|The Impact of Chaos on Science and Society (UNU, 1997, 394 p.)|
Bruce E. Kendall, W. M. Schaffer, C. W. Tidd, and Lars F. Olsen
Theoretical investigations have often found chaos in biological models, and laboratory experiments have revealed the potential for chaos in isolated biological subunits, but analyses of empirical data from outside the laboratory have typically been less conclusive. This leaves unanswered two important questions: "Is biology chaotic in the 'real world'?" and "If chaos is there, what is its biological meaning?" In this paper we present some approaches that begin to address these questions. Our examples are drawn from a variety of biological processes, including brain dynamics, human epidemiology, and extinction rates in ecological populations.
The title of this paper can be recast as a question: what is the impact of chaos on biology? But this question really has two parts: what impact has chaos had in the past? and what might it have in the future? This paper is primarily concerned with the latter question, but we will begin by reviewing the contributions chaos has made to biology so far.
The search for chaos in biology
We begin at a fundamental level: are there good reasons to expect to find chaos in biology? The ideas of chaos were first introduced to biology by theoreticians, and mathematical biologists have since found many examples of chaos in biological models (May 1976, 1985; May and Oster 1976; Gilpin 1979; Aihara et al. 1984; Chay 1985; Chay and Rinzel 1985; Caswell and Weeks 1986; an der Heiden and Mackey 1987; Samardzija and Greller 1988; Sleeman 1988; Gardini et al. 1989; Michaels et al. 1989; Alien 1990a, b; Glass and Hunter 1990; Glass and Malta 1990; Glass and Zeng 1990; Holden and Lab 1990; Hastings and Powell 1991). The ubiquity of chaos in biological models is perhaps best symbolized by the fact that the quadratic map, the simplest mathematical system capable of generating chaos and the basis of much of what we know about period doubling (Feigenbaum 1978; Collet and Eckmann 1980), is also well known in biology as the logistic map, which models the dynamics of a single population. Furthermore, nonlinear feedback mechanisms abound in biology, allowing organisms and populations to respond rapidly to changing environmental conditions. But this feedback is exactly the prerequisite for chaos, and so we should actually be surprised not to find chaos in biological models, at least for some combination of parameter values. From a theoretical standpoint, then, we can say, "Yes, biology is full of the potential for chaos."
However, biological models are abstractions of reality, and only the simplest biological systems are accurately described by them. Hence we turn to empirical work and ask, "Can chaos actually be found in biological data?" Laboratory experiments are the best place to start, for there one can control the parameters and measure the state variables to a reasonable degree of accuracy. Studies of biochemical oscillations (Olsen and Degn 1977; Geest et al. 1993), the dynamics of individual cells, both in vitro (Hayashi et al. 1982, 1985; Markus et al. 1984, 1985; Aihara et al. 1985, 1986; Lebrun and Atwater 1985; Hayashi and Ishizuka 1992) and in vivo (Rapp et al. 1985; Mpitsos et al. 1988), and oscillations of aggregates of cells (Guevara et al. 1981; Glass et al. 1983; Chialvo and Jalife 1987; Chialvo et al. 1990) suggest that there are circumstances in which the dynamics are chaotic. Numerous laboratory populations of insects have been examined for chaotic dynamics (Hassell et al. 1976; Thomas et al. 1980; Philippi et al. 1987); one (sheep blowflies: Nicholson 1954) was found to be chaotic. Again we have an affirmative: at least some biological subsystems are capable of displaying chaos.
Of course, there is a tremendous difference between isolated subsystems studied in the laboratory and the complexities of "real world" organisms and ecosystems. So the preceding results do not necessarily imply an answer for the last preliminary issue that needs to be addressed: is there biological chaos outside the laboratory? A number of studies claim to answer this affirmatively, presenting evidence for chaos in the electrical activity of the brain (Babloyantz et al. 1985; Babloyantz and Destexhe 1986; Dvorak and Siska 1986; Mayer-Kress et al. 1988; Frank et al. 1990; Ehlers et al. 1991; Gallez and Babloyantz 1991; Röschke and Aldenhoff 1991; Pritchard and Duke 1992; Fell et al. 1993), the dynamics of the heart (Babloyantz and Destexhe 1988; Mayer-Kress et al. 1988; Zbilut et al. 1988; Ravelli and Antolini 1992; Lefebvre et al. 1993; Yamamoto et al. 1993), and ecological and epidemiological dynamics (Schaffer 1984; Schaffer and Kot 1985; Olsen et al. 1988; Olsen and Schaffer 1990; Witteman et al. 1990; Hanski et al. 1993; Turchin 1993). However, we believe these results need to be viewed cautiously, for the methods used to determine the presence of chaos are not without problems (see appendix). Nevertheless, the large number of putative examples of chaos in biology suggests that at least some are real.
The search for meaning in chaos
That leaves one final hurdle: what is the biological meaning of chaos and its correlates (such as a non-integer dimension or positive Lyapunov exponent)? One calculates the dimension of a population dynamics experiment in exactly the same way as one does for a fluid dynamics experiment. This indicates that there are similarities between these apparently dissimilar systems, but it also means that calculating a fractal dimension really tells us very little about the biology. One way of addressing this issue is comparative. For example, researchers studying brain dynamics via the EEG1 have attempted to correlate dimension with levels of consciousness (Basar 1990). For the most part, however, these workers use "dimension" as an index of waveform complexity, concluding little about actual brain function. We believe that it is this lack of biological consequences that causes many biologists to feel that chaos really tells them nothing new about their subject.
The above discussion suggests that a rigorous quantitative approach to identifying and classifying chaos may be possible only in the laboratory. Even in such ideal situations, the relevance of chaos to the "real world" remains unclear. Outside the laboratory, a more productive approach may be to assume that the dynamics are chaotic, using a more or less specific model as seems appropriate, and attempt to infer unique, observable consequences. In addition to providing an end run around the limitations of most biological time series, this approach begins to answer the question, "What impact does the chaos have on the biology?" This is the central question that needs to be addressed if chaos is to become more than a plaything of theoretical biologists. In what follows, we examine some recent studies, which suggest how this question may be answered.
Persistence and extinction in animal populations
Although some of the earliest investigations of chaos were motivated by ecological models (May 1976), many ecologists continue to believe that chaos should not be observed in natural populations. In most simple population models, chaotic dynamics are characterized by high amplitude fluctuations, including frequent visitations to the neighbourhood of the origin. One might expect, then, that in the presence of the inevitable environmental and demographic stochasticity, chaos should increase the probability of population extinction (Thomas et al. 1980; Berryman and Millstein 1989). Thus any species unfortunate enough to have evolved life history traits leading to chaos will be strongly selected against, and we will see few, if any, such species in nature.
