Effects of correlated variability on information entropies in nonextensive systems

Abstract

We have calculated the Tsallis entropy and Fisher information matrix (entropy) of spatially correlated nonextensive systems, by using an analytic non-Gaussian distribution obtained by the maximum entropy method. The effects of the correlated variability on the Fisher information matrix are shown to be different from those on the Tsallis entropy. The Fisher information is increased (decreased) by a positive (negative) correlation, whereas the Tsallis entropy is decreased with increasing absolute magnitude of the correlation, independently of its sign. This fact arises from the difference in their characteristics. It implies from the Cramér-Rao inequality that the accuracy of an unbiased estimate of fluctuation is improved by a negative correlation. A critical comparison is made between the present study and previous ones employing the Gaussian approximation for the correlated variability due to multiplicative noise.

Generalized Fisher information matrix in nonextensive systems with spatial correlation

“…The point (iii) shows that if input information is carried by synchrony within the population code hypothesis [50,51], its decoding accuracy may be improved either by small or large correlation, independently of q [the point (v)]. Our calculation concerns the long-standing controversy on a role of the synchrony in neuronal ensembles [5]- [12].…”

“…(1) [24,25]. We derive the PDF, p(x), by using the OLM-MEM [29] for the Tsallis entropy, imposing the constraints given by [49] …”

“…The calculated GFI is expected to provide us with a clear insight to the controversy on a role of the spatial correlation discussed above. Quite recently, we have pointed out the possibility that input information to neuronal ensembles may be carried not only by mean but also by variance and/or correlation in firing rate within the population code hypothesis [50,51]. The inverse of the calculated GFI matrix expresses an accuracy of decoding when input information is carried by such population codes.…”

“…II. Section VI is devoted to conclusion with the relevance of our calculation to decoding in neuronal population code [50,51].…”

“…(4) reduces to the conventional one. In a previous paper [49], we discussed the effect of the spatial correlation on the Tsallis entropy and the GFI, calculating S q and g θθ for θ = µ q , where µ q stands for mean value [Eq. (6)].…”