Detecting and quantifying stochastic and coherence resonances via information-theory complexity measurements

Statistical complexity measures are used to detect noise-induced order and to quantify stochastic and coherence resonances. We illustrate the method with two paradigmatic models, one of a Brownian particle in a sinusoidally modulated bistable potential, and the other, the FitzHugh-Nagumo model of excitable systems. The method can be employed for the precise detection of subtle signatures of noise-induced order in real-world complex signals.

Figure 1

(Color online) Quantifying stochastic resonance in the bistable system for varying noise intensity and fixed modulation frequency $\left(\omega =0.0078\right)$. (a) The standard indicators in arbitrary units (the signal-to-noise ratio: squares; and the amplitude of the periodic component of the system response, $⟨x⟩$: triangles) are plotted vs the noise intensity, $\mathcal{D}$. (b) Amplitude of the first three peaks of the residence times distribution ($P_1$: squares; $P_2$: circles; and $P_3$: triangles) plotted vs $\mathcal{D}$. (c) MPR statistical complexity, $\mathcal{C}$, (filled triangles) and the normalized Shannon entropy, $\mathcal{H}$, (filled squares) both in arbitrary units plotted vs $\mathcal{D}$. The empty symbols display $\mathcal{C}$ and $\mathcal{H}$ calculated using the data rearranged randomly, thus demonstrating that the statistical complexity measures $\mathcal{C}$ and $\mathcal{H}$ quantify temporal correlations. (d) $\mathcal{C}$ vs $\mathcal{H}$ (dots, red online). The solid lines indicate the boundary values, ${\mathcal{C}}_{\text{max}}$ and ${\mathcal{C}}_{\text{min}}$.

Figure 2

(Color online) As Fig. 1, but the modulation frequency is varied while the noise level is kept fixed $\left(\mathcal{D}=0.05\right)$.

Figure 3

(Color online) Quantifying coherence resonance in the FitzHugh-Nagumo model. (a) Normalized variance of the interspike intervals, $R_p$, vs the noise intensity for two sets of parameters (filled symbols: $ϵ=0.01$ and $a=1.05$; empty symbols: $ϵ=0.1$ and $a=1.005$). (b) MPR statistical complexity (triangles) and the normalized Shannon entropy (squares) vs the noise intensity, $\mathcal{D}$, for the same parameters.