Desynchronization of stochastically synchronized neural populations through phase distribution control: a numerical simulation approach

Abstract

Stochastic synchrony, also known as noise-induced synchronization that leads to phase coherence, arises when a set of uncoupled neurons synchronizes to a common white noise input or other types of non-Gaussian noise. Evidence of abnormally high noise-induced synchronization, or impairment in synchronous activity, has been found in several pathologies such as epilepsy. Therefore, controlling the stochastic synchronization of neurons can have a significant effect on preventing seizures in epilepsy. The main aim of this study is to develop a fast and reliable numerical method to simulate controlling synchronization in a population of noisy and uncoupled neural oscillators. The control algorithm is based on phase reduction and uses the probability phase distribution partial differential equation to change the distribution of oscillators to the desired one. The accuracy and power consumption are two main issues that should be considered in the simulations. In this paper, a new numerical method called the generalized Lagrange Jacobi Gauss–Lobatto collocation method in space and backward-Euler scheme in time is applied to overcome the difficulties of the problem effectively. The resulting full-discrete scheme of the partial differential equation is a linear system of algebraic equations per time step which is solved via QR algorithm. Finally, the proposed algorithm is applied to various neural dynamical models with different phase response curves and investigates them in different factors such as computing time, energy consumption, and accuracy to confirm the applicability of the developed numerical method.

Appendices

Useful mathematical definitions and theorems

1.1 Roots of Jacobi polynomials

The Jacobi polynomial $J_n^{\alpha ,\beta }(x)$ has exactly n real roots on $[-1,1]$, and they can be obtained by computing the eigenvalues of the following three-diagonal matrix [92]

$$\begin{aligned} {\mathcal {K}}_n= \begin{bmatrix} \lambda _1 &{} \gamma _2 &{} &{} &{} &{} &{} \\ \gamma _1&{} \lambda _2 &{} \gamma _3 &{} &{} &{} &{}\\ &{} \gamma _3 &{} \lambda _3 &{} \gamma _4 &{} &{} &{}\\ &{} ~~\ddots &{} ~~\ddots &{} ~~\ddots &{} &{} &{}\\ &{} &{} &{} &{} \gamma _{n-1} &{} \lambda _{n-1}&{} \gamma _{n}\\ &{} &{} &{} &{} &{} \gamma _n &{} \lambda _{n} \end{bmatrix}, \end{aligned}$$

(A.1)

where

$$\begin{aligned} \lambda _{i+1}=\frac{(xJ_i^{\alpha ,\beta }(x),J_{i}^{\alpha ,\beta }(x))_{\omega ^{\alpha ,\beta }}}{(J_i^{\alpha ,\beta }(x),J_i^{\alpha ,\beta }(x))_{\omega ^{\alpha ,\beta }}},\nonumber \\ \gamma ^2_{i+1}={\left\{ \begin{array}{ll} 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i=0,\\ \frac{(J_i^{\alpha ,\beta }(x),J_i^{\alpha ,\beta }(x))_{\omega ^{\alpha ,\beta }}}{(J_{i-1}^{\alpha ,\beta }(x),J_{i-1}^{\alpha ,\beta }(x))_{\omega ^{\alpha ,\beta }}}~~~~~~~~~~~i\ge 1 \end{array}\right. }. \end{aligned}$$

(A.2)

1.2 Derivatives of generalized Lagrange Gauss–Lobatto functions

Theorem A.1

[93] Consider generalized Lagrange functions $L^{\phi }_j(x)$, the first-order derivative matrix of generalized Lagrange functions as

$$\begin{aligned} {\mathbf {D}}^{(1)}=\begin{bmatrix} d_{kj} \end{bmatrix}_{0\le j,k\le n}\in \mathrm I\!R ^{(n+1)\times (n+1)}, \end{aligned}$$

