A Brief Introduction to Some Simple Stochastic Processes

This chapter gives an overview of simple continuous, two-state, and point processes playing a role in theoretical neuroscience. First, various characteristics of these stochastic processes are introduced such as probability densities, moments, correlation functions, the correlation time, and the noise intensity of a process. Then analytical and numerical methods to calculate or compute these various statistics are explained and illustrated by means of simple examples (Ornstein–Uhlenbeck process, random telegraph noise, Poissonian shot noise). Further, useful relations among the different...

This chapter gives an overview of simple continuous, two-state, and point processes playing a role in theoretical neuroscience. First, various characteristics of these stochastic processes are introduced such as probability densities, moments, correlation functions, the correlation time, and the noise intensity of a process. Then analytical and numerical methods to calculate or compute these various statistics are explained and illustrated by means of simple examples (Ornstein–Uhlenbeck process, random telegraph noise, Poissonian shot noise). Further, useful relations among the different statistics (Wiener–Khinchin theorem, relations between spectral and interval statistics of point processes) are also discussed.