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1 Department of Neurology, Medical University of Lübeck, 23538 Lübeck, Germany; Departments of 2 Neurology, 4 Otolaryngology-Head and Neck Surgery, and 5 Biomedical Engineering, The Johns Hopkins University, School of Medicine, Baltimore, Maryland 21287; and 3 Defense Supply Center, Philadelphia, Pennsylvania 19111
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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Optokinetic nystagmus (OKN) is a reflexive eye movement with target-following slow phases (SP) alternating with oppositely directed fast phases (FP). We measured the following from OKN in three humans: FP beginning and ending positions, amplitudes, and intervals and SP amplitudes and velocities. We sought to predict future values of each parameter on the basis of past values, using state-space representation of the sequence (time-delay embedding) and local second-order approximation of trajectories. Predictability is an indication of determinism: this approach allows us to investigate the relative contributions of random and deterministic dynamics in OKN. FP beginning and ending positions showed good predictability, but SP velocity was less predictable. FP and SP amplitudes and FP intervals had little or no predictability. FP beginnings and endings were as predictable as randomized versions that retain linear autocorrelation; this is typical of random walks. Predictability of FP intervals did not change under random rearrangement, which is characteristic of a random process. Only linear determinism was demonstrated; nonlinear interactions may exist that would not be detected by our present approach.
vestibulo-ocular reflex; prediction; fractal scaling
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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OPTOKINETIC NYSTAGMUS (OKN) is a reflexive eye movement that can be elicited by a large visual surround that moves slowly and coherently around a subject (17). OKN is characterized by alternating phases of low eye velocity (slow phases) in which eye velocity nearly matches target velocity and fast phases in which position offset accumulated during a slow phase is reset. Our study addresses horizontal OKN, in which the visual scene rotates about an earth-vertical axis and the resulting eye movements are horizontal. OKN in this case may contain some "contamination" by the smooth pursuit eye movement system in addition to the desired (reflexive) optokinetic response (17). Smooth pursuit is an eye movement that is used to follow a single target as it moves in space; it is under more volitional control than is OKN. We attempt to reduce the influence of pursuit by instructing subjects to stare straight ahead as the visual scene moves ("stare nystagmus") rather than to actively follow the scene.
An alternative way to provide the visual stimulus for OKN is by rotating the subject within a stationary visual field. Initially, eye movements will be driven by the vestibular system (the vestibulo-ocular reflex, VOR). After several minutes of constant-velocity rotation, neural signals from the vestibular system that signal head rotation decay (the vestibular rotation sensors have poor response at very low frequencies and will eventually stop indicating rotation if it is prolonged and constant), and only the relative motion of the visual scene remains to stimulate OKN.
Consider as an elementary unit of OKN a slow phase and the following fast phase. This unit can be characterized by the eye positions at the beginning and at the end of the fast phase, the velocity during the slow phase (which we will for simplicity assume to be constant during a given slow phase), the amount the eyes travel during the slow phase and during the fast phase, and the time interval between the present fast phase and the preceding one. The temporal evolution of these quantities during a long segment of OKN shows seemingly random fluctuations. The aim of this study is to uncover possible nonrandom (deterministic) patterns in this temporal evolution that might shed light on the purpose and organization of the OKN reflex. We appreciate that much is known about the physiology of the systems that generate fast and slow phases of nystagmus, whereas relatively little is known about the processes that determine, for example, such aspects as the timing of fast phase generation. Even the best models in this area (e.g., Refs. 2, 3, 7) have a significant probabilistic component. The intent of our study is to see whether these properties are inherently random or whether there may be an underlying deterministic law that guides them. If the latter, then even though this study may not provide the details of that law, we will nevertheless know better whether there is such a rule to be sought after.
