Báo cáo khoa học: Observation of a chaotic multioscillatory metabolic attractor by real-time monitoring of a yeast continuous culture doc

Observation of a chaotic multioscillatory metabolic
attractor by real-time monitoring of a yeast continuous
culture
Marc R. Roussel
1,2
and David Lloyd
1
1 Microbiology Group, Cardiff School of Biosciences, Cardiff University, UK
2 Department of Chemistry and Biochemistry, University of Lethbridge, Canada
Organisms carry out processes necessary for the main-
tenance of life on many time scales [1]. Not all possible
cellular processes are compatible, so either temporal or
spatial separation of activity is required [2]. Temporal
coordination is provided by biological clocks such as
the circadian [2,3] and circahoralian (with periods, T,
of $ 1 h) [4–9], both of which are known to function
in a wide variety of organisms [10,11]. Other oscilla-
tory phenomena observed in yeast cultures include
cell-cycle-dependent oscillations [12–15], a collective
behavior, and the well-known glycolytic oscillations
[16–19]. There are other rhythms in eukaryotic cells
which have not thus far been observed in continuous
culture systems, such as mitochondrial ion transport
[20–24] and calcium oscillations [25,26]. Mitochondrial
oscillations have been observed in single yeast cells [27]
although, to our knowledge, calcium oscillations have
not. It is not clear if the former have any physiological
role although calcium oscillations are now known to
exercise a number of functions in metabolism [28], cell
division [29–31], and differentiation and development
[32–34].

The study of biological rhythms in continuous
culture systems has important advantages over other
techniques. First, oscillations can be studied under
constant chemical and physical conditions, the rhythm
itself notwithstanding. Second, long-term experiments
can be undertaken, which is particularly important for
slow rhythms, but also allows the very large amounts
of data required by some mathematical analyses to be
collected. Among possible continuous culture model
organisms, the yeast Saccharomyces cerevisiae stands
out due to its ability to synchronize its metabolic state
across the population in a relatively short period, and
Keywords
biochemical oscillations; chaos; continuous
culture; yeast
Correspondence
M. R. Roussel, Department of Chemistry
and Biochemistry, University of Lethbridge,
Lethbridge, Alberta, T1K 3M4, Canada
Fax: +1 403 329 2057
Tel: +1 403 329 2326
E-mail:
Website: />(Received 7 November 2006, revised 6
December 2006, accepted 14 December
2006)
doi:10.1111/j.1742-4658.2007.05651.x
We monitored a continuous culture of the yeast Saccharomyces cerevisiae
by membrane-inlet mass spectrometry. This technique allows very rapid
simultaneous measurements (one point every 12 s) of several dissolved
gases. During our experiment, the culture exhibited a multioscillatory mode

in which the dissolved oxygen and carbon dioxide records displayed period-
icities of 13 h, 36 min and 4 min. The 36- and 4-min modes were not vis-
ible at all times, but returned at regular intervals during the 13-h cycle.
The 4-min mode, which has not previously been described in continuous
culture, can also be seen when the culture displays simpler oscillatory
behavior. The data can be used to visualize a metabolic attractor of this
system, i.e. the set of dissolved gas concentrations which are consistent
with the multioscillatory state. Computation of the leading Lyapunov
exponent reveals the dynamics on this attractor to be chaotic.
Abbreviations
DO, dissolved oxygen; IBI, interbeat interval; MIMS, membrane-inlet mass spectrometry; PSD, power spectral density.
FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS 1011
often without the need for an initial kick to place the
culture in a synchronous state [8,16–18,35–37]. Also,
yeasts serve as useful experimental models for eukary-
otic cell biology [38] so that knowledge gained through
the study of these organisms often leads to advances in
our general understanding of eukaryotes.
We report here on the observation of chaotic oscilla-
tions in a continuous culture of the budding yeast S. ce-
revisiae which combined a slow, cell-cycle-dependent
mode (T $ 13 h), the circahoralian mode (T $ 40 min),
and a fast oscillatory mode with T $ 4 min, not previ-
ously reported in continuous cultures. The latter
rhythm may be a manifestation at the population level
of mitochondrial oscillations [21–23]. Because of our
observational technique (membrane-inlet mass spectr-
ometry; MIMS), we were able to simultaneously meas-
ure signals corresponding to several dissolved gases
and thus to directly observe the metabolic attractor of

