Understanding the rotational vestibular ocular reflex: From differential equations to Laplace transforms

The description of a biological system using a mathematical model is the most effective means to specify the functioning of a quantitative hypothesis, providing at the same time the ability to make predictions that can be further tested experimentally. The Vestibulo-Ocular Reflex (VOR), and more generally the ocular motor control system has been one of the first biological systems to be modeled mathematically and, thanks to contributions from anatomy, biology, biochemistry and information technology it is now the best understood sensory-motor system in humans. Basic science has made it possible to write the differential equations describing the functioning of the semicircular canals, of the otoliths, and of the ocular motor plant at different scales: from models explaining neurotransmitter behavior, to cell membranes and ionic currents, to individual neurons and entire populations, to those describing muscle contractions and eye movements. The differential equations are frequently represented in terms of Laplace transforms and provide a description of the input-output behavior of the system being considered as a function of frequency. Here we will review the input-output behavior of the rotational VOR to exemplify its mathematical modeling as a linear time-invariant dynamic system being stimulated by head rotations and producing eye movements as an output.