This paper investigates the interaction between two coupled neurons at the terminal end of a long chain of neurons. Specifically, we examine a bidirectional, two-cell FitzHugh–Nagumo neural model capable of exhibiting chaotic dynamics. Analysis of this model shows how mutual stabilization of the chaotic dynamics can occur through sigmoidal synaptic learning. Initially, this paper begins with a bifurcation analysis of an adapted version of a previously studied FitzHugh–Nagumo model that indicates regions of periodic and chaotic behaviors. Through allowing the synaptic properties to change dynamically via neural learning, it is shown how the system can evolve from chaotic to stable periodic behavior. The driving factor between this transition is representative of a stimulus coming down a long neural pathway. The result that two chaotic neurons can mutually stabilize via a synaptic learning implies that this may be a mechanism whereby neurons can transition from a disordered, chaotic state to a stable, ordered periodic state that persists. This approach shows that even at the simplest level of two terminal neurons, chaotic behavior can become stable, sustained periodic behavior. This is achieved without the need for a large network of neurons.

Recent work has highlighted the vast array of dynamics possible within both neuronal networks and individual neural models. In this work, we demonstrate the capability of interacting chaotic Hindmarsh–Rose neurons to communicate and transition into periodic dynamics through specific interactions which we call mutual stabilization, despite individual units existing in chaotic parameter regimes. Mutual stabilization has been seen before in other chaotic systems but has yet to be reported in interacting neural models. The process of chaotic stabilization is similar to related previous work, where a control scheme which provides small perturbations on carefully chosen Poincaré surfaces that act as control planes stabilized a chaotic trajectory onto a cupolet. For mutual stabilization to occur, the symbolic dynamics of a cupolet are passed through an interaction function such that the output acts as a control on a second chaotic system. If chosen correctly, the second system stabilizes onto another cupolet. This process can send feedback to the first system, replacing the original control, so that in some cases the two systems are locked into persistent periodic behavior as long as the interaction continues. Here, we demonstrate how this process works in a two-cell network and then extend the results to four cells with potential generalizations to larger networks. We conclude that stabilization of different states may be linked to a type of information storage or memory.

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.