The question of randomness vs. determinism in a physiological system is a fundamental one. The presence or absence of deterministic elements in OKN is of particular interest if a mathematical model of the underlying neural circuitry is sought. Various correlations between the parameters characterizing OKN can be studied. For VOR, a high correlation between slow-phase amplitude and the following fast-phase amplitude and a lower correlation between fast-phase amplitude and the following slow-phase amplitude have been reported (7). These correlations may be statistical in nature, or they may be the consequence of a nonlinear deterministic rule with important implications for the modeling process.
A new class of powerful tools for the analysis of seemingly random signals has emerged from the field of nonlinear dynamics (for an overview, see Refs. 1, 28). The development of these tools followed the observation that systems in which temporal evolution is described by simple deterministic equations can produce extremely complicated, seemingly random, time series (18). This class of systems has been termed "chaotic." One of the first measures derived to describe the complexity of these systems was the correlation dimension, which can characterize low-dimensional chaotic systems (as an approximation of the fractal dimension). Random systems have a high (theoretically infinite) dimension (9). It has proven difficult, however, to distinguish between signals from random systems that have undergone a filtering process and signals from deterministic systems on the basis of the correlation dimension alone (19). Another approach to decide between the deterministic or the random origin of a signal is to compare measures derived from the original sequence with those same measures derived from a randomized version of the sequence (a "surrogate"; Refs. 30-32). If the measure in question (dimension, predictability) does not change when the signal is randomized, then this is good evidence that the original signal had significant randomness to begin with. Different null hypotheses on the data structure lead to different methods to randomize the sequence; the null hypothesis can be rejected if a measure characterizing a sequence changes significantly under randomization.
As a second tool to identify deterministic systems, nonlinear prediction has been introduced by Farmer and Sidorowich (8). Whereas the evolution of a random system cannot be predicted, the behavior of a deterministic system can be predicted, at least for short time spans. One way to obtain a prediction at a given state of the system is to look at the past development of the system for states similar to the present one. The evolution from these past states can then serve as a model that predicts the development under consideration.
Comparisons of the predictability of various subsets of data (OKN under various pathologies, for example) may give insight into the "relative predictability" and thus the relative determinism/randomness of the OKN mechanism under different circumstances. One example of the insights that might be gained from the analysis of predictability can be found in a study of monthly data on epidemics (30), in which the dynamics of different epidemics were related to underlying mechanisms of disease transmission and reporting. In a study more along the lines of ours, Lefebvre et al. (16) analyzed predictability of heart beat intervals and found relations to cardiac pathology.
Shelhamer (27) used the correlation dimension to address the question of determinacy in OKN. In his study, some surrogate procedures produced time series that did not differ in correlation dimension from the original signal, and some surrogates showed higher correlation dimension than the original signal. Thus arguments for both random and deterministic structure in OKN were found. Nonlinear forecasting was then used to help settle the issue. As a preliminary result (29), a high predictability for fast-phase beginning and end positions could be demonstrated, along with little or no predictability for fast-phase amplitudes and fast-phase intervals. The present study extends that investigation by examining the predictability of surrogate data sets, and analyzing the scaling behavior (i.e., the decay of predictability as a function of the number of prediction steps) of the OKN parameters under study. The predictability of first differences is also used to help distinguish between random and deterministic behavior.
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METHODS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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Data collection. Horizontal OKN was elicited in three subjects who sat inside a cylinder of diameter 156 cm that was covered with a random black-and-white pattern. The drum was rotated with a constant angular velocity of 10, 30, or 60°/s around the subject (OKN condition), and in separate trials the subject was rotated at these speeds within the illuminated stationary drum (OKN invoked during head rotation; OKN-HR). Rotation lasted between 115 and 210 s. In one subject, we also collected 300 s of data at 20, 40, and 60°/s in the OKN condition. Eye movements were measured with the magnetic search coil technique (21) and were sampled at 500 Hz.
