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This paper investigates the chaotic behavior and synchronization of two different coupled chaotic FitzHugh-Nagumo (FHN) neurons with unknown parameters under external electrical stimulation (EES). The coupled FHN neurons of different parameters admit unidirectional and bidirectional gap junctions in the medium between them. Dynamical properties, such as the increase in synchronization error as a consequence of the deviation of neuronal parameters for unlike neurons, the effect of difference in coupling strengths caused by the unidirectional gap junctions, and the impact of large time-delay due to separation of neurons, are studied in exploring the behavior of the coupled system. A novel integral-based nonlinear adaptive control scheme, to cope with the infeasibility of the recovery variable, for synchronization of two coupled delayed chaotic FHN neurons of different and unknown parameters under uncertain EES is derived. Further, to guarantee robust synchronization of different neurons against disturbances, the proposed control methodology is modified to achieve the uniformly ultimately bounded synchronization. The parametric estimation errors can be reduced by selecting suitable control parameters. The effectiveness of the proposed control scheme is illustrated via numerical simulations.

In recent decades, behavior investigation of chaotic neurons including synchronization, particularly under external electrical stimulation (EES; e.g., deep brain stimulation), has become an important area of research in the study of clinical treatment mechanisms for neurodegenerative disorders [

The subject of FHN-neuronal synchronization as a potential application in cognitive engineering has been intensively examined in the literature [

The conventional techniques for synchronization of FHN neurons are based on either designed control laws for identical neurons of known or unknown parameters or developed control strategies for different neurons of known parameters. However, two coupled neurons cannot be completely identical, and the model parameters cannot be totally known, due to biological restrictions. Furthermore, whereas the traditional techniques assume bidirectional gap junctions for the interneuronal medium, they can in fact be unidirectional, resulting in different coupling strengths for each neuron [

This paper analyzes the behavior and synchronization of two different coupled distant FHN neurons under unidirectional gap junctions. The strengths of the gap junctions are assumed to be different for each neuron, owing to the presence of both unidirectional and bidirectional gap junctions in the interneuronal medium. Various dynamical aspects of coupled FHN neurons, such as the effects of parametric differences, time-delays, and unidirectional gap junctions on neuronal synchronization, are investigated. The design of robust adaptive control laws for synchronization of coupled chaotic distant FHN neurons under unidirectional gap junctions is also addressed. The resultant control approach represents a novel means for synchronizing different FHN neurons of unknown parameters subject to uncertain stimulation. By utilizing integral-based control and adaptation laws to deal with the unavailable neuronal state (i.e., the recovery variable), a new adaptive control scheme is developed for synchronization of different coupled chaotic FHN neurons of unknown parameters. Motivated by experimental results [

A model of coupled FHN neurons under both unidirectional and bidirectional gap junctions is investigated.

The complex behavior of two different coupled neurons in a medium containing gap junctions is studied through bifurcation analysis and Lyapunov-exponential investigation.

The idea that, by increasing the time-delay or the difference between the gap junction strengths for two neurons, the synchronization error can increase, which, further, can lead to nonsynchronous neuronal behavior, is explored.

Based on the experimental results, a biologically understandable synchronization tool ensuring convergence of the activation potential error to zero is offered, in contrast to the conventional approaches that consider unnecessary synchronization of the recovery variable [

The proposed synchronization control methodology fills the research gap on robust adaptive synchronization of FHN neurons of different and unknown parameters subject to disturbances.

The rest of this paper is organized as follows. Section

Consider two coupled chaotic delayed FHN neurons (see also [

In the present work, all of the physical quantities of FHN models (

It should be noted that all of the FHN model parameters associated with the master and slave neurons in (

In the next section, we examine the behavior of coupled FHN neurons (

Whereas, traditionally, studies have detailed the dynamical behavior of single FHN neurons, focusing on that, the dynamics of a coupled system of neurons is more significant to understanding the neuronal synchronization. Bifurcation analysis and studies on the largest Lyapunov exponent have been productive for biomedical systems such as magnetic resonance imaging of myocardial perfusion and snore classification [

Behavior of identical FHN neurons under EES: (a) bifurcation diagram of the first neuron; (b) bifurcation diagram of the second neuron; (c) largest Lyapunov exponent for the first neuron; (d) largest Lyapunov exponent for the second neuron; (e) bifurcation diagram of the synchronization error between the coupled neurons.

Next, the dynamics of different coupled FHN neurons are analyzed by changing the parameters of the first neuron to

Behavior of different FHN neurons under EES: (a) bifurcation diagram of the first neuron; (b) bifurcation diagram of the second neuron; (c) largest Lyapunov exponent for the first neuron; (d) largest Lyapunov exponent for the second neuron; (e) bifurcation diagram of the synchronization error between the coupled neurons.

Nonsynchronous behavior of the two different FHN neurons under EES: (a) phase portrait of the first neuron; (b) phase portrait of the second neuron; (c) phase portrait of the activation potentials for nonsynchronous behavior.

We now examine the effects of the strengths of gap junctions and of time-delays between two identical neurons. The model parameters are selected as

Effects of time-delay due to separation between the two neurons under different gap junction strengths: (a) bifurcation diagram of the synchronization error for

Effects of the unidirectional gap junctions in a medium between the two neurons: (a) bifurcation diagram of the synchronization error for

The present work proposes a control strategy that uses a single control input

The parameters (

Now, we develop a new control methodology for synchronization of master-slave neurons (

Here, we address a partial synchronization of two distinct FHN neurons according to their activation potentials, as supported by experimental results and theoretical reasoning. In order to construct a control law, the dynamics of synchronization error

The proposed controller takes the form

It is notable that the control and adaptation laws, containing integral terms in

Now, we provide a condition for synchronization of FHN neurons (

Consider the time-invariant FHN neural oscillators (

synchronization of the coupled FHN neurons under different and unknown parameters by guaranteeing the convergence of synchronization error

convergence of

Incorporating (

Using

In contrast to the traditional synchronization methodologies [

We now provide conditions for robust adaptive synchronization of different FHN neurons of unknown parameters under disturbances. First, we make the following assumption.

Assume that

Consider the time-invariant FHN neurons (

Consider Lyapunov function (

Using (

By application of Theorem

To demonstrate the effectiveness of the proposed methodology, we set the model parameters for FHN neurons (

Synchronization of the two different coupled chaotic FHN neurons under EES by application of the proposed control scheme. The controller was applied for the time

Although the simulation results provided herein represent a specific scenario of FHN neurons, the proposed methods in Theorems

This paper addressed the synchronization of two coupled chaotic FHN neurons for different and unknown parameters under uncertain external stimulation and disturbances. The dynamics of coupled FHN neurons of different parameters were studied in a medium containing both unidirectional and bidirectional gap junctions. The effects of the neuronal-parameter difference, the gap junction strength variation, and time-delay deviation on the synchronization error were investigated. Nonlinear adaptive and robust adaptive control strategies were developed to cope with synchronization of the FHN neurons under the circumstances of different and unknown parameters, the infeasibility of recovery-variable measurement, uncertainty of stimulation current, and disturbances. The proposed control scheme was successfully applied to the synchronization of coupled chaotic FHN neurons, the numerical simulation results of which were provided.

All the authors declare that there is no conflict of interests regarding publication of this paper.

This work was supported by Higher Education Commission (HEC) of Pakistan under Grant Agreement 074-2194-Eg4-036 and by the Ministry of Education, Science and Technology through the National Research Foundation of Korea (Grant no. MEST-2012-R1A2A2A01046411).