Quantifying the Transition from Spiral Waves to Spiral Wave Chimeras in a Lattice of Self-sustained Oscillators

Abstract

The present work is devoted to the detailed quantification of the transition from spiral waves to spiral wave chimeras in a network of self-sustained oscillators with two-dimensional geometry. The basic elements of the network under consideration are the van der Pol oscillator or the FitzHugh – Nagumo neuron. Both of the models are in the regime of relaxation oscillations. We analyze the regime by using the indices of local sensitivity, which enables us to evaluate the sensitivity of each oscillator at a finite time. Spiral waves are observed in both lattices when the interaction between elements has a local character. The dynamics of all the elements is regular. There are no pronounced high-sensitive regions. We have discovered that, when the coupling becomes nonlocal, the features of the system change significantly. The oscillation regime of the spiral wave center element switches to a chaotic one. Besides, a region with high sensitivity occurs around the wave center oscillator. Moreover, we show that the latter expands in space with elongation of the coupling range. As a result, an incoherence cluster of the spiral wave chimera is formed exactly within this high-sensitive area. A sharp increase in the values of the maximal Lyapunov exponent in the positive region leads to the formation of the incoherence cluster. Furthermore, we find that the system can even switch to a hyperchaotic regime when several Lyapunov exponents become positive.

APPENDIX

We also study the evolution of a spiral wave when the coupling range increases for the lattice of discrete-time oscillators (maps). A basic element of this lattice is the Nekorkin map described as follows:

$$\displaystyle x^{t+1}=x^{t}+F(x^{t})-y^{t}-\beta H(x^{t}-d),$$

(A.1)

$$\displaystyle y^{t+1}=y^{t}+\varepsilon(x^{t}-J),$$

where $x^{t}$ is a variable that describes the dynamics of the membrane potential of the nerve cell, $y^{t}$ is a variable that relates to the cumulative effect of all ion currents across the membrane, and the functions $F(x^{t})$ and $H(x^{t}-d)$ are given as follows:

$$\displaystyle F(x^{t})=x^{t}(x^{t}-a)(1-x^{t}),\qquad 0<a<1,$$

(A.2)

$$\displaystyle H(x^{t})=\begin{cases}1,\qquad x^{t}>0,\\ 0,\qquad\mbox{elsewhere}.\end{cases}$$

The parameter $\varepsilon>0$ determines the characteristic time scale of $y^{t}$, the parameter $J$ controls the level of the membrane depolarization $(J<d)$, the parameters $\beta>0$ and $d>0$ determine the excitation threshold of bursting oscillations, and $t=1,2,\ldots$ represents discrete time.

An $N\times N$ 2D lattice of the nonlocally coupled Nekorkin maps is described by the following system of equations:

$$\displaystyle x_{i,j}^{t+1}=x_{i,j}^{t}+F(x_{i,j}^{t})-y_{i,j}^{t}-\beta H(x_{i,j}^{t}-d)+$$

$$\displaystyle+\dfrac{\sigma_{x}}{B_{i,j}^{x}}\sum\limits_{m_{x},n_{x}}\left[f(x_{m_{x},n_{x}}^{t})-f(x_{i,j}^{t})\right],$$

(A.3)

$$\displaystyle y_{i,j}^{t+1}=y_{i,j}^{t}+\varepsilon(x_{i,j}^{t}-J),$$

where $m_{x},n_{x}\in\mathbb{N}$ are indices for nonlocal neighbors. The sum denotes nonlocal coupling of range $R_{x}$ in a square domain. The parameter $\sigma_{x}$ denotes the coupling strength between the elements in the $x$ variable, and $B_{i,j}^{x}$ gives the number of nonlocally coupled neighbors of node ($i,j$).

The numerical results show that, when the nonlocal coupling strength $\sigma_{x}$ and the coupling range $R_{x}$ are varied, the model (A.3) can exhibit all the typical spiral wave patterns, including spiral wave chimeras, which were observed earlier. Examples of these states are presented in Fig. 11 (left). It is seen that the evolution of a spiral wave with increasing values of the coupling range is the same as that of the systems described above. However, the region around the wave center has high sensitivity even when the coupling is local. This happens as the wave center element oscillates chaotically. Even a slight elongation of the coupling range leads to a significant expansion of the high-sensitive region around the wave center in space. The formation of the incoherence core has already taken place for $r=0.05$. The right column in Fig. 11 illustrates the spatial distributions of the ILS with the growth of the coupling strength.

The evolution of the first three Lyapunov exponents for the system (A.3) is illustrated in Fig. 12. The behavior of all the Lyapunov exponents is simpler than that in the models (1.2) and (1.5). The lattice is always in the hyperchaotic regime (even for the very short and very long coupling ranges) because all the three calculated Lyapunov exponents are positive. The elongation of the coupling range leads to the growth of the values of the Lyapunov exponents. The value of $\lambda_{1}(r)$ first monotonically increases and then undergoes almost no changes with increasing coupling range. The values of $\lambda_{2}(r)$ and $\lambda_{3}(r)$ also exhibit a monotonous increase. When the model (A.3) switches to the synchronous regime at $r=0.23$, the values of all the three Lyapunov exponents sharply decrease, but remain positive.