Analysis of a nonlinear dynamics model of the saccadic system

Models of the mechanisms of saccadic eye movements are typically described in terms of the block diagrams used in control theory. Recently, a nonlinear dynamics model of the saccadic system was developed. The model comprises a symmetric piecewise-smooth system of six first-order autonomous ordinary differential equations, which were obtained by combining parts of the existing control models with data from experimental observations of saccadic dynamics. A preliminary numerical investigation of the model revealed that in addition to generating normal saccades, it could also simulate inaccurate saccades, and an oscillatory instability known as congenital nystagmus (CN). By varying the parameters of the model, several types of CN oscillations could be produced, including jerk, bilateral jerk and pendular nystagmus. The aim of this study was to investigate the bifurcations and attractors of the nonlinear dynamics model, in order to obtain a classification of the simulated oculomotor behaviours. The application of standard local and global stability analysis techniques, together with numerical work, revealed that the equations have a rich bifurcation structure. In addi- tion to Hopf, homoclinic and saddlenode bifurcations organised by a Takens-Bogdanov point, the equations can undergo nonsmooth pitchfork bifurcations and nonsmooth glu- ing bifurcations. These nonsmooth bifurcations were observed to result from simultaneous transcritical and homoclinic bifurcations in a pair of related smooth systems. Evidence was also found for the existence of Hopf-initiated canards, and for a global bifurcation involving the catastrophic destruction of a symmetry-invariant limit cycle. Unlike the pitchfork and gluing bifurcations, this bifurcation could not be explained in terms of the related smooth systems. 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 equa- tions are a slow-fast system. The slow and fast phases of the simulated oscillations were attributed to the geometry of an underlying slow manifold. This provides an alternative explanation for the shape of the jerk oscillation, which contrasts with the prevalent control model view that CN is caused by structural abnormalities. The simulated bilateral jerk and pendular waveforms were attributed to a symmetry invariant limit cycle produced by the gluing of the asymmetric cycles. The bifurcation structure of the model suggests the possibility of moving between the differ- ent simulated behaviours by varying the parameters of the model. This was in agreement with experimental evidence showing that sub jects can exhibit several different types of behaviour in a single recording period. In addition, the bifurcation analysis places restric- tions on which kinds of behaviour are likely to be associated with each other in parameter space. On the basis of these restrictions, several experiments were suggested to assess the validity of the model as a predictor of saccadic behaviour. In particular, it was proposed that reducing the level of attention of a sub ject in a controlled way could induce a change from a jerk to a pendular oscillation