Population density methods in two spatial dimensions and application to neural networks with realistic synaptic kinetics.

摘要

We explore the extension of the population density methods to a two-dimensional state-space as a computationally efficient method for simulating networks of neurons. The method is applied to integrate-and-fire neurons with realistic synaptic kinetics. In this method neurons are grouped into populations of similar biophysical properties, and for each population a probability density function (PDF) is constructed. This PDF represents the distribution of neurons over state-space. The evolution equation of the probability density functions is a partial differential-integral equation. To begin with, we model neurons with only excitatory synaptic input. In the case where the unitary postsynaptic events are fast enough to be considered instantaneous (Nykamp and Tranchina, 2000), the PDF is one-dimensional, as the state of a neuron is completely determined by its transmembrane voltage. When synaptic kinetics are not assumed to be fast on the time-scale of the resting membrane time constant, and when the unitary postsynaptic conductance event has a single exponential time course, the state-space and the PDF are 2-dimensional; the state of the neuron is determined by its random membrane voltage and random excitatory postsynaptic conductance.; The population firing rate is given by the integral of the flux of probability per unit conductance across the threshold voltage over all possible excitatory postsynaptic conductance values.; We formulate a pair of coupled partial differential-integral equations, one for the neurons in their non-refractory state and the second one for the neurons in the refractory pool. The higher dimensionality causes an increase in computation time. We tackle this problem numerically by using an operator-splitting method to solve our partial-differential integral equations. We compare our two-dimensional results to Monte-Carlo simulations for simple neural networks and check for their speed and accuracy in such instances. We then extend the population density method by adding inhibitory synaptic input. We consider a method for reduction from three to two dimensions and we apply it to small sample neural networks, as well as to a model orientation selectivity network for 1 hypercolumn of the visual cortex.; Limitations of the method are discussed and possible improvements and directions for future study are suggested.