Strange Attractors that Govern Mammalian Brain Dynamics Shown by Trajectories of Electroencephalographic (EEG) Potential

WALTER J. FREEMAN

Reprinted from IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, Vol. 35, No. 7, July, 1988. Manuscript received September 10, 1987.

This work was supported by the National Institute of Mental Health, U.S. Public Health Service under Grant M1406686.

Fig. 1(a) (upper left) shows a 2-D view on a video screen of a 3-D display of a 4-second epoch of EEG from the olfactory system of the brain of a rat, awake but resting motionless. The trajectory of a point moving through this space in time traces the subspace that is occupied by a strange attractor. We have rotated and translated it to seek for structure, and we have colored it red when the fourth variable is negative and blue-white when it is positive. The Hausdorff dimension is 5.92 [4] and we are unable to find geometric order.

ONE OF THE most robust properties of the brain is its capacity to generate low-amplitude low-frequency fluctuations in electrical and magnetic potential that are detected within the brain and at the surface of the scalp as "brain waves" or the EEG. These persist so strongly during life that their absence is used as a sure sign of "brain death", and they are so sensitive to behavioral state that they serve as indicators of the levels of sleep and anesthesia, of abnormalities of consciousness and perception, and of emotional arousal. But they are very difficult to read, because they have the appearance and the statistical properties of random variables in the guise of "colored noise", that is, band-limited white noise.

Fig. 1(b) (upper right) shows the EEG trace from the same animal during an epileptic seizure. The same recording conditions hold, but the Hausdorff dimension is reduced to 2.52, and the structure of the strange attractor appears to be that of a 2-torus.

Recent developments in nonlinear dynamics and the theory of chaos have revealed that EEGs do not merely reflect stochastic processes, and instead that they manifest deterministic chaos. The bases for this conclusion lie in newly devised procedures for estimating "dimensions" of the underlying neural dynamics that generate EEG time series, [5], [6] and in recent studies on the spatial coherence of multiple EEG traces recorded simultaneously from arrays of up to 64 electrodes placed onto the brain surface [1], [3], showing what has been described as "spatially coherent turbulence".

Fig. 1(c) (lower right) shows the trajectory of a solution set of a dynamic model of the olfactory brain that simulates epileptic -activity [2], and which gives an estimate of the Hausdorff dimension of 3.76. It also has a form suggesting the 2-torus.

In principle, brain dynamics can be viewed as deterministic in the manner of the pendulum. When the potential and kinetic energies are plotted together for a simple pendulum, the time series is a spiral to an equilibrium point. If the pendulum has multiple hinges, and if it is driven with an outside source of energy, the trajectory may appear as a circle, a spiral on a torus, or as an erratic, intrinsically unpredictable trace, that ranges in complexity from the appearance of butterfly wings to a skein of wool and on to a bowl of spaghetti.

Fig. 1(d) (lower left) gives a view of the trajectory of a solution set from the same model but with a change in parameters to allow the genesis of normal EEG traces corresponding to the behavioral state of rest. Rotation of this model strange attractor shows that it may also have the structure of a 2-torus, but one that fluctuates in size and shape aperiodically, or as Otto Rössler [71 describes it, a torus that breathes". Its Hausdorff dimension is 5.46.

Yet there may be order in these trajectories, because the ways in which the system exchanges energy within itself are constrained. One means to reveal and explore the constraints is to examine the appearance of a trajectory viewed as an object -in state space. These novel phase portraits may inform us about the kinds of chaotic neural dynamics that may be the sources of our behavior, both normal and abnormal.

Some examples shown here are of the use of computer graphics to visualize chaotic attractors from the EEG. Recordings are made from multiple sites; factor analysis serves to partition the variance into orthogonal components, and the largest three serve best to represent the dynamics in the 3 Euclidean dimensions accessible to us with 'a flight simulator. Then we can "fly" the attractors in 3-space, using color or time slices to suggest yet higher dimensions.

These examples suffice to indicate the extraordinary opportunity provided by computer graphics for the display, comparison, and analysis of very large. and complex data sets of high dimension, revealing forms that are comprehensible and recognizable. We suggest that these methods may provide a useful clinical tool for the detection and description of the varieties of brain states underlying normal and pathological behavior in animals and man, that depend on chaotic dynamics [8].

ACKNOWLEDGMENT

The graphics displays were generated with the assistance of Peter Broadwell at Silicon Graphics Computer Services, Inc., Mountain View, CA.

REFERENCES

[1] W. J. Freeman, "Techniques used in the search for the physiological basis of the EEG," in Handbook of EEG and Clinical Neurophysiology, (A. Gevins and A. Remond, Eds.), Amsterdam, The Netherlands: Elsevier, 1987, vol. 3, part 2, chap. 18.

[2] W. J. Freeman, "Simulation of chaotic EEG patterns with a dynamic model of the olfactory system," Bio. Cybern., vol. 56, pp. 139-150, 1987.

[3] W. J. Freeman and B. Van Dijk, "Spatial patterns of visual cortical fast EEG during conditioned reflex in monkey," Brain Res., vol. 442, pp. 267-276, 1987.

[4] K. A. Grajski and W. J. Freeman, "Olfactory neural dynamics manifested in low-dimensional attractors," Physica D, submitted, 1988.

[5] P. Grassberger and 1. Procaccia, Measuring the strangeness of strange attractors, Physica, vol. 9D, pp. 189-208, 1983.

[6] J. Guckenheimer, "Dimension estimates for attractors," Contemporary Math., vol. 28, pp. 357-367, 1984.

[7] 0. E. Rössler, "The chaotic hierarchy," S. Nat., vol. 38a, pp.788-801, 1983.

[8] C. A. Skarda and W. J. Freeman, How brains make chaos In order to make sense of the world. Brain Behavioral Sci., vol. 10, pp. 161-195, 1987.