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Analysis of the Swim Central Pattern Generator of the Melibe: A Symmetry Approach

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CSSS 2006 Santa Fe

It has well been established that the central pattern generator (CPG), a network of neurons capable of generating a rhythmic output, (partly) controls animal locomotion. Motivated by the works of M. Golubitsky and I. Stewart ([1], [2]) on the model of the CPG for animal locomotion for quadrupeds, we give a mathematical analysis of the swim CPG network architecture of the nudibranch mollusc Melibe leonina given in [3]. Analysis of the model requires symmetry arguments (from symmetric bifurcation theory) in the context of coupled cell networks. We also give numerical results that show, in particular, that symmetry gives rise to phase relation and/or synchronization in the network.

  • By a cell, we mean a unit described by a system of ODEs defined on a Euclidean space. A coupled cell network N=(C,E,~_C,~_E) comprises (1) a finite set of nodes C, (2) an equivalence relation ~_C on C, (3) a finite set of edges E, (4) an equivalence relation ~_E on E, and (5) consistency and compatibility conditions.


References:

[1] M. Golubitsky and I. Stewart, The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Birkhauser, Basel, 2002.

[2] M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: the groupoid formalism, Bull. Amer. Math. Soc. 43 (2006) 305-364.

[3] S. Thompson and W. Watson III, The Journal of Experimental Biology, vol.208, 1347-1361, The Company of Biologists (2005).


Participants: Vivien, Jie, (insert your name if you're interested)