Department of Physics and Centre for Neural Dynamics, University of Ottawa, 598 King Edward, Ottawa K1N 6N5, Canada.
University of Ottawa Brain and Mind Research Institute, University of Ottawa, 451 Smyth Road, Ottawa, ON K1H 8M5, Canada.
Phys Rev E. 2017 May;95(5-1):052127. doi: 10.1103/PhysRevE.95.052127. Epub 2017 May 16.
The theoretical description of nonrenewal stochastic systems is a challenge. Analytical results are often not available or can be obtained only under strong conditions, limiting their applicability. Also, numerical results have mostly been obtained by ad hoc Monte Carlo simulations, which are usually computationally expensive when a high degree of accuracy is needed. To gain quantitative insight into these systems under general conditions, we here introduce a numerical iterated first-passage time approach based on solving the time-dependent Fokker-Planck equation (FPE) to describe the statistics of nonrenewal stochastic systems. We illustrate the approach using spike-triggered neuronal adaptation in the leaky and perfect integrate-and-fire model, respectively. The transition to stationarity of first-passage time moments and their sequential correlations occur on a nontrivial time scale that depends on all system parameters. Surprisingly this is so for both single exponential and scale-free power-law adaptation. The method works beyond the small noise and time-scale separation approximations. It shows excellent agreement with direct Monte Carlo simulations, which allow for the computation of transient and stationary distributions. We compare different methods to compute the evolution of the moments and serial correlation coefficients (SCCs) and discuss the challenge of reliably computing the SCCs, which we find to be very sensitive to numerical inaccuracies for both the leaky and perfect integrate-and-fire models. In conclusion, our methods provide a general picture of nonrenewal dynamics in a wide range of stochastic systems exhibiting short- and long-range correlations.
非更新随机系统的理论描述是一个挑战。分析结果通常不可用，或者只能在强条件下获得，这限制了它们的适用性。此外，数值结果主要通过特定的蒙特卡罗模拟获得，当需要高度准确性时，这些模拟通常计算成本很高。为了在一般条件下对这些系统进行定量分析，我们在这里引入了一种基于求解时变福克-普朗克方程（FPE）的数值迭代首次通过时间方法，以描述非更新随机系统的统计特性。我们分别使用触发神经元适应的渗漏和完美积分-触发模型来说明该方法。首次通过时间矩和它们的顺序相关的稳态过渡发生在一个非平凡的时间尺度上，该时间尺度取决于所有系统参数。令人惊讶的是，无论是单指数还是无标度幂律适应，都是如此。该方法适用于超越小噪声和时间尺度分离近似的情况。它与直接蒙特卡罗模拟非常吻合，蒙特卡罗模拟允许计算瞬态和稳态分布。我们比较了不同的方法来计算矩和序列相关系数（SCCs）的演化，并讨论了可靠计算 SCCs 的挑战，我们发现对于渗漏和完美积分-触发模型，SCCs 对数值不准确性非常敏感。总之，我们的方法为广泛的具有短程和长程相关性的随机系统中的非更新动力学提供了一个总体图景。
