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

Abstract

The study of eye movement control and oculomotor disorders has, for four decades, relied on control theoretic concepts for its theoretical foundation. 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. * Now at: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK 1The 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.

Nonlinear time series analysis of jerk congenital nystagmus

“…In the model, a fixed point representing stable gaze at the primary position loses stability in a pitchfork bifurcation as the braking strength is increased, producing a pair of stable fixed points corresponding to hypometric saccades. As the braking strength is increased still further, the fixed points undergo Hopf bifurcations, leading to a pair of limit cycle attractors corresponding to leftbeating and right-beating jerk oscillations (Akman et al, 2005). This picture is consistent with the proposition that the jerk instability is caused by a 1-dimensional bifurcation at a fixed point, followed by a secondary bifurcation which generates an oscillation.…”

“…This point has been further illustrated by a recent nonlinear dynamics model of the saccadic system which is able to generate CN waveforms with both slow and fast phases, despite having no slow eye movement components (Broomhead et al, 2000). A range of CN waveforms can be simulated by the model by varying parameters representing the strength of the saccadic braking pulse and the reaction time of saccadic burst neurons to the motor error signal that drives their firing (Broomhead et al, 2000;Akman et al, 2005). In the model, a fixed point representing stable gaze at the primary position loses stability in a pitchfork bifurcation as the braking strength is increased, producing a pair of stable fixed points corresponding to hypometric saccades.…”

“…As a consequence, the fast phases are forced to terminate close to the target position (Broomhead et al, 2000;Akman et al, 2005). Both types of models thus result in a fast phase which is a deterministic refixation of the target.…”

“…Current models of jerk CN based on both control theory ( (Optican and Zee, 1984;Tusa et al, 1992;Dell'Osso, 2002;Jacobs and Dell'Osso, 2004)) and nonlinear dynamics ((Broomhead et al, 2000;Akman et al, 2005)) produce fast phases that are a deterministic refixation of the target. As detailed in section 6.5, an analytical return map can be derived by combining the simplest model of such a fast phase -a near-homoclinic orbit -with a model of the slow phase dynamics obtained from the data.…”

“…A previous paper showed that one advantage of the nonlinear dynamics approach is the capacity of bifurcation analysis to reveal the full range of behaviour that a model is capable of producing (Akman et al, 2005). The present paper describes the use of nonlinear time series techniques to investigate hypotheses suggested by oculomotor models regarding the aetiology and mathematical characterisation of oscillatory ocular disorders.…”