Application of a generalised Levy residence time problem to neuronal dynamics
Abstract
The distribution of bursting lengths of neuron spikes, in a twocomponent integrateandfire 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 γ = 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, nonrelativistic Hamiltonian of quantum mechanics. The distribution of T depends on γt and varies from a Ushaped distribution for small γt to a bellshaped distribution for large γt. 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.
 Publication:

EPL (Europhysics Letters)
 Pub Date:
 February 2004
 DOI:
 10.1209/epl/i2003100915
 Bibcode:
 2004EL.....65..434W