A Tree-Like Model for Brain Growth and Structure

The Flory-Stockmayer theory for the polycondensation of branched polymers, modified for finite systems beyond the gel point, is applied to the connection (synapses) of neurons, which can be considered highly branched “monomeric” units. Initially, the process is a linear growth and tree-like branching between dendrites and axons of nonself-neurons. After the gel point and at the maximum “tree” size, the tree-like model prescribes, on average, one pair of twin synapses per neuron. About 13% of neurons, “unconnected” to the maximum tree, migrate to the surface to form cortical layers. The number of synapses in each neuron may reach 10000, indicating a tremendous amount of flexible, redundant, and neuroplastic loop-forming linkages which can be preserved or pruned by experience and learning. 1. Introduction The molecular weight distribution (MWD) of polycondensation of branched-chain monomers of the type has been derived classically by Flory [1] and generalized by Stockmayer [2]. Here, is the number of functional groups, or “functionality” of group A. Mathematically, this widely quoted distribution function has been treated as power-series distribution and compound distribution that provides a simple concept; that is, single-parameter expressions of number- and weight-average “degrees of polymerization” (DP) are sufficient to generate the entire MWD for branched polymers [3]. Moreover, using a cascade formulation involving functionals and probability generating functions (PGF), this distribution can be extended to finite systems [4]. Here, the previously derived properties of this finite distribution are applied to synapse formation in the brain. A neuron has multiple dendritic processes and an axon, which can also be branched. Neurons are generally three or more orders of magnitude greater in size than molecular units. However, the functionality of a neuron may be 103 times larger than that of typical branched molecules ( ). This large functionality also means there is great accessibility to the connection sites, and the long flexible axons offer a favorable condition for connection between neurons. The “tree-like,” or “ring-free,” assumption in the Flory-Stockmayer theory can be satisfied by the initial linkage of head-to-tail linear chains and followed by a “tree-like branching.” A neuron itself can be considered a small tree. Similarly, the peripheral nervous system (PNS) also resembles a tree made of the nerve bundles, which can be as large as 1.5 meters. The Finite Flory-Stockmayer theory (FFST) deals with numbers of highly branched repeat units and