(A.3)

where

$$\begin{aligned} d_{kj}=\frac{dL^{\phi }_j(x)}{\mathrm{d}x}\bigg |_{x=x_k}={\left\{ \begin{array}{ll} \nu _j\frac{w'(x)|_{x=x_j}}{p_k-p_j}~~~~~~~~~~~~~~~~~~~~~~~j\ne k\\ \nu _j\frac{p'(x)w''(x)-p''(x)w'(x)}{2(p'(x))^2}|_{x=x_j}~~~j=k \end{array}\right. }.\nonumber \\ \end{aligned}$$

(A.4)

Moreover, if we consider ${\mathbf {J}}^{(k)}=\mathrm{Diag}(\frac{d^k\phi _i(x)}{\mathrm{d}x^k})_{0\le i\le n}$, the second-order derivative matrix of generalized Lagrange functions can be computed as

$$\begin{aligned} {\mathbf {D}}^{(2)}=({\mathbf {J}}^{(2)}+{\mathbf {J}}^{(1)}{\mathbf {D}}^{(1)}) {\mathbf {J}}^{{(1)}^{-1}}{\mathbf {D}}^{(1)}. \end{aligned}$$

(A.5)

Theorem A.2

[93] The derivative matrix of generalized Lagrange Jacobi Gauss–Lobatto functions is expressed as follows

$$\begin{aligned} { {\mathbf {D}}^{(1)}=\begin{bmatrix} d_{kj}^{\alpha ,\beta }\end{bmatrix}_{0\le k,j\le n},} \end{aligned}$$

(A.6)

where

$$\begin{aligned} { d_{kj}^{\alpha ,\beta }= {\left\{ \begin{array}{ll} \frac{(-1)^{1-n}\phi '(x_k)J_{n-2}^{\alpha +2,\beta +2}(\phi (x_k)) \vartheta _{k,n}(n-1)!(\alpha +\beta + n+2)\Gamma (\beta +2)}{2\vartheta _{0,n} \Gamma (\beta +1+n)}~~~~~~~~j=0,~~~~~1\le k\le n-1,\\ \\ \frac{(-1)^{1-n}\phi '(x_n)\Gamma (\alpha +n+1)\Gamma (\beta +2)}{\vartheta _{0,n} \Gamma (\beta +n+1)\Gamma (\alpha +2)}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~j=0, ~~~~~ k=n,\\ \\ \frac{2(-1)^{n-1}\phi '(x_0)\vartheta _{0,n}\Gamma (\beta +n+1)}{\Gamma (\beta +2)(n-1)!(\alpha +\beta +n+2)J_{n-2}^{\alpha +2,\beta +2} (\phi (x_j))\vartheta _{0,n}^2\vartheta _{j,n}}~~~~~~~~~~~~~~~~~k=0,~~~~~~~~~1 \le j\le n-1,\\ \\ \frac{\phi '(x_0)\Gamma (\beta +n+1)\Gamma (\alpha +2)(-1)^{n-1}}{\vartheta _{n,0} \Gamma (\alpha +n+1)\Gamma (\beta +2)},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~k=0, ~~~~~j=n,\\ \\ \frac{\phi '(x_k)J_{n-2}^{\alpha +2,\beta +2}(\phi (x_k))\vartheta _{k,n}(n-1)! (\alpha +\beta +n+2)\Gamma (\alpha +2)}{2\vartheta _{n,0} \Gamma (\alpha +1+n)}~~~~~~~~~~~~~j=n,~~~~~1\le k\le n-1,\\ \\ \frac{\vartheta _{k,0}\vartheta _{k,n}\phi '(x_k)J_{N-2}^{\alpha +2,\beta +2}\phi (x_k)}{\vartheta _{j,0}\vartheta _{j,n} \vartheta _{k,j}J_{N-2}^{\alpha +2,\beta +2}(\phi (x_j))},~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~1\le j\ne k\le n-1,\\ \\ \frac{2\phi '(x_n)\vartheta _{n,0}\Gamma (\alpha +n+1)}{\vartheta _{0,j} \vartheta _{j,n}^2J_{n-2}^{\alpha +2,\beta +2}(\phi (x_j))(n+\alpha +\beta +2) (n-1)!\Gamma (\alpha +2)}~~~~~~~~~~~~~~~~1\le j\le n-1,~~~~~~~~ k=n,\\ \\ \frac{\phi '(x_j)}{\vartheta _{j,0}}+\frac{\phi '(x_j)}{\vartheta _{j,n}} +\phi '(x_j)(\alpha +\beta +n+3)\frac{J_{n-3}^{\alpha +3,\beta +3}(\phi (x_j))}{4 J_{n-2}^{\alpha +2,\beta +2}(\phi (x_j))},~~~~~~~~1\le k=j\le n-1,\\ \\ \frac{\phi ''(x_j)}{2\phi (x_j)}-\frac{\phi '(x_j)}{\vartheta _{n,0}} +\frac{\phi '(x_j)(n+\alpha +\beta +2)J_{n-2}^{\alpha +2,\beta +2}(\phi (x_j))}{2J_{n-1}^{\alpha +1,\beta +1}(\phi (x_j))},~~~~~~~~~~~~~~~~~~~~~~~k=j=0,\\ \\ \frac{\phi ''(x_j)}{2\phi (x_j)}+\frac{\phi '(x_j)}{\vartheta _{n,0}}+\frac{\phi '(x_j)(n+\alpha +\beta +2)J_{n-2}^{\alpha +2,\beta +2}(\phi (x_j))}{2J_{n-1}^{\alpha +1,\beta +1}(\phi (x_j))},~~~~~~~~~~~~~~~~~~~~~~~k=j=n, \end{array}\right. } } \end{aligned}$$

