Appendix A Original model transformation

Let us perform some transformations on Eq. (1) to design estimator that will satisfies imposed strict assumptions. First of all, we sum equations of every neuron in Eq. (1) with each other:

$$\begin \begin \sum \limits _^\dot_k = & \sum \limits _^\left( u_k - \frac - v_k + I_\right) \\ & + \sum \limits _^ U_k, \end\\ \sum \limits _^ \dot_k = \sum \limits _^\epsilon (u_k - a - bv_k) + \sum \limits _^ V_k. \end\right. } \end$$

(A1)

Here last coupling terms in right hand sides are equals to zero because \(A_ = A_\) for each \(k,\,j \in 1:\mathrm N\). Following this, based on the first assumption it is assumed that \(y_k=cu_k\), where \(c> 0\) is an a priori unknown scaling factor enabling to take into account systematic error occurring during the membrane potentials measurement and \(y_k\) for \(k \in 1:\mathrm N\) are new variables. Next this error c will be estimated with other unknown parameters.

$$\begin \sum \limits _^\dot_k = \sum \limits _^\left( y_k - \frac - cv_k + cI_\right) \\ \sum \limits _^ \dot_k = \sum \limits _^\epsilon (\frac - a - bv_k). \end\right. } \end$$

(A2)

Then, we have to get rid of unmeasured variables \(v_k\). For this purpose a standard transformation of a first order system of differential equations to one equation of higher order is performed: we differentiate the first equation of Eq. (A2) and then substitute the right hand side of second equation instead of \(\sum _^ \dot_k\). After that we express \(\sum _^ v_k\) from the first equation and substitute it into resulting second order equation:

$$\begin \begin \sum \limits _^\ddot_k =&\sum \limits _^\bigg [(1 - \epsilon b)\dot_k - \frac(3y_k^2\dot_k) \\ &+ \epsilon (b - 1)y_k - \fracy_k^3\bigg ] \\ &+ \mathrm Nc\epsilon (a + bI_). \end \end$$

(A3)

Appendix B Identification using the speed-gradient method based on the integral objective function

Let us consider a general state space nonlinear system model:

$$\begin \varvec} = \varvec(\varvec, \varvec^*, t), \end$$

(A4)

where \(\varvec(t) \in \mathbb ^n\) is a state vector, \(\varvec^* \in \mathbb ^m\) is a vector of true (but unknown) parameter values, \(t \geqslant 0\).

Assume that \(y^*(t) = y^*(\varvec(t)) \in \mathbb \), \(\varvec(t) = \varvec(\varvec(t)) \in \mathbb ^m\) is the observable output of the system Eq. (B4), for which the following equation holds:

$$\begin y^*(t) = \varvec^}\varvec(t). \end$$

(A5)

In other words, the system Eq. (B4) can be presented in a linear regression form Eq. (B5).

The problem of system Eq. (B4) identification is mathematically formulated in terms of Eq. (B5): build an adaptive system with the estimate of the output variable of Eq. (B4), \(y(t) \in \mathbb \), as the output variable and estimates of parameters of Eq. (B4), \(\varvec(t) \in \mathbb ^m\), as the parameters providing the identification goal:

$$\begin \begin 1)&\,\,y(t) - y^*(t) \rightarrow 0 \,\, \text\,\, t \rightarrow \infty ,\\ 2)&\,\,\varvec(t) - \varvec^* \rightarrow 0 \,\, \text\,\, t \rightarrow \infty . \end \end$$

(A6)

To solve the problem, an adaptive system structure is chosen as follows:

$$\begin y(t) = \varvec(t)^}\varvec(t). \end$$

(A7)

In order to tune the vector \(\varvec(t)\) of parameter estimates the speed-gradient method based on the following integral objective function, \(Q_t\), (Fradkov et al., 2013) is used:

$$\begin Q_t = \int \limits _0^t \frac \delta ^2(\varvec(s), \varvec(s)) ds, \end$$

(A8)

where \(\delta (\varvec(t), \varvec(t)) = y(t) - y^*(t) = (\varvec(t) - \varvec^*)^}\varvec(t)\). In this case the speed-gradient algorithm looks as follows:

$$\begin \varvec} = -\varvec\delta (\varvec, \varvec)\varvec(\varvec), \end$$

(A9)

where \(\varvec\in \mathbb ^\) is a symmetric positive-definite gain matrix.

Theorem 2

Consider the system Eq. (B4) with the algorithm of unknown parameters estimation Eq. (11) obtained with the speed-gradient method based on the integral objective function Eq. (B8). Assume that the following conditions are met:

1. Vector functions \(\varvec(\varvec, \varvec^*, t),\,\varvec(\varvec)\) are defined and continuous with their partial derivatives with respect to components of the vector \(\varvec\).