this experimental system.
Results
Cell-cycle-dependent oscillations
We monitored the state of a continuous culture of
S. cerevisiae using MIMS, a technique that allows for
the rapid simultaneous determination of several com-
ponents in solution [39]. Culture conditions were iden-
tical to those used to study the circahoralian clock in
S. cerevisiae [6,7]. Figure 1 shows the partial pressures
of O
2
and of CO
2
measured by MIMS relative to the
smoothed partial pressure of argon-40 used as a con-
trol, with the MIMS probe inserted directly into the
culture. The complex oscillations seen in the figure
started after a series of accidental disturbances, inclu-
ding prolonged periods of starvation (hours to days)
and, perhaps more importantly, loss of temperature
control. Indeed, we have typically observed large-
amplitude long-period oscillations overlaid with faster
oscillatory modes after temperature shocks. Complex
modes similar to this one, but with different phase
relationships of the slow and faster components, can
also be reached by pH jumps [40].
The most prominent feature of the traces in Fig. 1 is
a large-amplitude oscillation with a period of 13.1 h
(determined by Fourier analysis of the entire time ser-
ies). We find substantial cycle-to-cycle variability in

these oscillations, the cycle time varying from 11.7 to
15.5 h, with a mean of 13.6 ± 1.3 h (mean ± SD;
n ¼ 8). The dilution rate in this experiment was D ¼
0.0765 h
)1
. Because dilution and division must, on
average, be balanced, we can compute a mean doub-
ling time from the dilution rate [41] of ln2 ⁄ D ¼ 9.06 h.
The oscillatory period is thus significantly different
from the mean doubling time. Long-period oscillations
in yeast continuous cultures have been extensively
studied [12–15]. These long-period oscillations are a
collective phenomenon of the culture with a strong
dependence on the dilution rate, and have therefore
been described as a cell-cycle-dependent mode [35]. It
is thought that the oscillatory mechanism involves par-
tial cell-cycle synchronization [42,43].
Circahoralian bursts
Bursts of the circahoralian mode are obvious in the O
2
trace and can also be seen on close inspection of the
CO
2
data (Fig. 2). Three to six beats are clearly visible
in each major cycle. The beats were spaced by 27 min
at a minimum, ranging up to 52 min, with a mean of
36 ± 7min (n ¼ 30). The second burst (t ¼ 681–686 h,
$ 60 h after the last disturbance to the culture) was
much less regular than the others and probably repre-
sents transient behavior, long transients being well

known in yeast cultures [44]. It was thus excluded from
further analysis. (The inclusion of the second burst
does not sensibly affect the overall mean and standard
deviation of the interbeat intervals [IBI], but it does
have a significant effect on the statistics of the individ-
ual IBI. Most of the first burst, which occurred from
t ¼ 670 to 673 h, was acquired at a slower sampling
rate and is also excluded from any of our analyses.)
There was a marked tendency for the period to
lengthen during a circahoralian burst. The last IBI in
each burst, i.e. the time between the last two beats
before the large peak in the O
2
signal, averaged
Fig. 1. Relative MIMS signals of the m ⁄ z ¼ 32 and 44 components
versus time. These mass components correspond, respectively, to
O
2
and to CO
2
. Time is given in hours since the fermentor started
continuous operation.
A chaotic multioscillatory metabolic attractor M. R. Roussel and D. Lloyd
1012 FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS
42 ± 7 min (n ¼ 7), the penultimate IBI averaged
36 ± 4 min (n ¼ 7), whereas the mean of earlier IBIs
reached a plateau of 30.9 ± 2.4 min (when observed;
n ¼ 11). It is likely that the variation in the IBIs is
due, at least in part, to the superposition of the cell-
cycle-dependent and circahoralian oscillatory modes

rather than to variability in the underlying circahora-
lian clock. Indeed, the IBIs of a superposition of sine
waves also vary according to the relative phases of the
two waves.
Fast oscillations
There is at least one fast oscillatory component with a
period of $ 4 min which, like the circahoralian
rhythm, appears, disappears and reappears at regular
intervals. Figure 3 shows how the power spectral den-
sity (PSD) of the oxygen signal changes with time. The
value of the PSD at frequency f is essentially the
square of the magnitude of the corresponding Fourier
coefficient [45]. In other words, it tells us how strong a
particular periodic component is. We computed this
figure using 1024-point windows of the normalized O
2
data. Because our sampling rate was Dt ¼ 12 s, each
window covers $ 3.4 h, which is adequate to capture
both the circahoralian mode and faster components,
up to the Nyquist limit, T
min
¼ 2Dt ¼ 24 s [45]. The
recurrence at 13 h intervals of both the 40-min mode
and of a rhythm with a period of $ 4 min is quite clear
from the figure. This latter rhythm has not to our
knowledge been previously reported in yeast continu-
ous cultures.
The 4-min mode appears in both the O
2
and CO

