Paninski Liam
Department of Statistics, Columbia University, Columbia.
J Comput Neurosci. 2006 Aug;21(1):71-87. doi: 10.1007/s10827-006-7200-4. Epub 2006 Apr 22.
We develop theory and numerical methods for computing the most likely subthreshold voltage path of a noisy integrate-and-fire (IF) neuron, given observations of the neuron's superthreshold spiking activity. This optimal voltage path satisfies a second-order ordinary differential (Euler-Lagrange) equation which may be solved analytically in a number of special cases, and which may be solved numerically in general via a simple "shooting" algorithm. Our results are applicable for both linear and nonlinear subthreshold dynamics, and in certain cases may be extended to correlated subthreshold noise sources. We also show how this optimal voltage may be used to obtain approximations to (1) the likelihood that an IF cell with a given set of parameters was responsible for the observed spike train; and (2) the instantaneous firing rate and interspike interval distribution of a given noisy IF cell. The latter probability approximations are based on the classical Freidlin-Wentzell theory of large deviations principles for stochastic differential equations. We close by comparing this most likely voltage path to the true observed subthreshold voltage trace in a case when intracellular voltage recordings are available in vitro.
给定对神经元超阈值尖峰活动的观测结果，我们开发了理论和数值方法来计算有噪声的积分发放（IF）神经元最可能的亚阈值电压路径。这条最优电压路径满足一个二阶常微分（欧拉 - 拉格朗日）方程，在一些特殊情况下可以解析求解，一般情况下可以通过简单的“打靶”算法进行数值求解。我们的结果适用于线性和非线性亚阈值动力学，并且在某些情况下可以扩展到相关的亚阈值噪声源。我们还展示了如何使用这个最优电压来获得以下近似值：（1）具有给定参数集的IF细胞产生观测到的尖峰序列的可能性；（2）给定有噪声的IF细胞的瞬时发放率和峰间期分布。后一种概率近似基于随机微分方程的经典弗雷德林 - 温策尔大偏差原理理论。在体外可获得细胞内电压记录的情况下，我们通过将这条最可能的电压路径与真实观测到的亚阈值电压轨迹进行比较来结束本文。