2. \(\varvec(\varvec(t), \varvec^*, t),\,\varvec(\varvec(t))\) are bounded for \(0\leqslant t<\infty\).

3. \(\varvec(t)\) satisfies the so called persistent excitation (PE) condition, i.e. there exist positive numbers \(L, \alpha , t_0\) such that the following inequality holds for every \(t> t_0\):

$$\begin \varvec_L := \int \limits _t^\varvec(s)\varvec(s)^}ds \geqslant \alpha \varvec_m. \end$$

(A10)

Then, the identification goal Eq. (B6) in the system Eqs. (B4), (11) is achieved.

Proof

The fulfillment of the first condition ensures the existence and uniqueness of the system Eqs. (B4), (11) solution for any bounded initial conditions (Boyce et al., 1969).

Let us define the Lyapunov function as \(V_t\) = \(V(\varvec(t))\) = \(Q_t\) + \(\frac \Vert \varvec(t) - \varvec^*\Vert ^2_^}\) = \(Q_t\) + \(\frac (\varvec(t) - \varvec^*)^\varvec^\) \((\varvec(t) - \varvec^*)\). Its derivative with the respect to the system Eqs. (B4), (11): \(\dot_t\) = \(\frac \delta ^2(t)\) + \((\varvec(t) - \varvec^*)^\varvec^ \varvec}(t)\) = \(\frac \delta ^2(t)\) – \(\delta (t) (\varvec(t) - \varvec^*)^ \varvec(t)\) = –\(\frac\delta ^2(t) \leqslant 0\). Therefore, firstly, the equilibrium point of the system Eq. (11) \(\varvec^*\) is Lyapunov stable, secondly, \(\frac \Vert \varvec(t) - \varvec^*\Vert ^2_^} \leqslant V_t \leqslant V_0 = \frac \Vert \varvec(0) - \varvec^*\Vert ^2_^}\) and \(Q_t \leqslant V_0\), thus, \(\varvec(t)\) is bounded and the finite limit \(\lim \limits _\int \limits ^t_0 \delta ^2(s)ds\) exists.

The expression \(\frac\delta ^2(\varvec(t), \varvec(t))\) = \(2\delta (\varvec(t), \varvec(t))\dot(\varvec(t), \varvec(t))\) = \(2(\varvec(t) - \varvec^*)\varvec(\varvec(t))(\varvec}(t)\varvec(\varvec(t))\) + \((\varvec(t) - \varvec^*)\varvec}(\varvec(t)))\) is bounded because of the second condition and boundedness of \(\varvec(t)\). Therefore, the function \(\delta ^2(t)\) is uniformly continuous.

Thus, by the Barbalat lemma (Barbălat, 1959), \(\delta ^2(t) \rightarrow 0\) for \(t \rightarrow \infty\). It means that the first part of the identification goal Eq. (B6) is achieved: \(y(t) - y^*(t) \rightarrow 0 \,\, \text\,\, t \rightarrow \infty\).

Let us use a lemma from Rybalko and Fradkov (2023) to prove the second identification goal Eq. (B6) achievement.

Proposition 3

Consider the smooth vector function \(\varvec(t) \in R^\) and the PE function \(\varvec(t) \in R^\), which is defined on \([0, \infty )\). If \(\varvec}(t) \rightarrow 0\) and \(\varvec(t)^T\varvec(t) \rightarrow 0\) for \(t \rightarrow \infty\), then

$$\begin \lim _\varvec(t)=0. \end$$

(A11)

\(\varvec(t)=(\varvec_1(t)-\varvec_1^* \quad \dots \quad \varvec_5(t)-\varvec_5^*)^}\), z(t) = z(x(t)) in our case. We have that \(\varvec(t)^}\varvec(t) = \delta (t) \rightarrow 0\) for \(t\rightarrow \infty\); \(\varvec}(t) = \varvec}(t) = - \varvec\delta (t)\varvec(t) \rightarrow 0\) for \(t\rightarrow \infty\) because of the second condition. Thus, Proposition 3 conditions are fulfilled and, therefore, \(\varvec(t) - \varvec^* \rightarrow 0\) for \(t \rightarrow \infty\). \(\square\)

Appendix C Proof of Theorem 1

Proof of Theorem 1 is reduced to the check of the first two conditions of Theorem 2 for the system Eqs. (1), (6). First of all, let us transform the FHN model to the general form Eq. (B4):

$$\begin \begin \dot_1 = & \;y_1 + \theta _2^*y_1^3 -\frac}v_1+\frac}} \\ & + Y_1', \end\\ \begin \dot_1 = & \;(1-\theta _1^*-\theta _3^*)\sqrty_1 - \frac} \\ & +(1 - \theta _1)(I_ - v_1) + V_1', \end\\ \ldots \\ \begin \dot_ = & \;y_ + \theta _2^*y_^3 -\frac}v_ + \frac}} \\ & + Y_', \end\\ \begin \dot_ = & \;(1-\theta _1^*-\theta _3^*)\sqrty_ - \frac} \\ & +(1 - \theta _1)(I_ - v_1) + V_', \end \end\right. } \end$$

