# Synchronization Analysis of Multiple FitzHugh-Nagumo Noisy and Nonnoisy Neurobiological Networks

Authors: Malik Muhammad Ibrahim; Shazia Iram; Muhammad Ahmad Kamran; Malik Muhammad Naeem Mannan; Muhammad Umair Ali; Il Hyo Jung; Semin Oh; Sangil Kim
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In the biological neural bursting and firing synchronization plays a vital role in all neuronal activities that are utilized for making decisions, executing commands, and sending information by neurons and their complex networks in the biological complexed brain. Understanding how the biological brain functionality comes out from different patterns of neuronal transmission between the large group of neural networks stands as one of the enduring challenges of modern neuroscience. This study investigated a methodology for synchronization of multiple single/dual state gap junctions FitzHugh-Nagumo (FHN) drive and slave networks under the condition of external noise. The theory of control was utilized to propose simple and diverse controllers to examine the synchronization problem of the different single and dual state gap junctions coupled nonnoisy and noisy FHN neurobiological drive and slave networks. Control laws are designed to stabilize the error dynamics without direct cancelation and synchronize all the states of both FHN neurobiological drive and slave networks. Sufficient conditions for achieving synchronization in the multiple single/dual state gap junction FHN noisy and nonnoisy neurobiological drive and slave networks were derived analytically using the theory of Lyapunov stability. Furthermore, the proposed controllers have been verified by using five noisy/nonnoisy FHN neurobiological drive and slave networks through numerical simulations.


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## Body

## 1. Introduction

In the biological neural firing and bursting synchronization plays an essential role in all processing activities that are utilized for making decisions, executing commands, and sending information by neurons and their complex networks in the biological complexed brain [1–3]. Rapid constancy of synchronization may lead to an increase in the performance and functionality of neural brain processing or may decrease neuronal disorders such as epilepsy, schizophrenia, autism, and Parkinson’s disease, and sustainable treatment of these disorders is one of the major challenges of the modern neuroscience [1, 2, 4–6]. In past decades, synchronization of the nonlinear and control system has become a fascinating and challenging task for researchers due to the developing chaotic system applications in many scientific fields including image processing, neuroscience, biology, secure communication, physics, ecology, chemical reactions, parameters estimation, electrical circuits and systems, chemistry, mechanic, and control processing [7–11]. After the intellectual study of Pecora and Carroll on the chaotic system synchronization [12], the chaotic synchronization phenomenon has fascinated, attracted, and motivated researchers to investigate and explore the functionality of the biological complex brain to understand the synchronized neural firing [1].Different neuronal models have been broadly investigated to understand the phenomenon of synchronization of coupled neurons by many researchers [13–17]. In the past few years, neural networks and the study of their dynamical properties including synchronization, chaos, and bifurcation have gained tremendous interest and are greatly explored for their possible applications in many research fields [18–22]. Among these, the synchronization of neurons is contributory to the insight into the functionality of neurobiological networks and the core processing of information transfer/handling in the brain [23, 24]. Therefore, neural synchronization emerges as an interesting and at the same time essential research topic in the field of neuroscience and its applications.In the past few decades, there are several different mathematical models that have been developed/proposed to understand the dynamical features and functionality of single-neuron [1, 2]. In 1952, Alan Hodgkin and Andrew Huxley proposed the 1st quantitative model consisting of four coupled differential equations for estimation of the electrical information processing of the neurons [25]. Since then, several simplifications and modifications to reduce the complexity of their model have been offered. The two-dimensional mathematical model of FitzHugh-Nagumo (FHN) neuron developed by FitzHugh [26] and Nagumo et al. [27] is one of the best simplifications of the HH model to date. The FHN neuron model has been extensively utilized as a primary tool to investigate and explore the synchronization phenomenon in neurons [28–32]. In the past, researchers have incorporated gap junctions and external electrical stimulation (EES) in the dynamics of coupled FHN system to make it more realistic. For instance, gap junctions are introduced to compensate for the dynamics of protein channels by which neurons communicate with each other. These gap junctions play a crucial role in information processing and transmission between neurons and therefore have been the focal point in synchronization research [33, 34].In the last decade, many researchers have proposed/developed synchronization strategies for coupled FHN systems under the effect of various complexities including EES and gap junctions. For instance, Iqbal et al. proposed the control laws for the synchronization of uncertain and perturbed neurons [16]. Guadalupe Cascallares et al. investigated the capability of electrical synapses to transmit the noise-sustained network activity from one network to another [35]. Zhang investigated the synchronization of coupled FHN neurons with parameter disturbance [30]. Yu et al. studied the synchronization of two coupled FHN neurons with gap junction via sliding and backstepping control methodology [29]. Che et al. used the fuzzy approximator to estimate the nonlinear uncertain function of the synchronization error of the FHN system [36]. Jiang et al. studied the chaotic synchronization behavior of a coupled neural system of FHN neurons via a nonlinear control methodology [13]. Nugyen et al. explored the self-synchronization of coupled FHN systems by introducing Lyapunov-stability-based criteria [37]. Aqil et al. studied the synchronization issue in two coupled FHN neurons with gap junctions [15]. Rehan et al. [38] studied the synchronization problem of two FHN neurons with unknown parameters and designed a robust adaptive control strategy to guarantee synchronization; next, they addressed the synchronization problem of three FHN systems under EES and gap junctions [39]. Yang et al. studied the synchronization of space-clamped FHN neurons and developed an adaptive robust sliding-mode controller [40]. Ambrosio et al. [41] discussed the synchronization problem of coupled reaction-diffusion FHN systems, while a combined backstepping and sliding-mode control technique was proposed by Yu et al. to analyze the synchronization phenomenon in coupled FHN neurons [29]. Wang et al. explored the synchronization of two uncoupled FHN neurons under EES and proposed a nonlinear control scheme to tackle it [14].Previous experimental studies have discovered that the dynamical properties of a neural network are significantly influenced by coupling between neurons [20]. Accordingly, a neurobiological network composed of FHN neurons will display more interesting dynamical behavior and more complex dynamical characteristics than two or three coupled FHN systems [30, 42]. Therefore, it is a particularly challenging but important research direction to explore and address the synchronization issue of a neurobiological network composed of FHN neurons under EES with gap junctions for both practical and theoretical applications. The study of Ibrahim and Jung has investigated the single and dual state gap junction phenomenon in the network of multiple FHN neurons via feedback control [1]. They proposed different control schemes and derived sufficient conditions which guarantee the synchronization under different conditions. In past research, researchers have investigated the interaction between neurons and their networks in different brain regions using techniques like neuroimaging systems [43–47], and studies also revealed that neurons in different brain regions synchronize their activities during information processing and transmitting between inter- and intraneural networks [29, 48]. However, understanding how the biological brain functionality comes out from different patterns of neuronal transmission between the large group of neural networks stands as one of the enduring challenges of modern neuroscience. In numerous complex systems, the complex biological brain exhibits a very wide range of dynamic activity and connectivity patterns that are thought to be instrumental in enabling the integration and processing of information in the course of behavior and cognition [49]. Although much work has addressed the complex organization of the structural and functional networks of the brain [49–52], how this complexity supports synchronized communication processes that are fundamental to the brain’s computational capacities remains poorly understood. Indeed, investigation of synchronization properties of two FHN neurobiological networks is extremely rare in past research. To the best of our knowledge, only two studies investigated the synchronization phenomenon in two FHN neuronal networks. For instance, Zhou et al. proposed an adaptive feedback controller via a pinning mechanism to extend the concept of the synchronization of two FHN neurons to two FHN networks [42]. In another study, Zhang studied the synchronizing of two FHN neuronal networks with parameter disturbance by using sliding-mode control [30]. However, both these studies only considered networks composed of nonnoisy FHN neurons. In the last decade, research on the effects of noise on the dynamical properties of the nonlinear systems was the focus of many researchers that have discovered phenomena such as noise-sustained synchronization [53], coherent resonance [54], and stochastic resonance [55]. Research on neuron signal processing has revealed that neurons adjust their firing behavior and dynamics to transmit information and communicate optimally in the presence of noise [56]. Therefore, FHN neurons will depict more realistic neuronal behavior by the addition of noise but at the same time will make it more difficult and complex to explore the synchronization in neurobiological networks. Furthermore, these studies considered interneuronal couplings only through the membrane potential state of the FHN neurons whereas a recent investigation suggested the inclusion of interneuronal couplings in both states of FHN system foreshadows even more complex and real dynamical properties. These dynamical properties can facilitate understanding phenomena such as elliptic seizures since it is communications between different brain region neurons which give rise to these phenomena. Therefore, exploration of such dual state interneuronal couplings in FHN neurobiological network is an essential step in advancing our understanding of several complex phenomena present in brain communication [1, 57]. Therefore, investigation of the synchronization phenomena, in two coupled FHN neurobiological networks under EES in the presence and absence of external noise, with single- and dual state gap junctions, is extremely challenging, in terms of both practical and theoretical applications.In the light of the above, this study proposed a synchronized control methodology for multiple coupled drive and slave FHN neurobiological networks under the conditions of single/dual state gap junctions and external noise, as shown in Figure1. To our knowledge, this is the first study to explore and investigate the synchronization phenomenon for drive and slave FHN neurobiological networks under the condition of external noise. Control theory was utilized to design novel and diverse control strategies to examine the synchronization problem of the different single and dual state gap junctions coupled nonnoisy and noisy FHN neurobiological networks. Control laws are designed to stabilize the error dynamics without direct cancelation and synchronize all the states of both drive and slave FHN neurobiological networks. Sufficient conditions for achieving synchronization in the single/dual state gap junction FHN neurobiological noisy and nonnoisy networks were derived analytically using the theory of Lyapunov stability. Furthermore, the proposed controllers have been verified by using five noisy/nonnoisy FHN drive and slave neurobiological networks through numerical simulations.Figure 1
Schematic diagram of coupled neurobiological drive and slave FHN networks.The most important contributions of this study consist of (i) synchronization of single state gap junction FHN neurobiological drive and slave networks without noise; (ii) synchronization of single state gap junction FHN neurobiological drive and slave networks with noise; (iii) synchronization of dual state gap junction FHN nonnoisy neurobiological drive and slave networks; (iv) synchronization of dual state gap junctions FHN neurobiological drive and slave networks with noise; (v) the development of novel and diverse control laws, to synchronize both single/dual state gap junction FHN noisy and nonnoisy neurobiological drive and slave networks; (vi) synchronization of action potential and recovery variable states of single/dual state gap junctions FHN noisy and nonnoisy neurobiological drive and slave networks under the conditions of proposed control schemes.The remainder of the article is organized as follows. The mathematical model of a single FHN neuron is briefly described in Section2. FHN neurobiological drive and slave networks formulation, synchronization problem design, and the proposed control methodology are presented in Section 3. Numerical simulations showing the efficacy of the designed control laws are discussed in Section 4 and the conclusion has been shown in the last section.