Notice, however, the generalization from a single population to the whole species. The implicit assumption is that the populations comprising the species are independent, and will disappear one by one until the last is gone. In fact, many species are composed of sub-populations that are coupled to a greater or lesser extent by migration; such a species is called a "metapopulation." Traditional metapopulation theory (see Taylor 1990 for a brief review) holds that this spatial distribution will protect against extinctions due to local environmental perturbations. As long as the migration rate is not too small, local sub-populations will recover from extinction or near-extinction by having their ranks augmented by immigrants from other sub-populations. This says nothing, however, about the system's response to regional perturbations, such as droughts or cold snaps, to which natural populations are also subject. There is also no consideration of whether the protective effect of subdivision is affected by the nature of the intrinsic dynamics at the sub-population level.
Fig. 1 The Ricker map for three parameter values
Allen et al. (1993) have addressed these issues by modelling a metapopulation subject to both local and regional disturbance. The local dynamics are governed by the Ricker map2 (fig. 1), which is dynamically similar to the logistic map used by Berryman and Mill-stein (1989). Within a single sub-population, they found that extinction rates increase rapidly as the dynamics become more complex. In contrast, the extinction rates for the entire population drop as the local dynamics enter the chaotic region (fig. 2).
Fig. 2 Local and global extinction probabilities in the Allen-Schaffer model. There are 25 sub-populations; r is the bifurcation parameter in the Ricker map. (a) Bifurcation diagram for the Ricker map.
Fig. 2 Local and global extinction probabilities in the Allen-Schaffer model. There are 25 sub-populations; r is the bifurcation parameter in the Ricker map. (b) Local and global extinction probabilities for global perturbations only. Global perturbations are additive: x ® x + z, where z is drawn from a uniform distribution on [ -0.75, 0.75].
Fig. 2 Local and global extinction probabilities in the Allen-Schaffer model. There are 25 sub-populations; r is the bifurcation parameter in the Ricker map. (c) Local and global extinction probabilities for global and local perturbations combined. The local perturbations are also additive, with z drawn from [ -0.1, 0.1]. The perturbations are applied at every iteration
That this effect is due to the chaos itself, rather than some other feature of the model, can be seen by considering an example where there are only global perturbations (fig. 2b). The extinction rate abruptly returns to a high level in the period three window, and many of the other peaks in the extinction curve correspond to the smaller periodic windows in the bifurcation diagram.
When local perturbations, which are smaller but more frequent than global perturbations, are added to the simulation, the protective effect of chaos becomes even greater. Whereas the extinction rate for a single sub-population now rises extremely fast as a function of the bifurcation parameter, the probability of global extinction is actually lower than for the case of global perturbations alone. In the example illustrated in figure 2, the addition of local perturbations decreases the global extinction rate by a factor of 100 at r = 3.5. Sufficiently large local perturbations also destroy the periodic windows, and the detailed structures seen in figure 2b are lost.
The explanation for this counter-intuitive phenomenon is an elegantly simple example of how chaos can inform biological reasoning. The case of global perturbations alone reveals the underlying mechanism. First note that, since the maximum magnitude of the global perturbations is less than the maximum possible population size, a global extinction will only occur when all sub-populations are relatively small. In the regions of equilibrial and periodic dynamics, any populations that start in phase stay that way indefinitely, since they are subject to the same perturbations and migration effects. For low order periodicities, the "basins of attraction" of the various phases are rather large; thus the perturbations will often move two groups of populations, initially out of phase, into the same phase. Once all the populations are in phase, the whole system behaves like a single oscillator, and the probability that a given perturbation causes a global extinction is identical to the probability that it causes a local extinction. In the region of chaotic dynamics, however, any sub-populations that start out with similar population sizes diverge rapidly because of the sensitive dependence on initial conditions, which is characteristic of chaos (Ruelle 1979). Even if a perturbation brings all the sub-populations to a similar size (near zero, in particular), chaos ensures that they become decorrelated before the next perturbation. Thus it is unlikely that all the sub-populations will simultaneously be small enough for a single perturbation to destroy them all.
The enhanced protection in the chaotic region that results when local perturbations are added seems to reflect the fact that the local disturbance "jump starts" the sub-population separation. Although the deterministic divergence reflected by a positive Lyapunov exponent is exponential, it can initially be very slow in absolute terms. Local perturbations enhance the initial decorrelations, guarding against the extinctions caused by two global perturbations in rapid succession.
What do these results mean for the role of chaos in population dynamics? In simple nonspatial models, chaos is associated with a high reproductive rate. Individual selection for high fecundity, for example, might be expected to push a population into chaotic dynamics. Berryman and Millstein (1989) claimed that this would have the unfortunate side-effect of increasing the extinction rate; but in a subdivided population, it now appears that chaos actually enhances species survival. One reasonable prediction based on these results is that within a taxon (a group of closely related populations or species) populations with limited dispersal abilities should be more likely to demonstrate chaotic population dynamics than either populations that are well mixed over a large area or are almost entirely isolated. An animal that may fit this pattern is the vole (Microtus spp.). Most vole populations undergo dramatic fluctuations, and have high reproductive rates (Keller 1985; Taitt and Krebs 1985). There is also a fair amount of migration among local sub-populations (Lidicker 1985). One species of Microtus, however, is restricted to a small island off the coast of Massachusetts. Here, in the absence of migration, reproductive rates are lower than in mainland voles and the only fluctuations in population size are seasonal responses to resource levels (Tamarin 1977a, b). The question of why some animal populations fluctuate and others do not has long been an important one in ecology; if an explicit consideration of the effects of chaos in population dynamics can shed light on this problem, it will be an important advance.
Periodicity in chaos
Despite the classic description of chaos as "deterministic nonperiodic flow" (Lorenz 1963), chaotic trajectories can exhibit strong statistical periodicities. In some systems, this is because the stretching and folding occur in directions perpendicular to the flow (Farmer et al. 1980). This sort of periodicity permeates the entire time series, and can usually be removed by taking a section of the flow. Episodes of periodic motion can also occur when the trajectory comes near a nonstable periodic orbit and "shadows" it for a while (Auerbach et al. 1987; Cvitanovic 1988). More extended episodes of near-periodic motion (called "transient periodicity" by Kendall et al. 1993) can occur when the trajectory gets temporarily trapped in the neighbourhood of a "semi-periodic semi-attractor." The last two types of periodic behaviour are the result of local features of the attractor, and occur at sporadic intervals in the time series. Nevertheless, an episode of transient periodicity can be quite long lived. As biologists, we find this observation intriguing, for there are numerous circumstances in which biological systems switch erratically between episodes of periodic and irregular dynamics.
From the point of view of looking for chaos at the population level, epidemiological records supply perhaps the best data to be found. There are extensive records of various childhood diseases, including measles, rubella, poliomyelitis, and chickenpox, from various cities in North America and Europe; many of these extend from the beginning of this century until the implementation of mass vaccination programmes in the 1960s and 1970s. Although not all cases were reported (for measles, the reporting rate has been estimated to range from 10 to 35 per cent in various cities (London and Yorke 1973)), sampling error cannot conceal the tremendous variability in the magnitudes of the annual outbreaks.
