| |
ISSN 1320-0682 | |||
| Volume 02 | April 1995 | |||
William L. Ditto
Mark L. Spano
School of Physics
Georgia Institute of Technology
Atlanta, Georgia 30332, USA
Email: wditto@acl.gatech.edu
U.S. Naval Surface Warfare Center
Silver Spring, Maryland 20903, USA
Email: mark@chaos.nswc.navy.mil
There was a time when scientists knew that simple systems beget simple behaviour and complex systems beget complex behaviour. They also knew that complex behaviour was merely the accumulation of multiple simple behaviours. Twenty years ago, these views formed the foundation of our view of nature. Today, these illusions have been shattered by the discovery of chaos. We are discovering that nature prizes and exploits change and complexity. The existence of chaos has definitively shown that even simple systems can behave in very complex ways. As usual, nature has exhibited a richer, more elegant structure than our scientific paradigms allowed. From this identification came the recognition that chaos is pervasive in our world [1]. It is easy to make simple electronic circuits that are chaotic [2]. Mechanical systems on such wildly different scales as laboratory pendula and orbiting planets have been shown to exhibit chaos [3]. Laser emissions can fluctuate chaotically [4]. The human heart shows evidence of beating to chaotic rhythms [5].
The discovery of chaotic behaviour in nature initiated a rapid revolution in the sciences. Chaos was discovered, accepted and assimilated into the scientific community in the last fifteen years. Until recently, however, it was viewed as a very mathematical and theoretical discipline. More practical people were asking "what good is chaos?" and answering that it will never be more than a nuisance - something to be avoided in our attempt to use and apply nonchaotic systems. Recently, researchers have demonstrated that chaos admits possibilities and opportunities that simple behaviour cannot.
In what follows, we will discuss the concept of chaos, from both a theoretical and a laboratory viewpoint, always with an eye towards new methods for the exploitation and control of chaos in biological systems.
Chaos is inevitably confused with randomness and indeterminacy. Because many systems appeared random, they were actually thought to be random. This was true despite the fact that many of these systems seemed to fall into periods of almost periodic behaviour before returning to more "random" motion. Indeed, this observation leads to one of the definitions of chaos: the superposition of a very large number of unstable periodic motions. A chaotic system may dwell for a brief time on a motion that is very nearly periodic and then may change to another motion that is periodic with a period that is perhaps five times that of the original motion, and so on [1]. This constant evolution from one (unstable) periodic motion to another produces a long-term impression of randomness, while showing short-term glimpses of order. These glimpses are not misleading, since chaos is deterministic and not random in nature.
A second definition of chaos is the sensitivity of a chaotic system to small changes in its initial conditions. Thus, if a system is chaotic, these small perturbations quickly (indeed, exponentially) grow until they completely change the behaviour of the system. This is both the hope and the despair of those who have to deal with chaotic systems. It is the despair because it effectively renders long-term prediction of these systems impossible. Paradoxically, the cause of the despair is also the reason to hope. Because if a system is so sensitive to small changes, could not small changes be used to control it? This realisation led Ott, Grebogi and Yorke (OGY) [6] to propose an ingenious and versatile method for the control of chaos.
The starting point for the control of chaos is the phase space of the
system. A useful representation of the phase space in biological systems
that exhibit discrete events such as beating (in the heart) or bursting
(in the brain) can be obtained by plotting the current interval between
events against the previous interval. Such a
or return
plot reduces our information to a manageable level.
By way of contrast, a truly random system will behave in such a way that the points in phase space wander over the entire volume of space available to the system. Its corresponding section will be filled uniformly and densely.
Chaos falls between these two extremes. Since chaos is a superposition of a number of periodic motions, one might expect to see a finite number of points indicating several periodic motions characteristic of the chaotic section. This is true, as far as it goes. However, since chaos is the superposition of a large (read infinite) number of periodic motions, the number of points in the section is also infinite. In general, these points form an extended geometric structure, called the system's chaotic attractor, which is not a finite set of points (that is, does not represent periodic motion) and which also does not fill space (that is, is not random). (Generally a chaotic attractor is a fractal object.) Knowledge of this attractor and of its response to small perturbations of the system are the only ingredients that are necessary for the control of chaos.
In order to control chaos in biological systems, it is only necessary to identify an unstable periodic point in the attractor, to characterise the shape of the attractor locally around that point and put the system onto the stable direction inwardly forcing the system onto the unstable motion desired to be stabilised. Let's look at each of these three steps in detail.
The identification of unstable periodic motions is a fairly straightforward
process of looking for close returns in the
. (Of course
it is easier to find points with low order periodicity (small n),
but
chaos control has been successfully implemented to select periodic motions
of order up to about 90!) This is one of the strengths of chaos control:
the ability to control on any periodic motion from the infinity of motions
present in the system. This capability would allow an engineer to design
highly flexible systems employing chaos control.
Characterising the shape of the attractor is also straightforward. Once we have determined which unstable fixed point we wish to control about, we observe the motion of the point representing the current state of the system ( system state point) on the attractor. In low dimensional chaotic systems, this point will occasionally approach the vicinity of our chosen unstable fixed point and then move away again. It turns out that, in the neighbourhood of the unstable fixed point, the approach is consistently along the same direction (called the stable direction) and the departure along another direction (called the unstable direction). These two directions, one of which is stable (incoming) and the other of which is unstable (departing), form a saddle around the unstable fixed point (see Figure 1). These eigenvectors, along with the speed with which the system state point approaches or departs the vicinity of the unstable fixed point ( stable or unstable eigenvalues respectively), are all that is necessary to characterise the shape of the attractor locally around our chosen point.
