First passage time distributions for finite one-dimensional random walks

Khantha, M.; Balakrishnan, V.

doi:10.1007/bf02894735

Cited by 33 publications

(28 citation statements)

References 11 publications

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“…0 in order to keep those moments finite. Such relations have been previously proposed for certain random processes in discrete space [14,15] and have been anticipated without proof in [16] also for the present continuous dynamics (1). A proof will be given at the end of this Letter (see also [17]).…”

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confidence: 66%

Giant Acceleration of Free Diffusion by Use of Tilted Periodic Potentials

et al. 2001

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The effective diffusion coefficient for the overdamped Brownian motion in a tilted periodic potential is calculated in closed analytical form. Universality classes and scaling properties for weak thermal noise are identified near the threshold tilt where deterministic running solutions set in. In this regime the diffusion may be greatly enhanced, as compared to free thermal diffusion with, for a realistic experimental setup, an enhancement of up to 14 orders of magnitude.

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confidence: 66%

Giant Acceleration of Free Diffusion by Use of Tilted Periodic Potentials

et al. 2001

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“…To derive this expression, we needed to correct an error in Khantha and Balakrishnan (1983) in going from their equation 6 to equation 7: the inner expression in their equation 7 should read (m − m 0 )…”

Section: Appendix D: Analysis Of the Dlif Modelmentioning

confidence: 99%

“…The membrane potential V(t) for this model is a continuous-time Markov random walk, and the output spike trains are renewal processes. The Laplace transform of the first passage time densities for reflected random walks are obtained by Khantha and Balakrishnan (1983). The moments of the first passage times can be found from the derivatives of these Laplace transforms (Feller, 1991).…”

Section: Appendix D: Analysis Of the Dlif Modelmentioning

confidence: 99%

Mechanisms That Modulate the Transfer of Spiking Correlations

Rosenbaum

Josićć

2011

Neural Computation

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Correlations between neuronal spike trains affect network dynamics and population coding. Overlapping afferent populations and correlations between presynaptic spike trains introduce correlations between the inputs to downstream cells. To understand network activity and population coding, it is therefore important to understand how these input correlations are transferred to output correlations.Recent studies have addressed this question in the limit of many inputs with infinitesimal postsynaptic response amplitudes, where the total input can be approximated by gaussian noise. In contrast, we address the problem of correlation transfer by representing input spike trains as point processes, with each input spike eliciting a finite postsynaptic response. This approach allows us to naturally model synaptic noise and recurrent coupling and to treat excitatory and inhibitory inputs separately.We derive several new results that provide intuitive insights into the fundamental mechanisms that modulate the transfer of spiking correlations.

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“…Paul L evy was a French mathematician who was interested in probability distributions with certain properties, most notably that the probability distribution for a sum of independent, identically distributed random variables takes the same form as the probability distribution for each component variable (L evy 1937). His work found application within physics, especially since about the early 1980s, when it was realized that various kinds of random walk, considered to reflect certain physical processes, could lead to outcomes with L evy probability distributions (Hughes, Shlesinger & Montroll 1981;Khantha & Balakrishnan 1983;Mukamel, Stern & Ronis 1983). In this context, the L evy walk has proven particularly useful in studies of processes involving 'super diffusion', where diffusion occurs at a faster rate than the 'normal diffusion' resulting from Brownian motion (Eliazar & Klafter 2011;Eliazar & Shlesinger 2013).…”

Section: The L Evy and Other 'Random Walks'mentioning

confidence: 99%

Understanding movements of organisms: it's time to abandon the Lévy foraging hypothesis

2014

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Summary1. Interest in Levy walks within the context of movement of organisms has recently soared, with some now referring to this approach as the Levy walk/flight paradigm. The principal assumptions, taken from the world of physics, have been that organisms searching for food or some other resource adopt random walks, whereby the direction of each successive step in the walk is chosen at random from the complete circle and the length of each step, unless terminated through resource encounter, is chosen from a Levy probability distribution with a particular exponent l. The additional assumption that organisms forage optimally, such that l maximizes the rate of resource encounter, has led to the so-called Levy foraging hypothesis, with many attempts to test it. 2. However, the Levy walk model is unrealistic, especially as it omits directionality between successive steps, a typical feature of movements of individual organisms at spatial scales relevant to their movement decisions. It also results in lower foraging efficiency than other more realistic models and the evidence that organisms actually 'do the Levy walk' is weak to non-existent, despite claims to the contrary. Early optimal foraging studies of movements of organisms and a new generation of movement models avoid these problems. 3. It is therefore time to divorce the Levy walk model from optimal foraging theory, revisit some of the early optimal foraging studies of movements and pursue the new generation of movement models. However, the Levy approach may still prove useful at relatively large spatial scales, in terms of both theory and observations, especially in relation to distribution, dispersal and other population-level phenomena, and in this way biology, physics and mathematics may yet work well together.

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First passage time distributions for finite one-dimensional random walks

Cited by 33 publications

References 11 publications

Giant Acceleration of Free Diffusion by Use of Tilted Periodic Potentials

Giant Acceleration of Free Diffusion by Use of Tilted Periodic Potentials

Mechanisms That Modulate the Transfer of Spiking Correlations

Understanding movements of organisms: it's time to abandon the Lévy foraging hypothesis

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