Canards of Folded Saddle-Node Type I

2015 ◽  
Vol 47 (4) ◽  
pp. 3235-3283 ◽  
Author(s):  
Theodore Vo ◽  
Martin Wechselberger
Keyword(s):  
Type I ◽  
Saddle Node ◽  
Node Type

Time-Averaged Quantity of a Low-Temperature Semiconductor Experiment Reflects Scaling Behavior of Saddle-Node Bifurcation to Chaos

1994 ◽  
Vol 49 (9) ◽  
pp. 838-842 ◽  
Author(s):  
R. Richter ◽  
A. Kittel ◽  
J. Paris
Keyword(s):  
Magnetic Field ◽  
Low Temperature ◽  
Impact Ionization ◽  
Scaling Behavior ◽  
Type I ◽  
Saddle Node Bifurcation ◽  
Time Resolved ◽  
Saddle Node ◽  
Temperature Impact ◽  
P Type

Abstract Low-temperature impact ionization breakdown in p-type germanium crystals gives rise to spontaneous oscillations of the current flow. We demonstrate experimental evidence of a particularly high-conducting dynamical state that is limited to a finite parameter regime of the current versus magnetic field characteristic. After bifurcation from a coexisting nonoscillatory state to periodicity, one observes a type-I intermittent transition to chaos and, eventually, a jump back to the nonoscil­latory branch upon increasing the magnetic field control parameter. The scaling behavior of the underlying saddle-node bifurcation, already found in time-resolved measurements, also becomes visible in a square-root dependence of the time-averaged current developing both prior to and after the critical point. Our result might be of interest for time-averaged information is accessible.

scholarly journals Periodic Orbits as a Probe to Reveal Exotic States in Vibrationally Excited Molecules: The Saddle-Node States

1993 ◽  
Vol 13 (2) ◽  
pp. 87-99 ◽  
Author(s):  
Stavros C. Farantos
Keyword(s):  
Periodic Orbits ◽  
Vibrationally Excited ◽  
Simultaneous Generation ◽  
Potential Barriers ◽  
Saddle Node ◽  
Families Of Periodic Orbits ◽  
Molecular Potentials ◽  
Localized Quantum States ◽  
Theoretical Results ◽  
Node Type

We present theoretical results which show the existence of isomerizing localized quantum states above potential barriers for the excited 1B2 states of SO2 and O3. These states are assigned by periodic orbits which emerge from saddle-node bifurcations, the characteristic of which is the simultaneous generation of two families of periodic orbits one stable and one unstable. Similar isomerizing states and bifurcations have been found for other molecules, and this leads to the conclusion, that the appearance of saddle-node type states may be a generic phenomenon for molecular potentials with barriers.

scholarly journals Analytic Expressions for Rate and CV of a Type I Neuron Driven by White Gaussian Noise

2003 ◽  
Vol 15 (8) ◽  
pp. 1761-1788 ◽  
Author(s):  
Benjamin Lindner ◽  
André Longtin ◽  
Adi Bulsara
Keyword(s):  
White Noise ◽  
Neuron Model ◽  
First Passage Time ◽  
Input Parameter ◽  
Gaussian White Noise ◽  
Quadrature Formulas ◽  
Type I ◽  
Saddle Node ◽  
Type I Neuron ◽  
Firing Statistics

We study the one-dimensional normal form of a saddle-node system under the influence of additive gaussian white noise and a static “bias current” input parameter, a model that can be looked upon as the simplest version of a type I neuron with stochastic input. This is in contrast with the numerous studies devoted to the noise-driven leaky integrate-and-fire neuron. We focus on the firing rate and coefficient of variation (CV) of the interspike interval density, for which scaling relations with respect to the input parameter and noise intensity are derived. Quadrature formulas for rate and CV are numerically evaluated and compared to numerical simulations of the system and to various approximation formulas obtained in different limiting cases of the model. We also show that caution must be used to extend these results to the neuron model with multiplicative gaussian white noise. The correspondence between the first passage time statistics for the saddle-node model and the neuron model is obtained only in the Stratonovich interpretation of the stochastic neuron model, while previous results have focused only on the Ito interpretation. The correct Stratonovich interpretation yields CVs that are still relatively high, although smaller than in the Ito interpretation; it also produces certain qualitative differences, especially at larger noise intensities. Our analysis provides useful relations for assessing the distance to threshold and the level of synaptic noise in real type I neurons from their firing statistics. We also briefly discuss the effect of finite boundaries (finite values of threshold and reset) on the firing statistics.

