The stretch to stray on time: Resonant length of random walks in a transient

Abstract

First-passage times in random walks have a vast number of diverse applications in physics, chemistry, biology, and finance. In general, environmental conditions for a stochastic process are not constant on the time scale of the average first-passage time or control might be applied to reduce noise. We investigate moments of the first-passage time distribution under an exponential transient describing relaxation of environmental conditions. We solve the Laplace-transformed (generalized) master equation analytically using a novel method that is applicable to general state schemes. The first-passage time from one end to the other of a linear chain of states is our application for the solutions. The dependence of its average on the relaxation rate obeys a power law for slow transients. The exponent ν depends on the chain length N like ν=-N/(N+1) to leading order. Slow transients substantially reduce the noise of first-passage times expressed as the coefficient of variation (CV), even if the average first-passage time is much longer than the transient. The CV has a pronounced minimum for some lengths, which we call resonant lengths. These results also suggest a simple and efficient noise control strategy and are closely related to the timing of repetitive excitations, coherence resonance, and information transmission by noisy excitable systems. A resonant number of steps from the inhibited state to the excitation threshold and slow recovery from negative feedback provide optimal timing noise reduction and information transmission.

Models of stochastic $$\hbox {Ca}^{2+}$$ spiking

“…If the CV value is between 0 (deterministic) and 1 (pure randomness), a slow process changes the spike probability without rendering spike generation deterministic. Note, the average ISI is not simply the inverse of the recovery rate in that regime [39]. Measured CVs are between 0.2 (e.g., stimulated hepatocytes) and 0.98 (e.g., spontaneous spiking in microglia).…”

“…with a single cluster closing rate λ − . The first and second moments of the first passage time distribution from 0 to N cr open clusters can be calculated for very general f i,i±1 or Ψ i,i±1 with the method described in Falcke and Friedhoff [39]. The only requirement is that the f i,i±1 or Ψ i,i±1 can be Laplace transformed.…”

“…The only requirement is that the f i,i±1 or Ψ i,i±1 can be Laplace transformed. This is possible for the Ψ i,i±1 despite their dependency on t and t − t if the t-dependency is exponential like [39]. Therefore, this method provides a basis for investigating a large variety of positive feedbacks by the choice of i-dependency of rates and waiting times, puff duration properties by the choice of left-going rates and t−t -dependency, pathway properties by the choice of [IP 3 ]-dependency, etc.…”

“…The state scheme presented in Fig. 6 and its transition rate functions (or waiting time distributions) define a (generalized) master equation, which can be solved using Laplace transforms to determine the moments of the first passage time distribution to reach state N cr [39]. The only requirement towards the waiting time distributions is that their Laplace transform exists.…”

“…where A, B, E, and G are matrices that depend on the length of the chain of states N and the transition rates or waiting time distributions, f i,i±1 and Ψ i,i±1 , respectively, and r i andq contain the initial conditions, as explained in [39]. Application of this theory to a chain with state independent transitions, as it might result from a random walk, found that the CV has a minimum for a certain resonant lengthN .…”