Generalized Fisher information matrix in nonextensive systems with spatial correlation.

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 gthetanthetam for (theta1,theta2,theta3)=(muq,sigmaq2,s) , where muq, sigmaq2, 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 muq is improved (degraded) by a negative (positive) correlation s, (2) that of sigmaq2 is worsen with increasing s, and (3) that of s is much improved for s approximately= -1/(N-1) or s approximately 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.