Synchronization in Periodically Driven and Coupled Stochastic Systems--A Discrete State Approach

We investigate the behavior of stochastic bistable and excitable dynamics based on a discrete state modeling. In addition to the well known Markovian two state model for bistable dynamics we introduce a non Markovian three state model for excitable systems. Its relative simplicity compared to stochastic models of excitable dynamics with continuous phase space allows to obtain analytical results in different contexts. First, we study the joint influence of periodic signals and noise, both based on a characterization in terms of spectral quantities and in terms of synchronization properties with the driving signal. We present expressions for the spectral power amplification and signal to noise ratio for renewal processes driven by weak periodic signals. Applying these results to the discrete model for excitable systems allows to estimate signal frequencies which are optimally amplified by the stochastic system. Stochastic synchronization of the system to the driving signal is investigated based on diffusion properties of the transition events between the discrete states. We derive general results for the mean frequency and effective diffusion coefficient which, beyond the application to the discrete models considered in this work, provide a new tool in the study of periodically driven renewal processes. Applied to the dichotomically driven Markovian two state model for bistable system exact analytical results are obtained. While this system only exhibits one to one synchronization the three state model for excitable system shows different m : n synchronization regimes. The very same discrete model for excitable systems can also be considered as a simple model for a molecular motor. We show that an appropriate periodic modulation of the concentration of the fuel molecules can lead to a very regular motion of the motor. Finally the behavior of globally coupled excitable and bistable units is investigated based on the discrete state description. In contrast to the bistable systems, the excitable system exhibits synchronization and thus coherent oscillations. A non vanishing refractory period as well as an excitatory coupling are shown to be necessary conditions for synchronous firing to occur. All investigations of the non Markovian three state model are compared with the prototypical continuous model for excitable dynamics, the FitzHugh-Nagumo system. They reveal a good agreement between both models, rendering the non Markovian discrete state model for excitable systems an appropriate simplification of continuous phase space dynamics commonly used in the modeling of excitable dynamics.