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Phan NA, Wang Y. Mixed-mode oscillations in a three-timescale coupled Morris-Lecar system. CHAOS (WOODBURY, N.Y.) 2024; 34:053119. [PMID: 38717416 PMCID: PMC11087137 DOI: 10.1063/5.0181308] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2023] [Accepted: 04/19/2024] [Indexed: 05/12/2024]
Abstract
Mixed-mode oscillations (MMOs) are complex oscillatory behaviors of multiple-timescale dynamical systems in which there is an alternation of large-amplitude and small-amplitude oscillations. It is well known that MMOs in two-timescale systems can arise either from a canard mechanism associated with folded node singularities or a delayed Andronov-Hopf bifurcation (DHB) of the fast subsystem. While MMOs in two-timescale systems have been extensively studied, less is known regarding MMOs emerging in three-timescale systems. In this work, we examine the mechanisms of MMOs in coupled Morris-Lecar neurons with three distinct timescales. We investigate two kinds of MMOs occurring in the presence of a singularity known as canard-delayed-Hopf (CDH) and in cases where CDH is absent. In both cases, we examine how features and mechanisms of MMOs vary with respect to variations in timescales. Our analysis reveals that MMOs supported by CDH demonstrate significantly stronger robustness than those in its absence. Moreover, we show that the mere presence of CDH does not guarantee the occurrence of MMOs. This work yields important insights into conditions under which the two separate mechanisms in two-timescale context, canard and DHB, can interact in a three-timescale setting and produce more robust MMOs, particularly against timescale variations.
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Affiliation(s)
- Ngoc Anh Phan
- Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, USA
| | - Yangyang Wang
- Department of Mathematics, Brandeis University, Waltham, Massachusetts 02453, USA
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2
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Fazli M, Bertram R. Conversion of spikers to bursters in pituitary cell networks: Is it better to disperse for maximum exposure or circle the wagons? PLoS Comput Biol 2024; 20:e1011811. [PMID: 38289902 PMCID: PMC10826967 DOI: 10.1371/journal.pcbi.1011811] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2023] [Accepted: 01/08/2024] [Indexed: 02/01/2024] Open
Abstract
The endocrine cells of the pituitary gland are electrically active, and in vivo they form small networks where the bidirectional cell-cell coupling is through gap junctions. Numerous studies of dispersed pituitary cells have shown that typical behaviors are tonic spiking and bursting, the latter being more effective at evoking secretion. In this article, we use mathematical modeling to examine the dynamics of small networks of spiking and bursting pituitary cells. We demonstrate that intrinsic bursting cells are capable of converting intrinsic spikers into bursters, and perform a fast/slow analysis to show why this occurs. We then demonstrate the sensitivity of network dynamics to the placement of bursting cells within the network, and demonstrate strategies that are most effective at maximizing secretion from the population of cells. This study provides insights into the in vivo behavior of cells such as the stress-hormone-secreting pituitary corticotrophs that are switched from spiking to bursting by hypothalamic neurohormones. While much is known about the electrical properties of these cells when isolated from the pituitary, how they behave when part of an electrically coupled network has been largely unstudied.
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Affiliation(s)
- Mehran Fazli
- Department of Mathematics, Florida State University, Tallahassee, Florida, United States of America
| | - Richard Bertram
- Department of Mathematics, Florida State University, Tallahassee, Florida, United States of America
- Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, Florida, United States of America
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3
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Bertram R, Marinelli I, Fletcher PA, Satin LS, Sherman AS. Deconstructing the integrated oscillator model for pancreatic β-cells. Math Biosci 2023; 365:109085. [PMID: 37802364 PMCID: PMC10991200 DOI: 10.1016/j.mbs.2023.109085] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2023] [Revised: 09/12/2023] [Accepted: 09/26/2023] [Indexed: 10/10/2023]
Abstract
Electrical bursting oscillations in the β-cells of pancreatic islets have been a focus of investigation for more than fifty years. This has been aided by mathematical models, which are descendants of the pioneering Chay-Keizer model. This article describes the key biophysical and mathematical elements of this model, and then describes the path forward from there to the Integrated Oscillator Model (IOM). It is both a history and a deconstruction of the IOM that describes the various elements that have been added to the model over time, and the motivation for adding them. Finally, the article is a celebration of the 40th anniversary of the publication of the Chay-Keizer model.
