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Doubly periodic traveling waves in a cellular neural network with linear reaction. (English) Zbl 1188.37084

Summary: Szekeley observed that the dynamic pattern of the locomotion of salamanders can be explained by periodic vector sequences generated by logical neural networks. Such sequences can mathematically be described by “doubly periodic traveling waves” and therefore it is of interest to propose dynamic models that may produce such waves. One such dynamic network model is built here based on reaction-diffusion principles and a complete discussion is given for the existence of doubly periodic waves as outputs. Since there are 2 parameters in our model and 4 a priori unknown parameters involved in our search of solutions, our results are nontrivial. The reaction term in our model is a linear function and hence our results can also be interpreted as existence criteria for solutions of a nontrivial linear problem depending on 6 parameters.

MSC:

37N25 Dynamical systems in biology
92B20 Neural networks for/in biological studies, artificial life and related topics
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References:

[1] Szekely G: Logical networks for controlling limb movements in Urodela.Acta Physiologica Hungarica 1965, 27: 285-289.
[2] Cheng SS, Lin JJ: Periodic traveling waves in an artificial neural network.Journal of Difference Equations and Applications 2009,15(10):963-999. 10.1080/10236190802350631 · Zbl 1178.39009 · doi:10.1080/10236190802350631
[3] Cheng, SS; Chen, C-W; Wu, TY, Linear time discrete periodic diffusion networks, No. 42, 131-151 (2004), Providence, RI, USA · Zbl 1109.91044
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