Critical Analysis of Dimension Reduction by a Moment Closure Method in a Population Density Approach to Neural Network Modeling

Abstract

Computational techniques within the population density function (PDF) framework have provided time-saving alternatives to classical Monte Carlo simulations of neural network activity. Efficiency of the PDF method is lost as the underlying neuron model is made more realistic and the number of state variables increases. In a detailed theoretical and computational study, we elucidate strengths and weaknesses of dimension reduction by a particular moment closure method (Cai, Tao, Shelley, & McLaughlin, 2004; Cai, Tao, Rangan, & McLaughlin, 2006) as applied to integrate-and-fire neurons that receive excitatory synaptic input only. When the unitary postsynaptic conductance event has a single-exponential time course, the evolution equation for the PDF is a partial differential integral equation in two state variables, voltage and excitatory conductance. In the moment closure method, one approximates the conditional kth centered moment of excitatory conductance given voltage by the corresponding unconditioned moment. The result is a system of k coupled partial differential equations with one state variable, voltage, and k coupled ordinary differential equations. Moment closure at k = 2 works well, and at k = 3 works even better, in the regime of high dynamically varying synaptic input rates. Both closures break down at lower synaptic input rates. Phase-plane analysis of the k = 2 problem with typical parameters proves, and reveals why, no steady-state solutions exist below a synaptic input rate that gives a firing rate of 59 s(1) in the full 2D problem. Closure at k = 3 fails for similar reasons. Low firing-rate solutions can be obtained only with parameters for the amplitude or kinetics (or both) of the unitary postsynaptic conductance event that are on the edge of the physiological range. We conclude that this dimension-reduction method gives ill-posed problems for a wide range of physiological parameters, and we suggest future directions.

Spatiotemporal dynamics of continuum neural fields

“…It is not straightforward to determine how noise at the single cell level translates into noise at the population or network level. One approach is to formulate the dynamics of a population of spiking neurons in terms of the evolution of the probability density of membrane potentialsthe so-called population density method [51,52,70,[191][192][193][194][195][196][197]. Typically, a very simple model of a spiking neuron is used such as the integrate-and-fire (IF) model [48] and the network topology is assumed to be either fully connected or sparsely connected.…”

Stochastic Neural Field Theory and the System-Size Expansion

“…That is, when there are multiple fixed points, the truncated moment equations fail to take into account exponentially small transitions between fixed points, which underlie the asymptotically slow approach to the true steady state of the full probabilistic model (assuming that it exists). It would be interesting to explore the issue of rare event or large deviation statistics within the context of population density methods, where failure of moment closure even occurs in the absence of multiple fixed points [52]. neurons in the ith population can be extracted by operating on a state vector with the number operator Φ † i Φ i and using the commutation relations,…”

“…On the other hand, as the complexity of the individual neuron model increases, the gain in efficiency of the population density method decreases, and this has motivated the development of moment closure schemes. However, as recently shown by Ly and Tranchina [52], considerable care must be taken when carrying out the dimension reduction, since it can lead to an ill-posed problem over a wide range of physiological parameters. That is, the truncated moment equations may not support a steady-state solution even though a steady-state probability density exists for the full system.…”

Practical approximation method for firing-rate models of coupled neural networks with correlated inputs

“…Such approaches have a long history in the physical sciences [47,48] and recently in the life sciences [36,45,49]. Here we provide an alternative approach based on the probability density (or FokkerPlanck) equation of the stochastic neural network, rather than the stochastic integrals we considered in the main text.…”