Mapping the Brain

By Arnold J. Mandell and Karen A. Selz

Drs. Mandell and Selz are with the Laboratory of Experimental and Constructive Mathematics of the Departments of Mathematics and Psychology, Florida Atlantic University, Boca Raton, Florida, USA.

The work in our laboratory on a variety of neurophysiological reactions in the early 1980s suggested that the electroencephalographic (EEG) record of human and animal brains might represent a "strange attractor." The EEG is an important diagnostic tool that maps the brain's electrical impulses. These recordings posed the possibility that the natural fractal structures of an attractor might be the instrument responsible for important integrations of brain mechanisms. Could the structure of "chaos" help to explain how the brain monitors and coordinates the constant movements of billions of bits of information?

The EEG "diagnosis" of a strange attractor was consistent with another hypothesis that we were pursuing at that time: Why was it that a number of disease states in biological dynamical systems manifested themselves via a loss in complexity? It seemed natural that a deterministic, information-generating mathematical object such as a strange attractor would be consistent with the healthy EEG and the use of this positive entropy supply by the brain for information-bearing formation.

Is the Chaotic Mind Creative?

Since that time, it has been shown that petit mal epilepsy and a case of hereditary cerebral degeneration with dementia were associated with the expected decrease in complexity of the EEG signal. Other studies seemed to support the notion that a complexity loss is associated with decreases in alertness and intellectual performance by the brain. On the one hand, it has been demonstrated that the deeper the sleep, the lower is the measure of complexity of the EEG. At the other end of central nervous system functions are the findings in which the complexity of the EEG is increased in some subjects during the initial phases of performing cognitive tasks that require recognition and response to novelty.

The theories promulgated by Smale, Ornstein and Mandelbrot, concerning strange attractors, fractals and the creation of structures, lead us to ask whether these new techniques could add to our understanding of the electrical impulses that trigger the brain. When we begin to move deeper into the implications of a strange attractor theory of the EEG to explain neurobiological mechanisms yet another theory of its dynamics emerges which may be impossible to disentangle from the strange attractor hypothesis

The arguments concerning an appropriate theoretical model for the EEG stem from around those involved with the original challenge, by Ruelle and Takens, to the Landau-Hopf theory of the onset of hydrodynamic turbulence. In this previously widely-accepted concept, turbulence arose essentially from the superimposition of three or four periodic motions - or "wobbles" -which become larger and larger. These motions or frequencies take the form of a torus, the surface generated by a circle rotated about an axis in a plane that does not intersect the circle.

The Ruelle-Takens argument was that three such superimposed frequencies actually combined to create a new object - which they called a "strange attractor." They further argued that as more than three frequencies (greater than 3-tori in mathematical jargon) came into play, chaotic solutions became both more frequent and more structurally stable.

Three "Symmetries" of the Brain

The implications of this are particularly interesting for the brain, which has three spatial-functional symmetries: anterior-posterior (of future and past, of imagining vs. seeing); right-left (analytic-algebraic vs. geometric); and up and down the neural axis. These symmetries oscillate, which make issues of expected behaviour vs. exceptional departures different than they would be if studied only in time. The spatial symmetries of the brain may make it more likely that this organ possesses some form of "nongeneric" - that is to say, non-typical - stability.

One of the persistent mysteries of the EEG is the source of the so-called "slow" rhythms, which range from 1 to just over 50 cycles per second (Hz). The most characteristic are: (a) under 4 cycles per second, deep sleep or coma; (the) 4-7 Hz, day dreaming and light sleep; (c) 10-12 Hz, relaxed, eyes closed, awake; (d) 18-30 Hz, very alert, aroused or anxious; and (e) 35-50 Hz, periods thought to be associated with instances of spatial coherence in perceptual tasks.

These "facts" from over 60 years of research on the human EEG have never been disputed, yet they have never been well understood. The strange attractor theory has led to serious questions concerning the usual analytic approach to spectra with multiple broad band modes. The strange attractor theorist views such spectra in another way - as the instabilities and aperiodic actions of expanding and folding dynamical systems. They are the result of the post-Landau-Hopf global bifurcation phenomenon - the scenario of Ruelle and Takens - and are undecomposable. Ruelle refers to these complex singularities of the power spectrum as "resonances" of the dynamical system.