Biol. Cybern. 72, 69 79 (1994) Biological cybemet Springer-Verlag 1994 The fractional-order dynamics of brainstem vestibulo-oculomotor neurons Thomas J. Auastasio Beckman Institute and Department of Physiology and Biophysics, University of Illinois, 405 North Mathews Avenue, Urbana, IL 61801, USA Received: 18 January 1994/Accepted in revised form: 13 June 1994 Abstract. The vestibulo-ocular reflex (VOR) and other oculomotor subsystems such as pursuit and saccades are ultimately mediated in the brainstem by premotor neurons in the vestibular and prepositus nuclei that relay eye movement commands to extraocular motoneurons. The premotor neurons receive vestibular signals from canal afferents. Canal afferent frequency responses have a component that can be characterized as a fractional- order differentiation (dkx/dt k where k is a nonnegative real number). This article extends the use of fractional calculus to describe the dynamics of motor and premotor neurons. It suggests that the oculomotor integrator, which converts eye velocity into eye position commands, may be of fractional order. This order is less than one, and the velocity commands have order one or greater, so the resulting net output of motor and premotor neurons can be described as fractional differentiation relative to eye position. The fractional derivative dynamics of motor and premotor neurons may serve to compensate frac- tional integral dynamics of the eye. Fractional differenti- ation can be used to account for the constant phase shift across frequencies, and the apparent decrease in time constant as VOR and pursuit frequency increases, that are observed for motor and premotor neurons. Frac- tional integration can reproduce the time course of motor and premotor neuron saccade-related activity, and the complex dynamics of the eye. Insight into the nature of fractional dynamics can be gained through simulations in which fractional-order differentiators and integrators are approximated by sums of integer-order high-pass and low-pass filters, respectively. Fractional dynamics may be applicable not only to the oculomotor system, but to motor control systems in general. 1 Introduction Relative conceptual simplicity and a wealth of neuro- physiological data have enabled substantial progress to be made in understanding the dynamics of the neurons that mediate the horizontal vestibulo-ocular reflex (VOR). The function of VOR is to keep the retinal image stable by producing eye rotations that counterbalance head rotations. The VOR is mediated by premotor neu- rons in the vestibular and prepositus nuclei, which relay head rotation signals from vestibular canal afferent neu- rons to the motoneurons of the eye muscles (Wilson and Melvill Jones 1979). This premotor neuron to mo- toneuron pathway is also utilized by other oculomotor subsystems such as smooth pursuit and saccades, the functions of which in foveate animals are to track moving targets and quickly reorient gaze, respectively. The neurodynamics of the VOR have been elegantly summarized in a model by Robinson (1981). At lower frequencies (below about 0.3 Hz), the dynamics of canal afferents (A) and vestibular and prepositus nuclei neu- rons (V) reflect those of the canal receptors, and the frequency response of neural discharge rate relative to head angular (rotational) velocity (/:/) can be described approximately as a first-order high-pass filter: A(s) V(s) SZv t:I(s) = t:I(s) - (SVv + 1~ (1) where s, the Laplace variable, stands for complex fre- quency (s = j~o where j = .,f- 1 and o) in radians/s equals 2rtf with f in Hz), and Zv is the vestibular time constant. The value of Zv can be larger for premotor neurons than for canal afferents due to the action of the velocity storage integrator. At higher frequencies (above about 0.3 Hz), motoneuron (M) dynamics would offset the mechanical lag of the eye, and the frequency response of neural discharge rate relative to eye angular position (E) can be described approximately as a first-order lead: M(s) - (szc + 1) (2) E(s) where ze is the eye time constant. In Robinson's model, the lead is generated by the oculomotor neural integrator in parallel with a direct pathway. This higher-frequency behavior at the motoneuron level should be common to all eye movements including pursuit and saccades. The first-order transfer functions given in (1) and (2) approximate time and frequency domain data from canal afferents, vestibular and prepositus nuclei neurons, and motoneurons, but deviations from first-order dynamics are also apparent. For the canal afferent response relative