Neuroscience: Diffusion Models of Single Neurones' Activity

Neuroscience: Diffusion Models of Single Neurones' Activity

This chapter presents a survey of one-dimensional stochastic diffusion models for the membrane potential of a single neuron, with emphasis on the probabilistic properties of the models and on the related first-passage-time (FPT) problems, namely, on the determination of the neuronal output. It discusses how the much celebrated Ornstein-Uhlenbeck (OU) neuronal model can be obtained as the limit of a Markov process with discrete state space in continuous time. The stochastic models of neuronal activity rank among the most advanced applications of the theory of stochastic processes to biology. There are several reasons why many of these neuronal models stem out of the theory of diffusion processes: (1) the theory of diffusion processes is well developed and hence, it allows the neural modelers to apply many general results to the specific application area, (2) for neurons with many synaptic inputs, there is a rather good correspondence between the models and the biological reality, (3) even when some of the assumptions underlying the model may be somewhat questionable, the output behavior of the neuron can often be well approximated by that of a suitably chosen diffusion process, which is, for instance, strikely true for the well-known Gerstein-Mandelbrot diffusion model, and (4) from a biophysical point of view, the neuronal models of single cells reflect the electrical properties of the membrane via electric circuit models that contain energy storage elements.