The First Passage Time Problem for Gauss-Diffusion Processes: Algorithmic Approaches and Applications to LIF Neuronal Model

Buonocore, Aniello; Caputo, Luigia; Pirozzi, Enrica; Ricciardi, Luigi M.

doi:10.1007/s11009-009-9132-8

Cited by 36 publications

(19 citation statements)

References 29 publications

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“…We use as our model neuron the "leaky integrate-and-fire neuron," a simple yet widely used (36,(48)(49)(50)(51)(52)(53)(54)(55)(56)(57)(58)(59)(60) model of neuronal function defined by the following:…”

Section: First Passage Through a Rough Boundarymentioning

confidence: 99%

A phase transition in the first passage of a Brownian process through a fluctuating boundary with implications for neural coding

Taillefumier

Magnasco

2013

Proc. Natl. Acad. Sci. U.S.A.

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Finding the first time a fluctuating quantity reaches a given boundary is a deceptively simple-looking problem of vast practical importance in physics, biology, chemistry, neuroscience, economics, and industrial engineering. Problems in which the bound to be traversed is itself a fluctuating function of time include widely studied problems in neural coding, such as neuronal integrators with irregular inputs and internal noise. We show that the probability p(t) that a Gauss-Markov process will first exceed the boundary at time t suffers a phase transition as a function of the roughness of the boundary, as measured by its Hölder exponent H. The critical value occurs when the roughness of the boundary equals the roughness of the process, so for diffusive processes the critical value is H c = 1/2. For smoother boundaries, H > 1/2, the probability density is a continuous function of time. For rougher boundaries, H < 1/2, the probability is concentrated on a Cantor-like set of zero measure: the probability density becomes divergent, almost everywhere either zero or infinity. The critical point H c = 1/2 corresponds to a widely studied case in the theory of neural coding, in which the external input integrated by a model neuron is a white-noise process, as in the case of uncorrelated but precisely balanced excitatory and inhibitory inputs. We argue that this transition corresponds to a sharp boundary between rate codes, in which the neural firing probability varies smoothly, and temporal codes, in which the neuron fires at sharply defined times regardless of the intensity of internal noise.first-passage time | neural code | random walk A Brownian process W(t) that starts at t = 0 from W(t = 0) = 0 will fluctuate up and down, eventually crossing the value 1 infinitely many times: for any given realization of the process W, there will be infinitely many different values of t for which W(t) = 1. Finding the very first such time,known as the "first passage" of the process through the boundary B = 1, is easier said than done, one of those classical problems whose concise statements conceal their difficulty (1-4). For general fluctuating random processes, the first-passage time problem is both extremely difficult (5-9) and highly relevant, due to its manifold practical applications: it models phenomena as diverse as the onset of chemical reactions (10-14), transitions of macromolecular assemblies (15-19), time-to-failure of a device (20)(21)(22), accumulation of evidence in neural decision-making circuits (23), the "gambler's ruin" problem in game theory (24), species extinction probabilities in ecology (25), survival probabilities of patients and disease progression (26-28), triggering of orders in the stock market (29-31), and firing of neural action potentials (32-37).Much attention has been devoted to two extensions of this basic problem. One is the first passage through a stationary boundary within a complex spatial geometry, such as diffusion in porous media or complex networks. These models are used to describe foragi...

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Section: First Passage Through a Rough Boundarymentioning

confidence: 99%

A phase transition in the first passage of a Brownian process through a fluctuating boundary with implications for neural coding

Taillefumier

Magnasco

2013

Proc. Natl. Acad. Sci. U.S.A.

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“…When degradation process y ( t ) reaches a pre‐set threshold level Y L , the device is considered to be of low reliability or non‐usable; therefore, it is natural to view the lifetime termination as the point when the degradation process crosses this threshold level for the first time , . From the concept of FHT , the lifetime T is defined as T = inf( t : y ( t ) ≥ Y L | y (0) < Y L ), so the probability density function (PDF) of lifetime T at time t i is given by as:

f ((), (|, t) λ, y_{i}) = \frac{Y_{L} - y_{i}}{\sqrt{2 π {((), t - t_{i})}^{3} σ_{w}^{2}}} normalexp ((), prefix- \frac{{((), Y_{L} - y_{i} - λ ((), t - t_{i}))}^{2}}{2 σ_{w}^{2} ((), t - t_{i})})

where y i = y ( t i ). In our model, the drift coefficient λ is random; therefore, we view λ as a variable in the lifetime function.…”

Section: Methodsmentioning

confidence: 99%

“…When degradation process y(t) reaches a pre-set threshold level Y L , the device is considered to be of low reliability or non-usable; therefore, it is natural to view the lifetime termination as the point when the degradation process crosses this threshold level for the first time [15], [23]. From the concept of FHT [24][25][26], the lifetime T is defined as T = inf(t : y(t) ≥ Y L |y(0) < Y L ), so the probability density function (PDF) of lifetime T at time t i is given by [27] as:…”

Section: Brownian Motion-based Modelingmentioning

confidence: 99%

A Model for Residual Life Prediction Based on Brownian Motion in Framework of Similarity

Zhang

Kong

et al. 2015

Asian Journal of Control

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Residual lifetime (RL) estimation is a key part in prognostics and health management. This paper addresses the problem of estimating the RL from observed degradation data. A Brownian motion in the framework of a similarity‐based model utilizing degradation histories with failure and suspension events is developed to achieve this aim. A novel contribution of this paper is the use of observed degradation data from both failed and suspended historical devices, that is, reference devices, to predict the RL of the operating device. This is performed by comparing the similarity between the operating device and reference devices. An extensive numerical investigation is provided to substantiate the superiority of the proposed model over the classical similarity‐based approach. We also fit the model to different groups of simulated data to show the adaptability of the model under a small sample condition. The results show that our developed model can provide better residual life estimation accuracy.

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“…One can reset the process V(t) in the sense that one poses V(T−) = V th and V(T) = V r for some constant V r < V th , then one can consider the process V 2 (t) = V(t − T) to be solution of (1) with V 2 (0) = V r and I 2 (t) := I(t + T) and study the first passage time of this new process through V th (see [4,5,[26][27][28][29][30][31] and references therein); •…”

Section: The Semi-markov Leaky Integrate-and-fire Modelmentioning

confidence: 99%

“…It has been shown in [27] that under suitable assumptions on I(t), the probability survival functions of A n t → P(A n > t) are asymptotically exponential. However, in some particular settings this behavior is in contradiction with the experimental data.…”

Section: The Semi-markov Leaky Integrate-and-fire Modelmentioning

confidence: 99%

A Semi-Markov Leaky Integrate-and-Fire Model

Ascione

Toaldo

2019

Mathematics

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In this paper, a Leaky Integrate-and-Fire (LIF) model for the membrane potential of a neuron is considered, in case the potential process is a semi-Markov process. Semi-Markov property is obtained here by means of the time-change of a Gauss-Markov process. This model has some merits, including heavy-tailed distribution of the waiting times between spikes. This and other properties of the process, such as the mean, variance and autocovariance, are discussed.

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The First Passage Time Problem for Gauss-Diffusion Processes: Algorithmic Approaches and Applications to LIF Neuronal Model

Cited by 36 publications

References 29 publications

A phase transition in the first passage of a Brownian process through a fluctuating boundary with implications for neural coding

A phase transition in the first passage of a Brownian process through a fluctuating boundary with implications for neural coding

A Model for Residual Life Prediction Based on Brownian Motion in Framework of Similarity

A Semi-Markov Leaky Integrate-and-Fire Model

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