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Affiliation(s)
- Richard Bertram
- Department of Mathematics and Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, FL, United States.
| | - Isabella Marinelli
- Centre for Systems Modeling and Quantitative Biomedicine, University of Birmingham, United Kingdom
| | - Patrick A Fletcher
- Laboratory of Biological Modeling, National Institutes of Health, Bethesda, MD, United States
| | - Leslie S Satin
- Department of Pharmacology and Brehm Center for Diabetes Research, University of Michigan Medical School, Ann Arbor, MI, United States
| | - Arthur S Sherman
- Laboratory of Biological Modeling, National Institutes of Health, Bethesda, MD, United States
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4
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Fazli M, Vo T, Bertram R. Fast-slow analysis of a stochastic mechanism for electrical bursting. CHAOS (WOODBURY, N.Y.) 2021; 31:103128. [PMID: 34717336 DOI: 10.1063/5.0059338] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2021] [Accepted: 09/03/2021] [Indexed: 06/13/2023]
Abstract
Electrical bursting oscillations in neurons and endocrine cells are activity patterns that facilitate the secretion of neurotransmitters and hormones and have been the focus of study for several decades. Mathematical modeling has been an extremely useful tool in this effort, and the use of fast-slow analysis has made it possible to understand bursting from a dynamic perspective and to make testable predictions about changes in system parameters or the cellular environment. It is typically the case that the electrical impulses that occur during the active phase of a burst are due to stable limit cycles in the fast subsystem of equations or, in the case of so-called "pseudo-plateau bursting," canards that are induced by a folded node singularity. In this article, we show an entirely different mechanism for bursting that relies on stochastic opening and closing of a key ion channel. We demonstrate, using fast-slow analysis, how the short-lived stochastic channel openings can yield a much longer response in which single action potentials are converted into bursts of action potentials. Without this stochastic element, the system is incapable of bursting. This mechanism can describe stochastic bursting in pituitary corticotrophs, which are small cells that exhibit a great deal of noise as well as other pituitary cells, such as lactotrophs and somatotrophs that exhibit noisy bursts of electrical activity.
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Affiliation(s)
- Mehran Fazli
- Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA
| | - Theodore Vo
- School of Mathematics, Monash University, Clayton, Victoria 3800, Australia
| | - Richard Bertram
- Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA
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5
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Ghosh S, Mondal A, Ji P, Mishra A, Dana SK, Antonopoulos CG, Hens C. Emergence of Mixed Mode Oscillations in Random Networks of Diverse Excitable Neurons: The Role of Neighbors and Electrical Coupling. Front Comput Neurosci 2020; 14:49. [PMID: 32581757 PMCID: PMC7294985 DOI: 10.3389/fncom.2020.00049] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2020] [Accepted: 05/04/2020] [Indexed: 11/21/2022] Open
Abstract
In this paper, we focus on the emergence of diverse neuronal oscillations arising in a mixed population of neurons with different excitability properties. These properties produce mixed mode oscillations (MMOs) characterized by the combination of large amplitudes and alternate subthreshold or small amplitude oscillations. Considering the biophysically plausible, Izhikevich neuron model, we demonstrate that various MMOs, including MMBOs (mixed mode bursting oscillations) and synchronized tonic spiking appear in a randomly connected network of neurons, where a fraction of them is in a quiescent (silent) state and the rest in self-oscillatory (firing) states. We show that MMOs and other patterns of neural activity depend on the number of oscillatory neighbors of quiescent nodes and on electrical coupling strengths. Our results are verified by constructing a reduced-order network model and supported by systematic bifurcation diagrams as well as for a small-world network. Our results suggest that, for weak couplings, MMOs appear due to the de-synchronization of a large number of quiescent neurons in the networks. The quiescent neurons together with the firing neurons produce high frequency oscillations and bursting activity. The overarching goal is to uncover a favorable network architecture and suitable parameter spaces where Izhikevich model neurons generate diverse responses ranging from MMOs to tonic spiking.
