Stochastic Physiological Gaze-Evoked Nystagmus With Slow Centripetal Drift During Fixational Eye Movements at Small Gaze Eccentricities

Abstract

Involuntary eye movement during gaze (GZ) fixation, referred to as fixational eye movement (FEM), consists of two types of components: a Brownian motion like component called drifts-tremor (DRT) and a ballistic component called microsaccade (MS) with a mean saccadic amplitude of about 0.3° and a mean inter-MS interval of about 0.5 s. During GZ fixation in healthy people in an eccentric position, typically with an eccentricity more than 30°, eyes exhibit oscillatory movements alternating between centripetal drift and centrifugal saccade with a mean saccadic amplitude of about 1° and a period in the range of 0.5-1.0 s, which has been known as the physiological gaze-evoked nystagmus (GEN). Here, we designed a simple experimental paradigm of GZ fixation on a target shifted horizontally from the front-facing position with fewer eccentricities. We found a clear tendency of centripetal DRT and centrifugal MS as in GEN, but with more stochasticity and with slower drift velocity compared to GEN, even during FEM at GZ positions with small eccentricities. Our results showed that the target shift-dependent balance between DRT and MS achieves the GZ bounded around each of the given targets. In other words, GZ relaxes slowly with the centripetal DRT toward the front-facing position during inter-MS intervals, as if there always exists a quasi-stable equilibrium posture in the front-facing position, and MS actions pull GZ intermittently back to the target position in the opposite direction to DRT.

Keywords: centripetal drift; fixational eye movements; inter-microsaccadic interval; microsaccade; ocular drift; physiological gaze-evoked nystagmus; small gaze eccentricity.

Figure 1

Schematic diagram of the experiment. (A) Five target positions from the left to right, which were referred to as L2, L1, C, R1, and R2, respectively. (B) An experimental protocol. A small target was projected at 0 in the vertical direction and at −15.2°, −7.76°, 0°, 7.76°, and 15.2° in the horizontal direction. Fixational eye movement (FEM) was measured to obtain gaze (GZ) time series for each of these five positions of the fixation target. Single trials lasted 35 s, which were interrupted by several eye blinks. The target position for each trial was chosen pseudo-randomly from the five locations. Each subject performed 56 trials in total.

Figure 2

A representative example of the raw-GZ time series X_rGZ(t) (black curve, mostly behind the blue curve), the GZ time series X_GZ(t) (blue curve), which was decomposed into the drifts-tremor (DRT) time series X_DRT(t) (green curve), and the microsaccade (MS) time series X_MS(t) (red line). The vertical dotted lines indicate the time instances when MS events (MS onset and offset) occurred. The enlarged view compares the details of the raw-GZ time series X_rGZ(t) (small black filled circles) with the transient and overshoot ballistic components of two MS events and the GZ time series X_GZ(t) (blue open circles). The transient and overshoot ballistic components were eliminated from the raw GZ to obtain the GZ time series.

Figure 3

Detection and simplification of MS events. The upper panels show an example of raw-GZ data with an MS event detected by the Engbert and Kliegl (E&K) method in the v_x − v_y plane (left panel) and the corresponding time profile (right panel), where the red curves are MS components and the green curves are DRT components. The lower panels are for the corresponding GZ data, in which MS events were simplified, referred to simply as MS in this paper, by eliminating the transient and overshoot ballistic components from the raw-MS waveforms for further analysis performed in this paper.

Figure 4

Horizontal dependence of the target position on GZ, DRT, and MS time series and the slopes on their trend. Upper panels: All sample paths of GZ (left, blue), DRT (middle, green), and MS (right, red) time series of a representative subject were superimposed for each of the five target positions. Black dotted lines are the least-squares linear regression lines, representing the linear trend of the time series for each of the five target positions. Lower panels: box plots of the mean value of the subject-wise means of slopes for each time series across all subjects. Single and double asterisks indicate that there was a significant difference between groups at the 5 and 1% significance level, respectively.

Figure 5

Mean square displacement (MSD) analysis of the GZ and DRT time series. The blue and green curves represent, respectively, the MSDs of the GZ and DRT time series. The black lines are the MSD of the Brownian motion for comparison. The MSD curves depicted in each panel are the ensemble average of MSD for each sample path of a representative subject.

Figure 6

Relationships between the onset position of MS and its amplitude for each of the five target positions. Color map: two-dimensional histograms of the onset position and amplitude of MS with 0.25° and 0.20° bin widths, respectively, representing the joint frequency of onset positions and amplitudes of MS events. Upper histograms: one-dimensional histograms with a bin width of 0.10° for each of the rightward (red) and leftward (blue) MS events. Right-side histograms: one-dimensional histograms with a bin width of 0.10° for each of the rightward (red) and leftward (blue) MS events. In the right-side histograms, N_p and N_n are the rightward and leftward MS counts, respectively. Similarly, A_p and A_n are the mean amplitude of rightward and leftward MS. For each target condition, the black dotted line represents the mean value of the GZ time series, and the red dotted lines on both sides of the black line represent ±SD away from the mean value of the GZ time series. The dashed white line represents the fixation target position in each target condition.

Figure 7

Semi-log plot of the probability of inter-microsaccadic intervals (IMSI) of all left/right MS across all subjects, with a regression line (black line) in each condition. IMSI₊ plotted by the orange circles are the IMSI of rightward MS, and IMSI₋ plotted by the blue triangles are the IMSI of leftward MS.

Figure 8

Boxplots of the estimated amplitude and occurrence frequency of right/left MS, and the total trend generated by total MS across all subjects. ŵ₊, ŵ₋, ${\widehat{\lambda }}_{+}$, and ${\widehat{\lambda }}_{-}$ represent the subject-wise mean values of the right/left MS amplitude and occurrence frequency, respectively. $ŝ\triangleq {ŵ}_{+}{\widehat{\lambda }}_{+}+{ŵ}_{-}{\widehat{\lambda }}_{-}$ is the estimated total trend of MS reconstructed from ŵ₊, ŵ₋, ${\widehat{\lambda }}_{+}$, and ${\widehat{\lambda }}_{-}$. Single and triple asterisks indicate a significant difference between the groups at the 5 and 0.1% significance level, respectively.

Figure 9

Scatter plot of a DRT trend μ_{DRT, k} and the reconstructed MS trend s_k for the kth sample paths across all samples of all subjects for each target position. The green vertical line and the red horizontal line represent the mean value of μ_{DRT, k} and s_k, respectively. The dotted black line is the complete balanced condition defined by Equation (8).