**Optokinetic nystagmus (OKN) is a fundamental oculomotor response to retinal slip generated during natural movement through the environment. The timing and amplitude of the compensatory slow phases (SPs) alternating with saccadic quick phases (QPs) are remarkably variable, producing a characteristic irregular sawtooth waveform. We have previously found three stochastic processes that underlie OKN: the processes that determine QP and SP amplitude and the update dynamics of SP velocity. SP and QP parameters are interrelated and dependent on SP velocity such that changes in stimulus speed can have a seemingly complex effect on the nystagmus waveform. In this study we investigated the effect of stimulus spatial frequency on the stochastic processes of OKN. We found that increasing the spatial frequency of suprathreshold stimuli resulted in a significant increase in SP velocity with a corresponding reduction in retinal slip. However, retinal slip rarely reached values close to 0, indicating that the OKN system does not or cannot always minimize retinal slip. We deduce that OKN gain must be less than unity if extraretinal gain is lower than unity (as empirically observed), and that the difference between retinal and extraretinal gain determines the Markov properties of SP velocity. As retinal gain is reduced with stimuli of lower spatial frequency, the difference between retinal and extraretinal gain increases and the Markov properties of the system can be observed.**

*V*component determines the SP velocity of the

*i*th cycle, which we describe as a linear first-order autoregressive equation with the update dynamics where

*e*is a constant and

*ε*(

_{v}*i*) is a stochastic process with mean

*v̂*and standard deviation

*σ*. This accounts for the remarkable degree of variability observed in SP velocity from one cycle to the next. The

_{v}*S*component determines the amplitude of the

*i*th SP (

*S*) according to where

_{i}*x*is the start position of the SP,

_{i}*ε*(

_{s}*i*) is a second stochastic process with mean

*ŝ*and standard deviation

*σ*, and

_{s}*a*and

*b*are constants. The

*Q*component determines QP amplitude (

*Q*) according to where

_{i}*y*is the start position of the QP (end position of the previous SP),

_{i}*ε*(

_{q}*i*) is a third stochastic process with mean

*q̂*and standard deviation

*σ*, and

_{q}*c*and

*d*are constants. The stochastic processes

*ε*(

_{v}*i*),

*ε*(

_{s}*i*), and

*ε*(

_{q}*i*) are uncorrelated. However, because

*V*loads onto multiple components, the variables

_{i}*V*,

*S*,

*Q*,

*x*, and

*y*are mutually correlated (and autocorrelated) in a complex sequence of cycles (see Figure 1 and Table 1 for a full description of the system).

**Figure 1**

**Figure 1**

**Table 1**

*S*/

*V*↠

*T*) in which the denominator and numerator variables are correlated. We derived the probability density function (pdf) for this model in our previous study and compared it to the pdfs predicted from five other models of the QP trigger to demonstrate its superior fit to the data.

*SD*= 4) years, participated in the study and had no self-reported neurological, ophthalmological, or vestibular impairments. The visual acuity of each participant was measured using a Snellen chart before consent to participate was elicited, and only participants with uncorrected binocular visual acuity of 6/6 or higher on the Snellen scale were included in this study. All protocols were approved by the Plymouth University Faculty of Science Human Research Ethics Committee. Participants gave informed written consent and were made aware of their right to withdraw at any time.

^{2}. Peak velocity was then recorded from the time 1 ms before velocity first decreased, if velocity remained at a lower magnitude for more than 4 ms. The starting and ending point of each QP was determined by respective backward and forward passes of the data from the time of peak velocity to the time when eye velocity returned to a value between 0°/s and the stimulus speed for a period of 2 ms or more. This procedure allowed us to collect QPs that were made in the direction of optic flow as well as QPs made in the opposite direction.

