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"Networking Opportunity," by Ian Stewart. Nature, 12 February 2004, pages601-604.
Leeches, network dynamics, and groupoids. Don't see the connection? The leech'sheart, which consists of a series of tubes on either side of its body, beats ina very unusual way: on one side, the tubes contract simultaneously, while onthe other side, one tube contracts after another, moving from back to front.Approximately every 50 beats, this pattern shifts sides. The network of neuronsthat cause this phenomenon has been mapped. But a greater mathematicalunderstanding of the occurrence of patterns like synchrony and "phase locking,"here and in networks generally, results from the application of groupoidtheory.
Stewart offers some illustrations, including an example of three types ofnetworks: a ring, a chain, and a modified chain with feedback, where each noderepresents a system of differential equations, and each directed edge a"coupling." While only the ring has symmetry, the modified chain with feedbackhas a "hidden 'rotational' symmetry." Symmetry between specific subsets of thenetwork causes the dynamics in the modified chain to be equivalent to those ofthe ring. In Stewart's words, "The groupoid formalism makes it possible toapproach network dynamics within a coherent theoretical framework. Just asgroup theory has illuminated pattern formation in symmetric systems, sogroupoid theory can illuminate pattern formation in systems with repeatedsubunits." This application of mathematics to biology is important since natureis filled with networks, and it is an example of the way in which the fields ofmathematics and biology are offering new insights and presenting new challengesto each other.
--- Claudia Clark