The informational entropy endowed in cortical oscillations

Abstract

A two-dimensional shadow may encompass more information than its corresponding three-dimensional object. Indeed, if we rotate the object, we achieve a pool of observed shadows from different angulations, gradients, shapes and variable length contours that make it possible for us to increase our available information. Starting from this simple observation, we show how informational entropies might turn out to be useful in the evaluation of scale-free dynamics in the brain. Indeed, brain activity exhibits a scale-free distribution that leads to the variations in the power law exponent typical of different functional neurophysiological states. Here we show that modifications in scaling slope are associated with variations in Rényi entropy, a generalization of Shannon informational entropy. From a three-dimensional object's perspective, by changing its orientation (standing for the cortical scale-free exponent), we detect different two-dimensional shadows from different perception angles (standing for Rényi entropy in different brain areas). We show how, starting from known values of Rényi entropy (easily detectable in brain fMRIs or EEG traces), it is feasible to calculate the scaling slope in a given moment and in a given brain area. Because changes in scale-free cortical dynamics modify brain activity, this issue points towards novel approaches to mind reading and description of the forces required for transcranial stimulation.

Keywords: Central nervous system; Nervous system; Power laws; Rényi entropy; Scale-free; Shadows.

Fig. 1

A toy under a light source. In a, the 2D shadow offers more details of the 3D object than the view on the toy from above. b, c Show how toy rotations lead to shadows with different shapes, and, consequently, with different information content. Therefore, a complete 360° rotation of the object allows a complete evaluation of the shadows’ shapes

Fig. 2

Renyi entropy’ circle perimeter and its values (computed by the Renyi.z function in version 3.2.3 of R software) are superimposed, respectively, to the moving toy and its shadow. Left part: Cosine theorem for the evaluation of travel length Rényi entropies in a model of two-dimensional brain (modified from Van Essen 2005). Right part: simulated Rényi values for each circle’s perimeter values for n = 60 and a Rényi exponent = 4; $\theta$ is generated from a uniform distribution in (0, 1) and multiplied by 360. The Rényi entropy values’ bandwidth is around 4.6