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Affiliation(s)
- Subrata Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata, India
| | - Argha Mondal
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata, India
| | - Peng Ji
- The Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, China
| | - Arindam Mishra
- Department of Mathematics, Centre for Mathematical Biology and Ecology, Jadavpur University, Kolkata, India
| | - Syamal K Dana
- Department of Mathematics, Centre for Mathematical Biology and Ecology, Jadavpur University, Kolkata, India.,Division of Dynamics, Faculty of Mechanical Engineering, Lodz University of Technology, Lodz, Poland
| | - Chris G Antonopoulos
- Department of Mathematical Sciences, University of Essex, Colchester, United Kingdom
| | - Chittaranjan Hens
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata, India
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6
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Rodrigues CDS, dos Santos CGP, de Miranda RCC, Parma E, Varela H, Nagao R. A numerical investigation of the effect of external resistance and applied potential on the distribution of periodicity and chaos in the anodic dissolution of nickel. Phys Chem Chem Phys 2020; 22:21823-21834. [DOI: 10.1039/d0cp04238b] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
Abstract
Low density, elongation, and suppression of the shrimp-like structures in the resistance-potential phase diagrams have been observed in the oscillatory dissolution of nickel.
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Affiliation(s)
| | | | | | - Eduardo Parma
- Institute of Chemistry
- University of Campinas
- 13083-970 Campinas
- Brazil
| | - Hamilton Varela
- Institute of Chemistry of São Carlos
- University of São Paulo
- 13560-970 São Carlos
- Brazil
- School of Earth Sciences and Environmental Engineering
| | - Raphael Nagao
- Institute of Chemistry
- University of Campinas
- 13083-970 Campinas
- Brazil
- Center for Innovation on New Energies
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7
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Vo T, Bertram R. Why pacing frequency affects the production of early afterdepolarizations in cardiomyocytes: An explanation revealed by slow-fast analysis of a minimal model. Phys Rev E 2019; 99:052205. [PMID: 31212514 DOI: 10.1103/physreve.99.052205] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2019] [Indexed: 12/28/2022]
Abstract
Early afterdepolarizations (EADs) are pathological voltage oscillations in cardiomyocytes that have been observed in response to a number of pharmacological agents and disease conditions. Phase-2 EADs consist of small voltage fluctuations during the plateau of an action potential, typically under conditions in which the action potential is elongated. Although a single-cell behavior, EADs can lead to tissue-level arrhythmias. Much is currently known about the biophysical mechanisms (i.e., the roles of ion channels and intracellular Ca^{2+} stores) for the various forms of EADs, due partially to the development and analysis of mathematical models. This includes the application of slow-fast analysis, which takes advantage of timescale separation inherent in the system to simplify its analysis. We take this further, using a minimal three-dimensional model to demonstrate that phase-2 EADs are canards formed in the neighborhood of a folded node singularity. This allows us to predict the number of EADs that can be produced for a given parameter set, and provides guidance on parameter changes that facilitate or inhibit EAD production. With this approach, we demonstrate why periodic stimulation, as occurs in intact heart, preferentially facilitates EAD production when applied at low frequencies. We also explain the origin of complex alternan dynamics that can occur with intermediate-frequency stimulation, in which varying numbers of EADs are produced with each pulse. These revelations fall out naturally from an understanding of folded node singularities, but are difficult to glean from knowledge of the biophysical mechanism for EADs alone. Therefore, understanding the canard mechanism is a useful complement to understanding of the biophysical mechanism that has been developed over years of experimental and computational investigations.
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Affiliation(s)
- Theodore Vo
- Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA
| | - Richard Bertram
- Department of Mathematics and Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, USA
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8
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Abstract
Excitable biological cells, such as cardiac muscle cells, can exhibit complex patterns of oscillations such as spiking and bursting. Moreover, it is well known that an enhancement in calcium currents may yield certain kind of cardiac arrhythmia, so-called early afterdepolarisations (EADs). The presence of EADs strongly correlates with the onset of dangerous cardiac arrhythmia. In this paper we study mathematically and numerically the dynamics of a cardiac muscle cell with respect to the calcium current by investigating a simplistic system of differential equations. For the study of this phenomena, we use bifurcation theory, numerical bifurcation analysis, geometric singular perturbation theory and computational methods to investigate a nonlinear multiple time scales system. It will turn out that EADs related to an enhanced calcium current are canard–induced and that we have to combine these theories to derive a better understanding of the dynamics behind EADs. Moreover, a suitable time scale separation argument determines the important and sensitive system parameters which are related to the occurrence of EADs.
