Analysing nystagmus waveforms: a computational framework

Abstract

We present a new computational approach to analyse nystagmus waveforms. Our framework is designed to fully characterise the state of the nystagmus, aid clinical diagnosis and to quantify the dynamical changes in the oscillations over time. Both linear and nonlinear analyses of time series were used to determine the regularity and complexity of a specific homogenous phenotype of nystagmus. Two-dimensional binocular eye movement recordings were carried out on 5 adult subjects who exhibited a unilateral, uniplanar, vertical nystagmus secondary to a monocular late-onset severe visual loss in the oscillating eye (the Heimann-Bielschowsky Phenomenon). The non-affected eye held a central gaze in both horizontal and vertical planes (± 10 min. of arc). All affected eyes exhibited vertical oscillations, with mean amplitudes and frequencies ranging from 2.0°-4.0° to 0.25-1.5 Hz, respectively. Unstable periodic orbit analysis revealed only 1 subject exhibited a periodic oscillation. The remaining subjects were found to display quasiperiodic (n = 1) and nonperiodic (n = 3) oscillations. Phase space reconstruction allowed attractor identification and the computation of a time series complexity measure-the permutation entropy. The entropy measure was found to be able to distinguish between a periodic oscillation associated with a limit cycle attractor, a quasiperiodic oscillation associated with a torus attractor and nonperiodic oscillations associated with higher-dimensional attractors. Importantly, the permutation entropy was able to rank the oscillations, thereby providing an objective index of nystagmus complexity (range 0.15-0.21) that could not be obtained via unstable periodic orbit analysis or attractor identification alone. These results suggest that our framework provides a comprehensive methodology for characterising nystagmus, aiding differential diagnosis and also permitting investigation of the waveforms over time, thereby facilitating the quantification of future therapeutic managements. In addition, permutation entropy could provide an additional tool for future oculomotor modelling.

Figure 1

A computational framework for analysing a nystagmus time series that combines linear and nonlinear investigative methods to supplement the standard clinical assessment. The linear and nonlinear systems analyses are based on spectral decomposition and attractor reconstruction respectively.

Figure 2

A comparison between stable (first row (a)) and unstable fixation (second row (b) and third row (c)). Columns 2, 3 and 4 show the corresponding time series, attractor type and frequency spectra, respectively. Note that all phase trajectories converge to the attractor over time: (a) A system in a state of stable equilibrium (e.g. steady fixation) is represented by a fixed point. Unstable systems (giving rise to unsteady fixation) are illustrated by ((b)-top) a one-dimensional limit cycle (periodic oscillation), ((b)-bottom) a two-dimensional torus (quasiperiodic oscillation) and (c) a higher-dimensional chaotic attractor (nonperiodic oscillation).

Figure 3

Time series used for the permutation entropy calculations (see text for further details).

Figure 4

Eye position recordings during binocular viewing of a stationary target located in primary gaze. (a) Unilateral vertical nystagmus in the right eye of subject 1. (b) Unilateral vertical nystagmus in the left eye of subject 2. R_H = right horizontal, R_V = right vertical, L_H = left horizontal and L_V = left vertical. Arrows indicate saccadic intrusions (see text for further details).

Figure 5

Periodicity analysis. (a) 5 s samples of the time series from each subject. See Fig. 3 for the full time courses. (b) Frequency spectra. (c) Unstable periodic orbits (UPOs). UPOs were determined by comparing the relative frequency of the transformed interval data (thin lines) with surrogate data (bold lines). See text for further details.

Figure 6

Phase space reconstructions. Left column: Time series segments for each subject showing the unstable periodic orbits (UPO) extracted in each case (red lines). Right column: The attractors reconstructed from the UPOs using delay embedding (see text for further details).

Figure 7

Distributions of permutation entropies obtained by sliding a window across the time series shown in Fig. 3 (see text for further details). In each box plot, the central line denotes the median and the bottom and top edges indicate the 25th and 75th percentiles, respectively. The whiskers show the most extreme data points that are not considered outliers, with outliers plotted as crosses.