Competitive exclusion between axons dependent on a single trophic substance: a mathematical analysis

A mathematical model is presented of competition between axons for a trophic substance, such as is believed to occur particularly during development. The model is biologically realistic. The growth-stimulating activity of the trophic molecules is assumed to result from their binding to high-affinity receptors on neurons and their axons, but the model also incorporates uptake by nonneuronal cells possessing only lower affinity receptors. Plausible and fairly general assumptions are made concerning the kinetics of binding and internalization and the effects on axonal growth. The model takes into account the possibility that trophic factor production may be regulated by the afferent axons or autoregulated. The variables specified are the "axonal vigor" of each axon, representing the ability of each axon to take up trophic molecules, and the concentration of trophic molecules in the extracellular space of the axonal target region. Of the several parameters introduced, the most important turns out to be the "zero vigor-growth parameter," which is defined as the concentration of trophic molecules that gives zero growth of the vigor of a given axon. By means of a Lyapunov function, it is shown that the system will approach asymptotically to a stable equilibrium characterized by the survival of only the axon whose zero-growth parameter is lowest. Or, if several axons share the same lowest zero-growth parameter, these will all survive. The model may be particularly relevant to the elimination of polyneuronal innervation from developing muscle fibers and from autonomic ganglion cells.