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Köksal Ersöz E, Wendling F. Canard solutions in neural mass models: consequences on critical regimes. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2021; 11:11. [PMID: 34529192 PMCID: PMC8446153 DOI: 10.1186/s13408-021-00109-z] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2020] [Accepted: 08/17/2021] [Indexed: 05/06/2023]
Abstract
Mathematical 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.
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Affiliation(s)
- Elif Köksal Ersöz
- Univ Rennes, INSERM, LTSI-U1099, Campus de Beaulieu, F - 35000, Rennes, France
| | - Fabrice Wendling
- Univ Rennes, INSERM, LTSI-U1099, Campus de Beaulieu, F - 35000, Rennes, France.
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2
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Boeri J, Meunier C, Le Corronc H, Branchereau P, Timofeeva Y, Lejeune FX, Mouffle C, Arulkandarajah H, Mangin JM, Legendre P, Czarnecki A. Two opposite voltage-dependent currents control the unusual early development pattern of embryonic Renshaw cell electrical activity. eLife 2021; 10:62639. [PMID: 33899737 PMCID: PMC8139835 DOI: 10.7554/elife.62639] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2020] [Accepted: 04/24/2021] [Indexed: 11/25/2022] Open
Abstract
Renshaw cells (V1R) are excitable as soon as they reach their final location next to the spinal motoneurons and are functionally heterogeneous. Using multiple experimental approaches, in combination with biophysical modeling and dynamical systems theory, we analyzed, for the first time, the mechanisms underlying the electrophysiological properties of V1R during early embryonic development of the mouse spinal cord locomotor networks (E11.5–E16.5). We found that these interneurons are subdivided into several functional clusters from E11.5 and then display an unexpected transitory involution process during which they lose their ability to sustain tonic firing. We demonstrated that the essential factor controlling the diversity of the discharge pattern of embryonic V1R is the ratio of a persistent sodium conductance to a delayed rectifier potassium conductance. Taken together, our results reveal how a simple mechanism, based on the synergy of two voltage-dependent conductances that are ubiquitous in neurons, can produce functional diversity in embryonic V1R and control their early developmental trajectory.
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Affiliation(s)
- Juliette Boeri
- INSERM, UMR_S 1130, CNRS, UMR 8246, Neuroscience Paris Seine, Institute of Biology Paris Seine, Sorbonne Univ, Paris, France
| | - Claude Meunier
- Centre de Neurosciences Intégratives et Cognition, CNRS UMR 8002, Institut Neurosciences et Cognition, Université de Paris, Paris, France
| | - Hervé Le Corronc
- INSERM, UMR_S 1130, CNRS, UMR 8246, Neuroscience Paris Seine, Institute of Biology Paris Seine, Sorbonne Univ, Paris, France.,Univ Angers, Angers, France
| | | | - Yulia Timofeeva
- Department of Computer Science and Centre for Complexity Science, University of Warwick, Coventry, United Kingdom.,Department of Clinical and Experimental Epilepsy, UCL Queen Square Institute of Neurology, University College London, London, United Kingdom
| | - François-Xavier Lejeune
- Institut du Cerveau et de la Moelle Epinière, Centre de Recherche CHU Pitié-Salpétrière, INSERM, U975, CNRS, UMR 7225, Sorbonne Univ, Paris, France
| | - Christine Mouffle
- INSERM, UMR_S 1130, CNRS, UMR 8246, Neuroscience Paris Seine, Institute of Biology Paris Seine, Sorbonne Univ, Paris, France
| | - Hervé Arulkandarajah
- INSERM, UMR_S 1130, CNRS, UMR 8246, Neuroscience Paris Seine, Institute of Biology Paris Seine, Sorbonne Univ, Paris, France
| | - Jean Marie Mangin
- INSERM, UMR_S 1130, CNRS, UMR 8246, Neuroscience Paris Seine, Institute of Biology Paris Seine, Sorbonne Univ, Paris, France
| | - Pascal Legendre
- INSERM, UMR_S 1130, CNRS, UMR 8246, Neuroscience Paris Seine, Institute of Biology Paris Seine, Sorbonne Univ, Paris, France
| | - Antonny Czarnecki
- INSERM, UMR_S 1130, CNRS, UMR 8246, Neuroscience Paris Seine, Institute of Biology Paris Seine, Sorbonne Univ, Paris, France.,Univ. Bordeaux, CNRS, EPHE, INCIA, Bordeaux, France
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3