scholarly journals Multiple and Complex Codimension-2 Bifurcations underlying Post-inhibitory Rebound Spike and Excitability Transition

2021 ◽  
Author(s):  
Xianjun Wang ◽  
Huaguang Gu ◽  
Yuye Li ◽  
Bo Lu
Keyword(s):  
Hopf Bifurcation ◽  
Homoclinic Orbit ◽  
Neuronal Model ◽  
Type I ◽  
Type Ii ◽  
Neural Firing ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Snic Bifurcation ◽  
Degenerate Bifurcation

Abstract Neuron exhibits nonlinear dynamics such as excitability transition and post-inhibitory rebound (PIR) spike related to bifurcations, which are associated with information processing, locomotor modulation, or brain disease. PIR spike is evoked by inhibitory stimulation instead of excitatory stimulation, which presents a challenge to the threshold concept. In the present paper, 7 codimension-2 or degenerate bifurcations related to 10 codimension-1 bifurcations are acquired in a neuronal model, which presents the bifurcations underlying the excitability transition and PIR spike. Type I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC) bifurcation, and type II excitability to saddle-node (SN) bifurcation or sub-critical Hopf (SubH) bifurcation or sup-critical Hopf (SupH) bifurcation. The excitability transition from type I to II corresponds to the codimension-2 bifurcation, Saddle-Node Homoclinic orbit (SNHO) bifurcation, via which SNIC bifurcation terminates and meanwhile big homoclinic orbit (BHom) bifurcation and SN bifurcation emerge. A degenerate bifurcation via which BHom bifurcation terminates and fold limit cycle (LPC) bifurcation emerges is responsible for spiking transition from type I to II, and the roles of other codimension-2 bifurcations (Cusp, Bogdanov-Takens, and Bautin) are discussed. In addition, different from the widely accepted viewpoint that PIR spike is mainly evoked near Hopf bifurcation rather than SNIC bifurcation, PIR spike is identified to be induced near SNIC or BHom or LPC bifurcations, and threshold curves resemble that of Hopf bifurcation. The complex bifurcations present comprehensive and deep understandings of excitability transition and PIR spike, which are helpful for the modulation to neural firing activities and physiological functions.

A NONAUTONOMOUS SADDLE-NODE BIFURCATION PATTERN

2004 ◽  
Vol 04 (03) ◽  
pp. 335-350 ◽  
Author(s):  
ROBERTA FABBRI ◽  
RUSSELL JOHNSON ◽  
FRANCESCA MANTELLINI
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Modern Theory ◽  
Saddle Node Bifurcation ◽  
Averaging Technique ◽  
Saddle Node ◽  
Nonautonomous Differential Equations ◽  
Node Type

In this paper we study certain differential equations depending on a small parameter ε which exhibit a bifurcation of saddle-node type as ε passes through zero. We use a classical averaging technique together with methods and results from the modern theory of nonautonomous differential equations.

Heat-Etching with a Bullivant Type II Simple Freeze-Cleave Device

1970 ◽  
Vol 28 ◽  
pp. 106-107 ◽  
Author(s):  
Ronald S. Weinstein ◽  
N. Scott McNutt
Keyword(s):  
New Zealand ◽  
General Hospital ◽  
Massachusetts General Hospital ◽  
Type I ◽  
Type Ii ◽  
Collaborative Effort ◽  
Temperature And Pressure ◽  
The University

The Type I simple cold block device was described by Bullivant and Ames in 1966 and represented the product of the first successful effort to simplify the equipment required to do sophisticated freeze-cleave techniques. Bullivant, Weinstein and Someda described the Type II device which is a modification of the Type I device and was developed as a collaborative effort at the Massachusetts General Hospital and the University of Auckland, New Zealand. The modifications reduced specimen contamination and provided controlled specimen warming for heat-etching of fracture faces. We have now tested the Mass. General Hospital version of the Type II device (called the “Type II-MGH device”) on a wide variety of biological specimens and have established temperature and pressure curves for routine heat-etching with the device.

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