The phenomenon of coherence resonance

First encountered in oscillatory systems as stociiastic resonance without periodic forcing [17], and later as interiiai stochastic resonance [23], it 

To characterize the level of coherence of noise-induced excitations we analyze the time evolution of the activator concentration x in the FitzHugh-Nagumo model, see Fig. 1.6. In this representation the excitation loops shown previously in Fig. 1.4 become spikes spaced out by intervals during which the system performs noisy relaxation oscillations aroimd its stable state. The phenomenon of coherence resonance manifests itself in the three realizations of x t) for different noise intensities given in Fig. 1.6. For very low noise intensity (upper panel) an excitation is a rare event which happens at random times. In the panel at the bottom, for high noise intensity, the systems fires more easily but still rather randomly. In the panel in the center instead, at an optimal noise intensity, the system fires almost periodically. 

The typical oscillation period for the system is given by the mean interspike time interval (ISI) tp) between two successive noise-induced excitations over many realizations, see enlargements in Fig. 1.6. To it we associate as error the standard deviation. If the system fires regularly, say for simplicity periodically, then the error associated to tp is zero and consequently the ratio of the standard deviation srd tp) to its mean value tp), i.e. the normalized fluctuations 

The phenomenon of CR is due to the presence of two different characteristic time scales in the system which are affected by noise in different manner. One is the time during which the system just fluctuates around the stationary state, which is needed to activate an excitation. We call this the activation time The second time scale, tlie excur.sion time te, is the typical duration of an excitation loop, compare Fig. 1.0. Noise of low intensity does not affect tg, compare panels B.l and B.2 of Fig. 1.4. Consequently, assuming std[te) small and ta tg excitations are I are events), we can write that c sTd Q)/ (Q). In this regime, the spikes are completely random events, so that sjd(ta)/(f ) = 1, and, as noise in- 

Normalized fluctuations of the inter-spike interval versus the noise intensity for the FitzHugh-Nagmno model. Black curve reproduces the result shown in 40). Parameters 6o = 1.05, e = 0.01, 7 = 1,4, o = 1/3 in Eqs. 1.31. The fixed point is a stable focus. Same results with iip = 1.2 shown by the red curve, where the fixed point is a stable node. 

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