(A.7)

in which

$$\begin{aligned} \vartheta _{i,j}=\phi (x_i)-\phi (x_j). \end{aligned}$$

1.3 Convergence and stability analysis

In this part, we raise some theorems for convergence and stability analysis.

Theorem A.3

[77](Error analysis) Let u(x) be analytic in a region $\Omega$ containing distinct points $\phi (x_0), \ldots , \phi (x_n)$ where $x_j$s are also distinct points. Moreover, let $\Gamma$ be a contour in $\Omega$ enclosing these points in the positive direction. The interpolation function of $u_n(x)$ to u(x) at $x_i$ is

$$\begin{aligned} u_n(x)=\frac{1}{2\pi i}\int _{\Gamma }\frac{\phi '(s)u(s)(w_{\phi }(s)-w_{\phi }(x))}{w_{\phi }(s)(\phi (s)-\phi (x))}ds. \end{aligned}$$

(A.8)

If x is enclosed by $\Gamma$, the error in the interpolant is

$$\begin{aligned} u(x)-u_n(x)=\frac{1}{2\pi i}\int _{\Gamma }\frac{\phi '(s)w_{\phi }(x)u(s)}{w_{\phi }(s)(\phi (x)-\phi (s))}ds.\nonumber \\ \end{aligned}$$

(A.9)

Theorem A.4

Let ${\hat{\rho }}^i(\theta )$ be the best approximation of $\rho ^i(\theta ) \in [0,2\pi ]$ using the generalized Lagrange Jacobi polynomials. So, there is an upper bound of the absolute error $\Vert \rho ^i(\theta )-{\hat{\rho }}^i(\theta )\Vert _{\infty }$.

Proof

Consider the following space of generalized Lagrange Jacobi polynomials

$$\begin{aligned} {\mathcal {L}}_{\phi }^{\alpha ,\beta ,n}=span\lbrace L^{\phi }_j:~~0 \le j\le n\rbrace . \end{aligned}$$

(A.10)

Assume that ${\hat{\rho }}^i(\theta ) \in {\mathcal {L}}_{\phi }^{\alpha ,\beta ,n}$ be the best approximation of $\rho ^i(\theta )$. Afterward, by the definition of the best approximation, we have

$$\begin{aligned} \forall v \in {\mathcal {L}}_{\phi }^{\alpha ,\beta ,n}~~\Vert \rho ^i(\theta )-{\hat{\rho }}^i(\theta ) \Vert _{\infty } \le \Vert \rho ^i(\theta )-v \Vert _{\infty }.\nonumber \\ \end{aligned}$$

(A.11)