Childhood epidemic data have been examined for evidence of chaos (Schaffer and Kot 1985; Olsen and Schaffer 1990), and at least in the case of measles, the evidence for chaos is reasonably convincing. Furthermore, the measles data can be mimicked quite closely, both in "appearance" and in the value of dynamical invariants, such as dimension and Lyapunov exponent, by a standard epidemiological model, the seasonally forced SEIR equations3 (Dietz 1976; Olsen and Schaffer 1990). For parameter values appropriate to measles, small changes in the magnitude of seasonal variation in the contact rate, b1, give rise to three types of dynamics. The lowest values of b1 generate simple periodicity. As b1 is increased, there is a period-doubling cascade, which gets as far as a four-piece chaotic attractor, or "semi-periodic" attractor (Lorenz 1980), in which low amplitude chaos is superimposed upon a period four oscillation. At b1 » 0.271, there is a crisis (Grebogi et al. 1983), and a large amplitude chaotic attractor suddenly appears (fig. 3). Coexisting with the semiperiodic attractor is a chaotic repeller, topologically similar to the large amplitude chaotic attractor, which is only visible when the deterministic dynamics of the semiperiodic attractor are perturbed (Rand and Wilson 1991).
Turning to the data, we see that there is often a mix of periodicity and disorder. Measles epidemics in New York City, for example, showed an apparently biennial cycle from 1945 to 1963, and Detroit and Baltimore County had shorter episodes of the same pattern (fig. 4), but other parts of these time series are highly irregular. The biennial segments are not truly periodic, but they are extremely regular in comparison with the "typical" dynamics. Indeed, anyone looking at just the second half of the New York City data would be hard pressed to claim it was not a noisy limit cycle.
The SEIR model generates similar patterns. For b1 = 0.28, episodes of nearly biennial dynamics (similar to those seen in the data) are interspersed with more irregular fluctuations (fig. 5). One way to visualize a high-dimensional phase space is to extract a two-dimensional next amplitude map - a plot of each maximum in the time series against the previous maximum - from the time series of "infectives" (corresponding to reported cases in the data; fig. 6). In this representation, the "biennial" dynamics appear as two large V-shaped regions (fig. 7).
Fig. 3 Bifurcation diagram for the seasonally forced SEIR equations with parameters appropriate to measles simulations
Where do these episodes of periodic motion come from? With b1 = 0.27, just before the onset of large amplitude chaos, the model generates a semiperiodic attractor, which closely matches the "biennial" parts of the chaotic attractor (fig. 7). This similarity suggests that the two objects may be related. Kendall et al. (1993) have shown, using a two-dimensional map, that regions of transient periodicity are neighbourhoods of semiperiodic semiattractors (Kantz and Grassberger 1985) formed when a semiperiodic attractor is destroyed by an interior crisis (Grebogi et al. 1983). In practical terms, this has the following consequences: a trajectory enters a region of transient periodicity through a well-defined set of pre-images; remains trapped therein for a variable length of time; and exits via a well-defined "escape hatch."
Fig. 4 Monthly reported cases of measles in three North American cities (Sources: New York City data from Yorke and London (1973); Baltimore County data from Hedrich (1933))
Fig. 5 Transient periodicity in the seasonally forced SEIR equations, (a) A time series showing both large-amplitude irregular oscillations and small-amplitude "biennial" dynamics,
Fig. 5 Transient periodicity in the seasonally forced SEIR equations, (b) Magnification of the biennial episode
Fig. 6 A next amplitude map for the SEIR equations. Each maximum in the time series of infectives is plotted against the previous one
In this light, one does not require any special factors to explain the "discrepancies" among various parts of the measles data: the "biennial" episodes reflect transient periodicity, which is a natural part of the chaotic motion. An alternative is that the four-piece semi-periodic attractor defines the asymptotic dynamics, and the episodes of irregular motion result from perturbations on to the chaotic repeller discussed by Rand and Wilson (1991). When there is very little noise, the dynamics follow a noisy limit cycle. With somewhat higher noise levels there are occasional large-amplitude excursions as the trajectory undergoes an excursion on the chaotic semiattractor (fig. 8). We have termed this phenomenon "irruptive intermittency." Finally, with more (but still not very much) noise, the trajectory almost never has the opportunity for sustained limit-cycle behaviour (fig. 9), and the trajectory is essentially indistinguishable from the large-amplitude chaotic attractor associated with the larger values of b1. This scenario is vastly different from what is usually considered to be "noisy periodicity" (i.e. random variations about a limit cycle): here a single perturbation can lead to an extended chaotic excursion. While the initial perturbation is stochastic, the "erratic" fluctuations that follow are fully deterministic.
Fig. 7 (a) The regions of transient periodicity extracted from the next amplitude map of the SEIR equations (b1 = 0.28).
Fig. 7 (b) Next amplitude map of the four-piece attractor (b1 = 0.27)
Although understanding the dynamics of measles epidemics is not so important in this era of widespread vaccination, this example does have two interesting implications. The first is that it is possible to detect chaos in population-level phenomena, where one might expect stochastic influences to dominate. The second is the method itself: the measles study involved a dynamic interplay between model and the actual dynamics, permitting the recognition of dynamical features not readily apparent in the data alone.
Fig. 8 Irruptive intermittency in the SEIR model
In the preceding example, the underlying biological mechanisms are well understood. Unfortunately, such understanding is rare. If we are limited to such cases, then the approach presented above is not of widespread value. Moreover, periodic dynamics often emerge from a high-dimensional background, and the question arises as to whether we can understand the phenomenon without explicitly knowing the overall dynamics.
Fig. 9 Noise-sustained chaos in the SEIR model
A dramatic - and medically important - example is epilepsy. This disease is characterized by sporadically occurring seizures of varying duration, often accompanied by unconsciousness.4 Seizures themselves are generally not life threatening.5 Nonetheless, sudden and unpredictable lapses in consciousness prevent many epileptics from leading normal lives. Seizures are accompanied by dramatic changes in the EEG. Typically one observes an abrupt onset of high amplitude, nearly periodic dynamics (fig. 10).6 Although certain types of seizures can be induced by various external stimuli (reflex epilepsies: Forster 1977), it is more often the case that the onset of seizure activity is spontaneous. Similarly, most seizures are self-terminating (Dreifuss 1990; Niedermeyer 1990).
