Marty Golubitsky

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  Mathematical Homeostasis Motivated by Nijhout, Reed, and Best
  Marty Golubitsky
  We say that an input-output map xo(I) has infinitesimal homeostasis at I0 if x′o(I0) = 0. A consequence of infinitesimal homeostasis is that xo(I) is approximately constant on a neighborhood of I0. An input-output network is a network that has a designated input node , a designated output node o, and a set of regulatory nodes  = (i, . . . , n). We assume that the system of network differential equations Ë™X = F(X, I) has a stable equilibrium at X0. The implicit function theorem implies that there exists a family of equilibria X(I) = (x(I), x(I), xo(I)), where xo(I) is the network input-output map. We use the network architecture of input-output networks to classify infinitesimal homeostasis into three types: structural homeostasis, Haldane homeostasis, and appendage homeostasis. The first two types generalize feedforward excitation and substrate inhibition. The third type appears to be a new form of homeostasis. This research is a joint project with Yangyang Wang, Ian Stewart, Joe Huang, and Fernando Antoneli.
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  Animal gaits and symmetries of periodic solutions
  Marty Golubitsky
  In the first part of this talk I will briefly describe previous work on quadruped gaits (which distinguishing gaits by their spatio-temporal symmetries). In the second part, I will discuss how the application to gaits has led to results about phase-shift synchrony in periodic solutions of coupled systems of differential equations. This work is joint with David Romano, Yunjiao Wang, and Ian Stewart.
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  Wilson's Rivalry Networks and Derived Patterns
  Marty Golubitsky
  Wilson's Rivalry Networks and Derived Patterns
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  Synchrony and Symmetry
  Marty Golubitsky

  This talk reviews work on quadrupedal gaits and on a generalized model for binocular rivalry (proposed by Hugh Wilson). Both applications show how rigid phase-shift synchrony in periodic solutions of coupled systems of differential equations can help understand high level collective behavior in the nervous system.

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  Properties of Solutions of Coupled Systems
  Marty Golubitsky

  Networks of differential equations can be defined by directed graphs. The graphs (or network architecture) indicate who is talking to whom and when they are saying the same thing. We ask: Which properties of solutions of coupled equations follow from network architecture. Answers include "patterns of synchrony" for equilibria and "patterns of phase-shift synchrony" for time-periodic solutions. We show how these properties can be used to explain surprising results in binocular rivalry experiments and we discuss how homeostasis can be thought of as a network phenomenon.

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  Phase-Shift Synchrony and Symmetry
  Marty Golubitsky
  A coupled cell system is a network of interacting dynamical systems. Coupled cell models assume that the output from each node is important and that signals from two or more nodes can be compared so that patterns of synchrony can emerge. The principal question is: How does network architecture (who is talking to whom) affect the kinds of synchronous solutions that are expected in network equations. This talk will discuss a classification of rigid phase-shift synchrony in time-periodic solutions in such systems, as well as some curious synchrony-breaking bifurcations.