A Fast Algorithm for the First-Passage Times of Gauss-Markov Processes with Hölder Continuous Boundaries

Taillefumier, Thibaud; Magnasco, Marcelo O.

doi:10.1007/s10955-010-0033-6

Cited by 28 publications

(16 citation statements)

References 35 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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“…Some other researchers use Volterra integral equations to approximate the FPD for more general diffusion processes (Ricciardi et al 1984;Buonocore et al 1987;Lehmann 2002). More recently, Taillefumier and Magnasco (Taillefumier and Magnasco 2010) proposed a discrete simulation-based algorithm to approximate the FPD of Gauss-Markov processes and Hölder continuous boundaries, while Molini et al (Molini et al 2011) obtained the approximation by solving a Fokker-Planck equation subject to an absorbing boundary and deriving the transition probability using the method of image. Besides heavy computational cost, all these methods apply to continuous or differentiable boundaries only.…”

Section: P (T; C) = P (τ C > T) = P (W S < C(s) ∀S ∈ [0 T])mentioning

confidence: 99%

First Passage Time for Brownian Motion and Piecewise Linear Boundaries

Jin

Wang

2015

Methodol Comput Appl Probab

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We propose a new approach to calculating the first passage time densities for Brownian motion crossing piecewise linear boundaries which can be discontinuous. Using this approach we obtain explicit formulas for the first passage densities and show that they are continuously differentiable except at the break points of the boundaries. Furthermore, these formulas can be used to approximate the first passage time distributions for general nonlinear boundaries. The numerical computation can be easily done by using the Monte Carlo integration, which is straightforward to implement. Some numerical examples are presented for illustration. This approach can be further extended to compute two-sided boundary crossing distributions.

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“…We then construct a finite-dimensional representation of our driving current that is adapted to the sLIF integration scheme while allowing us to control the Hölder regularity of the resulting effective barrier. We finally depict the principles of a probabilistic dichotomic search algorithm (Taillefumier & Magnasco, 2010) that computes efficiently and accurately first-passage times in the case of rough boundaries.…”

Section: Monte Carlo Numerical Frameworkmentioning

confidence: 99%

“…We refer to Taillefumier and Magnasco (2010) for a detailed account of the algorithm. Here, we recall just the essential tenets of the method.…”

Section: Probabilistic Dichotomic-search Algorithm Using a Well-chosenmentioning

confidence: 99%

A Transition to Sharp Timing in Stochastic Leaky Integrate-and-Fire Neurons Driven by Frozen Noisy Input

Taillefumier

Magnasco

2014

Neural Computation

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The firing activity of intracellularly stimulated neurons in cortical slices has been demonstrated to be profoundly affected by the temporal structure of the injected current (Mainen & Sejnowski, 1995 ). This suggests that the timing features of the neural response may be controlled as much by its own biophysical characteristics as by how a neuron is wired within a circuit. Modeling studies have shown that the interplay between internal noise and the fluctuations of the driving input controls the reliability and the precision of neuronal spiking (Cecchi et al., 2000 ; Tiesinga, 2002 ; Fellous, Rudolph, Destexhe, & Sejnowski, 2003 ). In order to investigate this interplay, we focus on the stochastic leaky integrate-and-fire neuron and identify the Hölder exponent H of the integrated input as the key mathematical property dictating the regime of firing of a single-unit neuron. We have recently provided numerical evidence (Taillefumier & Magnasco, 2013 ) for the existence of a phase transition when [Formula: see text] becomes less than the statistical Hölder exponent associated with internal gaussian white noise (H=1/2). Here we describe the theoretical and numerical framework devised for the study of a neuron that is periodically driven by frozen noisy inputs with exponent H>0. In doing so, we account for the existence of a transition between two regimes of firing when H=1/2, and we show that spiking times have a continuous density when the Hölder exponent satisfies H>1/2. The transition at H=1/2 formally separates rate codes, for which the neural firing probability varies smoothly, from temporal codes, for which the neuron fires at sharply defined times regardless of the intensity of internal noise.

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A Fast Algorithm for the First-Passage Times of Gauss-Markov Processes with Hölder Continuous Boundaries

Cited by 28 publications

References 35 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

First Passage Time for Brownian Motion and Piecewise Linear Boundaries

A Transition to Sharp Timing in Stochastic Leaky Integrate-and-Fire Neurons Driven by Frozen Noisy Input

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