## 2. FitzHugh-Nagumo Neuronal Model

The coupled differential equations of single-neuron as FHN model under EES and external noise can be formulated as follows [1, 32]:(1)x˙11=x11x11−11−rx11−x2+I+d+ξ,x˙12=bx11−cx12,where x1,x2∈ℜ represent membrane potential and the recovery variable, respectively, r, b, and c are positive parameters, I=A/ωcosωt is the EES of frequency ω and amplitude, d is the ionic gate disturbance, and ξ is the external noise modeled as Gaussian noise. The dynamics of the state variables and their phase portrait for the parameter values [1], r=10, b=1, c=0.001, d=0.01sin0.2t, Α=0.1, and ω=2πf with initial conditions x110=0, x120=0, are shown in Figure 2.Figure 2
Dynamical response of an FHN neuron with ionic gate disturbance and Gaussian noise: (a) Temporal dynamics of the membrane potential state. (b) Recovery variable state. (c) Phase plan analysis.
(a)(b)(c)


## 3. Model Formulation for FHN Networks with Single and Dual State Gap Junctions and Control Design

This section discusses the formulation of the synchronization problem for two coupled nonnoisy and noisy FHN neurobiological networks under EES with gap junctions in either membrane potential state or in both membrane potential and recovery variable states. Consequently, four different neurobiological networks with nonnoisy and noisy FHN neurons are formulated and utilized for the exploration of synchronization phenomena in coupled neural networks. Each network formulated in each case consists ofn neurons.


### 3.1. Noisy and Nonnoisy FHN Drive and Slave Networks with Single State Gap Junction

Single state gap junction FHN drive and slave networks are modeled by formulating gap junction dynamics only in the membrane potential state of each neuron in the network.


#### 3.1.1. Networks without Noise

Mathematically, neurobiological FHN drive and slave neural networks under EES with single state gap junctions without external noise can be formulated as follows [42]:(2)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+uxi1,x˙i2=bxi1−cxi2,i=1,2,3,⋯,n,(3)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+uyi1,y˙i2=byi1−cyi2,where uxi1 and uyi1 are the control parameters. G=gijN×N represents the topology of the network and is known as coupling matrix whose elements are defined as below.(4)gij=gji≥0,ifi↔j,∀i≠j;gji=0,ifi↮j,∀i≠j;gii=∑j=1j≠1Ngij,∀i=j.


#### 3.1.2. Networks with Noise

Mathematically, noisy neurobiological FHN drive and slave neural networks under EES with single state gap junctions can be formulated as follows:(5)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+ξ+uxi1,x˙i2=bxi1−cxi2,i=1,2,3,⋯,n,(6)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+ξ+uyi1,y˙i2=byi1−cyi2,where ξi1 are zero mean Gaussian noise [32] having correlation function:(7)ξ1tξ1t′=2Dδt−t′.


