Eye movement instabilities and nystagmus can be predicted by a nonlinear dynamics model of the saccadic system

Abstract

The study of eye movements and oculomotor disorders has, for four decades, greatly benefitted from the application of control
theoretic concepts. This paper is an example of a complementary approach based on the theory of nonlinear dynamical systems.
Recently, a nonlinear dynamics model of the saccadic system was developed, comprising a symmetric piecewise-smooth system
of six first-order autonomous ordinary differential equations. A preliminary numerical investigation of the model revealed
that in addition to generating normal saccades, it could also simulate inaccurate saccades, and the oscillatory instability
known as congenital nystagmus (CN). By varying the parameters of the model, several types of CN oscillations were produced,
including jerk, bidirectional jerk and pendular nystagmus.

The aim of this study was to investigate the bifurcations and attractors of the model, in order to obtain a classification
of the simulated oculomotor behaviours. The application of standard stability analysis techniques, together with numerical
work, revealed that the equations have a rich bifurcation structure. In addition to Hopf, homoclinic and saddlenode bifurcations
organised by a Takens-Bogdanov point, the equations can undergo nonsmooth pitchfork bifurcations and nonsmooth gluing bifurcations.
Evidence was also found for the existence of Hopf-initiated canards.

The simulated jerk CN waveforms were found to correspond to a pair of post-canard symmetry-related limit cycles, which exist
in regions of parameter space where the equations are a slow-fast system. The slow and fast phases of the simulated oscillations
were attributed to the geometry of the corresponding slow manifold. The simulated bidirectional jerk and pendular waveforms
were attributed to a symmetry invariant limit cycle produced by the gluing of the asymmetric cycles. In contrast to control
models of the oculomotor system, the bifurcation analysis places clear restrictions on which kinds of behaviour are likely
to be associated with each other in parameter space, enabling predictions to be made regarding the possible changes in the
oscillation type that may be observed upon changing the model parameters. The analysis suggests that CN is one of a range
of oculomotor disorders associated with a pathological saccadic braking signal, and that jerk and pendular nystagmus are the
most probable oscillatory instabilities. Additionally, the transition from jerk CN to bidirectional jerk and pendular nystagmus
observed experimentally when the gaze angle or attention level is changed is attributed to a gluing bifurcation. This suggests
the possibility of manipulating the waveforms of subjects with jerk CN experimentally to produce waveforms with an extended
foveation period, thereby improving visual resolution.