Within-burst synchrony changes for coupled elliptic bursters
Authors: Abul kalam al Azad, Peter Ashwin
Abstract
We study the appearance of a novel phenomenon for linearly coupled identical bursters: synchronized bursts where there are changes of spike synchrony within each burst. The examples we study are for normal form elliptic bursters where there is a periodic slow passage through a Bautin (codimension two degenerate Andronov-Hopf) bifurcation. This burster has a subcritical Andronov-Hopf bifurcation at the onset of repetitive spiking while end of burst occurs via a fold limit cycle bifurcation. We study synchronization behavior of two and three Bautin-type elliptic bursters for a linear direct coupling scheme. Burst synchronization is known to be prevalent behavior among such coupled bursters, while spike synchronization is more dependent on the details of the coupling. We note that higher order terms in the normal form that do not affect the behavior of a single burster can be responsible for changes in synchrony pattern; more precisely, we find within-burst synchrony changes associated with a turning point in the spiking frequency.
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Papers around this one
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Closest papers
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Synchronization of Elliptic BurstersE. Izhikevich · 2001 · SIAM Review · cited 28 times · Not summarized yet
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Shared Inputs, Entrainment, and Desynchrony in Elliptic Bursters: From Slow Passage to Discontinuous Circle MapsGuillaume Lajoie, Eric T. Shea-Brown · 2010 · SIAM Journal on Applied Dynamical Systems · cited 12 times · Not summarized yet
Abstract
What input signals will lead to synchrony vs. desynchrony in a group of biological oscillators? This question connects with both classical dynamical systems analyses of entrainment and phase locking and with emerging studies of stimulation patterns for controlling neural network activity. Here, we focus on the response of a population of uncoupled, elliptically bursting neurons to a common pulsatile input. We extend a phase reduction from the literature to capture inputs of varied strength, leading to a circle map with discontinuities of various orders. In a combined analytical and numerical approach, we apply our results to both a normal form model for elliptic bursting and to a biophysical…
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Bursting oscillations, bifurcation and synchronization in neuronal systemsHaixia Wang, Qingyun Wang, Qi-Shao Lu · 2011 · Chaos Solitons & Fractals · cited 63 times · Not summarized yet
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Burst and spike synchronization of coupled neural oscillatorsJ. Schwarz, G. Dangelmayr, A. Stevens et al. · 2001 · Dynamical Systems · cited 6 times · Not summarized yet
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Bursting and Synchrony in Networks of Model NeuronsChristian Geier, Alexander Rothkegel, Christian E. Elger et al. · 2016 · R. Tetzlaff and C. E. Elger and K. Lehnertz (2013), Recent Advances in Predicting and Preventing Epileptic Seizures, pag · Not summarized yet
Abstract
Bursting neurons are considered to be a potential cause of over-excitability and seizure susceptibility. The functional influence of these neurons in extended epileptic networks is still poorly understood. There is mounting evidence that the dynamics of neuronal networks is influenced not only by neuronal and synaptic properties but also by network topology. We investigate numerically the influence of different neuron dynamics on global synchrony in neuronal networks with complex connection topologies.
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Canonical models for torus canards in elliptic bursters.Emre Baspinar, Daniele Avitabile, Mathieu Desroches · 2020 · Chaos · cited 6 times · Not summarized yet
Abstract
We revisit elliptic bursting dynamics from the viewpoint of torus canard solutions. We show that at the transition to and from elliptic burstings, classical or mixed-type torus canards may appear, the difference between the two being the fast subsystem bifurcation that they approach: saddle-node of cycles for the former and subcritical Hopf for the latter. We first showcase such dynamics in a Wilson-Cowan-type elliptic bursting model, then we consider minimal models for elliptic bursters in view of finding transitions to and from bursting solutions via both kinds of torus canards. We first consider the canonical model proposed by Izhikevich [SIAM J. Appl. Math. 60, 503-535 (2000)] and adapte…
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Subcritical Elliptic Bursting of Bautin TypeE. Izhikevich · 1999 · SIAM Journal on Applied Mathematics · cited 69 times · Not summarized yet
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Effects of noise on elliptic burstersJian-Zhong Su, J. Rubin, D. Terman · 2004 · Nonlinearity · cited 62 times · Not summarized yet
Prior work it builds on
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Neural excitability, Spiking and burstingE. Izhikevich · 2000 · International Journal of Bifurcation and Chaos in Applied Sciences and Engineering · cited 1,931 times · Not summarized yet
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The slow passage through a Hopf bifurcation: delay, memory effects, and resonanceS. Baer, T. Erneux, J. Rinzel · 1989 · Siam Journal on Applied Mathematics · cited 341 times · Not summarized yet
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Dissection of a model for neuronal parabolic burstingJ. Rinzel, Young-Seek Lee · 1987 · Journal of Mathematical Biology · cited 262 times · Not summarized yet
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A Formal Classification of Bursting Mechanisms in Excitable SystemsJ. Rinzel · 1987 · cited 507 times · Not summarized yet
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Bursting: The genesis of rhythm in the nervous systemS. Coombes, P. Bressloff · 2005 · cited 248 times · Not summarized yet
Papers that cite it
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A model for the collective synchronization of flashing in Photinus carolinusMadeline McCrea, B. Ermentrout, J. Rubin · 2022 · Journal of the Royal Society Interface · cited 18 times · Not summarized yet
Abstract
Recent empirical investigations have characterized the synchronized flashing behaviours of male Photinus carolinus fireflies in their natural habitat in Great Smoky Mountain National Park as well as in controlled environments. We develop a model for the flash dynamics of an individual firefly based on a canonical elliptic burster, a slow-fast dynamical system that produces a repeating pattern of multiple flashes followed by a quiescent period. We show that a small amount of noise renders that oscillation very irregular, but when multiple model fireflies interact through their flashes, the behaviour becomes much more periodic. We show that the aggregate behaviour is qualitatively similar to t…
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Multiple oscillatory States in Models of Collective neuronal DynamicsS. Kalitzin, M. Koppert, G. Petkov et al. · 2014 · International Journal of Neural Systems · cited 45 times · Not summarized yet
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Bursting mechanism in a time-delayed oscillator with slowly varying external forcingYue Yu, Hong-Ji Tang, Xiu-Jing Han et al. · 2014 · Communications in nonlinear science & numerical simulation · cited 24 times · Not summarized yet
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Mathematical Frameworks for Oscillatory Network Dynamics in NeuroscienceP. Ashwin, S. Coombes, R. Nicks · 2015 · Journal of Mathematical Neuroscience · cited 304 times · Not summarized yet
Abstract
The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for a…
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Localized numerical impulses solutions in diffuse neural networks modeled by the complex fractional Ginzburg-Landau equationA. Mvogo, Antoine Tambue, G. Ben-Bolie et al. · 2014 · arXiv: Numerical Analysis · preprint, not peer reviewed · cited 50 times · Not summarized yet
Abstract
We investigate localized wave solutions in a network of Hindmarsh-Rose neural model taking into account the long-range diffusive couplings. We show by a specific analytical technique that the model equations in the infrared limit (wave number $k\rightarrow 0$) can be governed by the complex fractional Ginzburg-Landau (CFGL) equation. According to the stiffness of the system, we propose both the semi and the linearly implicit Riesz fractional finite-difference schemes to solve efficiently the CFGL equation. The obtained fractional numerical solutions for the nerve impulse reveal localized short impulse properties. We also show the equivalence between the continuous CFGL and the discrete Hindm…
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