The fractal brain: scale-invariance in structure and dynamics

Abstract

The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.

Networks' modulation: How different structural network properties affect the global synchronization of coupled Kuramoto oscillators

“…From a neuroscience point of view Scale-Free networks play an important role in the structure and function of mammal brains, see for example [30] (a study on the scale-free dynamics and the emergence of collective organisation occurs in rodents) or [31] (investigating the fractal structure of the human brain and its dynamics). Furthermore, in Alzheimer patients' brain, the functional connectivity structure is found to exhibit properties similar to Random network graphs (see e.g.…”

Multiscale Approaches to Music and the Embodied Brain

“…A succinct definition for the presence of scaling laws can be provided as; "An abstract object is scale invariant or scale free if relevant features of the object remain invariant under dilations of the object along a set of dimensions, such as space, time, node degree, distance etc." [13], p.3. Put more simply, scaling phenomena are events at different scales that show topological or statistical self-similarity and statistical dependence on the measurement apparatus [14], and it has been insisted elsewhere that it is only in virtue of such scaling laws that a system can be said to be self-organising [5].…”

“…An important distinction that is common practice to make in the literature on scaling laws in the life sciences is between scale free structure and scale free dynamics [13]. Scale free structures in complex systems are characterised by geometric, topological, or connective regularities at different levels that are invariant at temporal or spatial scales.…”

“…Common to music, speech and song is hierarchical temporal structure [22,23]. It has been suggested that similarly organised structures and principles among brains, bodies and environments enable organisms to thrive in their environment [13], and musical activity and performance may be one such manifestation of this similarly organised thriving through mediating multiscale and scaling mechanisms. Investigations into hierarchical structures in music have been carried out in numerous registers, from the fundamental organisational principles across music and song [24] to the specific statistical treatments of nested properties within musical structures [25,26].…”

“…1/f scaling has been demonstrated at the micro-scale of the brain in neuronal activity where it is suggested that this 'optimal complexity' exhibited by 1/f signals allows for efficient neuronal coding and adaptation to natural stimuli [42]. It has reliably been shown to be a multiscale dynamic insofar as it spans the micro-, meso-and macro-scales of the brain [13], fluctuations across the body [72,73], and across multiple bodies [74]. Given its universality across systems and social phenomena, it is reasonable to turn specific attention from natural stimuli that exhibit 1/f activity to specific musical stimuli and how such scaling is made manifest in these musical contexts.…”