Europhys. Lett.
Volume 65, Number 3, February 2004
Page(s) 434 - 439
Section Interdisciplinary physics and related areas of science and technology
Published online 01 January 2004
Europhys. Lett., 65 (3) , pp. 434-439 (2004)
DOI: 10.1209/epl/i2003-10091-5

Application of a generalised Levy residence time problem to neuronal dynamics

D. Waxman1 and J. F. Feng2

1  Centre for the Study of Evolution, University of Sussex Brighton BN1 9QG, Sussex, UK
2  Department of Informatics, University of Sussex Brighton BN1 9QG, Sussex, UK

(Received 22 September 2003; accepted in final form 21 November 2003)

The distribution of bursting lengths of neuron spikes, in a two-component integrate-and-fire model, is investigated. The stochastic process underlying this model corresponds to a generalisation of the Brownian motion underlying Levy's arcsine law of residence times. The generalisation involves the inclusion of a quadratic potential of strength $\gamma$ and $\gamma=0$ corresponds to Levy's original problem. In the generalised problem, the distribution of the residence times, T, over a time window t, is related to spectral properties of a complex, non-relativistic Hamiltonian of quantum mechanics. The distribution of T depends on $\gamma t$ and varies from a U-shaped distribution for small $\gamma t$ to a bell-shaped distribution for large $\gamma t$. The first two moments of T of the generalised problem are explicitly calculated and the crossover point between the two forms of the distribution is calculated. The distribution of residence times is shown to be independent of the magnitude of the stochastic force. This corresponds, in the neuron model, to exactly balanced synaptic inputs and, in this case, the distribution of residence times contains no information on synaptic inputs.

87.19.La - Neuroscience.
05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.
87.18.Sn - Neural networks.

© EDP Sciences 2004