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
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
We prove that the first passage time density for an
Ornstein-Uhlenbeck process obeying to reach
a fixed threshold from a suprathreshold initial condition
has a lower bound of the form for positive constants and for times exceeding some
positive value . We obtain explicit expressions for and in terms
of , , and , 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
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?
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 and noise intensity . Different IF models
have been proposed, the firing statistics of which depends nontrivially on the
input parameters and . 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 (, ) 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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