## Abstract

To compensate for neural delays, organisms require predictive motor control. We investigated the transition between reaction and prediction in saccades (rapid eye movements) to periodically paced targets. Tracking at low frequencies (0.2–0.3 Hz) is reactive (eyes lag target) and at high frequencies (0.9–1.0 Hz) is predictive (eyes anticipate target); there is an abrupt rather than smooth transition between the two modes (a “phase transition,” as found in bistable physical systems). These behaviors represent stable modes of the oculomotor control system, with attendant rapid switching between the neural pathways underlying the different modes. Furthermore, predictive saccades exhibit long-term correlations (slow decay of the autocorrelation function, manifest as a 1/*f* ^{α} spectrum). This indicates that predictive trials are not independent. The findings have implications for the understanding of predictive motor control: predictive performance during a given trial is influenced by a feedback process that takes into account the latency of previous trials.

## INTRODUCTION

The saccadic system produces eye movements that rapidly redirect the fovea to visual targets. Reaction time (latency) is approximately 200 ms for typical saccades of random amplitude, direction, and timing (Becker 1989). Under some cases of regular target stepping, predictive saccades are made, which have a much shorter latency. Previous studies have shown that periodically paced targets presented in different frequency ranges can promote either reactive or predictive behavior (Ross and Ross 1987; Stark et al. 1962; Zambarbieri et al. 1987). When targets are presented at low frequencies (e.g., 0.2 Hz), latency is normally between 150 and 220 ms and is considered a reactive response to the stimulus. At higher pacing frequencies (e.g., 0.9 Hz) saccade tracking switches to a predictive response with a reduced latency, usually <80 ms.

An analogy to these two different behaviors of the saccadic system can be found in finger tapping. In one experiment (Haken et al. 1985; Kelso 1981, 1995), subjects were requested to tap the index fingers of both hands in synchrony with a pacing metronome. At low frequencies, tapping is in phase (“tapping together”) and in time with the metronome. When the frequency of the metronome increases, tapping behavior becomes 180° out of phase (“tapping alternately”). This change in behavioral modes occurs with an abrupt rather than smooth transition. This abrupt change in tapping behavior was interpreted as a *phase transition* (a rapid change between steady states) similar to such physical transitions as that of liquid to solid. Following this transition, the metronome's frequency was decreased toward the initial tapping frequency; tapping remained in the out-of-phase condition beyond the initial transition point, before switching back to in-phase. This *hysteresis* represents the system's desire to remain in its present state and implies stability of the two tapping modes.

In this study, we wished to determine if the saccadic eye movement system exhibits similar properties: a phase transition when tracking periodically paced targets at different frequencies. Earlier work suggests that such a transition may be present but had gone unnoticed (Ross and Ross 1987; Stark et al. 1962). Furthermore, existing studies of saccade generation have not explicitly addressed the generation of predictive behavior or the transition from reaction to prediction. Here, we provide evidence that a phase transition occurs in the saccadic system as it switches between reactive and predictive behavior. In addition, using spectral analysis and scaling properties of the latency time series, we demonstrate that these behaviors exhibit different structures of time correlation.

## METHODS

The eye movements of eight subjects were recorded while they performed one of two saccade tasks. Informed consent, according to the local institutional research board, was obtained from each participant. Subjects A–D performed task 1 (phase transition); A and D were naïve as to the goals of the study. Subjects E–H performed task 2 (scaling); only F was non-naïve. (Subject C is the same as subject F). Subject I performed a modified version of task 1 as a control experiment.

Data were acquired on a PC compatible Pentium 166-MHz computer running real-time experiment control software developed inhouse. Horizontal movements of the eyes were recorded with a Series 1000 Binocular Infrared Recording System (Microguide), sampled at 500 or 1,000 Hz. The system was calibrated prior to data acquisition by having subjects fixate known targets. Subjects were seated in a stationary chair, and the head was fixed with a chin rest. (There was some tendency for subjects to move the head in time with the target. In one case we recorded head movements with a search coil attached to a biteboard that was held in the subject's mouth but not attached to any stationary apparatus. The head moved in a somewhat sinusoidal manner at the higher pacing frequencies, with some occasional larger movements at lower frequencies—probably due simply to the subject fidgeting during the longer duration of the low-frequency segments. The amplitude of the approximately sinusoidal movements was about 0.1° peak-to-peak, with the larger movements at about 0.5°.)

