Comparison of methods to calculate exact spike times in integrate-and-fire neurons with exponential currents

Discrete-time neuronal network simulation strategies typically constrain spike times to a grid determined by the computational step size. This approach can have the effect of introducing artificial synchrony [1]. However, time-continuous approaches can be computationally demanding, both with respect to calculating future spike times and to event management, particularly for large network sizes. To address this problem, Morrison et al. [2] presented a general method of handling off-grid spiking in combination with exact subthreshold integration in discrete time driven simulations [3, 4]. Within each time step an event-driven environment is emulated to process incoming spikes, whereas the timing of outgoing spikes is based on interpolation. Therefore, the computation step size is a decisive factor for both integration error and simulation time.

An alternative approach for calculating the exact spike times of integrate-and-fire neurons with exponential currents was recently published by Brette [5]. The problem of accurate detection of the first threshold crossing of the membrane potential is converted into finding the largest root of a polynomial. Common numerical means like Descartes' rule and Sturm's theorem are applicable. Although this approach was developed in the context of event-driven simulations, we take advantage of its ability to predict future threshold crossings in the time-driven environment of NEST [3]. We compare the accuracy of the two approaches in single-neuron simulations and the efficiency in a balanced random network of 10,000 neurons [6]. We show that the network simulation time when using the polynomial method depends only weakly on the computational step size, and the single neuron integration error is independent of it. Although the polynomial method attains the maximum precision expected from double numerics for all input rates and computation step sizes, the interpolation method is more efficient for input rates above a critical value. For applications where a lesser degree of precision is acceptable, the interpolation method is more efficient for all input rates.