Lag synchronization of coupled time-delayed FitzHugh–Nagumo neural networks via feedback control

Synchronization plays a significant role in information transfer and decision-making by neurons and brain neural networks. The development of control strategies for synchronizing a network of chaotic neurons with time delays, different direction-dependent coupling (unidirectional and bidirectional), and noise, particularly under external disturbances, is an essential and very challenging task. Researchers have extensively studied the synchronization mechanism of two coupled time-delayed neurons with bidirectional coupling and without incorporating the effect of noise, but not for time-delayed neural networks. To overcome these limitations, this study investigates the synchronization problem in a network of coupled FitzHugh–Nagumo (FHN) neurons by incorporating time delays, different direction-dependent coupling (unidirectional and bidirectional), noise, and ionic and external disturbances in the mathematical models. More specifically, this study investigates the synchronization of time-delayed unidirectional and bidirectional ring-structured FHN neuronal systems with and without external noise. Different gap junctions and delay parameters are used to incorporate time-delay dynamics in both neuronal networks. We also investigate the influence of the time delays between connected neurons on synchronization conditions. Further, to ensure the synchronization of the time-delayed FHN neuronal networks, different adaptive control laws are proposed for both unidirectional and bidirectional neuronal networks. In addition, necessary and sufficient conditions to achieve synchronization are provided by employing the Lyapunov stability theory. The results of numerical simulations conducted for different-sized multiple networks of time-delayed FHN neurons verify the effectiveness of the proposed adaptive control schemes.

Synchronization plays a tremendous role in information transfer and decision-making in different fields of science and technology 1 . In the past decade, many researchers have investigated different synchronization schemes, methodologies, and regimes for chaotic systems such as lag synchronization 2 , generalized synchronization 3 , phase synchronization 4 , projective synchronization 5 , anticipating synchronization 6 , cluster synchronization 7 , and consecutive synchronization 8 . Lag synchronization, introduced for drive and response systems and extensively studied, is defined as an overlapping of the shifted-in-time states of two systems with a positive constant.
In neuroscience, the interactions of neurons and their networks have been the primary targets for investigating the differences in the functionality and dynamics of healthy and diseased brains [9][10][11] . For example, the brain responses of stroke patients have been analyzed and compared with those of healthy subjects to develop an effective rehabilitation system 12,13 . Neurons in the brain are connected to other neurons in the same or other regions to form a network of neurons for communication and efficient information processing. In the past decades, the internal dynamics behind the coordination and communication mechanism of neurons and their networks have been explored by developing different strategies and systems [14][15][16][17][18] . The processing of cognitive information in the brain is based upon the synchronized interactions between large numbers of neurons distributed within and across different specialized brain regions 19 . Researchers have found that synchronization between individual neurons or networks of neurons is an important factor for the stable and efficient working of the brain 20 www.nature.com/scientificreports/ of control strategies for synchronization of a network of chaotic neurons with time delays, different directiondepend coupling (unidirectional and bidirectional), and noise, particularly under external disturbances, is very important and challenging. This paper examines the synchronization phenomena of unidirectional and bidirectional ring-structured FHN networks with and without noise under ionic and EES with different gap junctions and time-delay dynamics between the neurons. Each neuron of the unidirectional network is connected to the next neuron via synapses, and the first neuron of the network is a master neuron for connecting networks. However, each neuron in a bidirectional network is connected to the next neuron as well as the previous neuron behaves as a master and slave neuron simultaneously. The configuration of both networks is shown in Fig. 1a, b. The separation between each neuron in both networks is incorporated through different time-delay parameters.
This paper also investigates the influence of the time delay between connected neurons on synchronization conditions. Unique and different adaptive control laws are proposed for both unidirectional and bidirectional neuronal networks, which guarantee the synchronization of time-delayed FHN networks in the absence and presence of noise. Necessary and sufficient conditions were driven using the Lyapunov theory of stability, which also assures the synchronization of delayed FHN networks. Furthermore, the proposed control schemes were verified using multiple networks consisting of five, ten, fifty, one hundred, two hundred and fifty, five hundred, and one thousand time-delayed FHN neurons with and without noise through numerical simulations.
The main contributions of this paper include (1) investigating the synchronization of a unidirectional network of n-FHN neurons in the absence and presence of noise with different gap-junctions and time delays, (2) investigating the synchronization of a bidirectional network of n-FHN neurons in the absence and presence of noise with different gap-junctions and time delays, (3) the development of unique and different adaptive control laws for both unidirectional and bidirectional networks, and (4) achieving the synchronization of membrane and recovery states for both unidirectional and bidirectional FHN neurons using the proposed adaptive schemes.

