Coexistence of firing patterns and its control in two neurons coupled through an asymmetric electrical synapse

Abstract

In this paper, the effects of asymmetry in an electrical synaptic connection between two neuronal oscillators with a small discrepancy are studied in a 2D Hindmarsh–Rose model. We have found that the introduced model possesses a unique unstable equilibrium point. We equally demonstrate that the asymmetric electrical couplings as well as external stimulus induce the coexistence of bifurcations and multiple firing patterns in the coupled neural oscillators. The coexistence of at least two firing patterns including chaotic and periodic ones for some discrete values of coupling strengths and external stimulus is demonstrated using time series, phase portraits, bifurcation diagrams, maximum Lyapunov exponent graphs, and basins of attraction. The PSpice results with an analog electronic circuit are in good agreement with the results of theoretical analyses. Of most/particular interest, multistability observed in the coupled neuronal model is further controlled based on the linear augmentation scheme. Numerical results show the effectiveness of the control strategy through annihilation of the periodic coexisting firing pattern. For higher values of the coupling strength, only a chaotic firing pattern survives. To the best of the authors' knowledge, the results of this work represent the first report on the phenomenon of coexistence of multiple firing patterns and its control ever present in a 2D Hindmarsh–Rose model connected to another one through an asymmetric electrical coupling and, thus, deserves dissemination.

Control of Coexisting Attractors with Preselection of the Survived Attractor in Multistable Chua’s System: A Case Study

“…e same result was found when the same research team was addressing the control of multistability (involving three disconnected attractors) in simplified canonical Chua's oscillator with smooth hyperbolic sine nonlinearity using the linear augmentation scheme [3,44]. Much recently, Tabekoueng Njitacke et al [50] investigated the coexistence of firing patterns and their control in two neurons coupled through an asymmetric electrical synapse. eir numerical results show the effectiveness of the control strategy through annihilation of the periodic coexisting firing pattern.…”

“…e choice of paradigmatic Chua's oscillator system within this work is based on the fact that it possesses three equilibrium points and thus opens the possibility to target or select a specific coexisting attractor located around unstable equilibria. Remark that the previous studies which focused on the multistability control were done on the system having three, four, and five coexisting attractors and only one equilibrium point [2,3,44,50]. Based on linear augmentation, the unique equilibrium point excludes the possibility to target a coexisting attractor.…”

“…Since the coexistence of attractors has been commonly used in image processing [16,38], it becomes very urgent to control this phenomenon when, sometimes, periodic and chaotic orbits exist simultaneously. Up to date, the prominent methods reported in the relevant literature which enable to turn a multistable system to a monostable system are the noise selection [39], pseudo-forcing [12], short pulses [40], harmonic perturbation [41], intermittent feedback [42], temporal feedback [43], and linear augmentation [2,3,[44][45][46][47][48][49][50]. Except for the temporal feedback and linear augmentation methods, in almost all other existing methods, the control is applied to one parameter of the system to remove on the attractors for all initial points.…”

“…From these results, it can be seen that intensive works have been done on the multistability control of the nonlinear oscillators based on the linear augmentation method. Remark that all these successful results on the control of multistability were performed so far only on systems with unique equilibrium point [2,3,44,50]. is unique equilibrium point excludes the possibility to target/select a desired attractor during the multistability control process.…”

Effects of Symmetric and Asymmetric Nonlinearity on the Dynamics of a Third-Order Autonomous Duffing–Holmes Oscillator

“…e study of nonlinear dynamic systems has been attracting researchers from various fields of science. anks to analytical, numerical, and experimental tools, striking nonlinear phenomena involved for instance in the evolution of stock market [1], neural networks [2][3][4][5][6], and population dynamics [7] have been uncovered. However, systems exhibiting chaotic dynamics are of particular interest owing to their potential applications in various fields of science and engineering such as image encryption [8], random bits generation [9], and secure communications [10].…”