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Behbood M, Lemaire L, Schleimer JH, Schreiber S. The Na+/K+-ATPase generically enables deterministic bursting in class I neurons by shearing the spike-onset bifurcation structure. PLoS Comput Biol 2024; 20:e1011751. [PMID: 39133755 PMCID: PMC11383233 DOI: 10.1371/journal.pcbi.1011751] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2023] [Revised: 09/09/2024] [Accepted: 08/02/2024] [Indexed: 09/10/2024] Open
Abstract
Slow brain rhythms, for example during slow-wave sleep or pathological conditions like seizures and spreading depolarization, can be accompanied by oscillations in extracellular potassium concentration. Such slow brain rhythms typically have a lower frequency than tonic action-potential firing. They are assumed to arise from network-level mechanisms, involving synaptic interactions and delays, or from intrinsically bursting neurons. Neuronal burst generation is commonly attributed to ion channels with slow kinetics. Here, we explore an alternative mechanism generically available to all neurons with class I excitability. It is based on the interplay of fast-spiking voltage dynamics with a one-dimensional slow dynamics of the extracellular potassium concentration, mediated by the activity of the Na+/K+-ATPase. We use bifurcation analysis of the complete system as well as the slow-fast method to reveal that this coupling suffices to generate a hysteresis loop organized around a bistable region that emerges from a saddle-node loop bifurcation-a common feature of class I excitable neurons. Depending on the strength of the Na+/K+-ATPase, bursts are generated from pump-induced shearing the bifurcation structure, spiking is tonic, or cells are silenced via depolarization block. We suggest that transitions between these dynamics can result from disturbances in extracellular potassium regulation, such as glial malfunction or hypoxia affecting the Na+/K+-ATPase activity. The identified minimal mechanistic model outlining the sodium-potassium pump's generic contribution to burst dynamics can, therefore, contribute to a better mechanistic understanding of pathologies such as epilepsy syndromes and, potentially, inform therapeutic strategies.
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Affiliation(s)
- Mahraz Behbood
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience, Berlin, Germany
| | - Louisiane Lemaire
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience, Berlin, Germany
| | - Jan-Hendrik Schleimer
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience, Berlin, Germany
| | - Susanne Schreiber
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience, Berlin, Germany
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2
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Abstract
Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach. Bursting is ubiquitous in cellular excitable rhythms and comes in a plethora of patterns, both experimentally recorded and reproduced through models. As these different patterns may reflect different coding or information properties, it is therefore crucial to develop modeling frameworks that can both capture them and understand their characteristics. In this review, we propose a comprehensive account of the main bursting classification systems that have been developed over the past 40 years, together with recent developments allowing us to extend these classifications. Based upon bifurcation theory and heavily reliant on timescale separation, these schemes take full advantage of the fast subsystem analysis, obtained when slow variables are frozen and considered as bifurcation parameters. We complement this classical view by showing that nontrivial slow subsystem may also encode key informations important to classify bursting rhythms, due to the presence of so-called folded-node singularities. We provide minimal idealized models as well as one generic conductance-based example displaying bursting oscillations that require our extended classification in order to be fully characterized. We also highlight examples of biological data that could be suitably revisited with the lenses of this extended classifications and could lead to new models of complex cellular activity.
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Affiliation(s)
- Mathieu Desroches
- MathNeuro Team, Inria Sophia Antipolis Méditerranée Research Centre, Sophia Antipolis, France
- MCEN Team, Basque Centre for Applied Mathematics (BCAM), Bilbao, Bizkaia, Spain
- * E-mail: (MD); (SR)
| | - John Rinzel
- Center for Neural Science, New York University, New York, New York, United States of America
- Courant Institute for Mathematical Sciences, New York University, New York, New York, United States of America
| | - Serafim Rodrigues
- MCEN Team, Basque Centre for Applied Mathematics (BCAM), Bilbao, Bizkaia, Spain
- Ikerbasque, The Basque Science Foundation, Bilbao, Bizkaia, Spain
- * E-mail: (MD); (SR)
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Somatostatin-Positive Interneurons Contribute to Seizures in SCN8A Epileptic Encephalopathy. J Neurosci 2021; 41:9257-9273. [PMID: 34544834 DOI: 10.1523/jneurosci.0718-21.2021] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022] Open
Abstract
SCN8A epileptic encephalopathy is a devastating epilepsy syndrome caused by mutant SCN8A, which encodes the voltage-gated sodium channel NaV1.6. To date, it is unclear if and how inhibitory interneurons, which express NaV1.6, influence disease pathology. Using both sexes of a transgenic mouse model of SCN8A epileptic encephalopathy, we found that selective expression of the R1872W SCN8A mutation in somatostatin (SST) interneurons was sufficient to convey susceptibility to audiogenic seizures. Patch-clamp electrophysiology experiments revealed that SST interneurons from mutant mice were hyperexcitable but hypersensitive to action potential failure via depolarization block under normal and seizure-like conditions. Remarkably, GqDREADD-mediated activation of WT SST interneurons resulted in prolonged electrographic seizures and was accompanied by SST hyperexcitability and depolarization block. Aberrantly large persistent sodium currents, a hallmark of SCN8A mutations, were observed and were found to contribute directly to aberrant SST physiology in computational modeling and pharmacological experiments. These novel findings demonstrate a critical and previously unidentified contribution of SST interneurons to seizure generation not only in SCN8A epileptic encephalopathy, but epilepsy in general.SIGNIFICANCE STATEMENT SCN8A epileptic encephalopathy is a devastating neurological disorder that results from de novo mutations in the sodium channel isoform Nav1.6. Inhibitory neurons express NaV1.6, yet their contribution to seizure generation in SCN8A epileptic encephalopathy has not been determined. We show that mice expressing a human-derived SCN8A variant (R1872W) selectively in somatostatin (SST) interneurons have audiogenic seizures. Physiological recordings from SST interneurons show that SCN8A mutations lead to an elevated persistent sodium current which drives initial hyperexcitability, followed by premature action potential failure because of depolarization block. Furthermore, chemogenetic activation of WT SST interneurons leads to audiogenic seizure activity. These findings provide new insight into the importance of SST inhibitory interneurons in seizure initiation, not only in SCN8A epileptic encephalopathy, but for epilepsy broadly.