*m*SP–QP cycles, where

*m*ranged from 63 to 342. Six measurements were taken from each OKN cycle:

*x*,

_{i}*S*,

_{i}*V*,

_{i}*T*,

_{i}*y*, and

_{i}*Q*(

_{i}*i*= 1, … ,

*m*), according to the scheme in Figure 1 (where

*x*is the SP start position and

*y*is the QP start position). When calculating the mean, median, and standard deviation of variables from each trial, the first 5 s was removed from the analysis to ignore early OKN behavior.

**Figure 2**

**Figure 2**

*mle*with the exception of the correlation coefficient, which was estimated using the sample Pearson's correlation coefficient between SP amplitude and SP velocity from the respective trial. The chi-square test statistic was calculated using the MATLAB function

*chi2gof*, using the fitted pdf to give the expected frequency and the SP duration histogram to give the observed frequency. Each SP duration histogram was originally covered by 45 bins of equal size, and the expected and observed frequencies were determined from the midpoint of each bin. When the expected frequency from a bin in either tail of a histogram was less than 5, the

*chi2gof*function automatically merged neighboring bins until there was a minimum expected frequency of 5 or more in every bin, to maintain the reliability of the test.

*x*,

_{i}*i*= 1, … , 5, and

*S*,

_{i}*V*,

_{i}*T*,

_{i}*y*, and

_{i}*Q*,

_{i}*i*= 1, … , 4, to generate a 25 × 25 element correlation matrix. Cycles separated by blinks were not included in generating the correlation matrix. During one trial, blinks were so frequent that a correlation matrix including all variables from four consecutive cycles of OKN could not be created, so only 95 of 96 correlation matrices were analyzed using PCA.

*pcacov*to diagonalize the correlation matrices and yield the underlying eigenvectors and eigenvalues. We discarded the eight components with the smallest eigenvalues from each trial to retain 13 principal components. The remaining eigenvectors explained over 99.3% of the variability in the data, indicating that the data were well explained by the (linear and stochastic) components. After discarding redundant components, we performed factor rotation using the varimax strategy to obtain orthogonal rotated components using the MATLAB function

*rotatefactors*. After factor rotation, similar loading patterns were observed across trials but expressed in a different eigenvalue order. The 13 principal components from each trial were sorted into categories according to their loading pattern using numerical heuristics. As eigenvectors can be rotated to face in the opposite direction, it was necessary to flip the sign of all loadings in these mirrored components so that they could be sorted correctly. Each loading pattern was then displayed as a line plot of loading value against OKN variable, where components placed in the same category were averaged across trials and error bars were used to show the variability between trials. This sorting was exhaustive, and we found that all loading patterns clearly fell into only three qualitatively different categories (see Results).

**Figure 3**

**Figure 3**

*F*= 6.8,

*p*= 0.009, and consequently a decrease in retinal slip (Figure 4A), although retinal slip did not fall to zero. Increasing stimulus speed also resulted in a significant increase in mean SP velocity as expected,

*F*= 14.5,

*p*= 0.007. Remarkably, we found that at stimulus speeds of 10°/s and 30°/s, retinal slip reached values greater than 3°/s in over 30% and 87% of SPs, respectively.

**Figure 4**

**Figure 4**

*F*= 4.8,

*p*= 0.025, and mean QP magnitude,

*F*= 6.4,

*p*= 0.01. Essentially, increasing spatial frequency resulted in an increase in both mean SP and QP magnitude, but the effect of increasing spatial frequency was greater at 30°/s than at 10°/s (Figure 4B). We did not find an effect of spatial frequency or stimulus speed on the mean angle of contraversion, which was held at −3.9° when averaged across all trials.

*F*= 12.5,

*p*= 0.009 (Figure 4C).

*p*< 0.004 for 85 trials; Holm, 1979). Histograms of reciprocal SP duration (QP rate) were also usually positively skewed, and 85% were significantly different from Gaussian (Holm–Bonferroni-corrected Lilliefors test,

*p*< 0.003 for 82 trials).

