Synchronization of Coupled Different Chaotic FitzHugh-Nagumo Neurons with Unknown Parameters under Communication-Direction-Dependent Coupling

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.


Introduction
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 [1,2]. The many published reports of fascinating outcomes have, since the beginnings, attracted and inspired many additional researchers to find effectual ways of improving external therapies for patients suffering from cognitive diseases [3][4][5][6]. The famous FitzHugh-Nagumo (FHN) neuronal model has been given extensive consideration for its utility in symbolizing the dynamical behavior of neurons and complex neuronal networks under EES [7].
The subject of FHN-neuronal synchronization as a potential application in cognitive engineering has been intensively examined in the literature [8][9][10][11][12][13][14][15][16][17][18][19][20]. Integration of the gap junction strength in FHN neurons renders the synchronization dilemma nontrivial. And the synchronization problem becomes still more complex, once the delay terms owing to distant communication are entertained in the coupled models [21][22][23][24]. To synchronize various chaotic FHN systems, researchers have utilized different control strategies including backstepping [25], active control [26], nonlinear control [8,9,27], and adaptive control [20,28,29]. Synchronization of two identical FHN neurons of known or unknown parameters, by means of nonlinear adaptive control schemes based on fuzzy logic, neural networks, uncertainty estimator, and feedback linearization, has been investigated [8,9,12,13,20]. Recently, robust adaptive control schemes for synchronization of two or three FHN neurons of unknown model parameters have been developed as well. Synchronization of two identical neurons of unknown parameters, under uncertain stimulation currents caused by medium losses and phase shifts, has been explored by application of an adaptive control scheme [14]. Another recent work has combined the ideas of parametric adaptation and 2 gain reduction for synchronization of multiple but slightly different neurons with respect to multiple communication pathways and unknown parameters and disturbances [15]. A simple methodology for synchronization of two different coupled neurons of known parameters by application of a reference-signal-based control approach has been evaluated as well [20].
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 [30][31][32]. The effects of unidirectional gap junctions in the presence of time-delay (due to neuronal separation) cannot be ignored. All in all, the development of control strategies for synchronization of two coupled delayed chaotic neurons of different and unknown parameters, particularly under disturbances, is very challenging. 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 [33], the proposed control scheme, unlike the traditional synchronization approaches, ensures partial synchronization of neurons in terms of their activation potentials (or membrane voltages). By utilizing the ideas of the standard Lyapunov theorem [13,34], the proposed adaptive control scheme is modified to ensure uniformly ultimately bounded synchronization and parametric estimation errors for robust synchronization of neurons against disturbances. The results of the proposed robust adaptive control scheme for chaos synchronization of FHN neurons of different and unknown parameters are verified through numerical simulation. The main contributions of this paper can be summarized as follows.
(i) A model of coupled FHN neurons under both unidirectional and bidirectional gap junctions is investigated.
(ii) The complex behavior of two different coupled neurons in a medium containing gap junctions is studied through bifurcation analysis and Lyapunovexponential investigation.
(iii) 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.
(v) 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 2 presents the model of two coupled chaotic FHN neurons for different and unknown parameters. Section 3 analyzes the behavior of the coupled FHN neurons. Section 4 presents the design of nonlinear adaptive and robust adaptive control schemes for synchronization of coupled chaotic FHN neurons of different and unknown parameters. Section 5 provides the relevant numerical simulation results. Section 6 draws conclusions. Standard notation is used throughout the paper. The notation ‖ ⋅ ‖ symbolizes the Euclidian norm of a vector.

Model Description
Consider two coupled chaotic delayed FHN neurons (see also [7,14]) of different and unknown parameters under uncertain EES, given by

Computational and Mathematical Methods in Medicine
where 1 and 1 are the states of the master FHN neuron in terms of the activation potential and the recovery variable, respectively, and 2 and 2 are the corresponding states of the slave FHN neuron. The FHN model parameters ( 1 , 2 ) and ( 1 , 2 ) are linked with the neurons' nonlinear part and recovery variable, respectively. The parameters 1 and 2 denote the amplitude of the external stimulation current for the master and the slave neurons, respectively, while 1 and 2 represent their phase shifts. Time and the angular frequency of the stimulation current are indicated by and = 2 , respectively, where denotes frequency. The strength of the gap junctions for communication from the master neuron to the slave neuron is represented by 1 . Correspondingly, 2 represents the strength of the gap junctions for transmission of an electrochemical signal from the slave neuron to the master neuron. The time-delay between the master and slave neurons is represented by . Disturbances at the master and slave neurons are denoted by 1 and 2 .
In the present work, all of the physical quantities of FHN models (1) are assumed to be dimensionless. In modeling most of biological processes, we often know only the nominal parametric values, not the true ones, owing to the biological restrictions. The amplitudes and phases associated with the stimulation current are taken to be different due to the medium losses and different path lengths occurring during the current flow from an electrode to both of the coupled neurons. The strengths of the gap junctions differ owing to the fact that some of the communication channels are unidirectional while others are bidirectional.

