Generalized Rate-Code Model for Neuron Ensembles with Finite Populations
  We have proposed a generalized Langevin-type rate-code model subjected to multiplicative noise, in order to study stationary and dynamical properties of an ensemble containing {\it finite} $N$ neurons. Calculations using the Fokker-Planck equation (FPE) have shown that owing to the multiplicative noise, our rate model yields various kinds of stationary non-Gaussian distributions such as gamma, inverse-Gaussian-like and log-normal-like distributions, which have been experimentally observed. Dynamical properties of the rate model have been studied with the use of the augmented moment method (AMM), which was previously proposed by the author with a macroscopic point of view for finite-unit stochastic systems. In the AMM, original $N$-dimensional stochastic differential equations (DEs) are transformed into three-dimensional deterministic DEs for means and fluctuations of local and global variables. Dynamical responses of the neuron ensemble to pulse and sinusoidal inputs calculated by the AMM are in good agreement with those obtained by direct simulation. The synchronization in the neuronal ensemble is discussed. Variabilities of the firing rate and of the interspike interval (ISI) are shown to increase with increasing the magnitude of multiplicative noise, which may be a conceivable origin of the observed large variability in cortical neurons. 
Hideo Hasegawa (Tokyo Gakugei Univ.)