2
data (Fig. 2), although it is detectable at a different
time and has a more complex waveform in the latter.
Fourier analysis of windows of the data set where this
oscillatory mode is particularly easily resolved in the
normalized O
2
record from the mass analyzer, i.e.
between the large excursions and the circahoralian
bursts, gives a period of 3.58 ± 0.15 min (average
from eight windows of the data set, each between 5
and 8.5 h long). In the O
2
record, the 4-min mode dis-
appears when the oxygen level in the culture medium
is high, at which time this mode is evident in the CO
2
data (Fig. 2). It continues in the latter time series until
roughly the midway point between circahoralian
bursts, at which point it gives way to large amplitude,
apparently random fluctuations. The 4-min mode is,
however, not evident in our recording at m ⁄ z ¼ 34
(Fig. 4), which is diagnostic for H
2
S.
Although we analyze just one data set here, we have
seen this 4-min rhythm repeatedly in this experimental
system, often in combination with the circahoralian
oscillations. We have even observed it in the off-gases
when the MIMS probe was placed in the fermentor’s

headspace. This rhythm is highly robust and reappears
after inevitable disturbances to the fermentor during
long-term operation (e.g. failures in the medium feed
system). By contrast, it does disappear for a time after
such disturbances, indicating that it is not a simple
electrical or mechanical artifact. Moreover, although
we emphasize the data from the MIMS measurements
in this report, the 4-min oscillation is also visible in
the recordings from the dissolved oxygen (DO) elec-
trode (data not shown). Differences in the instrumental
responses make the oscillation observable over a
Fig. 2. One period of the oscillation shown in Fig. 1. The inset
shows the individual data points for m ⁄ z ¼ 32 (O
2
) for a 30-min
span starting at 730 h.
Fig. 3. Time evolution of the PSD of the normalized m ⁄ z ¼ 32 data.
Equally spaced 1024-point (3.4 h) windows of the data set were
Fourier transformed and normalized so that the area under each
PSD versus f curve was the same. Colors represent the relative
intensities of the frequency components of the signal in each of
the time windows, except that all PSD values > 0.3 have been
mapped to red to enhance the contrast.
M. R. Roussel and D. Lloyd A chaotic multioscillatory metabolic attractor
FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS 1013
greater part of the cycle with the mass analyzer than
with the DO electrode, which no doubt explains in
part why this 4-min rhythm has not previously been
reported. Also, we have found no correlation between
these oscillations, on the one hand, and pH, NaOH

pump activation or heating cycles, on the other hand,
the latter two having been recorded manually for a
few hours to rule out such artifacts. In particular, the
heater turns on every 1–2 min to maintain a tempera-
ture of 30 °C, and the pH, and hence the alkali addi-
tion rate, fluctuate much less regularly than the
oscillations seen in the dissolved oxygen. We conclude
that this is a real biochemical rhythm.
Metabolic attractor
Figure 5 shows the ‘metabolic attractor’, i.e. the set of
biochemical states, as reflected in the concentrations of
dissolved oxygen, carbon dioxide and hydrogen sulfide,
through which this experimental system passes during
the complex oscillations. We measured the capacity
dimension of the attractor, one of several measures of
fractal dimension [46], directly from the 3D data set.
We found this dimension to be 2.09 ± 0.07 (95% con-
fidence). This value very near two implies that the
attractor is neither a simple cycle, which would give a
dimension near one, nor does it fill 3D space the way
a cycle fattened by a substantial level of noise would.
To go further with our analysis, we need to recon-
struct the attractor using a single time series because
methods for dealing directly with multidimensional
time series are not well developed. A standard time-
delay embedding was constructed from the O
2
signal.
The O
2