It is also true if v be an approximation for the interpolating polynomial for $\rho ^i(\theta )$ at points $x_i$, where $x_i(0 \le i \le n)$ are the roots of $J_{n+1}^{\alpha ,\beta }(\phi (x))$. So, we can write that [94]:

$$\begin{aligned}&\rho ^i(\theta )-v=\frac{\partial ^{n+1}\rho ^i({\hat{\theta }})}{\partial x^{n+1}(n+1)!}\prod _{j=0}^n(x-x_j)\nonumber \\&\quad -\frac{\partial ^{n+1} \rho ^i({\tilde{\theta }})}{\partial x^{n+1}(n+1)!}\prod _{j=0}^n(x-x_j), \end{aligned}$$

(A.12)

where ${\hat{\theta }}, {\tilde{\theta }} \in \Lambda =[0,2\pi ]$. As a result, it is obtained that

$$\begin{aligned}&\Vert \rho ^i(\theta )-v\Vert _{\infty }\le \max _{x \in \Lambda }\bigg \vert \frac{\partial ^{n+1}\rho ^i({\hat{\theta }})}{\partial x^{n+1}}\bigg \vert \frac{\bigg \Vert \prod _{j=0}^n(x-x_j)\bigg \Vert _{\infty }}{(n+1)!}\nonumber \\&\quad +\max _{x \in \Lambda }\bigg \vert \frac{\partial ^{n+1}\rho ^i({\tilde{\theta }})}{\partial x^{n+1}}\bigg \vert \frac{\bigg \Vert \prod _{j=0}^n(x-x_j)\bigg \Vert _{\infty }}{(n+1)!}. \end{aligned}$$

(A.13)

Since $\rho ^i(\theta )$ is a smooth function on $\Lambda$, there exists constant ${\mathcal {C}}$ such that [88]

$$\begin{aligned} \max _{x \in \Lambda }\bigg \vert \frac{\partial ^{n+1}\rho ^i({\tilde{\theta }})}{\partial x^{n+1}}\bigg \vert \le {\mathcal {C}}. \end{aligned}$$

In order to minimize $\bigg \Vert \prod _{j=0}^n(x-x_j)\bigg \Vert _{\infty }$, we use $x=\pi (\eta +1)$ to shift interval $[-1,1]$ to $[0,2\pi ]$. Therefore

$$\begin{aligned}&\min _{x_j \in \Lambda }\max _{x \in \Lambda }\bigg \vert \prod _{j=0}^n(x-x_j)\bigg \vert \nonumber \\&=\min _{\eta _j \in [-1,1]}\max _{\eta \in [-1,1]}\bigg \vert \prod _{j=0}^n\pi (\eta -\eta _j)\bigg \vert \nonumber \\&\quad =\pi ^{n+1}\min _{\eta _j \in [-1,1]}\max _{\eta \in [-1,1]}\bigg \vert \frac{J^{\alpha ,\beta }_{n+1}(\eta )}{\kappa _n^{\alpha ,\beta }} \bigg \vert , \end{aligned}$$

(A.14)

in which $\kappa _n^{\alpha ,\beta }=\frac{\Gamma (2n+\alpha +\beta +1)}{2^nn!\Gamma (n+\alpha +\beta +1)}$ and $\eta _j$ are the leading coefficients and the roots of $J_{n+1}^{\alpha ,\beta }(\eta )$, respectively. There exists a positive constant K such that Jacobi polynomials satisfy the following equation [95]

$$\begin{aligned} \max _{\eta \in [-1,1]}\vert J_{n+1}^{\alpha ,\beta }(\eta )\vert \le K(1+n)^\mu , \end{aligned}$$

(A.15)

where $\mu =\max (\alpha , \beta , -0.5)$, and reach the maximum of their absolute value on the interval $[-1, 1]$, at $\eta = -1$ provided that $\beta \le \alpha$ and $-0.5 \le \alpha$ [94].

$$\begin{aligned} \max _{\eta \in [-1,1]}\vert J_{n+1}^{\alpha ,\beta }(\eta )\vert =\frac{\Gamma (n+\alpha +2)}{(n+1)!\Gamma (\alpha +1)}\nonumber \\ ={\mathcal {O}}((n+1)^{\alpha }).\nonumber \\ \end{aligned}$$

