We present a discrete model of stochastic excitability by a low-dimensional set of delayed integral equations governing the probability in the rest state, the excited state, and the refractory state. The process is a random walk with discrete states and nonexponential waiting time distributions, which lead to the incorporation of memory kernels in the integral equations. We extend the equations of a single unit to the system of equations for an ensemble of globally coupled oscillators, derive the mean field equations, and investigate bifurcations of steady states. Conditions of destabilization are found, which imply oscillations of the mean fields in the stochastic ensemble. The relation between the mean field equations and the paradigmatic Kuramoto model is shown.

中文翻译：

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随机分层系统：可兴奋动力学

我们通过控制静止状态、激发状态和难熔状态的概率的一组低维延迟积分方程来提出随机兴奋性的离散模型。该过程是具有离散状态和非指数等待时间分布的随机游走，这导致在积分方程中加入内存内核。我们将单个单元的方程扩展到全局耦合振荡器集合的方程组，推导出平均场方程，并研究稳态的分岔。发现了不稳定的条件，这意味着随机集合中平均场的振荡。显示了平均场方程和范式 Kuramoto 模型之间的关系。