Bistable system doublewell

One possibility to simplify a continuous stochastic system is the reduction to a description in terms of a few discrete states. The system s behavior is then specified by the transition times between these discrete states. For example when investigating a neurons behavior, the important aspect are often only the times when a spike is emitted and not the complex evolution of the membrane potential [16]. In a doublewell potential system, depending on the questions asked, it may be sufficient to know in which of the two wells the system is located, neglecting the fluctuations in the wells as well as the actual dynamics when crossing from one well to the other. In these cases a reduction to a discrete description can be considered as an appropriate simplification. We first review the two state description of bistable systems [10] and then introduce a phenomenological discrete model for excitable dynamics. 

Another possibility to quantify the response of a stochastic system to periodic signals is to generalize the notion of synchronization, which is known from deterministic nonlinear oscillators. We will pursue this idea in what follows. To this end we review in section 2.2 the notion of effective synchronization in stochastic systems. The mean number of synchronized system cycles turns out to be an appropriate quantity to characterize the synchronization properties of the system to the periodic signal. However the task remains to calculate this quantity. This calculation will be based on discrete renewal models for bistable and excitable dynamics. These discrete models are introduced in section 2.3. We first recapitulate the well known two state model for the stochastic dynamics of an overdamped particle in a doublewell system [10] and afterwards introduce a phenomenological discrete model for excitable dynamics. In section 2.4 a theory to calculate the mean frequency and effective diffusion coefficient in periodically driven renewal processes is presented. These two quantities allow to calculate the mean number of synchronized cycles. Finally in section 2.5 we apply this theory to investigate synchronization in bistable and excitable systems. 

The remaining part of this paper is devoted to the calculation of the mean frequency v and the effective diffusion coefficient Deff as defined in eqs. (2.2) for the discrete model of bistable and excitable systems. To this end we introduce a general theory to calculate these quantities for periodic renewal processes in section 2.4. This theory is then applied to the driven renewal model eqs. (2.9), (2.10) for the doublewell system and eq. (2.12) for the excitable system. 

Another typical example of the stochastic resonance system is the nonlinear bistable doublewell dynamic system, which describes the overdamped motion of a Brownian jjartide in a symmetric double-well potential in the presence of noise and periodic forcing as shown in... 

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See also in sourсe #XX -- [ , ]

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