#### 3.1.3. Control Law Design

The error dynamics of the coupled neurobiological drive and slave networks described by equations (2-3) and (5-6) can be defined as(8)exi1=xi1−yi1,eyi1=xi2−yi2.Time derivative of equation (8) yields the following equation: (9)e˙x11e˙x21⋮e˙xn1=−rx113−y113x213−y213⋮xn13−yn13+1+rx112−y112x212−y212⋮xn12−yn12−ex11ex21⋮exn1−eyn1eyn1⋮eyn1+∑j=1Ng1jxi1−yi1∑j=1Ng2jxi1−yi1⋮∑j=1Ngnjxi1−yi1+u11u21⋮un1,e˙y11e˙y21⋮e˙yn1=bex11ex21⋮exn1−ceyn1eyn1⋮eyn1,where ui1=uxi1−uyi1. Now let us define the following terms for simplification:(10)ex1=ex11ex21⋯exn1Tey1=ey11ey21⋯eyn1Tcd=x113−y113x213−y213⋯xn13−yn13Tsd=x112−y112x212−y212⋯xn12−yn12TG=giju=u11u21⋯un1T,where ex1,ey1,cd,sd,u∈ℜNx1,G∈ℜNxN and T represents the transpose operator. As a result (10) can be written as(11)e˙x1=−rcd+1+rsd−ex1−ey1+Gex1+u,e˙y1=bex1−cey1,It can be concluded that the problem of synchronization is substituted with the synchronized error system (11) with the requirement to stabilize by using a suitable control inputu. Synchronization can be achieved by designing the controller u such that for any initial conditions ex1 and ey1 converge to zero. This implies that the dynamics of the slave system (3/6) can converge to that of the master system (2/5).Next, Lyapunov stability and control theories are utilized to design a unique and simple control scheme that will guarantee the synchronization of the single state gap junction coupled neurobiological networks.Theorem 1.
Consider the FHN neurobiological drive and slave networks described in equations (2) and (3) and (5) and (6) with the error dynamics described by equation (8). If the control laws in the error system described by equation (10) are designed as u=Gex1+ey1−b−cex1+ey1 then the synchronization of single state coupled FHN neurobiological drive and slave networks can be achieved if the synchronized error systems converge to zero.proof.
The proof of this theorem and sufficient conditions that ensured the synchronization can be obtained using the theory of Lyapunov stability. According to the theory of Lyapunov stability, the stability of a system can be proved by choosing a positive definite function called Lyapunov function candidate V. In this study it is chosen as(12)V=12ex12+ey12,dv/dt<0 for all t.
It is easy to verify that functionV is a positive definite function. After taking the derivative of equation (12) with respect to time, we obtain(13)V˙=ex1Te˙x1+ey1Te˙y1.
Incorporating synchronized error system (10) and the control law into (12) yields(14)V˙=ex1T−rcd+1+rsd−ex1+2Gex1−b−cex1+ey1+ey1Tbex1−cey1,or(15)V˙=−rex1Tcd+1+rex1Tsd−ex1Tex1+2ex1TGex1−bex1Tex1+cex1Tex1−bex1Tey1,+cex1Tey1+bey1Tex1−cey1Tey1.
Now(16)ex1Tsd=ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1≤ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1,and it is known that neurons have bounded trajectories; i.e., system (2) and (3) and (5) and (6) are bounded with some positive constants qxi and qyi satisfying xi1≤qxi and yi1≤qyi, ∀i=1,2,3⋯,n. Additionally, assuming that q1=maxqxi+qyi;i=1,2,3⋯,n, (16) results into(17)ex1Tsd≤q1ex1Tex1.
Correspondingly,(18)ex1Tcd≤q2ex1Tex1.
Incorporating (17) and (18) into (19) gives(19)V˙=−ex1Trq2−1+rq1+I−2G+b−cex1+b−cex1TIey1−bey1Tex1+cey1TIey1,where I∈ℜN×N is the identity matrix,(20)V˙t≤−EPET,where E=ex1ey1, and P=rq2−1+rq1+I−2G+b−cb−cI−bIcI.
It can be concluded that the matrixP should be a positive definite to ensure the asymptotic stability of the synchronized error system (11) at the origin. The positive definiteness of P could be easily derived for considered networks of neurons (e.g., a network of five neurons) using the method of determinants; i.e., determinants of all leading principal minors have positive values (see [1]). Accordingly, for any value 0≤gij<0.1, the origin of the error system (11) is asymptotically stable. Consequently, these networks of n-identical, single state gap junction FHN noisy and nonnoisy neurobiological networks, under ionic gates disturbance, and EES, will achieve synchronization. This completes the proof.


### 3.2. Noisy and Nonnoisy FHN Drive and Slave Networks with Dual State Gap Junctions

Dual state gap junction FHN networks are modeled by formulating gap junction dynamics in both the membrane potential and recovery variable states of each neuron in the network.


#### 3.2.1. Networks without Noise

Mathematically, neurobiological FHN drive and slave neural networks under EES with dual state gap junctions without external noise can be formulated as follows:(21)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+uxi1,x˙i2=bxi1−cxi2+∑j=1ngijxj1,i=1,2,3,⋯,n,(22)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+uyi1,y˙i2=byi1−cyi2+∑j=1ngijxj1,where uxi1 and uyi1 are the control parameters.


#### 3.2.2. Networks with Noise

Mathematically, noisy neurobiological FHN drive and slave neural networks under EES with dual state gap junctions can be formulated as follows:(23)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+ξ+uxi1,x˙i2=bxi1−cxi2+∑j=1ngijxj1+ξi=1,2,3,⋯,n,(24)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+ξ+uyi1,y˙i2=byi1−cyi2+∑j=1ngijxj1+ξ.


#### 3.2.3. Control Law Design

The time derivative of equation (8) yields equation (25):(25)e˙x11e˙x21⋮e˙xn1=−rx113−y113x213−y213⋮xn13−yn13+1+rx112−y112x212−y212⋮xn12−yn12−ex11ex21⋮exn1−eyn1eyn1⋮eyn1+∑j=1Ng1jxi1−yi1∑j=1Ng2jxi1−yi1⋮∑j=1Ngnjxi1−yi1+u11u21⋮un1,e˙y11e˙y21⋮e˙yn1=bex11ex21⋮exn1−ceyn1eyn1⋮eyn1+∑j=1Ng1jxi1−yi1∑j=1Ng2jxi1−yi1⋮∑j=1Ngnjxi1−yi1.Now let us define the following terms for simplification:(26)ex1=ex11ex21⋯exn1Tey1=ey11ey21⋯eyn1Tcd=x113−y113x213−y213⋯xn13−yn13Tsd=x112−y112x212−y212⋯xn12−yn12TG=giju=u11u21⋯un1T,where ex1,ey1,cd,sd,u∈ℜNx1,G∈ℜNxN and T is the transpose operator. Thus (27) becomes(27)e˙x1=−rcd+1+rsd−ex1−ey1+Gex1+u,e˙y1=bex1−cey1+Gex1.It can be concluded that the problem of synchronization is substituted with the synchronized error system (27) with the requirement to stabilize by using a suitable control input u. Synchronization can be achieved by designing the controller u such that for any initial conditions ex1 and ey1 converge to zero. This implies that the dynamics of the slave system (22/24) can converge to that of the master system (21/23).Next, Lyapunov stability and control theories are utilized to design a unique and simple control scheme that will guarantee the synchronization of the dual state gap junction coupled neurobiological networks.Theorem 2.
Consider the FHN neurobiological drive and slave networks described in equations (21)–(24) with the error dynamics described by equation (8). If the control laws in the error system described by equation (27) are designed as u=Gex1−Gey1−b+cex1+ey1 then the synchronization of dual state coupled FHN neurobiological drive and slave networks can be achieved if the synchronized error systems converge to zero.proof.
The Lyapunov function candidateV for this proof is chosen as follows:(28)V=12ex12+ey12,dvdt<0,∀t.
It is easy to verify that the chosen candidate functionV is a positive definite function. After taking the derivative of equation (28) with respect to time, we obtain equation (29)V˙=ex1Te˙x1+ey1Te˙y1.
Incorporating the synchronized error system (26) and the control law into (29) yields(30)V˙=ex1T−rcd+1+rsd−ex1+2Gex1+Gey1−b+cex1+ey1+ey1Tbex1−cey1+Gex1,or(31)V˙=−rex1Tcd+1+rex1Tsd−ex1Tex1+2ex1TGex1+ex1TGey1−bex1Tex1−cex1Tex1−bex1Tey1−cex1Tey1+bey1Tex1−cey1Tey1+ey1TGex1.
Now(32)ex1Tsd=ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1≤ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1,and it is known that neurons have bounded trajectories; i.e., systems (21)–(24) are bounded with some positive constants qxi and qyi satisfying xi1≤qxi and yi1≤qyi, ∀i=1,2,3⋯,n. Additionally, assuming that q1=maxqxi+qyi;i=1,2,3⋯,n, (32) results into(33)ex1Tsd≤q1ex1Tex1.
Correspondingly,(34)ex1Tcd≤q2ex1Tex1.
Incorporating (33) and (34) into (31) gives(35)V˙=−ex1Trq2−1+rq1+I−2Gex1+b+cex1−ex1T−G+b+cey1−ey1T−b−Gex1+cey1TIey1,where I∈ℜN×N is the identity matrix,(36)V˙t≤−EPET,where E=ex1ey1, and P=rq2−1+rq1+I−2Gex1+b+c−−G+b+c−−b−GcI.
It can be concluded that the matrixP should be a positive definite to ensure the asymptotic stability of the synchronized error system (27) at the origin. The positive definiteness of P could be easily derived for considered networks of neurons (e.g., a network of five neurons) using the method of determinants; i.e., determinants of all leading principal minors have positive values (see [1]). Accordingly, the origin of the error system (27) is asymptotically stable. Consequently, these networks of n-identical, dual state gap junction coupled FHN noisy and nonnoisy neurobiological drive and slave networks, under ionic gates disturbance, and EES will achieve synchronization. This completes the proof.


