Comparison of two nonlinear models for fitting saccadic eye movement data

Saccadic eye movements are rapid shifts in the direction of gaze which are being studied increasingly for clinical and pharmacological purposes. The evaluation of the relationship between amplitude and peak velocity of these ocular movements (the so-called 'main sequence' plot) is particularly useful for characterising the saccade pattern in individual patients. This relationship is nonlinear and the peak velocity tends to achieve an asymptote for high values of amplitude. Since a standard parametrisation of the main sequence based on specific mathematical models has not yet been achieved, in the present study two simple models based on the Michaelis-Menten equation and on an exponential equation are proposed together with their implementation on a microcomputer. Two microcomputer programs are described which estimate the model parameters from the experimental data of the patients using a weighted nonlinear least-squares fit. The two procedures have been tested and compared in a series of 23 healthy volunteers. The following results (mean +/- S.D.) were obtained: Michaelis-Menten model. Km (degrees) = 31.2 +/- 7.7, Vmax (degrees/s) = 841.0 +/- 165.5, root-mean-squared error(%) = 6.0 +/- 1.6; exponential model. K (degrees) = 23.4 +/- 4.6, Vmax (degrees/s) = 578.0 +/- 97.4, root-mean-squared error(%) = 5.4 +/- 1.6. The two techniques of parametrisation provided similar indices of intra-individual variability in 4 healthy volunteers. In conclusion, our methods for saccade parametrisation can be regarded as simple but efficient tools for facilitating research on these ocular movements.