Slow-fast control of eye movements: an instance of Zeeman's model for an action.

The rapid eye movements (saccades) used to transfer gaze between targets are examples of an action. The behaviour of saccades matches that of the slow-fast model of actions originally proposed by Zeeman. Here, we extend Zeeman's model by incorporating an accumulator that represents the increase in certainty of the presence of a target, together with an integrator that converts a velocity command to a position command. The saccadic behaviour of several foveate species, including human, rhesus monkey and mouse, is replicated by the augmented model. Predictions of the linear stability of the saccadic system close to equilibrium are made, and it is shown that these could be tested by applying state-space reconstruction techniques to neurophysiological recordings. Moreover, each model equation describes behaviour that can be matched to specific classes of neurons found throughout the oculomotor system, and the implication of the model is that build-up, burst and omnipause neurons are found throughout the oculomotor pathway because they constitute the simplest circuit that can produce the motor commands required to specify the trajectories of motor actions.

Components of the neural signal underlying congenital nystagmus.

Congenital nystagmus is an involuntary bilateral horizontal oscillation of the eyes that develops soon after birth. In this study, the time constants of each of the components of the neural signal underlying congenital nystagmus were obtained by time series analysis and interpreted by comparison with those of the normal oculomotor system. In the neighbourhood of the fixation position, the system generating the neural signal is approximately linear with 3 degrees of freedom. The shortest time constant was in the range of 7-9 ms and corresponds to a normal saccadic burst signal. The other stable time constant was in the range of 22-70 ms and corresponds to the slide signal. The final time constant characterises the unidentified neural mechanism underlying the unstable drift component of the oscillation cycle and ranges between 31 and 32 ms across waveforms. The characterisation of this unstable time constant poses a challenge for the modelling of both the normal and abnormal oculomotor control system. We tentatively identify the unstable component with the eye position signal supplied to the superior colliculus in the normal eye movement system and explore some of the implications of this hypothesis.

Analysing nystagmus waveforms: a computational framework.

We present a new computational approach to analyse nystagmus waveforms. Our framework is designed to fully characterise the state of the nystagmus, aid clinical diagnosis and to quantify the dynamical changes in the oscillations over time. Both linear and nonlinear analyses of time series were used to determine the regularity and complexity of a specific homogenous phenotype of nystagmus. Two-dimensional binocular eye movement recordings were carried out on 5 adult subjects who exhibited a unilateral, uniplanar, vertical nystagmus secondary to a monocular late-onset severe visual loss in the oscillating eye (the Heimann-Bielschowsky Phenomenon). The non-affected eye held a central gaze in both horizontal and vertical planes (± 10 min. of arc). All affected eyes exhibited vertical oscillations, with mean amplitudes and frequencies ranging from 2.0°-4.0° to 0.25-1.5 Hz, respectively. Unstable periodic orbit analysis revealed only 1 subject exhibited a periodic oscillation. The remaining subjects were found to display quasiperiodic (n = 1) and nonperiodic (n = 3) oscillations. Phase space reconstruction allowed attractor identification and the computation of a time series complexity measure-the permutation entropy. The entropy measure was found to be able to distinguish between a periodic oscillation associated with a limit cycle attractor, a quasiperiodic oscillation associated with a torus attractor and nonperiodic oscillations associated with higher-dimensional attractors. Importantly, the permutation entropy was able to rank the oscillations, thereby providing an objective index of nystagmus complexity (range 0.15-0.21) that could not be obtained via unstable periodic orbit analysis or attractor identification alone. These results suggest that our framework provides a comprehensive methodology for characterising nystagmus, aiding differential diagnosis and also permitting investigation of the waveforms over time, thereby facilitating the quantification of future therapeutic managements. In addition, permutation entropy could provide an additional tool for future oculomotor modelling.

Fixed point analysis of nystagmus.

Motor disorders frequently contain a rhythmic component, but the associated oscillations are not usually precisely periodic. This lack of strict periodicity can make it difficult to identify the effects of experimental manipulations on the oscillation. In this report, we describe the application of a numerical technique for identifying fixed points of a nonlinear map to the recovery of underlying periodicities of the eye movement disorder of nystagmus. The technique is illustrated by application to two different types of nystagmus. In addition we use a local analysis of the behaviour at the fixed points to distinguish between different bifurcations in the two examples with changes in gaze angle. We conclude that the technique reveals consistent effects of experimental manipulations, which may be useful for quantitative characterisation of experimental and therapeutic manipulations of motor disorders.

