Diversity-Induced Resonance

We present conclusive evidence showing that different sources of diversity, such as those represented by quenched disorder or noise, can induce a resonant collective behavior in an ensemble of coupled bistable or excitable systems. Our analytical and numerical results show that when such systems are subjected to an external subthreshold signal, their response is optimized for an intermediate value of the diversity. These findings show that intrinsic diversity might have a constructive role and suggest that natural systems might profit from their diversity in order to optimize the response to an external stimulus.

Figure 1

Spectral amplification factor $\eta$ of the globally coupled bistable model, Eq. (1). The values of the $a_i$’s are drawn from a Gaussian distribution of zero mean and variance ${\sigma }^2$. Some system parameters are $N={10}^3$, $C=1$, $A=0.20$. In the main plot, we observe that the amplification factor exhibits a maximum as a function of the diversity in the case of both a period $T=50$ (black circles) and $T={10}^3$ (open circles) of the external forcing. The inset plots the spectral amplification factor as a function of the period of the forcing (the diversity is fixed at $\sigma =0.55$). In both plots, symbols correspond to numerical simulations, and the lines are the corresponding theoretical predictions of a simplified theory (see the text for details).

Figure 2

(a) Spectral amplification factor $\eta$ of the globally coupled bistable model, Eq. (1) as a function of diversity for different values of the amplitude $A$ of the forcing. (b) Dependence of $\eta$ on the forcing amplitude for fixed values of diversity. Both figures use a period $T=200$ and other parameters as in Fig. 1. Symbols represent the results coming from a numerical simulation of the system’s equations, and lines correspond to the analytical approximation (see text for details).

Figure 3

Spectral amplification factor $\eta$ of the globally coupled FitzHugh-Nagumo model, Eqs. (5), where the $a_i$’s have been drawn from a Gaussian distribution of mean $a$ and variance ${\sigma }^2$. Some system parameters are $N={10}^3$, $ϵ={10}^{-2}$, $a=1.12$, $C=1$, $A=0.05$. (a) Plot of $\eta$ as a function of the diversity $\sigma$ for different periods of the external forcing. (b) Plot of $\eta$ as a function of the period $T$ for different values of the diversity $\sigma$. In both cases, the symbols represent the results coming from a numerical simulation of the system’s equations (the solid line is a guide to the eye).