Fast-phase beginning and ending positions were identified by use of an interactive computer program. Between 857 and 998 fast phases were recorded for the longer (300-s) files, and the shorter files contained 231-638 fast phases. Slow-phase gains (slow-phase velocity/stimulus velocity) ranged from 0.9 for low-stimulation velocities to 0.2 for the fastest stimulation condition (the latter value is low but still within the broad distribution of normal values; see for example Ref. 6). The following parameters were extracted from each data set (Fig. 1): eye positions at the beginning and end of each fast phase, fast-phase amplitude, interval between adjacent fast phases, mean velocity of each slow phase, and position traversed during each slow phase.
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Prediction procedure.
Successive values of each OKN parameter form a data sequence
{p1, ... , pn}
(Fig. 2A). In the engineering
analysis of many types of systems, it is common to examine the behavior
of the system in a state space, in which various state
variables (e.g., position, velocity, etc.) are plotted with respect to
each other. In general, we may not know the state variables of the
system, may not have access to them, and may not know how many state
variables are needed (i.e., the dimension of the space) to fully
characterize the behavior. To avoid these issues, time-delay embedding
can be used to reconstruct the state space, using the time series of a
single measured quantity. Consequently, as the first step in nonlinear
prediction of each sequence, a time-delay embedding (24)
was performed (Fig. 2B). Through time-delay embedding in an
embedding dimension of M, the state of the system at the
nth point in the sequence is characterized by a point in an
M-dimensional space, with coordinates of that point given by
the parameter under consideration for fast phases {n,
n 
1, ... , n (M 1)} (i.e., the present value and the
M 1 previous values of the parameter). The resulting
topological object on which these trajectories exist is usually termed
an "attractor." (A true attractor in fact only exists if
trajectories return to the same region of state space after
perturbations of the system. Certainly, our subjects made occasional
voluntary eye movements, closed their eyes, and in other ways
interrupted the ongoing OKN, and the trajectories in state space
continued to overlay one another, as demonstrated empirically in Ref.
27. Therefore, although not verified rigorously, we use
the term attractor to describe the trajectories in the state space.)
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50, ... , n 1} (i.e., the immediately preceding 50 values). For the longer files,
a template length of 100 points was used. A second-order polynomial fit
was made to the coordinates of the points at
{n1 + d, ... , nM + 1 + d} as a
function of the coordinates of points at {n1,
... , nM + 1}. From this model
polynomial, estimates of the coordinates of the point at
(n + d) (the prediction) were made by
plugging in the coordinates of point n. Note that a new
polynomial is derived for each point in the series from which
predictions are desired, forming a local topological fit.
To verify the appropriateness of our template size (i.e., the number of
earlier points from which the K nearby points were selected), we computed all possible interpoint distances in the entire
reconstructed attractor. This represents the extreme case of a template
the size of the entire data set. Then we compared these distances to
the median of the distances actually used for each set of predictions,
with template sizes of 50 and 100 points. In the case of a template of
50 points, only 9.4% of the distances taken from the entire attractor
were less than the median of the distances actually used. With a
template of 100 points, only 5.6% of all distances were less than the
median of the distances among those points used. From this we conclude
that, even if nearest neighbors were drawn from the entire attractor,
only a small proportion of them would be smaller than the distances
that we in fact used on the basis of our actual templates.
After calculating predictions starting from every fast phase of the
data set, we assessed the prediction quality by calculating the
coefficient of correlation r between the predicted values and the actual future values (Fig. 2, D and E).
Prediction was carried out d = 1 to d = 10 steps ahead. Following Tsonis and Elsner (33),
log(1 r) was plotted as a function of prediction step d and also as a function of log(d).
Prediction error for a deterministic system will scale approximately as
a decaying exponential with time step, so that log(1 r) vs. d will be a straight line. Prediction
error for a random system [more specifically, fractional Brownian
motion (FBM)] will scale as a synthesis of power laws, such that
log(1 r) vs. log(d) will appear as a
straight line.