signal was chosen because it shows the three
periodicities identified above most clearly. In the fol-
lowing, we work with data which has been interpolated
so that the points are separated by equal time intervals
(see Experimental procedures for details). The points
in the time series can thus be labeled by an index i.
The analysis starts with a computation of the
mutual information I(k) between points in the time ser-
ies separated by k time intervals, i.e. between points i
and i + k for all values of i. The mutual information
I(k) measures the amount of information about point
i + k we have if we know point i [46]. The mutual
information curve is shown in Fig. 6. The oscillations
in the mutual information are due to the strong 4-min
mode of the time series: these have a mean period of
19.1 ± 0.9 points, or 3.82 ± 0.19 min.
Construction of a time-delay embedding requires
both a delay and an embedding dimension. The first
minimum in the mutual information curve was used to
set the delay [46] at 12 points (2.4 min). An embedding
dimension of at least twice the capacity dimension is
A B
Fig. 4. Relative m ⁄ z ¼ 34 signal (H
2
S) versus time. Insets each
show 1 h of data starting, respectively, at (A) 712 h a period of high
H
2
S and (B) 730 h (low H
2

S).
Fig. 5. Metabolic attractor. The values of the relative O
2
and CO
2
signals (m ⁄ z ¼ 32 and 44) are plotted on the axes, whereas the rel-
ative H
2
S(m ⁄ z ¼ 34) signal is mapped onto the color scale. Circula-
tion around the attractor is in the clockwise direction.
Fig. 6. Mutual information I as a function of the dephasing k.
A chaotic multioscillatory metabolic attractor M. R. Roussel and D. Lloyd
1014 FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS
known to be sufficient to guarantee a good embedding
[47], which suggests an embedding dimension of 5 or
higher. However, this theoretical lower bound is often
unnecessarily large. Moreover, our estimate of the
capacity dimension based directly on the multidimen-
sional time series is somewhat higher than estimates
calculated from time-delay embeddings. These values
depend weakly on the embedding dimension and delay,
and cluster around 1.9. We therefore carried out the
bulk of our analysis with an embedding dimension of
4, and spot-checked the results in five dimensions. We
also checked our results with a delay of 25 points. Sta-
tistics computed with all combinations of delays and
embedding dimensions tried are in good agreement
with each other.
For our optimal delay of 12 points and an embed-
ding dimension of 4, we found the capacity dimension

to be 1.90 ± 0.08. Note that this value is in reasonable
agreement with that found directly from the full 3D
data set. Furthermore, the reconstructed attractor
(Fig. 7) is qualitatively similar to the metabolic attrac-
tor of Fig. 5. Both of these observations imply that the
reconstruction has been successful, i.e. that most of the
information carried by our 3D data set is preserved in
the reconstructed attractor.
A capacity dimension near 2 could either result from
quasiperiodicity (multiple independent frequencies) or
chaotic dynamics. To resolve this question, we calcula-
ted the leading Lyapunov exponent, a measure of the
mean rate of divergence of neighboring points on the
attractor. A positive Lyapunov exponent indicates sen-
sitive dependence on initial conditions, a hallmark of
chaos [46]. By contrast, the leading Lyapunov expo-
nent for quasiperiodic evolution would be 0. We found
the leading Lyapunov exponent to have a value of
0.752 ± 0.004 h
)1
(95% confidence). Because the lead-
ing Lyapunov exponent is bounded well away from 0,
our analysis implies that the O
2
time series is chaotic.
By extension, we can conclude that the culture dynam-
ics is chaotic in the regime studied in our experiment.
Discussion
The fact that the 4-min mode is continually visible
during this mixed-mode oscillation, although not

always in the same dissolved gas, strongly supports the
hypothesis that we are observing an intrinsic cellular
rhythm rather than a collective behavior due mainly to
interactions between members of the population. As
with all rhythms observed in bulk measurements, this
one has to be synchronized across some portion of the
population. Because the rhythm is robustly observed in
this system and does not fade away with time, persist-
ent chemical synchronization by a diffusible factor is
indicated, rather than synchronization by a single initi-
ating event such as the disturbances to the reactor
which initiated the complex oscillations.
What is the biochemical basis of the 4-min rhythm?
The most attractive hypothesis is that we are observing
a manifestation at the population level of mitochond-
rial oscillations. Oscillations associated with ion-trans-
port processes in mitochondria with periods of a few
minutes have been observed in a variety of experimen-
tal preparations [21–23], including studies with the
same yeast strain as used in our experiments [27].
These oscillations are typically synchronized across a
population of mitochondria [20,24], although complex
spatial patterns can also be seen [23]. The ability of
some strains of S. cerevisiae to spontaneously syn-
chronize their metabolic state across a population to
reveal the circahoralian [8,35,37] and glycolytic oscilla-
tions [16–18,36] evidently creates conditions propitious
to the synchronization of mitochondrial states, making
the mitochondrial oscillations observable in the bulk
measurements.

