scholarly journals Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

2017 ◽  
Author(s):  
Randolph J. Leiser ◽  
Horacio G. Rotstein
Keyword(s):  
Piecewise Linear ◽  
Slow Variable ◽  
Rate Of Change ◽  
Fast Variable ◽  
Feedback Effects ◽  
Relaxation Oscillators ◽  
Mixed Mode Oscillations ◽  
Coupled Networks ◽  
Localized Patterns ◽  
Canard Phenomenon

AbstractRelaxation oscillators may exhibit small amplitude oscillations (SAOs) in addition to the typical large amplitude oscillations (LAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. We investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear (PWL) relaxation oscillators where global feedback acting on the rate of change of the activator (fast variable) involves the inhibitor (slow variable). We also investigate of these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry breaking global feedback effects.

CLOSED-LOOP CONTROL OF THE THALAMOCORTICAL RELAY NEURON'S PARKINSONIAN STATE BASED ON SLOW VARIABLE

2013 ◽  
Vol 23 (04) ◽  
pp. 1350017 ◽  
Author(s):  
CHEN LIU ◽  
JIANG WANG ◽  
YING-YUAN CHEN ◽  
BIN DENG ◽  
XI-LE WEI ◽  
...  
Keyword(s):  
Feedback Control ◽  
Control Strategy ◽  
Closed Loop ◽  
Slow Variable ◽  
Closed Loop Control ◽  
Fast Variable ◽  
Qualitative And Quantitative Analysis ◽  
Feedback Controllers ◽  
Loop Control ◽  
Qualitative And Quantitative

A novel closed-loop control strategy is proposed to control Parkinsonian state based on a computational model. By modeling thalamocortical relay neurons under external electric field, a slow variable feedback control is applied to restore its relay functionality. Qualitative and quantitative analysis demonstrates the performance of feedback controller based on slow variable is more efficient compared with traditional feedback control based on fast variable. These findings point to the potential value of model-based design of feedback controllers for Parkinson's disease.

A Surface of Heteroclinic Connections Between Two Saddle Slow Manifolds in the Olsen Model

2020 ◽  
Vol 30 (16) ◽  
pp. 2030048
Author(s):  
Elle Musoke ◽  
Bernd Krauskopf ◽  
Hinke M. Osinga
Keyword(s):  
Periodic Orbits ◽  
Slow Variable ◽  
Connecting Orbits ◽  
Stable And Unstable Manifolds ◽  
Small Oscillations ◽  
One Dimensional ◽  
Mixed Mode Oscillations ◽  
Slow Manifolds ◽  
Connecting Orbit ◽  
Crucial Ingredient

The Olsen model for the biochemical peroxidase-oxidase reaction has a parameter regime where one of its four variables evolves much slower than the other three. It is characterized by the existence of periodic orbits along which a large oscillation is followed by many much smaller oscillations before the process repeats. We are concerned here with a crucial ingredient for such mixed-mode oscillations (MMOs) in the Olsen model: a surface of connecting orbits that is followed closely by the MMO periodic orbit during its global, large-amplitude transition back to another onset of small oscillations. Importantly, orbits on this surface connect two one-dimensional saddle slow manifolds, which exist near curves of equilibria of the limit where the slow variable is frozen and acts as a parameter of the so-called fast subsystem. We present a numerical method, based on formulating suitable boundary value problems, to compute such a surface of connecting orbits. It involves a number of steps to compute the slow manifolds, certain submanifolds of their stable and unstable manifolds and, finally, a first connecting orbit that is then used to sweep out the surface by continuation. If it exists, such a surface of connecting orbits between two one-dimensional saddle slow manifolds is robust under parameter variations. We compute and visualize it in the Olsen model and show how this surface organizes the global return mechanism of MMO periodic orbits from the end of small oscillations back to a region of phase space where they start again.

Synchronization in Relaxation Oscillator Networks with Conduction Delays

2001 ◽  
Vol 13 (5) ◽  
pp. 1003-1021 ◽  
Author(s):  
Jeffrey J. Fox ◽  
Ciriyam Jayaprakash ◽  
DeLiang Wang ◽  
Shannon R. Campbell
Keyword(s):  
Numerical Simulations ◽  
Coupled Oscillators ◽  
Initial Conditions ◽  
Relaxation Oscillator ◽  
Phase Lag ◽  
Relaxation Oscillators ◽  
Oscillator Networks ◽  
Coupled Networks ◽  
Parameter Values ◽  
Zero Phase

We study locally coupled networks of relaxation oscillators with excitatory connections and conduction delays and propose a mechanism for achieving zero phase-lag synchrony. Our mechanism is based on the observation that different rates of motion along different nullclines of the system can lead to synchrony in the presence of conduction delays. We analyze the system of two coupled oscillators and derive phase compression rates. This analysis indicates how to choose nullclines for individual relaxation oscillators in order to induce rapid synchrony. The numerical simulations demonstrate that our analytical results extend to locally coupled networks with conduction delays and that these networks can attain rapid synchrony with appropriately chosen nullclines and initial conditions. The robustness of the proposed mechanism is verified with respect to different nullclines, variations in parameter values, and initial conditions.

scholarly journals Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems

SIAM Review ◽  
2016 ◽  
Vol 58 (4) ◽  
pp. 653-691 ◽  
Author(s):  
Mathieu Desroches ◽  
Antoni Guillamon ◽  
Enrique Ponce ◽  
Rafael Prohens ◽  
Serafim Rodrigues ◽  
...  
Keyword(s):  
Mixed Mode ◽  
Piecewise Linear ◽  
Mixed Mode Oscillations

Rule Dynamics of Electronic Turbulence in Semiconductors Caused by Impact Ionization Avalanche

1997 ◽  
Vol 07 (05) ◽  
pp. 1059-1064 ◽  
Author(s):  
Kazunori Aoki
Keyword(s):  
Impact Ionization ◽  
Period Doubling ◽  
Transverse Field ◽  
Slow Variable ◽  
Dissipative Structures ◽  
Fast Variable ◽  
Diffusion Term ◽  
Periodically Driven ◽  
Rule Dynamics ◽  
Route To Chaos

Complex dissipative structures and turbulence of filamentary currents have been studied numerically in a model of one-level impact ionization avalanche in semiconductors. The model includes the important parts of (1) transverse diffusion term of impact-ionized carriers in the equation of the generation-recombination process and (2) drift/diffusion currents in the dielectric relaxation of the transverse field. Under a periodically-driven regime using a dc + ac field E0 + Eac sin (2πf0t) the spatiotemporal patterns of the electron density (fast variable) and subsequently the current density exhibit period doubling bifurcation route to chaos as a function of E0, which is ruled by the pattern of the transverse field (slow variable).

scholarly journals PID Controller Singularly Perturbing Impulsive Differential Equations and Optimal Control Problem

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Wichai Witayakiattilerd
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Approximation Method ◽  
Pid Controller ◽  
Impulsive Differential Equations ◽  
Integral Form ◽  
Impulsive System ◽  
Slow Variable ◽  
Fast Variable

We study singular perturbation of impulsive system with a proportional-integral-derivative controller (PID controller) and solve an optimal control problem. The perturbation system comprises two important variables, a fast variable and a slow variable. Because of the complexity of the system, it is difficult to find its exact solution. This paper presents an approximation method for solving it. The aim of the approximation method is to reduce the complexity of the system by eliminating the fast variable. The solution of the method is expressed in an integral form, and it is called an approximated mild solution of the perturbed system. An example is provided to illustrate our result.

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