Surrogate data sets. Predictions of the OKN parameters were compared with predictions of randomized, surrogate data sets. A shuffled surrogate (S-surrogate) was obtained by random rearrangement of the sequence {p1, ... , pn} itself. A S-surrogate tests the null hypothesis that the data sequence can be modeled as a sequence of independent identically distributed random variates. A phase-shuffled surrogate (PS-surrogate) was also obtained, as follows. In the Fourier transform of the parameter sequence under study, the phases were randomly rearranged (respecting the symmetry required for a real-valued signal), and the inverse Fourier transform was applied to the resulting sequence. With a PS-surrogate we can test whether the parameter sequence can be modeled as linearly correlated Gaussian noise. Finally, an amplitude-adjusted phase-shuffled surrogate (AAPS-surrogate, see Ref. 32) was generated by arranging a sequence of Gaussian random numbers such that the rank order of their amplitudes in the sequence matched that of the original parameter sequence, obtaining a PS-surrogate from the result, and rearranging the original parameter sequence according to the rank of this PS-surrogate. With the AAPS-surrogate, the hypothesis tested is that the parameter sequence is linearly correlated Gaussian noise with a static monotonic nonlinear scaling.
For each parameter, 10 surrogate data sets of each type were obtained. The correlation coefficient for prediction of the original parameter sequence must be outside of 99% of the distribution of correlation coefficients from the surrogates to reject the null hypothesis that the surrogates are identical to the original (i.e., that the original sequence has a dominant random component). This statistical determination was made as follows. The correlation coefficients (prediction qualities) were analyzed by taking the Fisher z-transforms of the coefficients and finding the mean and standard deviation of the result. Confidence intervals were obtained as the inverse z-transform of the mean of the transformed coefficients ±2.576 × standard deviation of the transformed coefficients (22). This corresponds to a cutoff of p = 0.01 for significance in comparing predictions of a parameter with its surrogates. The z-transformation adjusts for the fact that, as r approaches 1, small changes in the correlation coefficient correspond to large changes in prediction quality.Supplementary analyses.
First differences {(p2 p1), ... , (pN pN
1)} of fast-phase beginning
and ending positions were also examined for predictability, as
described above. For a nonlinear deterministic signal, the first
differences should be as predictable as the original sequence (13). If the signal is a random walk (i.e.,
pn + 1 = pn + xn with {xn} an
uncorrelated random sequence), then the first differences should
exhibit no predictability.
n} are chosen as random numbers in
{0,2
}. To produce an FBM with Hurst scaling exponent of
H, for 0 < H < 1, the amplitude of each
component {an} is further scaled by the
factor n(0.5H) to form a series
with a power spectrum of 1/f2H + 1. The FBM sequence {pn} is then
calculated as the inverse Fourier transform of
{anexp(I
n)}, ignoring the imaginary part of the result.
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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We will present our results separately for fast-phase beginning and ending positions (group 1 parameters) and all other parameters (group 2 parameters).
Nonlinear prediction of fast-phase beginning and ending positions.
The predictabilities (correlation coefficients) for fast-phase
beginning and ending positions were relatively high (see Fig. 3). Notably, these predictabilities did
not change significantly under the AAPS-surrogate procedure (no
significant change for any prediction step d in 11 of 21 files, no change for d = 1 in 19 of 21 files for both
parameters, Fig. 4). There was an
unexpected increase of predictability under the PS procedure relative
to the original OKN. [This increase in predictability is
counterintuitive, but Schiff and Chang (25) found a
similarly unexpected decrease in the correlation dimension of
(averaged) H-reflex data under phase randomization.] In a
log-log plot, (1 r) was a linear function of
prediction step d in most cases (indicative of a random sequence), whereas the graph was curved in the log-linear plot (Fig.
4). A comparison of the predictability of fast-phase beginning and
ending positions in each file with a paired t-test showed no
significant difference (P = 0.39).
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Nonlinear prediction of other OKN parameters.