One dynamical interpretation of epilepsy is that a seizure represents a bifurcation into a periodic window as the brain "moves" through parameter space. This is an attractive model for a non-terminating seizure: the medication administered to end the fit can be viewed as inducing a parameter shift back out of the window. Alternatively, Schaffer et al. (1993) point out that if the brain is near a periodic window, then it will be subject to episodes of transient periodicity. Finally, there may also be regions of parameter space in which the brain is subject to intermittency (Pomeau and Manneville 1980), which, although different from transient periodicity, also have the effect of inducing episodes of nearly periodic motion. The latter two phenomena could explain both the spontaneous onset and termination of most seizures. Since seizure frequency and severity often varies throughout a patient's history, one plausible scenario is that there are long-term (weeks to years) movements through parameter space, in which the system drifts closer or further away from a periodic window.7
Fig. 10 An EEG trace of an epileptic seizure (petit mal) (Source: Avoli et al. 1990)
A problem with viewing epilepsy as transient periodicity is that whereas in transient periodicity the nearly periodic orbits are contained within the "normal" attractor, the oscillations associated with a seizure are in an entirely different part of the phase space from the background oscillations of the EEG. This suggests that a model more like irruptive intermittency might be more appropriate. Speelman et al. (1995) have been exploring a model of the hippocampus that can produce "bursts" of large-amplitude oscillations over a range of parameter values. They suggest that if such bursts could be spatially coherent over large areas of the brain, they might represent a reasonable model for epilepsy.
Ideally, one would like a spatially explicit model of brain dynamics, but the number of neurons in the brain makes this unpractical. However, there are some results from simpler models of spatially extensive dynamical systems that might shed light on the problem. Kaneko (1993, and references therein) has studied large arrays of chaotic logistic maps with either global or nearest-neighbour coupling. Studies of simpler systems of just two coupled logistics (Kaneko 1983; Gyllenberg et al. 1993; Hastings 1993; Kendall and Fox 1995) have shown the possibility of multiple coexisting periodic and chaotic attractors, with different spatial patterns; as the number of oscillators is increased, this multiplicity increases accordingly. One possible state of the large array is "fully developed spatio-temporal chaos," in which all of the oscillators are chaotic, and there is little apparent correlation among them. This produces a mean signal with low-amplitude fluctuations that are nevertheless chaotic and high dimensional, much like the normal EEG. An opposite extreme of possible dynamics is when all of the oscillators are in phase: there is no spatial variation in the state of the system. In this case the mean signal has a large amplitude and either periodic or low-dimensional chaotic dynamics, much like the EEG during a seizure.
If the dominant Lyapunov exponent of the in-phase dynamics is small enough, then the in-phase dynamics is locally attracting: if the system is nearly in-phase, it will remain that way, and will not spontaneously escape. However, the size of the local basin of attraction for the in-phase attractor declines exponentially with increasing system size (Kaneko 1993). In a system as large as the brain, any small perturbation would knock the system out of the in-phase state.
There is an added twist: often, after a long period of spatio-temporal chaos, the system will suddenly collapse into in-phase dynamics. Even though the local basin of attraction is small, there are long-lived chaotic transients that eventually lead into it, just like the chaotic transients associated with the periodic windows in simple maps. Thus we have an alternate, explicitly spatial, model of epilepsy: a seizure begins when a long transient of apparently normal dynamics decays to the in-phase state. The seizure ends when a perturbation, either internal to the brain or from an external stimulus, knocks the brain out of the narrow local basin of attraction of the in-phase state. For healthy individuals, in contrast, the in-phase state is not even locally stable.
These views of epilepsy require that the brain activity actually transpire in a phase space. Accordingly, we believe it is significant that some epileptics experience a sensory forewarning of their seizure (the "developing aura"), and further, that there are reports of seizures being triggered by specific thoughts (Paulson 1963; Symonds 1970). Both observations suggest that there are well-defined pre-images of the epileptic state.
Under the first model, there is no fundamental difference between healthy individuals and epileptics; the latter are merely in an unfortunate region of parameter space (close to a periodic window, or with more stable in-phase dynamics). Anti-seizure medications typically change either the ion transport capacity of neural membranes or the concentration of various neurotransmittors (Prichard 1980; Suria and Killam 1980; Woodbury 1980), and often take a period of weeks or months to reach full effectiveness; this could also be viewed as movement in parameter space. The insights provided by this model may also help guide the search for new treatments. For example, the first model suggests that we try to map out pre-images of the seizure state. Sometimes, seizures can be aborted by applying a sudden stimulus just before or after its onset (Symonds 1970; Forster 1977; Rajna and Loma 1989; Iragui and McCutchen 1991). The therapeutic possibilities of automating this process by monitoring the brain for pre-images of seizures is potentially tremendous. Not only would epileptics be able to live normal lives (many activities that we take for granted, such as driving automobiles, are closed to individuals who might go into a seizure at any time), but the sometimes severe side effects of anti-seizure medication could be avoided. If nonlinear dynamics can inform not only our understanding but also our treatment of a disease such as epilepsy, it will have proved to be a useful tool indeed.
Despite the many efforts to find evidence for complex dynamics in biology, the functional implications of chaos have seldom been examined. As a result, chaos has had little impact on most biologists. The studies presented here attempt to address this question directly; we feel that they are indicative of profitable approaches. We hope they inspire further creative approaches to the problem.
This work was supported in part by NIH grant R01 AI2354-03 to WMS and a grant from the Danish Research Council to LFO. We thank G. A. Fox for making helpful comments on the manuscript.
1. The electroencephalogram (EEG) is the most common measure of overall brain activity. It is obtained by measuring the voltage difference between electrodes placed at various points on the scalp, and represents a spatially averaged recording of the electrical activity in the outer layers of the cortex.
2. xt+1 = xt exp[r(1 - xt)]
This is based on a model created by Ricker (1954) to describe the dynamics of populations (such as certain fish species) in which density dependent mortality in juveniles is the result either of cannibalism by adults or competition with adults. However, the functional form relating to subsequent generations is fairly general (see Ricker 1954 for numerous examples). Its main distinction from the logistic map is the exponential tail: there are no positive values of r for which deterministic dynamics can lead to extinction.
3. dS/dt = m(N - S) -
dE/dt = b(t)SI - (m+a)E
dI/dt = aE - (m+ g)I
dR/dt = gI - mR
b(t) = b0(1 + b1 cos 2pt)
See Olsen and Schaffer (1990) for a description of the variables and parameters. The "measles" values for the parameters, used in the simulations discussed here, are m = 0.02 y-1; a = 35.84 y-1; g = 100 y-1; b0 = 1800 y-1; b1 is varied from 0.2 to 0.3.
4. One useful classification of epilepsy divides it into two broad types, focal and generalized. A focal seizure starts at a specific site in the brain, often corresponding to some sort of physiological damage, and the seizure spreads to other parts of the brain rather gradually. A generalized seizure begins simultaneously in large parts of the brain, including both hemispheres, and is seldom traceable to any specific injury (Niedermeyer 1990). The dynamical paradigms discussed in this section are primarily applicable to generalized epilepsy.
