Pattern Formation in Excitable Neuronal Maps

TL;DR

The study investigates pattern formation in two-dimensional excitable neuronal maps using two coupled Chialvo maps and two coupling schemes. A discretized Okubo-Weiss-like discriminant is introduced, and the sign-persistence of this parameter across the lattice serves as a high-signal metric to distinguish ring versus spiral patterns. Ring patterns exhibit slower-than-exponential persistence decay, with intermediate regimes showing power-law scaling, while spiral patterns show stretched-exponential decay with possible early-time oscillations and saturation. The results reveal a direct link between the dynamical evolution of patterns and the persistence of the discriminant, offering a diagnostic approach for identifying and understanding pattern formation and transitions to turbulence in excitable media.

Abstract

Coupled excitable systems can generate a variety of patterns. In this work, we investigate coupled Chialvo maps in two dimensions under two types of nearest-neighbor couplings. One coupling produces ringlike patterns, while the other produces spirals. The rings expand with increasing coupling, whereas spirals evolve into turbulence and dissipate at stronger coupling. To quantify these patterns, we introduce an analogue of the discriminant of the velocity gradient tensor and examine the persistence of its sign. For ring-type patterns, the persistence decays more slowly than exponentially, often following a power law or stretched exponential. When spiral structures remain intact, persistence saturates asymptotically and can exhibit superposed periodic oscillations, suggesting complex exponents at early times. These behaviors highlight deep connections with the underlying dynamics.

Figures (6)

• Figure 1: Field of $x(i,j)$ at $t=8\times10^5$ for $\epsilon=0.2,0.3 \dots0.8$ for nonlinear coupling.
• Figure 2: Persistence $P(t)$ as a function of $(t/t_c-1)^\beta$ on a log-normal scale for $\epsilon=0.2, 0.3, 0.7, 0.8$ with $t_c=125$. It is stretched exponential $P(t)=B\exp(A(t/t_c-1)^\beta)$. An appropriate fit is shown by the red line. a) For $\epsilon=0.2$, $\beta=0.36$, $A=5.7$ and $B=5.9$. b) For $\epsilon=0.3$, $\beta=0.12$, $A=11.7$ and $B=6500$ c) For $\epsilon$= 0.7, $\beta=0.2$, $A=6.1$ and $B=33$ d)For $\epsilon$= 0.8, $\beta=0.1$, $A=16.6$ and $B=700000$.
• Figure 3: Persistence $P(t)$ as a function of $t$ for $\epsilon=0.4,0.5$ and 0.6 on a log-log scale. The Y-axis is multiplied by arbitrary constants for better visibility.
• Figure 4: Field of $x(i,t)$ for a) $\epsilon=0.05$, b)$\epsilon=0.1$, c)$\epsilon=0.2$, d)$\epsilon=0.3$, e)$\epsilon=0.4$, e) $\epsilon=0.5$, f) $\epsilon=0.6$ g) $\epsilon=0.7$ and h) $\epsilon=0.8$ for nonlinear quadratic coupling at $t=10^5$. Spirals start breaking for $\epsilon>0.4$
• Figure 5: Persistence $P(t)$ as a function of $t$ for a) $\epsilon=0.05$ with $\beta=0.8$, b)$\epsilon=0.1$ with $\beta=1$, c)$\epsilon=0.2$ with $\beta=0.8$ d)$\epsilon=0.3$ with $\beta=0.9$, e)$\epsilon=0.4$ with $\beta=0.2$ on semilogarithmic scale.