The group 2 parameters showed low correlation coefficients
(fast- and slow-phase amplitude and fast-phase interval) or correlation coefficients comparable to those for the group 1 parameters
(slow-phase velocity). In general, there was only a slight reduction in
predictability under phase shuffling or amplitude-adjusted phase
shuffling; this was most evident at low values of d. The
original parameter sequences of the slow phase and fast-phase
amplitudes and slow-phase velocities were significantly more
predictable than surrogates generated via (plain) shuffling, whereas
the predictability of the fast-phase intervals often did not change
under this procedure (no change for d = 1 in 13 of 21 files; see Fig. 5).
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Comparison of parameters, stimulation conditions. We performed a two-way repeated-measures ANOVA on the z-transformed r values for d = 1, with stimulation condition (OKN-HR vs. OKN) and parameter type (fast-phase beginning position, fast-phase ending position, fast-phase interval, fast-phase amplitude, slow-phase amplitude, slow-phase velocity) as factors. Stimulus velocity was included in the between-subjects variable. The influence of stimulation condition was not significant (F1,8 = 0.338): there is no difference in predictability between the OKN and the OKN-HR conditions. There was no significant interaction of parameter type and stimulation condition (F5,8 = 0.497). Parameter type was significant (F5,8 = 21.78, P < 0.01): as already noted, there is a difference in predictability of the different parameters.
Variance scaling, Hurst exponent.
The variance of fast-phase beginning and ending positions, plotted as a
function of the length of the time interval considered, showed a
power-law scaling (indicated by a straight line in the log-log plot,
Fig. 6). Hurst exponents were between
0.05 and 0.30 for beginning and ending positions and below 0.10 for
slow-phase and fast-phase amplitudes and fast-phase intervals.
Slow-phase velocity exhibited exponents up to 0.20. For the specific
values, see Fig. 7. No statistically
significant difference between the exponents for fast-phase beginning
and ending positions or between the stimulation conditions could be
demonstrated.
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FBM.
For FBMs with lengths comparable to the fast-phase parameter sequences,
high predictabilities were obtained with nonlinear prediction. The
log-log plot of (1 r) as a function of the
prediction step d exhibited a straight line (as expected for
a random process); this behavior was not changed by phase shuffling.
First differences of FBM showed only a small amount of predictability
and only for low and high Hurst exponents (approaching 0 and 1, respectively). Significant predictability occurred only for
d = 1.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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Predictability of different parameters. The interpretation of our findings has to account for the fact that there seem to be two groups of parameters, one group being somewhat predictable (fast-phase beginning and ending positions) and one group being less predictable (slow- and fast-phase amplitudes, fast-phase intervals, slow-phase velocity).
As one can see from the example of FBM, random processes can often give rise to significant predictability (high correlation coefficients) with a nonlinear prediction approach. Brownian motion (i.e., FBM with Hurst exponent H = 0.5) is a model for the one-dimensional diffusion of a particle in a solution; the dynamics of such a system are described by position increments within a time interval. These increments are independent of previous positions. This can be expressed by conditional expectations as E{x[(n + 1)
] x(n)/ x(), ... , x(n)} = E{x[(n + 1)] x(n)}. Therefore, the past history of the diffusion process provides no information on future increments (5), and thus increments of a Brownian motion
are not predictable. However, this is not the case for future
positions: E{x[(n + 1)]/x(n)} = x(n),
where x(n) is random variates of a Brownian
process at discrete times n (n = 1, 2, ...) and E is expectation of the random variate. Therefore,
future positions can be predicted, and the best prediction at each time
step into the future is simply the position at the immediately
preceding time. This is exactly reflected in the prediction results for artificially generated FBM with H = 0.5, which showed
high predictability of the sequence itself and no predictability of the
first differences (in which the position information is destroyed and
only changes are preserved). With H
0.5, the situation is
slightly more complicated, because this implies long-range correlations
between future and past first differences in the sense of persistence
(past increase implies future increase, H > 0.5) and
antipersistence (past increase implies future decrease,
H < 0.5). Regardless of the value of H, FBM
shows a power law scaling of (1 r) as a function of
the prediction step, which is manifest as a straight line in the
log-log plot of (1 r) vs. d
(33). Finally, correlation coefficients obtained in
nonlinear prediction of FBM are identical to those from their
PS-surrogate; this is to be expected because the phases of the Fourier
transform of FBM are random by definition.