**Table 2**

*autocorr*and

*parcorr*. We plotted the mean autocorrelogram of SP velocity across all trials and found that autocorrelation of SP velocity extended back as far as five or six cycles (Figure 5A). However, the mean partial autocorrelogram of SP velocity showed a distinct cutoff at a lag of 1 (Figure 5B), implying that the correlation between SP velocity in one cycle and the cycle before could explain all the higher order autocorrelation observed in the autocorrelogram, identifying the time series as a first-order autoregressive process.

**Figure 5**

**Figure 5**

**Figure 6**

**Figure 6**

*regress*and Holm–Bonferroni correction for multiple comparisons. We found that regression of

*S*against

_{i}*x*and

_{i}*V*gave a mean

_{i}*R*

^{2}= 0.34 (

*p*< 0.006 for 91 trials),

*Q*against

_{i}*y*and

_{i}*V*gave a mean

_{i}*R*

^{2}= 0.28 (

*p*< 0.005 for 95 trials), and

*V*

_{i}_{+1}against

*V*gave a mean

_{i}*R*

^{2}= 0.27 (

*p*< 0.003 for 78 trials). The similarity of these results to the results from our previous study indicated that our model could extend to account for spatial-frequency effects on the OKN waveform.

*robustfit*, and assessed their dependency on stimulus spatial frequency and stimulus speed using repeated-measures analysis of variance.

*a*,

*b*,

*c*, or

*d*; the mean values of these parameters were −0.25, 0.12, −0.38, and −0.18, respectively. We noted that these were approximately the same values as found in our previous study (−0.25, 0.16, −0.48 and −0.17; Waddington and Harris, 2012)—see Table 3—which indicated that these parameters may represent fundamentally invariant relationships between OKN variables.

**Table 3**

*v̂*, from 4.6 to 6.1 and 7.7,

*F*= 4.2,

*p*= 0.037, but did not have a significant effect on

*e*or

*σ*. Conversely, increasing stimulus speed from 10°/s to 30°/s resulted in a significant increase in

_{v}*e*from 0.38 to 0.60,

*F*= 19.5,

*p*= 0.003, and an increase in

*σ*from 1.6 to 3.8,

_{v}*F*= 95.4,

*p*< 0.001, but did not have a significant effect on

*v̂*. Effectively, increasing the stimulus spatial frequency resulted in the mean SP velocity being shifted by a constant value to be more positive, without having any other effects on the update dynamics of SP velocity.

*q̂*,

*F*= 4.2,

*p*= 0.037, but not

*σ*. When the stimulus speed was 30°/s, increasing spatial frequency resulted in

_{q}*q̂*becoming more negative, from −1.9 to −2.4 and −3.4, but there was no clear trend at 10°/s. This may reflect the interaction effect observed between stimulus speed and spatial frequency on the mean SP and QP amplitude in our initial analysis of the stimulus effects on OKN variables. Spatial frequency did not have a significant effect on

*ŝ*or

*σ*. However, increasing stimulus speed did cause an increase in both

_{s}*σ*,

_{s}*F*= 18.2,

*p*= 0.004, and

*σ*,

_{q}*F*= 26.2,

*p*= 0.001.

*v̂*to be more positive. Additionally, increasing spatial frequency increased the magnitude of QPs and SPs (dependent on stimulus speed) by shifting

*q̂*to be more negative.

*V*component described how SP velocity is updated on every cycle (Equation 1). We found that it is independent of the other processes but dependent on stimulus velocity and spatial frequency (Equation 12). We also found that the process that determines QP amplitude (the

*Q*component) is dependent on stimulus velocity and spatial frequency (Equation 14), and confirmed that both SP and QP amplitudes depend on SP velocity and each other (Equations 1 and 2). The three underlying stochastic processes that drive the system are uncorrelated, but because individual OKN variables load onto multiple components (and multiple variables load onto each component), many variables are dependent on more than one of the underlying sources of variance. This shared variance gives rise to a complex sequence of cycles in which OKN variables are mutually correlated and autocorrelated (Equations 1 and 7–11).

