Effective generators for superpositions of non-Poissonian spike trains
Moritz Deger (Bernstein Center Freiburg, University of Freiburg), Moritz Helias (Inst of Neuroscience and Medicine (INM-6), Research Center Juelich), Stefan Rotter (Bernstein Center Freiburg & Faculty of Biology, University of Freiburg)
If all external spike trains are Poisson processes (PP), their superposition is again a PP, with a rate equal to the sum of the individual rates. To represent the sum of all external inputs, it is, therefore, only necessary to generate a single spike train with a higher rate. In most areas of the neocortex, however, neural spike trains are either more or less regular than a PP . In this case, the superposition (pooled input) is not a PP any more . In fact, our analyses of statistical properties of superpositions of non-Poissonian (NPP) processes, and of the dynamics of leaky-integrate-and-fire neurons driven by such inputs, showed that NPP superpositions exhibit profound differences to the PP, to which neurons are sensitive .
Suppose we can model the external input as N independent and identical renewal processes. To generate the superposition, the naive approach is to generate N realizations of the renewal process, and then collect all the spikes in a pooled spike train. Since this has to be repeated for each of M simulated neurons, the procedure results in computational costs proportional to M*N.
Depending on the details of the modeled system, N can be on the order of 1000. In contrast, in the case of external PP inputs, it suffices to generate a single PP only. Using NPP external inputs thus can slow down a simulation by a factor of N, which is why PPs are commonly used.
Here, we present two optimised algorithms to generate superpositions of NPP spike trains directly : One for gamma processes with integer shape parameter, and one for PPs with dead time. Both generators have a computational cost which is independent of N. The generators exploit a population description of the superimposed processes, require time-discrete simulation, and have been implemented in NEST .
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