The fractional-order dynamics of brainstem vestibulo-oculomotor neurons

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/dtk 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 fractional integral dynamics of the eye. Fractional differentiation 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. Fractional 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.