Generalized Fisher information matrix in nonextensive systems with spatial correlation

Abstract

By using the $q$-Gaussian distribution derived by the maximum entropy method for spatially-correlated $N$-unit nonextensive systems, we have calculated the generalized Fisher information matrix of $g_{{\theta }_n{\theta }_m}$ for $({\theta }_1,{\theta }_2,{\theta }_3)=({\mu }_q,{\sigma }_q^2$, $s$), where ${\mu }_q$, ${\sigma }_q^2$ 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 ${\mu }_q$ is improved (degraded) by a negative (positive) correlation $s$, (2) that of ${\sigma }_q^2$ is worsen with increasing $s$, and (3) that of $s$ is much improved for $s\simeq -1/(N-1)$ or $s\simeq 1.0$ 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 spatially-correlated inputs.

18 pages, 3 figures: revised version accepted in Phys. Rev. E

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