title: Ill-Posed Point Neuron Models.
abstract: We show that point-neuron models with a Heaviside firing rate function can be ill posed. More specifically, the initial-condition-to-solution map might become discontinuous in finite time. Consequently, if finite precision arithmetic is used, then it is virtually impossible to guarantee the accurate numerical solution of such models. If a smooth firing rate function is employed, then standard ODE theory implies that point-neuron models are well posed. Nevertheless, in the steep firing rate regime, the problem may become close to ill posed, and the error amplification, in finite time, can be very large. This observation is illuminated by numerical experiments. We conclude that, if a steep firing rate function is employed, then minor round-off errors can have a devastating effect on simulations, unless proper error-control schemes are used.
authors.affiliation: Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, P.O. Box 5003, Ås, 1432, Norway. bjorn.f.nielsen@nmbu.no.
authors.forename: Bjørn Fredrik
authors.initials: BF
authors.lastname: Nielsen
authors.affiliation: Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, P.O. Box 5003, Ås, 1432, Norway.
authors.forename: John
authors.initials: J
authors.lastname: Wyller
keywords: Ill posed; Numerical solution; Point-neuron models