Population density equations for stochastic processes with memory kernels

Abstract

We present a method for solving population density equations (PDEs)-a mean-field technique describing homogeneous populations of uncoupled neurons-where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation-a recent result from random network theory-describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky-and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process.

Computational geometry for modeling neural populations: From visualization to simulation

“…It suggests that the right-hand side of Eq 5—representing the master equation of a Poisson process—can be replaced by more general forms without affecting the left-hand side of the equation that allows use of the method of characteristics. Indeed, recently we have considered a generalization to spike trains generated by non-Markov processes [23]. This generalizes the right-hand side of Eq 2, but leaves the left-hand side unchanged, and in [23] we show explicitly that for one dimensional densities the method discussed here extends to non-Markov renewal processes.…”

“…Indeed, recently we have considered a generalization to spike trains generated by non-Markov processes [23]. This generalizes the right-hand side of Eq 2, but leaves the left-hand side unchanged, and in [23] we show explicitly that for one dimensional densities the method discussed here extends to non-Markov renewal processes. The generalization of Eq 2 requires a convolution over the recent history of the density, using a kernel whose shape is dependent on the renewal process.…”

Computational Geometry for Modeling Neural Populations: from Visualization to Simulation

“…This generalizes 725 the right-hand side of Eq. 10, but leaves the left-hand side unchanged, and in [23] we 726 show explicitly that for one dimensional densities the method discussed here extends to 727 non-Markov renewal processes. The generalization of Eq.…”

“…For a given bin k it amounts to finding out what bins are covered by its boundaries after a translation by an amount of h, the synaptic efficacy. Figure reprinted from [23] with permission.…”

“…13 -representing the master equation of a Poisson process -can 722 be replaced by more general forms without affecting the left-hand side of the equation 723 that allows use of the method of characteristics. Indeed, recently we have considered a 724 generalization to spike trains generated by non-Markov processes [23]. This generalizes 725 the right-hand side of Eq.…”