A random walk model of fast-phase timing during optokinetic nystagmus

Most vertebrate animals produce optokinetic nystagmus in response to rotation of their visual surround. Nystagmus consists of an alternation of slowphase eye rotations, which follow the surround, and fast-phase eye rotations, which quickly reset eye position. The time intervals between fast phases vary stochastically, even during optokinetic nystagmus produced by constant velocity rotation of a uniform surround. The inter-fast-phase interval distribution has a long tail, and intervals that are long relative to the mode become even more likely as constant surround velocity is decreased. This paper provides insight into fast-phase timing by showing that the process of fast-phase generation during constant velocity optokinetic nystagmus is analogous to a random walk with drift toward a threshold. Neurophysiologically, the output of vestibular nucleus neurons, which drive the slow phase, would approximate a random walk with drift because they integrate the noisy, constant surround velocity signal they receive from the visual system. Burst neurons, which fire a burst to drive the fast phase and reset the slow phase, are brought to threshold by the vestibular nucleus neurons. Such a nystagmic process produces stochastically varying inter-fastphase intervals, and long intervals emerge naturally because, as drift rate (related to surround velocity) decreases, it becomes more likely that any random walk can meander for a long time before it crosses the threshold. The theoretical probability density function of the first threshold crossing times of random walks with drift is known to be that of an inverse Gaussian distribution. This probability density function describes well the distributions of the intervals between fast phases that were either determined experimentally, or simulated using a neurophysiologically plausible neural network model of fast-phase generation, during constant velocity optokinetic nystagmus.