purpose. To investigate the proper usage of wavelet analysis in infantile nystagmus syndrome (INS) and determine its limitations and abilities.

methods. Data were analyzed from accurate eye-movement recordings of INS patients. Wavelet analysis was performed to examine the foveation characteristics, morphologic characteristics and time variation in different INS waveforms. Also compared were the wavelet analysis and the expanded nystagmus acuity function (NAFX) analysis on sections of pre- and post-tenotomy data.

results. Wavelet spectra showed some sensitivity to different features of INS waveforms and reflected their variations across time. However, wavelet analysis was not effective in detecting foveation periods, especially in a complicated INS waveform. NAFX, on the other hand, was a much more direct way of evaluating waveform changes after nystagmus treatments.

conclusions. Wavelet analysis is a tool that performs, with difficulty, some things that can be done faster and better by directly operating on the nystagmus waveform itself. It appears, however, to be insensitive to the subtle but visually important improvements brought about by INS therapies. Wavelet analysis may have a role in developing automated waveform classifiers where its time-dependent characterization of the waveform can be used. The limitations of wavelet analysis outweighed its abilities in INS waveform-characteristic examination.

^{ 1 }has prompted researchers to use advanced mathematical tools such as spectral analysis and wavelet analysis for the purpose of understanding the etiology of this disorder. INS is not a stationary process. The waveforms of INS may become worse with gaze position or during stress or visual demand.

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^{ 5 }Conversely, loss of attention to an object being viewed may lead to the nystagmus’ diminishing. For this reason, studies of nystagmus generally analyze data segments lasting only 5 to 10 seconds.

*foveation periods*facilitate better acuity.

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^{ 11 }These key features of the INS waveform have durations measured in tens of milliseconds. Identifying features at this time scale is beyond the resolution of even the short-time Fourier transform (STFT, see the Discussion section). Foveation characteristics have historically been examined by differentiating the INS position signal and identifying those periods in the waveform that meet specific position and velocity criteria.

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^{ 19 }These criteria have been incorporated into the expanded nystagmus acuity function (NAFX),

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^{ 21 }which may be used to predict the best potential visual acuity possible for a given waveform in the absence of a visuosensory abnormality.

^{ 22 }to detect features in the electrocardiogram (ECG),

^{ 23 }and to identify the time of muscle contraction in surface electromyograms (EMGs),

^{ 24 }among other physiological applications. It has also been used to identify seizure activity and other features of the electroencephalogram (EEG).

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^{ 28 }) is small compared with the nonfunctional parts outside of the foveation window. Although wavelet analysis is well-suited to identifying time-varying changes in a waveform, it is not at all clear that, when applied to an INS eye movement recording, it would offer any advantage over the more direct application of velocity and position criteria for foveation, which have been used for many years. INS has been examined using this approach only once before, by Miura et al.

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^{ 30 }However, they averaged the wavelet coefficients obtained over the entire several-minutes’ duration of the recordings. Although wavelet analysis may be useful in identifying some time-associated changes in nystagmus, it is not obvious how such a “global” parameter calculation can be used to detect the foveation parts of the waveforms.

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^{ 33 }first described in 1979.

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^{ 36 }This surgery for nystagmus consists of detaching the muscle at the insertion end of the tendon and reattaching it at the same place with absorbable sutures. Tenotomy effectively decreases the gain of the ocular motor plant to small, nonsaccadic signals

^{ 37 }and elevates foveation quality. It has been shown to be efficacious for INS, acquired nystagmus (pendular and jerk, horizontal and vertical) and to reduce the associated oscillopsia.

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^{ 41 }Since the tenotomy procedure improves the foveation periods, not necessarily the intensity of the nystagmus, it is crucial to examine only the foveation periods for the surgical effects, rather than the nystagmus intensity or the spectral average holistically calculated.

^{ 42 }It is computationally efficient and, after empiric testing, was better than alternative wavelets at generating coefficients that, on some scales, resembled familiar functions such as eye velocity. Use of the power-2 mode offered computational advantages over the step-by-step mode that were analogous to those the fast Fourier transform offers over the continuous Fourier transform. The use of a power of 6 was determined empirically, when adding additional levels yielded coefficients containing mostly noise. We have also included several of the computationally more intensive step-by-step examples (see 1 2 3 4 5 Figs. 6 7 8 9 10 ) to show the nature of the more continuous contour plots. However, a given coefficient line for the same data is the same when computed using either approach.

^{ 9 }was used to demonstrate the difference in a pre- and a postsurgical fixation segment from one patient. The NAFX is an objective measure of waveform foveation quality. It predicts potential visual acuity in nystagmus patients, assuming no sensory deficits. It is a direct measurement of the eye-movement effects of nystagmus therapies. It could be applied to any nystagmus waveform with a position and velocity variability lying within the maximum foveation window of ±6 and ±10°/s. All the NAFX analysis was performed in the MatLab environment using OMLAB (Ocular Motility Laboratory) software (OMtools, downloadable from http://www.omlab.org, provided in the public domain by the The Daroff-Dell'Osso Ocular Motility Laboratory, Louis Stokes Cleveland Department of Veterans Affairs Medical Center, Case School of Medicine, Case Western Reserve University, Cleveland, OH). Details of how to perform NAFX analysis properly can be found on http://www.omlab.org/OMLAB_page/Teaching/Using_NAFX.html.