Subjects performed two eye movement tasks. The first was designed to look for phase transitions and hysteresis. The second was designed to assess the statistical structure of the latency time series in the reactive and the predictive regimens. In both paradigms, subjects followed target jumps between two LEDs at ±15° on either side of the vertical midline in a dark room. In the first task, targets jumped at a series of frequencies that increased monotonically from 0.2 to 1.0 Hz. There were six major frequencies (0.2, 0.3, 0.5, 0.7, 0.9, and 1.0 Hz) and five minor frequencies between them (0.25, 0.4, 0.6, 0.8, and 0.95 Hz). Fifty target jumps were presented at each major frequency and 10 jumps at each minor frequency. (Analysis was carried out on data at each major frequency. Minor frequencies were interspersed to make the change between major frequencies less dramatic for the subjects.) When 1.0 Hz was reached, targets were presented at the same frequencies as they decreased monotonically in the opposite order. Target pacing was carried out continuously during the test session (approximately 9 min). In the second task, the targets jumped for 1,000 trials at 0.2 Hz, and following a break, for 1,000 trials at 0.9 Hz. In each task, subjects were asked to follow the targets and were given no explicit instructions as to timing or accuracy; they were told simply to “look at the targets.”

As a control, subject I performed a modified version of task 1. Only the major pacing frequencies were presented, in random order. Eye movements and head movements were recorded with a scleral search coil. This experiment allowed us to address three subsidiary issues: head motion during tracking, anticipatory slow eye movements, and whether or not a randomized presentation of frequencies produces a phase transition as with the ordered presentation described previously.

Analysis of eye tracking data was done off-line with an interactive computer program that determined the onset time of each saccade. Saccade latency was determined by comparing the onsets of the primary saccade and the target in each trial. The first 10 saccades at each major frequency (0.2, 0.3, 0.5, 0.7, 0.9, and 1.0 Hz) were eliminated to remove transients, as were the 10 saccades made at each minor frequency. (These transient responses were not analyzed because we are interested in the steady-state behavior at each frequency—in particular, whether or not the tracking settles into a stable unimodal state.)

For the first task, statistical tests were carried out as described below to confirm that the latency data could be described in terms of a phase transition.

Two computational procedures were used to assess the correlation structure of the latency time series from the second task. (The “latency time series” is the sequence of latencies for all trials, indexed by consecutive trial number.) The first method quantifies the fractal structure of a time series, in a statistical sense. A fractal (Mandelbrot 1983) is an object that is self-similar on different scales. We quantify statistical self-similarity (in terms of fluctuations) via rescaled range analysis (Bassingthwaighte et al. 1994) as a function of time window. The Hurst exponent *H* quantifies the magnification factor needed to produce statistically identical fluctuations as the time scale is changed (Beran 1994; Taqqu et al. 1995). In the case of the exponent *H*, it is desired to see if the “rescaled range” is a power law function of the time duration of the data. The rescaled range is defined as the range of the data (span from minimum to maximum values) divided by the SD: *R/S*. For certain types of time series, this value changes systematically as the length of the series: *R*/*S* ∝ (*T*)* ^{H}*, where

*T*is the duration of the time series. Computation of

*H*is straightforward:

*R/S*is found for a range of values of

*T*(i.e., the data series is segmented in windows of progressively longer length

*T*), and

*R/S*is plotted as a function of

*T*on a log-log plot. If the resulting graph is a straight line, the slope (here determined by linear regression) is the exponent

*H*. Brownian noise (a random walk) has

*H*= 0.5. Values of

*H*in the interval (0.5,1] indicate persistence, in the sense that large values are in general followed by large values, and vice versa, on different time scales. Values of

*H*< 0.5 indicate anti-persistence, such that large values are likely to be followed by small values, and vice versa, on the average over different time scales. These different values of

*H*indicate that the time series exhibits long-term correlations.

