Human optokinetic nystagmus and spatial frequency.

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.

The distribution of quick phase interval durations in human optokinetic nystagmus.

There is an interesting dichotomy between models that predict the quick phase interval durations (QPIDs) of human optokinetic nystagmus (OKN). Accumulator models describe a stochastic signal in a neural network that triggers a response once the signal reaches a fixed threshold value. However, it is also possible that quick phases are triggered after eye position reaches a variable amplitude threshold. In this study, we fitted a range of probability density functions previously predicted by stochastic models of OKN (including those of the reciprocal truncated Normal, inverse Gaussian, gamma, lognormal and the mixture of two reciprocal truncated Normal distributions) to individual QPID histograms. We compared the goodness of fit between these models, and a model where the distribution of QPIDs is determined by the ratio of two correlated and truncated Normal random variables. The ratio distribution gave the best fit to the data, and we propose this is due to the approximately linear trajectory of slow phases (SPs) and that QPIDs are given by the ratio of a variable SP amplitude threshold and variable SP velocity.

Human optokinetic nystagmus: a stochastic analysis.

Optokinetic nystagmus (OKN) is a fundamental gaze-stabilizing response found in almost all vertebrates, in which eye movements attempt to compensate for the optic flow caused by self-motion. It is an alternating sequence of slow compensatory eye movements made in the direction of stimulus motion and fast eye movements made predominantly in the opposite direction. The timing and amplitude of these slow phases (SPs) and quick phases (QPs) are remarkably variable, and the cause of this variability is poorly understood. In this study principal components analysis was performed on OKN data to illustrate that the variability in correlation matrices across individuals and recording sessions reflected changes in the noise in the system while the linear relationships between variables remained predominantly the same. Three components were found that could explain the variance in OKN data, and only variables from within a single cycle contributed highly to any given component. A linear stochastic model of OKN was developed based on these results that describes OKN as a triple first order Markov process, with three sources of noise affecting SP velocity, the QP trigger, and QP amplitude. This model was used to predict the degree of signal dependent noise in the system, the duration of the transient state of SP velocity, and an apparent undershoot bias to the QP target location.

Manual choice reaction times in the rate-domain.

Over the last 150 years, human manual reaction times (RTs) have been recorded countless times. Yet, our understanding of them remains remarkably poor. RTs are highly variable with positively skewed frequency distributions, often modeled as an inverse Gaussian distribution reflecting a stochastic rise to threshold (diffusion process). However, latency distributions of saccades are very close to the reciprocal Normal, suggesting that "rate" (reciprocal RT) may be the more fundamental variable. We explored whether this phenomenon extends to choice manual RTs. We recorded two-alternative choice RTs from 24 subjects, each with 4 blocks of 200 trials with two task difficulties (easy vs. difficult discrimination) and two instruction sets (urgent vs. accurate). We found that rate distributions were, indeed, very close to Normal, shifting to lower rates with increasing difficulty and accuracy, and for some blocks they appeared to become left-truncated, but still close to Normal. Using autoregressive techniques, we found temporal sequential dependencies for lags of at least 3. We identified a transient and steady-state component in each block. Because rates were Normal, we were able to estimate autoregressive weights using the Box-Jenkins technique, and convert to a moving average model using z-transforms to show explicit dependence on stimulus input. We also found a spatial sequential dependence for the previous 3 lags depending on whether the laterality of previous trials was repeated or alternated. This was partially dissociated from temporal dependency as it only occurred in the easy tasks. We conclude that 2-alternative choice manual RT distributions are close to reciprocal Normal and not the inverse Gaussian. This is not consistent with stochastic rise to threshold models, and we propose a simple optimality model in which reward is maximized to yield to an optimal rate, and hence an optimal time to respond. We discuss how it might be implemented.

On the convergence of time interval moments: caveat sciscitator.

The frequency distributions of biological time-intervals (TIs) have been measured in innumerable behavioural and neurophysiological studies including, for example, reaction time experiments and studies of neuronal spike timing. Regardless of context, TI distributions tend to be significantly positively skewed, and many studies have attempted to characterise these distributions by their mean or higher moments (variance, skewness, and kurtosis). It is, however, not widely appreciated in the neural/behavioural literature that highly skewed distributions may not have moments: they may not converge to a finite value. For example, the reciprocal Normal distribution, frequently used as a model of saccade latency and manual reaction time, does not have a finite mean or higher moments. We explore this non-trivial phenomenon. We introduce 'cumulative' moments and show that moment convergence can be easily related to the distribution of the reciprocal of time-intervals (rate). The behaviour of the rate distribution near zero determines which moments in the time domain converge. Non-convergence may be very slow leading to the appearance of convergence, but after infinite time they become infinite (pseudo-convergence). Experiments take place in finite time and cannot reconcile pseudo-convergence. Nevertheless, cumulative sample moments can provide insight into the convergence of the parent moments, but this depends on sample size. We illustrate convergence issues with three empirical examples: manual reaction times, neural inter-spike intervals, and optokinetic inter-saccadic intervals. We conclude that estimating moments of skewed TI distributions is at best questionable and propose that rate (reciprocal TI) usually (but not always) provides more information.