Two underrated players shed new light on brain function

The Road from Spike Frequency to Mental States is paved with Intermediate Energetic Steps which Can be Defined and Quantified

Published (april 2015) on: The Society for Chaos Theory in Psychology & Life Sciences, Newsletter

DOI: 10.13140/2.1.2181.1527


Here you can find an abridged version: 





The brain electric activity exhibits a power law distribution which appears as a straight line when plotted on logarithmic scales in a log power versus log frequency plot.  The slope of the line is given by a single constant, the power law exponent.  Since a variation in slope may occur during different functional states, the brain currents are said to be multifractal, i.e. characterized by a spectrum of multiple possible exponents.  A role for such non-stationary scaling properties in neural coding has scarcely been taken into account.  However, changes in fractal slopes and in the message content are correlated: in view of energetic arguments, it turns out that modifications in power law exponents are associated with variations in the Rényi’s entropy, which is a generalization of the Shannon’s entropy.  Changes in the Rényi’s entropy, in turn, are able to modify the information transmitted by spikes.  Thus, multifractal systems lead to different probability outcomes based solely on increases or decreases of fractal exponents.   




The brain activity observed at many spatiotemporal scales exhibits fluctuations with complex scaling behavior (Newman), including not only cortical electric oscillations, but also membrane potentials and neurotransmitter release (Milstein, Linkenkaer-Hansen).  In particular, the frequency spectrum of cerebral electric activity displays a scale-invariant behaviour S(f)= 1/fn, where S(f) is the power spectrum, f is the frequency and n is an exponent that equals the negative slope of the line in a log power versus log frequency plot (Van de Ville; Pritchard).  Pink noise can be regarded as an intrinsic property of the brain characterizing a large class of neuronal processes (de Arcangelis); moreover, power law distributions contain information about how large-scale physiological and pathological outcomes (Jirsa) arise from the interactions of many small-scale processes.  It must be emphasized that the fractal slope is not invariant in brain, but is rather characterized by multiple possible exponents (He).  It has been demonstrated that different functional states - spontaneous fluctuations, task-evoked, perceptual and motor activity (Buszaki), cognitive demands (Fetterhoff), ageing (Suckling) - account for variations in power law exponents across cortical regions (Tinker; Wink).  Accordingly, we may view the multifractal cortex as an ensemble of intertwined (mono)fractals, each with its own dimension and scaling slope: the brain is thus regarded as a system of fractal geometry with a complex spectrum of self-exact similarity breakdown, in which scaling exponents mark dynamical transitions between different response regimes (Papo).

Do cortical fluctuations in power law exponents modify the energy of the system?  The answer is positive.  The metabolic activity of the brain is high, accounting for 20% of the energy consumed.  Much of the brain’s vast energy budget is reserved for spontaneous neuronal activity, but perceptual and motor activity, task performance and cognitive demands account for an additional energy consumption of 5%, often confined to small cortical areas (Sengupta).  Indeed, local boosts in spike frequency (in particular beta and gamma waves) cause a transitory increase of energy consumption and free energy production, with a metabolic cost of 6.5 μmol/ATP/gr/min for each spike (Attwell).  Throughout the increases in free energy, the 1/fn exponent varies across brain regions.  Recent papers start to uncover connections between the exponent of a fractal scaling in escape paths from energy basins and the activation free energy (Perkins).  The ongoing fluctuations with complex scale-free properties can thus be absorbed into a free energy symmetry (Friston): the critical slowing implicit in power law scaling of dynamics is mandated by any system that minimizes its energetic expenditure.  The aim of our theoretical paper was to evaluate if increases or decreases in the 1/fn power slope in multifractal systems, besides the above mentioned impact on energy efficiency, might play a role in information processing. 



We build a system equipped with a random variable p in the case of two possibilities with probabilities p and (1-p) (Shannon) and calculated the theoretical values of Rényi’s entropy on the X-axis plotted as a function of p on the Y-axis (Figure A).  Indeed, the basic thermodynamic properties of multifractal systems may be discussed by extending the notion of the thermodynamical Gibbs’ entropy and the information Shannon’s entropy into the more general framework of the Rényi’s entropy.  If p is a probability distribution on a finite set, its Rényi’s entropy of order β is defined to be (Baez):

Where   The term X is a random variable with n possible outcomes and and pi = P (X=i), for i= 1, 2, 3, …n.   

The Rényi’s entropy approaches the Shannon’s entropy as β approaches 1, so that β=1 is defined to be the Shannon’s entropy. 

The Rényi’s entropy is also closely related to the thermodynamic free energy F through the formula: 

Mathematically, it is expressed as follows: the Rényi’s entropy of a system is minus the “1/β-derivative” of its free energy with respect to a quantity. 

Because of its build–in predisposition to account for self–similar systems, such a generalized entropy is an effective tool to describe (multi)fractal systems (Jizba).  The Rényi’s entropy and the scale-free dimensions indeed exhibit a straight relation, because the power law exponent n and the Rényi’s parameter β are correlated: changes in n lead to changes in β, and vice versa (Słomczynski).  Thus, modifications in brain power law exponents are linked with different Rényi’s exponents, which in turn are linked with different curves of the Rényi’s entropy. 

Furthermore, the difference Δp between the curves β=2 and β=1 was plotted against a range of values of Rényi’s entropy (Figure B). 



The solid curves in Figure A illustrate three cases of Rényi’s entropy, with exponents β=1, β=2 and β=.  The curve β=1 stands for the Shannon’s entropy (under ergodic conditions).  The curves of the Rényi’s entropy show that changes in β exponent and in probability distribution occur together, because a variation in β exponent leads to a different probability: as an example, the left right arrow shows the difference in p between the curves β=1 and β=2, at the fixed point of Rényi’s entropy=.55.  As a consequence, if we keep the values of Rényi’s entropy on the Y-axis constant, the different curves display different probabilities on the X-axis.

Figure B points up that different rates of Rényi’s entropy are correlated with a wide spectrum of probabilities p: at each given value of entropy, the shift of the exponent from β=2 to β=1 causes a change in Δp.





Our analysis leads to an unexpected conclusion: the road from spike frequency to mental states and vice versa is paved with intermediate energetic steps which can be defined and quantified.  In order to optimize perceptions and thoughts, the brain is equipped with an intrinsic mechanism of fluctuations with complex temporal and spatial scaling properties: in this context, changes in power law exponents play a crucial role in information processing.  The relation between stimulus, spike trains and transmitted inputs forms the neural code, the crucial tool by which neurons recognize and store the data.  But where is the neural code?  A range of different theories has been offered over the years: rate or temporal codes, latency, relational, synchrony codes, mixtures of them (Gollisch).  We hypothesize that another underrated mechanism might play a role in neural communication: the power law exponent.  Based on energetic constrains, we indeed propose that variations in fractal slope are correlated with changes in the content of the message.  In such a framework, complex scale-free statistics are fixed points of a renormalization flow and can be understood as asymptotic behaviors emerging as the system is rescaled (Fraiman), while cognitive tasks modulate the 1/fn exponents of the brain fluctuation probability function, leading to a shrinking of multifractal spectrum and/or transitions from mono- to multi-fractal distributions (Popivanov). 




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