Isochrons (and applications to Neuroscience)

Abstract

Many biological systems exhibit a periodic behaviour. From a mathematical point of view they can be considered as systems moving along a stable limit cycle, that can be parametrised by its phase. The phase can be extended to the whole basin of attraction of the limit cycle via the asymptotic phase and the set of all points having the same asymptotic phase is called isochron. Isochrons were introduced in 1974 by Winfree in order to understand the behaviour of an oscillatory system under a brief stimulus, namely, the phase advance or delay that the system would experience when sent away from the periodic orbit. This helps understanding, for example, the synchronisation in neural nets. Soon after Winfree’s paper, Guckenheimer showed that isochrons are in fact the leaves of the stable foliation of the stable manifold of a periodic orbit. Different techniques have been developed to compute the isochrons. An important part of this undergraduate thesis consists on understanding the mathematical concept underlying the idea of isochron. In chapter 2 a definition is given and we describe some properties of isochrons with the objective of being able to find an approximation to first order. We also formulate a functional equation for the parametrisation of the invariant cycle and the tangent vector to the isochrons and we show how it can be solved using a quasi-Newton method. Our arguments are based in, where the parametrisation of the whole isochron is found. With some trans- formations we can simplify our equations and easily solve them. We end the chapter by giving some hints on how to proof the convergence of the method using KAM arguments. In chapter 3 we describe briefly a simplified version of the well-known Hodgkin-Huxley model for the neuron and try to get some insight on what isochrons can tell us about this model. Another important part of this thesis has been writing a program in language C to implement the algorithm described in chapter 2 in order to be able to apply it to the reduced Hodgkin-Huxley model. The program is described in chapter 4 and the source code is appended. In chapter 5 the results obtained with the program are discussed.