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9
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Kügler P, Erhardt AH, Bulelzai MAK. Early afterdepolarizations in cardiac action potentials as mixed mode oscillations due to a folded node singularity. PLoS One 2018; 13:e0209498. [PMID: 30596698 PMCID: PMC6312222 DOI: 10.1371/journal.pone.0209498] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2018] [Accepted: 12/06/2018] [Indexed: 12/27/2022] Open
Abstract
Early afterdepolarizations (EADs) are pathological voltage oscillations during the repolarization phase of cardiac action potentials. They are considered as potential precursors to cardiac arrhythmias and have recently gained much attention in the context of preclinical drug safety testing under the Comprehensive in vitro Proarrhythmia Assay (CiPA) paradigm. From the viewpoint of multiple time scales theory, the onset of EADs has previously been studied by means of mathematical action potential models with one slow ion channel gating variable. In this article, we for the first time associate EADs with mixed mode oscillations in dynamical systems with two slow gating variables and present a folded node singularity of the slow flow as a novel mechanism for EADs genesis. We derive regions of the pharmacology parameter space in which EADs occur using both the folded node analysis and a full system bifurcation analysis, and we suggest the normal distance to the boundary of the EADs region as a mechanism-based risk metric to computationally estimate a drug’s proarrhythmic liability.
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Affiliation(s)
- Philipp Kügler
- Institute of Applied Mathematics and Statistics, University of Hohenheim, Stuttgart, Germany
- * E-mail:
| | | | - M. A. K. Bulelzai
- Departmeny of Basic Sciences and Related Studies, Quaid-e-Awam University of Engineering, Science and Technology, Nawabshah, Pakistan
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10
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Zhou Y, Vo T, Rotstein HG, McCarthy MM, Kopell N. M-Current Expands the Range of Gamma Frequency Inputs to Which a Neuronal Target Entrains. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2018; 8:13. [PMID: 30519798 PMCID: PMC6281550 DOI: 10.1186/s13408-018-0068-6] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2018] [Accepted: 11/18/2018] [Indexed: 05/25/2023]
Abstract
Theta (4-8 Hz) and gamma (30-80 Hz) rhythms in the brain are commonly associated with memory and learning (Kahana in J Neurosci 26:1669-1672, 2006; Quilichini et al. in J Neurosci 30:11128-11142, 2010). The precision of co-firing between neurons and incoming inputs is critical in these cognitive functions. We consider an inhibitory neuron model with M-current under forcing from gamma pulses and a sinusoidal current of theta frequency. The M-current has a long time constant (∼90 ms) and it has been shown to generate resonance at theta frequencies (Hutcheon and Yarom in Trends Neurosci 23:216-222, 2000; Hu et al. in J Physiol 545:783-805, 2002). We have found that this slow M-current contributes to the precise co-firing between the network and fast gamma pulses in the presence of a slow sinusoidal forcing. The M-current expands the phase-locking frequency range of the network, counteracts the slow theta forcing, and admits bistability in some parameter range. The effects of the M-current balancing the theta forcing are reduced if the sinusoidal current is faster than the theta frequency band. We characterize the dynamical mechanisms underlying the role of the M-current in enabling a network to be entrained to gamma frequency inputs using averaging methods, geometric singular perturbation theory, and bifurcation analysis.
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Affiliation(s)
- Yujia Zhou
- Department of Mathematics and Statistics, Boston University, Boston, USA
| | - Theodore Vo
- Department of Mathematics, Florida State University, Tallahassee, USA
| | - Horacio G. Rotstein
- Department of Biological Sciences, New Jersey Institute of Technology, Newark, USA
| | | | - Nancy Kopell
- Department of Mathematics and Statistics, Boston University, Boston, USA
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11
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Leng G, MacGregor DJ. Models in neuroendocrinology. Math Biosci 2018; 305:29-41. [DOI: 10.1016/j.mbs.2018.07.008] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/16/2018] [Revised: 07/20/2018] [Accepted: 07/24/2018] [Indexed: 12/18/2022]
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12
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Yamakou ME, Jost J. Weak-noise-induced transitions with inhibition and modulation of neural oscillations. BIOLOGICAL CYBERNETICS 2018; 112:445-463. [PMID: 29995240 PMCID: PMC6153713 DOI: 10.1007/s00422-018-0770-1] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2017] [Accepted: 06/27/2018] [Indexed: 05/29/2023]
Abstract
We analyze the effect of weak-noise-induced transitions on the dynamics of the FitzHugh-Nagumo neuron model in a bistable state consisting of a stable fixed point and a stable unforced limit cycle. Bifurcation and slow-fast analysis give conditions on the parameter space for the establishment of this bi-stability. In the parametric zone of bi-stability, weak-noise amplitudes may strongly inhibit the neuron's spiking activity. Surprisingly, increasing the noise strength leads to a minimum in the spiking activity, after which the activity starts to increase monotonically with an increase in noise strength. We investigate this inhibition and modulation of neural oscillations by weak-noise amplitudes by looking at the variation of the mean number of spikes per unit time with the noise intensity. We show that this phenomenon always occurs when the initial conditions lie in the basin of attraction of the stable limit cycle. For initial conditions in the basin of attraction of the stable fixed point, the phenomenon, however, disappears, unless the timescale separation parameter of the model is bounded within some interval. We provide a theoretical explanation of this phenomenon in terms of the stochastic sensitivity functions of the attractors and their minimum Mahalanobis distances from the separatrix isolating the basins of attraction.
