A Generalized Rate Model for Neuronal Ensembles

There has been a long-standing controversy whether information in neuronal networks is carried by the firing rate code or by the firing temporal code. The current status of the rivalry between the two codes is briefly reviewed with the recent studies such as the brain-machine interface (BMI). Then we have proposed a generalized rate model based on the {\it finite} $N$-unit Langevin model subjected to additive and/or multiplicative noises, in order to understand the firing property of a cluster containing $N$ neurons. The stationary property of the rate model has been studied with the use of the Fokker-Planck equation (FPE) method. Our rate model is shown to yield various kinds of stationary distributions such as the interspike-interval distribution expressed by non-Gaussians including gamma, inverse-Gaussian-like and log-normal-like distributions. The dynamical property of the generalized rate model has been studied with the use of the augmented moment method (AMM) which was developed by the author [H. Hasegawa, J. Phys. Soc. Jpn. 75 (2006) 033001]. From the macroscopic point of view in the AMM, the property of the $N$-unit neuron cluster is expressed in terms of {\it three} quantities; $\mu$, the mean of spiking rates of $R=(1/N) \sum_i r_i$ where $r_i$ denotes the firing rate of a neuron $i$ in the cluster: $\gamma$, averaged fluctuations in local variables ($r_i$): $\rho$, fluctuations in global variable ($R$). We get equations of motions of the three quantities, which show $\rho \sim \gamma/N$ for weak couplings. This implies that the population rate code is generally more reliable than the single-neuron rate code. Our rate model is extended and applied to an ensemble containing multiple neuron clusters.