Mean-Field Approximations for Coupled Populations of Generalized Linear Model Spiking Neurons with Markov Refractoriness

• Published in: Neural computation Vol. 21; no. 5; pp. 1203 - 1243
• Main Authors: Toyoizumi, Taro, Rad, Kamiar Rahnama, Paninski, Liam
• Format: Journal Article
• Language: English
• Published: One Rogers Street, Cambridge, MA 02142-1209, USA MIT Press 01.05.2009; MIT Press Journals, The
• Subjects: Action Potentials - physiology; Animals; Applied sciences; Approximation; Artificial intelligence; Biological and medical sciences; Computer science; control theory; systems; Computer Simulation; Correlation analysis; Exact sciences and technology; Fundamental and applied biological sciences. Psychology; General aspects; Learning and adaptive systems; Linear inference, regression; Linear Models; Markov Chains; Mathematics; Mathematics in biology. Statistical analysis. Models. Metrology. Data processing in biology (general aspects); Miscellaneous; Models, Neurological; Monte Carlo simulation; Nerve Net - physiology; Neural networks; Neurons; Neurons - physiology; Probability and statistics; Sciences and techniques of general use; Statistics; Time Factors; Input output; Spiking neuron; Neural computation; Neuronal discharge; Differential equation; Linear model; Delay; Graph connectivity; Behavior; Coupling; Spike; Monte Carlo method; Mean field approximation; Stimulus; Filtering; Fitting; Interaction; Firing rate; Markov model; Neural network; Mean estimation; Steady state; Statistical method; Linear process; Correlation analysis; Linear approximation; Linear filter; Non linear model; Point process; Cross correlation
• ISSN: 0899-7667; 1530-888X
• DOI: 10.1162/neco.2008.04-08-757

There has recently been a great deal of interest in inferring network connectivity from the spike trains in populations of neurons. One class of useful models that can be fit easily to spiking data is based on generalized linear point process models from statistics. Once the parameters for these models are fit, the analyst is left with a nonlinear spiking network model with delays, which in general may be very difficult to understand analytically. Here we develop mean-field methods for approximating the stimulus-driven firing rates (in both the time-varying and steady-state cases), auto- and cross-correlations, and stimulus-dependent filtering properties of these networks. These approximations are valid when the contributions of individual network coupling terms are small and, hence, the total input to a neuron is approximately gaussian. These approximations lead to deterministic ordinary differential equations that are much easier to solve and analyze than direct Monte Carlo simulation of the network activity. These approximations also provide an analytical way to evaluate the linear input-output filter of neurons and how the filters are modulated by network interactions and some stimulus feature. Finally, in the case of strong refractory effects, the mean-field approximations in the generalized linear model become inaccurate; therefore, we introduce a model that captures strong refractoriness, retains all of the easy fitting properties of the standard generalized linear model, and leads to much more accurate approximations of mean firing rates and cross-correlations that retain fine temporal behaviors.