Propagation and Its Failure in Coupled Systems of Discrete Excitable Cells

Keener, James P.

doi:10.1137/0147038

Cited by 563 publications

(433 citation statements)

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“…(7) has an unstable unpolarised steady state L = 1 2 and a stable polarised steady state L * > 1 2 . Thus, we expect waves taking L j = 1 2 to L j = L * similar to the travelling wave solutions of Fisher's equation [9], but in a discrete system like the ones studied by Elmer and Van Vleck [5], Keener [6] and Owen [10]. Introduction of the travelling wave coordinate…”

Section: Because Of (11) the Homogeneous Unpolarised Steady Statementioning

confidence: 87%

“…The rest of the wild-type cells are unpolarised. This is due to the fact that for this choice of g and d the polarised state is no longer a steady state of system (6). In this case, the unpolarised state is the only stable steady state (see also Figure 15(b)).…”

Section: Analysis Of Clonesmentioning

confidence: 97%

“…We chose g(x) = 2x 6 1.5 6 +x 6 and d = 0. Two waves are initiated and the one propagating polarity to the right finally overcomes the one initiated The analysis reveals that whether or not a wrongly polarised cell can be corrected by a polarising wave depends on the initial distance between this anomaly and the wave front and the strength of the anomaly.…”

Section: Behaviour For Irregularities In the Initial Conditionsmentioning

confidence: 99%

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Modelling and Analysis of Planar Cell Polarity

et al. 2009

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Access from the University of Nottingham repository:http://eprints.nottingham.ac.uk/1002/1/ms_schamberg_PCP.pdf Copyright and reuse:The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions. This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence. For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdf A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Abstract Planar cell polarity (PCP) occurs in the epithelia of many animals and can lead to the alignment of hairs, bristles and feathers; physiologically, it can organise ciliary beating. Here we present two approaches to modelling this phenomenon. The aim is to discover the basic mechanisms that drive PCP, while keeping the models mathematically tractable. We present a feedback and diffusion model, in which adjacent cell sides of neighbouring cells are coupled by a negative feedback loop and diffusion acts within the cell. This approach can give rise to polarity, but also to period two patterns. Polarisation arises via an instability provided a sufficiently strong feedback and sufficiently weak diffusion. Moreover, we discuss a conservative model in which proteins within a cell are redistributed depending on the amount of proteins in the neighbouring cells, coupled with intracellular diffusion. In this case polarity can arise from weakly polarised initial conditions or via a wave provided the diffusion is weak enough. Both models can overcome small anomalies in the initial conditions. Furthermore, the range of the effects of groups of cells with different properties than the surrounding cells depends on the strength of the initial global cue and the intracellular diffusion.

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Section: Because Of (11) the Homogeneous Unpolarised Steady Statementioning

confidence: 87%

Section: Analysis Of Clonesmentioning

confidence: 97%

Section: Behaviour For Irregularities In the Initial Conditionsmentioning

confidence: 99%

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Modelling and Analysis of Planar Cell Polarity

et al. 2009

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“…A similar phenomenon has been analysed by Keitt et al (2000) in an ecological context for the prey-only model. It also arises as so-called 'propagation failure' in models for excitable systems in physiology (Keener, 1987(Keener, , 1993.…”

Section: Noninvasive Solutionsmentioning

confidence: 99%

How Predation can Slow, Stop or Reverse a Prey Invasion

Owen

Lewis

2001

Bulletin of Mathematical Biology

180

141

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Observations on Mount St Helens indicate that the spread of recolonizing lupin plants has been slowed due to the presence of insect herbivores and it is possible that the spread of lupins could be reversed in the future by intense insect herbivory [Fagan, W. F. and J. Bishop (2000). Trophic interactions during primary sucession: herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155, [238][239][240][241][242][243][244][245][246][247][248][249][250][251]. In this paper we investigate mechanisms by which herbivory can contain the spatial spread of recolonizing plants. Our approach is to analyse a series of predatorprey reaction-diffusion models and spatially coupled ordinary differential equation models to derive conditions under which predation pressure can slow, stall or reverse a spatial invasion of prey. We focus on models where prey disperse more slowly than predators. We comment on the types of functional response which give such solutions, and the circumstances under which the models are appropriate.

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“…They are also used to model the propagation of pulses in myelinated axons where the membrane is excitable only at spatially discrete sites. In this case, u i represents the potential at the i-th active site; see for example, [7], [8], [39], [36], [30,31]. Lattice differential equations can also be found in chemical reaction theory [24,28,33].…”

Section: Introductionmentioning

confidence: 99%