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Köksal Ersöz E, Desroches M, Guillamon A, Rinzel J, Tabak J. Canard-induced complex oscillations in an excitatory network. J Math Biol 2020; 80:2075-2107. [PMID: 32266428 DOI: 10.1007/s00285-020-01490-1] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2019] [Revised: 03/25/2020] [Indexed: 10/24/2022]
Abstract
In Neuroscience, mathematical modelling involving multiple spatial and temporal scales can unveil complex oscillatory activity such as excitable responses to an input current, subthreshold oscillations, spiking or bursting. While the number of slow and fast variables and the geometry of the system determine the type of the complex oscillations, canard structures define boundaries between them. In this study, we use geometric singular perturbation theory to identify and characterise boundaries between different dynamical regimes in multiple-timescale firing rate models of the developing spinal cord. These rate models are either three or four dimensional with state variables chosen within an overall group of two slow and two fast variables. The fast subsystem corresponds to a recurrent excitatory network with fast activity-dependent synaptic depression, and the slow variables represent the cell firing threshold and slow activity-dependent synaptic depression, respectively. We start by demonstrating canard-induced bursting and mixed-mode oscillations in two different three-dimensional rate models. Then, in the full four-dimensional model we show that a canard-mediated slow passage creates dynamics that combine these complex oscillations and give rise to mixed-mode bursting oscillations (MMBOs). We unveil complicated isolas along which MMBOs exist in parameter space. The profile of solutions along each isola undergoes canard-mediated transitions between the sub-threshold regime and the bursting regime; these explosive transitions change the number of oscillations in each regime. Finally, we relate the MMBO dynamics to experimental recordings and discuss their effects on the silent phases of bursting patterns as well as their potential role in creating subthreshold fluctuations that are often interpreted as noise. The mathematical framework used in this paper is relevant for modelling multiple timescale dynamics in excitable systems.
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Affiliation(s)
- Elif Köksal Ersöz
- MathNeuro Team, Inria Sophia Antipolis Méditerranée, Valbonne, France. .,Université Côte d'Azur, Nice, France. .,LTSI-U1099, INSERM, 35000, Rennes, France.
| | - Mathieu Desroches
- MathNeuro Team, Inria Sophia Antipolis Méditerranée, Valbonne, France.,Université Côte d'Azur, Nice, France
| | - Antoni Guillamon
- Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain
| | - John Rinzel
- Center for Neural Science, New York University, New York, USA.,Courant Institute for Mathematical Sciences, New York University, New York, USA
| | - Joël Tabak
- University of Exeter Medical School, University of Exeter, Exeter, UK
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4
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Wang Y, Rubin JE. Complex bursting dynamics in an embryonic respiratory neuron model. CHAOS (WOODBURY, N.Y.) 2020; 30:043127. [PMID: 32357647 DOI: 10.1063/1.5138993] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2019] [Accepted: 03/23/2020] [Indexed: 06/11/2023]
Abstract
Pre-Bötzinger complex (pre-BötC) network activity within the mammalian brainstem controls the inspiratory phase of the respiratory rhythm. While bursting in pre-BötC neurons during the postnatal period has been extensively studied, less is known regarding inspiratory pacemaker neuron behavior at embryonic stages. Recent data in mouse embryo brainstem slices have revealed the existence of a variety of bursting activity patterns depending on distinct combinations of burst-generating INaP and ICAN conductances. In this work, we consider a model of an isolated embryonic pre-BötC neuron featuring two distinct bursting mechanisms. We use methods of dynamical systems theory, such as phase plane analysis, fast-slow decomposition, and bifurcation analysis, to uncover mechanisms underlying several different types of intrinsic bursting dynamics observed experimentally including several forms of plateau bursts, bursts involving depolarization block, and various combinations of these patterns. Our analysis also yields predictions about how changes in the balance of the two bursting mechanisms contribute to alterations in an inspiratory pacemaker neuron activity during prenatal development.
