"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.

More Musings on Computational Neuroscience Paradigms

A couple months ago I posted a brief description of two overarching paradigms in theoretical computational neuroscience. During the course of writing my final project report, I addressed the same subject in slightly more detail. Since I seem to have strayed from my purported task of publishing pertinent computational neuroscience posts, I thought I would reproduce the two paragraphs in question here. I aready sent them to a friend of mine in biophysics who I know from one of my physiology courses, and he mentioned that I didn't address a couple things that I had never heard of before... so please keep in mind that this is all relatively new content for me, and the paragraphs I post here might simply be the pedestrian musings of an undergraduate amateur. Of course, they could also be brilliantly insightful, but I think the amateur option is a little more likely.

Anyway, here are the paragraphs:

Within the field of theoretical computational neuroscience, there are two general forms in which the problem of cognitive function is mathematically cast: as an adaptive control system and as a dynamical system on the edge of chaos. As with many competing fields of academic thought, disdain from adherents of one mode is often expressed for the ideas of those in the other camp. Fundamentally, the two interpretations are quite similar, as an adaptive controller functions on a dynamical system. However, proponents of the view that the brain functions as a system on the verge of chaos argue that the well-behaved systems generally analysed within the context of control theory fail to take into account the entire activity of the brain and therefore fall short of the goal of generating an accurate physiological model for cognitive function. These proponents also point to the efficacy of mathematical techniques from chaotic and dynamical system analysis to interpretations of electroencephalogram (EEG) readings, which serves as support for the near-chaotic dynamical system interpretation of the brain.

I would argue, however, that while an adaptive control experiment such as the one being implemented here seeks to isolate and investigate a specific cognitive task irrespective of the rest of the neuronal activity (or, in the case of the simulated robots used in this study, assuming no other neuronal activity), such a blinkered approach is not necessarily done out of ignorance of the larger issues of overall cognitive interconnectivity. Rather, I posit that the near-chaotic nature of the global brain behaviour arises out of the necessity of having many simultaneous well-behaved and sometimes contradictory control loops operating as one. The phase transitions apparent in EEG readings could arise from the necessity of transitioning from one set of precedent control loops to another, and a full understanding of the underlying control loops themselves can thus still further our overall understanding of cognitive function. While admittedly ad hoc, I hope this reasoning may serve to at least somewhat mollify those detractors who would dismiss adaptive control as a convenient tool of engineering misapplied to neuroscience. Continued exploration of adaptive control and implicit supervision can therefore have benefits for the field of theoretical computational neuroscience in addition to direct practical benefits in robotics.

I have removed the references, but if anyone is interested in what I am basing the discussion on, let me know and I will send you the appropriate articles.

A Brief Introduction to Computational Neuroscience Paradigms

Within computational neuroscience there seem to be two main theoretical paradigms. In the first, the brain is viewed as an elaborate and nested control system. This branch of investigation tends to use many of the same mathematical models as those used in the engineering discipline of control, albeit with an eye on the biological feasibility and possible neuronal configurations necessary for attaining such a control system. In the second, the brain is viewed as a dynamical system on the edge of chaos, and thus utilizes the mathematical tools found in dynamical system analysis. I have to admit that the latter of these two paradigms I am rather fuzzy on, despite having taken (and done rather well in) a course on Chaos, Fractals, and Dynamics. I am not sure if my inability to fathom what a 'dynamical system on the verge of chaos' means is due to a lack of intellectual capacity on my part or a lack of substance underlying the fancy terms being thrown around on the part of those championing the dynamical system interpretation. My guess is that the two paradigms are not as entirely exclusive as some claim them to be, but I think I will have to gain a better understanding of the application of dynamics to physiology before I can be sure. In the meantime, the control systems approach speaks quite clearly to the (former) engineer in me, and I find the control theory approach rather appealing. It is simple, elegant, and powerful.

Before I continue in this vein, however, I should mention a brief caveat. There is a third branch of thought which I have not included in this description known as machine learning. While it could also be argued to be a paradigm of computational neuroscience (or at least my interpretation of what computational neuroscience ought to be), I have not included it in this discussion because, to me, it is much more a branch of traditional approaches to artificial intelligence. Machine learning tends to focus more on function modeling through stochastic methods. While this provides many powerful tools (some of which are even utilized within the control systems approach), there is a lack of emphasis on physiological feasibility which might provide for a general theory of intelligence. Of course, I think many of the mathematical tricks used in machine learning (like principle components analysis (PCA)) will likely have neuronal correlates found in which our brains somehow provide a system to achieve similar results, machine learning does not tend to be devoted to uncovering methods of cognition as its primary goal.

Now that I have rambled about machine learning, I shall return to control theory. A control system is essentially any system designed to control a variable through time. The actual form the control system takes can be quite varied, including electronic control systems, mechanical ones, and, as I surmise our brains might be, electrochemical. They usually utilize some form of feedback (most often negative), since an open control system (as those without feedback are called) are not really much good at controlling anything. However, I will go into more detail about control theory in another post. This post was simply meant to introduce the idea of the different paradigms, as well as the fact that I am currently more focused on control theory.