The dependence of four firing statistics in neuronal excitable systems is studied as a function of noise intensity and sinusoidal forcing period. For a range of biologically relevant frequencies, we find that the noise amplitude optimizing these statistics depends on the forcing period T, and that stochastic resonance with time-scale matching occurs. Results are explained by the interplay of generic static and dynamic threshold properties of excitable systems. [S0031-9007(98)07562-0] PACS

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... s: 87.22.Jb, 05.40. + j Excitable dynamics underlie the behavior of many systems ranging from Josephson junctions, to chemical reactions to cardiac and nerve cells [1, 2] . In these systems, a large perturbation can elicit a large amplitude spike or "firing," followed by a quick return to a globally attracting fixed point. The dynamical response of these systems to periodic deterministic forcing has been extensively described in both experimental and model studies. Periodic responses include n:m phase locking patterns with m firings for n forcing cycles [3] . In the space of forcing parameters, the generic arrangement of isoperiodic regions has received particular attention [2, 4] . Forcing these systems with sufficiently small periodic perturbations yields trivial 1:0 steady state responses having no "spikes." This deterministically uninteresting "subthreshold" regime has received much recent attention in the context of noise-induced oscillations in neuronal networks (see, e.g., [5] and references therein) and of stochastic resonance (SR) in neurons [6] . Recent theoretical work [7-9] on SR in excitable systems has focused on the regime in which the system simply behaves as a static threshold element. For example, in "aperiodic SR" [8], signal fluctuations are slower than all system time scales; the noise intensity D optimizing the linear correlation between signal and firing rate is then independent of signal frequencies. However, many systems are driven by higher frequency signals, and do not fully recover their resting state between firings. The effect of this recovery time scale, which by interacting with the signal time scale produces deterministic phase locking patterns, is poorly understood in the context of SR. Further, SR in its restricted sense is described as a match between signal period and an "interevent" time scale due to noise alone, and assessed by the optimization of some signal-firing correlation for some D . 0 [6,10]. Here we show that, for a large range of biologically relevant frequencies, the optimization of many firing statistics used in the SR context closely follows such a time-scale matching notion. Our approach focuses on the timeaveraged phase locking that results from forcing with both subthreshold periodic signals and additive noise. We use the Fitzhugh-Nagumo neuron model, but our results are relevant to general noise-perturbed excitable systems [1, 2] , in particular to their sensitivity and tuning properties for arbitrary signals [11] . The Fitzhugh-Nagumo model with additive periodic and stochastic forcing is [8, 9, 12]