Application of a generalised Levy residence time problem to neuronal dynamics

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

Corresponding author: D.Waxman@sussex.ac.uk

Received: 22 September 2003
Accepted: 21 November 2003

Abstract

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 γ and 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 and varies from a U-shaped distribution for small to a bell-shaped distribution for large . 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.