Another way to assess this correlation structure is with spectral analysis. The power spectrum *S*_{xx}(*f*) of a time series *x*(*t*) can be obtained from the Fourier transform of its autocorrelation function *R*_{xx}(τ) (Papoulis 1984). [*f* is frequency in Hz, τ is the time lag of the autocorrelation function, which quantifies the linear correlation of a time series *x*(*t*) with a time-shifted version of itself: *x*(*t* – τ).] Many “conventional” time series have autocorrelations that decay with an exponential envelope: *R*_{xx}(τ) ∼ e^{–t/τ}. A system with long-term correlations has an autocorrelation that decays much more slowly, and for a wide class of systems, this is as a power law function of the autocorrelation time lag τ: *R*_{xx}(τ) ∼ τ^{–β} (Rangarajan and Ding 2000). The power spectrum of the latter type of time series also has a power-law form: *S*_{xx}(*f*) ∼ *f*^{–α}. [Note that the relationship between power spectrum and autocorrelation can be used to demonstrate the connection between long-term correlation in the time domain and power-law scaling in the frequency domain, while the actual power spectrum can be obtained directly from the time series by finding the magnitude of its Fourier spectrum. The latter is the approach we use, with exponent α determined from linear regression on a log-log plot of *S*_{xx}(*f*) vs. *f*.]

Thus power-law scaling is indicative of a process with long memory (in the statistical sense) (Beran 1994) and is a way to characterize the temporal structure of a data set or process (Ding et al. 2002). It is possible to derive the fact that *H* = (1 + α)/2, which expresses the relation between time-series scaling exponent *H* and frequency scaling exponent α. To avoid artifactual “false-positive” indications of such scaling behavior that can arise from the use of either *H* or α alone, both values are computed and compared for agreement with the expression above (Rangarajan and Ding 2000).

## RESULTS

Figure 1 presents data from control subject I (randomized frequency presentation) at pacing frequencies of 0.2, 0.5, and 1.0 Hz (reactive, transition, and predictive ranges). Saccades lag target motion at 0.2 Hz and are frequently in time with target motion at 1.0 Hz; tracking at 0.5 Hz is a mixture of these two behaviors. Latency data are shown as histograms at each major pacing frequency, for one subject, in the six small graphs at the bottom of Fig. 2. (In this example, stimulus frequency was increasing.) The individual latency values are plotted as a function of frequency in the top graph along with mean latency at each frequency. Mean latency clearly varies in a systematic manner with stimulus frequency. At the two lowest frequencies (0.2 and 0.3 Hz), mean latency is almost constant at 180 ms, indicating that the subject is reacting to each target jump with a “normal” saccade latency. Mean latency is almost constant at –50 ms at the two highest frequencies (0.9 and 1.0 Hz), indicating that the subject is making saccades in a predictive manner, in anticipation of each target jump. Mean latency is near zero (but with high variability) at 0.7 Hz. At 0.5 Hz, the latency is close to that at the lowest frequencies; there is a clear lag rather than prediction at this frequency. (See below for explanation of the lines in the upper graph.) [One may note that the latency distributions at 0.2 and 0.3 Hz are much sharper with increasing frequency (Fig. 2) than with decreasing frequency (Fig. 3). The latter broader distribution may reflect some “residue” of the predictive tracking, when frequency has decreased from the predictive to the reactive range; after the subject has experienced predictive tracking, there may be a tendency occasionally to try to predict the target motion even when the system is (nominally) back in the reactive range. This is similar to the notion that prior experience affects the expectation of a sensorimotor context (Vetter and Wolpert 2000).]

Rather than a change in latency that is *proportional* to stimulus frequency, there instead is an *abrupt* and *qualitative* change in latency over this range of stimulus frequencies. At low frequencies there is a delay, and at high frequencies there is anticipation. In Fig. 3, are data from the same subject, with the pacing frequency decreasing. The latency data exhibit the same general properties as with the frequency increasing, but the transition frequency is now 0.5 rather than 0.7 Hz. An increase in variability occurs near the transition frequencies, 0.5 and 0.7 Hz. The histograms at these frequencies are indeed broad, but they also appear in some cases to be bimodal. This suggests that the subject is actually switching between the two tracking modes, reactive and predictive.

