142 research outputs found
The Morse Lemma in Infinite Dimensions via Singularity Theory
Get PDFAn 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
Get PDFWe 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
Get PDFThe 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 âcoupled cell network,â 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 âpush-pull pairsâ 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
Get PDFHugh 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. 399â417, 2009; The Constitution of Visual Consciousness: Lessons from Binocular Rivalry, pp. 281â304, 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 KovĂĄcs et al. (Proc. Natl. Acad. Sci. USA 93:15508â15511, 1996), Shevell and Hong (Vis. Neurosci. 23:561â566, 2006; Vis. Neurosci. 25:355â360, 2008), and Suzuki and Grabowecky (Neuron 36:143â157, 2002). We also analyze an experiment with four colored dots (a simplified version of a 24-dot experiment performed by KovĂĄcs), 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
Get PDFA coupled cell system is a network of dynamical systems, or âcells,â 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 âmultiarrow formalism,â 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
Get PDFA coupled cell system is a network of dynamical systems, or âcells,â 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 graphâthe dynamical systems with
the corresponding internal dynamics and couplingsâare precisely those that are equivariant under
the symmetry groupoid. A pattern of synchrony is ârobustâ if it arises for all admissible vector
fields. The first main result shows that robust patterns of synchrony (invariance of âpolydiagonalâ
subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an
equivalence relation on cells is âbalanced.â 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 âquotient network.â The existence of quotient networks has surprising implications
for synchronous dynamics in coupled cell systems
Homeostasis in Input-Output Networks: Structure, Classification and Applications
Full text linkHomeostasis is concerned with regulatory mechanisms, present in biological
systems, where some specific variable is kept close to a set value as some
external disturbance affects the system. Mathematically, the notion of
homeostasis can be formalized in terms of an input-output function that maps
the parameter representing the external disturbance to the output variable that
must be kept within a fairly narrow range. This observation inspired the
introduction of the notion of infinitesimal homeostasis, namely, the derivative
of the input-output function is zero at an isolated point. This point of view
allows for the application of methods from singularity theory to characterize
infinitesimal homeostasis points (i.e. critical points of the input-output
function). In this paper we review the infinitesimal approach to the study of
homeostasis in input-output networks. An input-output network is a network with
two distinguished nodes `input' and `output', and the dynamics of the network
determines the corresponding input-output function of the system. This class of
dynamical systems provides an appropriate framework to study homeostasis and
several important biological systems can be formulated in this context.
Moreover, this approach, coupled to graph-theoretic ideas from combinatorial
matrix theory, provides a systematic way for classifying different types of
homeostasis (homeostatic mechanisms) in input-output networks, in terms of the
network topology. In turn, this leads to new mathematical concepts, such as,
homeostasis subnetworks, homeostasis patterns, homeostasis mode interaction. We
illustrate the usefulness of this theory with several biological examples:
biochemical networks, chemical reaction networks (CRN), gene regulatory
networks (GRN), Intracellular metal ion regulation and so on.Comment: 45 pages, 26 figures, submitted to the MBS special issue "Dynamical
Systems in Life Sciences
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