A mathematical and comparative study on cerebellar control of vestibular reflexes
The first aim of this thesis is an introduction to some basic aspects of multivariate control theory which are relevant to the question of how the brain controls movements. A regulator is a device which forces a system to follow a specified trajectory in the presence of perturbations which might cause it to diverge from that trajectory. Regulation involves constructing an additional control input which depends upon the difference between the actual system state and the desired state. This requires the construction of a state estimate from raw data about system input and output. For effective state estimation, the sensor input gain to the state estimator needs to be time-varying. Under certain assumptions, the appropriate input gain can be specified analytically. The feedback regulation signal can then be constructed as a function of the state estimate. For effective regulation, the gain of the feedback function has to vary during maneuvers. Under certain assumptions an appropriate feedback gain can be specified analytically. The state observer input gain equations have a simple relationship to the feedback gain equations, so that gain specification is essentially the same task in each case. Cerebellar research has been dominated for the past 25 years by the theories of James Albus and David Marr. These mathematicians proposed similar models in which certain synapses in the cerebellar cortex are continuously modified by experience in such a way that movements which are consistently repeated under a given set of circumstances come to be performed automatically by the cerebellum. Much experimental work has focussed on the role of the vestibulo-cerebellum in fine control and learning of the vestibulo-ocular reflex. The state of the art along this line is formally described by Fujita's adaptive filter model of the cerebellar cortex. In chapter 4 it is shown that a basic feature of Fujita's model is inconsistent with available evidence. The 'Tensorial theory of brain function' is discussed in chapter 5. This is a novel theory of brain function which has been used in an attempt two explain cerebellar function. The attempt is a failure, based on sophistocated misconceptions and flawed by poor reasoning and clumsy analysis. The approach serves to confuse rather than clarify the question of cerebellar function. The final chapter of the first part of the thesis presents a basis for a new approach to cerebellar function based on the engineering theory of control of multivariate dynamical systems. It is proposed that the cerebellum is involved in movement regulation by controlling the gains of brainstem motor pathways, and in mapping the animal's environment by controlling the gains of sensory inputs to the midbrain. While learning undoubtedly does occur in the cerebellar cortex, this is not specifically a 'learning device', as commonly conceived. The second part of the thesis is concerned with the development and application of a method of system identification for characterising the dynamics of the vestibulo-ocular reflex and its components in an elasmobranch. The chosen method involves pulse-rate modulated bilateral electrical stimulation of the horizontal semicircular canal ampullary nerves. This produces a synthetic vestibulo-ocular reflex in a stationary preparation. The stimulus pattern is a pseudorandom binary sequence of pulse rates, so that cross-correlation of the stimulus pattern with the response signal gives a Unit Impulse Response dynamic signature for the system. Computer software for signal generation, recording, analysis and display was written by the author. The identification system was applied first to characterise the dynamics of the eye movement response to horizontal canal ampullary nerve stimulation, and compare this to the dynamics of the eye motor plant alone. The eye motor preparation acts as a first order low-pass filter with a time constant of about 0.2 seconds (16°C), while the ampullary preparation acts as a second order low-pass filter with a dominant time constant of about 0.75 seconds (16°C). Central pathways of the elasmobranch vestibulo-ocular reflex extend the time constant of the motor plant by a factor of 3-4, as in other animals. Eye movements predicted by fitted linear models accurately mimic eye movements recorded during experiments, suggesting both that central pathways of the reflex operate normally during this somewhat un-naturally evoked response and that the identification procedure is effective. Furthermore, combination of the ampullary nerve to eye movement transfer function obtained in this study, with head rotation to ampullary nerve transfer functions obtained by other workers, gives a consistent picture of elasmobranch vestibulo-ocular reflex function predicting compensatory eye movements in the band 0.2 - 4.0 Hz., and perhaps higher. The identification method has also been applied to produce models of vestibulocerebellar Purkinje cell dynamics during electrically evoked vestibular eye movements. Linear identification gives a poor characterisation of Purkinje cell activity during the high frequency vestibulo-ocular reflex. This is incompatible with linear phase-compensator models of the cerebellar cortex, but consistent with the reflex gain modulation theory of cerebellar function advocated in the first part of the thesis.