(A.16)

Now, according to the aforementioned equations, we can conclude that

$$\begin{aligned} \Vert \rho ^i(\theta )-{\hat{\rho }}^i(\theta )\Vert _{\infty } \le {\mathfrak {C}}\frac{\pi ^{n+1}(n+1)^{\mu }}{\kappa _n^{\alpha ,\beta }(n+1)!} \end{aligned}$$

(A.17)

$\square$

Remark A.5

In chapter 10 of [96], Canuto et. al. discuss stability analysis of spectral methods in details. In section 4.10.3, they examine the stability analysis of collocation method, especially for basic functions that have the property of the Kronecker delta and the choice of the points of Gaussian type. Since we stated, GLFs have Kronecker delta property and the collocation points are Gauss–Lobatto nodes [97]; thus, the theorems discussed in [96] can be expressed for the proposed method. In order to avoid to express duplicate contents, for further interested readers be referred to chapter 10 in [96].

In addition Solomonoff and Turkel [98] and Gottlieb [99, 100] studied the impact of boundary conditions and collocation points on stability and convergence of pseudospectral method.

1.4 Numerical integration of 2.7 and 2.8

In order to compute I(t) and G(t) integrals in Eq.(2.7) and Eq.(2.8), we use shifted Legendre–Gauss–Lobatto integration formula.

Legendre polynomials are one of the special forms of Jacobi polynomials which are denoted by $P_i(x)$. So, these polynomials are orthogonal with respect to weight function $w(x)=1$ on the interval $[-1,1]$ and can be defined by the following recursive formula [101]

$$\begin{aligned} P_0(x)= & {} 1,~~~~~P_1(x)=x,\nonumber \\ P_{n+1}(x)= & {} \bigg (\frac{2n+1}{n+1}\bigg )xP_{n}(x)\nonumber \\&-\bigg (\frac{n}{n+1} \bigg )P_{n-1}(x)~~~n\ge 1. \end{aligned}$$

(A.18)

Let ${\mathcal {H}}_n[0,2\pi ]$ indicate the space of algebraic polynomials of degree $\le n$; then, for every function $f \in {\mathcal {H}}_{2n-1}[0,2\pi ]$ we have [102]

$$\begin{aligned} \int _0^{2\pi }f(x)\mathrm{d}x\simeq \sum _{j=0}^n\omega _jf(x_j), \end{aligned}$$

(A.19)

where $\lbrace x_j\rbrace _{j=0}^n$ are the shifted Legendre–Gauss–Lobatto nodes, and $\omega _j$ are the shifted Legendre–Gauss–Lobatto weights, given in

$$\begin{aligned} \omega _j=\frac{2}{n(n+1)\bigg (P^s_n(x_j)\bigg )^2}, \end{aligned}$$

(A.20)

in which $P^s_n(x_j)$ is the $n^{th}$ shifted Legendre polynomials on interval $[0,2\pi ]$. Therefore, for $G(t_k)$ and $I(t_k)$ in the $k^{th}$ time step, we have

$$\begin{aligned}&I(t_k)\simeq 2\sum _{j=0}^n\bigg (\frac{\partial \rho (\theta ,t_k)}{\partial \theta }-\frac{\partial \rho _f(\theta ,t_k)}{\partial \theta }\bigg )\nonumber \\&\quad \bigg \vert _{\theta =x_j}{\mathcal {Z}}(x_k)\rho (x_k,t_k)\omega _j, \end{aligned}$$

(A.21)

$$\begin{aligned}&G(t_k)\simeq -{\mathcal {B}}\sum _{j=0}^n\Big (\frac{\partial \rho (\theta ,t_k)}{\partial \theta }-\frac{\partial \rho _f(\theta ,t_k)}{\partial \theta }\Big )\bigg \vert _{\theta =x_j}\nonumber \\&\quad \frac{\partial \rho (\theta ,t_k)}{\partial \theta }\bigg \vert _{\theta =x_j}\omega _j. \end{aligned}$$

(A.22)

