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Updated: Jan 1, 2026

Generation of Local CA1 γ Oscillations by Tetanic Stimulation
Published on: August 14, 2015
Continuous stochastic processes with nonlocal memory
S S Melnyk1, V A Yampol'skii2, O V Usatenko2
1A. Ya. Usikov Institute for Radiophysics and Electronics NASU, 61085 Kharkiv, Ukraine.
Abstract:
We study the non-Markovian random continuous processes described by the Mori-Zwanzig equation. As a starting point, we use the Markovian Gaussian Ornstein-Uhlenbeck process and introduce an integral memory term depending on the past of the process into an expression for the higher-order transition probability function and the stochastic differential equation. We show that the proposed processes can be considered as continuous-time interpolations of discrete-time higher-order autoregressive sequences. An equation connecting the memory function (the kernel of integral term) and the two-point correlation function is obtained. A condition for stationarity of the process is established. We suggest a method to generate stationary continuous stochastic processes with a prescribed pair correlation function. As an illustration, some examples of numerical simulation of the processes with nonlocal memory are presented.
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