"Bob Loblaw"

As I have been mentioned a few times over the last couple of days, I gave a talk yesterday morning. I mentioned it so often because I was fairly nervous about it, and I was fairly nervous about it not because it was a talk of any great importance, but because I don't have a lot of experience giving technical talks to highly academic people. The talk was titled Phase Response as a Function of Graph Structure, and was essentially an overview of what I have spent my last month doing. The first half of the talk was a mathematical and intuitive development of the concept of phase response, and then the second half was how that related to dynamical networks (primarily of identical weakly coupled oscillators). Robert left a comment to my post about being nervous, pointing out no one was likely to remember my talk in ten years. As this was an intra-departmental talk to an audience of about ten people, I would be surprised if memories lasted even half that long. What was nice, though, was that I received several compliments on the talk, including from the head of the research group. What was less nice was the Ph.D. student I've been working with and I discovered a handful of minor mistakes on the slides the morning of the talk during my last practice run-through, and my audience managed to spot all but one of them (at least it means they were paying attention). I guess that is what happens when you give a talk to an audience primarily composed of mathematicians and physicists... they actually pay attention to the equations you have on your slides!

I will now try to give a brief overview of what the subject Phase Response as a Function of Graph Structure actually means. If you take an arbitrary dynamical system (which is essentially a fancy word for saying a system that evolves through time) that has a stable periodic limit cycle (which means the system has a state that repeats after a period of time T, and small perturbations to that system will disappear over time and it will settle back to the periodic motion), then you can define something called the phase of the system as how far along in the period the system is. Phase is usually parameterized to be between either 0 and 1 or 0 and 2π by convention. I find the 0 and 1 parameterization more intuitive (it essentially translates to what percentage of the period has already passed, with 0.5 being 50% of the way through from whatever point is defined as the period beginning). The idea of phase can then be generalized to the basin of attraction around the limit cycle (which is essentially the region of your dynamical system's feature space which eventually settles onto your limit cycle), such that a point on the limit cycle and point within the basin of attraction are considered to have the same phase if they evolve through time to the same point on the limit cycle. A rough picture of this idea is shown in Figure 1. This leads to the idea of an isochron (the dotted lines in Figure 1), which is the collection of points in your feature space that all share the same phase.

Figure 1: A point on the limit cycle and off that have the same phase. The mustard yellow curve represents the time evolution of a point off the limit cycle as the moves back to the cycle, while the green curve represents the evolution of a point that starts on the limit cycle. When the mustard yellow curve rejoins the limit cycle, it does so at the same point that the green curve reaches in an equivalent length of time. The two starting points are therefore said to have the same phase.

With phase now defined both on and off the limit cycle, one is able to develop the idea of phase response. If a perturbation (essentially, some sort of externally applied influence that drives the system away from its normal time evolution) is applied to a dynamical system with a stable limit cycle, the phase of the unperturbed system and the perturbed system are both defined (assuming the perturbation is small enough that your system remains within the basin of attraction of the limit cycle), and the change in phase resulting from the perturbation is the phase response of the system (see Figure 2).

Figure 2: The phase response (Δφ, where φ is the phase of the system) to a perturbation ε.

Until now, I have left the discussion fairly open-ended about the properties of the dynamical system under analysis. The idea of phase response is usually applied to the analysis of single oscillators. An example of such a system would be the Hodgkin-Huxley model of a neuron exposed to a constant ambient current such that it is tonically firing at a set period. The feature space of the system is then the voltage across the membrane, the applied current, and the ionic concentrations (both intracellularly and extracellularly) of several key ions (such as potassium and sodium). What we have been investigating is the phase response of networks of oscillators coupled together, at which point the coupling relationship between oscillators becomes part of your feature space. A perturbation applied to one element of the network might ellicit a different phase response than a perturbation applied to another element.

On the surface, one might wonder what the point of all of this is. The thing is, coupled dynamical systems are found in all sorts of areas. Networks of neurons are an obvious example, but gene expression is another area of biological research where there are large systems of interacting biochemical pathways. There are examples outside of biology as well, but I am having a hard time thinking of one off the top of my head since my group tends to focus on the biological tie-in of our research. Therefore, having a better understanding of the phase response of networks will lead to a better understanding of these exceedingly complex systems.

Note: Figure 2 was pulled from Christoph Kirst's diploma thesis, Dynamics of Pulse-Coupled Neuronal Oscillators with Partial Reset. Figure 1 was a (rather shoddy) edit of Figure 2 that I made over the weekend using GIMP.