The fractional order (FO) calculus has a very long mathematical history and has gained extensive attention in the areas of science and engineering with the advent of high computational devices recent years. The FO calculus can be seen as the comprehensive and generalized version of the conventional integer-order calculus, which encompassed both fractional and integer-order differential and integral equations (Dar et al., 2022). Correspondingly, the FO derivative possesses complex or real arbitrary order, for which various mathematical operators have been proposed. Among those operators, the Grunwald-Letnikov (GL) (Huang, 2016), Liouvill (L) (Huang, 2016), Riemamn-Liouville (RL) (Efe, 2009), and Caputo (C) (Gorenflo and Mainardi, 1997) are most commonly used. Compared with the L, RL, and C operators, the GL operator pays more attention to the numerical calculation of fractional-order differentiation. Since the fractional derivative describes memory and hereditary properties in such an appropriate manner that it demonstrates much advantages in system representation compared to the integer-order models, the GL-based fractional order definition (Huang, 2016) is introduced in this study:

where, G means the GL based fractional calculus, nh = t−a, if q < 0, the Equation (1) is the G-L based fractional integral definition; on the contrary, if q>0, the Equation (1) is the G-L based differential definition.

The main and commonly used properties of fractional derivatives are given as follows:

holds under some reasonable constraints on the function f(t). In particular, there is

In 2009, the deterministic learning (DL) theory was proposed for the problem of learning in uncertain dynamic environments (Wang et al., 2009). It mainly focuses on the dynamic process of knowledge learning, representation, and utilization in unknown dynamic environment. With deterministic learning, fundamental knowledge on system dynamics can be accumulated, stored, and represented by constant RBF networks in a deterministic manner. Moreover, in a scenario whereby an adaptive neural network (NN) controller achieves tracking of a periodic or periodic-like reference orbit, the deterministic learning mechanism is shown capable of achieving closed-loop identification of partial system dynamics during tracking control.

In detail, for any unknown continuous non-linear function f(X): Ω_X→R with recurrent system trajectories ψ(x₀), in which ΩX⊂Rq is a compact set, an ideal constant weight vector W^* of the RBF networks exists, that is, f(X)=W*Tϕ(X)+ε*,∀X∈ΩX, where ε^*>0 is the approximation error and X∈ΩX⊂Rq denotes the input vector of the radial basic function networks (RBFNs), W*=[w1*,⋯,wn*]T∈RN is the ideal RBFNs weight with N being the number of neurons. ϕ(X)=[φ1(||X-c1||),⋯,φn(||X-cn||)]T is the regression vector of RBFs with φ_i(·) being one of the radial basic function, and c_i is the center of neurons distributed in the input space. For the radial basic function, the Gaussian function is one of the most commonly used kernel RBFs given as φi(||X-ci||)=exp[-(X-ci)T(X-ci)ηi2], in which η_i is the adjacent width of the radial base kernel. It satisfies the Schoenberg theorem (Schoenberg, 1938) and is localized basis function in the sense that φ_i(||X−c_i||) → 0 as ||X|| → ∞. All these properties of the Gaussian function provide a rich source of RBFs that are suitable for interpolation of data in Euclidean spaces. The conditional non-singularity property is essential in proving the partial persistent excitation (PE) condition of RBF networks, which is the key to the accurate identification ability for the deterministic learning theory.

With the development of neuroscience, various differential models have been proposed for describing the neuron system, including the Hodgin-Huxley (HH) model, the FitzHugh-Nagumo (FHN) model, the Hindmarsh-Rose (HR) model, and the Ermentrout neuronal model. Among those models, the HR model, possessing the simplest system form, can accurately describe the signal transmission process across the nerve fiber membrane. Thus, the HR model is commonly used for neuron dynamic describing and analysis. The classical three-variable HR neuronal model can be described by the following equations:

where x is the membrane potential, y is the recovery variable standing for the gating dynamics of the potassium (K⁺) channel, and z represents the adaptation current corresponding to the dynamics of calcium (Ca²⁺) channel. Moreover, the model parameters a, b, c, r, and s₀ are positive constants, while the parameter q₀ stands for the resting potential, and I represents the external stimulation input.