## 3.1. Noisy and Nonnoisy FHN Drive and Slave Networks with Single State Gap Junction

Single state gap junction FHN drive and slave networks are modeled by formulating gap junction dynamics only in the membrane potential state of each neuron in the network.


### 3.1.1. Networks without Noise

Mathematically, neurobiological FHN drive and slave neural networks under EES with single state gap junctions without external noise can be formulated as follows [42]:(2)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+uxi1,x˙i2=bxi1−cxi2,i=1,2,3,⋯,n,(3)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+uyi1,y˙i2=byi1−cyi2,where uxi1 and uyi1 are the control parameters. G=gijN×N represents the topology of the network and is known as coupling matrix whose elements are defined as below.(4)gij=gji≥0,ifi↔j,∀i≠j;gji=0,ifi↮j,∀i≠j;gii=∑j=1j≠1Ngij,∀i=j.


### 3.1.2. Networks with Noise

Mathematically, noisy neurobiological FHN drive and slave neural networks under EES with single state gap junctions can be formulated as follows:(5)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+ξ+uxi1,x˙i2=bxi1−cxi2,i=1,2,3,⋯,n,(6)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+ξ+uyi1,y˙i2=byi1−cyi2,where ξi1 are zero mean Gaussian noise [32] having correlation function:(7)ξ1tξ1t′=2Dδt−t′.


### 3.1.3. Control Law Design

The error dynamics of the coupled neurobiological drive and slave networks described by equations (2-3) and (5-6) can be defined as(8)exi1=xi1−yi1,eyi1=xi2−yi2.Time derivative of equation (8) yields the following equation: (9)e˙x11e˙x21⋮e˙xn1=−rx113−y113x213−y213⋮xn13−yn13+1+rx112−y112x212−y212⋮xn12−yn12−ex11ex21⋮exn1−eyn1eyn1⋮eyn1+∑j=1Ng1jxi1−yi1∑j=1Ng2jxi1−yi1⋮∑j=1Ngnjxi1−yi1+u11u21⋮un1,e˙y11e˙y21⋮e˙yn1=bex11ex21⋮exn1−ceyn1eyn1⋮eyn1,where ui1=uxi1−uyi1. Now let us define the following terms for simplification:(10)ex1=ex11ex21⋯exn1Tey1=ey11ey21⋯eyn1Tcd=x113−y113x213−y213⋯xn13−yn13Tsd=x112−y112x212−y212⋯xn12−yn12TG=giju=u11u21⋯un1T,where ex1,ey1,cd,sd,u∈ℜNx1,G∈ℜNxN and T represents the transpose operator. As a result (10) can be written as(11)e˙x1=−rcd+1+rsd−ex1−ey1+Gex1+u,e˙y1=bex1−cey1,It can be concluded that the problem of synchronization is substituted with the synchronized error system (11) with the requirement to stabilize by using a suitable control inputu. Synchronization can be achieved by designing the controller u such that for any initial conditions ex1 and ey1 converge to zero. This implies that the dynamics of the slave system (3/6) can converge to that of the master system (2/5).Next, Lyapunov stability and control theories are utilized to design a unique and simple control scheme that will guarantee the synchronization of the single state gap junction coupled neurobiological networks.Theorem 1.
Consider the FHN neurobiological drive and slave networks described in equations (2) and (3) and (5) and (6) with the error dynamics described by equation (8). If the control laws in the error system described by equation (10) are designed as u=Gex1+ey1−b−cex1+ey1 then the synchronization of single state coupled FHN neurobiological drive and slave networks can be achieved if the synchronized error systems converge to zero.proof.
The proof of this theorem and sufficient conditions that ensured the synchronization can be obtained using the theory of Lyapunov stability. According to the theory of Lyapunov stability, the stability of a system can be proved by choosing a positive definite function called Lyapunov function candidate V. In this study it is chosen as(12)V=12ex12+ey12,dv/dt<0 for all t.
It is easy to verify that functionV is a positive definite function. After taking the derivative of equation (12) with respect to time, we obtain(13)V˙=ex1Te˙x1+ey1Te˙y1.
Incorporating synchronized error system (10) and the control law into (12) yields(14)V˙=ex1T−rcd+1+rsd−ex1+2Gex1−b−cex1+ey1+ey1Tbex1−cey1,or(15)V˙=−rex1Tcd+1+rex1Tsd−ex1Tex1+2ex1TGex1−bex1Tex1+cex1Tex1−bex1Tey1,+cex1Tey1+bey1Tex1−cey1Tey1.
Now(16)ex1Tsd=ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1≤ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1,and it is known that neurons have bounded trajectories; i.e., system (2) and (3) and (5) and (6) are bounded with some positive constants qxi and qyi satisfying xi1≤qxi and yi1≤qyi, ∀i=1,2,3⋯,n. Additionally, assuming that q1=maxqxi+qyi;i=1,2,3⋯,n, (16) results into(17)ex1Tsd≤q1ex1Tex1.
Correspondingly,(18)ex1Tcd≤q2ex1Tex1.
Incorporating (17) and (18) into (19) gives(19)V˙=−ex1Trq2−1+rq1+I−2G+b−cex1+b−cex1TIey1−bey1Tex1+cey1TIey1,where I∈ℜN×N is the identity matrix,(20)V˙t≤−EPET,where E=ex1ey1, and P=rq2−1+rq1+I−2G+b−cb−cI−bIcI.
It can be concluded that the matrixP should be a positive definite to ensure the asymptotic stability of the synchronized error system (11) at the origin. The positive definiteness of P could be easily derived for considered networks of neurons (e.g., a network of five neurons) using the method of determinants; i.e., determinants of all leading principal minors have positive values (see [1]). Accordingly, for any value 0≤gij<0.1, the origin of the error system (11) is asymptotically stable. Consequently, these networks of n-identical, single state gap junction FHN noisy and nonnoisy neurobiological networks, under ionic gates disturbance, and EES, will achieve synchronization. This completes the proof.


## 3.1.1. Networks without Noise

Mathematically, neurobiological FHN drive and slave neural networks under EES with single state gap junctions without external noise can be formulated as follows [42]:(2)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+uxi1,x˙i2=bxi1−cxi2,i=1,2,3,⋯,n,(3)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+uyi1,y˙i2=byi1−cyi2,where uxi1 and uyi1 are the control parameters. G=gijN×N represents the topology of the network and is known as coupling matrix whose elements are defined as below.(4)gij=gji≥0,ifi↔j,∀i≠j;gji=0,ifi↮j,∀i≠j;gii=∑j=1j≠1Ngij,∀i=j.


## 3.1.2. Networks with Noise

Mathematically, noisy neurobiological FHN drive and slave neural networks under EES with single state gap junctions can be formulated as follows:(5)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+ξ+uxi1,x˙i2=bxi1−cxi2,i=1,2,3,⋯,n,(6)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+ξ+uyi1,y˙i2=byi1−cyi2,where ξi1 are zero mean Gaussian noise [32] having correlation function:(7)ξ1tξ1t′=2Dδt−t′.


