Cupolets in a Chaotic Neuron Model

TL;DR

This paper discovers and stabilizes cupolets, a type of unstable periodic orbit, in a chaotic neural model, demonstrating control methods and exploring implications for biological neurons.

Contribution

First identification of cupolets in a chaotic neuron model, with a novel control scheme to stabilize these orbits and insights into neural dynamics.

Findings

Multiple cupolets identified with specific control sequences

Control scheme successfully stabilizes unstable orbits

Differences observed compared to double scroll system cupolets

Abstract

This paper reports the first finding of cupolets in a chaotic Hindmarsh-Rose neural model. Cupolets (chaotic, unstable, periodic, orbit-lets) are unstable periodic orbits that have been stabilized through a particular control scheme applying a binary control sequence. We demonstrate different neural dynamics (periodic or chaotic) of the Hindmarsh-Rose model through a bifurcation diagram where the external input current, $I$, is the bifurcation parameter. We select a region in the chaotic parameter space and provide the results of numerical simulations. In this chosen parameter space, a control scheme is applied when the trajectory intersects with either of the two control planes. The size of the control is determined by a bit in a binary control sequence. The control is either a small microcontrol (0) or a large macrocontrol (1) that adjusts the future dynamics of the trajectory . We report the discovery of many cupolets with corresponding control sequences and comment on the differences with previously reported cupolets in the double scroll system. We provide some examples of the generated cupolets and conclude by discussing potential implications for biological neurons.