On December 19-20, 2011 the Brain-Mind workshop took place at Fudan University in Shanghai. In the workshop I presented one of the invited keynotes. The title of my talk was “Applications of dynamical systems in biology and synchronization”.
The concept of synchronization plays a very important role in biology. In the talk I presented two systems that exhibit synchronization. The first such system is a network of pulse coupled oscillators with delay. Such networks are used for modelling, for example, the activity in biological neuron networks or the synchronization processes in networks of interacting agents. Because of the non-zero delay the state space of such systems is infinite dimensional. Important questions here are the existence of unstable attractors, i.e., of saddle periodic orbits whose stable set has non-empty interior. In an earlier work we showed that for any number \(n\) of oscillators with \(n \ge 3\) there is an open parameter region in which the system has unstable attractors. Moreover, in the case of \(n = 4\) oscillators we showed that there exist unstable attractors with heteroclinic cycles between them. The second such system is a model for circadian rhythms. We have studied how a single pacer cell synchronizes to a periodic signal. This signal includes the effect of the external environment (light-dark cycle) but also the effect of the rest of the pacer cells. It turns out that such system can be described by a family of circle maps. In the presentation I discussed the properties of this family (emphasizing resonances and Arnol’d tongues) and their biological significance.