Runge–Kutta scheme for stochastic ODEs

It is clear that Eq. (2.5) should be solved to find phase oscillators at each step. We utilize a well-known fourth-order Runge–Kutta approach to solve this stochastic ODE. We have

$$\begin{aligned} \mathrm{d}\theta _j^i=\delta \mathrm{d}t+{\mathcal {Z}}(\theta )\bigg [u^{i-1}+\sqrt{2D}dW_j(t)\bigg ],~~~\nonumber \\j=1, \cdots , M, \end{aligned}$$

(B.1)

in which $W_j(t)$ is the standard Weiner process. We define functions f and g from Eq. B.1

$$\begin{aligned} f(\theta )=\delta +u{\mathcal {Z}}(\theta ), \end{aligned}$$

(B.2)

$$\begin{aligned} g(\theta )=\sqrt{2D}+{\mathcal {Z}}(\theta ) \end{aligned}$$

(B.3)

The main problem is discretized as follows ($k=0, \cdots , r$ where r is the number of Runge–Kutta steps and $h=\frac{\Delta t}{r}$).

$$\begin{aligned} {\hat{\theta }}^{k+1}_j= & {} {\hat{\theta }}^{k}_j+\frac{1}{6}\bigg (h({\mathcal {K}}_1 +2{\mathcal {K}}_2+2{\mathcal {K}}_3+{\mathcal {K}}_4)\nonumber \\&+W_j({\mathcal {S}}_1 +2{\mathcal {S}}_2+2{\mathcal {S}}_3+{\mathcal {S}}_4)\bigg ), \nonumber \\ {\mathcal {K}}_1= & {} f({\hat{\theta }}^{k}_j),~ {\mathcal {S}}_1=g({\hat{\theta }}^{k}_j),\nonumber \\ \sigma _1= & {} {\hat{\theta }}^{k}_j+0.5h{\mathcal {K}}_1+0.5W_j{\mathcal {S}}_1,\nonumber \\ {\mathcal {K}}_2= & {} f(\sigma _1),~ {\mathcal {S}}_2=g(\sigma _1),\nonumber \\ \sigma _2= & {} {\hat{\theta }}^{k}_j+0.5h{\mathcal {K}}_2+0.5W_j{\mathcal {S}}_2,\nonumber \\ {\mathcal {K}}_3= & {} f(\sigma _2),~ {\mathcal {S}}_3=g(\sigma _2),\nonumber \\ \sigma _3= & {} {\hat{\theta }}^{k}_j+h{\mathcal {K}}_3+W_j{\mathcal {S}}_3,\nonumber \\ {\mathcal {K}}_4= & {} f(\sigma _3),~ {\mathcal {S}}_4=g(\sigma _3). \end{aligned}$$

(B.4)

Note that ${\hat{\theta }}^0_j=\theta _j^i$ and $\theta _j^{i+1}={\hat{\theta }}^{r}_j$.

Neuronal models

Here, the details of mathematical models used in the examples of this paper are explained.

1.1 Hindmarsh–Rose (HR) model

The HR model which is considered in Example 1 is as follows: [85]

$$\begin{aligned}&{\dot{v}}=(I-g_{Na}m_{\infty }^3(v)(0.85-3(q-\lambda \mu _{\infty }(v)))\\&\quad (v-v_{Na})-g_Kq(v-v_K)\\&\quad -g_L(v-v_L)c^{-1}+u(t), \\&{\dot{q}}=\frac{q_{\infty }(v)-q}{\eta _q(v)}, \\&q_{\infty }(v)=n_{\infty }^4(v)\lambda \mu _{\infty }(v), \\&\mu _{\infty }(v)=\Bigg (\frac{1}{1+\exp (\gamma (v+53.3))}\Bigg )^4, \\&m_{\infty }(v)=\frac{\alpha _m(v)}{\alpha _m(v)+\beta _m(v)}, \\&n_{\infty }(v)=\frac{\alpha _n(v)}{\alpha _n(v)+\beta _n(v)}, \\&\eta _q(v)=\frac{\eta _b(v)+\eta _n(v)}{2}, \\&\eta _b(v)=\theta _b\Bigg (1.24+\frac{2.678}{1+\frac{\exp (v+50)}{16.027}}\Bigg ), \\&\eta _n(v)=\frac{\theta _n}{\alpha _n(v)+\beta _n(v)}, \\&\alpha _n(v)=\frac{0.01(v+45.7)}{1-\frac{\exp (-(v+45.7))}{10}}, \\&\alpha _m(v)=\frac{0.1(v+29.7)}{1-\frac{\exp (-(v+29.7))}{10}}, \\&\beta _n(v)=0.125\exp \bigg (\frac{-(v+55.7)}{80}\bigg ), \\&\beta _m(v)=4\exp \bigg (\frac{-(v+54.7)}{18}\bigg ), \end{aligned}$$