The FO differential model has more advantages in neuronal dynamic description compared to that of the integer-order model. In addition, the FO system has a wider stability region. Thus, in this study, the following fractional order HR (FOHR) neuronal model is introduced, that is,

in which, the operator Dtq represents the GL fractional derivative as shown in Equation (1).

The state variables and model parameters of the FOHR model possess the same physical meaning with the integer-order HR model. Through bifurcation analysis under different values of the external stimulation input I and fractional order q, the FOHR model demonstrates a wealth of dynamic behaviors, such as the subthreshold oscillations, spiking, bursting as well as chaotic behaviors.

In detail, when taking the fractional order q = 1, the FOHR model degenerates to an integer-order HR model. By setting q = 1 and the corresponding system parameters a = 1.0, b = 3.0, c = 1.0, d = 6.0, r = 0.013, s = 4.0, and q₀ = −1.56, diverse non-linear dynamics under different external stimulus I of the HR model are generated. By changing the control parameter I, the membrane potential x presents different state characteristics, which can be seen from Figure 2, in which the initial system state (x₀, y₀, z₀) is set as (0.1, 1.0, 0.2).

Precisely, when setting I = 1.5, the neuron produces slow-peak regular spiking (single-cycle spiking) state as given in Figure 2A. Gradually increasing I to 1.8, 2.3, 2.8, the HR system exhibits regular bursting state, shown as the period-2, period-3, and period-4 bursting behaviors, respectively, which can be seen from Figures 2B–D. When I increased to 3.2, the state x of the HR system becomes chaotic as shown in Figure 2E. Further increasing I to 3.58, the system regresses to a fast single-cycle spiking state as demonstrated in Figure 2F, in which the period interval is significantly shorter and the rate of the dynamic activity is much faster than that of the interval demonstrate in Figure 2A.

Except for the non-linear behavior of the time response of the membrane potential x, the inter-spike interval (ISI) (Rabinovich and Abarbanel, 1998) is one of most commonly used physiological indicators, which carries important information of neuronal firing. In the following discussion, the bifurcation diagram of the peak membrane potential x_max and the ISI sequence of the FOHR neural system with different bifurcation parameters are considered.

First, the dynamic non-linearity of the FOHR model under different fractional orders q with the external excitation I = 3 is considered. As shown in Figure 3A, the bifurcation diagram of the ISI sequence exhibits a comb-shaped region with the increase of fractional order q. Correspondingly, the bifurcation diagram of the peak of the membrane potential x (denoted as x_max) demonstrates that the discharge characteristics of the system varies more obviously according to the fractional order. That is, with the increase of the fractional order q within a certain range, the system as a whole shows the tendency of periodic decline, and the periodic bursting phenomenon occurs as demonstrated in Figure 3B. In other words, the firing behavior of neurons becomes more complex and unstable with the increase of the fractional order of the neuronal system, exhibiting richer dynamic activity characteristics.

Second, take the external excitation I as the control parameter for analyzing the dynamic behaviors of the FOHR model with a certain fractional order. The parameter I is taken within the interval [1.2, 4.3] and r = 0.013.

The simulation results of the bifurcation diagram of the ISI sequence (shown in Figure 4) exhibit that the dynamic characteristics of the FOHR system become more complex with the increase of I. Taking the integer order as an example (given in Figure 4A), the ISI sequence experiences the process of period-2 bifurcation to chaos and then back to single period by the period-doubling bifurcation process. Correspondingly, the ISI sequence of the 0.98-order FOHR model indicates similar discharge behaviors with that of the integer-order model, but the chaos duration is reduced. In addition, from the amplitude of the ISI sequence, it can be seen that the effect of external stimulus current on the system dynamics was much obvious.