5. An exception is "status epilepticus," in which seizures occur repeatedly at intervals of minutes or less, and the patient never regains consciousness (Niedermeyer 1990).
6. The electrical activity measured during a seizure appears to actually be chaotic - Babloyantz and Destexhe (1986) calculated a dimension Dc = 2.05 ± 0.09 dominant Lyapunov exponent l =2.1 ± 0.6s-1, and Prank et al. (1990) found Dc = 5.6 and l » 1.0s-1. It is clearly "approximately periodic," however, just as are the semiperiodic attractors generated by the SEIR equations.
7. Episodes of intermittency and transient periodicity are both longer and more frequent the closer the system is to the periodic window.
Aihara, K., G. Matsumoto, and M. Ichikawa. 1985. "An Alternating Periodic-chaotic Sequence Observed in Neural Oscillators." Physics Letters A111: 251-255.
Aihara, K., G. Matsumoto, and Y. Ikegaya. 1984. "Periodic and Non-Periodic Responses of a Periodically Forced Hodgkin-Huxley Oscillator." Journal of Theoretical Biology 109: 249-269.
Aihara, K., T. Numajiri, G. Matsumoto, and M. Kotani. 1986. "Structures of Attractors in Periodically Forced Neural Oscillators." Physics Letters A116: 313-317.
Allen, J. C. 1990a. "Chaos and Phase-locking in Predator-Prey Models in Relation to the Functional Response." Florida Entomologist 73: 100-110.
Allen, J. C. 1990b. "Factors Contributing to Chaos in Population Feedback Systems." Ecological Modelling 51: 281-298.
Allen, J. C., W. Schaffer, and D. Rosko. 1993. "Chaos Reduces Species Extinction by Amplifying Local Population Noise." Nature 364: 229-232.
Auerbach, D., P. Cvitanovic, J.-P. Eckmann, G. Gunaratne, and I. Procaccia. 1987. "Exploring Chaotic Motion through Periodic Orbits." Physical Review Letters 58: 2387-2389.
Avoli, M., P. Gloor, G. Kostopoulos, and R. Naquet. 1990. Generalized Epilepsy: Neurobiological Approaches (Birkhäuser, Boston).
Babloyantz, A., and A. Destexhe. 1986. "Low-dimensional Chaos in an Instance of Epilepsy." Proceedings of the National Academy of Sciences of the USA 83: 3513-3517.
Babloyantz, A., and A. Destexhe. 1988. "Is the Normal Heart a Periodic Oscillator?" Biological Cybernetics 58: 203-211.
Babloyantz, A., J. M. Salazar, and C. Nicolis. 1985. "Evidence of Chaotic Dynamics of Brain Activity during the Sleep Cycle." Physics Letters A111: 152-156.
Basar, E. 1990. "Chaotic Dynamics and Resonance Phenomena in Brain Function: Progress, Perspectives, and Thoughts," pp. 1 30. In E. Basar, ed. Chaos in Brain Function (Springer-Verlag, Berlin).
Berryman, A. A., and J. A. Millstein. 1989. "Are Ecological Systems Chaotic - and If Not, Why Not?" Trends in Ecology and Evolution 4: 26-28.
Bier, M., and T. C. Bountis. 1984. "Remerging Feigenbaum Trees in Dynamical Systems." Physics Letters A104: 239-244.
Caswell, H., and D. E. Weeks. 1986. "Two-sex Models: Chaos, Extinction, and Other Dynamic Consequences of Sex." The American Naturalist 128: 707-735.
Chay, T. R. 1985. "Chaos in a Three-variable Model of an Excitable Cell." Physica D16:233-242.
Chay, T. R., and J. Rinzel. 1985. "Bursting, Beating, and Chaos in an Excitable Membrane Model." Biophysical Journal 47: 357-366.
Chialvo, D. R., R. F. Gilmour, Jr., and J. Jalife. 1990. "Low-Dimensional Chaos in Cardiac Tissue." Nature 343: 653-657.
Chialvo, D. R., and J. Jalife. 1987. "Non-linear Dynamics of Cardiac Excitation and Impulse Propagation." Nature 330: 749-752.
Collet, P., and J.-P. Eckmann. 1980. Iterated Maps On the Interval As Dynamical Systems (Birkhäuser, Basel).
Cvitanovic, P. 1988. "Invariant Measurement of Strange Sets in Terms of Cycles." Physical Review Letters 61: 2729-2732.
Destexhe, A., J. A. Sepulchre, and A. Babloyantz. 1988. "A Comparative Study of the Experimental Quantification of Deterministic Chaos." Physics Letters A132: 101-106.
Dietz, K. 1976. "The Incidence of Infectious Diseases under the Influence of Seasonal Fluctuations," pp. 1-15. In: J. Berger, W. Blihier, R. Repges, and P. Tauto, eds. Mathematical Models in Medicine (Springer-Verlag, Berlin).
Dreifuss, F. E. 1990. "The Syndromes of Generalized Epilepsy," pp. 19-29. In: M. Avoli, P. Gloor, G. Kostopoulos, and R. Naquet, eds. Generalized Epilepsy: Neurobiological Approaches (Birkhäuser, Boston).
Dvorak, I., and J. Siska. 1986. "On Some Problems Encountered in the Estimation of the Correlation Dimension of the EEG." Physics Letters A118: 63-66.
Eckmann, J.-P., and D. Ruelle. 1985. "Ergodic Theory of Chaos and Strange Attractors." Reviews of Modem Physics 57: 617-656.
Ehlers, C. L., J. W. Havstad, A. Garfinkel, and D. J. Kupfer. 1991. "Nonlinear Analysis of EEG Sleep States." Neuropsychopharmacology 5:167-176.
Ellner, S., A. R. Gallant, D. McCaffrey, and D. Nychka. 1991. "Convergence Rates and Data Requirements for Jacobian-based Estimates of Lyapunov Exponents from Data." Physics Letters A153: 357-363.
Farmer, D., J. Crutchfield, H. Froehling, N. Packard, and R. Shaw. 1980. "Power Spectra and Mixing Properties of Strange Attractors." Annals of the New York Academy of Sciences 357: 453-472.
Farmer, J. D., and J. J. Sidorowich. 1987. "Predicting Chaotic Time Series." Physical Review Letters 59: 845-848.
Feigenbaum, M. J. 1978. "Quantitative Universality for a Class of Nonlinear Transformations." Journal of Statistical Physics 19: 25-52.
Fell, J., J. Röschke, and P. Beckmann. 1993. "Deterministic Chaos and the First Positive Lyapunov Exponent: A Nonlinear Analysis of the Human Electroencephalogram during Sleep." Biological Cybernetics 69: 139-146.
Forster, F. M. 1977. Reflex Epilepsy, Behavioral Therapy and Conditional Reflexes (Charles C. Thomas, Springfield, Illinois).
