Integrate-and-fire neurons with stochastic thresholds

Wilhelm Braun

We consider a leaky integrate-and-fire neuron with deterministic subthreshold
dynamics and a firing threshold that evolves as an Ornstein-Uhlenbeck process.
The formulation of this minimal model is motivated by the experimentally
observed widespread variation of neural firing thresholds.  We show
numerically that the mean first-passage time can depend nonmonotonically on
the noise amplitude.  For sufficiently large values of the correlation time of
the stochastic threshold the mean first-passage time is maximal for
nonvanishing noise.  We provide an explanation for this effect by analytically
transforming the original model into a first-passage-time problem for Brownian
motion.  This transformation also allows for a perturbative calculation of the
first-passage-time histograms.  In turn this provides quantitative insights
into the mechanisms that lead to the nonmonotonic behavior of the mean
first-passage time.  The perturbation expansion is in excellent agreement with
direct numerical simulations.  The approach developed here can be applied to
any deterministic subthreshold dynamics and any Gauss-Markov processes for the
firing threshold.  This opens up the possibility to incorporate biophysically
detailed components into the subthreshold dynamics, rendering our approach a
powerful framework that sits between traditional integrate-and-fire models and
complex mechanistic descriptions of neural dynamics.