The large-amplitude, apparently noisy fluctuations
in the carbon dioxide data, which obscure the 4-min
rhythm through part of the cycle, might also be
worthy of investigation. Their amplitude far exceeds
the noise level of the instrument. Moreover, the regu-
larity with which they appear and disappear in the
record again suggests coordinated action among the
x
i +2t
x
i +t
Fig. 7. Reconstructed attractor of dimension 4 with a time delay of
12 points. Here, x is the normalized m ⁄ z ¼ 32 signal. The first
three coordinates of the embedding are plotted as dots on the axes
and the fourth coordinate is rendered using the color scale. The teal
‘shadows’ are projections of the attractor onto the (x
i
,x
i+s
) and
(x
i
,x
i+2s
) coordinate planes, in this case with the points connected
by lines. The projection onto the (x
i+s
,x
i+2s
) plane is identical to that

onto the (x
i
,x
i+s
) plane. Note that the topology of the reconstructed
attractor is very similar to that of the directly observed metabolic
attractor (Fig. 5).
M. R. Roussel and D. Lloyd A chaotic multioscillatory metabolic attractor
FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS 1015
cells. It is possible that these fluctuations are due to
other components whose mass spectra include frag-
ments with m ⁄ z ¼ 44 such as volatile fatty acids [39].
Mixing several oscillators with periods of a few min-
utes could conceivably produce fluctuations of this
sort. However, it seems unlikely that we are observing
a mixed signal in this case: The high concentration of
carbon dioxide in the culture medium and its high per-
meability in silicone mean that one or two other dis-
solved species contributing to the signal at m ⁄ z ¼ 44
would have to display oscillations of very large ampli-
tude in order to obscure the 4-min oscillation in CO
2
.
Moreover, a very similar mode is seen at m ⁄ z ¼ 34
(H
2
S; Fig. 4). It thus seems likely that these fast fluctu-
ations are due to the biochemical dynamics of the sys-
tem, and not to an observational artifact. These
fluctuations may in fact turn out to be a yet faster

rhythm. Note also that these fluctuations may be
responsible for the slightly lower estimate of the capa-
city dimension from embeddings of the O
2
data versus
the full 3D data set. Unfortunately, our instrument
cannot collect data fast enough to decide these issues.
The finding of chaotic dynamics in this system can
be due to one of two possible factors. First, one par-
ticular oscillator whose existence was revealed by this
experiment could be chaotic of itself. The second and
perhaps more likely possibility is that these oscillators
(and perhaps others) interact. Biochemical pathways
in a cell always interact so that the completely isola-
ted functioning of any oscillator would at best be an
approximate description. These data can thus be seen
as supporting the view that multiple interacting oscil-
lators are involved in and perhaps critical to cell
function [11,48]. Such phenomena are certainly not
confined to yeast cells. Consider for instance the opti-
cal measurements of Visser et al. [49], which also
revealed a complex evolution of the frequency spec-
trum in suspensions of murine erythroleukemia cells.
It seems likely that complex oscillations will ulti-
mately be detected in most cell types when experi-
ments of sufficient temporal resolution and duration
are carried out.
Rapid sampling technologies like MIMS now
enable us to measure several variables from a single
system simultaneously. In our analysis, we were, how-

