The Simulation and Approximation of the First Passage Time of the Ornstein-uhlenbeck Process of Neuron

JIANG, LIQIU. The Simulation and Approximation of the First Passage Time of the Ornstein–Uhlenbeck Process of Neuron. (Under the direction of Charles Eugene Smith.) Neurons communicate with each other via sequences of action potentials. The purpose of this study is to approximate the interval between action potentials which is also called the First Passage Time (FPT), the first time the membrane voltage passes a threshold. The subthreshold depolarization of a neuron receiving a multitude of random synaptic inputs has often been modelled as the Ornstein–Uhlenbeck (OU) process. This model provides an analytically tractable formalism of neuronal membrane voltage mean and variance in terms of a neuron’s membrane time constant and the mean of input voltage. Some authors obtained an approximate mean and variance of the FPT for Stein’s model with a constant threshold for firing by using Stein’s method. They approximated the mean and variance of FPT by using the first term of the Taylor’s series expansion. We expect this procedure works for the OU process, a diffusion process. This study finds that Stein’s method works well for the OU process with the small Wiener process parameter. After adding a few other terms of the Taylor’s series, the parameter range in which the approximation works well are almost the same as the range in which the first term does. The relationship between the approximation results and the confidence band of the mean and variance of the simulated FPT gives evidence that their parameter range is the same; but, the approximation by two terms of the Taylor’s series gives less approximation error. The goodness– of–fit–test shows that the lognormal distribution is close to the distribution of FPT for all the Wiener parameters we used. We compared a lognormal distribution of the FPT, estimated from simulation of the OU process, with the probability density function (pdf) of the FPT, approximated from a transformation of the marginal distribution of membrane voltage at the time at which the mean of membrane voltage passes the threshold. We found that the approximation pdf and the lognormal pdf are almost equally close to the true and unknown pdf when the parameter of the Wiener process is small. THE SIMULATION AND APPROXIMATION OF THE FIRST PASSAGE TIME OF THE ORNSTEIN-UHLENBECK PROCESS OF NEURON