where

$$\begin{aligned} v_{Na}= & {} 55 mV, v_K=-72 mV, v_L=-17 mV, \\ g_Na= & {} 120 mS/cm^2,g_K=20 mS/cm^2,\\ g_L= & {} 0.3 mS/cm^2, c=1 \mu F/cm^2, I=5 \mu A/cm^2, \\ \gamma= & {} 0.069 mV^{-1}, \theta _b=1 ms, \theta _n=0.52 ms,\\&\quad \lambda =0.21. \end{aligned}$$

1.2 Fitzhugh–Nagumo (FHN) model

Here, we represent FHN neural dynamical model [85].

$$\begin{aligned} {\dot{v}}= & {} (-w-v(v-1)(v-a)+I)c^{-1}+u(t), \\ {\dot{w}}= & {} \epsilon (v-g_aw), \end{aligned}$$

where

$$\begin{aligned} g_a=1, \epsilon =0.05, a=0.1 mV, c=1 \mu F/cm^2. \end{aligned}$$

1.3 Leaky itegrate-and-fire (LIF) model

The LIF mathematical model is defined as follows:

$$\begin{aligned} {\dot{v}}=I+g_L(v_L-v)c^{-1}, \end{aligned}$$

where

$$\begin{aligned} c=1\mu F/cm^2,~ g_L=0.11 mS/cm^2 \end{aligned}$$

1.4 Kuromoto model

The transformed Kuromoto model details are as follows:

$$\begin{aligned} {\dot{\theta }}_i=w_i+Kr\sin (\theta _i)+u(t), \end{aligned}$$

in which

$$\begin{aligned} w=2\pi , ~K=-30.4095. \end{aligned}$$

1.5 Hodgkin–Huxley (HH) model

In the last example of Sect. 4, we use reduced HH model with the following details [53]:

$$\begin{aligned} {\dot{v}}= & {} I-g_{Na}(m_{\infty }(v))^3(0.8-n)(v-v_{Na})\\&-g_Kn^4(v-v_K)-g_L(v-v_L) c^{-1}+u(t), \\ {\dot{n}}= & {} a_n(v)(1-n)-b_n(v)n, \\ {\dot{m}}= & {} a_m(v)(1-m)-b_m(v)m, \\ {\dot{h}}= & {} a_h(v)(1-h)-b_h(v)h, \\ a_n(v)= & {} \frac{0.01(v+55)}{1-\exp (\frac{-(v+55)}{10})}, \\ b_n(v)= & {} 0.125\exp (\frac{-(v+65)}{80}), \\ a_m(v)= & {} \frac{0.1(v+40)}{1-\exp (\frac{-(v+40)}{10})}, \\ b_m(v)= & {} 4\exp (\frac{-(v+65)}{18}), \\ a_h(v)= & {} 0.07\exp (\frac{-(v+65)}{20}), \\ b_h(v)= & {} \frac{1}{\exp (\frac{-(v+35)}{10})}, \end{aligned}$$

with the following parameters

$$\begin{aligned} v_{Na}= & {} 50 mV, v_K=-77 mV, v_L=-54.4 mV,\\ g_{Na}= & {} 120 mS/cm^2,\\ g_K= & {} 36 mS/cm^2, g_L=0.3 mS/cm^2, c=1 \mu F/cm^2. \end{aligned}$$