^{2}and 0.16 elements/°

^{2}, at a speed of 4.6°/s). However, they also noted that Schor and Narayan (1981) had found a decrease in SP velocity at high spatial frequencies and high speeds (drifting gratings with a spatial frequency of 0.5–8 c/° and speeds of 3°/s–48°/s).

*V*depends on stimulus velocity

_{i}*V*

_{S}, but

*V*

_{S}is not known directly and can only be estimated from retinal information and extraretinal information (efference copy and other proprioceptive cues). At the start of each SP, retinal information can only arise from previous SPs:

*R*=

_{j}*V*

_{S}−

*V*for

_{j}*j*<

*i*. Thus the current estimate of stimulus velocity

*v̂*

_{S,}

*must depend somehow on actual previous SP velocities*

_{i}*V*

_{j}_{<}

*, and hence must be intrinsically Markov when estimates are stochastic. In principle, dependencies on previous SPs could stretch far back, but our empirical findings indicate a dependency on only the last SP (*

_{i}*j*=

*i*− 1).

*v̂*

_{S,}

*=*

_{i}*R̂*

_{i}_{−1}+

*v̂*

_{i}_{−1}. Psychophysical evidence suggests that the gains of

*R̂*

_{i}_{−1}and

*v̂*

_{i}_{−1}are different (Freeman, 2001; Freeman & Banks, 1998) and possibly less than unity (Pola & Wyatt, 1989), which would lead to an underestimate of stimulus velocity. We denote the retinal slip estimate by the nonlinear equation where

*g*

_{r}is a nonconstant gain that depends on the spatiotemporal contrast sensitivity of the retina (and hence spatial frequency and retinal slip),

*n*

_{r}is a standard normal noise process, and

*σ*

_{r}is the standard deviation of the noise. We denote the SP velocity estimate by the nonlinear equation where

*g*

_{e}is extraretinal gain and may also depend on eye velocity

*V*,

_{i}*n*

_{e}is a standard normal noise process, and

*σ*

_{e}is the standard deviation of the noise. If we assume that current SP velocity attempts to match the internal estimate of the current stimulus speed, then

*v̂*

_{i}_{+1}= (

*R̂*+

_{i}*v̂*); and substituting Equations 12 and 13, we obtain the first-order Markov relationship

_{i}**Figure 7**

**Figure 7**

*g*

_{r}and

*g*

_{e}are not directly observable, but comparing Equation 14 to Equation 1, we have

*e*=

*g*

_{e}−

*g*

_{r},

*v̂*=

*g*

_{r}

*V*

_{S}, and . Freeman, Champion, and Warren (2010) have proposed that the perceived speed of visual stimuli during SPEM is based on Bayesian estimates of retinal slip velocity and extraretinal eye velocity. Their key assumption is that prior expectation of stimulus speed is zero, causing shifts to lower velocities for the posterior estimates. Further, because the combined signal is less certain than the retinal signal alone ( ), the perceptual estimates of pursued stimuli will be lowered by prior expectations more than moving stimuli observed during fixation, which they use to explain a number of perceptual illusions. However, it is not clear whether priors could be updated over time, eventually leading to an unbiased asymptote if OKN stimulus parameters remain constant. We will thus explore some alternative explanations for reduced SP gain.

*e*=

*g*

_{e}−

*g*

_{r}and

*v̂*=

*g*

_{r}

*V*

_{S}. This indicates a relationship between

*v̂*and

*e*that is linear if extraretinal gain remains constant:

*v̂*/

*V*

_{S}=

*g*

_{e}–

*e*. Plotting

*v̂*/

*V*

_{S}against

*e*(Figure 8) revealed an intercept of 0.82 and a slope of −0.92. The nearly (negative) unity slope supports the scheme in Figure 7, although we should exercise some caution, as it is slightly lower than expected.

