When presented with a moving image, the eyes respond with a movement in the same direction as the image, interrupted by quick resetting phases [1-3]. These reflexive, rhythmic eye movements, which are called optokinetic nystagmus (OKN), interact with the vestibulo-ocular reflex and the smooth pursuit function to hold objects steady on the retina.

The optokinetic nystagmus amplitude (OKNA) sequence is the subsequent slow and fast phases of OKN amplitudes.

The scaling coefficient H, the Hurst exponent, is a measure of memory of a time series with the three properties [4-8].

A time-integrated white noise time series (i.e., a random walk series) has a Hurst exponent of H = 0.5.

A Hurst exponent that is higher than 0.5 (H > 0.5) corresponds to a process which is trending (persistent), also known as positive autocorrelation.

A Hurst exponent that is lower than 0.5 (H < 0.5) corresponds to a mean reverting process (anti-persistent) (i.e., negative autocorrelation).

Surrogate data testing is a statistical technique that can be used to test if a sequence of data differs significantly from a random sequence; for example, a random permutation of the same data/numbers. New data sequences are generated from the original sequence by shuffling, randomly changing the order of the sequence (the same numbers but where the serial dependency is broken [9 , 10]). A parameter is calculated from the original data sequence and tested statistically against the distribution of the parameters from the shuffled sequences. The null hypothesis H₀: Data_original = mean Data_surrogates, that the original data sequence belongs to the same distribution as the surrogates can then be rejected at, for example, 1% probability level.

OKN was recorded from a healthy subject for four conditions: stripes moving 30˚/s left and right and 60˚/s left and right.

Horizontal eye movements were recorded with two electrodes (Ag-AgCl skin electrodes), which were placed lateral to each eye, along with a reference electrode at the center of the forehead. The signal was amplified (10 s time constant and an upper cut-off frequency of 30 Hz) and digitized into a computer using 12 bit A/D resolution and 100 Hz sampling frequency (sampling time τs = 0.01).

OKN was obtained by stimulating the visual field with 3.75˚ width vertical light stripes separated by 11.25˚ width dark stripes. A slit projector presented the stripes on the inside of a hemispherical screen (100 cm in diameter). The subjects were sitting in front of a screen in a darkened room with the head restrained. The subjects were instructed to not follow the stripes with the eyes but to focus their vision on the screen, allowing the optokinetic reflex to control the eye movements. Recordings were performed with the movement of the stripes at a velocity of 30˚/s and at 60˚/s, which are below and above the normal threshold for smooth pursuit function [11 , 12]. Each recording lasted for 1 min. Figure 1 shows a 1 s recording of OKN.

Analyzing methods for distinguishing random behavior from long-term correlation/memory in time series have been described (Mandelbrot and Van Ness [13]). It has been shown that the mean square

displacement exhibit scaling laws proportional to Δ t 2 H [14].

〈 Δ x 2 〉 = 〈 ( x i − x i − Δ t ) 2 〉 ~ Δ t 2 H (1)

〈 Δ x 2 〉 is the mean square displacement, Δ t is the time interval and H is the Hurst exponent. If data are independent, i.e. no memory, the displacement will increase with the square root of time and H = 1/2.

The algorithm was applied to the integrated—the cumulative sum—of the nystagmus amplitude sequence series, A_i, adjusted for the mean.

{ x } i = 1 n = ∑ i = 1 n ( A i – mean A i ) (2)

where

mean A i = ∑ i = 1 n A i n (3)

The time series, x_i, represents the integrated OKNA sequence.

First, we calculated the mean square displacement 〈 Δ x 2 〉 in measure of number of nystagmus phases.

〈 Δ x i 2 〉 k = { ∑ i = 1 − k + n ( x i + k − x i ) 2 − k + n } k = 1 β = 54 (4)

Then, from the scaling properties, we found the slope, s i .

s i = 1 2 log ( 〈 Δ x i 2 〉 k ) log ( k ) (5)

The scaling, the Hurst exponents for 50 time scales, was then calculated using the method of least square to fit straight lines for 1 to n (n = 4 to 54) nystagmus phases.

The Hurst exponents for increasing time horizons for the integrated original OKNA sequences were statistically tested against 100 shuffled sequences (i.e., the serial dependency of the original data sequences is broken by changing the order of the sequence) [9 , 10].

The statistical Z score is

Z = H orig − mean H surr SD H surr (6)

where H_orig is the Hurst exponent of the original integrated OKNA sequence, mean H_surr is the mean value for the 100 surrogates

mean H surr = ∑ i = 1 n H surr i n (7)

and SD H_surr is the standard deviation of the distributed Hurst exponents from the 100 surrogate data sets.

SD H surr = ( H − mean H surr ) 2 n (8)