Application of a generalised Levy residence time problem to neuronal dynamics

– 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 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, non-relativistic Hamiltonian of quantum mechanics. The distribution of T depends on γt and varies from a U-shaped distribution for small γt to a bell-shaped 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. Introduction. – After over a century of neurophysiological research [1], we still do not understand the principles by which a stimulus such as an odour, image or sound is represented within the nervous system by a distributed set of neural states. There is little doubt that much of the information processing power of the nervous system resides in the activities of neural spikes (electrical pulses in the temporal domain). While a large numbers of detailed analyses of completely deterministic or random spikes have been made, such results cannot address issues of the functional roles of various bursting patterns of neurons that have been widely observed in experiments [2]. If we can unlock the principles, by which information is encoded within these various patterns of spikes, we may actually be able to understand how the nervous system works. (∗) E-mail: [email protected]