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, 9 (1), e1002872

Dynamic Finite Size Effects in Spiking Neural Networks

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Dynamic Finite Size Effects in Spiking Neural Networks

Michael A Buice et al. PLoS Comput Biol.

Abstract

We investigate the dynamics of a deterministic finite-sized network of synaptically coupled spiking neurons and present a formalism for computing the network statistics in a perturbative expansion. The small parameter for the expansion is the inverse number of neurons in the network. The network dynamics are fully characterized by a neuron population density that obeys a conservation law analogous to the Klimontovich equation in the kinetic theory of plasmas. The Klimontovich equation does not possess well-behaved solutions but can be recast in terms of a coupled system of well-behaved moment equations, known as a moment hierarchy. The moment hierarchy is impossible to solve but in the mean field limit of an infinite number of neurons, it reduces to a single well-behaved conservation law for the mean neuron density. For a large but finite system, the moment hierarchy can be truncated perturbatively with the inverse system size as a small parameter but the resulting set of reduced moment equations that are still very difficult to solve. However, the entire moment hierarchy can also be re-expressed in terms of a functional probability distribution of the neuron density. The moments can then be computed perturbatively using methods from statistical field theory. Here we derive the complete mean field theory and the lowest order second moment corrections for physiologically relevant quantities. Although we focus on finite-size corrections, our method can be used to compute perturbative expansions in any parameter.

Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Figure 1
Figure 1. Phase model.
A. Numerical computations (green line) and analytical predictions (black line) for formula image (top), formula image (middle), formula image (bottom) of formula image for formula image, formula image, formula image. B. Numerical computations (green line) and analytical predictions (black line) for formula image (top), formula image (middle), formula image (bottom) of formula image for formula image, formula image, formula image. C. Numerical computations (green line) and analytical predictions (black line) for formula image (top), formula image (middle), formula image (bottom) of formula image for formula image, formula image, formula image. D. Numerical computations (green line) and analytical predictions (black line) for formula image (top), formula image (middle), formula image (bottom) of formula image for formula image, formula image, formula image, where the “Poisson” contribution has been subtracted. E. Two-time correlator formula image for formula image, formula image, formula image, and formula image. F. Equal time correlators in a heterogeneous network; formula image and formula image for formula image, formula image, formula image and formula image. formula image is taken from the interval formula image for each neuron. Ensemble averages for all simulations are taken over formula image samples.
Figure 2
Figure 2. Quadratic integrate-and-fire model.
A. Numerical computations (green line) and analytical predictions (black line) for formula image for formula image, formula image, formula image for formula image (top), formula image (middle), formula image (bottom) neurons. B. Numerical computations (green line) and analytical predictions (black line) for formula image for formula image, formula image, formula image for formula image (top), formula image (middle), formula image (bottom) neurons. C. Numerical computations (green line) and analytical predictions (black line) for formula image (top) and formula image (bottom) for formula image, formula image, formula image, formula image. D. formula image (top) and formula image (bottom) for formula image, formula image, formula image, formula image, where the Poisson contribution has been subtracted. E. Two-time correlator formula image for formula image, formula image, formula image, and formula image. F Equal time correlators in a heterogeneous network; formula image and formula image for formula image, formula image, formula image and formula image. formula image is taken from the interval formula image for each neuron. Ensemble average for all simulations are taken over formula image samples.
Figure 3
Figure 3. Numerical computations (green line) and analytical predictions (black line) of the firing rate fluctuations for the quadratic integrate-and-fire model for , , for (top), (middle), (bottom) neurons with Poisson contribution subtracted.
Ensemble average is taken over formula image samples.
Figure 4
Figure 4. Vertices of the Feynman Rules for the neural models.
Figure 5
Figure 5. Feynman diagrams for the connected two point correlation functions in the neural field models.
By row they are formula image, formula image, and formula image.

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