(A12)

where

 \(Y_k'\) = \(\sigma\sum_^A_\) \(\left[B_(y_j-y_k)+\frac}}(v_j-v_k)\right]\), \(V_k'\) = \(\sigma\sum_^A_\) \(\left[\sqrtB_(y_j-y_k)+B_(v_j-v_k)\right]\), \(k \in 1:\mathrm N\).

The first condition of Theorem 2 is obviously met. Taking into account Remark 2 and the structure of Eq. (C12), fulfilment of the second condition is followed from boundedness of FHN trajectories.

Proposition 4

If \(\sigma < \varepsilon b/r\) then the solution of the system of \(\mathrm N\) diffusively coupled FHN models is bounded.

Proof

Let us define a non-negative function of \(\varvec = (y_1 \quad y_2 \quad \ldots \quad y_)^\textrm\) and \(\varvec = (v_1 \quad v_2 \quad \ldots \quad v_)^\textrm\) as follows:

$$\begin H(\varvec, \varvec) = \frac\sum _^ (y_k^2 + v_k^2). \end$$

(A13)

Its derivative with respect to the system Eq. (C12) is as follows:

$$\begin \begin&\dot(\varvec, \varvec) = \sum _^\Bigl (-\alpha _2y_k^4 - \alpha _1 v_k^2 + y_k^2\\ &+ c_1y_kv_k +c_2y_k + c_3v_k\Bigl ) + \sigma R(\varvec, \varvec), \end \end$$

(A14)

where

$$\begin \begin&\alpha _2 = -\theta _2^* = c^/3> 0,\\&\alpha _1 = 1 - \theta _1^* = \varepsilon b> 0,\\&c_1 = - (3\theta _2^*(1 - \theta _1^* - \theta _3^*)+1)/\sqrt,\\&c_2 = I_/\sqrt,\\&c_3 = I_\theta _1^*-\theta _5^*\sqrt/\mathrm N. \end \end$$

(A15)

Last three terms in parenthesis in Eq. (C14) can be estimated by standard quadratic inequalities:

$$\begin \begin&c_1y_kv_k \leqslant \fracy_k^2 + \fracv_k^2,\\&c_2y_k \leqslant \fracy_k^2 + \frac,\;c_3v_k \leqslant \fracv_k^2 + \frac, \end \end$$

(A16)

where \(\delta _1\) is an arbitrary positive constant. Similarly for any \(\delta _2> 0\) the following inequality holds:

$$\begin y_k^2 \leqslant \fracy_k^4 + \frac. \end$$

(A17)

Multiplying both sides of Eq. (C17) by \(2\alpha _2/\delta _2> 0\) leads to the following estimate of first term in parenthesis in Eq. (C14):

$$\begin -\alpha _2y_k^4 \leqslant -\fracy_k^2 + \frac. \end$$

(A18)

From Eqs. (12), (C16), (C18) it implies that Eq. (C14) can be estimated as follows:

$$\begin \begin&\dot(\varvec, \varvec) \leqslant -\beta _1^y\sum _^y_k^2 -\beta _1^v\sum _^v_k^2 + \beta _2, \end \end$$

(A19)

where

$$\begin \begin&\beta _1^y = \frac - 1 - \frac - \sigma r - \frac,\\&\beta _1^v = \alpha _1 - \sigma r - \delta _1,\\&\beta _2 = \mathrm N\left( \frac + \frac\right) . \end \end$$

(A20)

By the Lemma condition \(\sigma < \varepsilon b/r = \alpha _1/r\), therefore there are such \(\delta _1> 0\), \(\delta _2> 0\) that \(\beta _1^y\) and \(\beta _1^v\) are positive. Equation (C19) can be rewritten as follows:

$$\begin \dot(\varvec, \varvec) \leqslant -\beta _1H(\varvec, \varvec) + \beta _2, \end$$

(A21)

where \(\beta _1 = \textrm\> 0\). Integrating of the inequality Eq. (C21) leads to the following estimate:

$$\begin H(\varvec(t), \varvec(t)) \leqslant H(\varvec(0), \varvec(0))\textrm^ + \frac. \end$$

(A22)

Therefore, \(H(\varvec, \varvec)\) is a bounded function, which means that the norm of the vector \((\varvec\quad \varvec)^\) is bounded. \(\square\)