Macro- and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons

Abstract

We study a plethora of chaotic phenomena in the Hindmarsh-Rose neuron model with the use of several computational techniques including the bifurcation parameter continuation, spike-quantification, and evaluation of Lyapunov exponents in bi-parameter diagrams. Such an aggregated approach allows for detecting regions of simple and chaotic dynamics, and demarcating borderlines-exact bifurcation curves. We demonstrate how the organizing centers-points corresponding to codimension-two homoclinic bifurcations-along with fold and period-doubling bifurcation curves structure the biparametric plane, thus forming macro-chaotic regions of onion bulb shapes and revealing spike-adding cascades that generate micro-chaotic structures due to the hysteresis.

Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models

“…These bifurcation lines have been obtained using the continuation software AUTO and the SPC technique. In [5,21,22], the role of these bifurcations in the spike-adding process and their influence in the creation of the chaotic structures has been shown.…”

“…Fig. 3 depicts schematically the global organization of the chaotic region as being "onion-like" [21], with different lobes that are concentric and going deeper moving from right to left but in a exponentially small scale from left to right beginning on the other side. This structure governs directly the systematic evolution in the spectrum of UPOs and the topological structure of the chaotic attractors in squarewave bursters ( Bifurcation analysis provides several insights into the spike-adding process and the creation of chaotic lobes.…”

Stochastic Generation of Bursting Oscillations in the Three-dimensional Hindmarsh–Rose Model

“…Then, with the increase of I, a sequence of period-adding bifurcations occurs: bursts with 2,3,4, ..., 13 spikes appear. A transition from burst with 12 spikes to burst with 13 spikes is accompanied with the transition to chaos [5,6,8]. Fig.…”

“…The analysis of the deterministic dynamics of the HR system in various parametric zones was provided in many works [3][4][5][6][7][8].…”

Synchronization of two coupled Hindmarsh-Rose neurons

“…Some models for bursting behaviors, repetitive and patterned activities of molluscan neurones can be mathematically represented as the Hindmarsh-Rose (HR) model [1,5,6,7,27]. Synchronization of coupled HR neurons has played a key role in the neuronal information processing and communication within the brain area [2,7,21,26,27].…”