Fast-phase beginning and ending positions share properties with FBM;
future values in each sequence are best predicted by the immediately
preceding values, and their first differences form (mostly)
unpredictable random sequences. The latter has been used as a criterion
to distinguish nonlinear chaotic systems from random systems
(13). Furthermore, beginning and ending positions have the
appropriate scaling law for the correlation coefficient of nonlinear
prediction, which differs from the result expected for a chaotic
(deterministic) system (34). Note, however, that in
contrast to FBM, the appropriate surrogate for beginning and ending
positions is the AAPS-surrogate, because the distribution of fast-phase
beginning and ending positions is not Gaussian. In fact, application of
the PS procedure results in a significant increase in predictability in
data sets that violate the Gaussian distribution. This could lead to
false negative results, because a significant decrease in true
predictability could be masked by an artifactual increase due to an
incorrect surrogate.
The variance scaling behavior of FBM is described by its Hurst
exponent. Hurst exponents of up to 0.3 for fast-phase beginning and
ending positions place them between classical Brownian motion, which
corresponds to H = 0.5, and a random sequence. The
deviation from 0.5 causes a small amount of antipersistence, which
implies that a leftward drift is on average followed by a rightward
drift. Thus the correlation between adjacent drifts is negative.
However, the correlations thereby induced are only on the order of
22H1 1 = 0.25, for (increments of)
fast-phase beginning and ending positions (4). This
correlation accounts for the predictability of the first differences
for d = 1.
The second group of variables showed markedly smaller predictabilities
(correlation coefficients). Of these, the sequence of fast-phase
intervals is easily identified as a sequence of independent random
variates, because its predictability often does not change (does not
decrease, in particular) even under random rearrangement of the
parameter sequence. This also demonstrates that the fast-phase interval
sequences are stationary.
The other three parameters in this group (fast-phase amplitude,
slow-phase amplitude, and slow-phase velocity) show a change in
predictability under the plain shuffling procedure but not under PS
shuffling and AAPS shuffling. This suggests that they are FBM
sequences, because preserving the power spectrum (which is
f
for FBM, with 
= 2H + 1 and frequency f) does not change the predictability. Hurst
exponents of the sequences of fast- and slow-phase amplitude were near
zero, however. The variance scaling behavior defining FBM (i.e.,
increase ~t2H) approaches the
variance scaling of a random sequence (i.e., constant variance) as
H approaches 0. At the same time, the correlation coefficient between (xn + 1 xn) and (xn xn 1) (equal to
22H1 1; see Ref. 4)
approaches 0.5, which is to be expected if xn
is a sequence of independent Gaussian random variates. Thus, for
H approaching 0, the FBM approaches a random sequence. The low degree of predictability that is revealed by the change under shuffling is therefore only due to nonstationarity in the data sequence, that is, by long-term fluctuations and not by putative short-term interdependencies that reflect the dynamics of the system.
We conclude therefore that slow- and fast-phase amplitudes also form
random sequences.
The sequences of slow-phase velocities have a Hurst exponent of ~0.2.
This corresponds to antipersistent behavior that allows a certain
degree of prediction. Slow-phase velocity is the only parameter that is
not a random walk or a random sequence, but is somewhere between these extremes.
Comparison with previous studies of OKN correlation dimension and predictability. Shelhamer (27) looked for deterministic components in OKN with the correlation dimension. A comparison with the present study is limited by the fact that embedding the complete time series, as in the dimension study, mixes fast and slow phases. In particular, this holds true for PS or APS shuffling. A finding that can be compared with the results presented here, however, is that under shuffling of fast- or slow-phase segments, the correlation dimension did not change. This is compatible with the assumption that fast- and slow-phase amplitudes form a random sequence and is in accord with our findings, if one keeps in mind that the dimension was estimated in shorter 10-s segments that could be considered stationary.
