M. Golubitsky, L-J. Shiau and I. Stewart
Spatiotemporal symmetries in the disynaptic canal-neck projection
SIAM J Appl Math. 67 (5) (2007) 1396-1417.
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 neck motoneurons and activate eight corresponding muscle groups. These signals may be either excitatory or inhibitory, depending on the direction of acceleration. McCollum and Boyle have observed that in the cat the network of neurons concerned possesses octahedral symmetry, a structure that they deduce from the known innervation patterns (connections) from canals to muscles.
We re-derive 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 that network also has octahedral symmetry. The network and its symmetries imply 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.