A discrete-time continuous-space neural model for shell patterns in mollusks

We introduce a discrete-time continuous-space neural model to produce diverse shell structures and pigmentation patterns observed in aquatic mollusks. The model builds on an earlier neural model for shell patterns by incorporating the inhibition as a separate population and thus eliminates the need for a "refractory" substance, yet is still able to produce many varieties of molluscan pigmentation patterns. The model utilizes a system of neural excitation and inhibition to conduct secretory activity and successfully replicates various natural shell patterns found in these organisms. Through an analysis of local stability around equilibria and an analysis of bifurcation, we establish the critical role of parameters involved in our system on the bifurcations in governing the emergence of spatial, temporal, and spatio-temporal patterns.