**Figure 8**

**Figure 8**

*g*

_{r}=

*v̂*/

*V*

_{S}(see Table 3), which showed a significant decrease with stimulus speed and a significant increase with spatial frequency, whereas

*g*

_{e}=

*g*

_{r}+

*e*only showed a slight decrease with speed and a slight increase with spatial frequency. These results mirror the findings that perceived retinal speed is dependent on spatial frequency (Campbell & Maffei, 1981; Diener, Wist, Dichigans, & Brandt, 1976; Ferrera & Wilson, 1991; Freeman & Banks, 1998; Smith & Edgar, 1990) and support the findings of Sumnall et al. (2003) that extraretinal signals do not appear to be significantly affected by spatial frequency.

*g*

_{e}= 1 the mean OKN gain becomes unity regardless of retinal gain. Thus, in principle, it is possible to track the stimulus perfectly at steady state. Indeed, efference copy was introduced implicitly by Young, Forster, and van Houtte (1968) for a discrete-time smooth-pursuit model and explicitly by Robinson, Gordon, and Gordon (1986) for a continuous-time smooth-pursuit model to explain how the smooth-pursuit system could track a moving target with large open-loop gain and long loop delays.

*g*

_{e}has been shown to be considerably less than unity (Pola & Wyatt, 1989), and we surmise that the report by Spering et al. (2005) of low SPEM gain to low contrast stimuli could only occur if

*g*

_{e}< 1. Our deduction that

*g*

_{e}is less than unity for discrete-time OKN is consistent with these results (see Table 3), although we perhaps find a higher extraretinal gain for OKN SPs than some SPEM estimates.

*g*

_{e}=

*g*

_{r}, variance is at a minimum ( ) and the system becomes zero order.

*e*=

*g*

_{e}−

*g*

_{r}affects the steady-state mean and variance of most OKN variables that we have measured, including QP and SP amplitude, duration, and position (see Table 4). It is possible that variance constraints or costs on OKN variables other than SP velocity could lower the optimal

*g*

_{e}to reduce the difference

*e*=

*g*

_{e}−

*g*

_{r}and hence reduce . For example, increasing will lead to more extremely long SP durations (a property of reciprocal and ratio distributions; Harris & Waddington, 2012; also see Equation 27 in Table 4) and more variability in the end position of QPs (Equations 23 and 26). Too much variability in the end position of QPs could be extremely costly to the visual system, as contrast sensitivity is dependent on the retinal location of the visual stimulus and visual acuity decreases rapidly outside the central 2° of foveal vision. It is not difficult to imagine a situation where a compromise needs to be made between minimizing positional variance on QPs and minimizing retinal slip.

**Table 4**

*i*= 0 with velocity

*V*

_{0}; then from Equation A3, the velocity of the

*i*th cycle is which clearly takes time to reach mean velocity

*V̄*. That is, any deviation from the mean takes time to recover depending on

*g*

_{e}−

*g*

_{r}. In natural OKN, stimulus velocity may change rapidly when gaze is shifted from one region of optic flow to another, so it seems plausible that responding quickly to any changes could be important. If

*g*

_{e}−

*g*

_{r}→ 1, response time becomes infinite (a random walk) and the system would be trapped by its history and unable to change. On the other hand, when

*g*

_{e}=

*g*

_{r}response is instantaneous, but

*V̄*, would never reach

*V*

_{S}if

*g*

_{r}< 1. Presumably some compromise is needed, but how speed and accuracy trade off is unknown.

*g*

_{e}to reduce the difference

*g*

_{e}−

*g*

_{r}will also have visual consequences that cannot be directly inferred from Equations 17–28. Recently, Harrison, Freeman, and Sumner (2015) have shown that the horizontal component of saccades made to visual targets flashed during ongoing OKN falls well short of the target. That is, the saccades do not compensate for the excursion of the current SP. This undershoot error was proportional to the distance traveled by the eye during the saccade latency period, which is expected if the error was due to extraretinal underestimation. Therefore, if we assume that QPs are visually guided, lower extraretinal gain will lead to QPs that undershoot (or overshoot, depending on location) their target, as well as lower SP gain. We should also recognize that the SP amplitude threshold is likely to be estimated by an extraretinal signal of eye position. If this signal is also an underestimate, then QPs will tend to be triggered before they reach the target threshold, and any variance in the end position of QPs could not be fully compensated for during the SP. Indeed, this would lead to a partial negative correlation between the amplitude and start position of SPs (and QPs), and to the first-order Markov properties that we have observed.

