Spatiotemporal symmetries in the disynaptic canal-neck projection

The vestibular system in almost all vertebrates, and in particular in humans, controls balance by employing a set of six semicircular canals, three in each inner ear, to detect angular accelerations of the head in three mutually orthogonal coordinate planes. Signals from the canals are transmitted to eight (groups of) neck motoneurons, which activate the eight corresponding muscle groups. These signals may be either excitatory or inhibitory, depending on the direction of head acceleration. McCollum and Boyle have observed that in the cat the relevant network of neurons possesses octahedral symmetry, a structure that they deduce from the known innervation patterns (connections) from canals to muscles. We rederive the octahedral symmetry from mathematical features of the probable network architecture, and model the movement of the head in response to the activation patterns of the muscles concerned. We assume that connections between neck muscles can be modeled by a “coupled cell network,” a system of coupled ODEs whose variables correspond to the eight muscles, and that this network also has octahedral symmetry. The network and its symmetries imply that these ODEs must be equivariant under a suitable action of the octahedral group. It is observed that muscle motoneurons form natural “push-pull pairs” in which, for given movements of the head, one neuron produces an excitatory signal, whereas the other produces an inhibitory signal. By incorporating this feature into the mathematics in a natural way, we are led to a model in which the octahedral group acts by signed permutations on muscle motoneurons. We show that with the appropriate group actions, there are six possible spatiotemporal patterns of time-periodic states that can arise by Hopf bifurcation from an equilibrium representing an immobile head. Here we use results of Ashwin and Podvigina. Counting conjugate states, whose physiological interpretations can have significantly different features, there are 15 patterns of periodic oscillation, not counting left-right reflections or time-reversals as being different. We interpret these patterns as motions of the head, and note that all six types of pattern appear to correspond to natural head motions

Rigid patterns of synchrony for equilibria and periodic cycles in network dynamics

We survey general results relating patterns of synchrony to network topology, applying the formalism of coupled cell systems. We also discuss patterns of phase-locking for periodic states, where cells have identical waveforms but regularly spaced phases. We focus on rigid patterns, which are not changed by small perturbations of the differential equation. Symmetry is one mechanism that creates patterns of synchrony and phase-locking. In general networks, there is another: balanced colorings of the cells. A symmetric network may have anomalous patterns of synchrony and phase-locking that are not consequences of symmetry. We introduce basic notions on coupled cell networks and their associated systems of admissible differential equations. Periodic states also possess spatio-temporal symmetries, leading to phase relations; these are classified by the H/K theorem and its analog for general networks. Systematic general methods for computing the stability of synchronous states exist for symmetric networks, but stability in general networks requires methods adapted to special classes of model equations

Vestibulo-Spinal Pathways in Tetrapods

The vestibulospinal system provides the spinal motor circuits controlling head/neck and limb movements and body posture with rapid reflex adjustments to maintain equilibrium and stability and with a continuous essential excitatory drive, called tonus, to enhance reactive responses to perturbations that force the animal off normal posture. The sensory signals to these reflex circuits originate from hair cells in the inner ear of otolith structures, namely the utricle and saccule, that transduce inertial acceleration and orientation of the head with respect to gravity and in the three orthogonally arranged semicircular canals that transduce angular head rotation. The principal vestibulospinal pathways are 1) the medial vestibulospinal tract that descends in the ventromedial funiculus and innervates inter- and motoneurons located mainly in lamina VII, VIII, and dorsomedial IX throughout the cervical segments; and 2) the lateral vestibulospinal tracts that course in the lateral to ventrolateral funiculi and are distinguished by two divisions: i) a cervical-projecting tract that overlaps many of the targets of medial vestibulospinal tract neurons including the motoneurons in ventromedial IX and also contributes to reflex control of shoulder and forelimb (arm) muscles; and ii) a lumbosacral-projecting tract that provides a rapid input to maintain stable posture and reflex control of the lower body. A striking observation in understanding the functional organization of this sensory-motor system is both that the driving sensory input can be dynamically modified by the behavioral context in which the sensation is made and that it remains able to quickly respond to an external force during self-generated head movements. The structural basis for vestibulospinal inputs to spinal motor control circuits in quadrupeds and bipeds rely in part on the animal's need for coordination between fore- and hind-limb reflex movements. Understanding the sensory-to-motor transformations in the diverse species rely on the correlations of the conserved and unique species behavior, morphology and physiologic function

The effects of symmetry on the dynamics of antigenic variation

In the studies of dynamics of pathogens and their interactions with a host immune system, an important role is played by the structure of antigenic variants associated with a pathogen. Using the example of a model of antigenic variation in malaria, we show how many of the observed dynamical regimes can be explained in terms of the symmetry of interactions between different antigenic variants. The results of this analysis are quite generic, and have wider implications for understanding the dynamics of immune escape of other parasites, as well as for the dynamics of multi-strain diseases.Comment: 21 pages, 4 figures; J. Math. Biol. (2012), Online Firs

Phase-Linking and the Perceived Motion during Off-Vertical Axis Rotation

Human off-vertical axis rotation (OVAR) in the dark typically produces perceived motion about a cone, the amplitude of which changes as a function of frequency. This perception is commonly attributed to the fact that both the OVAR and the conical motion have a gravity vector that rotates about the subject. Little-known, however, is that this rotating-gravity explanation for perceived conical motion is inconsistent with basic observations about self-motion perception: (a) that the perceived vertical moves toward alignment with the gravito-inertial acceleration (GIA) and (b) that perceived translation arises from perceived linear acceleration, as derived from the portion of the GIA not associated with gravity. Mathematically proved in this article is the fact that during OVAR these properties imply mismatched phase of perceived tilt and translation, in contrast to the common perception of matched phases which correspond to conical motion with pivot at the bottom. This result demonstrates that an additional perceptual rule is required to explain perception in OVAR. This study investigates, both analytically and computationally, the phase relationship between tilt and translation at different stimulus rates—slow (45°/s) and fast (180°/s), and the three-dimensional shape of predicted perceived motion, under different sets of hypotheses about self-motion perception. We propose that for human motion perception, there is a phase-linking of tilt and translation movements to construct a perception of one’s overall motion path. Alternative hypotheses to achieve the phase match were tested with three-dimensional computational models, comparing the output with published experimental reports. The best fit with experimental data was the hypothesis that the phase of perceived translation was linked to perceived tilt, while the perceived tilt was determined by the GIA. This hypothesis successfully predicted the bottom-pivot cone commonly reported and a reduced sense of tilt during fast OVAR. Similar considerations apply to the hilltop illusion often reported during horizontal linear oscillation. Known response properties of central neurons are consistent with this ability to phase-link translation with tilt. In addition, the competing “standard” model was mathematically proved to be unable to predict the bottom-pivot cone regardless of the values used for parameters in the model