Abstract
By guiding cell and chemical migration and coupling with genetic mechanisms, bioelectric networks of potentials influence biological pattern formation and are known to have profound effects on growth processes. An abstract model that is amenable to exact analysis has been proposed in the circuit tile assembly model (cTAM) to understand self-assembled and self-controlled growth as an emergent phenomenon that is capable of complex behaviors, like self-replication. In the cTAM, a voltage source represents a finite supply of energy that drives growth until it is unable to overcome randomizing factors in the environment, represented by a threshold. Here, the cTAM is extended to the axon or alternating cTAM model (acTAM) to include a circuit similar to signal propagation in axons, exhibiting time-varying electric signals and a dependence on frequency of the input voltage. The acTAM produces systems of circuits whose electrical properties are coupled to their length as growth proceeds through self-assembly. The exact response is derived for increasingly complex circuit systems as the assembly proceeds. The model exhibits complicated behaviors that elucidate the interactive role of energy, environment, and noise with electric signals in axon-like circuits during biological growth of complex patterns and function.