Abstract
Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker-Planck equations for the n-time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker-Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.
Highlights
Introduction to spiking neural networksInformation processing, learning and applications.Acta neurobiologiae experimentalis, 71(4):409–433, 2011.46
Delay dynamics have been applied in climate models [34], and across scales in biology: from gene regulatory networks at the single cell level [35,36], to organism-scale neural networks [37], and even multi-organism collective movement [38], consensus [39] and population dynamics [40]
A clear focus to study non-Markov processes is present in biological neural networks, where communication delays can be caused by the time it takes for signals to propagate down different axon lengths
Summary
For linear processes such as (3.1), it is possible to use the simplified Langevin form (3.5), so that each element of the FP hierarchy (2.12) is governed by. To understand how the dynamics is affected by this first term it suffices to consider the evolution of P1, or P2, in the absence of diffusion, i.e. σ = 0, and zero history Ψ (t) = 0 Eliminating these terms from the P1 equation results in the FP solution P1(x, t) = δ(x − x0λ(t)), which implies that the system is localised at x = x0λ(t) for all times t > 0. To interpret the final term in Eq (4.2), it is helpful to consider the meaning of a mixed partial derivative of a function, f with respect to x and x This represents how the slope of f changes along the x direction as one moves along the x direction (and vice-versa). The drift term and the time-dependent diffusion constant are not anymore, respectively, the derivative of the mean and of the MSD of the original Langevin equation. The functional (5.1) is a multivariate Gaussian distribution in the Fourier domain, whose inverse Fourier transform is the full time-dependent probability distribution
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