## 3.1.3. Control Law Design

The error dynamics of the coupled neurobiological drive and slave networks described by equations (2-3) and (5-6) can be defined as(8)exi1=xi1−yi1,eyi1=xi2−yi2.Time derivative of equation (8) yields the following equation: (9)e˙x11e˙x21⋮e˙xn1=−rx113−y113x213−y213⋮xn13−yn13+1+rx112−y112x212−y212⋮xn12−yn12−ex11ex21⋮exn1−eyn1eyn1⋮eyn1+∑j=1Ng1jxi1−yi1∑j=1Ng2jxi1−yi1⋮∑j=1Ngnjxi1−yi1+u11u21⋮un1,e˙y11e˙y21⋮e˙yn1=bex11ex21⋮exn1−ceyn1eyn1⋮eyn1,where ui1=uxi1−uyi1. Now let us define the following terms for simplification:(10)ex1=ex11ex21⋯exn1Tey1=ey11ey21⋯eyn1Tcd=x113−y113x213−y213⋯xn13−yn13Tsd=x112−y112x212−y212⋯xn12−yn12TG=giju=u11u21⋯un1T,where ex1,ey1,cd,sd,u∈ℜNx1,G∈ℜNxN and T represents the transpose operator. As a result (10) can be written as(11)e˙x1=−rcd+1+rsd−ex1−ey1+Gex1+u,e˙y1=bex1−cey1,It can be concluded that the problem of synchronization is substituted with the synchronized error system (11) with the requirement to stabilize by using a suitable control inputu. Synchronization can be achieved by designing the controller u such that for any initial conditions ex1 and ey1 converge to zero. This implies that the dynamics of the slave system (3/6) can converge to that of the master system (2/5).Next, Lyapunov stability and control theories are utilized to design a unique and simple control scheme that will guarantee the synchronization of the single state gap junction coupled neurobiological networks.Theorem 1.
Consider the FHN neurobiological drive and slave networks described in equations (2) and (3) and (5) and (6) with the error dynamics described by equation (8). If the control laws in the error system described by equation (10) are designed as u=Gex1+ey1−b−cex1+ey1 then the synchronization of single state coupled FHN neurobiological drive and slave networks can be achieved if the synchronized error systems converge to zero.proof.
The proof of this theorem and sufficient conditions that ensured the synchronization can be obtained using the theory of Lyapunov stability. According to the theory of Lyapunov stability, the stability of a system can be proved by choosing a positive definite function called Lyapunov function candidate V. In this study it is chosen as(12)V=12ex12+ey12,dv/dt<0 for all t.
It is easy to verify that functionV is a positive definite function. After taking the derivative of equation (12) with respect to time, we obtain(13)V˙=ex1Te˙x1+ey1Te˙y1.
Incorporating synchronized error system (10) and the control law into (12) yields(14)V˙=ex1T−rcd+1+rsd−ex1+2Gex1−b−cex1+ey1+ey1Tbex1−cey1,or(15)V˙=−rex1Tcd+1+rex1Tsd−ex1Tex1+2ex1TGex1−bex1Tex1+cex1Tex1−bex1Tey1,+cex1Tey1+bey1Tex1−cey1Tey1.
Now(16)ex1Tsd=ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1≤ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1,and it is known that neurons have bounded trajectories; i.e., system (2) and (3) and (5) and (6) are bounded with some positive constants qxi and qyi satisfying xi1≤qxi and yi1≤qyi, ∀i=1,2,3⋯,n. Additionally, assuming that q1=maxqxi+qyi;i=1,2,3⋯,n, (16) results into(17)ex1Tsd≤q1ex1Tex1.
Correspondingly,(18)ex1Tcd≤q2ex1Tex1.
Incorporating (17) and (18) into (19) gives(19)V˙=−ex1Trq2−1+rq1+I−2G+b−cex1+b−cex1TIey1−bey1Tex1+cey1TIey1,where I∈ℜN×N is the identity matrix,(20)V˙t≤−EPET,where E=ex1ey1, and P=rq2−1+rq1+I−2G+b−cb−cI−bIcI.
It can be concluded that the matrixP should be a positive definite to ensure the asymptotic stability of the synchronized error system (11) at the origin. The positive definiteness of P could be easily derived for considered networks of neurons (e.g., a network of five neurons) using the method of determinants; i.e., determinants of all leading principal minors have positive values (see [1]). Accordingly, for any value 0≤gij<0.1, the origin of the error system (11) is asymptotically stable. Consequently, these networks of n-identical, single state gap junction FHN noisy and nonnoisy neurobiological networks, under ionic gates disturbance, and EES, will achieve synchronization. This completes the proof.


## 3.2. Noisy and Nonnoisy FHN Drive and Slave Networks with Dual State Gap Junctions

Dual state gap junction FHN networks are modeled by formulating gap junction dynamics in both the membrane potential and recovery variable states of each neuron in the network.


### 3.2.1. Networks without Noise

Mathematically, neurobiological FHN drive and slave neural networks under EES with dual state gap junctions without external noise can be formulated as follows:(21)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+uxi1,x˙i2=bxi1−cxi2+∑j=1ngijxj1,i=1,2,3,⋯,n,(22)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+uyi1,y˙i2=byi1−cyi2+∑j=1ngijxj1,where uxi1 and uyi1 are the control parameters.


### 3.2.2. Networks with Noise

Mathematically, noisy neurobiological FHN drive and slave neural networks under EES with dual state gap junctions can be formulated as follows:(23)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+ξ+uxi1,x˙i2=bxi1−cxi2+∑j=1ngijxj1+ξi=1,2,3,⋯,n,(24)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+ξ+uyi1,y˙i2=byi1−cyi2+∑j=1ngijxj1+ξ.


