Stochastic Dynamic Bifurcations and Excitability

Berglund, Nils; Gentz, Barbara

doi:10.1093/acprof:oso/9780199235070.003.0003

Cited by 10 publications

(11 citation statements)

References 36 publications

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“…As a second step we introduce stochasticity into the dynamics. Our approach is closest to the work by Berglund and Gentz [19,14]; they demonstrated the applicability of stochastic multiscale differential equations in a variety of contexts such as climate modeling and neuroscience. Here we point out what their approach implies for critical transitions.…”

Section: Introductionmentioning

confidence: 84%

A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics

Kuehn

2011

Physica D: Nonlinear Phenomena

316

320

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Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find earlywarning signs. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Based on this elementary classification, we analyze stochastic fluctuations and calculate scaling laws of the variance of stochastic sample paths near critical transitions for fast subsystem bifurcations up to codimension two. The theory is applied to several models: the Stommel-Cessi box model for the thermohaline circulation from geoscience, an epidemic-spreading model on an adaptive network, an activator-inhibitor switch from systems biology, a predator-prey system from ecology and to the Euler buckling problem from classical mechanics. For the Stommel-Cessi model we compare different detrending techniques to calculate early-warning signs. In the epidemics model we show that link densities could be better variables for prediction than population densities. The activator-inhibitor switch demonstrates effects in three time-scale systems and points out that excitable cells and molecular units have information for subthreshold prediction. In the predator-prey model explosive population growth near a codimension two bifurcation is investigated and we show that early-warnings from normal forms can be misleading in this context. In the biomechanical model we demonstrate that early-warning signs for buckling depend crucially on the control strategy near the instability which illustrates the effect of multiplicative noise.

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Section: Introductionmentioning

confidence: 84%

A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics

Kuehn

2011

Physica D: Nonlinear Phenomena

316

320

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“…Borowski and Kuske [146] consider a similar stochastic slow-fast equation of FitzHugh-Nagumo type and find MMOs due to noise as well; see also [147]. Closely related is the work by Berglund and Gentz [24,25], who study spike generation in slow-fast neural models with noise in the framework of SDEs. The common ingredient in these examples is excitability: while small noise only leads to small irregular oscillations, a sufficiently large noise perturbation can kick the system beyond a threshold that results in a large excursion.…”

Section: Mmos Beyondmentioning

confidence: 97%

Mixed-Mode Oscillations with Multiple Time Scales

Desroches¹,

Guckenheimer

Krauskopf³

et al. 2012

SIAM Rev.

500

635

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Abstract. Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale "mechanism." Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

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“…Observe that we can view V s also as a function of y, and write V = V (y), since the mapping between y s and s is bijective when restricting to C a− 0 . In the relaxation oscillation regime (see Figure 1(a1)-(a2)) and if (ǫ, σ) are sufficiently small it can be shown [40,6] that…”

Section: Single Neuronsmentioning

confidence: 99%

“…The data in Figure 4 have been generated using a simple model for a Hopf critical transition [39,40]: where ξ j,τ are independent white noise processes that satisfy ξ j,τ 1 ξ j,τ 2 = δ(τ 1 − τ 2 ) for j = 1, 2. The model (6) was first analyzed in the context of delayed Hopf bifurcation [61,62]. Observe that the deterministic part of the fast variables (x 1 , x 2 ) is the normal form of a (subcritical) Hopf bifurcation [42].…”

Section: Local Data and Clustersmentioning

confidence: 99%

Scaling Effects and Spatio-Temporal Multilevel Dynamics in Epileptic Seizures

Meisel

Kuehn

2012

PLoS ONE

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Epileptic seizures are one of the most well-known dysfunctions of the nervous system. During a seizure, a highly synchronized behavior of neural activity is observed that can cause symptoms ranging from mild sensual malfunctions to the complete loss of body control. In this paper, we aim to contribute towards a better understanding of the dynamical systems phenomena that cause seizures. Based on data analysis and modelling, seizure dynamics can be identified to possess multiple spatial scales and on each spatial scale also multiple time scales. At each scale, we reach several novel insights. On the smallest spatial scale we consider single model neurons and investigate early-warning signs of spiking. This introduces the theory of critical transitions to excitable systems. For clusters of neurons (or neuronal regions) we use patient data and find oscillatory behavior and new scaling laws near the seizure onset. These scalings lead to substantiate the conjecture obtained from mean-field models that a Hopf bifurcation could be involved near seizure onset. On the largest spatial scale we introduce a measure based on phase-locking intervals and wavelets into seizure modelling. It is used to resolve synchronization between different regions in the brain and identifies time-shifted scaling laws at different wavelet scales. We also compare our wavelet-based multiscale approach with maximum linear cross-correlation and mean-phase coherence measures.

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Stochastic Dynamic Bifurcations and Excitability

Cited by 10 publications

References 36 publications

A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics

A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics

Mixed-Mode Oscillations with Multiple Time Scales

Scaling Effects and Spatio-Temporal Multilevel Dynamics in Epileptic Seizures

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