Probabilistic methods for a linear reaction-hyperbolic system with constant coefficients

Brooks, Elizabeth

doi:10.1214/aoap/1029962811

Cited by 22 publications

(19 citation statements)

References 16 publications

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“…The main differences between their setting and ours are: (a) we have two moving populations with different velocities and different directions of movement, and (b) the initial conditions and the boundary conditions (at the axon soma) are different. The argument of Reed et al (1990) was subsequently made rigorous by Brooks (1999) in the case n = 1, by using a probabilistic method which resembles the stochastic setup of Brown et al (2005). It would be interesting to extend the asymptotic results of Reed et al (1990) and Brooks (1999) to the case of an arbitrary number of species and velocities.…”

Section: Discussionmentioning

confidence: 99%

A dynamical system model of neurofilament transport in axons

Crăciun

Brown

Friedman

2005

Journal of Theoretical Biology

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We develop a dynamical system model for the transport of neurofilaments in axons, inspired by Brown's "stop-and-go" model for slow axonal transport. We use fast/slow time-scale arguments to lower the number of relevant parameters in our model. Then, we use experimental data of Wang and Brown to estimate all but one parameter. We show that we can choose this last remaining parameter such that the results of our model agree with pulse-labeling experiments from three different nerve cell types, and also agree with stochastic simulation results.

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Section: Discussionmentioning

confidence: 99%

A dynamical system model of neurofilament transport in axons

Crăciun

Brown

Friedman

2005

Journal of Theoretical Biology

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“…the telegraph equation (3) becomes the planar heat equation and the density (2) tends to the transition density of the planar standard Brownian motion, as happens in the one-dimensional case (see Kac (1974), Orsingher (1990), or Brooks (1999). The density (2) takes an especially interesting form in polar coordinates, namely…”

Section: We Also Obtain the Following Conditional Distributions For Tmentioning

confidence: 97%

A planar random motion with an infinite number of directions controlled by the damped wave equation

Kolesnik¹,

Orsingher²

2005

J. Appl. Probab.

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We consider the planar random motion of a particle that moves with constant finite speed c and, at Poisson-distributed times, changes its direction θ with uniform law in [0, 2π). This model represents the natural two-dimensional counterpart of the wellknown Goldstein-Kac telegraph process. For the particle's position (X(t), Y (t)), t > 0, we obtain the explicit conditional distribution when the number of changes of direction is fixed. From this, we derive the explicit probability law f (x, y, t) of (X(t), Y (t)) and show that the density p(x, y, t) of its absolutely continuous component is the fundamental solution to the planar wave equation with damping. We also show that, under the usual Kac condition on the velocity c and the intensity λ of the Poisson process, the density p tends to the transition density of planar Brownian motion. Some discussions concerning the probabilistic structure of wave diffusion with damping are presented and some applications of the model are sketched.

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“…the telegraph equation 3becomes the planar heat equation and the density (2) tends to the transition density of the planar standard Brownian motion, as happens in the one-dimensional case (see Kac (1974), Orsingher (1990), or Brooks (1999)). The density (2) takes an especially interesting form in polar coordinates, namely…”

mentioning

confidence: 94%

A planar random motion with an infinite number of directions controlled by the damped wave equation

Kolesnik

Orsingher

2005

Journal of Applied Probability

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We consider the planar random motion of a particle that moves with constant finite speed c and, at Poisson-distributed times, changes its direction θ with uniform law in [0, 2π). This model represents the natural two-dimensional counterpart of the well-known Goldstein–Kac telegraph process. For the particle's position (X(t), Y(t)), t > 0, we obtain the explicit conditional distribution when the number of changes of direction is fixed. From this, we derive the explicit probability law f(x, y, t) of (X(t), Y(t)) and show that the density p(x, y, t) of its absolutely continuous component is the fundamental solution to the planar wave equation with damping. We also show that, under the usual Kac condition on the velocity c and the intensity λ of the Poisson process, the density p tends to the transition density of planar Brownian motion. Some discussions concerning the probabilistic structure of wave diffusion with damping are presented and some applications of the model are sketched.

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Probabilistic methods for a linear reaction-hyperbolic system with constant coefficients

Cited by 22 publications

References 16 publications

A dynamical system model of neurofilament transport in axons

A dynamical system model of neurofilament transport in axons

A planar random motion with an infinite number of directions controlled by the damped wave equation

A planar random motion with an infinite number of directions controlled by the damped wave equation

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