### 3.2.3. Control Law Design

The time derivative of equation (8) yields equation (25):(25)e˙x11e˙x21⋮e˙xn1=−rx113−y113x213−y213⋮xn13−yn13+1+rx112−y112x212−y212⋮xn12−yn12−ex11ex21⋮exn1−eyn1eyn1⋮eyn1+∑j=1Ng1jxi1−yi1∑j=1Ng2jxi1−yi1⋮∑j=1Ngnjxi1−yi1+u11u21⋮un1,e˙y11e˙y21⋮e˙yn1=bex11ex21⋮exn1−ceyn1eyn1⋮eyn1+∑j=1Ng1jxi1−yi1∑j=1Ng2jxi1−yi1⋮∑j=1Ngnjxi1−yi1.Now let us define the following terms for simplification:(26)ex1=ex11ex21⋯exn1Tey1=ey11ey21⋯eyn1Tcd=x113−y113x213−y213⋯xn13−yn13Tsd=x112−y112x212−y212⋯xn12−yn12TG=giju=u11u21⋯un1T,where ex1,ey1,cd,sd,u∈ℜNx1,G∈ℜNxN and T is the transpose operator. Thus (27) becomes(27)e˙x1=−rcd+1+rsd−ex1−ey1+Gex1+u,e˙y1=bex1−cey1+Gex1.It can be concluded that the problem of synchronization is substituted with the synchronized error system (27) with the requirement to stabilize by using a suitable control input u. Synchronization can be achieved by designing the controller u such that for any initial conditions ex1 and ey1 converge to zero. This implies that the dynamics of the slave system (22/24) can converge to that of the master system (21/23).Next, Lyapunov stability and control theories are utilized to design a unique and simple control scheme that will guarantee the synchronization of the dual state gap junction coupled neurobiological networks.Theorem 2.
Consider the FHN neurobiological drive and slave networks described in equations (21)–(24) with the error dynamics described by equation (8). If the control laws in the error system described by equation (27) are designed as u=Gex1−Gey1−b+cex1+ey1 then the synchronization of dual state coupled FHN neurobiological drive and slave networks can be achieved if the synchronized error systems converge to zero.proof.
The Lyapunov function candidateV for this proof is chosen as follows:(28)V=12ex12+ey12,dvdt<0,∀t.
It is easy to verify that the chosen candidate functionV is a positive definite function. After taking the derivative of equation (28) with respect to time, we obtain equation (29)V˙=ex1Te˙x1+ey1Te˙y1.
Incorporating the synchronized error system (26) and the control law into (29) yields(30)V˙=ex1T−rcd+1+rsd−ex1+2Gex1+Gey1−b+cex1+ey1+ey1Tbex1−cey1+Gex1,or(31)V˙=−rex1Tcd+1+rex1Tsd−ex1Tex1+2ex1TGex1+ex1TGey1−bex1Tex1−cex1Tex1−bex1Tey1−cex1Tey1+bey1Tex1−cey1Tey1+ey1TGex1.
Now(32)ex1Tsd=ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1≤ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1,and it is known that neurons have bounded trajectories; i.e., systems (21)–(24) are bounded with some positive constants qxi and qyi satisfying xi1≤qxi and yi1≤qyi, ∀i=1,2,3⋯,n. Additionally, assuming that q1=maxqxi+qyi;i=1,2,3⋯,n, (32) results into(33)ex1Tsd≤q1ex1Tex1.
Correspondingly,(34)ex1Tcd≤q2ex1Tex1.
Incorporating (33) and (34) into (31) gives(35)V˙=−ex1Trq2−1+rq1+I−2Gex1+b+cex1−ex1T−G+b+cey1−ey1T−b−Gex1+cey1TIey1,where I∈ℜN×N is the identity matrix,(36)V˙t≤−EPET,where E=ex1ey1, and P=rq2−1+rq1+I−2Gex1+b+c−−G+b+c−−b−GcI.
It can be concluded that the matrixP should be a positive definite to ensure the asymptotic stability of the synchronized error system (27) at the origin. The positive definiteness of P could be easily derived for considered networks of neurons (e.g., a network of five neurons) using the method of determinants; i.e., determinants of all leading principal minors have positive values (see [1]). Accordingly, the origin of the error system (27) is asymptotically stable. Consequently, these networks of n-identical, dual state gap junction coupled FHN noisy and nonnoisy neurobiological drive and slave networks, under ionic gates disturbance, and EES will achieve synchronization. This completes the proof.


## 3.2.1. Networks without Noise

Mathematically, neurobiological FHN drive and slave neural networks under EES with dual state gap junctions without external noise can be formulated as follows:(21)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+uxi1,x˙i2=bxi1−cxi2+∑j=1ngijxj1,i=1,2,3,⋯,n,(22)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+uyi1,y˙i2=byi1−cyi2+∑j=1ngijxj1,where uxi1 and uyi1 are the control parameters.


## 3.2.2. Networks with Noise

Mathematically, noisy neurobiological FHN drive and slave neural networks under EES with dual state gap junctions can be formulated as follows:(23)x˙i1=xi1xi1−11−rxi1−xi2+∑j=1ngijxj1+I+d+ξ+uxi1,x˙i2=bxi1−cxi2+∑j=1ngijxj1+ξi=1,2,3,⋯,n,(24)y˙i1=yi1yi1−11−ryi1−yi2+∑j=1ngijyj1+I+d+ξ+uyi1,y˙i2=byi1−cyi2+∑j=1ngijxj1+ξ.


## 3.2.3. Control Law Design

The time derivative of equation (8) yields equation (25):(25)e˙x11e˙x21⋮e˙xn1=−rx113−y113x213−y213⋮xn13−yn13+1+rx112−y112x212−y212⋮xn12−yn12−ex11ex21⋮exn1−eyn1eyn1⋮eyn1+∑j=1Ng1jxi1−yi1∑j=1Ng2jxi1−yi1⋮∑j=1Ngnjxi1−yi1+u11u21⋮un1,e˙y11e˙y21⋮e˙yn1=bex11ex21⋮exn1−ceyn1eyn1⋮eyn1+∑j=1Ng1jxi1−yi1∑j=1Ng2jxi1−yi1⋮∑j=1Ngnjxi1−yi1.Now let us define the following terms for simplification:(26)ex1=ex11ex21⋯exn1Tey1=ey11ey21⋯eyn1Tcd=x113−y113x213−y213⋯xn13−yn13Tsd=x112−y112x212−y212⋯xn12−yn12TG=giju=u11u21⋯un1T,where ex1,ey1,cd,sd,u∈ℜNx1,G∈ℜNxN and T is the transpose operator. Thus (27) becomes(27)e˙x1=−rcd+1+rsd−ex1−ey1+Gex1+u,e˙y1=bex1−cey1+Gex1.It can be concluded that the problem of synchronization is substituted with the synchronized error system (27) with the requirement to stabilize by using a suitable control input u. Synchronization can be achieved by designing the controller u such that for any initial conditions ex1 and ey1 converge to zero. This implies that the dynamics of the slave system (22/24) can converge to that of the master system (21/23).Next, Lyapunov stability and control theories are utilized to design a unique and simple control scheme that will guarantee the synchronization of the dual state gap junction coupled neurobiological networks.Theorem 2.
Consider the FHN neurobiological drive and slave networks described in equations (21)–(24) with the error dynamics described by equation (8). If the control laws in the error system described by equation (27) are designed as u=Gex1−Gey1−b+cex1+ey1 then the synchronization of dual state coupled FHN neurobiological drive and slave networks can be achieved if the synchronized error systems converge to zero.proof.
The Lyapunov function candidateV for this proof is chosen as follows:(28)V=12ex12+ey12,dvdt<0,∀t.
It is easy to verify that the chosen candidate functionV is a positive definite function. After taking the derivative of equation (28) with respect to time, we obtain equation (29)V˙=ex1Te˙x1+ey1Te˙y1.
Incorporating the synchronized error system (26) and the control law into (29) yields(30)V˙=ex1T−rcd+1+rsd−ex1+2Gex1+Gey1−b+cex1+ey1+ey1Tbex1−cey1+Gex1,or(31)V˙=−rex1Tcd+1+rex1Tsd−ex1Tex1+2ex1TGex1+ex1TGey1−bex1Tex1−cex1Tex1−bex1Tey1−cex1Tey1+bey1Tex1−cey1Tey1+ey1TGex1.
Now(32)ex1Tsd=ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1≤ex11ex21⋯exn1x11+y11ex11x21+y21ex21⋮xn1+yn1exn1,and it is known that neurons have bounded trajectories; i.e., systems (21)–(24) are bounded with some positive constants qxi and qyi satisfying xi1≤qxi and yi1≤qyi, ∀i=1,2,3⋯,n. Additionally, assuming that q1=maxqxi+qyi;i=1,2,3⋯,n, (32) results into(33)ex1Tsd≤q1ex1Tex1.
Correspondingly,(34)ex1Tcd≤q2ex1Tex1.
Incorporating (33) and (34) into (31) gives(35)V˙=−ex1Trq2−1+rq1+I−2Gex1+b+cex1−ex1T−G+b+cey1−ey1T−b−Gex1+cey1TIey1,where I∈ℜN×N is the identity matrix,(36)V˙t≤−EPET,where E=ex1ey1, and P=rq2−1+rq1+I−2Gex1+b+c−−G+b+c−−b−GcI.
It can be concluded that the matrixP should be a positive definite to ensure the asymptotic stability of the synchronized error system (27) at the origin. The positive definiteness of P could be easily derived for considered networks of neurons (e.g., a network of five neurons) using the method of determinants; i.e., determinants of all leading principal minors have positive values (see [1]). Accordingly, the origin of the error system (27) is asymptotically stable. Consequently, these networks of n-identical, dual state gap junction coupled FHN noisy and nonnoisy neurobiological drive and slave networks, under ionic gates disturbance, and EES will achieve synchronization. This completes the proof.


