Bifurcation Analysis and Synapse Pattern Formation with Bidirectional Motor-Mediated Transports

Abstract

This study investigates the development of intricate pattern formation in synaptogenesis, which describes the transfer of synapses between neurons via bidirectional active and passive motor-mediated transport in the nervous system. The trafficking model, which consists of three strongly coupled partial differential equations, exhibits remarkable spatiotemporal dynamics that contribute to both static and constantly changing spatial patterns. We perform and detail the steady-state and Hopf bifurcation to present the formation of nonhomogeneous stationary or time-periodic solutions from the homogeneous one. In particular, the steady-state reduces to that of a two-component PDE system, independent of the initial condition and the system parameters. Moreover, we analyze the stability of these nontrivial patterns and our results find a wave mode selection mechanism, which in turn suggests that a large interval length admits time-periodic solutions, while a small interval length most likely drives spatially monotonous nonhomogeneous stationary solutions. Finally, we provide numerical experiments to illustrate and support our theoretical findings.