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Affiliation(s)
- Marius E. Yamakou
- Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22, 04103 Leipzig, Germany
- Fakultät für Mathematik und Informatik, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany
| | - Jürgen Jost
- Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22, 04103 Leipzig, Germany
- Fakultät für Mathematik und Informatik, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany
- Santa Fe Institute for the Sciences of Complexity, Santa Fe, NM 87501 USA
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13
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Kaper TJ, Vo T. Delayed loss of stability due to the slow passage through Hopf bifurcations in reaction-diffusion equations. CHAOS (WOODBURY, N.Y.) 2018; 28:091103. [PMID: 30278640 DOI: 10.1063/1.5050508] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2018] [Accepted: 09/11/2018] [Indexed: 06/08/2023]
Abstract
This article presents the delayed loss of stability due to slow passage through Hopf bifurcations in reaction-diffusion equations with slowly-varying parameters, generalizing a well-known result about delayed Hopf bifurcations in analytic ordinary differential equations to spatially-extended systems. We focus on the Hodgkin-Huxley partial differential equation (PDE), the cubic Complex Ginzburg-Landau PDE as an equation in its own right, the Brusselator PDE, and a spatially-extended model of a pituitary clonal cell line. Solutions which are attracted to quasi-stationary states (QSS) sufficiently before the Hopf bifurcations remain near the QSS for long times after the states have become repelling, resulting in a significant delay in the loss of stability and the onset of oscillations. Moreover, the oscillations have large amplitude at onset, and may be spatially homogeneous or inhomogeneous. Space-time boundaries are identified that act as buffer curves beyond which solutions cannot remain near the repelling QSS, and hence before which the delayed onset of oscillations must occur, irrespective of initial conditions. In addition, a method is developed to derive the asymptotic formulas for the buffer curves, and the asymptotics agree well with the numerically observed onset in the Complex Ginzburg-Landau (CGL) equation. We also find that the first-onset sites act as a novel pulse generation mechanism for spatio-temporal oscillations.
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Affiliation(s)
- Tasso J Kaper
- Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215, USA
| | - Theodore Vo
- Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA
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14
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Abstract
The stochastic nature of neuronal response has lead to conjectures about the impact of input fluctuations on the neural coding. For the most part, low pass membrane integration and spike threshold dynamics have been the primary features assumed in the transfer from synaptic input to output spiking. Phasic neurons are a common, but understudied, neuron class that are characterized by a subthreshold negative feedback that suppresses spike train responses to low frequency signals. Past work has shown that when a low frequency signal is accompanied by moderate intensity broadband noise, phasic neurons spike trains are well locked to the signal. We extend these results with a simple, reduced model of phasic activity that demonstrates that a non-Markovian spike train structure caused by the negative feedback produces a noise-enhanced coding. Further, this enhancement is sensitive to the timescales, as opposed to the intensity, of a driving signal. Reduced hazard function models show that noise-enhanced phasic codes are both novel and separate from classical stochastic resonance reported in non-phasic neurons. The general features of our theory suggest that noise-enhanced codes in excitable systems with subthreshold negative feedback are a particularly rich framework to study.