## 4. Numerical Results

In this section, the results of the numerical simulations are discussed to analyze and validate the efficacy of the designed control laws for the synchronization of single and dual state gap junction coupled FHN neural networks. To perform this, we considered five neurons of two FHN networks with single and dual state gap junctions and external noise. The parameter values used in this study arer11  = 10, b11  = 1, c11  = 0.001/0.003, d11  = 0.001 sin0.2×t, A  = 0.1, f  = 0.129, ω  =  2πf and initial conditions are x110  = 0, y110  = 0, x120  = 0.1, y120  = 0.1, x210  = 0.2, y210  = 0.2, x220  = 0.3, y220  = 0.3, x310  = 0.4, y310  = 0.4, x320  = 0.5, y220  = 0.5, x410  = 0.6, y410  = 0.6, x420  = 0.7, y420  = 0.7, x510  = 0.8, y510  = 0.8, x520  = 0.9, y520  = 0.9. The values of the gap junctions are listed in the matrix G(37)G=−2.4745×10−036.5574×10−043.5712×10−058.4913×10−049.3399×10−046.5574×10−04−2.8353×10−036.7874×10−047.5774×10−047.4313×10−043.5712×10−056.7874×10−04−1.7621×10−033.9223×10−046.5548×10−048.4913×10−047.5774×10−043.9223×10−04−2.1702×10−031.7119×10−049.3399×10−047.4313×10−046.5548×10−041.7119×10−04−2.5037×10−03.Figures3(a) and 3(b) illustrate the errors in temporal dynamics of the state variables for single state gap junction FHN networks without noise, respectively. The nonconverging behavior observed through spiked errors for both membrane potential and recovery states of each network revealed the unsynchronized activities of the networks. Now, we will show simulation results for the drive and slave networks by using the control law designed for the single state gap junction FHN networks without noise. Figure 4 shows the results of these simulations. We used the strategy of controller switch off and on for critically analyzing and evaluating the performance of the proposed control scheme. As shown in Figures 4(a) and 4(b), the proposed controller was not applied until t = 450 and the temporal dynamics of the neurons is highly abrupt, but all the errors converged to zero, proving synchronization between both states of all neurons in both networks, as soon as the designed control laws are activated, showing the efficacy of the proposed scheme. Furthermore, the unsynchronized activities of the network could also be observed through the abrupt behavior of the phase plane diagrams for both states as shown in Figures 5(a) and 5(c). In addition, the straight lines in phase plane diagrams (Figures 5(b) and 5(d)) after the application designed control laws indicate that both membrane potential and recovery variable states of each neuron in both networks have synchronized activities.Figure 3
Synchronization error dynamics of single state gap junction coupled nonnoisy FHN networks without any controller. (a) Membrane potential temporal dynamics. (b) Recovery variable temporal dynamics.
(a)(b)Figure 4
Synchronization error dynamics of single state gap junction coupled nonnoisy FHN networks with proposed control scheme. (a) Membrane potential temporal dynamics. (b) Recovery variable temporal dynamics.
(a)(b)Figure 5
Synchronization error dynamics of single state gap junction coupled nonnoisy FHN networks with and without proposed control scheme. (a) Phase plane analysis of membrane potential states without control. (b) Phase plane analysis of membrane potential states with proposed control. (c) Phase plane analysis of recovery states without control. (d) Phase plane analysis of recovery states with proposed control.
(a)(b)(c)(d)Figures6–8 demonstrate the simulation results for the single state gap junction noisy FHN neurobiological networks. Like the nonnoisy networks, the time dynamics shown in Figures 6(a) and 6(b) for both networks result in spiky behavior indicating that, at this point, the firing of all the neurons is not synchronized. Similarly, the nonlinear behavior in the phase plane diagrams as illustrated in Figures 8(a) and 8(c) further certifies that the firing activities of all the neurons in both neurobiological noisy FHN networks are nonsynchronized. Next, we will discuss the application of the designed control laws for the coupled networks of noisy FHN neurons. The quick convergence of the error temporal dynamics to zero after the activation of the designed control laws at t = 450 in contrast to the spiky behavior before the activation of the designed control laws in Figures 7(a) and 7(b) validates the effectiveness of the proposed controllers in achieving synchronization for the noisy FHN neurobiological networks with single state gap junction. It can be visualized that both states of all the neurons in both noisy networks converged to zero, showing that the firing activities of the neurons are synchronized. Furthermore, the phase plane analysis in Figures 8(b) and 8(d) of the noisy FHN networks revealed the same results as the subsequent plots demonstrate a linear line passing across the origin with a slope of 1.Figure 6
Synchronization error dynamics of single state gap junction coupled noisy FHN networks without any controller. (a) Membrane potential temporal dynamics. (b) Recovery variable temporal dynamics.
(a)(b)Figure 7
Synchronization error dynamics of single state gap junction coupled noisy FHN networks with proposed control scheme. (a) Membrane potential temporal dynamics. (b) Recovery variable temporal dynamics.
(a)(b)Figure 8
Synchronization error dynamics of single state gap junction coupled noisy FHN networks with and without proposed control scheme. (a) Phase plane analysis of membrane potential states without control. (b) Phase plane analysis of membrane potential states with proposed control. (c) Phase plane analysis of recovery states without control. (d) Phase plane analysis of recovery states with proposed control.
(a)(b)(c)(d)Figures9–11 and 12–14 demonstrate the simulation results for dual state gap junctions nonnoisy and noisy FHN neurobiological networks, respectively. Temporal analysis of the membrane potential and recovery variable dynamics of dual state nonnoisy FHN networks results in spiked errors without the activation of the designed control laws as illustrated in Figures 9(a) and 9(b). Furthermore, the phase plane analysis demonstrated in Figures 11(a) and 11(c) revealed the abrupt nonlinear behavior. All these results indicate that the firing of neurons in both networks is not synchronized. Next, by applying the designed control laws in the dual state gap junction FHN network without noise, we will show the simulation results for the synchronization of two networks. Figures 10(a) and 10(b) demonstrate the results of the temporal dynamics in response to the application of the designed control scheme. It can be visualized that the errors time dynamics have resulted in nonzero spikes, before the activation of the designed control laws at t = 450, indicating nonsynchronized firing activities of the neurons in the two networks. However, the error dynamics converged to zero right after the application of the designed control scheme indicating synchronization between the networks. Furthermore, in Figure 11 the linear lines in phase plane analysis also revealed that both networks have successfully achieved synchronization after the implementation of the designed control scheme. Finally, we will discuss the simulation results for dual state gap junction noisy FHN networks (Figures 12–14). Like previous cases, the firing activities of all the neurons in both networks are nonsynchronized without the application of any controller as evident from the error spikes in the temporal dynamics as shown in Figures 12(a) and 12(b) and the nonlinear abrupt behavior in the phase plane analysis in Figures 14(a) and 14(c). However, both noisy FHN networks showed synchronized firing activities after the activation of the proposed control scheme as demonstrated in Figure 13. Figures 13(a) and 13(b) revealed the efficacy of the designed control laws as both states of all neurons converged to zero as soon as the designed control scheme is applied at t = 450 in contrast to before the application of the control scheme. Moreover, the linear lines passing across the origin with a slope of 1 in Figures 14(b) and 14(d) also indicate the synchronized behavior of the drive and slave networks. In summary, all these simulation results validate the efficacy and effectiveness of the proposed control schemes to successfully achieve synchronization in different cases of nonnoisy and noisy FHN neurobiological networks.Figure 9
Synchronization error dynamics of dual state gap junction coupled nonnoisy FHN networks without any controller. (a) Membrane potential temporal dynamics. (b) Recovery variable temporal dynamics.
(a)(b)Figure 10
Synchronization error dynamics of dual state gap junction coupled nonnoisy FHN networks with proposed control scheme. (a) Membrane potential temporal dynamics. (b) Recovery variable temporal dynamics.
(a)(b)Figure 11
Synchronization error dynamics of dual state gap junction coupled nonnoisy FHN networks with and without proposed control scheme. (a) Phase plane analysis of membrane potential states without control. (b) Phase plane analysis of membrane potential states with proposed control. (c) Phase plane analysis of recovery states without control. (d) Phase plane analysis of recovery states with proposed control.
(a)(b)(c)(d)Figure 12
Synchronization error dynamics of dual state gap junction coupled noisy FHN networks without any controller. (a) Membrane potential temporal dynamics. (b) Recovery variable temporal dynamics.
(a)(b)Figure 13
Synchronization error dynamics of dual state gap junction coupled noisy FHN networks with proposed control scheme. (a) Membrane potential temporal dynamics. (b) Recovery variable temporal dynamics.
(a)(b)Figure 14
Synchronization error dynamics of dual state gap junction coupled noisy FHN networks with and without proposed control scheme. (a) Phase plane analysis of membrane potential states without control. (b) Phase plane analysis of membrane potential states with proposed control. (c) Phase plane analysis of recovery states without control. (d) Phase plane analysis of recovery states with proposed control.
(a)(b)(c)(d)Furthermore, we calculated the mean errors for both membrane potential states and recovery variable states in each single and dual state neuorobiological network with and without presence of noise. The results of this analysis are listed in Tables1 and 2. It can be visualized that the error for both states in each network is almost zero, indicating the synchronized behavior of the network.Table 1
Mean errors in the synchronization dynamics of the membrane potential states and recovery variable states for single state FHN neurobiological networks with and without noise.
Membrane potential statesRecovery variable statesWithout controller and without noise0.00380.0021With controller and without noise1.4931×10−141.1648×10−11Without controller and with noise−0.0016−0.0151With controller and with noise−4.0322×10−09−3.3136×10−07Table 2
Mean errors in the synchronization dynamics of the membrane potential states and recovery variable states for dual state FHN neurobiological networks with and without noise.
Membrane potential statesRecovery variable statesWithout controller and without noise0.0011−0.0196With controller and without noise1.7513×10−09−5.4067×10−09Without controller and with noise−0.0021−0.0053With controller and with noise−3.2866×10−07−3.7260×10−07


