1,721,000 research outputs found
A continuous Markovian model for neuronal activity
No full textA diffusion model for the description of neurons' membrane potential fluctuations is proposed. Though retaining the well known feature consisting of the spontaneous exponential decay of the membrane potential to its resting value, the model discussed differs substantially from the ones in the current literature. Moreover, the Fokker-Planck equation now describing the membrane potential fluctuations is singular. The neuron's firing times probability density function is calculated in closed form as in a first passage time problem, and its expectation value and variance are evaluated. A detailed study of the mode of the firing times probability density function as related to the noise's intensity is performed. Some other auxiliary results are also obtained
A diffusion model for population growth in random environment
No full textThe growth of a population in a randomly varying environment is modeled by replacing the Malthusian growth rate with a delta-correlated normal process. The population size is then shown to be a random process, lognormally distributed, obeying a diffusion equation of the Fokker-Planck type. The first passage time p.d.f. through any arbitrarily assigned value and the probability of absorption are derived. The asymptotic behavior of the population size is investigated
A note on growth processes in random environment
No full textThe purpose of this note is to discuss certain features of population growth models carlier proposed and to construct an alternative diffusion model for regulated growth in random environment. This model is shown to be the analogue of the Malthusian one, although it is a generalization of the latter due to the presence of regulation
Diffusion approximation and first passage time problem for a model neuron
No full textA diffusion equation for the transition p.d.f. describing the time evolution of the membrane potential for a model neuron, subjected to a Poisson input, is obtained, without breaking up the continuity of the underlying random function. The transition p.d.f. is calculated in a closed form and the average firing interval is determined by using the steady-state limiting expression of the transition p.d.f. The Laplace transform of the first passage time p.d.f. is then obtained in terms of Parabolic Cylinder Functions as solution of a Weber equation, satisfying suitable boundary conditions. A continuous input model is finally investigated
Growth with regulation in random environment
No full textThe diffusion model for a population subject to Malthusian growth is generalized to include regulation effects. This is done by incorporating a logarithmic term in the regulation function in a way to obtain, in the absence of noise, an S-shaped growth law retaining the qualitative features of the logistic growth curve. The growth phenomenon is modeled as a diffusion process whose transition p.d.f. is obtained in closed form. Its steady state behavior turns out to be described by the lognormal distribution. The expected values and the mode of the transition p.d.f. are calculated, and it is proved that their time course is also represented by monotonically increasing functions asymptotically approaching saturation values. The first passage time problem is then considered. The Laplace transform of the first passage time p.d.f. is obtained for arbitrary thresholds and is used to calculate the expected value of the first passage time. The inverse Laplace transform is then determined for a threshold equal to the saturation value attained by the population size in the absence of random components. The probability of absorption for an arbitrary barrier is finally calculated as the limit of the absorption probability in a two-barrier problem
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