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Affiliation(s)
- Cheng Ly
- Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284, United States of America
| | - Brent Doiron
- Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, United States of America
- Center for the Neural Basis of Cognition, Pittsburgh, PA 15213, United States of America
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15
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Bertram R, Rubin JE. Multi-timescale systems and fast-slow analysis. Math Biosci 2016; 287:105-121. [PMID: 27424950 DOI: 10.1016/j.mbs.2016.07.003] [Citation(s) in RCA: 42] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2016] [Revised: 07/01/2016] [Accepted: 07/10/2016] [Indexed: 11/28/2022]
Abstract
Mathematical models of biological systems often have components that vary on different timescales. This multi-timescale character can lead to problems when doing computer simulations, which can require a great deal of computer time so that the components that change on the fastest time scale can be resolved. Mathematical analysis of these multi-timescale systems can be greatly simplified by partitioning them into subsystems that evolve on different time scales. The subsystems are then analyzed semi-independently, using a technique called fast-slow analysis. In this review we describe the fast-slow analysis technique and apply it to relaxation oscillations, neuronal bursting oscillations, canard oscillations, and mixed-mode oscillations. Although these examples all involve neural systems, the technique can and has been applied to other biological, chemical, and physical systems. It is a powerful analysis method that will become even more useful in the future as new experimental techniques push forward the complexity of biological models.
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Affiliation(s)
- Richard Bertram
- Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University, Florida State University, Tallahassee, FL, United States.
| | - Jonathan E Rubin
- Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, United States
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16
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Fletcher P, Bertram R, Tabak J. From global to local: exploring the relationship between parameters and behaviors in models of electrical excitability. J Comput Neurosci 2016; 40:331-45. [PMID: 27033230 DOI: 10.1007/s10827-016-0600-1] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2015] [Revised: 03/02/2016] [Accepted: 03/07/2016] [Indexed: 01/25/2023]
Abstract
Models of electrical activity in excitable cells involve nonlinear interactions between many ionic currents. Changing parameters in these models can produce a variety of activity patterns with sometimes unexpected effects. Further more, introducing new currents will have different effects depending on the initial parameter set. In this study we combined global sampling of parameter space and local analysis of representative parameter sets in a pituitary cell model to understand the effects of adding K (+) conductances, which mediate some effects of hormone action on these cells. Global sampling ensured that the effects of introducing K (+) conductances were captured across a wide variety of contexts of model parameters. For each type of K (+) conductance we determined the types of behavioral transition that it evoked. Some transitions were counterintuitive, and may have been missed without the use of global sampling. In general, the wide range of transitions that occurred when the same current was applied to the model cell at different locations in parameter space highlight the challenge of making accurate model predictions in light of cell-to-cell heterogeneity. Finally, we used bifurcation analysis and fast/slow analysis to investigate why specific transitions occur in representative individual models. This approach relies on the use of a graphics processing unit (GPU) to quickly map parameter space to model behavior and identify parameter sets for further analysis. Acceleration with modern low-cost GPUs is particularly well suited to exploring the moderate-sized (5-20) parameter spaces of excitable cell and signaling models.
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Affiliation(s)
- Patrick Fletcher
- Currently at the Laboratory of Biological Modeling, National Institutes of Health, Bethesda, MD, 20892, USA
| | - Richard Bertram
- Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA.
| | - Joel Tabak
- Currently at the University of Exeter Medical School, Biomedical Neuroscience Research Group, EX4 4PS, Exeter, UK
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17
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18
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Talla Mbé JH, Talla AF, Chengui GRG, Coillet A, Larger L, Woafo P, Chembo YK. Mixed-mode oscillations in slow-fast delayed optoelectronic systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:012902. [PMID: 25679677 DOI: 10.1103/physreve.91.012902] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2014] [Indexed: 06/04/2023]
Abstract
In this article, we investigate the dynamical behavior of breathers in optoelectronic oscillators from the standpoint of mixed-mode oscillations. In the phase space, these breathers are composite oscillations that are damped to the attractive branches of an invariant manifold. Our study shows that the emergence of breather dynamics is linked to the apparition of inflection points in the phase space, and we develop an analytical framework based on the Liénard reduction form in order to provide an analytical insight into this phenomenology. Our theoretical results are in excellent agreement with experimental measurements.