Frank, G. W., T. Lookman, M. A. H. Nerenberg, C. Essex, J. Lemieux, and W. Blume. 1990. "Chaotic Time Series Analyses of Epileptic Seizures." Physica D46:427-438.
Gallez, D., and A. Babloyantz. 1991. "Predictability of Human EEG: A Dynamical Approach." Biological Cybernetics 64: 381-391.
Gardini, L., R. Lupini, and M. G. Messia. 1989. "Hopf Bifurcation and Transition to Chaos in Lotka-Volterra Equation." Journal of Mathematical Biology 27: 259-272.
Geest, T., L. F. Olsen, C. G. Steinmetz, R. Larter, and W. M. Schaffer. 1993. "Nonlinear Analyses of Periodic and Chaotic Time Series from the Peroxidase - Oxidase Reaction." Journal of Physical Chemistry 97: 8431-8441.
Gilpin, M. E. 1979. "Spiral Chaos in a Predator-Prey Model." The American Naturalist 113: 306-308.
Glass, L., M. R. Guevara, A. Shrier, and R. Perez. 1983. "Bifurcation and Chaos in a Periodically Stimulated Cardiac Oscillator." Physica D7: 89-101.
Glass, L., and P. Hunter. 1990. "There Is a Theory of Heart." Physica D43:1-16.
Glass, L., and C. P. Malta. 1990. "Chaos in Multi-looped Negative Feedback Systems." Journal of Theoretical Biology 145: 217-223.
Glass, L., and W.-Z. Zeng. 1990. "Complex Bifurcations and Chaos in Simple Theoretical Models of Cardiac Oscillations." Annals of the New York Academy of Sciences 591: 316-327.
Grassberger, P., and I. Procaccia. 1983. "Characterization of Strange Attractors." Physical Review Letters 50: 346-349.
Grebogi, C., E. Ott, and J. A. Yorke. 1983. "Crises, Sudden Changes in Strange Attractors, and Transient Chaos." Physica D7: 181-200.
Guevara, M. R., L. Glass, and A. Shrier. 1981. "Phase Locking, Period-doubling Bifurcations, and Irregular Dynamics in Periodically Stimulated Cardiac Cells." Science 214: 1350-1353.
Gyllenberg, M., G. S. Soderbacka, and S. Ericsson. 1993. "Does Migration Stabilize Local Population Dynamics? Analysis of a Discrete Metapopulation Model." Mathematical Biosciences 118: 25-49.
Hanski, I., P. Turchin, E. Korpimaki, and H. Henttonen. 1993. "Population Oscillations of Boreal Rodents: Regulation by Mustelid Predators Leads to Chaos." Nature 364: 232-235.
Hassell, M. P., J. H. Lawton, and R. M. May. 1976. "Patterns of Dynamical Behavior in Single-species Populations." Journal of Animal Ecology 45: 471-486.
Hastings, A. 1993. "Complex Interactions between Dispersal and Dynamics: Lessons from Coupled Logistic Equations." Ecology 74: 1362-1372.
Hastings, A., and T. Powell. 1991. "Chaos in a Three-species Food Chain." Ecology 72:896-903.
Hayashi, H., and S. Ishizuka. 1992. "Chaotic Nature of Bursting Discharges in the Onchidium Pacemaker Neuron." Journal of Theoretical Biology 156: 269 291.
Hayashi, H., S. Ishizuka, and K. Hirakawa. 1985. "Chaotic Response of the Pace-maker Neuron." Journal of the Physical Society of Japan 54: 2337-2346.
Hayashi, H., S. Ishizuka, M. Ohta, and K. Hirakawa. 1982. "Chaotic Behavior in the Onchidium Giant Neuron under Sinusoidal Stimulation." Physics Letters ASS: 435-438.
Hedrich, A. W. 1933. "Monthly Estimates of the Child Population 'Susceptible' to Measles, 1900-1931, Baltimore, Md." American Journal of Hygiene 17: 613-636.
an der Heiden, U., and M. C. Mackey. 1987. "Mixed Feedback: A Paradigm for Regular and Irregular Oscillations," pp. 30-46. In L. Rensing, U. an der Heiden, and M. C. Mackey, eds. Temporal Disorder in Human Oscillatory Systems (Springer-Verlag, Berlin).
Holden, A. V., and M. J. Lab. 1990. "Chaotic Behavior in Excitable Systems." Annals of the New York Academy of Sciences 591: 303-315.
Holling, C. S. 1959. "Some Characteristics of Simple Types of Predation and Parasitism." Canadian Entomologist 91: 385-398.
Iragui, V. J., and C. B. McCutchen. 1991. "Self-Abatement of Simple Partial Epileptic Seizures." European Neurology 31: 21-22.
Kaneko, K. 1983. "Transition from Torus to Noise Accompanied by Frequency Lockings with Symmetry Breaking." Progress of Theoretical Physics 69: 1427-1442.
Kaneko, K. 1993. "The Coupled Map Lattice: Introduction, Phenomenology, Lyapunov Analysis, Thermodynamics and Applications," pp. 1-49. In K. Kaneko, ed. Theory and Applications of Coupled Map Lattices (Wiley, Chichester).
Kantz, H., and P. Grassberger. 1985. "Repellers, Semi-attractors, and Long-lived Chaotic Transients." Physica D17: 75-86.
Keller, B. L. 1985. "Reproductive Patterns," pp. 725-778. In R. H. Tamarin, ed. Biology of New World Microtus (The American Society of Mammalogists, Shippenburg, PA).
Kendall, B. E., and G. F. Fox. 1995. "The Impact of Spatial Structure on Population Dynamics: Analysis of the Coupled Logistic Map," preprint.
Kendall, B. E., W. M. Schaffer, and C. W. Tidd. 1993. "Transient Periodicity in Chaos." Physics Letters A177: 13-20.
Lebrun, P., and I. Atwater. 1985. "Chaotic and Irregular Bursting Electrical Activity in Mouse Pancreatic B-cells." Biophysical Journal 48: 529-531.
Lefebvre, J. H., D. A. Goodings, M. V. Kamath, and E. L. Fallen. 1993. "Predictability of Normal Heart Rhythms and Deterministic Chaos." Chaos 3: 267-276.
Lidicker, W. Z. 1985. "Dispersal," pp. 420-454. In R. H. Tamarin, ed. Biology of New World Microtus (The American Society of Mammalogists, Shippenburg, PA).
London, W. P., and J. A. Yorke. 1973. "Recurrent Outbreaks of Measles, Chickenpox and Mumps I. Seasonal Variation in Contact Rates." American Journal of Epidemiology 98: 453-468.
Lorenz, E. N. 1963. "Deterministic Nonperiodic Flow." Journal of the Atmospheric Sciences 20: 130-141.