Remark 1.
It should be noted that all of the FHN model parameters associated with the master and slave neurons in (1) are fairly different and unknown owing to the physical limitations and the biological restrictions. Furthermore, usually, the majority of gap junctions allow bidirectional communication between two neurons. However, some permit only unidirectional transmission of a signal [30][31][32], responsible for different strengths of gap junctions. To capture this property, the strengths of gap junctions are taken as 1 and 2 . This direction-dependent selection of gap junction strength, in contrast to the schemes available in the literature [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], enables a more realistic model and, as such, is a superior synchronization-study tool for coupled FHN neurons.
In the next section, we examine the behavior of coupled FHN neurons (1) in exploring the effects of neuronalparameter difference, gap junction strength variation, and time-delay deviation on synchronization.

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 [35,36]. The bifurcation diagrams show the qualitative change in the dynamical behavior of neurons by changing amplitude of stimulation current over a range of 0 < < 2, while the maximum Lyapunov exponent informs about how much and for which range of stimulation amplitude the neuronal behaviors are chaotic. The method of Lyapunov exponent analysis is specifically used to avoid the ambiguity that the complicated behavior is occurring either due to a strange or a chaotic attractor. Furthermore, the degree of synchronization of neurons can be quantified by utilizing bifurcation diagrams. To this end, we first select the model parameters  Figure 1(e) depicts a more interesting phenomenon, that is, synchronization of FHN neurons by means of a rare bifurcation diagram treatment of synchronization error = 1 − 2 . It is evident that the identical neurons possess synchronous behavior, except in the regions 0.05 < < 0.74 and 0.9 < < 1.1. This means that both of the neurons can be synchronized by selecting a proper stimulation amplitude, either in the region 0.74 < < 0.9 or > 1.1, without utilizing any control signal.
Next, the dynamics of different coupled FHN neurons are analyzed by changing the parameters of the first neuron to 1 = 10, 1 = 1, 1 = , and 1 = 0.1. The amplitudes of stimulation are taken to be different, and the difference is fixed to 2 − 1 = 0.04. Figure 2 plots the bifurcation diagrams and largest Lyapunov exponents for the system of different coupled FHN neurons. Similarly, to the identical neurons case, both neurons exhibit oscillatory behavior for most of the amplitude values, as shown in Figures 2(a) and 2(b). The first neuron shows chaotic behavior in the regions 0.03 < 1 < 0.37 and 0.69 < 1 < 0.95, as depicted in Figure 2(c), while the second neuron shows chaotic behavior in the regions 0.12 < 2 < 0.72 and 1.33 < 2 < 1.77, as indicated in Figure 2(d). The bifurcation diagram of synchronization error = 1 − 2 is shown in Figure 2(e). Surprisingly, neither of the neurons are at all synchronous for any stimulation amplitude within the entire region 0 < 1 < 2 (and 0.4 < 2 < 2.4), due to the different parameters as compared with the case of identical neurons. We can conclude that the two different FHN neurons can be nonsynchronous, owing to variations in model parameters and, oscillators with a small synchronization error, as depicted in Figure 5(a). As we decrease the value of 2 to 0.8 and, further, to 0.5, the synchronization error increases, as shown in Figures 5(b) and 5(c), respectively. At 2 = 0.01, the FHN neurons' degree of nonsynchronization is the worst, as apparent in Figure 5

