Yamanobe Takanobu
Hokkaido University School of Medicine, North 15, West 7, Kita-ku, Sapporo 060-8638, Japan and PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Nov;88(5):052709. doi: 10.1103/PhysRevE.88.052709. Epub 2013 Nov 13.
Nonlinear oscillators have been used to model neurons that fire periodically in the absence of input. These oscillators, which are called neuronal oscillators, share some common response structures with other biological oscillations such as cardiac cells. In this study, we analyze the dependence of the global dynamics of an impulse-driven stochastic neuronal oscillator on the relaxation rate to the limit cycle, the strength of the intrinsic noise, and the impulsive input parameters. To do this, we use a Markov operator that both reflects the density evolution of the oscillator and is an extension of the phase transition curve, which describes the phase shift due to a single isolated impulse. Previously, we derived the Markov operator for the finite relaxation rate that describes the dynamics of the entire phase plane. Here, we construct a Markov operator for the infinite relaxation rate that describes the stochastic dynamics restricted to the limit cycle. In both cases, the response of the stochastic neuronal oscillator to time-varying impulses is described by a product of Markov operators. Furthermore, we calculate the number of spikes between two consecutive impulses to relate the dynamics of the oscillator to the number of spikes per unit time and the interspike interval density. Specifically, we analyze the dynamics of the number of spikes per unit time based on the properties of the Markov operators. Each Markov operator can be decomposed into stationary and transient components based on the properties of the eigenvalues and eigenfunctions. This allows us to evaluate the difference in the number of spikes per unit time between the stationary and transient responses of the oscillator, which we show to be based on the dependence of the oscillator on past activity. Our analysis shows how the duration of the past neuronal activity depends on the relaxation rate, the noise strength, and the impulsive input parameters.
非线性振荡器已被用于对在无输入情况下周期性放电的神经元进行建模。这些振荡器被称为神经元振荡器，它们与其他生物振荡（如心脏细胞）具有一些共同的响应结构。在本研究中，我们分析了脉冲驱动的随机神经元振荡器的全局动力学对极限环弛豫率、内在噪声强度和脉冲输入参数的依赖性。为此，我们使用一个马尔可夫算子，它既反映了振荡器的密度演化，又是相变曲线的扩展，相变曲线描述了单个孤立脉冲引起的相移。此前，我们推导了描述整个相平面动力学的有限弛豫率的马尔可夫算子。在此，我们构建了描述限于极限环的随机动力学的无限弛豫率的马尔可夫算子。在这两种情况下，随机神经元振荡器对时变脉冲的响应都由马尔可夫算子的乘积来描述。此外，我们计算两个连续脉冲之间的尖峰数量，以将振荡器的动力学与单位时间内的尖峰数量和尖峰间隔密度联系起来。具体而言，我们基于马尔可夫算子的性质分析单位时间内尖峰数量的动力学。每个马尔可夫算子可根据特征值和特征函数的性质分解为稳态和瞬态分量。这使我们能够评估振荡器稳态和瞬态响应之间单位时间内尖峰数量的差异，我们表明这是基于振荡器对过去活动的依赖性。我们的分析表明过去神经元活动的持续时间如何取决于弛豫率、噪声强度和脉冲输入参数。