According to the bifurcation diagram of the x_max sequence shown in Figure 5, the dynamical behaviors correspond to the same fractional order has similar and abundant dynamic characteristics with that of the ISI sequence. In addition, some hidden information contained in the integer order can be clearly demonstrated in the 0.98-order HR model as shown in Figure 5B, such as the period-4 cluster bursting under I = 3.0, the period-5 cluster discharge when I = 3.3, the comb-shaped region and the chaotic region. If a further decrease in the fractional order q to 0.96 and 0.95 as can be seen from Figures 5C, D, the dynamic structure of the FOHR system changes qualitatively. Precisely, with the increase of parameter I, the dynamic behavior of the FOHR model becomes more complex. The structure and stability of the system is influenced correspondingly.

To further analyze the dynamic characteristics of the FOHR model, another important parameter r which relates to the calcium (Ca²⁺) concentration and significant to many neurological disorders, is considered as the control parameter in this part. All the other parameters are kept as the same as mentioned above, while parameter I is fixed to 3.5. When ranging the parameter r from 0.0015 to 0.06, a variety of dynamic behaviors of the FOHR neuron system are presented. As shown in Figures 6, 7, the bifurcation diagram of the ISI sequence and the x_max sequence demonstrate similar non-linear characteristics. Moreover, compared to the integer-order HR model, the 0.98-order HR model presents a more detailed and clear dynamic process.

In conclusion, the dynamic simulations given above suggest that compared to the integer-order HR model, the fractional-order HR model can describe the numerous computational features and the non-linear dynamics of the neuron model more accurately, which help enrich the functional neuron mechanisms and further ensures more accurate dynamic analysis. Thus, it is necessary and important to introduce the FOHR model, and the FOHR model with fractional order q = 0.98 is taken into consideration in the following study.

In this sub-section, the model-based sliding mode control problem of two FOHR neuronal models is considered. The two neurons interconnect in a master-slave configuration. The master FOHR neuronal model is given as follows:

and the slave FOHR neuronal model under control is denoted by

where d_i, i = 1, 2, 3 represents the bounded unknown external disturbance; that is, |di|≤dī,i=1,2,3 and the terms u_i and i = 1, 2, 3 denote the control inputs of the state variables.

For the convenience of discussion, the simplified master-slave neuron system models are presented as follows:

where xm=[xm,1,xm,2,xm,3]T and xs=[xs,1,xs,2,xs,3]T are the state vectors of the master and slave neuronal system, respectively. f_m represents the known system dynamics vectors of the master FOHR model. Correspondingly, f_s represents the unknown system dynamics vectors of the slave FOHR model. Precisely, f_s is smooth, but unknown non-linear dynamics of the slave system. d_i and u_i have the same meaning as the formula given in Equation (13). The main task in this part is to realize the synchronization of the master-slave system with proper amount of calculation and correct the synchronization error by adjusting the parameters.

The synchronization of the master-slave neuronal system is to drive the slave neuron system to track the state as well as the trajectory of the master system under certain external disturbance in unknown dynamic environment by properly designed controller. In order to achieve ideal stability effect of the control system, the gain parameters of the traditional sliding mode control algorithm are usually set too large, which leads to serious chattering problem. In this part, the obtained system dynamics W̄i stored in the pattern base χ is applied for the sliding mode control to achieve fast synchronization performance for the master-slave neuron system. In addition, the accurate modeling of the system dynamics help reduce the synchronization error of the master-slave system without large gain, thus reducing the chattering caused by sliding mode gain.

The synchronization error of the master-slave FOHR system is defined as follows;

where i = 1, 2, 3. To achieve fast synchronization of the master-slave FOHR system, the identified and stored model-based sliding mode control method is proposed. First, the fractional order proportional integral sliding surface is designed as follows:

where s_i, (i = 1, 2, 3) is the fractional order proportional integral sliding surface. The derivative of the sliding mode surface can be achieved according to the properties of fractional order models discussed in Section 3.2, that is,

where the corresponding constant rate of convergence is designed as

The following sliding mode control rate is designed according to the Equations (17) and (18)

For the unknown system dynamics f_m,i(x) of the slave system, the rapid recognition process is introduced, that is,

in which x̄ik represents the state of the dynamic model and the corresponding dynamic information of the system has been identified and stored in the pattern base χ as mentioned above, x_i is the ith state of the unknown slave system, and b_i>0 is a design parameter.

