An EEG represents the time series that maps the voltage corresponding to neurological activity as a function of time (examining a specific neural network, as described in the background information). Yet this does not present much information relating to the dynamics of the system. Because time is always increasing, it is difficult to tell whether there is any attractors in the system, or the general dynamics: the graph appears quite random. (I refer you to Figure 1.14 (a) of a standard cat’s EEG on p. 27 of “Encounters with Chaos” for clarification. ) Yet a relative new analysis for EEG’s converts it to a two-dimensional representation. This is done by plotting the voltage at time t on one axis, as before, but on the other, the voltage at time t+tau, where tau is constant and represents the time lag being represented by this plot, called a phase portrait. If actually viewing the data is unimportant, this can be extended to n dimensions. For comparison purposes, a Fast Fourier Transform (FFT) can be used to convert the data from the time domain to the frequency domain, in order to get a power spectrum which will also not be dependent on the time, and thus can be used in the same regard as the phase portrait. It is unknown as to whether the phase portrait reveals more information that the FFT.Go BackBut what does this give you? It would seem that this would just change the representation of the same data, but in fact, this gives you a simple way to categorize the system. Because any nth-order differential equation can be converted to n independent 1st-order differential equations (as stated in a friend’s Calculus 4 notes), it is feasible to assume that the brain activity in question is based on n variables satisfying a set of n first-order differential equations. Thus by determining the phase portrait of the system, the attractors can be much more readily seen: because time itself is not being plotted, the graph is fixed to a specific domain, that from the minimum to maximum value of the range. This space that contains all the points in (x = {xmin, xmax}, y = {xmin, xmax}) is called the phase space.
Take random noise, for example, where the voltage from one time point has no bearing on another. In this scenario, there would be no attractors, and eventually as the number of time points increases, the entirety of the phase space will eventually be covered. Just imagine drawing connected lines on a piece of paper – after doing this long enough, the entire paper will be covered. If, on the other hand, there is an attractor, then only a subset of the phase space will be covered and the set of points plotted will have fractional dimension (or dimension less than two, at any rate – it could be possible that the points fall on the x=y line, and then the dimension would be equal to 1). Then the box-counting or capacity dimension of these points could be calculated to determine the extent of the attractor.
Another type of dimension, termed the correlation dimension, can be calculated as described by Roschke and Baser, in “The EEG is Not a Simple Noise”. Because experimental data is being used as opposed to specific mathematical formulae, it is necessarily impossible to take the limit as required in the standard box-counting dimension, and this would thus give a crude approximation. The correlation dimension gets around this problem by looking at the probability that two points will be separated by distance R given the data in question, and taking the limit as R decreases. I was unable to get any good pictures off the Internet of phase-space represented EEG’s, and hence was unable to analyze them using my box-counting approximator.
But what do attractors in the mind represent? It has been determined that as information travels to deeper levels of the brain that the dimension of the EEG decreases. (Roschke & Baser, 1988) Neurological signals from the acoustical cortex which is responsible for processing the direct input of sounds coming into the brain, has a higher correlation dimension (mean value 5.06) than the hippocampus (mean value 4.58) which is thought to be used in higher functioning such as the determination of spatial location (i.e. trying to navigate using a map). These attractors were shown to be statistically relevant in multiple studies. This could imply a compression or reduction of the information used to store the original input: since a fewer number of states are possible because of the lower dimensional attractor at the higher-level processing centres, information must be being reduced as it is being processed.
A more interesting result of inherent chaotic attractors in the mind is a potentially valid explanation of creativity and sudden insight. The suddenness of insight has been extremely difficult to explain using standard functional approaches (as the majority of cognitive scientists use): no algorithm to date has been able to measure up well on divergent thinking exercises, where entirely new problems must be solved. (Some artificial intelligence systems are able to extrapolate problem solving approaches, but in a very restricted manner.) Yet, thinking of insight in terms of a dynamic systems approach comes very easily. Oftentimes the mind seems to be focussed on a limited set of possible heuristics for problem solving and it seems impossible to get out of that state. In terms of dynamic systems theory, it could be hypothesized that the brain is focussed on a specific attractor. The flash of insight could be a phase shift, where the brain suddenly and unpredictably shifts to another, different, attractor state. This attractor state would represent the possible solution of the problem. This use of dynamic systems theory would also explain findings that show a similar sudden effect, even when the ‘sudden insight’ is incorrect. This would just have the mind traverse to a different attractor state as well, but an incorrect one.
The potential in looking at the mind using Dynamic Systems Theory is enormous. Already it has been found to provide explanations that can not be had using other approaches. As more in depth research is performed in the analysis of EEG’s using fractal and chaos theory concepts, this is bound to provide more explanations in cognition, as it has already in meteorology, physics, and biology.