The final step for the control of chaos in biological systems is to inject a stimuli to force the system onto the stable direction of the chosen unstable fixed point and thus allow the chaos of the system to exponentially force the motion of the system onto the unstable fixed point.
Figure 1:
The return map of a typical ouabain-induced arrhythmia heart preparation
which clearly demonstrates a local approach and divergence to an unstable
fixed point
(see [1]).
Such typical points are candidates for our
control of chaos algorithm. 
is the nth interbeat interval.
We have implemented chaos control in an experiment in which a section of a rabbit's heart was induced to beat chaotically by the injection of the drug ouabain. The measurable quantity here is the interval between heartbeats, which is about 0.8 sec in the healthy tissue. The effect of the drug is to accelerate the heartbeat and to cause the interbeat intervals to vary chaotically. The chaotic attractor for this with an approach to an unstable fixed point is shown in Figure 1.
To calculate these chaos control interventions, it was necessary to use the observed chaotic attractor to predict the time of the next heartbeat. An electrical stimulus was then used to induce a heartbeat before the predicted natural beat could occur, thereby shortening the next interbeat interval. The amount of time to advance the next beat was calculated so as to place the system state point directly onto the stable direction of the attractor. Thus, the succeeding beat would tend to naturally move toward the desired unstable fixed point rather than away from it. One should contrast this with on-demand pacing in which a simple "no-intervals-above-a-fixed-limit" strategy is employed. In chaos control, stimuli were not applied every cycle, but were only applied as often as necessary to nudge the wandering system onto the stable manifold and hence onto the unstable fixed point. The relevant parameter here is the magnitude of the unstable eigenvalue associated with the unstable fixed point which governs the local rate of expansion along the unstable manifold. Since we could only shorten long beats, we were forced to inject control stimuli not every beat but every third or fourth beat, as dictated by the presence of short beat mapping to a long beat along the unstable manifold associated with the desired unstable fixed point.
Figure 2:
Interbeat interval 
versus beat number demonstrating the response of the
oubain-induced arrhythmia to chaos control interventions (see
[1]
).
The results of a typical control run are presented in Figure 2. It was not possible in these experiments to achieve a good period 1 (that is, "normal") heartbeat. But it was possible to control the chaos consistently into a period 3 beat which, while not optimal, is better for pumping blood than chaotic beating. As for the relevance to human heart arrhythmias, there is evidence that atrial and ventricular fibrillation may be examples of chaos. Thus, future work may be able to present strategies for dealing with these serious and widespread arrhythmias.
Figure 3:
Diagram of the transverse hippocampal slice and arrangement of recording
electrodes (see [9] ).
Figure 4:
Demonstration of anti-chaos control and chaos control in a hippocampal
slice of a rat brain exposed to artificial cerebrospinal fluid containing
high [
]
and undergoing spontaneous chaotic population burst-firing or
spiking.
Our success in controlling chaos in the rabbit heart tissue preparation
led us to see if a similar strategy could control chaotic behaviour in
brain tissue. One of the hallmarks of the human epileptic brain during
periods in between seizures is the presence of brief bursts of focal
neuronal activity known as interictal spikes. Often such spikes emanate
from the
same region of the brain from which the seizures are generated
[7].
Several types of in vitro brain slice preparations, usually after
exposure to convulsant drugs that reduce neuronal inhibition, exhibit
population burst-firing activity similar to the interictal spike. One of
these preparations is the high potassium [
]
model, where slices from the
hippocampus of the temporal lobe of the rat brain (a frequent site of
epileptogenesis in the human) are exposed to artificial cerebrospinal
fluid containing high [
]
which causes spontaneous bursts of synchronised
neuronal activity which originate in a region known as the third part of
the cornu ammonis or CA3 [8] as shown in Figure 3.
If one observes the timing of these bursts, clear evidence for unstable fixed points are seen in the return map. As reported [9], we were able to regularise the timing of such bursts through intervention with stimuli delivered by micropipette, with timing as dictated by chaos control to put the system onto the stable direction. As shown in Figure 4, not only were we able to regularise the intervals between spikes but we were also able through an anti-chaos control strategy [8] to make the intervals more chaotic. It is the latter which might serve a useful purpose in breaking up seizing activity through the prevention or eradication of pathological order in the timing of the spikes.
Two fundamental questions dominate future chaos control theories. The first is the problem of controlling higher dimensional chaos. Almost certainly our control and anti-control of chaotic biological systems will be tremendously enhanced by taking into account the true higher dimensionality of these systems. Just such a higher dimensional chaos control theory has been made by Auerbach, Grebogi, Ott and Yorke [10]. This method can be "implemented directly from time series data, irrespective of the overall dimension of the phase space". Although applied in numerical studies, this method has yet to be tested experimentally.
The second question that has yet to be addressed is the problem of control in a spatiotemporal system. Indeed, any true model of biological systems needs to account for the spatial extent of the system. Such systems exhibit both spatial as well as temporal chaos. The study of such systems, while of obvious importance for real world applications, is as yet in its infancy. The challenge for experimentalists now lies not only in achieving control of higher dimensional and spatiotemporal chaos, but also in guiding the achievements of the past three years out of the laboratory and into our lives.
Control and Anti-Control of Chaos in Hearts and Brains
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