For the unknown slave FOHR system, the recognition error system is given as

where x~ik=x̄ik-xi is the state tracking error between the empirical pattern stored in the base and the unknown slave system.

Commonly, without identifying the unknown dynamics of the unknown slave FOHR system, the differences between the dynamic systems stored in the pattern base and the slave pattern denoted as |W̄ikTφi(x)-fs,i(x)| shown in Equation (21) is unavailable for direct computation. However, as presented in Wang et al. (2009), the state tracking error |x~ik| can be explicitly measured.

For any unknown slave FOHR system with regression system trajectory φ(x_d0), the tracking error |x~ik| can be achieved within finite time by properly selecting the design parameters; that is, by introducing the average L₁-norm based dynamic similarity measure, that is given as

where T>0 is a design parameter, and the difference between system dynamics can be explicitly measured. Based on the similarity measure, the smallest tracking error between certain unknown slave system and the system identified as well as stored in the pattern base χ can be obtained, that is

in which m denotes the number of models stored in the pattern base χ.

Remark 2: According to the recognition process discussed above, the dynamic differences between the slave system and those systems stored in the pattern base can be accurately measured without identifying of the dynamic information of the slave system. This process is therefore referred to as rapid recognition. In particular, the most similar dynamic pattern χk0 can be selected from the pattern base according to the minimum recognition error, and the dynamic information of the selected model denoted as W̄ik0T can be used to replace the unknown dynamics f_s,i(x) of the slave FOHR model in the following control process.

Based on the recognition process, the unknown system dynamics f_s,i(x) of the slave system can be locally accurately identified as well as stored by the constant weight NNs along the system trajectory, that is,

Substituting Equation (24) into Equation (19), the following control rate is obtained:

where W̄ik0Tφi(x) denotes the most similar dynamic model recognized from the pattern base to the unknown slave system by using the localized RBFNNs located close to the system trajectory.

Remark 3: The mode-based sliding mode control is designed to fit the unknown dynamics of the slave system quickly by calling the acquired dynamic information of the neurons, and the experience is applied to the control process. During this process, the generalization ability of the rapid recognition mechanism based on deterministic learning provides the right decisions for invoking right dynamic patterns for better control performance. Put it another way, the empirical dynamic information learned and stored in the pattern base is so sufficiently utilized that the on-line control time is reduced and the fast synchronization is achieved. Compared with the traditional sliding mode control method, the model based sliding mode control algorithm can effectively reduce the sliding mode gain so as to reduce the chattering problem of the system.

To verify the stability of the master-slave synchronization control system, consider the following Lyapunov function candidate:

The derivative of V is

By taking the differential equation of the sliding surface given in Equation (17) and the sliding mode rate given in Equation (25) to Equation (27), we have

as shown in Equation (28), the external disturbance d_i and the identification error ε_i2 have an upper bound. Therefore, to ensure that the function V_i is negative definite, just need to set appropriate sliding mode gain η_i to make the equation η_i > ε_i2 + d_i work, which will further ensure the convergence of synchronization error.

As discussed above, the robustness and generalization ability of the recognition system are greatly related to the richness of the patterns in the dynamic pattern database. When considering the condition that there is no ideal similar dynamic pattern in the pattern base for the unknown slave system, that is,even if the smallest tracking error exists, the corresponding constant system dynamics W̄ik0Tφi(x) utilized in the control rate may result in large synchronization error and affects the stability and convergence of the control process. This analysis suggests that it is necessary to further explore how to improve the synchronization effect under limited off-line pattern base.

