integrates parameter continuation analysis, newly developed symbolic techniques, and Lyapunov exponents, collectively unveiling the intricate dynamical and bifurcation structures present in the system.