Cupolets in a chaotic neuron model

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 by 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, [Formula: see text], 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 type 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 by a perturbation determined by the coding function [Formula: see text]. 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.

Cupolets: History, Theory, and Applications

“…The dynamics of the HR system are quite similar to those of biological neurons, and typically the x variable goes through a sequence of consecutive spikes (action potentials) before entering the refractory period. The system will produce chaotic dynamics when r = 0.006 and I = 3.25, and that is the regime studied in [93,94] and the three-dimensional attractor of the HR neuron appears in Figure 15. The bursting behavior, particularly of the x variable time series, is illustrated in Figure 16.…”

“…Since the HR neuron model can behave chaotically, Parker and Short chose it for the first study of cupolets outside of the standard chaotic attractors found in the chaos/nonlinear dynamics literature. In this investigation, they found that cupolets exist in the HR neuron model [93], and further, there is evidence that mutual stabilization (e.g., chaotic entanglement) is possible within the HR model [94]. This recent development is potentially quite important since communication between biological neurons is via impulsive action potentials, and it is natural to equate the impulsive kicks applied along control planes in cupolet research with idealizations of the action potentials passed over the synapses of neurons.…”

“…Figure 15. Numerical integration of Equation (5) with r = 0.006 and I = 3.25 so that the model is in a chaotic region[93]. (A) 3-Dimensional phase space plot of the x, y, and z dynamics.…”

Mutual Stabilization in Chaotic Hindmarsh–Rose Neurons

“…The neuron model chosen was the Hindmarsh-Rose (HR) model [8], since it is a three-dimensional system and has parameter regimes where the behavior is fully chaotic. The first paper on the HR study [9] looked at whether a particular control scheme could be used to stabilize the HR neuron onto a set of its unstable periodic orbits, which are generally dense within a chaotic system, but are all unstable so are not readily accessible. The control approach closely followed the work in Morena and Short [10,11], which was an extension of the initial work in Parker [12], Hayes et al [13,14].…”

“…As a first step toward determining if stabilization of unstable periodic orbits was possible in a neuron model, the work in Parker and Short [9] examined the HR neuron and set up a similar control structure as used for the other chaotic systems listed in the previous paragraph. In the HR case, the control planes were positioned so that one control plane intersected the region where the HR model spikes and the other where the HR model moves into a refractory period after spiking.…”