scholarly journals Tipping Cascades in a Multi-patch System with Noise and Spatial Coupling

2021 ◽  
Vol 83 (11) ◽  
Author(s):  
Abhishek Mallela ◽  
Alan Hastings
Keyword(s):  
Environmental Stochasticity ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Tipping Points ◽  
Isolated Populations ◽  
Population Extinction ◽  
Spatial Coupling ◽  
Area Of Interest ◽  
Spatially Extended ◽  
Analytic Expressions

AbstractForecasting tipping points in spatially extended systems is a key area of interest to ecologists. A slowly declining spatially distributed population is an important example of an ecological system that could exhibit a cascade of tipping points. Here, we develop a spatial two-patch model with environmental stochasticity that is slowly forced through population collapse, in the presence of changing environmental conditions. We begin with a basic spatial model, then introduce a fast–slow version of the model using geometric singular perturbation theory, followed by the inclusion of stochasticity. Using the spectral density of the fluctuating subpopulation in each patch, we derive analytic expressions for candidate indicators of population extinction and evaluate their performance through a simulation study. We find that coupling and spatial heterogeneity decrease the magnitude of the proposed indicators in coupled populations relative to isolated populations. Moreover, the degree of coupling dictates the trends in summary statistics. We conclude that this theory may be applied to other contexts, including the control of invasive species.

Periodic solutions for equations with distributed delays

2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
General Theorem ◽  
Linear Chain ◽  
Distributed Delays ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Existence Of Periodic Solutions

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

scholarly journals Canard solutions in neural mass models: consequences on critical regimes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Elif Köksal Ersöz ◽  
Fabrice Wendling
Keyword(s):  
Temporal Dynamics ◽  
Spatial Scales ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Critical Transitions ◽  
Intracerebral Eeg ◽  
Neural Mass ◽  
Mesoscopic Level ◽  
Eeg Recordings ◽  
Physiological Background

AbstractMathematical models at multiple temporal and spatial scales can unveil the fundamental mechanisms of critical transitions in brain activities. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neuronal subpopulations without explicitly representing the underlying cellular activity. The mesoscopic level offered by the neural mass formulation has been used to model electroencephalographic (EEG) recordings and to investigate various cerebral mechanisms, such as the generation of physiological and pathological brain activities. In this work, we consider a NMM widely accepted in the context of epilepsy, which includes four interacting neuronal subpopulations with different synaptic kinetics. Due to the resulting three-time-scale structure, the model yields complex oscillations of relaxation and bursting types. By applying the principles of geometric singular perturbation theory, we unveil the existence of the canard solutions and detail how they organize the complex oscillations and excitability properties of the model. In particular, we show that boundaries between pathological epileptic discharges and physiological background activity are determined by the canard solutions. Finally we report the existence of canard-mediated small-amplitude frequency-specific oscillations in simulated local field potentials for decreased inhibition conditions. Interestingly, such oscillations are actually observed in intracerebral EEG signals recorded in epileptic patients during pre-ictal periods, close to seizure onsets.

Firing patterns of the CA1 pyramidal neuron with geometric singular perturbation: a model study

2020 ◽  
Vol 34 (32) ◽  
pp. 2050316
Author(s):  
Yaru Liu ◽  
Shenquan Liu
Keyword(s):  
Singular Perturbation ◽  
Pyramidal Neurons ◽  
Sodium Current ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
External Stimulation ◽  
Model Study ◽  
Geometric Singular Perturbation ◽  
Current Activation ◽  
Ca1 Pyramidal Neuron

An investigation of CA1 pyramidal model is an important issue for applications, which is intimately related to the composition of ions in the extracellular environment and external stimulation. In this paper, it is demonstrated that the effects of different electrophysiological parameters such as muscarinic-sensitive potassium current activation variable and sustained sodium current inactivation variable on the firing sequence of model by numerical simulations. Furthermore, the paper also discusses that the temperature affects the firing of the CA1 model from direct current (DC) and alternating current (AC) stimuli. It is found that the model exhibits excellent spiking and bursting patterns, even chaotic patterns occur. Meanwhile, generalized mixed oscillations emerge in the model. Additionally, the firing modes are depicted by providing the response curve (RC), inter-spike interval curve (ISI), phase diagram curve (PDC) and the number of spikes per burst curve (NC). Mathematically, the paper elaborates the results which are presented to obtain two lower dimensional subsystems, which govern the fast and slow dynamics for giving insight into the dynamic behaviors of the full 5D system based on the geometric singular perturbation theory (GSPT). Particularly, we analyse the phase diagrams of the CA1 model to understand the properties better. The present results may contribute to further understand the information processing of the CA1 pyramidal neurons.

Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems

2020 ◽  
pp. 2150003
Author(s):  
Ali Atabaigi
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Limit Cycles ◽  
Heteroclinic Orbits ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Generalist Predator ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Geometric Singular Perturbation

This paper studies the dynamics of the generalist predator–prey systems modeled in [E. Alexandra, F. Lutscher and G. Seo, Bistability and limit cycles in generalist predator–prey dynamics, Ecol. Complex. 14 (2013) 48–55]. When prey reproduces much faster than predator, by combining the normal form theory of slow-fast systems, the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan [Relaxation oscillation and canard explosion, J. Differential Equations 174(2) (2001) 312–368; Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal. 33(2) (2001) 286–314], we provide a detailed mathematical analysis to show the existence of homoclinic orbits, heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles. Moreover, on global stability of the unique positive equilibrium, we provide some new results. Numerical simulations are also carried out to support the theoretical results.

scholarly journals Mode coupling instability mitigation in friction systems by means of nonlinear energy sinks: Numerical highlighting and local stability analysis

2017 ◽  
Vol 24 (15) ◽  
pp. 3487-3511 ◽  
Author(s):  
Baptiste Bergeot ◽  
Sébastien Berger ◽  
Sergio Bellizzi
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Mode Coupling ◽  
Passive Control ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Complete Suppression ◽  
Slow Flow ◽  
Mode Coupling Instability ◽  
Nonlinear Energy

In this paper, we study the problem of passive control of friction-induced vibrations due to mode coupling instability in braking systems. To achieve that, the well-known two degrees of freedom Hultén’s model, which reproduces the typical dynamic behavior of friction systems, is coupled to two ungrounded nonlinear energy sinks (NES). The NES involves an essential cubic restoring force and a linear damping force. First, using numerical simulations it is shown that the suppression or the mitigation of the instability is possible and four steady-state responses are highlighted: complete suppression, mitigation through periodic response, mitigation through strongly modulated response, and no suppression of the mode coupling instability. Then the system is analyzed applying a complexification-averaging method and the resulting slow-flow is finally analyzed using geometric singular perturbation theory. This analysis allows us to explain the observed steady-state response regimes and predict some of them. The boundary values of the friction coefficient for some of the transitions between these regimes are predicted. However, the appearance of a three-dimensional super-slow flow subsystem highlights the limitation of the local linear stability analysis of the slow-flow to predict all these boundaries.

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