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.

The deep layers of the superior colliculus (SC) integrate multisensory inputs and initiate an orienting response toward the source of stimulation (target). Multisensory enhancement, which occurs in the deep SC, is the augmentation of a neural response to sensory input of one modality by input of another modality. Multisensory enhancement appears to underlie the behavioral observation that an animal is more likely to orient toward weak stimuli if a stimulus of one modality is paired with a stimulus of another modality. Yet not all deep SC neurons are multisensory. Those that are exhibit the property of inverse effectiveness: combinations of weaker unimodal responses produce larger amounts of enhancement. We show that these neurophysiological findings support the hypothesis that deep SC neurons use their sensory inputs to compute the probability that a target is present. We model multimodal sensory inputs to the deep SC as random variables and cast the computation function in terms of Bayes' rule. Our analysis suggests that multisensory deep SC neurons are those that combine unimodal inputs that would be more uncertain by themselves. It also suggests that inverse effectiveness results because the increase in target probability due to the integration of multisensory inputs is larger when the unimodal responses are weaker.

Cross-modal enhancement (CME) occurs when the neural response to a stimulus of one modality is augmented by another stimulus of a different modality. Paired stimuli of the same modality never produce supra-additive enhancement but may produce modality-specific suppression (MSS), in which the response to a stimulus of one modality is diminished by another stimulus of the same modality. Both CME and MSS have been described for neurons in the deep layers of the superior colliculus (DSC), but their neural mechanisms remain unknown. Previous investigators have suggested that CME involves a multiplicative amplifier, perhaps mediated by N-methyl D-aspartate (NMDA) receptors, which is engaged by cross-modal but not modality-specific input. We previously postulated that DSC neurons use multisensory input to compute the posterior probability of a target using Bayes' rule. The Bayes' rule model reproduces the major features of CME. Here we use simple neural implementations of our model to simulate both CME and MSS and to argue that multiplicative processes are not needed for CME, but may be needed to represent input variance and covariance. Producing CME requires only weighted summation of inputs and the threshold and saturation properties of simple models of biological neurons. Multiplicative nodes allow accurate computation of posterior target probabilities when the spontaneous and driven inputs have unequal variances and covariances. Neural implementations of the Bayes' rule model account better than the multiplicative amplifier hypothesis for the effects of pharmacological blockade of NMDA receptors on the multisensory responses of DSC neurons. The neural implementations also account for MSS, given only the added hypothesis that input channels of the same modality have more spontaneous covariance than those of different modalities.

According to the amyloid hypothesis, Alzheimer Disease results from the accumulation beyond normative levels of the peptide amyloid-β (Aβ). Perhaps because of its pathological potential, Aβ and the enzymes that produce it are heavily regulated by the molecular interactions occurring within cells, including neurons. This regulation involves a highly complex system of intertwined normative and pathological processes, and the sex hormone estrogen contributes to it by influencing the Aβ-regulation system at many different points. Owing to its high complexity, Aβ regulation and the contribution of estrogen are very difficult to reason about. This report describes a computational model of the contribution of estrogen to Aβ regulation that provides new insights and generates experimentally testable and therapeutically relevant predictions. The computational model is written in the declarative programming language known as Maude, which allows not only simulation but also analysis of the system using temporal-logic. The model illustrates how the various effects of estrogen could work together to reduce Aβ levels, or prevent them from rising, in the presence of pathological triggers. The model predicts that estrogen itself should be more effective in reducing Aβ than agonists of estrogen receptor α (ERα), and that agonists of ERβ should be ineffective. The model shows how estrogen itself could dramatically reduce Aβ, and predicts that non-steroidal anti-inflammatory drugs should provide a small additional benefit. It also predicts that certain compounds, but not others, could augment the reduction in Aβ due to estrogen. The model is intended as a starting point for a computational/experimental interaction in which model predictions are tested experimentally, the results are used to confirm, correct, and expand the model, new predictions are generated, and the process continues, producing a model of ever increasing explanatory power and predictive value.

The oculomotor integrator is a network that is composed of neurons in the medial vestibular nuclei and nuclei prepositus hypoglossi in the brainstem. Those neurons act approximately as fractional integrators of various orders, converting eye velocity commands into signals that are intermediate between velocity and position. The oculomotor integrator has been modeled as a network of linear neural elements, the time constants of which are lengthened by positive feedback through reciprocal inhibition. In this model, in which each neuron reciprocally inhibits its neighbors with the same Gaussian profile, all model neurons behave as identical, first-order, low-pass filters with dynamics that do not match the variable, approximately fractional-order dynamics of the neurons that compose the actual oculomotor integrator. Fractional-order integrators can be approximated by weighted sums of first-order, low-pass filters with diverse, broadly distributed time constants. Dynamic systems analysis reveals that the model integrator indeed has many broadly distributed time constants. However, only one time constant is expressed in the model due to the uniformity of its network connections. If the model network is made nonuniform by removing the reciprocal connections to and from a small number of neurons, then many more time constants are expressed. The dynamics of the neurons in the nonuniform network model are variable, approximately fractional-order, and resemble those of the neurons that compose the actual oculomotor integrator. Completely removing the connections to and from a neuron is equivalent to eliminating it, an operation done previously to demonstrate the robustness of the integrator network model. Ironically, the resulting nonuniform network model, previously supposed to represent a pathological integrator, may in fact represent a healthy integrator containing neurons with realistically variable, approximately fractional-order dynamics.