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ARTICLE Communicated by Carmen Canavier The Shape of PhaseResetting Curves in Oscillators with a Saddle Node on an Invariant Circle Bifurcation
"... We introduce a simple twodimensional model that extends the Poincaré oscillator so that the attracting limit cycle undergoes a saddle node bifurcation on an invariant circle (SNIC) for certain parameter values.Arbitrarily close to this bifurcation, the phaseresetting curve (PRC) continuously de ..."
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We introduce a simple twodimensional model that extends the Poincaré oscillator so that the attracting limit cycle undergoes a saddle node bifurcation on an invariant circle (SNIC) for certain parameter values.Arbitrarily close to this bifurcation, the phaseresetting curve (PRC) continuously
POSTER PRESENTATION Open Access Why are all phase resetting curves bimodal?
"... Neurons are excitable cells, which means that they operate close to a bifurcation points that allows them to switch back and forth between stable steady states and periodic attractors [1]. Repetitive firing neurons belong to either Class 1, i.e., characterized by a continuous frequency versus bias ..."
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bias current (fI) curve and arbitrarily low frequency of oscillation, or Class 2, i.e., discontinuous fI curve and firing starts with a relatively high frequency at threshold stimulation [2]. The bifurcation mechanism leading to repetitive firing is related to excitability classes, i.e., saddle node
Synchronization of firing in cortical fastspiking interneurons at gamma frequencies: a phaseresetting analysis
 PLoS Comput. Biol
, 2010
"... Fastspiking (FS) cells in the neocortex are interconnected both by inhibitory chemical synapses and by electrical synapses, or gapjunctions. Synchronized firing of FS neurons is important in the generation of gamma oscillations, at frequencies between 30 and 80 Hz. To understand how these synaptic ..."
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these synaptic interactions control synchronization, artificial synaptic conductances were injected in FS cells, and the synaptic phaseresetting function (SPRF), describing how the compound synaptic input perturbs the phase of gammafrequency spiking as a function of the phase at which it is applied
INTERACTION OF TWO SYSTEMS WITH SADDLENODE BIFURCATIONS ON INVARIANT CIRCLES I. FOUNDATIONS AND THE MUTUALISTIC CASE.
"... Abstract. The saddlenode bifurcation on an invariant circle (SNIC) is one of the codimensionone routes to creation or destruction of a periodic orbit in a continuoustime dynamical system. It governs the transition from resting behaviour to periodic spiking in many class I neurons, for example. He ..."
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Abstract. The saddlenode bifurcation on an invariant circle (SNIC) is one of the codimensionone routes to creation or destruction of a periodic orbit in a continuoustime dynamical system. It governs the transition from resting behaviour to periodic spiking in many class I neurons, for example
Bifurcations of Relaxation Oscillations near Folded Saddles ∗
, 2004
"... Relaxation oscillations are periodic orbits of multiple time scale dynamical systems that contain both slow and fast segments. The slowfast decomposition of these orbits is defined in the singular limit. Geometric methods in singular perturbation theory classify degeneracies of these decompositions ..."
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of these decompositions that occur in generic one parameter families of relaxation oscillations. This paper investigates the bifurcations that are associated with one type of degeneracy that occurs in systems with two slow variables, namely orbits that become homoclinic to a folded saddle. 1
Nonautonomous saddlenode bifurcations: random and deterministic forcing
 J. Differ. Equations
"... ar ..."
POSTER PRESENTATION Open Access Are phase resetting curves tunable?
"... Neurons are excitable cells capable of sustaining high amplitude membrane potential oscillations called action potential (AP). There are two classes of repetitively firing neurons: Class 1 characterized by a continuously and arbitrarily low tunable firing frequency, or Class 2 characterized by a mi ..."
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phenomenological understanding of neural activity suffices. In such cases, the phase response curve (PRC) rather than a detailed biophysical model of the neuron provides accurate and quick results. A PRC tabulates the transient changes in
Quadratic VolumePreserving Maps: Invariant Circles and Bifurcations
, 2008
"... We study the dynamics of the fiveparameter quadratic family of volumepreserving diffeomorphisms of R3. This family is the unfolded normal form for a bifurcation of a fixed point with a tripleone multiplier and also is the general form of a quadratic threedimensional map with a quadratic inverse. ..."
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, at least in its normal form, an elliptic invariant circle. We develop a simple algorithm to accurately compute these elliptic invariant circles and their longitudinal and transverse rotation numbers and use it to study their bifurcations, classifying them by the resonances between the rotation numbers
Results 1  10
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28,291