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![Structure and location of the semicircular canals (right ear). From Vilis [17].](profile/Ian-Stewart-21/publication/37142991/figure/fig1/AS:276487799361546@1442931236776/Structure-and-location-of-the-semicircular-canals-right-ear-From-Vilis-17.png)
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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 activa...
Contexts in source publication
Context 1
... ear contains three semicircular canals (henceforth 'canals') arranged in three approximately mutually orthogonal planes. See Figure 1. A similar arrangement occurs in most vertebrates. ...
Context 2
... 3 , Z 3 ): If we take H = V 1 , E 15 and K = V 1 , then this case turns out to be exactly like the previous one, except that the time series of the direct muscled motoneurons u(t) is unequal to the time series of the central dial muscle motoneurons z(t). Since the z(t) motions cancel out, the motion again looks like 'no' and is reproduced as the first image in Figure 10. (Here angle brackets indicate the subgroup generated by their contents.) ...
Context 3
... patterns for the other two conjugates of this motion can be deduced in a similar manner and are visualized in Figure 10. In pattern 3 the head is inclined alternately down to the left and up to the right, while the nose oscillates from side to side. ...
Context 4
... Table 3, and noting that F 1 induces a phase shift of ± 1 4 , we obtain the pattern listed in Table 4. (We also use the (−I, 1 2 ) symmetry of all periodic states.) The motions are visualized in Figure 12. In pattern 9, the head moves in an ellipse with long axis pointing towards the front. ...
Context 5
... here replace V 1 by V 3 , V 5 , V 7 , noting that V 3 and V 5 are mirror images. The motions are visualized in Figure 13. In pattern 11 the head rotates in a rounded hexagonal curve, while the nose oscillates slightly. ...
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Background
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Citations
... However, the literature we are aware of identifies symmetry-breaking in structured networks dominated by deterministic behavior. For example, symmetry breaking has been hypothesized to underlie the dynamics of visual hallucinations [37] and ambiguous visual percepts [38], central pattern generators that govern rhythmic behaviors of breathing, eating, and swimming [39][40][41], and periodic head/limb motions [42][43][44]. Most recently, Kriener et al. [45] investigate a Dale's law-conforming orientation model and find that the dynamics are affected by a translation symmetry imposed by the regularity of the cell grid. ...
We examine a family of random firing-rate neural networks in which we enforce the neurobiological constraint of Dale's Law --- each neuron makes either excitatory or inhibitory connections onto its post-synaptic targets. We find that this constrained system may be described as a perturbation from a system with non-trivial symmetries. We analyze the symmetric system using the tools of equivariant bifurcation theory, and demonstrate that the symmetry-implied structures remain evident in the perturbed system. In comparison, spectral characteristics of the network coupling matrix are relatively uninformative about the behavior of the constrained system.
... The vertical canal planes also have an approximate symmetry about the vertical axis, which leaves the horizontal plane approximately invariant. These transformations and symmetries of the canals themselves correspond to transformations and symmetries of their projections (McCollum and Boyle 2004;Foster et al. 2007;McCollum 2007;Golubitsky et al. 2007;McCollum and Hanes 2010), as transformations of the utricle are likely to do. ...
... It has been shown that canal projections to the neck and to the uvula-nodulus via the CVS display functionally-relevant symmetries (McCollum and Boyle 2004;Foster et al. 2007;McCollum 2007;Golubitsky et al. 2007;McCollum and Hanes 2010). The CVS is a sensorimotor integration center, concerned with functions such as posture, gaze, and balance (Goldberg and Fernández 1984;Barmack 2003;Wilson et al. 1995;Straka and Dieringer 2004;Blümle et al. 2006). ...
... For these sensorimotor functions, the geometry of physical space is crucial. The disynaptic projection from the canal nerves to neck motor neurons has the symmetry group O of the cube (McCollum and Boyle 2004;Golubitsky et al. 2007). Symmetries of the cube form a discrete framework for three-dimensional movements. ...
