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The first passage time density of a diffusion process to a time varyingthreshold is of primary interest in different fields. Here we consider aBrownian motion in presence of an exponentially decaying threshold to model theneuronal spiking activity. Since analytical expressions of the first passagetime density are not available, we propose to approximate the curved boundaryby means of a continuous two-piecewise linear threshold. Explicit expressionsfor the first passage time density towards the new boundary are provided. Thenwe introduce different approximating linear threshold and describe how tochoose the optimal one minimizing the distance to the curved boundary and hencethe error in the corresponding passage time density. Theoretical means,variances and coefficients of variation given by our method are then comparedwith empirical quantities from simulated data as well as other firingstatistics derived under the assumption of a small amplitude of thetime-dependent change in the threshold. Finally maximum likelihood and momentestimators of the parameters of the Wiener process are also proposed andcompared on simulated data.

mathematical biosciences and engineering

Approximation of the first passage time density of a Wiener process to  an exponentially decaying threshold by two-piecewise linear threshold.  Application to neuronal spiking activity