Time-delayed FHN neuronal networks and control design.
This section presents the mathematical formulation of the synchronization problem for unidirectional and bidirectional neuronal networks of n timedelayed FHN neurons with and without noise connected in a ring structure, under ionic and EES, and with different time delays and gap junctions. Based on the literature, we hypothesized that both unidirectional and bidirectional time-delayed FHN neuronal networks will show very complex and unpredictable behaviors and dynamics with delays in gap junctions. Furthermore, the inclusion of the noise and dynamic effects of ionic gate disturbances in the network will make it more realistic but more challenging to analyze.   www.nature.com/scientificreports/ where x i and y i are stated action potential and recovery variables, respectively; r , b , and c are positive constants; d = 0.01 sin(0.2t) is the ionic gate disturbance;I = A ω cos(ωt) is the stimulus current; g i represents the coupling strength of the gap junction between master and slave neurons; τ i is non-negative delay parameter; u i represents the control; and i = 1, 2 . . . , n is the number of FHN neurons. Next, we propose a unique control scheme by using Lyapunov stability and adaptive control theories.   where g 2j−1 and g 2j represent the coupling strength of the gap junction between the master and slave neurons, and τ j−1 and τ j are the delay parameters between the neurons, for j = 1, 2, 3, …, n.

(b) Network with noise
Let us consider a bidirectional ring-structured time-delayed FHN network with different gap junctions and external noise. Mathematically, is the Gaussian noise 70 source having zero mean and the following correlation function:

(c) Control laws design
Taking the derivative of Eq. (4) with respect to time, the error system for the time-delayed FHN networks described in Eqs. (6) and (7) can be expressed as shown by Eq. (9).
Next, we propose a unique control scheme by using Lyapunov stability and adaptive control theories.
Theorem 2 Consider bidirectional time-delayed FHN networks as described in Eqs. (6) and (7) with the dynamical error system of Eq. (9). If the controllers u j in the error system are defined as then this will ensure the synchronization of the bidirectional networks of the time-delayed FHN systems described in Eqs. (6) and (7) by converging the error of the synchronized system to zero.
Proof Please see the supplementary information for the proof.

Results
After creating an accurate model for the dynamics of the ring-structured network of n-identical FHN neurons with and without noise, we establish a synchronization control scheme for achieving the coherent behavior of the neurons. Numerical simulations were executed to verify the proposed control laws and examine their impact on the synchronization of time-delayed unidirectional and bidirectional ring-structured FHN networks composed of five, ten, fifty, one hundred, two hundred and fifty, five hundred, and one thousand neurons. The parameter values used in this study are listed in Table 1. The values for g, τ , and initial conditions are randomly chosen from (0-0.1), , and ((0-0.5), (0-0.5)), respectively.

Analysis of unidirectional time-delayed FHN networks. (a) Networks without noise
The proposed unidirectional networks of five, ten, fifty, one hundred, two hundred and fifty, five hundred, and one thousand neurons with different gap-junctions and time delays can be modeled as presented in Eq. (1) with i = 1, 2, . . . , 1000.
In the case of the unidirectional network without any noise, the unsynchronized error dynamics for the membrane potential states and recovery variable states for the network of five neurons (blue line), ten neurons (red line), fifty neurons (black line), one hundred neurons (green line), two hundred and fifty neurons (magenta line), five hundred neurons (brown line), and one thousand neurons (cyan line) are shown in Figs. 2a and 3a.
(7) www.nature.com/scientificreports/ These figures show the error dynamics of five randomly selected pairs of neurons from each network. These plots reveal that the error dynamics are not convergent for both states and the activity of neurons in all networks is highly non-synchronized. Next, we analyzed the results shown in Figs. 2b and 3b after the implementation of the proposed control laws for the unidirectional time-delayed network without noise. We analyzed the effectiveness of the proposed control scheme by examining the synchronization dynamics of the ring-structured delayed FHN neuronal network, with and without the adaptive control. The outcomes of this examination are shown in Figs. 2b and 3b. The proposed controllers for the unidirectional network were activated at t = 130. The results show oscillatory and nonsynchronized behavior for all networks before the application of the proposed controller. In contrast, when the controller was activated at t = 130, the errors between the membrane potential and recovery variables converged to zero, indicating the synchronization between both states of the delayed unidirectional FHN neuronal networks. Furthermore, we calculated the mean errors for both membrane potential states and recovery variable states in each network to show the overall effectiveness of the proposed control scheme for the non-noisy unidirectional network of time-delayed FHN networks. The results of this analysis are listed in Table 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 3. The results in Figs. 4a and 5a show the unexpected behavior of the neurons in each network and indicate that the primary neuronal networks of delayed unidirectional FHN neurons with noise are not synchronized. It can be concluded at this stage that the non-synchronized behavior is present during the firing of the unidirectional neurons of the delayed networks. The results in Figs. 4b and 5b illustrate the effectiveness of the proposed control scheme for noisy networks of FHN neurons. Initially, the controller was switched off until t = 130, and it can be seen that five randomly selected error dynamics show non-zero and non-convergent spikey behavior, indicating that the activity of the neurons in each network is non-synchronized. In contrast, the error dynamics converged to zero and all the networks achieved synchronization as soon as the controller is switched on at t = 130. Furthermore, the mean errors for each state and each  Table 3-show that the proposed control scheme is successful in guaranteeing the synchronization of noisy FHN neuronal networks regardless of the size of the network.

