Growth-driven percolations: the dynamics of connectivity in neuronal systems

Abstract

The quintessential property of neuronal systems is their intensive patterns of selective synaptic connections. The current work describes a physics-based approach to neuronal shape modeling and synthesis and its consideration for the simulation of neuronal development and the formation of neuronal communities. Starting from images of real neurons, geometrical measurements are obtained and used to construct probabilistic models which can be subsequently sampled in order to produce morphologically realistic neuronal cells. Such cells are progressively grown while monitoring their connections along time, which are analysed in terms of percolation concepts. However, unlike traditional percolation, the critical point is verified along the growth stages, not the density of cells, which remains constant throughout the neuronal growth dynamics. It is shown, through simulations, that growing beta cells tend to reach percolation sooner than the alpha counterparts with the same diameter. Also, the percolation becomes more abrupt for higher densities of cells, being markedly sharper for the beta cells.

A framework for modeling the growth and development of neurons and networks

“…The prototypical example of the construction approach is L-Systems, a recursive procedure initially invented for modeling plant branching structures (Lindenmayer, 1968 ), which has been successfully applied to neural morphologies (Hamilton, 1993 ; Mulchandani, 1995 ; Ascoli et al, 2001 ). Other methods have been proposed like probabilistic branching models (van Pelt and Verwer, 1983 ; Kliemann, 1987 ), Markov models (Samsonovich and Ascoli, 2005 ), Monte Carlo processes (da Fontoura Costa and Coelho, 2005 ), or diffusion limited aggregation (Luczak, 2006 ). But as successful as they are in reproducing neuronal shapes, these models provide very little insight into the fundamental growth mechanisms leading to cortex formation.…”

Percolation may explain efficiency, robustness, and economy of the brain

“…The only difference lies in that the space where connections can form changes from the lattice space of the stone to the random graph characterized by static connectivity. In decades, the equivalence relation between brain connectivity formation and the percolation on random graphs have attracted extensive explorations in biology ( Bordier, Nicolini, & Bifone, 2017 ; Carvalho et al, 2020 ; Del Ferraro et al, 2018 ; Kozma & Puljic, 2015 ; Lucini, Del Ferraro, Sigman, & Makse, 2019 ; Zhou, Mowrey, Tang, & Xu, 2015 ) and physics ( Amini, 2010 ; Breskin et al, 2006 ; Cohen et al, 2010 ; Costa, 2005 ; da Fontoura Costa & Coelho, 2005 ; da Fontoura Costa & Manoel, 2003 ; Eckmann et al, 2007 ; Stepanyants & Chklovskii, 2005 ), serving as a promising direction to study brain criticality, neural collective dynamics, optimal neural circuitry, and the relation between brain anatomy and functions.…”

“…Perhaps because of these advantages, percolation theory has attracted emerging interest in neuroscience. From early explorations that combine limited neural morphology data with computational simulations to analyze percolation ( Costa, 2005 ; da Fontoura Costa & Coelho, 2005 ; da Fontoura Costa & Manoel, 2003 ; Stepanyants & Chklovskii, 2005 ) to more recent works that study percolation directly on the relatively small-scale and coarse-grained brain connectome and electrically stimulated neural dynamics data captured from living neural networks (e.g., primary neural cultures in rat hippocampus) ( Amini, 2010 ; Breskin et al, 2006 ; Cohen et al, 2010 ; Eckmann et al, 2007 ), the efforts from physics have inspired numerous follow-up explorations in neuroscience ( Bordier et al, 2017 ; Carvalho et al, 2020 ; Del Ferraro et al, 2018 ; Kozma & Puljic, 2015 ; Lucini et al, 2019 ; Zhou et al, 2015 ). These studies capture an elegant and enlightening view about optimal neural circuitry, neural collective dynamics, criticality, and the relation between brain connectivity and brain functions.…”

An Algorithm for Finding Candidate Synaptic Sites in Computer Generated Networks of Neurons with Realistic Morphologies

“…Cuntz et al (2010) apply a minimal spanning tree principle in generating neuronal morphologies. Costa and Coelho (2005, 2008) generate 2D neuronal morphologies by statistically sampling a probabilistic model of neuronal geometry based on branch probabilities per branch level using a Monte Carlo approach, and form connections when neuronal trees overlap in 2D. The simulator NETMORPH (Koene et al, 2009) is based on biological growth principles of neurons by stochastically modeling the elongation and branching of growth cones in developmental time.…”

“…Other studies emphasize the hierarchy in neural network connectivity (e.g., Kaiser et al, 2010) or different spatial scales (e.g., Passingham et al, 2002; Sporns et al, 2005; Stam and Reijneveld, 2007). An interesting aspect during development of network connectivity is the critical phenomenon of percolation when the largest cluster of connected neurons makes an abrupt transition in size toward a giant cluster as studied by Costa and Manoel (2002) and Costa and Coelho (2005, 2008), who also showed the dependence of this phenomenon on the morphology of the developing neurons. Many of these studies rely on methods for estimating synaptic connectivity between axonal and dendritic arborizations, underscoring the need for algorithms that estimate the connectivity as realistic as possible.…”