There is an apparent discrepancy between the results of our earlier study of OKN prediction (29) and this one. The previous study claimed to show evidence for determinism in OKN fast-phase starting and ending positions, on the basis of improved prediction using a local nonlinear approximation (as here) vs. a global linear (ARMA, autoregressive moving average) predictor. Surrogate data techniques were not used in that study. In the present study, there is a slight reduction in predictability for surrogates formed from these sequences, but the reduction is small, indicating that the predictability is due in large part to random correlations as for FBM. One explanation for the discrepancy between that result and the ones presented here may be that the dynamic behavior of OKN on a global scale (i.e., over the duration of our data records) is nonstationary, with a parameter or parameters that vary slowly and in a random fashion, whereas the short-term behavior is more (but not completely) deterministic. This can also be seen in the correlation dimension results (27), which show that the dimension of 10-s OKN segments decreases over the course of 2 min. By using data over a long time span, the ARMA predictor unavoidably includes the apparently random nonstationarity or drift, leading to poor predictions. The nonlinear predictor, on the other hand, makes a local topological fit. This is also effectively a local time fit if the attractor is drifting in state space because of some slowly time-varying parameter(s). Thus the localized nonlinear predictor is better able to characterize any short-term determinism (either linear or nonlinear) in the data series. An obvious point for further study would be the development of a local linear (ARMA) predictor and time-localized surrogates to help resolve this issue further, as well as an exploration of prediction based on OKN parameters that may be coupled in some nonlinear fashion. The predictability seen in the previous study, although found with a local nonlinear predictor, may in fact be linear predictability, the result of a stochastic process with significant short-term autocorrelation. We might also interpret the long-range correlations present in an FBM sequence (as the sequence of fast-phase beginning and ending positions seem to be) as a form of "determinism" in that these correlations impose a structure on the data; this structure may have been detected as predictability in the previous study.Nature of the prediction method. Predictions in this study were carried out for a given OKN parameter sequence by using only previous values of the same sequence. For example, predictions of future fast-phase amplitudes were based solely on previous fast-phase amplitudes. There is a relationship (the well-known "main sequence" for saccadic eye movements) between these amplitudes and fast-phase velocity and duration (although the latter two values were not studied here), and in that sense the fast-phase amplitudes are not entirely random. Although our analysis does not explicitly consider relationships between variables, this is implicit in the state-space approach because all parameters are presumably coupled through the dynamics of OKN, and the time-delay embedding and reconstruction, if done properly, will preserve these interrelations. It is certainly true that the properties of a given fast phase are somewhat constrained by the preceding slow phase, but this does not help to predict subsequent fast phases.
Physiological implications. The most unambiguous findings in our study are the stationarity and the random character of fast-phase intervals. This justifies random approaches for both describing and modeling this parameter. A random parameter is adequately described by its distribution. Stationarity implies in particular that the fast-phase intervals during a trial did not change their distribution.