One might question if the latency data truly exhibit an abrupt transition between reactive and predictive tracking or if latency might better be described in terms of a smooth and linear function of frequency. To address this question, we conducted a statistical analysis to decide between these two alternatives, which are represented by the broken lines (abrupt transition) and the solid lines (smooth transition) in Figs. 2 and 3 (fitting of these functions was performed separately for each subject and for each direction of frequency change). The straight-line fit was determined by linear regression of the latency and frequency data. The abrupt-transition fit consists of two straight lines with zero slope— one across the mean latency of all reactive saccades and one across the mean latency of all predictive saccades. (The exact nature of the transition is not critical; the model we chose is the simplest one that demonstrates that an abrupt transition is a better fit than is a smooth transition.) To delineate between reactive and predictive saccades, the distributions of saccades in the pure-reactive and pure-predictive ranges were found (typically 0.2–0.3 Hz and 0.9–1.0 Hz, respectively, as determined by inspection of histograms and verified by statistical test as noted below), and the latency that equally divided these distributions was used as a threshold. Latencies in the transition range were divided into reactive and predictive based on this threshold. Reactive saccades were then defined as those in the pure-reactive range, plus those in the transition range with latencies greater than the threshold value. Similarly, predictive saccades were defined as those in the pure-predictive range, plus those in the transition range with latencies less than the threshold value. This allowed for overlap of the two lines in the abrupt-transition fit (not shown in the graphs), which corresponds to the hypothesis that both modes of tracking are present in the transition-frequency range. (The different ranges were determined by a set of *t*-tests that confirmed that mean latency was the same at each frequency in the pure-reactive range and in the pure-predictive range. For the case represented in Fig. 2, the reactive range was identified as 0.2 and 0.3 Hz and the predictive range as 0.9 and 1.0 Hz. Notwithstanding these statistical tests, the distinction between the two modes of tracking was always obvious by inspection.) The mean-squared error was found for each of the two fitted functions (linear and abrupt), and these errors were compared with an *F*-test (ratio of the mean squared error of the linear fit divided by the mean squared error of the transition fit). The abrupt-transition fit was significantly better than the straight-line fit in all cases (*P* values for subjects A–D, increasing frequency: 0.009, 4 × 10^{–9}, 0.00002, 0.01; decreasing frequency: 0.004, 0.009, 0.005, 0.0006).

Further *t*-tests verified the phase-transition nature of the tracking. In most cases, mean latency in the reactive range as frequency increased was identical to mean latency in the reactive range as frequency decreased (*P* > 0.5 except *P* = 0.001 for subject A). This was less consistent for latency in the predictive range (*P* = 0.05, 5 × 10^{–9}, 0.4, 2 × 10^{–4}). (The cases here in which latency does not return to its earlier value, after the subject has traversed the range of pacing frequencies, may be related to the issue discussed above of prior experience and expectation of the stimulus behavior.) However, mean latencies in the reactive and the predictive ranges were significantly different from each other, in all subjects (*P* ≤ 1.2 × 10^{–12}). All subjects exhibited high variability in the mid-frequency range (0.5 and 0.7 Hz); this variability was significantly larger than that in the reactive range both for increasing and decreasing frequency (*P* < 4 × 10^{–4}) and also generally larger than that in the predictive range (*P* < 0.005 except for subject D increasing frequency and B decreasing frequency). Latency in this range was highly dependent on the trend of the pacing: mean latency as frequency increased was significantly different from mean latency as frequency decreased (*P* < 5 × 10^{–5}). In subjects A and C (the naïve subjects), mean latency at the transition frequencies was identical to that in the immediately preceding frequency range (i.e., mean latency in the transition range was the same as in the predictive range as frequency decreased and the same as in the reactive range as frequency increased; *P* = 0.8, 0.02). This is an indication of hysteresis: tracking tends to stay in its existing mode through the transition range.

A similar relationship between pacing frequency and behavior (except for hysteresis) was found for subject I in the randomized-pacing control experiment. Reactive behavior (mean latency 140 ms) was present at low frequencies (0.2 and 0.3 Hz) and predictive behavior (mean latency –90 ms) at high frequencies (0.9 and 1.0 Hz). Although the phase transition was less dramatic in this case, the abrupt-transition fit was significantly better than the straight-line fit (*P* < 0.01). Latencies in the reactive and predictive ranges (defined as above) were significantly different from each other and were significantly different from those in the transition range (*P* < 0.01).