In order to solve the above problems to ensure a stable and rapid control effect, further identification of the unknown slave system is considered. Based on the selected dynamics W̄ik0Tφi(x) according to the smallest recognition error, the improved control rate is proposed below:

Based on the Equation (29) and Equation (17), the time derivative of the sliding mode variable is given as

where -W~iTφi(x)=fs,i(x)-ŴiTφi(x), |d_i(t)| ≤ D_i, and |εi|≤ε̄ are external excitation and identification error with upper bound, respectively. For convenience of presentation, define Di+ε̄i=κi, the derivation of the sliding mode surface is given as follows:

In addition, the NN adaptive update law of the sliding mode control is designed as

where Γ_i and σ_i are positive adjustable parameters. Since κi=Di+ε̄, the synchronization error is precisely related to the identification accuracy; that is, the higher the identification accuracy of the unknown slave system, the better the synchronization effect of the master-slave neuronal system.

Theorem 1 Consider the master-slaver neuron FOHR system as shown in Equation (14), the learning-based controller Equation (29), and the NN weight updating law Equation (32). For initial condition x_d(0) which generates the recurrent orbit φ_d(x₀), and with corresponding initial condition x(0) selected in a close vicinity of the recurrent orbit, the control error of the master-slave system described by Equation (15) converges exponentially to a small neighborhood around zero.

Proof: For the sliding mode-based control system, consider the following Lyapunov function:

The derivative of V is

By introducing the designed sliding mode surface and the adaptive update rate equation, there is,

in which,

with W̄̄i being the upper bound of the ideal identification NN weight Wi*. Thus, it follows that

It is clear that V. is negative definite when the following conditions are met:

Since the ideal identification NN weight Wi*, the external excitation d_i and the estimate error ε_i are all upper bounded; therefore, all signals in a closed-loop control system remain bounded, including the estimate NN weight Ŵ_i and the sliding mode variable s_i.

In addition, to the convergence of the sliding mode variable, the following Lyapunov function is given as

The corresponding derivative is given as

Considering that the Gauss function φ_i(x) and -W~iTφi(x)+κi are both bounded, when the gain η_i satisfies the condition that ηi>-W~iTφi(x)+κi, there is

where γi=ηi+W~iTφi(x)-κi. As long as the parameter η_i is reasonably designed, the convergence of the tracking error is ensured, and the sliding mode variable do converge to some neighborhood of zero. In addition, the size of the convergence neighborhood depends on the control parameter; that is, by properly design the control parameters, ideal synchronization control performance can be achieved.

Remark 4: According to the relearning-based sliding mode control algorithm given above, if there is no dynamic pattern that is sufficiently similar to the unknown slave system in the pattern base χ, the on-line identification process for the unknown slave system is started. Different from the initial identification process, the initial weights of the neural network during the identification process for unfamiliar synchronization objects are taken from the constant weight of the dynamic system corresponding to the minimum recognition error rather than iterating from zero. Thus, the learned and stored dynamic information help reduce the on-line identification time. Additionally, the identified dynamic information of the slave system will be restored in the form of constant weights and can further utilized to new synchronization problems. This process will help enrich the empirical dynamics information of the pattern base to improve the accuracy and efficiency of the new synchronization tasks.

In this part, the identification of the FOHR system shown in Equation (6) under unknown dynamic environment is considered. For the convenience of presentation, the system state x, y, and z are denoted as x₁, x₂, and x₃, respectively. The corresponding state vector x=[x1,x2,x3]T∈R3 of the FOHR model is available from measurement and the parameter μ=[a,b,c,d,r,s0,q0]T is taken as a constant vector and chosen as a = 1, b = 3, c = 1, d = 6, r = 0.013, s₀ = 4, q₀ = −1.56. As demonstrated in Section 3, by varying the fractional order parameter q and fixing all the other parameters unchanged, the FOHR system presents diverse non-linear behaviors. Moreover, the 0.98-order HR model can best describe the abundant non-linear dynamic characteristic of neurons. Thus, the 0.98-order HR system is considered for dynamic identification with the external excitation I being taken as the control parameter.

