343 research outputs found

    First-passage-time statistics of a Brownian particle driven by an arbitrary unidimensional potential with a superimposed exponential time-dependent drift

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    In one-dimensional systems, the dynamics of a Brownian particle are governed by the force derived from a potential as well as by diffusion properties. In this work, we obtain the first-passage-time statistics of a Brownian particle driven by an arbitrary potential with an exponential temporally decaying superimposed field up to a prescribed threshold. The general system analyzed here describes the sub-threshold signal integration of integrate-and-fire neuron models, of any kind, supplemented by an adaptation-like current, whereas the first-passage-time corresponds to the declaration of a spike. Following our previous studies, we base our analysis on the backward Fokker Planck equation and study the survival probability and the first-passage-time density function in the space of the initial condition. By proposing a series solution we obtain a system of recurrence equations, which given the specific structure of the exponential time-dependent drift, easily admit a simpler Laplace representation. Naturally, the present general derivation agrees with the explicit solution we found previously for the Wiener process in (2012 JPhysA 45 185001). However, to demonstrate the generality of the approach, we further explicitly evaluate the first-passage-time statistics of the underlying Ornstein Uhlenbeck process. To test the validity of the series solution, we extensively compare theoretical expressions with the data obtained from numerical simulations in different regimes. As shown, agreement is precise whenever the series is truncated at an appropriate order. Beyond the fact that both the Wiener and Ornstein Uhlenbeck processes have a direct interpretation in the context of neuronal models, given their ubiquity in different fields, our present results will be of interest in other settings where an additive state-independent temporal relaxation process is being developed as the particle diffuses.Comment: 22 pages (20 pages in the journal version), 3 figures, published in J. Phys.

    A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process

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    We prove that the first passage time density ρ(t)\rho(t) for an Ornstein-Uhlenbeck process X(t)X(t) obeying dX=βXdt+σdWdX=-\beta X dt + \sigma dW to reach a fixed threshold θ\theta from a suprathreshold initial condition x0>θ>0x_0>\theta>0 has a lower bound of the form ρ(t)>kexp[pe6βt]\rho(t)>k \exp\left[-p e^{6\beta t}\right] for positive constants kk and pp for times tt exceeding some positive value uu. We obtain explicit expressions for k,pk, p and uu in terms of β\beta, σ\sigma, x0x_0 and θ\theta, and discuss application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.Comment: 15 pages, 1 figur

    Towards the Modeling of Neuronal Firing by Gaussian Processes

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    This paper focuses on the outline of some computational methods for the approximate solution of the integral equations for the neuronal firing probability density and an algorithm for the generation of sample-paths in order to construct histograms estimating the firing densities. Our results originate from the study of non-Markov stationary Gaussian neuronal models with the aim to determine the neuron's firing probability density function. A parallel algorithm has been implemented in order to simulate large numbers of sample paths of Gaussian processes characterized by damped oscillatory covariances in the presence of time dependent boundaries. The analysis based on the simulation procedure provides an alternative research tool when closed-form results or analytic evaluation of the neuronal firing densities are not available.Comment: 10 pages, 3 figures, to be published in Scientiae Mathematicae Japonica

    Are the input parameters of white-noise-driven integrate-and-fire neurons uniquely determined by rate and CV?

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    Integrate-and-fire (IF) neurons have found widespread applications in computational neuroscience. Particularly important are stochastic versions of these models where the driving consists of a synaptic input modeled as white Gaussian noise with mean μ\mu and noise intensity DD. Different IF models have been proposed, the firing statistics of which depends nontrivially on the input parameters μ\mu and DD. In order to compare these models among each other, one must first specify the correspondence between their parameters. This can be done by determining which set of parameters (μ\mu, DD) of each model is associated to a given set of basic firing statistics as, for instance, the firing rate and the coefficient of variation (CV) of the interspike interval (ISI). However, it is not clear {\em a priori} whether for a given firing rate and CV there is only one unique choice of input parameters for each model. Here we review the dependence of rate and CV on input parameters for the perfect, leaky, and quadratic IF neuron models and show analytically that indeed in these three models the firing rate and the CV uniquely determine the input parameters
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