Stochastic calculus:
application to dynamic bifurcations and threshold crossings


Journal of Statistical Physics 100 1097 (2000)

Kalvis Jansons

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

G.D. Lythe

GISC, Departamento de Matemáticas
Universidad Carlos III de Madrid
Avenida de la Universidad 30
28911 Leganés, Madrid, Spain


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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