Lorenz, E. N. 1980. "Noisy Periodicity and Reverse Bifurcation." Annals of the New York Academy of Sciences 357: 282-291.
Mackey, M. C., and L. Glass. 1977. "Oscillations and Chaos in Physiological Control Systems." Science 197: 287-289.
Markus, M., D. Kuschmitz, and B. Hess. 1984. "Chaotic Dynamics in Yeast Glycolysis under Periodic Substrate Input Flux." FEES Letters 172: 235-238.
Markus, M., D. Kuschmitz, and B. Hess. 1985. "Properties of Strange Attractors in Yeast Glycolysis." Biophysical Chemistry 22: 95-105.
May, R. M. 1976. "Simple Mathematical Models with Very Complicated Dynamics." Nature 261: 459-467.
May, R. M. 1985. "When Two and Two Do Not Make Four: Nonlinear Phenomena in Ecology." Proceedings of the Royal Society of London B228: 241-266.
May, R. M., and G. P. Oster. 1976. "Bifurcations and Dynamic Complexity in Simple Ecological Models." The American Naturalist 110: 573-599.
Mayer-Kress, G., F. E. Yates, L. Benton, M. Keidel, W. Tirsch, S. J. Poppi, and K. Geist. 1988. "Dimensional Analysis of Nonlinear Oscillations in Brain, Heart, and Muscle." Mathematical Biosciences 90: 155-182.
McCaffrey, D., S. Ellner, D. W. Nychka, and A. R. Gallant. 1992. "Estimating the Lyapunov Exponent of a Chaotic System with Nonlinear Regression." Journal of the American Statistical Association 87: 682-695.
Michaels, D. C., D. R. Chialvo, E. P. Matyas, and J. Jalife. 1989. "Chaotic Activity in a Mathematical Model of the Vagally Driven Sinoatrial Node." Circulation Research 65: 1350-1360.
Mpitsos, G. J., H. C. Creech, C. S. Cohan, and M. Mendelson. 1988. "Variability and Chaos: Neurointegrative Principles in Self-organization of Motor Patterns," pp. 162-190. In J. A. S. Kelso, A. J. Mandell, and M. F. Shiesinger, eds. Dynamic Pattens in Complex Systems (World Scientific, Singapore).
Nerenberg, M. A. H., and C. Essex. 1990. "Correlation Dimension and Systematic Geometric Effects." Physical Review A42: 7065-7074.
Nicholson, A. J. 1954. "An Outline of the Dynamics of Animal Populations." Australian Journal of Zoology 2: 9-65.
Niedermeyer, E. 1990. The Epilepsies: Diagnosis and Management (Urban & Schwarzenberg, Baltimore).
Olsen, L. F. 1983. "An Enzyme Reaction with a Strange Attractor." Physics Letters A94:454-457.
Olsen, L. F., and H. Degn. 1977. "Chaos in an Enzyme Reaction." Nature 267: 177-178.
Olsen, L. F., and W. M. Schaffer. 1990. "Chaos versus Noisy Periodicity: Alternative Hypotheses for Childhood Epidemics." Science 249: 499-504.
Olsen, L. F., G. L. Truty, and W. M. Schaffer. 1988. "Oscillations and Chaos in Epidemics: A Nonlinear Dynamic Study of Six Childhood Diseases in Copenhagen, Denmark." Theoretical Population Biology 33: 344-370.
Olsen, L. F., K. R. Valeur, T. Geest, C. W. Tidd, and W. M. Schaffer. 1995. "Predicting Nonuniform Chaotic Attractors in an Enzyme Reaction," pp. 161-174. In H. Tong, ed. Chaos and Forecasting (World Scientific, Singapore).
Paulson, G. W. 1963. "Inhibition of Seizures." Diseases of the Nervous System 24: 657-664.
Philippi, T. E., M. P. Carpenter, T. J. Case, and M. E. Gilpin. 1987. "Drosophila Population Dynamics: Chaos and Extinction." Ecology 68: 154-159.
Pomeau, Y., and P. Manneville. 1980. "Intermittent Transition to Turbulence in Dissipative Dynamical Systems." Communications in Mathematical Physics 74: 189-197.
Prichard, J. W. 1980. "Phenobarbitol: Proposed Mechanisms of Antiepileptic Action," pp. 553-562. In G. H. Glaser, J. K. Penry, and D. M. Woodbury, eds. Antiepileptic Drugs: Mechanisms of Action (Raven, New York).
Pritchard, W. S., and D. W. Duke. 1992. "Dimensional Analysis of No-task Human EEG Using the Grassberger-Procaccia Method." Psychophysiology 29: 182-192.
Rajna, P., and C. Loma. 1989. "Sensory Stimulation for Inhibition of Epileptic Seizures." Epilepsia 30: 168-174.
Rand, D. A., and H. Wilson. 1991. "Chaotic Stochasticity." Proceedings of the Royal Society of London B246: 179-184.
Rapp, P. E., I. D. Zimmerman, A. M. Albano, G. C. deGuzman, and N. N. Greenbaun. 1985. "Dynamics of Spontaneous Neural Activity in the Simian Motor Cortex: The Dimension of Chaotic Neurons." Physics Letters A110: 335-338.
Ravelli, F., and R. Antolini. 1992. "Complex Dynamics Underlying the Human Electrocardiogram." Biological Cybernetics 67: 57-65.
Ricker, W. E. 1954. "Stock and Recruitment." Journal of the Fisheries Research Board of Canada 11: 559-623.
Röschke, J., and J. Aldenhoff. 1991. "The Dimensionality of Human's Electroencephalogram during Sleep." Biological Cybernetics 64: 307-313.
Rössler, O. E. 1976. "An Equation for Continuous Chaos." Physics Letters A35: 397-298.
Ruelle, D. 1979. "Sensitive Dependence on Initial Conditions and Turbulent Behavior." Annals of the New York Academy of Sciences 316: 408-416.
Ruelle, D. 1990. "Deterministic Chaos: The Science and the Fiction." Proceedings of the Royal Society of London Mil: 241-248.
Samardzija, N., and L. D. Greller. 1988. "Explosive Route to Chaos through a Fractal Torus in a Generalized Lotka-Volterra Model." Bulletin of Mathematical Biology 50: 465-491.
Schaffer, W. M. 1984. "Stretching and Folding in Lynx Fur Returns: Evidence for a Strange Attractor in Nature?" The American Naturalist 124: 798-820.
Schaffer, W. M., B. Kendall, C. W. Tidd, and L. F. Olsen. 1993. "Transient Periodicity and Episodic Predictability in Biological Dynamics." IMA Journal of Mathematics Applied in Medicine and Biology 10: 227-247.
Schaffer, W. M., and M. Kot. 1985. "Nearly One Dimensional Dynamics in an Epidemic." Journal of Theoretical Biology 112: 403-427.
