1,721,013 research outputs found
The Ornstein-Uhlenbeck process as a model for neuronal activity
No full textMean and variance of the first passage time through a constant boundary for the Ornstein-Uhlenbeck process are determined by a straight-forward differentiation of the Laplace transform of the first passage time probability density function. The results of some numerical computations are discussed to shed some light on the input-output behavior of a formal neuron whose dynamics is modeled by a diffusion process of Ornstein-Uhlenbeck type
On the transformation of diffusion equations and boundaries into the Kolmogorov equation for the Wiener process
No full textThe transformation problem of Kolmogorov equations for one-dimensional diffusion processes to the Kolmogorov equation for the standard Wiener process analysed by I. D. Cherkasov [Teor. Veroyatn. Primen. 2, 384–388 (1957; Zbl. 81, 135)] and L. M. Ricciardi [J. Math. Anal. Appl. 54, 185–199 (1976; Zbl. 361.60043)] is reconsidered in the light of the contributions successively due to G. W. Bluman [SIAM J. Appl. Math. 39, 239–247 (1980; Zbl. 448.60056)] and I. D. Cherkasov [Sov. Math., Dokl. 21, 175–180 (1980; Zbl. 458.60075)]. In particular, the problem of boundary transformations is raised. Two examples are discussed in which the standard Wiener process is transformed first into itself and then into the Feller process
On the transformation of diffusion equations and boundaries into the Kolmogorov equation for the Wiener process
No full textThe transformation problem of Kolmogorov equations for one-dimensional diffusion processes to the Kolmogorov equation for the standard Wiener process analysed by I. D. Cherkasov [Teor. Veroyatn. Primen. 2, 384–388 (1957; Zbl. 81, 135)] and L. M. Ricciardi [J. Math. Anal. Appl. 54, 185–199 (1976; Zbl. 361.60043)] is reconsidered in the light of the contributions successively due to G. W. Bluman [SIAM J. Appl. Math. 39, 239–247 (1980; Zbl. 448.60056)] and I. D. Cherkasov [Sov. Math., Dokl. 21, 175–180 (1980; Zbl. 458.60075)]. In particular, the problem of boundary transformations is raised. Two examples are discussed in which the standard Wiener process is transformed first into itself and then into the Feller process
On the probability densities of an Ornstein-Uhlenbeck process with reflecting boundaries
No full textWe show that the transition p.d.f. of the Ornstein-Uhlenbeck process with a reflection condition at an assigned state S is related by integral-type equations to the free transition p.d.f., to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to S and to the probabi1ity current. Such equation, that are seen to be useful also for computationa1 purposes, yield as an immediate consequence all known closed form results for Ornstein-Uhlenbeck process
On the probability densities of an Ornstein-Uhlenbeck process with reflecting boundaries
No full textWe show that the transition p.d.f. of the Ornstein-Uhlenbeck process with a reflection condition at an assigned state S is related by integral-type equations to the free transition p.d.f, to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to S and to the probability current. Such equations, which are also useful for computational purposes, yield as an immediate consequence all known closed-form results for Wiener and Ornstein-Uhlenbeck processes
On the evaluation of first-passage-time densities for diffusion processes
No full textUse of a Volterra second-kind integral equation is made to evaluate first passage time probability density functions through time varying boundaries for diffusion processes. The solutions are constructed in the form of infinite series whose terms are expressed as multidimensional integrals. An evaluation of such solutions is provided for the cases of Wiener and Ornstein-Uhlenbeck processes by standard numerical procedures, and by a Monte Carlo method. Results are discussed with reference to other existing computational methods
On an integral equation for first-passage-time probability densities
No full textWe prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(t)=a+bt1/p (p ≥ 2, a, b ∈ R)) for which no explicit analytical results have previously been available
Diffusion approximation and first passage time problem for a model neuron. II. Outline of a computational method
No full textTo account for nonstationary effects the customary diffusion approximations in neuro-biology have to be modified in order to include time-varying terms in the expressions of drift and infinitesimal variance. The evaluation of the statistics of the firing time thus generally requires the use of numerical procedures. In this paper it is shown how one can extend a method due to Durbin (1971) to the case of temporally inhomogeneous diffusion equations by employing a transformation technique. A few simple examples are briefly discussed
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