Fast numerical methods for simulating large-scale integrate-and-fire neuronal networks

We discuss numerical methods for simulating large-scale, integrate-and-fire (I&F) neuronal networks. Important elements in our numerical methods are (i) a neurophysiologically inspired integrating factor which casts the solution as a numerically tractable integral equation, and allows us to obtain stable and accurate individual neuronal trajectories (i.e., voltage and conductance time-courses) even when the I&F neuronal equations are stiff, such as in strongly fluctuating, high-conductance states; (ii) an iterated process of spike-spike corrections within groups of strongly coupled neurons to account for spike-spike interactions within a single large numerical time-step; and (iii) a clustering procedure of firing events in the network to take advantage of localized architectures, such as spatial scales of strong local interactions, which are often present in large-scale computational models - for example, those of the primary visual cortex. (We note that the spike-spike corrections in our methods are more involved than the correction of single neuron spike-time via a polynomial interpolation as in the modified Runge-Kutta methods commonly used in simulations of I&F neuronal networks.) Our methods can evolve networks with relatively strong local interactions in an asymptotically optimal way such that each neuron fires approximately once in InlineEquation O(N) operations, where N is the number of neurons in the system. We note that quantifications used in computational modeling are often statistical, since measurements in a real experiment to characterize physiological systems are typically statistical, such as firing rate, interspike interval distributions, and spike-triggered voltage distributions. We emphasize that it takes much less computational effort to resolve statistical properties of certain I&F neuronal networks than to fully resolve trajectories of each and every neuron within the system. For networks operating in realistic dynamical regimes, such as strongly fluctuating, high-conductance states, our methods are designed to achieve statistical accuracy when very large time-steps are used. Moreover, our methods can also achieve trajectory-wise accuracy when small time-steps are used.