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Affiliation(s)
- Yangyang Wang
- Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242, USA
| | - Jonathan E Rubin
- Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
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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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Osinga HM, Tsaneva-Atanasova KT. Geometric analysis of transient bursts. CHAOS (WOODBURY, N.Y.) 2013; 23:046107. [PMID: 24387586 DOI: 10.1063/1.4826655] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We consider the effect of a brief stimulation from the rest state of a minimal neuronal model with multiple time scales. Such transient dynamics brings out the intrinsic bursting capabilities of the system. Our main goal is to show that a minimum of three dimensions is enough to generate spike-adding phenomena in transient responses, and that the onset of a new spike can be tracked using existing continuation packages. We take a geometric approach to illustrate how the underlying fast subsystem organises the spike adding in much the same way as for spike adding in periodic bursts, but the bifurcation analysis for spike onset is entirely different. By using a generic model, we further strengthen claims made in our earlier work that our numerical method for spike onset can be used for a broad class of systems.
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Affiliation(s)
- Hinke M Osinga
- Department of Mathematics, the University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
| | - Krasimira T Tsaneva-Atanasova
- College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter EX4 4QF, United Kingdom
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8
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Desroches M, Kaper TJ, Krupa M. Mixed-mode bursting oscillations: dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster. CHAOS (WOODBURY, N.Y.) 2013; 23:046106. [PMID: 24387585 DOI: 10.1063/1.4827026] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
This article concerns the phenomenon of Mixed-Mode Bursting Oscillations (MMBOs). These are solutions of fast-slow systems of ordinary differential equations that exhibit both small-amplitude oscillations (SAOs) and bursts consisting of one or multiple large-amplitude oscillations (LAOs). The name MMBO is given in analogy to Mixed-Mode Oscillations, which consist of alternating SAOs and LAOs, without the LAOs being organized into burst events. In this article, we show how MMBOs are created naturally in systems that have a spike-adding bifurcation or spike-adding mechanism, and in which the dynamics of one (or more) of the slow variables causes the system to pass slowly through that bifurcation. Canards are central to the dynamics of MMBOs, and their role in shaping the MMBOs is two-fold: saddle-type canards are involved in the spike-adding mechanism of the underlying burster and permit one to understand the number of LAOs in each burst event, and folded-node canards arise due to the slow passage effect and control the number of SAOs. The analysis is carried out for a prototypical fourth-order system of this type, which consists of the third-order Hindmarsh-Rose system, known to have the spike-adding mechanism, and in which one of the key bifurcation parameters also varies slowly. We also include a discussion of the MMBO phenomenon for the Morris-Lecar-Terman system. Finally, we discuss the role of the MMBOs to a biological modeling of secreting neurons.
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Affiliation(s)
- Mathieu Desroches
- INRIA Paris-Rocquencourt Research Centre, MYCENAE Project-Team, Domaine de Voluceau, Rocquencourt BP 105, 78153 Le Chesnay cedex, France
| | - Tasso J Kaper
- Department of Mathematics and Statistics, Center for BioDynamics, Boston University, 111 Cummington Mall, Boston, Massachusetts 02215, USA
| | - Martin Krupa
- INRIA Paris-Rocquencourt Research Centre, MYCENAE Project-Team, Domaine de Voluceau, Rocquencourt BP 105, 78153 Le Chesnay cedex, France
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9
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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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Nowacki J, Osinga HM, Tsaneva-Atanasova KT. Continuation-Based Numerical Detection of After-Depolarization and Spike-Adding Thresholds. Neural Comput 2013; 25:877-900. [DOI: 10.1162/neco_a_00425] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
The changes in neuronal firing pattern are signatures of brain function, and it is of interest to understand how such changes evolve as a function of neuronal biophysical properties. We address this important problem by the analysis and numerical investigation of a class of mechanistic mathematical models. We focus on a hippocampal pyramidal neuron model and study the occurrence of bursting related to the after-depolarization (ADP) that follows a brief current injection. This type of burst is a transient phenomenon that is not amenable to the classical bifurcation analysis done, for example, for periodic bursting oscillators. In this letter, we show how to formulate such transient behavior as a two-point boundary value problem (2PBVP), which can be solved using well-known continuation methods. The 2PBVP is formulated such that the transient response is represented by a finite orbit segment for which onsets of ADP and additional spikes in a burst can be detected as bifurcations during a one-parameter continuation. This in turn provides us with a direct method to approximate the boundaries of regions in a two-parameter plane where certain model behavior of interest occurs. More precisely, we use two-parameter continuation of the detected onset points to identify the boundaries between regions with and without ADP and bursts with different numbers of spikes. Our 2PBVP formulation is a novel approach to parameter sensitivity analysis that can be applied to a wide range of problems.