A new framework for investigating both normal and abnormal eye movements.

Any comprehensive framework for understanding eye movements has to include both normal and abnormal eye movement behaviour. One approach which is applicable to the entire range of oculomotor behaviour is provided by the techniques of nonlinear dynamics. The stability of models of the oculomotor system can be analysed in terms of the characteristics of their fixed points and periodic orbits, and the method of delays can be used to recover such parameters from measurements of eye position. Within this framework, quantitative comparisons can be made between the predictions of different models, and both normal and clinical eye movement recordings.

Periodic orbit analysis reveals subtle effects of atropine on epileptiform activity in the guinea-pig hippocampal slice.

Epileptiform activity is a state often induced in vitro in order to study seizures and antiepileptic/anticonvulsant drugs. Traditional methods of evaluating drug effects have commonly relied upon measuring changes in the frequency and duration of such events. We have used a recently developed mathematical technique based on periodic orbit analysis to investigate the effect of atropine (a muscarinic antagonist) on epileptiform activity induced by soman (an irreversible acetylcholinesterase inhibitor), 4-aminopyridine (a K+ channel blocker) and 8-cyclopentyl-1,3-dipropylxanthine (an adenosine A1 receptor antagonist) in the guinea-pig hippocampal slice. This technique showed that significant changes in periodic orbits can occur without an accompanying change in burst rate. These results suggest that periodic orbit analysis may be useful in detecting and predicting novel actions of anticonvulsant drugs.

Characterisation of congenital nystagmus waveforms in terms of periodic orbits.

Because the oscillatory eye movements of congenital nystagmus vary from cycle to cycle, there is no clear relationship between the waveform produced and the underlying abnormality of the ocular motor system. We consider the durations of successive cycles of nystagmus which could be (1) completely determined by the lengths of the previous cycles, (2) completely independent of the lengths of the previous cycles or (3) a mixture of the two. The behaviour of a deterministic system can be characterised in terms of a collection of (unstable) oscillations, referred to as periodic orbits, which make up the system. By using a recently developed technique for identifying periodic orbits in noisy data, we find evidence for periodic orbits in nystagmus waveforms, eliminating the possibility that each cycle is independent of the previous cycles. The technique also enables us to identify the waveforms which correspond to the deterministic behaviour of the ocular motor system. These waveforms pose a challenge to our understanding of the ocular motor system because none of the current extensions to models of the normal behaviour of the ocular motor system can explain the range of identified waveforms.

Dynamics of saccadic oscillations.

The brainstem circuitry underlying saccades is symmetrical with respect to the midline. The oculomotor behaviour generated by the circuitry depends on a combination of signals passed along fibre tracts and less easily identifiable connections, such as those across the midline. The midline crossing connections are often affected by developmental disorders which give rise to unstable eye movements (see J. Jen, this volume). The connections at the levels of the colliculus, pause cells, and neural integrator generate different dynamical mechanisms for the development of instabilities, which can be identified in eye movement recordings using phase space analysis techniques.

Stability of the saccadic oculomotor system.

A variety of different types of instability has been found in the saccadic system of humans. Some of the instabilities correspond to clinical conditions, whereas others are inherent in the normal saccadic system. How can these instabilities arise within the mechanism of normal saccadic eye movements? A physiologically-based model of the saccadic system predicts that horizontal saccadic oscillations will occur with excessive mutual inhibition between the left and right burst cells and with underaction of the pause cells. The amplitudes and frequencies of the oscillations had ranges of 0-6 degrees and 6-20 cycles per second, respectively. Application of stability analysis techniques to the model reveals that development of the oscillations can be explained by the Hopf bifurcation mechanism. Future development of this approach will involve classifying pathological instabilities of the saccadic system according to the bifurcation involved in their generation.

A quantitative description of lens eye morphology and its implications.

If it is assumed that the differences between eyes constitute adaptations to different visual environments, then it should be possible to identify the visual environments for which the eye is optimised from these differences. Two different classes of simple eye shapes can be distinguished from the ratio of maximum pupil size to the axial length of an eye. The quantitative description of the two different eye shapes can be used to identify the relative importance of image illumination and resolution to an animal from a cross-section of its eye.