ever, mostly forced to treat each variable as a separ-
ate time series due to the lack of suitable methods
for analyzing the dynamical properties of multidimen-
sional data sets. We would encourage our mathemat-
ical colleagues to turn their attention to these
problems. A richer understanding of data sets such as
ours will no doubt emerge once such methods become
available.
Experimental procedures
Strain and culture conditions
A continuous culture of the yeast S. cerevisiae IFO 0233
was studied in an LH Engineering 500 series fermentor with
a working volume of $ 800 mL. The fermentor was stirred
at 900 rpm and aerated at 180 mL Æmin
)1
, the aeration rate
being controlled by a GEC-Elliott model 1100 air flow
meter. The feed pump (Watson-Marlow 101 U) was calib-
rated to deliver 1 mLÆmin
)1
of a standard medium whose
composition is described elsewhere [7]. The pH was main-
tained at 3.4 by controlled addition of 2.5 m NaOH solu-
tion. The temperature controller held the culture at 30 °C.
Monitoring
The state of the fermentor was monitored using oxygen
and pH electrodes, as well as a mass analyzer (Hiden Ana-
lytical, model HAL 301⁄ 3F) fitted with a membrane-inlet
probe [50]. The probe is a closed stainless-steel tube with a
small aperture drilled into its side wall near the closed end.

This aperture was covered with silicone tubing (the mem-
brane). The probe was inserted into the fermentor at a
depth sufficient to ensure that it would be covered by the
culture medium during operation. The mass analyzer was
set to record partial pressures at m ⁄ z ¼ 32, 34, 40 and 44.
Data analysis
The m ⁄ z ¼ 40 signal (Ar) was smoothed by averaging
a moving window of 300 points ($ 1 h of data). This
smoothed signal, which we denote by

P
40
, was used to nor-
malize the other signals from the mass analyzer in order to
correct for long-term drift in the response of the instru-
ment. The quantity P
i
=

P
40
, the relative signal of mass com-
ponent i, is thus used in all further analyses.
To determine the period of the large-amplitude oscilla-
tion, we used a technique based on Poincare
´
sections [46].
We looked for pairs of points in the time series where the
threshold P
32

=

P
40
¼ 7 was crossed in the increasing direc-
tion. The cycle time is then the time between crossings of
this section. Noise sometimes caused the appearance of a
cluster of repeated crossings of the section. In these cases,
we averaged the crossing times in a cluster. Calculation of
the cycle time based on the absolute maximum of each
cycle is a little less accurate since the relative m ⁄ z ¼ 32 sig-
nal versus time is relatively flat near the maximum (Fig. 2)
but gives very similar results. To analyze the circahoralian
periodicity, however, we simply used the absolute maximum
of each beat to compute the IBI.
The Hiden mass analyzer adapts its dwell times automat-
ically in order to keep the error in measurements within
acceptable limits. Accordingly, the points collected are not
uniformly spaced in time. Prior to further analysis, we
A chaotic multioscillatory metabolic attractor M. R. Roussel and D. Lloyd
1016 FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS
therefore preprocessed the data, using linear interpolation
to estimate the values of the normalized signal at equally
spaced times covering the data window of interest. For
Fourier analysis, we further applied a linear transformation
which makes the two endpoints equal to each other and
which sets the mean of the transformed time series to zero
in order to reduce low-frequency artifacts [46]: For a time
series with points x
i

, i ¼ 1,2, . , N, the transformed time
series was computed by y
i
¼ x
i
) (A+Bi), where B ¼
(x
N
) x
1
)/(N ) 1), and A ¼

x À BðN þ 1Þ=2,

x being the
mean of the time series. The PSD was then computed from
the fast Fourier transform of the y
i
in the usual way [45].
For Fig. 3, we defined a series of 1024-point equally
spaced and overlapping windows of the normalized O
2
data
(points 1–1024, 230–1253, 459–1482, , 35 351–36 374).
The PSD was computed for each window and normalized
to make the area under each of these curves identical.
Some of the analysis relies on a time-delay embedding of
the O
2
data, i.e. on analysis of data in the space (x

i
, x
i+s
,
x
i+2s
, ,x
i+(d)1)s
), where s is an appropriate delay and d
is the embedding dimension [46]. The mutual information
and leading Lyapunov exponent were calculated using
standard algorithms [46]. The Lyapunov exponent was cal-
culated as the slope of the longest linear section in the
graph of the logarithmic separation as a function of time
for nearest neighbors in the time-delay embedding. The
determination of this linear segment was done by eye, but
the results are not greatly sensitive to the choice of the win-
dow chosen. The capacity dimension of the attractor was
calculated using the algorithm of Liebovitch and Toth [51].
Acknowledgements
We thank C. J. Roussel for technical assistance. MRR’s
research is supported by the Natural Sciences and
Engineering Research Council of Canada.
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