Maximum Entropy Principle Underlies Wiring Length Distribution in Brain Networks

Abstract

A brain network comprises a substantial amount of short-range connections with an admixture of long-range connections. The portion of long-range connections in brain networks is observed to be quantitatively dissimilar across species. It is hypothesized that the length of connections is constrained by the spatial embedding of brain networks, yet fundamental principles that underlie the wiring length distribution remain unclear. By quantifying the structural diversity of a brain network using Shannon’s entropy, here we show that the wiring length distribution across multiple species—including Drosophila, mouse, macaque, human, and C. elegans—follows the maximum entropy principle (MAP) under the constraints of limited wiring material and the spatial locations of brain areas or neurons. In addition, by considering stochastic axonal growth, we propose a network formation process capable of reproducing wiring length distributions of the 5 species, thereby implementing MAP in a biologically plausible manner. We further develop a generative model incorporating MAP, and show that, for the 5 species, the generated network exhibits high similarity to the real network. Our work indicates that the brain connectivity evolves to be structurally diversified by maximizing entropy to support efficient interareal communication, providing a potential organizational principle of brain networks.

The fractal brain: scale-invariance in structure and dynamics

“…For example, neural wiring incorporates fractal principles ( Hofman 2016 ; Schröter et al. 2017 ) because the brain is dominated by a tradeoff between wiring economy, on one hand, and efficient communication between neurons, on the other ( Bullmore and Sporns 2012 ; Song et al. 2021 ).…”

“…2012 ; Stevens 2012 ; Ercsey-Ravasz et al. 2013 ; Wang and Clandinin 2016 ; Song et al. 2021 ) or mechanical constraints ( Garcia et al.…”

Information encoded in volumes and areas of dendritic spines is nearly maximal across mammalian brains

“…It was shown that minimization of axonal and dendritic processes leads to an efficient use of metabolic energy in the brain (Attwell and Laughlin 2001; Chklovskii et al 2002), especially when the brain is scaled up in the evolutionary process (Herculano-Houzel 2011; Karbowski 2007; Yu et al 2014). A recent, interesting result shows that the entropy associated with the length of neuronal wiring is maximized (Song et al 2021). This result is complementary to ours, and both results suggest an efficient encoding of long-term information in the whole neuron.…”

The specificity of different-distance connections in human structural connectomes

“…The spatial embedding of the brain network imposes constraints on axonal outgrowth, drives the expression of significant topological features, and ultimately impinges on brain function [15, 16, 17]. Growth trajectories of most axons are confined to small spheres of space due to the spatial distribution of molecules [18, 19, 20, 21], leading to the probability of brain regions being connected decreasing rapidly with the interregional distance [22, 23]. This dominance of short-range connections reduces the material and metabolic consumption required to construct and maintain connections, facilitating the propagation and preservation of network architecture within surviving species under the pressure of limited time, space, and resources [24].…”