Enhancement of synchronized chaotic state in a delay-coupled complex neuronal network

Abstract

Synchronized firing of neurons is considered as one of the crucial characteristics for signal transmission and neuronal coding. In this article, spatiotemporal synchronization of coupled chaotic Chialvo maps is studied over a network where the connection topology changes at every time instance with chaotically changing rewiring probability. However, the main objective of this study is to investigate the effect of time delay in the ensemble behavior of neurons. As it is well known that delay is unavoidable in neuronal system, we introduce a delay parameter in the coupling term. Examining the bifurcation diagrams, we observe interesting changes in the synchronization behavior with the increment of delay. These results are complemented by the linear stability analysis of the synchronized state of the system using master stability function technique. Moreover, effects of time delay on synchronization behavior of the coupled system are investigated by evaluating average synchronization error and synchronization time. We observe non-monotonic variation of synchronization time with respect to coupling strength for non-delayed system, whereas this characteristic does not persist with the increment of delay. It is observed that the synchronized region of the considered model appears as a combination of homogeneous steady state and homogeneous chaotic state for both delayed and non-delayed systems. Our key finding in this study is the enhancement of synchronized chaotic region and consequently reduction of synchronized steady state with the increasing values of time delay in the system. Global stability of the synchronized state is examined by basin stability measure. It is also observed that depending on different values of delay, the system exhibits various spatiotemporal patterns.

Appendix A: Stability analysis for the fixed point of the non-delayed system

We consider system (4) with $\tau =0$ to check the stability of the fixed point of the non-delayed system with the corresponding time-averaged adjacency matrix, obtained from Laplacian (3) as,

$$\begin{aligned} \bar{{\mathcal {A}}}= {\left\{ \begin{array}{ll} 1-p, &{} \hbox {if } i-1\le j \le i+1 \hbox { and } i\ne j \\ 0, &{} \hbox {if }\ i=j \\ \frac{2p}{N-3}, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(A.1)

The corresponding Jacobian matrix is defined as,

$$\begin{aligned} J=\begin{pmatrix} R &{} S &{} T &{} T &{} . &{} . &{} . &{} T &{} S\\ S &{} R &{} S &{} T&{} . &{} .&{}.&{} T&{} T\\ . &{} .&{} .&{} .&{} .&{} .&{} .&{}. &{} . \\ . &{} .&{} .&{} .&{} .&{} .&{} .&{}.&{} . \\ S &{} T &{} T &{} T&{}.&{} .&{}.&{} S &{}R \end{pmatrix}, \end{aligned}$$

(A.2)

where

$$\begin{aligned} R= & {} \begin{pmatrix} (1-\epsilon )x_*(2-x_*)e^{(y_*-x_*)} &{} (1-\epsilon ){x_*}^2e^{(y_*-x_*)}\\ -b &{} a \end{pmatrix} , \\ S= & {} \begin{pmatrix} \frac{\epsilon }{2}(1-p) &{} 0\\ 0 &{} 0 \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned} T=\begin{pmatrix} \frac{\epsilon }{2}.\frac{2p}{N-3} &{} 0\\ 0 &{} 0 \end{pmatrix}. \end{aligned}$$

Specifically, we get the nonzero fixed point as $(x_*,y_*)$ $\approx (0.9634,0.969)$. It is well known that if the modulus of all the eigen values of $2n\times 2n$ Jacobian matrix (A.2) is less than unity, it guarantees the stability of the fixed point. Consequently, in the process of obtaining the eigen values, we observe that above matrix (A.2) is a block circulant matrix and it is easily reduced to a block diagonal matrix as,

$$\begin{aligned} D=\begin{pmatrix} M_0 &{} O &{} O &{} O &{} . &{} . &{} . &{} O&{} O\\ O &{} M_1 &{} O &{} O &{} . &{} . &{} . &{} O&{} O\\ . &{} .&{} .&{} .&{} .&{} .&{} .&{} . &{}. \\ . &{} .&{} .&{} .&{} .&{} .&{} .&{} . &{}. \\ O &{} O &{} O &{} O &{} . &{} .&{}. &{} O &{}M_{N-1} \end{pmatrix}, \end{aligned}$$

where all $2\times 2$ diagonal blocks $M_r$ are defined as,

$$\begin{aligned} M_r= & {} R+\omega _rS+{\omega _r}^2T+\ldots +{\omega _r}^{N-2}T+{\omega _r}^{N-1}S \nonumber \\= & {} R+S(\omega _r+{\omega _r}^{N-1})+T({\omega _r}^2+{\omega _r}^3+\ldots +{\omega _r}^{N-2}) \nonumber \\= & {} R+2Scos{\theta _r}+T({\omega _r}^2+{\omega _r}^3+\ldots +{\omega _r}^{N-2}), \nonumber \\&\quad r=0,1,2,\ldots ,N-1, \end{aligned}$$

(A.3)

where $\omega _r=e^{i\theta _{r}}$ and ${\theta _r=\frac{2\pi r}{N}}$. Therefore ${\omega _r}^{N-1}=e^{-i\theta r}$. Now, we investigate the modulus of the eigen values of $M_r$ for $0\le \theta _r\le 2\pi$ .

Fig. 10

Maximum modulus of the eigen values of matrix (A.3) under the variation of coupling strength

We plot the maximum modulus of eigen values of the matrices $M_r$ for $r=0,1,2,\ldots ,(N-1)$ in Fig. 10. It is observed that the maximum modulus turns into less than unity for $\epsilon >0.36$. Consequently, it ensures the global attracting fixed point is stable for this range of coupling strength $(0.36<\epsilon <1)$ which also validates the result we observed from Fig. 2.

Now, we consider system (4) for unit delay:

$$\begin{aligned} x_{t+1}(i)= & {} (1-\epsilon )(x_{t}(i)^2e^{(y_t(i)-x_t(i))}+k)\nonumber \\&+\frac{\epsilon }{2}\sum _{j=1}^{N}\bar{{\mathcal {A}}}_{ij}x_{t-1}(j), \nonumber \\ y_{t+1}(i)= & {} ay_t(i)-bx_t(i)+c, \quad i=1,2,\ldots ,N. \end{aligned}$$

(A.4)

This system can be transformed into a system of non-delayed equations by introducing extra variables as,

$$\begin{aligned} x_{t+1}(i)= & {} (1-\epsilon )(x_{t}(i)^2e^{(y_t(i)-x_t(i))}+k)\nonumber \\&+\frac{\epsilon }{2}\sum _{j=1}^{N}\bar{{\mathcal {A}}}_{ij}z_{t}(j), \nonumber \\ y_{t+1}(i)= & {} ay_t(i)-bx_t(i)+c,\nonumber \\ z_{t+1}(i)= & {} x_t(i), \quad i=1,2,\ldots ,N. \end{aligned}$$

(A.5)

This is a 3N-dimensional non-delayed system, and we can easily check the stability of the fixed point by the previously mentioned procedure.

Similarly, any homogeneous finite time-delayed system can be converted into a non-delayed system by introducing a suitable number of new variables, and following the same method, we can examine the stability of the fixed points of the original delay coupled system.