To verify the identification effects, four kinds of representative discharge models of the 0.98-order HR system with the parameters given above are chosen, that is, the slow-spiking model χ¹ with I = 1.5, the period-3 bursting model χ² with I = 2.5, the chaotic bursting model χ³ with I = 3.6 and the fast-spiking model χ⁴ with I = 4. The dynamic analysis about FOHR system have demonstrated that the corresponding state trajectories of the four dynamic models mentioned above possess regression properties. Thus, the DL algorithm is introduced for the unknown dynamic identification process.

According to the DL algorithm, the dynamical RBF network x^.=-A(x^-x)+Ŵϕ(x) is employed to identify the unknown system dynamics f_i(x; μ)(i = 1, 2, 3) as shown in Equation (6). For the space limitation, the unknown dynamic f₃(x; μ) = r(s₀(x−q₀)−z) is taken as an example to show the identification effects. The center of the neural network is evenly placed on [−2.1, 2.1] × [0.9, 5.1] and the widths are set as η_i = 0.3. The weights of the RBF networks are updated online according the equation Ŵ.i=W~.i=-Γiφi(x)x~i-σiΓiŴi, within which the parameters are chosen as Γ_i = diag{2, 2, 2}, σ_i = 0.0001, i = 1, 2, 3 and a₃ from A=[a1,a2,a3]T is set as a₃ = 10. The initial condition of the dynamical system is set as [x1(0),x2(0),x3(0)]T=[0.3,1,3]T, [x^1(0),x^2(0),x^3(0)]T=[0.2,0.3,0.0]T, and the initial weights are Ŵ_i(0) = 0.0.

First, the 0.98-order HR system with external excitation I = 1.5 denoted in a slow-spiking model as χ¹ is to be identified. Figure 8A is the projection of the state trajectory of the slow-spiking model on the x−z plane. In Figures 8B, C, it is seen that the state trajectory can be accurately identified by using the DL algorithm. More importantly, in addition to the state tracking, the NN approximation of the system dynamics f₃(x; μ) along the system trajectory is shown in Figure 8C. The convergence of the weights of the RBF neural network is further obtained from the Figure 8D. That is, by introducing the DL algorithm, the unknown dynamic information f₃(x; μ) of the FOHR model is locally accurately approximated by Ŵ_iφ_i(x), and the identified non-linear dynamic information can be further stored in the constant weights of networks given as W̄iφi(x).

Second, similar results are obtained for the identification of the non-linear dynamics of the 0.98-order HR system with I = 2.5 that exhibiting a period-3 bursting model denoted as χ². It can be seen from the Figure 9A that the non-linear dynamics of the period-3 bursting model are richer than that of the dynamics presented in Figure 8A. Even though, ideal approximation effects of both the system state and the unknown system function are obtained as demonstrated in Figures 9B, C. The parameters of the corresponding RBF networks also converge to an ideal value, which can be seen from Figure 9D.

Third, consider the identification of the dynamics of model χ³ with I = 3.6, as shown in Figure 10. The system state given in Figure 10A presents a complex state of chaos, which contains more dynamic information of the FOHR system. By properly designing the identification parameters, locally accurate NN approximations of the system state as well as the unknown system dynamics are achieved along the system trajectory, which can be seen from Figures 10B, C. In addition, it is noticed from the Figure 10D that more neurons are involved and activated in the identification of the chaotic bursting model χ³. Moreover, the oscillation of the NN weights during the convergence process is so obvious that more time is needed for it converge to the ideal values.

Finally, a further increase in the external excitation I to 4 (denoted as model χ⁴), the system returns back to a simple discharge state. As can be seen from Figure 11A, the state trajectory of model χ⁴ is a typical period-1 behavior, but the discharge rhythm is faster compared to that of the model χ¹ shown in Figure 8A. As for its identification simulations demonstrated in Figures 11B–D, it is shown that it achieves better state and dynamic tracking effects, and the parameter convergence process is much smooth and faster.

