The canard unchainedor how fast/slow dynamical systems bifurcate

Diener, Marc

doi:10.1007/bf03024127

Cited by 146 publications

(128 citation statements)

References 4 publications

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“…The name 'canard' originates in the study of relaxation oscillations in the Van der Pol equation and was introduced by Diener and Diener (1981). It refers to the shape of transitional limit cycles that appear in an exponentially small neighbourhood of a certain critical parameter value (Diener 1984;Eckhaus 1983;Braaksma 1993).…”

Section: Discussionmentioning

confidence: 99%

“…Clear explanations, including an application to the Hodgkin-Huxley based FitzHugh-Nagumo model (FitzHugh 1960;Nagumo et al 1962) (see also Murray 1989), are provided in Wechselberger (2005). Earlier successful attempts to understand this behaviour used nonstandard analysis (Bénoît et al 1981;Diener 1984) or matched asymptotic expansions (Eckhaus 1983).…”

Section: Application Of Fenichel's First Theoremmentioning

confidence: 99%

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Geometric singular perturbation theory in biological practice

2009

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Geometric singular perturbation theory is a useful tool in the analysis of problems with a clear separation in time scales. It uses invariant manifolds in phase space in order to understand the global structure of the phase space or to construct orbits with desired properties. This paper explains and explores geometric singular perturbation theory and its use in (biological) practice. The three main theorems due to Fenichel are the fundamental tools in the analysis, so the strategy is to state these theorems and explain their significance and applications. The theory is illustrated by many examples. Mathematics Subject Classification (2000)34e15 · 34c30 · 34c60 · 92b09

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Section: Discussionmentioning

confidence: 99%

Section: Application Of Fenichel's First Theoremmentioning

confidence: 99%

Geometric singular perturbation theory in biological practice

2009

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“…Such orbits are reminiscent of canard type solutions [56]. The trajectory is irregular and is suggestive of spatiotemporal chaotic nature of the peel front.…”

Section: Fig 5(c)mentioning

confidence: 90%

Intermittent peel front dynamics and the crackling noise in an adhesive tape

Kumar

Ananthakrishna

2008

Phys. Rev. E

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We report a comprehensive investigation of a model for peeling of an adhesive tape along with a nonlinear time series analysis of experimental acoustic emission signals in an effort to understand the origin of intermittent peeling of an adhesive tape and its connection to acoustic emission. The model represents the acoustic energy dissipated in terms of Rayleigh dissipation functional that depends on the local strain rate. We show that the nature of the peel front exhibits rich spatiotemporal patterns ranging from smooth, rugged and stuck-peeled configurations that depend on three parameters, namely, the ratio of inertial time scale of the tape mass to that of the roller, the dissipation coefficient and the pull velocity. The stuck-peeled configurations are reminiscent of fibrillar peel front patterns observed in experiments. We show that while the intermittent peeling is controlled by the peel force function, the model acoustic energy dissipated depends on the nature of the peel front and its dynamical evolution. Even though the acoustic energy is a fully dynamical quantity, it can be quite noisy for a certain set of parameter values suggesting the deterministic origin of acoustic emission in experiments. To verify this suggestion, we have carried out a dynamical analysis of experimental acoustic emission time series for a wide range of traction velocities. Our analysis shows an unambiguous presence of chaotic dynamics within a subinterval of pull speeds within the intermittent regime. Time series analysis of the model acoustic energy signals is also found to be chaotic within a subinterval of pull speeds. Further, the model provides insight into several statistical and dynamical features of the experimental AE signals including the transition from burst type acoustic emission to continuous type with increasing pull velocity and the connection between acoustic emission and stick-slip dynamics. Finally, the model also offers an explanation for the recently observed feature that the duration of the slip phase can be less than that of the stick phase.

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“…The trajectories continue to remain in an O( ) neighborhood of the now repelling equilibrium curve for an O(1/ ) time, however, until the expansion governed by the now positive real parts of the eigenvalues of the linearization accumulates sufficiently to counter the earlier contraction. The time needed to escape from this curve can be calculated using a way-in way-out function (Diener, 1984;Neishtadt, 1987Neishtadt, , 1988. Interestingly, this delayed escape effect gives rise to elliptic bursting (Rinzel, 1987;Wang and Rinzel, 1995;Hoppensteadt and Izhikevich, 1997;Rubin and Terman, 2002) when the drift direction of the slow variable switches after escape and the periodic orbits born from the Hopf bifurcation have appropriate characteristics (Baer et al, 1989;Izhikevich, 2000;Kuske and Baer, 2002;Su et al, 2004).…”

Section: Low-frequency Synchronized Firing With Synaptic Excitationmentioning

confidence: 99%

“…The first finding that I will discuss is that synaptic excitation, particularly with a slow decay constant, can yield a drastic slowing of tonic spiking or burst firing in certain types of networks (Drover et al, 2005), typified by a network of classical Hodgkin-Huxley neurons (Hodgkin and Huxley, 1952). The mechanism underlying this slowing relates to canards, which are solutions to systems of differential equations that spend a prolonged time near structures that are unstable in the corresponding phase space (Diener, 1984;Szmolyan and Wechselberger, 2001). The second result that I will present is that a network of intrinsically silent cells, with distance-dependent excitatory synaptic coupling and no inhibitory coupling, can support sustained, localized activity in the absence of sustained external input (Rubin and Bose, 2004).…”

Section: Introductionmentioning

confidence: 99%

Surprising Effects of Synaptic Excitation

Rubin

2005

J Comput Neurosci

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Abstract. Typically, excitatory synaptic coupling is thought of as an influence that accelerates and propagates firing in neuronal networks. This paper reviews recent results explaining how, contrary to these expectations, the presence of excitatory synaptic coupling can drastically slow oscillations in a network and how localized, sustained activity can arise in a network with purely excitatory coupling, without sustained inputs. These two effects stem from interactions of excitatory coupling with two different forms of intrinsic neuronal dynamics, and both serve to highlight the fact that the influence of synaptic coupling in a network depends strongly on the intrinsic properties of cells in the network.

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The canard unchainedor how fast/slow dynamical systems bifurcate

Cited by 146 publications

References 4 publications

Geometric singular perturbation theory in biological practice

Geometric singular perturbation theory in biological practice

Intermittent peel front dynamics and the crackling noise in an adhesive tape

Surprising Effects of Synaptic Excitation

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