Fractals in the nervous system: conceptual implications for theoretical neuroscience

Abstract

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power-law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review.

Neuronal long-range temporal correlations and avalanche dynamics are correlated with behavioral scaling laws

“…The metastability of critical systems maximizes their dynamic range (57), storage capacity (58), and computational power. Fractal selfsimilarity, power-law scaling behavior, and "1/f noise" at the phenomenological level are typical for systems exhibiting "avalanche dynamics" and operating in a critical (37) or self-organized critical state (59). Nevertheless, the fact that numerous complex systems exhibit similar dynamics raises the question of whether fractal neuronal dynamics are an epiphenomenon without functional relevance.…”

“…Neuronal avalanches characterize spontaneous neuronal network activity in organotypic cultures (32), brain slices in vitro (35), and monkey (34) and human cortex (36) in vivo. In monkey cortex, the avalanches are delimited by cycles of ongoing neuronal oscillations (34) showing that in addition to LRTCs (12), neuronal avalanches also coexist with neuronal oscillations.The power-law scaling behavior and fractal properties of neuronal LRTCs and avalanches strongly suggest that the brain operates near a critical state (12,30,32,33,37). Computational modeling predicts that LRTCs and neuronal avalanches are coupled (38) and suggests that they coemerge from neuronal interactions in a critical regime (30).…”

“…The power-law scaling behavior and fractal properties of neuronal LRTCs and avalanches strongly suggest that the brain operates near a critical state (12,30,32,33,37). Computational modeling predicts that LRTCs and neuronal avalanches are coupled (38) and suggests that they coemerge from neuronal interactions in a critical regime (30).…”

From brain states to mental phenomena via phase space transitions and renormalization group transformation: proposal of a theory

“…According to Critical Theory of Statistical Physics, a system's dynamics can encounter singularities along its path through state space, which trigger sudden phase transitions to new macroscopic configurations with distinctly novel properties. The occurrence of such singularities in neural dynamics and associated phase transitions is by now amply established for all levels of neural system organization, and is documented in recent reviews by Chialvo (2010), Tagliazucchi andChialvo (2011), andWerner (2010). Brain phase transitions between active-conscious and unconscious states are observed at critical values of anesthetic concentration (Steyn-Ross and Steyn-Ross 2010).…”

“…Their common features are inverse power-law statistical distributions, multiplicity of scales, manifestations of non-stationary, and non-ergodic statistical processes (West and Grigolini 2011; for review of neurobiological data, see : Werner 2010).…”