Sigmoidal synaptic learning produces mutual stabilization in chaotic FitzHugh–Nagumo model

Abstract

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.

Chaotic Entanglement: Entropy and Geometry

“…We are aware that several studies, such as [18] and the aforementioned [16], have also examined interactions between chaotic systems that have led to periodic states, but to our knowledge none have reported results consistent with our formulation of chaotic entanglement. For instance, the procedure outlined in [18] describes synchronizing two coupled laser systems first into chaotic states and then into a quasi-periodic coupling, but this is all performed by carefully tuning the parameters to a desired set of values.…”

“…For instance, the procedure outlined in [18] describes synchronizing two coupled laser systems first into chaotic states and then into a quasi-periodic coupling, but this is all performed by carefully tuning the parameters to a desired set of values. Similarly, the more recent study discussed in [16] reports on driving two interacting, chaotic neuron models into mutual stabilization via an external signal that adjusts a parameter shared by the two neurons. Chaotic entanglement is distinguished from studies of this nature because it describes how two chaotic systems mutually stabilize one another onto cupolets, all while the system parameters remain fully in the chaotic regime.…”

“…Recent work has addressed the question of whether cupolets could be found in mathematical models of neurons, since one of the main exchange functions under consideration is an "integrate-and-fire" model taken from neuroscience research. In [16], it is shown that persistent mutual stabilization could be achieved between a pair of Fitzhugh Nagumo neurons (mathematical neuron models where each individual model is two-dimensional, but the combined system exhibits chaos). The mutual stabilization occurs when the exchange function exhibits properties related to "synaptic learning," where the dynamics of the synapse change as a function of the passage of a neural firing.…”

Analysis of Turing Instability of the Fitzhugh-Nagumo Model in Diffusive Network

“…Zheng et al [12,14] explored the Turing instability phenomenon in the FHN model and its relevance to short-term memory [15]. Parker et al [16] studied synaptic learning in the FHN model.…”

A new locally active memristive synapse-coupled neuron model

“…With the construction of a threshold memristor, Bao proposed a memristor synapse-coupled neuron network and discovered abundant firing phenomena, then achieved the complete exponential synchronization between two identical HR neurons [29]. To investigate the characteristics of the new proposed bistable threshold locally active memristor based neuron network, this new memristor is introduced to the modified FitzHugh-Nagumo(FHN) nervous system, which was proposed by [30], as a coupled synapse.…”