Sensory contribution to perception and action depends on both sensory receptors and the organization of pathways (or projections) reaching the central nervous system. Unlike the semicircular canals that are divided into three discrete sensitivity directions, the utricle has a relatively complicated anatomical structure, including sensitivity directions over essentially [Formula: see text] of a curved, two-dimensional disk. The utricle is not flat, and we do not assume it to be. Directional sensitivity of individual utricular afferents decreases in a cosine-like fashion from peak excitation for movement in one direction to a null or near null response for a movement in an orthogonal direction. Directional sensitivity varies slowly between neighboring cells except within the striolar region that separates the medial from the lateral zone, where the directional selectivity abruptly reverses along the reversal line. Utricular primary afferent pathways reach the vestibular nuclei and cerebellum and, in many cases, converge on target cells with semicircular canal primary afferents and afference from other sources. Mathematically, some canal pathways are known to be characterized by symmetry groups related to physical space. These groups structure rotational information and movement. They divide the target neural center into distinct populations according to the innervation patterns they receive. Like canal pathways, utricular pathways combine symmetries from the utricle with those from target neural centers. This study presents a generic set of transformations drawn from the known structure of the utricle and therefore likely to be found in utricular pathways, but not exhaustive of utricular pathway symmetries. This generic set of transformations forms a 32-element group that is a semi-direct product of two simple abelian groups. Subgroups of the group include order-four elements corresponding to discrete rotations. Evaluation of subgroups allows us to functionally identify the spatial implications of otolith and canal symmetries regarding action and perception. Our results are discussed in relation to observed utricular pathways, including those convergent with canal pathways. Oculomotor and other sensorimotor systems are organized according to canal planes. However, the utricle is evolutionarily prior to the canals and may provide a more fundamental spatial framework for canal pathways as well as for movement. The fullest purely otolithic pathway is likely that which reaches the lumbar spine via Deiters' cells in the lateral vestibular nucleus. It will be of great interest to see whether symmetries predicted from the utricle are identified within this pathway.
... McCollum and Boyle [74] 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. Golubitsky et al. [37] derive octahedral symmetry from network architecture, and model the movement of the head in response to activation patterns of the muscles. ...
Many biological systems have aspects of symmetry. Symmetry is formalized using group theory. This theory applies not just to the geometry of symmetric systems, but to their dynamics. The basic ideas of symmetric dynamics and bifurcation theory are applied to speciation, animal locomotion, the visual cortex, pattern formation in animal markings and geographical location, and the geometry of virus protein coats.
... Under some generic assumptions on H and K, the H/K Theorem states that periodic states have spatio-temporal symmetry group pairs (H, K) only if H/K is cyclic, and K is an isotropy subgroup (Buono and Golubitsky 2001,Golubitsky and Stewart 2002). The H/K Theorem was originally derived in the context of equivariant dynamical systems byBuono and Golubitsky (2001), and it has subsequently been used to classify various types of periodic behaviours in systems with symmetry that arise in a number of contexts, from speciation(Stewart 2003) to animal gaits(Pinto and Golubitsky 2006) and vestibular system of vertebrates(Golubitsky et al 2007). ...
In mathematical studies of the dynamics of multi-strain diseases caused by antigenically diverse pathogens, there is a substantial interest in analytical insights. Using the example of a generic model of multi-strain diseases with cross-immunity between strains, we show that a significant understanding of the stability of steady states and possible dynamical behaviours can be achieved when the symmetry of interactions between strains is taken into account. Techniques of equivariant bifurcation theory allow one to identify the type of possible symmetry-breaking Hopf bifurcation, as well as to classify different periodic solutions in terms of their spatial and temporal symmetries. The approach is also illustrated on other models of multi-strain diseases, where the same methodology provides a systematic understanding of bifurcation scenarios and periodic behaviours. The results of the analysis are quite generic, and have wider implications for understanding the dynamics of a large class of models of multi-strain diseases.
... Under some generic assumptions on H and K, the H/K Theorem states that periodic states have spatio-temporal symmetry group pairs (H, K) only if H/K is cyclic, and K is an isotropy subgroup [12,25]. The H/K Theorem was originally derived in the context of equivariant dynamical systems by Buono and Golubitsky [12], and it has subsequently been used to classify various types of periodic behaviours in systems with symmetry that arise in a number of contexts, from speciation [50] to animal gaits [43] and vestibular system of vertebrates [22]. In the case of D 4 symmetry group acting on four elements, there are eleven pairs of subgroups H and K satisfying the above requirements [25]. ...
Effects of immune delay on symmetric dynamics are investigated within a model of antigenic variation in malaria. Using isotypic decomposition of the phase space, stability problem is reduced to the analysis of a cubic transcendental equation for the eigenvalues. This allows one to identify periodic solutions with different symmetries arising at a Hopf bifurcation. In the case of small immune delay, the boundary of the Hopf bifurcation is found in a closed form in terms of system parameters. For arbitrary values of the time delay, general expressions for the critical time delay are found, which indicate bifurcation to an odd or even periodic solution. Numerical simulations of the full system are performed to illustrate different types of dynamical behaviour. The results of this analysis are quite generic and can be used to study within-host dynamics of many infectious diseases.