*g*

_{e}−

*g*

_{r}(or

*g*

_{e}, if

*g*

_{r}is unknown). The overall process is complex, however, and at present we have no specific model of adaptive control of OKN. Indeed, it is even possible that there are three separate adaptive mechanisms each attempting to optimize some visual consequence of OKN, with competing and nonintuitive effects.

*g*

_{e}−

*g*

_{r}would decrease, leading to an overall increase in SP gain and also a decrease in SP velocity variance and a general reduction in cyclic variability. Should

*g*

_{e}=

*g*

_{r}, the Markov properties would disappear and zero-order statistics would set in.

*i*th cycle. This will increase retinal slip, and hence reduce retinal gain

*g*

_{r}(

*i*). This in turn will reduce SP velocity on the next cycle, causing a further increase in retinal slip, and so on in a positive-feedback fashion. Although a steady state may occur, it is also possible that OKN will simply cease if the random fluctuation is large enough. The opposite effect could occur with a fluctuation that increases SP velocity. A moving high-spatial-frequency grating may have minimal contrast for a stationary eye and generate no OKN. However, if the eye happened to move spontaneously in the direction of the stimulus, retinal slip would decrease, increasing contrast and potentially sustaining OKN for a while. This nonlinearity leads to a nonstationary Markov process, but whether it can be detected remains to be explored, although we expect it to be stronger for high-spatial-frequency stimuli because of the sharp dependency on retinal slip.

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*w*,

_{i}*i*≥ 0, where each is continuous. The simplest stochastic process is the zero-order process, where all

*w*are mutually independent. If the probability distribution of

_{i}*w*depends explicitly on the outcome of the previous random variable

_{i}*w*

_{i}_{−1}but not explicitly on the outcome of earlier random variables, the system is said to be a first-order Markov process. To describe

*w*requires the specification of the infinite dimensional probability transition matrix Pr(

_{i}*w*|

_{i}*w*

_{i}_{−1}). A special case is given by the stationary Gauss–Markov first-order autoregressive system where

*w*

_{0}is the initial value and may be a random variable,

*α*is a constant, and

*ε*(

*i*) is a sample from a continuous white-noise process that is normally distributed with a nonzero mean

*μ*and variance

*σ*

^{2}:

*ε*(

*i*) ∼

*N*(

*μ*,

*σ*

^{2}). Samples are mutually independent so that cov(

*ε*(

*i*),

*ε*(

*j*)) = 0 for

*i*≠

*j*. Equations 1 and 7–11 can be written in this form. Expanding Equation A1 gives where

*n*(

*i*) is a standard normal random variable

*n*(

*i*) ∼

*N*(0, 1). From Equation A2, we have

*w*

_{1}=

*αw*

_{0}+

*μ*+

*σn*(0);

*w*

_{2}=

*α*

^{2}

*w*

_{0}+

*αμ*+

*μ*+

*αn*

_{1}+

*n*(2);

*w*

_{3}=

*α*

^{3}

*w*

_{0}+

*α*

^{2}

*μ*+

*αμ*+

*μ*+

*α*

^{2}

*n*(1) +

*αn*(2) +

*n*(3); etc. Summing the power series yields the

*i*th term with mean and variance is the variance associated with the initial value

*w*

_{0}. Provided |

*α*| < 1, this series converges with an initial transient that depends on

*w*

_{0}, followed asymptotically by a steady-state behavior that is independent of

*w*

_{0}. Thus the steady-state mean is given by

*w̄*

_{∞}=

*αw̄*

_{∞}+

*μ*and hence Similarly, the steady-state variance is