Using Theodorou's method,
7 we decomposed the nystagmus waveform into two classes: dominantly pendular and dominantly jerk. Nearly all nystagmus waveforms with normalized amplitude can be considered a linear combination of a jerk and a pendular form, as illustrated in
Figure 2. We classified waveforms with a factor a above 0.5 as pendular and below 0.5 as a jerk. The only nystagmus that cannot be classified this way is the bidirectional jerk, which we classified manually.
In equation form, the linear combination of a jerk and pendular nystagmus looks as follows:
\begin{eqnarray}
{\rm{f}} = \left( {1 - {\rm{a}}} \right){\rm{sawtooth}}\left( {2{\rm{\pi t}},1 - {\rm{w}}} \right) + {\rm{a}}(\cos 2{\rm{\pi t}})\quad
\end{eqnarray}
with parameter
t as the time. Parameter
a, between 0 and 1, represents the amplitude of the pendular term, and (1-a) the amplitude of the jerk term. Parameter
w is the width of the rising ramp of the sawtooth function.
17 The width of the rising ramp is a proportion of the total cycle. The default is 1, producing a rising ramp, whereas 0 produces a falling ramp. The width = 0.5 produces a triangle wave. Only the two extreme ranges for w were considered: between 0.0 and 0.1, representing a right jerk, and between 0.9 and 1.0, representing a left jerk. We determined parameter
a as follows: first, with a Python program, we identified the different cycles of the measured nystagmus and for each cycle, the peaks and troughs were superposed by scaling the time axis. Subsequently, we discarded the outliers – cycles with log amplitudes that fell outside two times the standard deviation – and constructed an averaged and normalized waveform cycle. We then used Python's scipy.optimize.curve_fit,
18 nonlinear least squares to best fit the function,
f, to the data by varying parameters
a and
w (
a and
w only in the ranges specified above).
We defined the percentage of low velocity (PLOV) by calculating the horizontal eye rotation velocity during the nystagmus and determining the percentage of time within a specific interval during which the absolute velocity was less than 4.0 degrees/second.
9 For illustration see
Figure 3. A typical interval was around 20 seconds and was selected based on the quality of the measurement.