Phase Resetting in an Asymptotically Phaseless System: On the Phase Response of Limit Cycles Verging on a Heteroclinic Orbit

Shaw, KM. Park, YM, Chiel, HJ. Thomas, PJ. (2012) Phase Resetting in an Asymptotically Phaseless System: On the Phase Response of Limit Cycles Verging on a Heteroclinic Orbit. Journal of Applied Dynamical Systems 11(1): 350-391.

Ok, I'm not going to read through this whole paper, but Hillel Chiel and Peter Thomas came and visited our lab yesterday (1/9/13), and I thought they had some interesting things to say about dynamical systems, control, and neural theory.

They were discussing CPGs and noting the interesting fact that if you look at these CPGs they can stall out at certain points during their oscillations. The stalling is driven by other factors, but they were asking the basic question of how to design a dynamical system that could have this kind of property.

Their solution revolves around what is becoming a popular way of making CPG-like control systems, which is to use heteroclinic channels. The idea of heteroclinic channels is that you utilize simple saddle-nodes that have a stable and unstable manifolds. You point one unstable manifold of one node at the stable manifold of another node, and then you arrange several nodes around in a circle. In this configuration you create a stable, heteroclinic orbit. This orbit, however, goes through the equilibrium points of the saddle nodes, which means that the stable orbit has an infinite period. However, you'll never reach the stable orbit, so you slowly go around in circles (with your period increasing in size each time, exponentially).

Their insight was that if you point the manifolds slightly off of, then you can create stable limit cycles that have different oscillatory dynamics. One way they do it is by making the system piece-wise linear, which also allows them to do calculations analytically. It basically works like this:

So, a single paramter varies how much these grids are aligned. Each grid is basically a simple linear system that has a saddle node. You don't have to do it completely piece wise linear, but it  is useful to see. By adjusting the a value, they can alter the dwell time around the limit cycle, which is a hypothesis to suggest how neural systems can produce these CPGs which dwell in certain parts of the rhythm. Sensory feedback etc is modulating the CPG to adjust the dwells.

The other really nice thing about making these heteroclinic orbits is that it could be a useful engineering tool for designing arbitrary oscillatory control systems. Each of the saddle nodes can be placed in various locations and pointed at each other in a pretty simple way. Perhaps instead of being piece-wise linear there are more continuous ways of designing it as well, something like the nodes effect on the system fall of with distance.

It could be interesting just to explore how you could make a neural system that utilizes these properties, and how you could get it to learn certain types of oscillations. There would be some relation between the neural circuitry and the intrinsic neural dynamics that govern the parameters of the saddle nodes, and then you could make a learning rule to relate those parameters to patterns in the system. It would be like motor learning: as you learn to walk you begin by forcing neurons to do the basic pattern, and each time you do the basic pattern the learning adjusts the saddle nodes formed by the system. Eventually the saddle nodes converge to the oscillation that is basically the average.

But how do you craft the nodes? How do you decide, given an oscillation where to place the nodes? How would you learn to place new nodes? Can you adjust the stability of the nodes for more control? How do you make a neural system that has these properties, and can easily manipulate these properties?