Analysis of bidirectional time-delayed FHN networks.
(a) Networks without noise Similar to the unidirectional network's analysis, we considered bidirectional networks of five, ten, fifty, one hundred, two hundred and fifty, five hundred, and one thousand neurons with different gap-junctions and time delays. The results of this analysis are illustrated in Figs. 6 and 7 and Table 4. The non-synchronized activity of neurons in different bidirectional networks without noise can be visualized in Figs. 6a and 7a as the time dynamics of the errors are non-zero and non-convergent for both membrane potential states and recovery variable   Table 4. These results suggest that each state of the delayed neuronal network achieved synchronized behavior with the state of the corresponding slave neuron because the outcome of the time dynamics of error converged to zero.

(b) Networks with noise
The dynamical structure of the bidirectional delayed network with noise is more complex and complicated than that of the unidirectional delayed FHN neuronal networks, but the results of the numerical simulations suggest that synchronization can be achieved successfully with the activation of the proposed controller at t = 130. The time-error dynamics of different-sized networks revealed in Figs. 8a and 9a show the unsynchronized neuronal activities between all neurons for all networks. Next, we analyzed the results after the implementation of the proposed control law for the bidirectional delayed network in Figs. 8b and 9b. These results show the error dynamics before and after the application of the proposed adaptive controller. Similar to the previous cases, the delayed networks showed highly non-synchronized behavior until the proposed controller was switched on. Then, as soon as the controller was applied, all the errors converged to zero, indicating the effectiveness of the proposed scheme. Furthermore, the results listed in Table 5 with very low mean errors also show that different bidirectional networks of noisy time-delayed FHN neurons achieved synchronization, indicating the effectiveness of the proposed scheme.
The convergence of the error dynamics to zero in all cases guaranteed the synchronization of the differentsized ring-structured networks of FHN neurons in the absence and presence of noise with different gap-junctions and time delays under conditions of EES and ionic gate disturbance. It was also observed that the errors between neurons for unidirectional and bidirectional gap-junction networks converged to zero very rapidly in all cases, showing the efficiency of the proposed control schemes.

