COMPETITIVE-EXCLUSION BETWEEN AXONS DEPENDENT ON A SINGLE TROPHIC SUBSTANCE - A MATHEMATICAL-ANALYSIS

A mathematical model is presented of competition between axons for a t rophic substance, such as is believed to occur particularly during dev elopment. The model is biologically realistic. The growth-stimulating activity of the trophic molecules is assumed to result from their bind ing to high-affinity receptors on neurons and their axons, but the mod el 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 effect s on axonal growth. The model takes into account the possibility that trophic factor production may be regulated by the afferent axons or au toregulated, The variables specified are the ''axonal vigor'' of each axon, representing the ability of each axon to take up trophic molecul es, and the concentration of trophic molecules in the extracellular sp ace 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 giv es zero growth of the vigor of a given axon. By means of a Lyapunov fu nction, it is shown that the system will approach asymptotically to a stable equilibrium characterized by the survival of only the axon whos e zero-growth parameter is lowest. Or, if several axons share the same lowest zero-growth parameter, these will all survive. The model may b e particularly relevant to the elimination of polyneuronal innervation from developing muscle fibers and from autonomic ganglion cells.