Stochastic calculus:
application to dynamic bifurcations and threshold crossings


Journal of Statistical Physics 90 227-251 (1998)

Kalvis Jansons

Department of Mathematics, University College London.
Gower Street WC1E 6BT, ENGLAND.

G.D. Lythe

Optique nonlinéaire théorique, Université Libre de Bruxelles CP231,
Bruxelles 1050 BELGIUM

Abstract

For the dynamic pitchfork bifurcation in the presence of white noise, the statistics of the last time at zero are calculated as a function of the noise level and the rate of change of the parameter. The threshold crossing problem used, for example, to model the firing of a single cortical neuron is considered, concentrating on quantities that may be experimentally measurable but have so far received little attention. Expressions for the statistics of pre-threshold excursions, occupation density and last crossing time of zero are compared with results from numerical generation of paths.

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