Discussion
The processing of cognitive information in the brain is based upon the synchronized interactions between large numbers of neurons distributed within and across different specialized brain regions. Experimental and theoretical results from previous studies suggested that synchronization of neuronal activity is not only a fundamental property of cortical and subcortical networks within and across different brain regions but also serves a variety of functions in cognitive processes. It is evident from past research that certain brain disorders, such as Alzheimer's disease, epilepsy, Parkinson's, autism, and schizophrenia are associated with abnormal neural synchronization 19 .
Researchers have studied the synchronization problem in neurons by using different mathematical models of neurons [71][72][73] . Among those, FHN is the most commonly used model to investigate the synchronization of coupled neurons because of its wide applicability and complex dynamical aspects. In the literature, the subject of neuronal synchronization, using the FHN model, has been intensively examined as a potential application in cognitive engineering 1,20,28,47,57 . Researchers have developed adaptive 20,41 , nonlinear 28 , robust control 23 , neuralnetwork-, fuzzy 74 , and observer-based control schemes 63 to study the synchronization phenomenon in FHN    www.nature.com/scientificreports/ neurons under external electrical stimulations. However, these conventional methodologies were developed for two or three coupled neurons and cannot guarantee synchronization of distant neurons if used for synchronizing the activity of networks of neurons because the mathematical models ignore the time delays arising from the separation between coupled neurons, and hence cannot synchronize distant FHN neurons. Therefore, the addition of a delay term in mathematical models accounting for the distant communication between neurons makes it realistic but more complex to study the synchronization problem. Furthermore, the integration of the gap-junction strength in FHN neurons renders the synchronization dilemma nontrivial 75 . Moreover, previous studies assumed bidirectional coupling between the neurons whereas experimental observation suggests that the coupling between neurons could be unidirectional and may result in different coupling strengths for each neuron 75,76 . Therefore, the dynamical effects of unidirectional gap junctions while entertaining time delays (due to neuronal separation) in the mathematical models should not be ignored. Additionally, it is evident from past research that the presence of noise can affect the dynamical behavior of the neuronal system and neurons adjust their firing properties to transmit information optimally 70 . Therefore, researchers should entertain the noise effect in their mathematical models to study the synchronization problem.
To overcome the shortcomings of previous studies, this study investigated the synchronization problem in the network of coupled FHN neurons by incorporating time delays, different direction-dependent coupling  Although the present study proposed efficient control schemes that guarantee the synchronization of networks of time-delayed FHN neurons, it also has some limitations/drawbacks. One obvious of this study is that we considered the networks of identical FHN neurons with known fixed parameters in the present configurations. However, the possibility of two coupled neurons to be non-identical cannot be ignored, and the actual model parameters cannot be completely known, owing to the known dynamics of the brain and biological restrictions. Therefore, considering a network of large numbers of non-identical neurons having unknown parametric values where several hundred neurons communicate with each other under different directiondependent couplings (unidirectional and bidirectional), can enhance the complexity and analysis of neuronal synchronization. The understanding of the physical theory behind direction-dependent couplings, it's discrete  www.nature.com/scientificreports/ modeling, and simulations is a complex and tedious job; hence, it is a challenging task for the research community. Thus, it is our immediate future plan to study the synchronization of the network of non-identical FHN neurons with unknown parameters and disturbances. Moreover, in the current configurations, the coupling between the neurons of each network is assumed to be unidirectional or bidirectional, but in a real scenario, a network may have both types of direction-dependent coupling that will enhance the complexity of the network. Another possible future direction of the current study is that we could investigate the link between mathematical simulations and experimental data recorded from real neurons, as this could help to understand the dynamics of various brain disorders 19 . This could be done by estimating the parameters of FHN neurons to replicate the experimental data. For instance, Che et al. 77 developed an identification method to estimate the parameters of FHN models to replicate the experimental data recorded from real neurons. Furthermore, in the present study, only synchronization within a network of neurons is considered and how two or more networks with different dynamics and configurations (non-identical neurons, unknown parameters, external disturbances, and different direction-dependent coupling) communicate and synchronize their activities is yet to be explored in future work.

Conclusion
The development of control strategies for synchronization of a network of chaotic neurons with time delays, different direction-dependent coupling (unidirectional and bidirectional), and noise, particularly under external disturbances, is essential and very challenging. Most of the previous studies developed control strategies for two or three coupled neurons with bidirectional coupling and without incorporating the effect of noise, but not for time-delayed neural networks. To overcome these limitations, this study investigated the synchronization problem in the network of coupled FHN neurons by incorporating time delays, different direction-dependent coupling (unidirectional and bidirectional), noise, and ionic and external disturbances in mathematical models. Both unidirectional and bidirectional time-delayed FHN neuronal networks have very complex and unpredictable behavior and dynamics. In this study, the lag synchronization of a network of delayed FHN neurons with unidirectional and bidirectional coupling in the absence and presence of noise was addressed. Different gap junctions and time-delay parameters were used to incorporate the dynamics of time delays in neurons. Two different networks, one with unidirectional coupling between two neurons and the other with bidirectional coupling in membrane states, were considered. To achieve the synchronization between the states of the delayed neuronal networks, we designed two different adaptive control strategies, which compensated for the nonlinear dynamics without direct cancelation. Lyapunov stability theory was used to derive sufficient conditions that guarantee the synchronization of the delayed FHN neuronal networks. Numerical simulations with networks of five, ten, fifty, one hundred, two hundred and fifty, five hundred, and one thousand neurons were performed to demonstrate the efficiency of the proposed control schemes.