Synchronization of FHN Neurons
The present work proposes a control strategy that uses a single control input for synchronization of coupled FHN neurons of different and unknown parameters. Thus, model (1) takes the form Assumption 2. The parameters ( 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 , 1 , and 2 ) of FHN neurons (2) are unknown constants. Now, we develop a new control methodology for synchronization of master-slave neurons (2) of different and unknown parameters. Traditionally, synchronization of neurons is addressed in order to minimize the differences between all of the corresponding states of the master and slave neurons. In the literature [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], synchronization techniques for FHN neurons ensure convergence of both synchronization errors, the difference between the activation potentials and the error between the recovery variables for the masterslave systems, either to zero or in a small compact set. Nevertheless, various experimental studies (e.g., [33]) have demonstrated identical behavior of two synchronous neurons for their activation potentials only. In reality, the recovery variable is introduced in the model for membrane responses of potassium activation and sodium inactivation. Two different FHN neurons, with identical firing in terms of membrane (or activation) potentials, might not necessarily have the same (or similar) patterns for this hypothetical recovery variable. This property can also be verified from FHN neurons (2). Suppose that control law is designed to achieve 1 = 2 = and (2) reveal thaṫ1 = 1 anḋ2 = 2 . This implies that the recovery variables, due to different parametric values of In order to construct a control law, the dynamics of synchronization error = 1 − 2 for FHN neurons (2) can be expressed as where The synchronization error dynamics in (3) contain the recovery variables (i.e., 1 and 2 ). These terms can be canceled through the control law ; however, this requires measurement (or estimation) of the recovery variables, which may not be possible in the case of neuronal synchronization.
To deal with this problem, we integrate the recovery-variable dynamics in (2) to obtain Computational and Mathematical Methods in Medicine 9 According to (4)-(5), the alternate synchronization error dynamics becomes where Φ ∈ 10 is a vector of unknown constant parameters and Γ( 1 , 2 ) ∈ 10 is a vector of a known bases function given by Φ = [ 1 2 1 2 1 (0) 2 (0) 1 2 ] (7) The proposed controller takes the form whereΦ is the estimate of vector Φ and is a scalar quantity. The selected adaptation law forΦ iṡ where and are scalars.

Remark 3.
It is notable that the control and adaptation laws, containing integral terms in Γ( 1 , 2 ), do not require measurement of recovery variables for the master-slave neurons, owing to the utilization of (5). Now, we provide a condition for synchronization of FHN neurons (2) by application of control and adaptation laws (7)-(10) as follows. Proof. Incorporating (9) into (6) leads to Constructing a Lyapunov function candidate as with > 0, > 0, the time-derivative of (12) is given bẏ Note that (Φ −Φ)Φ =Φ (Φ −Φ). Incorporating (11) into (13), we obtaiṅ Application of the adaptation law (10) under = 0 yieldṡ Thus, convergence of synchronization error to zero is ensured, which completes proof of statement (i) in Theorem 4. In practice, it has been observed that the steady state is achieved within a finite time by application of an adaptive control law such as in Theorem 4. In the steady state, the synchronization error and the neuronal states satisfẏ Using = 0 in (10),Φ = 0 is obtained in the steady state. This further implies thatΦ =Φ * is satisfied in the steady state, whereΦ * is a constant. Similarly to the steady-state analysis in [14], applying the steady-state conditions (i.e.,̇= 0, = 0, 1 = 2 , andΦ =Φ * ) to (11), we obtain which completes the proof of statement (ii) in Theorem 4.

Remark 5.
In contrast to the traditional synchronization methodologies [9][10][11][12][13][14][15][16][17][18][19][20], the proposed adaptive control strategy in Theorem 4 can be used for synchronization of two different FHN neurons with all parameters unknown. This feature is achieved by incorporation of the knowledge acquired in experimental studies (e.g., [33]) on synchronization of a single state (i.e., the membrane potential) of neurons (rather than both the membrane potential and the recovery variable).
We now provide conditions for robust adaptive synchronization of different FHN neurons of unknown parameters under disturbances. First, we make the following assumption.

Theorem 7. Consider the time-invariant FHN neurons
where is a scalar, will ensure uniformly ultimately bounded synchronization error and parameter estimation error Φ −Φ, if ( +1) > 0 is satisfied. Moreover, the estimation error can be kept small by selecting a large value of for a given value of , and the synchronization error can be minimized by enlarging ( + 1) for a given value of .
Remark 8. By application of Theorem 7, synchronization of different chaotic FHN systems of unknown parameters under disturbances can be achieved, in contrast to the traditionalistic synchronization tools, by ensuring uniformly ultimately bounded synchronization and parametric estimation errors. Further, the effect of disturbances can be minimized by selecting suitable control parameters , , and .
Although the simulation results provided herein represent a specific scenario of FHN neurons, the proposed methods in Theorems 4 and 7 are applicable to a general form of FHN neurons with different parameters. Further, robustness against bounded perturbations has been ensured in Theorem 7. The results of Theorems 4 and 7 may not be applicable to FHN systems with fast time-varying parameters. Nevertheless, studies on time-varying FHN systems can be carried out in the future. To conclude, the proposed robust adaptive control methodology can be used for synchronization of distinct FHN neurons of unknown model parameters subject to disturbances.

Conclusions
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 neuronalparameter 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.