Sleeman, B. D. 1988. "Period-doubling Bifurcations Leading to Chaos in Discrete Models of Biology." IMA Journal of Mathematics Applied in Medicine and Biology 5:21-31.
Smith, L. A. 1988. "Intrinsic Limits on Dimension Calculations." Physics Letters A133:283-288.
Speelman, B. T., R. Larter, and R. M. Worth. 1995. "A Dynamical Systems Approach to the Modelling of Epileptic Seizures," preprint (submitted to Chaos).
Sugihara, G., and R. M. May. 1990. "Non-linear Forecasting as a Way of Distinguishing Chaos from Measurement Error in Time Series." Nature 344: 734-741.
Suria, A., and E. K. Killam. 1980. "Carbamazepine," pp. 563-575. In G. H. Glaser, J. K. Penry, and D. M. Woodbury, eds. Antiepileptic Drugs: Mechanisms of Action (Raven, New York).
Symonds, C. 1970. Studies in Neurology (Oxford University Press, London).
Taitt, M. J., and C. J. Krebs. 1985. "Population Dynamics and Cycles," pp. 567-620. In R. H. Tamarin, ed. Biology of New World Microtus (The American Society of Mammalogists, Shippenburg, PA).
Takens, F. 1981. "Detecting Strange Attractors in Turbulence," pp. 366-381. In D. A. Rand and L. S. Young, eds. Dynamical Systems and Turbulence (Springer-Verlag, Berlin).
Tamarin, R. H. 1977a. "Demography of the Beach Vole (Microtus breweri) and the Meadow Vole (Microtus pennsylvanicus) in Southeastern Massachusetts." Ecology 58:1310-1321.
Tamarin, R. H. 1977b. "Reproduction in the Island Beach Vole, Microtus breweri, and the Mainland Meadow Vole, Microtus pennsylvanicus, in Southeastern Massachusetts." Journal of Mammalogy 58: 536-548.
Taylor, A. D. 1990. "Metapopulations, Dispersal, and Predator-Prey Dynamics: An Overview." Ecology 71: 429-433.
Theiler, J. 1986. "Spurious Dimension from Correlation Algorithms Applied to Limited Time-series Data." Physical Review A34: 2427-2432.
Theiler, J. 1990. "Statistical Precision of Dimension Estimators." Physical Review A41:3038-3051.
Thomas, W. R., M. J. Pomerantz, and M. E. Gilpin. 1980. "Chaos, Asymmetric Growth and Group Selection for Dynamical Stability." Ecology 61: 1312-1320.
Turchin, P. 1993. "Chaos and Stability in Rodent Population Dynamics: Evidence from Nonlinear Time-series Analysis." Oikos 68: 167-172.
Wales, D. J. 1991. "Calculating the Rate of Loss of Information from Chaotic Time Series by Forecasting." Nature 350: 485-488.
Witteman, G. J., A. Redfearn, and S. L. Pimm. 1990. "The Extent of Complex Population Changes in Nature." Evolutionary Ecology 4:173-183.
Wolf, A., J. B. Swift, H. L. Swinney, and J. A. Vastano. 1985. "Determining Lyapunov Exponents from a Time Series." Physica D16: 285-317.
Woodbury, D. M. 1980. "Phenytoin: Proposed Mechanisms of Anticonvulsant Action," pp. 447-471. In G. H. Glaser, J. K. Penry, and D. M. Woodbury, eds. Antiepileptic Drugs: Mechanisms of Action (Raven, New York).
Yamamoto, Y., R. L. Hughson, J. R. Sutton, C. S. Houston, A. Cymerman, E. L. Fallen, and M. V. Kamath. 1993. "Operation Everest II: An Indication of Chaos in Human Heart Rate Variability at Simulated Extreme Altitude." Biological Cybernetics 69: 205-212.
Yorke, J. A., and W. P. London. 1973. "Recurrent Outbreaks of Measles, Chickenpox and Mumps II. Systematic Differences in Contact Rates and Stochastic Effects." American Journal of Epidemiology 98: 469-482.
Zbilut, J. P., G. Mayer-Kress, and K. Geist. 1988. "Dimensional Analysis of Heart Rate Variability in Heart Transplant Recipients." Mathematical Biosciences 90: 49-70.
Appendix. The difficulties of finding chaos in biological data
The most common technique for determining if an experimental system is chaotic uses the method of Takens (1981) to reconstruct a putative attractor from a scalar time series. Various dynamical invariants, usually the correlation dimension and the dominant Lyapunov exponent (Eckmann and Ruelle 1985), can be estimated from the reconstructed attractor. The algorithms commonly used for estimating these quantities from a time series (Grassberger and Procaccia 1983; Wolf et al. 1985) require enough points to cover the putative attractor with a reasonable density, which is typically a large number. Because many biological systems operate on time-scales of months or years, few data sets are long enough for these methods to work reliably. The exception to this is physiology, in which the time-scale is often minutes or seconds (this explains, at least in part, the predominance of physiological studies); but here there is commonly a problem of nonstationarity. Except when the subject is in deep sleep (or in cell cultures studied in vitro) physiological activity is often affected by changes in environmental factors (including movement of the subject); these can cause either simple perturbations of the time series or actually shift the system to a different parameter regime. While the response of the system to these perturbations is itself of considerable interest, the net result is that it is difficult to obtain an unperturbed data set long enough to analyse any particular state for chaos.
More recent techniques for identifying chaos include fitting to sophisticated nonlinear stochastic models (Ellner et al. 1991; McCaffrey et al. 1992) and interpretations of the prediction accuracy of various nonlinear forecasting algorithms when applied to the data (Farmer and Sidorowich 1987; Sugihara and May 1990; Wales 1991). These methods hold promise for working with shorter data sets than the older methods require, and they can directly address the interactions between nonlinearity and noise. However, these methods are still in relatively early stages of development, and their limitations and biases are not fully understood.
All of these methods perform best on low-dimensional systems. There is no a priori reason to believe that most biological attractors are low dimensional: the number of state variables, after all, is often very large. Efforts to calculate the correlation dimension of the brain's electrical activity, for example, commonly produces values of eight or more (Basar 1990). Dimensions of this magnitude greatly compound the difficulties of limited sample size, for the amount of data required to accurately characterize the dynamics increases rapidly with dimension (Theiler 1986, 1990; Destexhe et al. 1988; Smith 1988; Nerenberg and Essex 1990). For example, Ruelle (1990) suggests that when calculating dimensions with the Grassberger-Procaccia algorithm, we should view as spurious any results that are on the order of 2 log10 N or greater, where N is the number of data points. Another confounding effect that often plagues biological data is non-uniformity (Geest et al. 1993; Olsen et al. 1995): some parts of the attractor (usually the ones which are most informative about the nature of the dynamics) are rarely visited. Once again, a large number of datapoints are required to adequately reconstruct the dynamics.