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Affiliation(s)
| | - Hinke M. Osinga
- Department of Mathematics, University of Auckland, Auckland 1142, New Zealand
| | - Krasimira T. Tsaneva-Atanasova
- Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, U.K
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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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Nowacki J, Osinga HM, Tsaneva-Atanasova K. Dynamical systems analysis of spike-adding mechanisms in transient bursts. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2012; 2:7. [PMID: 22655748 PMCID: PMC3497719 DOI: 10.1186/2190-8567-2-7] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2011] [Accepted: 02/13/2012] [Indexed: 05/03/2023]
Abstract
Transient bursting behaviour of excitable cells, such as neurons, is a common feature observed experimentally, but theoretically, it is not well understood. We analyse a five-dimensional simplified model of after-depolarisation that exhibits transient bursting behaviour when perturbed with a short current injection. Using one-parameter continuation of the perturbed orbit segment formulated as a well-posed boundary value problem, we show that the spike-adding mechanism is a canard-like transition that has a different character from known mechanisms for periodic burst solutions. The biophysical basis of the model gives a natural time-scale separation, which allows us to explain the spike-adding mechanism using geometric singular perturbation theory, but it does not involve actual bifurcations as for periodic bursts. We show that unstable sheets of the critical manifold, formed by saddle equilibria of the system that only exist in a singular limit, are responsible for the spike-adding transition; the transition is organised by the slow flow on the critical manifold near folds of this manifold. Our analysis shows that the orbit segment during the spike-adding transition includes a fast transition between two unstable sheets of the slow manifold that are of saddle type. We also discuss a different parameter regime where the presence of additional saddle equilibria of the full system alters the spike-adding mechanism.
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Affiliation(s)
- Jakub Nowacki
- Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen’s Building, University Walk, Bristol, BS8 1TR, United Kingdom
| | - Hinke M Osinga
- Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand
| | - Krasimira Tsaneva-Atanasova
- Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen’s Building, University Walk, Bristol, BS8 1TR, 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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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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Burst firing transitions in two-compartment pyramidal neuron induced by the perturbation of membrane capacitance. Neurol Sci 2011; 33:595-604. [PMID: 22037696 DOI: 10.1007/s10072-011-0819-6] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2011] [Accepted: 10/06/2011] [Indexed: 10/15/2022]
Abstract
Neuronal membrane capacitance C (m) is one of the prominent factors in action potential initiation and propagation and then influences the firing patterns of neurons. Exploring the roles that C (m) plays in different firing patterns can facilitate the understanding of how different factors might influence neuronal firing behaviors. However, the impacts of variations in C (m) on neuronal firing patterns have been only partly explored until now. In this study, the influence of C (m) on burst firing behaviors of a two-compartment pyramidal neuron (including somatic compartment and dendritic compartment) was investigated by means of computer simulation, the value of C (m) in each compartment was denoted as C (m,s) and C (m,d), respectively. Two cases were considered, in the first case, we let C (m,s) =C (m,d), and then changed them simultaneously. While in the second case, we assumed C (m,s) ≠C (m,d), and then changed them, respectively. From the simulation results obtained from these two cases, it was found that the variation of C (m) in the somatic compartment and the dendritic compartment show much difference, simulated results obtained from the variation of C (m,d) have much more similarities than that of C (m,s) when comparing with the results obtained in the first case under which C (m,s) =C (m,d). These different effects of C (m,s) and C (m,d) on neuronal firing behaviors may result from the different topology and functional roles of soma and dendrites. Numerical results demonstrated in this paper may give us some inspiration in understanding the possible roles of C (m) in burst firing patterns, especially their transitions in compartmental neurons.
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