In Fig. 4, we show analysis of latency data from one subject in the second task, where 1,000 consecutive saccades were obtained at 0.3 Hz (reactive) and 1,000 saccades at 0.9 Hz (predictive). Frequency spectra of the two latency time series are overlaid. Reactive timing resembles white noise (flat spectrum), implying that the trials (saccades) are largely independent. Predictive timing exhibits an *f*^{–α} spectrum (α indicated on each plot), implying that these trials exhibit long-term correlations across saccades.

Scaling results for all subjects are shown in Table 1. Hurst scaling exponents *H* are also presented for each subject and tracking condition. The computed values of *H* correspond closely to those expected from the relation *H* = (1 + α)/2 for data in the predictive range (correlation coefficient *r* = 0.84) but not in the reactive range (*r* = 0.48). This point is crucial as it confirms that the scaling results for prediction are not artifactual (Rangarajan and Ding 2000). [As stated in Rangarajan and Ding (2000), where the required relationship between α and *H* is described: “consistency here is defined in a rather loose fashion.” In that report, values of *H* and (1 + α)/2 within approximately 20% are considered to be consistent. Our results are similarly consistent.] Scaling exponents α are larger for predictive tracking than for reactive tracking (mean values 0.13 and 0.41, difference between the groups is significant, paired *t*-test, *P* = 0.02). [To verify that fatigue at the lower frequency (1,000 saccades at 0.3 Hz take almost 28 min) was not the cause of the observed differences in power-law scaling, we also analyzed only the first 200 saccades, at 0.3 and 0.9 Hz, for each subject. The overall trend was not altered: spectral slopes were highly correlated between the 2 analyses (*r* = 0.93), and slopes in the reactive and predictive ranges were significantly different (*P* = 0.005) in each case.]

## DISCUSSION

At low pacing frequencies, subjects make saccades with latencies that are consistent with reactive behavior: each target jump produces a saccade with a latency typically associated with normal visually evoked saccades (Becker 1989). At high frequencies, latency is substantially smaller, consistent with predictive behavior. There is an abrupt change in this timing behavior as frequency changes monotonically across the range of 0.2–1.0 Hz, rather than a gradual change as might be expected. This is independent of whether the pacing frequency is increasing or decreasing. However, the frequency at which the transition occurs does depend on whether frequency is increasing or decreasing (compare Figs. 2 and 3); this is a form of hysteresis and suggests that the reactive and predictive tracking behaviors are stable states: tracking in either the reactive or the predictive regime tends to remain in its existing state as the transition frequency range is approached. This hysteresis is present in two of our subjects who were naïve to eye movement experiments. In addition, histograms of saccade latency near the transition frequency exhibit bimodality, which we interpret to mean that tracking is spontaneously switching between the two modes. (An analogy might be made to locomotion on a treadmill, where there is typically an intermediate gait speed where one finds it difficult to decide whether to walk or to make the transition to trotting.) Such bistability is typical of nonlinear systems that exhibit phase transitions, and in fact is implied in the definition of “phase transition.”

Studies have suggested that predictive and reactive saccades are two distinct modes of ocular tracking (Findlay 1981; Horrocks and Stark 1962), are generated by different neural pathways (Gaymard et al. 1998), and can be selected by stimulus timing (Ross and Ross 1987; Stark et al. 1962; Zambarbieri et al. 1987). However, no study has proven stability of the two modes, nor addressed how the transition between them occurs. The presence of both predictive and reactive saccades in the transition range in our study could be the result of alternating between the two neural pathways and might explain the timing behavior and performance variability in studies that used stimuli in this frequency range (Ross and Ross 1987; Stark et al. 1962; Zambarbieri et al. 1987). [Similar results to our were found by Zambarbieri et al. (1987), but the presentation of only mean latencies made a smooth fit appear reasonable in that study.]

