The Morse Lemma in Infinite Dimensions via Singularity Theory

An infinite dimensional Morse lemma is proved using the deformation lemma from singularity theory. It is shown that the versions of the Morse lemmas due to Palais and Tromba are special cases. An infinite dimensional splitting lemma is proved. The relationship of the work here to other approaches in the literature in discussed

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

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

Network Symmetry and Binocular Rivalry Experiments

Hugh Wilson has proposed a class of models that treat higher-level decision making as a competition between patterns coded as levels of a set of attributes in an appropriately defined network (Cortical Mechanisms of Vision, pp. 399–417, 2009; The Constitution of Visual Consciousness: Lessons from Binocular Rivalry, pp. 281–304, 2013). In this paper, we propose that symmetry-breaking Hopf bifurcation from fusion states in suitably modified Wilson networks, which we call rivalry networks, can be used in an algorithmic way to explain the surprising percepts that have been observed in a number of binocular rivalry experiments. These rivalry networks modify and extend Wilson networks by permitting different kinds of attributes and different types of coupling. We apply this algorithm to psychophysics experiments discussed by Kovács et al. (Proc. Natl. Acad. Sci. USA 93:15508–15511, 1996), Shevell and Hong (Vis. Neurosci. 23:561–566, 2006; Vis. Neurosci. 25:355–360, 2008), and Suzuki and Grabowecky (Neuron 36:143–157, 2002). We also analyze an experiment with four colored dots (a simplified version of a 24-dot experiment performed by Kovács), and a three-dot analog of the four-dot experiment. Our algorithm predicts surprising differences between the three- and four-dot experiments

Patterns of synchrony in coupled cell networks with multiple arrows

A coupled cell system is a network of dynamical systems, or “cells,” coupled together. The architecture of a coupled cell network is a graph that indicates how cells are coupled and which cells are equivalent. Stewart, Golubitsky, and Pivato presented a framework for coupled cell systems that permits a classification of robust synchrony in terms of network architecture. They also studied the existence of other robust dynamical patterns using a concept of quotient network. There are two difficulties with their approach. First, there are examples of networks with robust patterns of synchrony that are not included in their class of networks; and second, vector fields on the quotient do not in general lift to vector fields on the original network, thus complicating genericity arguments. We enlarge the class of coupled systems under consideration by allowing two cells to be coupled in more than one way, and we show that this approach resolves both difficulties. The theory that we develop, the “multiarrow formalism,” parallels that of Stewart, Golubitsky, and Pivato. In addition, we prove that the pattern of synchrony generated by a hyperbolic equilibrium is rigid (the pattern does not change under small admissible perturbations) if and only if the pattern corresponds to a balanced equivalence relation. Finally, we use quotient networks to discuss Hopf bifurcation in homogeneous cell systems with two-color balanced equivalence relations

Symmetry groupoids and patterns of synchrony in coupled cell networks

A coupled cell system is a network of dynamical systems, or “cells,” coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only mechanism that can create such states in a coupled cell system and show that it is not. The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information about the input sets of cells. (The input set of a cell consists of that cell and all cells connected to that cell.) The admissible vector fields for a given graph—the dynamical systems with the corresponding internal dynamics and couplings—are precisely those that are equivariant under the symmetry groupoid. A pattern of synchrony is “robust” if it arises for all admissible vector fields. The first main result shows that robust patterns of synchrony (invariance of “polydiagonal” subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an equivalence relation on cells is “balanced.” The second main result shows that admissible vector fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled cell network, the “quotient network.” The existence of quotient networks has surprising implications for synchronous dynamics in coupled cell systems