Angular homeostasis VII: non-monotonic correction systems.
We extend our model of angular homeostasis to correction functions that have a single maximum at a discrepant angle less than pi radians. We find that there are stable, and asymptotically stable, solutions that in general consist of self-intersecting curves. We investigate conditions for these curves to be periodic, and describe their symmetries. One typical pattern of such a closed curve involves a finite number of loops, each having a reflection axis of symmetry, with the complete curve having a cyclic rotation group. These bear a close resemblance to patterns found in lobulated biological structures (such as the petals of a flower or the primitive fetal hand). We further discuss implications for morphogenesis.
Murphy
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EA
Division of Medical Genetics, Johns Hopkins University School of Medicine, Baltimore, Md 21205.
Berger
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KR
D000818:Animals; D006706:Homeostasis / Q000502:physiology*; D008433:Mathematics; D008954:Models, Biological; D009024:Morphogenesis / Q000502:physiology