Based on the acquisition and storage of the unknown dynamic information of the FOHR system, the rapid recognition of the FOHR model is demonstrated in this part. The dynamic models χ^{1, 2, 3, 4} mentioned above are taken as the training patterns. The testing patterns are generated from the FOHR system presented in Equation (7), with I = 1.43 denoted as χ⁵, I = 2.3 denoted as χ⁶, I = 3.4 denoted as χ⁷, and I = 4.2 denoted as χ⁸. The other parameters are set as the same to the training patterns, that is, q = 0.98, a = 1, b = 3, c = 1, d = 6, r = 0.013, s = 4, and q₀ = −1.56. For the recognition process, the dynamic NN network system is introduced, that is,

for which, the initial states is given as [x0,y0,z0]T=[0.3,1,3]T and [x~0,ỹ0,z~0]T=[0,0,0]T.

Based on the obtained dynamic pattern database χ, which contains the learned system dynamics as experience of the slaw peak regular spiking model, period-doubling, period-3, period-4 bursting model, and chaotic bursting model, the simulation of the learning-based sliding-mode control of the master-slave neural system is discussed in this part. The corresponding parameters are given as η_i = 1, c_i = 1, Γ_i = 2, and σ_i = 0.01, (i = 1, ⋯ , n). The external disturbances are set as d₁(t) = 0.6 + 0.2cos(t), d₂(t) = 0.0, d₃(t) = 0.01 + 0.05sin(t), and the other parameters of the master-slave system are given as the same as shown in the previous section. The external stimulus current of the master system is set as I = 1.5, while for the slave system, the external stimulus current is set as I = 3.8. The other parameters are designed as q = 0.98, a = 1, b = 3, c = 1, d = 6, r = 0.013, s = 4, and q₀ = −1.56, the initial state of the master-slave system is given as [x0,y0,z0]T=[0.313]T, and the control will be added at t = 300ms.

As can be seen from the Figures 12A, B, when the control quantity is added to the slave system at t = 300ms, the state of the master-slave neurons can quickly reach consistency, and the selected NN controller achieves good synchronization to the master neuron system. Moreover, the synchronization error demonstrated in Figures 12B–D shows that the FOHR master-slave neuronal system achieves fast synchronization performance.

Since the external excitation of the master and slave system are set as I = 1.5 and I = 3.8, respectively, it means that the master system is in slow-spiking state and the slave system is in a state of rapid-peak spiking, as described in the identification phase. For accurate synchronization effect, the rapid-peak spiking model shown be recalled from the pattern base χ⁴, which can be validated from the Figure 13.

In addition, through the simulation comparison by recalling the rapid-peak spiking model, the ideal known dynamic model corresponding to the slave system and the slow-spiking model, respectively, the synchronization errors are shown in Figure 14. It demonstrate that in terms of convergence speed, accuracy, and buffeting size, the more accurate the dynamic model is selected, the better the synchronization effect will be. It further indicates that the performance of the sliding-mode control algorithm is highly related to the dynamic information accuracy of the invoked dynamic models.

To verify the effectiveness of the relearning-based sliding-mode control performance to the master-slave neuronal system, the third dimension dynamics of the neuron system is taken as an example, and the sliding-mode controller is set as

in which the initial NN weight is set as Ŵ3(0)=W̄3, with W̄3 being the constant NN weight. During the control process, the model-based sliding mode controller is added to the system at 300 ms and at 500 ms switch to the relearning-based sliding mode controller. The synchronous response of the system can be seen from Figure 15A. Furthermore, it can be seen from Figure 15A, when the system switch to the relearning-based sliding mode control policy, the synchronization error is getting smaller because of more accurate identification of the dynamics of slave system, and the cusp error is obviously improved. In addition, the NN weight of the relearning process can convergence to ideal values as shown in Figure 15B.