Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, Haus 2, 10115, Berlin, Germany.
Physics Department of Humboldt University Berlin, Newtonstr. 15, 12489, Berlin, Germany.
Biol Cybern. 2022 Apr;116(2):235-251. doi: 10.1007/s00422-022-00920-1. Epub 2022 Feb 15.
Stochastic oscillations can be characterized by a corresponding point process; this is a common practice in computational neuroscience, where oscillations of the membrane voltage under the influence of noise are often analyzed in terms of the interspike interval statistics, specifically the distribution and correlation of intervals between subsequent threshold-crossing times. More generally, crossing times and the corresponding interval sequences can be introduced for different kinds of stochastic oscillators that have been used to model variability of rhythmic activity in biological systems. In this paper we show that if we use the so-called mean-return-time (MRT) phase isochrons (introduced by Schwabedal and Pikovsky) to count the cycles of a stochastic oscillator with Markovian dynamics, the interphase interval sequence does not show any linear correlations, i.e., the corresponding sequence of passage times forms approximately a renewal point process. We first outline the general mathematical argument for this finding and illustrate it numerically for three models of increasing complexity: (i) the isotropic Guckenheimer-Schwabedal-Pikovsky oscillator that displays positive interspike interval (ISI) correlations if rotations are counted by passing the spoke of a wheel; (ii) the adaptive leaky integrate-and-fire model with white Gaussian noise that shows negative interspike interval correlations when spikes are counted in the usual way by the passage of a voltage threshold; (iii) a Hodgkin-Huxley model with channel noise (in the diffusion approximation represented by Gaussian noise) that exhibits weak but statistically significant interspike interval correlations, again for spikes counted when passing a voltage threshold. For all these models, linear correlations between intervals vanish when we count rotations by the passage of an MRT isochron. We finally discuss that the removal of interval correlations does not change the long-term variability and its effect on information transmission, especially in the neural context.
随机振荡可以用相应的点过程来描述；这在计算神经科学中是一种常见的做法，在这种科学中，受噪声影响的膜电压的振荡通常根据峰间间隔统计来分析，特别是后续阈值穿越时间的间隔分布和相关性。更一般地说，可以为不同类型的随机振荡器引入穿越时间和相应的间隔序列，这些振荡器已被用于模拟生物系统中节律活动的可变性。在本文中，我们表明，如果我们使用所谓的平均返回时间（MRT）相位等时线（由 Schwabedal 和 Pikovsky 引入）来计数具有马尔可夫动力学的随机振荡器的周期，那么相间间隔序列不会显示任何线性相关性，即，相应的通过时间序列大致形成更新点过程。我们首先概述了这一发现的一般数学论据，并通过三种复杂度不断增加的模型进行了数值说明：（i）各向同性的 Guckenheimer-Schwabedal-Pikovsky 振荡器，如果通过轮辐旋转计数，则显示正峰间间隔（ISI）相关性；（ii）具有白色高斯噪声的自适应漏电积分和放电模型，如果以通常的方式通过电压阈值计数，则显示负峰间间隔相关性；（iii）具有通道噪声的 Hodgkin-Huxley 模型（在扩散近似中用高斯噪声表示），即使在通过电压阈值计数时，也表现出微弱但具有统计学意义的峰间间隔相关性。对于所有这些模型，当我们通过 MRT 等时线计数旋转时，间隔之间的线性相关性就会消失。我们最后讨论了间隔相关性的消除不会改变长期可变性及其对信息传输的影响，特别是在神经学背景下。