Anastasio (2), in a model of goldfish OKN, hypothesized a quantity that drifts under random influences toward a fixed threshold, which then triggers a fast phase, and thereby obtained an inverse Gaussian distribution (26) for the fast-phase intervals, which was compatible with his experimental findings. Balaban and Ariel (3), modeling turtle OKN, assumed that the ratio of adjacent fast-phase intervals could be produced by a random number and obtained a log-normal distribution of fast-phase intervals. The random walk nature of fast-phase beginning and ending positions implies that, at least within the range that the eyes moved in our study, all orbital positions are equivalent. (We stress that this is within the range of eye movements actually attained, as it is well known that average eye position during nystagmus is deviated in the direction of the fast phases, the so-called "beating field." Within this range, there is no apparent preference of one position over another.) The eyes tended to reach any arbitrary position (in this range) after completion of a slow phase and the following fast phase. Even the small amount of antipersistence intrinsic to these parameters reflects a reaction to the previous shift in eye position rather than the previous position itself. This is the meaning of the correlations between adjacent fast-phase and slow-phase amplitudes in cat VOR noted by Chun and Robinson (7). They reported relatively high correlation between the amplitude of each fast phase and the amplitude of the preceding slow phase. Therefore, a fast phase corrects appropriately for the eye position accumulated during a slow phase; thus the ending positions of the fast phases might be expected to exhibit little noise (randomness). Because the correlation of fast-phase amplitude with the amplitude of the subsequent slow phase was not as high, the ending positions of the slow phases (i.e., the fast-phase beginning positions) should show a larger variation. Therefore, the predictability of the fast-phase beginning positions should be higher than the predictability of the fast-phase ending positions. This is not reflected in our OKN and OKN-HR data, however, given that we could not demonstrate a significant difference in the predictability of these two parameter sequences. It is noteworthy that we could not demonstrate a difference in the predictability of any parameter in the comparison of OKN vs. OKN-HR stimulation. After several VOR time constants, no input from the semicircular canals is to be expected, and therefore both stimulation conditions are used interchangeably in routine clinical testing of the vestibular-optokinetic system (11). However, in the OKN-HR condition there is some otolith stimulation due to centrifugal stimulation. It mimics right tilt of the right utricle and left tilt of the left utricle, and therefore no net effect is expected. However, a modification of the fast-phase-generating algorithm is theoretically possible but is ruled out at least by the measures we applied in our study. Whereas the random walk character of the eye positions at the beginning and end of fast phases is not surprising if they arise by the summation of random shifts due to imprecise matching of fast- and slow-phase amplitudes, it is noteworthy that slow-phase velocity exhibited a dynamic behavior between random walk and random sequence. This implies that future slow-phase velocities depend on the past values of this parameter, but in a linear and stochastic manner, not in a nonlinear deterministic manner. This could be an interesting starting point for future modeling efforts in this field. It is tempting to speculate about the implications of our findings for the function of the neural mechanisms (vestibular and oculomotor) involved in the optokinetic response. There is undoubtedly a gross level of deterministic behavior in OKN, as for example fast-phase starting and ending positions are almost completely contained in the beating field (see above). Yet within this range, there is apparently significant random behavior, with a small amount of predictability. The random behavior has overlaid on it a form of long-term correlation in the form of antipersistence. This mixture of dynamics is intriguing and provides a challenge for mathematical modeling efforts. The physiological meaning of these dynamics is open to conjecture. One reasonable interpretation is simply that this system is inherently noisy (random), with only that amount of imposed determinism to accomplish effectively its desired function: if it is not detrimental to performance, no energy (metabolic or evolutionary) is expended in reining in any random aspect of the behavior. This does not rule out the notion that some amount of variability may indeed enhance the performance of the system, perhaps by aiding phase transitions (15), preventing mode-locking periodicities, or pushing the system to probe more of the environment through a form of random search. It seems more likely, in our view, that this variability is inherent and allowed to remain, rather than being deliberately created.| |
ACKNOWLEDGEMENTS |
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This work was supported by Deutsche Forschungsgemeinschaft (Grant Tr 449/1-1), by the National Science Foundation (Grant DBI-9630733), and by the Whitaker Foundation.
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FOOTNOTES |
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Address for reprint requests and other correspondence: P. Trillenberg, Medizinische Universität zu Lübeck, Klinik für Neurologie, Ratzeburger Allee 160, D-23538 Lübeck, Germany (E-mail: trillenberg_p{at}neuro.mu-luebeck.de).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 16 January 2001; accepted in final form 18 June 2001.
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DISCUSSION
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, OKN stimulus condition. 
,
OKN-HR condition.