## 5. Discussion and Limitations

Synchronization among the large number of neurons and their networks across different regions of the brain is one of the primary components of processing cognitive information. Findings from previous experimental and theoretical studies not only revealed that synchronization serves a variety of functions in cognitive information processing, but is also a major property of cortical and subcortical networks within and across different areas of the brain. It is evident from past studies that absence/abnormality in the synchronization of neurons plays important role in many brain disorders like Alzheimer’s disease, epilepsy, Parkinson’s, autism, and schizophrenia [58]. In the last decades, different mathematical models depicting the activity of biological neurons have been extensively deployed as a valuable tool to explore the synchronization phenomenon in neurons [59–61]. However, FHN is the most used model to explore the synchronization of coupled neurons because of its wide applicability and complex dynamical characteristics. In previous studies, the problem of neuronal synchronization, using the FHN model, has been intensively examined as a potential application in cognitive engineering [1, 13, 29, 32]. Researchers have provided insight on neural synchronization using FHN model under EES by designing robust controllers [62], nonlinear controllers [13], adaptive controllers [29, 63], observer [64] neural network, and fuzzy based control schemes [65]. However, these traditional approaches were designed for two or three coupled neurons and cannot ensure synchronization of the network of neurons. Furthermore, integration of the gap junction strength in FHN neurons renders the synchronization dilemma nontrivial [66]. Additionally, previous research revealed that the presence of noise can alter the dynamics and firing behavior of neurons; however, neurons adjust their firing properties to optimally transmit information [56]. Therefore, the effect of noise should be incorporated into the mathematical models to truly understand the dynamics and synchronization phenomenon of neurons. To address these issues, more recently, Ibrahim and coauthors investigated the synchronization of a network of neurons in the absence and presence of external noise under EES with unidirectional and bidirectional couplings between neurons [1, 32]. They designed different control schemes that guarantee the synchronization of a network of identical FHN neurons. As discussed above, cognitive information is processed because of synchronized activities of different neuronal networks across different brain regions; however, the problem of synchronization of neurobiological drive and slave networks has not been extensively explored yet. To overcome these shortcomings, this study explores the synchronization problem in two coupled neurobiological networks of FHN neurons by incorporating the effect of external noise, direction-dependent couplings (gap junctions), and ionic and external disturbances in mathematical models. In contrast to traditional techniques, we designed control schemes that guarantee the synchronization of two networks with single state and dual state direction-dependent couplings. Numerical simulations were performed to validate the efficiency of the designed control schemes. Comprehensive analysis illustrated through Figures 2–14 demonstrates the efficacy of the designed control schemes.Even though the present study designed efficient control schemes that guarantee the synchronization of two neurobiological FHN neuronal networks, it also has some drawbacks/limitations. One apparent shortcoming of the present investigation is that configurations of both neurobiological networks are formulated with identical FHN neurons with fixed system parameters. The likelihood of two coupled neurons being nonidentical cannot be ignored; however, it is difficult to completely know the real parameters of the neurons due to the biological restrictions and hidden dynamics of the brain. Therefore, networks with unknown parameters and nonidentical neurons in the presence of external noise with direction-dependent coupling can give further insight into the neuronal synchronization, but at the same time, it will enhance the complexity of the mathematical system. Furthermore, the absence of time-delays in the formulation of networks is another limitation of this study [67]. It is evident from past research that the information transmittance between distant neurons causes time-delays and hence can affect the synchronization properties of the networks. Therefore, the addition of delay term in mathematical models accounting for the distant communication between neurons makes it realistic but more complex to study the synchronization problem. Moreover, past studies also revealed that the execution of tasks has been achieved by the communication of neurons in multiple regions of the brain, i.e., communication between multiple networks. Thus, it is our immediate future plan to formulate a network of neurons incorporating nonidentical neurons with unknown parameters, time-delays, different direction-dependent couplings, and external disturbances and investigate the synchronization of multiple neurobiological drive and slave networks. Another possible extension of the current work is to investigate the link between experimental data recorded from real neurons and mathematical simulations as this could enhance our knowledge about the information processing between neurons and possibly the underlying dynamics behind several brain disorders [19]. One possible way to achieve this is by using experimental data from real neurons to estimate the parameters of the mathematical model of the neuron as it is done in [68] where the authors proposed a system identification methodology that approximates the parameters of FHN neurons to replicate the real neurons data.


## 6. Conclusion

This paper presents the design of different control schemes for the synchronization of single and dual sate gap junction FHN neurobiological drive and slave networks under EES with and without external noise. The proposed control laws for both drive and slave networks are simple and unique to achieve synchronization. Control theory was utilized to propose diverse and simple control strategies to investigate the synchronization problem of the different single and dual state gap junctions coupled nonnoisy and noise FHN neurobiological drive and slave networks. Control laws are designed to stabilize the error dynamics without direct cancelation and synchronize all the states of both FHN neurobiological networks. Sufficient conditions for achieving synchronization in the single and dual state gap junction coupled FHN neurobiological drive and slave networks were derived analytically using Lyapunov stability theory. The results of numerical simulations demonstrated the efficacy of the designed control schemes.



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