*y*-axis roughly corresponds to the inverse of frequency (i.e., the small numbers [scales] represent the equivalent of high frequencies). Unless otherwise stated, we used power level 6 in power-2 mode. Power 6 was chosen because higher levels did not show much additional information in these particular cases. Absolute-value mode for the contour plot was used because it made the foveation periods in the contour plot more evident. The bottom trace is the wavelet output at a certain selected coefficient (designated by the position of the horizontal cursor crossing the contour plot) without taking the absolute value.

*t*= 1750 seconds is prominent throughout the plot, reflecting the wavelet transform’s ability to localize transient events temporally.

^{ 43 }and the relationship between cardiovascular damage and mild cognitive impairment.

^{ 44 }The applicability of these techniques is based on a set of assumptions on the statistical properties of the signal being examined. One of them is that the time series being analyzed be stationary (i.e., that its statistical properties not change over time).

^{ 45 }In ocular motor studies, this assumption makes Fourier analysis well-suited for the study of clearly periodic signals such as smooth pursuit

^{ 46 }or pendular nystagmus.

^{ 47 }In such studies, the properties of the entire sample of the waveform are analyzed, with the underlying assumption being that these properties are stable over time. Such techniques are by design insensitive to transient changes in the waveform and can provide no information about the properties of the waveform at a particular instant in time. When transient events have been sought in the EEG, wavelet analysis has been used.

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^{ 48 }showed periodicity in some nystagmus waveforms. The ability to identify time-related changes in the signal, however, is limited by the duration of the windows into which it is divided. The frequency resolution of the STFT also drops as the window becomes briefer.

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^{ 50 }it appears that a more direct operation on the eye position time series itself gives the needed information more simply than does wavelet analysis. Wavelet analysis may be of some use in examining the morphologic characteristics of INS waveforms. The low-span (high-frequency) coefficients could be used as fast-phase detectors, although it remains to be seen if this is more efficient than simple differentiation of the initial waveform. An application that takes advantage of the ability of wavelet analysis to represent waveform morphology may be to use a fixed number of coefficients to identify waveforms automatically as they change in a given record, perhaps by training a neural network to recognize exemplars of each waveform. Although such waveforms are easy for experienced observers to identify by visual inspection, such a process may make identification available more widely. It may also be useful in attempting to quantify how much time is spent in various waveforms or how they change under experimental or clinical manipulation. Similarly, it may be possible to use it to quantify how APAN components and fast-phase directions change across time. Wavelet analysis could be combined with electrophysiological measures of arousal or stress, to provide quantitative, moment-by-moment information about how these changes in internal state affect the INS waveform. Averaging the coefficients in wavelet analysis of INS, as in Miura et al.,

^{ 30 }unfortunately obscures the key advantage of wavelet analysis—capturing temporal variations in waveform morphology. This averaging essentially treats the waveform as stationary and renders the wavelet analysis equivalent to Fourier analysis. It thus would have been unable to distinguish any morphologic or foveation characteristics of INS waveforms.

^{ 51 }of spectral analysis and intensity calculation in evaluating several surgical approaches to treating INS. Their justification for the “whole of signal” approach can be found in their statement, “We regarded each signal recording as a realization of a stochastic process, supposed stationary at least to the second moment. In fact, different samples of signal recorded at different times, for the same patient and angular position, resemble each other only in their average properties, owing to the presence of noise and small random changes in the nystagmus waveform pattern.”

^{ 51 }They identified the peak of the power spectrum and used its amplitude and frequency, as well as evaluation of the power plot, to characterize the effects of nystagmus surgery. Similarly, Miura et al.

^{ 30 }operated on the entire recorded datasets, averaging the log ratio of the wavelet spectra in a 4- to 10-minute “fixation” recording. There are several problems with doing this when evaluating the effects of an intervention. First, as noted by Roberti et al.,

^{ 51 }is the unavoidable presence of noise in the recording, both intrinsic to the recording system and arising from blinks, facial motion, and the like. Furthermore, by operating on the entire recording, the effects on the signal of changes in the fixating eye (in strabismic patients), of losses of fixation and of lapses in attention are not separated out. When a task consists of gazing at a target for minutes on end, these are all common, real influences on the nature of the nystagmus waveform. Blinks and inattention, shown in Figures 7 8 and 10 , are also frequently present in INS data. These contaminants of the signal are extremely likely to affect the outcome of any sort of global analysis far more than would an increase in the duration of foveation periods, which constitute in many patients only a small percentage of the total signal. Thus, averaging segments containing these artifacts would include data with no functional (foveation) importance.

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^{ 54 }The NAFX analysis,

^{ 9 }used in several studies,

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^{ 56 }inspects each fixation segment to determine the foveation quality in that segment; several NAFX values are averaged to reflect the subject’s ability to fixate. Any blink or inattention data are excluded from the NAFX analysis. Therefore, NAFX is a much more direct and objective methodology to perform INS analysis, especially one that correlates with visual acuity. The use of wavelets does not appear to be sensitive enough for all waveforms to detect the changes in foveation quality brought about by current therapies.

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