Noise-induced excitation wave and its size distribution in coupled FitzHugh-Nagumo equations on a square lattice

There are various research topics such as stochastic resonance, coherent resonance, and neuroavalanche in excitable systems under external noises. We perform numerical simulation of coupled noisy FitzHugh-Nagumo equations on the square lattice. Excitation waves are generated most efficiently at an intermediate noise strength. The cluster size distributions obey a power-law-like distribution at a certain parameter range. However, we consider that this is not a self-organized critical phenomenon, partly because the exponent of the power law is not constant. We have studied the propagation of excitation waves in the coupled noisy FitzHugh-Nagumo equations with a one-dimensional pacemaker region and found that there is a phase-transition-like phenomenon from the short-range propagation to the whole-system propagation by changing the noise strength $T$. The power-law distribution is observed most clearly near the phase transition of the propagation of excitation waves in the coupled noisy FitzHugh-Nagumo equations without the one-dimensional pacemaker.

Figure 1

(a) Order parameter $S$ as a function of $T$ at $I=-0.135$ in a system of $L\times L=100\times 100$. (b) $S$ as a function of $T$ at $I=-0.15$ in a system of $100\times 100$. (c) $S$ as a function of $T$ at $I=-0.15$ in a system of $200\times 200$.

Figure 2

Parameter range where the temporal average of $S$ takes a value larger than 0.001 in the parameter space $(T,I)$ calculated in a system of $100\times 100$.

Figure 3

Time evolutions of lattice points satisfying $u_{i,j}>0.5$ at $j=L/2$, at (a) $T=0.0254$, and (b) $T=0.0390$ for $I=-0.135$. At the black dot positions, $u_{i,j}$ takes a value larger than 0.5.

Figure 4

Three snapshots of the regions of $u_{i,j}>0.5$ at (a) $t=5850$, (b) $t=5855$, and (c) $t=5870$ at $T=0.0390$ for $I=-0.135$.

Figure 5

Time evolutions of $S$ at (a) $T=0.0254$ and $I=-0.135$, (b) $T=0.0352$ and $I=-0.15$, and (c) $T=0.0547$ and $I=-0.15$.

Figure 6

Probability distribution of the cluster size $C$ at $T=0.0371$ and $I=-0.15$ for system size (a) $L\times L=100\times 100$ and (b) $200\times 200$ at $T=0.0371$ in a double-logarithmic scale. (c) Probability distribution of the cluster size $C$ of $s^{\prime }$ at $T=0.0371$ and $I=-0.15$ for system size $L\times L=100\times 100$.

Figure 7

(a) Cumulative distribution function $Q\left(C\right)={\int }_C^{\infty }P\left(C^{\prime }\right)d{C}^{\prime }$ of the cluster size $C$ at $T=0.0254$, 0.0371, and 0.0742 for $I=-0.15$. (b) Average cluster size $<C>$ as a function of $T$ for $I=-0.15$.

Figure 8

Propagation of excitation waves at $I=-0.15$ for $T=0.0293$ by Eq. (2). Five snapshots of lattice points satisfying $s_{i,j}=1$ at $t=6000$, 6200, 6400, 6600, and 6800 are plotted.

Figure 9

Propagation of excitation waves at (a) $T=0.0215$, (b) $T=0.0293$, and (c) $T=0.0371$ for $I=-0.15$. The $j$ coordinate of the excited lattice points $(i,j)$ is plotted at time $t=5n$ ($n$: integer).

Figure 10

(a) Average value of $j$ for the lattice points satisfying $s_{i,j}=1$ as a function of $T$ for $I=-0.15$. (b) The critical line of the propagation of the excited wave in the parameter space of $(T,I)$.

Figure 11

Cluster size distributions at (a) $T=0.0298,I=-0.15$ and (b) $T=0.0342,I=-0.14$. The dashed lines denote the power law of exponent $-2.1$. (c) Cluster size distributions at (a) $T=0.0298,I=-0.15$ in a larger system of $300\times 300$. The dashed line denotes the power law of exponent $-1.85$.

Figure 12

(a) Time evolutions of $I\left(t\right)$ by Eq. (3) at $T=0.0361,S_0=0.01767$ (solid line) and $T=0.0293,S_0=0.00635$ (dashed line) for ${\epsilon }^{\prime }=0.0001$. (b) Cluster size distribution at $T=0.0293,S_0=0.00635$, and ${\epsilon }^{\prime }=0.0001$. The dashed lines denote the power law of exponent $-2.1$.