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Affiliation(s)
- Jimmi H Talla Mbé
- Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, University of Yaoundé I, Department of Physics, P.O. Box 812, Yaoundé, Cameroon and African Center of Excellence for Information and Communication Technologies (CETIC), Polytechnic School of Yaoundé, P.O. Box 8390 Yaoundé, Cameroon
| | - Alain F Talla
- Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, University of Yaoundé I, Department of Physics, P.O. Box 812, Yaoundé, Cameroon and African Center of Excellence for Information and Communication Technologies (CETIC), Polytechnic School of Yaoundé, P.O. Box 8390 Yaoundé, Cameroon
| | - Geraud R Goune Chengui
- Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, University of Yaoundé I, Department of Physics, P.O. Box 812, Yaoundé, Cameroon and African Center of Excellence for Information and Communication Technologies (CETIC), Polytechnic School of Yaoundé, P.O. Box 8390 Yaoundé, Cameroon
| | - Aurélien Coillet
- FEMTO-ST Institute [CNRS UMR6174], Optics Department, 15B Avenue des Montboucons, 25030 Besançon Cedex, France
| | - Laurent Larger
- FEMTO-ST Institute [CNRS UMR6174], Optics Department, 15B Avenue des Montboucons, 25030 Besançon Cedex, France
| | - Paul Woafo
- Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, University of Yaoundé I, Department of Physics, P.O. Box 812, Yaoundé, Cameroon
| | - Yanne K Chembo
- FEMTO-ST Institute [CNRS UMR6174], Optics Department, 15B Avenue des Montboucons, 25030 Besançon Cedex, France
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A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells. J Comput Neurosci 2013; 36:259-78. [DOI: 10.1007/s10827-013-0470-8] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2013] [Revised: 04/25/2013] [Accepted: 05/29/2013] [Indexed: 12/13/2022]
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Teka W, Tabak J, Bertram R. The relationship between two fast/slow analysis techniques for bursting oscillations. CHAOS (WOODBURY, N.Y.) 2012; 22:043117. [PMID: 23278052 PMCID: PMC3523400 DOI: 10.1063/1.4766943] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2012] [Accepted: 10/04/2012] [Indexed: 06/01/2023]
Abstract
Bursting oscillations in excitable systems reflect multi-timescale dynamics. These oscillations have often been studied in mathematical models by splitting the equations into fast and slow subsystems. Typically, one treats the slow variables as parameters of the fast subsystem and studies the bifurcation structure of this subsystem. This has key features such as a z-curve (stationary branch) and a Hopf bifurcation that gives rise to a branch of periodic spiking solutions. In models of bursting in pituitary cells, we have recently used a different approach that focuses on the dynamics of the slow subsystem. Characteristic features of this approach are folded node singularities and a critical manifold. In this article, we investigate the relationships between the key structures of the two analysis techniques. We find that the z-curve and Hopf bifurcation of the two-fast/one-slow decomposition are closely related to the voltage nullcline and folded node singularity of the one-fast/two-slow decomposition, respectively. They become identical in the double singular limit in which voltage is infinitely fast and calcium is infinitely slow.
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Affiliation(s)
- Wondimu Teka
- Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA
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Desroches M, Burke J, Kaper TJ, Kramer MA. Canards of mixed type in a neural burster. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:021920. [PMID: 22463257 DOI: 10.1103/physreve.85.021920] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2011] [Revised: 12/16/2011] [Indexed: 05/31/2023]
Abstract
Canards are solutions of slow-fast systems that spend long times near branches of repelling equilibria, periodic orbits, or higher-dimensional invariant sets. Here, we report on the observation of a new type of canard orbit, labeled a canard of mixed type. This canard orbit is a hybrid of the classical limit cycle canards, which spend long times near attracting and repelling branches of equilibria, and torus canards, which spend long times near attracting and repelling branches of periodic orbits. The canards of mixed type arise in a model of neural bursting activity of fold-fold cycle type, and, as other canard phenomena, separate different dynamic states.