... The H/K theorem was originally derived in the context of equivariant dynamical systems by Buono and Golubitsky [14], and it has subsequently been used to classify various types of periodic behaviours in systems with symmetry that arise in a number of contexts, from speciation [59] to animal gaits [51] and vestibular system of vertebrates [28]. Now we can proceed with the analysis of symmetry properties of the system (6) as represented by its adjacency matrix A. In the case of two minor epitopes with m variants in the first epitope and n variants in the second, the system (6) is equivariant with respect to the following symmetry group [10] ...
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.
... One of the important conclusions of the theory of coupled cell systems is that the network itself imposes constraints on the possible behaviors of the system even when we lack detailed knowledge of the behavior of the cells within the network. A recent application to head movement that illustrates the importance of this is [6]. For further applications see [10]. ...
We give an simple criterion for ODE equivalence in identical edge homogeneous
coupled cell networks. This allows us to give a simple proof of Theorem 10.3 of
Aquiar and Dias "Minimal Coupled Cell Networks", which characterizes minimal
identical edge homogeneous coupled cell networks. Using our criterion we give a
formula for counting homogeneous coupled cell networks up to ODE equivalence.
Our criterion is purely graph theoretic and makes no explicit use of linear
algebra.
... Although spontaneous oscillatory behavior is often associated with the properties of individual neurons in a population [33][34][35], it is not limited to that level of analysis. Given a pathway inducing a network of relationships on a target population, the symmetry group of the pathway can present a menu of geometrically-appropriate patterns for the nervous system to produce spontaneously and constructively [36]. Depending on the behavior of interacting networks, these patterns can be reflected in behaviors such as gaits and head movements. ...
... Permutations of the muscle types form rotations like those of the eight vertices of a cube [15]. While the nervous system may segregate sub-patterns in ways as yet unforeseen, the pattern shown in Figure 1B has the symmetry group of the cube, the 11-element octahedral group O [36] ( Figure 1C). The eight muscle types (L1-4, R1-4) are displayed on the vertices of the cube, and the six canals (LA, LH, LP, RA, RH, RP) occupy one face each. ...
... [15]. C is modified from [36]). ...
Intrinsic dynamics of the central vestibular system (CVS) appear to be at least partly determined by the symmetries of its connections. The CVS contributes to whole-body functions such as upright balance and maintenance of gaze direction. These functions coordinate disparate senses (visual, inertial, somatosensory, auditory) and body movements (leg, trunk, head/neck, eye). They are also unified by geometric conditions. Symmetry groups have been found to structure experimentally-recorded pathways of the central vestibular system. When related to geometric conditions in three-dimensional physical space, these symmetry groups make sense as a logical foundation for sensorimotor coordination.
Symmetries in the external world constrain the evolution of neuronal circuits that allow organisms to sense the environment and act within it. Many small “modular” circuits can be viewed as approximate discretizations of the relevant symmetries, relating their forms to the functions they perform. The recent development of a formal theory of dynamics and bifurcations of networks of coupled differential equations permits the analysis of some aspects of network behavior without invoking specific model equations or numerical simulations. We review basic features of this theory, compare it to equivariant dynamics, and examine the subtle effects of symmetry when combined with network structure. We illustrate the relation between form and function through examples drawn from neurobiology, including locomotion, peristalsis, visual perception, balance, hearing, location detection, decision-making, and the connectome of the nematode Caenorhabditis elegans.
Patterns of synchrony in networks of coupled dynamical systems can be represented as colorings of the nodes, in which nodes of the same color are synchronous. Balanced colorings, where nodes of the same color have color-isomorphic input sets, correspond to dynamically invariant subspaces, which can have a significant effect on the typical bifurcations of network dynamical systems. Orbit colorings for subgroups of the automorphism (symmetry) group of the network are always balanced, although the converse is false. We compute the automorphism groups of all doubly periodic quotient networks of the square lattice with nearest-neighbor coupling, and classify the “exotic” cases where this quotient network has extra automorphisms not induced by automorphisms of the square lattice. These comprise five isolated exceptions and two infinite families with wreath product symmetry. We also comment briefly on implications for bifurcations to doubly periodic patterns in square lattice models.