Phase-transition behavior as identified here is not ubiquitous in the oculomotor system. As an example (Watanabe et al. 2000), if an adaptive gain increase of one size saccade is requested and at the same time a gain decrease of another saccade size, then intermediate-size saccades exhibit intermediate changes—they are not “captured” by either the high-gain or the low-gain adapted state. This means that such a “dual-state” adaptation is not readily described as a phase transition; there instead is a gradual change in behavior with saccade size. The lack of a phase transition in this behavior makes it notable that there is phase transition behavior in saccade timing and makes the study of timing behavior interesting to determine how it is similar to and different from other aspects of motor behavior.

Anticipatory drifts have been found to precede saccades in some cases, especially when target motions are known (Kowler and Steinman 1979). We examined the data from our control experiment for such drifts; they were present in approximately 5% of trials in the reactive and transition ranges (0.2–0.7 Hz). These slow eye movements began 200–250 ms before the saccade and had velocities ≤2°/s. The timing and velocity of these drifts are within the range of those reported by Kowler and Steinman with periodically paced targets. However, in that study, the target alternated between locations on or near the fovea (3.3° separation), while targets in our study were separated by 30°. Therefore although these drifts may signify some level of expectation of a target jump, saccades still occur after the target jump in the reactive and transition regions, confirming that subjects are truly reacting at these pacing frequencies.

We also found significant differences in power-law scaling between the reactive and the predictive regimens, indicating a difference in the correlation structure of the latency data between the two tracking conditions. Specifically, the results suggest that there is long-term correlation between latencies during predictive but not reactive tracking: predictive saccades separated by several trials are related to each other in terms of their timing performance. We may speculate on a physiological interpretation of this finding. At low frequencies, when the target moves, the subject reacts and makes a saccade; latency is positive (reactive). Power-law scaling suggests that this behavior is largely uncorrelated from trial to trial in terms of the reaction to each target jump, which in turn may reflect variations in alertness and vigilance which would naturally be associated with volitional control. At high frequencies, predictive behavior dominates. There is insufficient time to visually process and react to each target step, and so triggering of the current saccade must be based on perceived performance of past saccades. If earlier saccades appear to have been late (arriving after the target has already appeared), future ones may be reduced in latency. If earlier saccades appear to have been early (arriving before the target appears), future ones may be increased in latency. Thus history has an influence on subsequent trials, leading to long-range correlation.

The preceding argument might imply that the latency data exhibit anti-persistent behavior: long and short latencies tending to alternate (on average over different time scales). The actual values for *H* found here, on the other hand, indicate persistence (*H* > 0.5): a prevailing trend of increasing or decreasing latencies is likely to continue (on average). This might be interpreted as meaning that in fact saccades on average arrive at the target precisely in synchrony with its illumination, which serves to reinforce this timing behavior for future saccades. In fact the latency of saccades in the predictive range coincides closely with typical saccade duration, meaning that the eyes do on average arrive at or near each target “just in time” (i.e., the primary saccade is completed, although corrective saccades may be necessary to bring the eyes exactly to the target, as in Fig. 1, *right*).

Our results on both phase transitions and scaling have implications for mathematical models of saccade generation. For example, those existing models that focus on the generation of a saccade in response to a single target presentation or with unexpected presentations (Carpenter and Williams 1995; Fischer et al. 1995) will not reproduce long-term correlations in latency as found here for predictive saccades. Models that address the generation of sequences of saccades (Schmid and Ron 1986; van Loon et al. 2002) may produce correlated trials (although the authors of those studies have not identified such behavior in their models, an area we are now investigating), but do not address the transition between reactive and predictive control. Recent imaging results have demonstrated changes in neural activity associated with transitions between stable behaviors (Meyer-Lindenberg et al. 2002), and evidence for long-term correlations under certain circumstances in other motor behaviors (Roberts et al. 2000). Thus cortical regions responsible for the phase transition and the long-term correlations presented here may be revealed by similar techniques. The outcomes of these investigations may challenge current models of oculomotor control and have implications for understanding how the CNS organizes predictive behavior.

## DISCLOSURES

This study was supported by National Institute on Deafness and Other Communication Disorders Grant DC-05506 and a Whitaker Foundation training grant.

## Acknowledgments

We appreciate the helpful comments of two anonymous reviewers.

## Footnotes

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- Copyright © 2003 by the American Physiological Society