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Affiliation(s)
- M Desroches
- Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom
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Vo T, Bertram R, Wechselberger M. Bifurcations of canard-induced mixed mode oscillations in a pituitary Lactotroph model. ACTA ACUST UNITED AC 2012. [DOI: 10.3934/dcds.2012.32.2879] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Osinga HM, Sherman A, Tsaneva-Atanasova K. CROSS-CURRENTS BETWEEN BIOLOGY AND MATHEMATICS: THE CODIMENSION OF PSEUDO-PLATEAU BURSTING. ACTA ACUST UNITED AC 2012; 32:2853-2877. [PMID: 22984340 DOI: 10.3934/dcds.2012.32.2853] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
A great deal of work has gone into classifying bursting oscillations, periodic alternations of spiking and quiescence modeled by fast-slow systems. In such systems, one or more slow variables carry the fast variables through a sequence of bifurcations that mediate transitions between oscillations and steady states. A rigorous classification approach is to characterize the bifurcations found in the neighborhood of a singularity; a measure of the complexity of the bursting oscillation is then given by the smallest codimension of the singularities near which it occurs. Fold/homoclinic bursting, along with most other burst types of interest, has been shown to occur near a singularity of codimension three by examining bifurcations of a cubic Liénard system; hence, these types of bursting have at most codimension three. Modeling and biological considerations suggest that fold/homoclinic bursting should be found near fold/subHopf bursting, a more recently identified burst type whose codimension has not been determined yet. One would expect that fold/subHopf bursting has the same codimension as fold/homoclinic bursting, because models of these two burst types have very similar underlying bifurcation diagrams. However, no codimension-three singularity is known that supports fold/subHopf bursting, which indicates that it may have codimension four. We identify a three-dimensional slice in a partial unfolding of a doubly-degenerate Bodganov-Takens point, and show that this codimension-four singularity gives rise to almost all known types of bursting.
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Affiliation(s)
- Hinke M Osinga
- Bristol Centre for Applied Nonlinear Mathematics Department of Engineering Mathematics University of Bristol, Queen's Building, University Walk Bristol BS8 1TR, UK
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Teka W, Tabak J, Vo T, Wechselberger M, Bertram R. The dynamics underlying pseudo-plateau bursting in a pituitary cell model. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2011; 1. [PMID: 22268000 PMCID: PMC3261773 DOI: 10.1186/2190-8567-1-12] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns of electrical activity. Several types of pituitary cells produce short bursts of electrical activity which are more effective than single spikes in evoking hormone release. These bursts, called pseudo-plateau bursts, are unlike bursts studied mathematically in neurons (plateau bursting) and the standard fast-slow analysis used for plateau bursting is of limited use. Using an alternative fast-slow analysis, with one fast and two slow variables, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. The information gained from this one-fast/two-slow decomposition complements the information obtained from a two-fast/one-slow decomposition.
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Affiliation(s)
- Wondimu Teka
- Department of Mathematics; Florida State University, Tallahassee, FL, USA
| | - Joël Tabak
- Department of Biological Science; Florida State University, Tallahassee, FL, USA
| | - Theodore Vo
- School of Mathematics and Statistics; University of Sydney, Sydney, NSW, Australia
| | - Martin Wechselberger
- School of Mathematics and Statistics; University of Sydney, Sydney, NSW, Australia
| | - Richard Bertram
- Department of Mathematics, and Programs in Neuroscience and Molecular Biophysics; Florida State University, Tallahassee, FL, USA
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Brøns M. Canard explosion of limit cycles in templator models of self-replication mechanisms. J Chem Phys 2011; 134:144105. [DOI: 10.1063/1.3577998] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Harvey E, Kirk V, Osinga HM, Sneyd J, Wechselberger M. Understanding anomalous delays in a model of intracellular calcium dynamics. CHAOS (WOODBURY, N.Y.) 2010; 20:045104. [PMID: 21198116 DOI: 10.1063/1.3523264] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
In many cell types, oscillations in the concentration of free intracellular calcium ions are used to control a variety of cellular functions. It has been suggested [J. Sneyd et al., "A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations," Proc. Natl. Acad. Sci. U.S.A. 103, 1675-1680 (2006)] that the mechanisms underlying the generation and control of such oscillations can be determined by means of a simple experiment, whereby a single exogenous pulse of inositol trisphosphate (IP(3)) is applied to the cell. However, more detailed mathematical investigations [M. Domijan et al., "Dynamical probing of the mechanisms underlying calcium oscillations," J. Nonlinear Sci. 16, 483-506 (2006)] have shown that this is not necessarily always true, and that the experimental data are more difficult to interpret than first thought. Here, we use geometric singular perturbation techniques to study the dynamics of models that make different assumptions about the mechanisms underlying the calcium oscillations. In particular, we show how recently developed canard theory for singularly perturbed systems with three or more slow variables [M. Wechselberger, "A propos de canards (Apropos canards)," Preprint, 2010] applies to these calcium models and how the presence of a curve of folded singularities and corresponding canards can result in anomalous delays in the response of these models to a pulse of IP(3).
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Affiliation(s)
- Emily Harvey
- Department of Mathematics, University of Auckland, Private Bag, 92019 Auckland, New Zealand
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