Generalized Fisher information matrix in nonextensive systems with spatial correlation

Abstract

By using the q-Gaussian distribution derived by the maximum entropy method for spatiallycorrelated N -unit nonextensive systems, we have calculated the generalized Fisher information matrix of g θnθm for (θ 1 , θ 2 , θ 3 ) = (µ q , σ 2 q , s), where µ q , σ 2 q and s denote the mean, variance and degree of spatial correlation, respectively, for a given entropic index q. It has been shown from the Cramér-Rao theorem that (1) an accuracy of an unbiased estimate of µ q is improved (degraded)by a negative (positive) correlation s, (2) that of σ 2 q is worsen with increasing s, and (3) that of s is much improved for s ≃ −1/(N − 1) or s ≃ 1.0 though it is worst at s = (N − 2)/2(N − 1). Our calculation provides a clear insight to the long-standing controversy whether the spatial correlation is beneficial or detrimental to decoding in neuronal ensembles. We discuss also a calculation of the q-Gaussian distribution, applying the superstatistics to the Langevin model subjected to spatiallycorrelated inputs.