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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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Barrio R, Ibáñez S, Pérez L, Serrano S. Classification of fold/hom and fold/Hopf spike-adding phenomena. CHAOS (WOODBURY, N.Y.) 2021; 31:043120. [PMID: 34251261 DOI: 10.1063/5.0037942] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2020] [Accepted: 03/26/2021] [Indexed: 06/13/2023]
Abstract
The Hindmarsh-Rose neural model is widely accepted as an important prototype for fold/hom and fold/Hopf burstings. In this paper, we are interested in the mechanisms for the production of extra spikes in a burst, and we show the whole parametric panorama in an unified way. In the fold/hom case, two types are distinguished: a continuous one, where the bursting periodic orbit goes through bifurcations but persists along the whole process and a discontinuous one, where the transition is abrupt and happens after a sequence of chaotic events. In the former case, we speak about canard-induced spike-adding and in the second one, about chaos-induced spike-adding. For fold/Hopf bursting, a single (and continuous) mechanism is distinguished. Separately, all these mechanisms are presented, to some extent, in the literature. However, our full perspective allows us to construct a spike-adding map and, more significantly, to understand the dynamics exhibited when borders are crossed, that is, transitions between types of processes, a crucial point not previously studied.
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Affiliation(s)
- Roberto Barrio
- Departamento de Matemática Aplicada and IUMA, Computational Dynamics Group, University of Zaragoza, Zaragoza E-50009, Spain
| | - Santiago Ibáñez
- Departamento de Matemáticas, University of Oviedo, E-33007 Oviedo, Spain
| | - Lucía Pérez
- Departamento de Matemáticas, University of Oviedo, E-33007 Oviedo, Spain
| | - Sergio Serrano
- Departamento de Matemática Aplicada and IUMA, Computational Dynamics Group, University of Zaragoza, Zaragoza E-50009, Spain
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6
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Darki F, Rankin J. Methods to assess binocular rivalry with periodic stimuli. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2020; 10:10. [PMID: 32542516 PMCID: PMC7295892 DOI: 10.1186/s13408-020-00087-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/2019] [Accepted: 06/04/2020] [Indexed: 05/29/2023]
Abstract
Binocular rivalry occurs when the two eyes are presented with incompatible stimuli and perception alternates between these two stimuli. This phenomenon has been investigated in two types of experiments: (1) Traditional experiments where the stimulus is fixed, (2) eye-swap experiments in which the stimulus periodically swaps between eyes many times per second (Logothetis et al. in Nature 380(6575):621-624, 1996). In spite of the rapid swapping between eyes, perception can be stable for many seconds with specific stimulus parameter configurations. Wilson introduced a two-stage, hierarchical model to explain both types of experiments (Wilson in Proc. Natl. Acad. Sci. 100(24):14499-14503, 2003). Wilson's model and other rivalry models have been only studied with bifurcation analysis for fixed inputs and different types of dynamical behavior that can occur with periodically forcing inputs have not been investigated. Here we report (1) a more complete description of the complex dynamics in the unforced Wilson model, (2) a bifurcation analysis with periodic forcing. Previously, bifurcation analysis of the Wilson model with fixed inputs has revealed three main types of dynamical behaviors: Winner-takes-all (WTA), Rivalry oscillations (RIV), Simultaneous activity (SIM). Our results have revealed richer dynamics including mixed-mode oscillations (MMOs) and a period-doubling cascade, which corresponds to low-amplitude WTA (LAWTA) oscillations. On the other hand, studying rivalry models with numerical continuation shows that periodic forcing with high frequency (e.g. 18 Hz, known as flicker) modulates the three main types of behaviors that occur with fixed inputs with forcing frequency (WTA-Mod, RIV-Mod, SIM-Mod). However, dynamical behavior will be different with low frequency periodic forcing (around 1.5 Hz, so-called swap). In addition to WTA-Mod and SIM-Mod, cycle skipping, multi-cycle skipping and chaotic dynamics are found. This research provides a framework for either assessing binocular rivalry models to check consistency with empirical results, or for better understanding neural dynamics and mechanisms necessary to implement a minimal binocular rivalry model.
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Affiliation(s)
- Farzaneh Darki
- Department of Mathematics, College of Engineering, Mathematics & Physical Sciences, University of Exeter, Exeter, UK.
| | - James Rankin
- Department of Mathematics, College of Engineering, Mathematics & Physical Sciences, University of Exeter, Exeter, UK
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Barrio R, Ibáñez S, Pérez L. Homoclinic organization in the Hindmarsh-Rose model: A three parameter study. CHAOS (WOODBURY, N.Y.) 2020; 30:053132. [PMID: 32491901 DOI: 10.1063/1.5138919] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2019] [Accepted: 04/24/2020] [Indexed: 06/11/2023]
Abstract
Bursting phenomena are found in a wide variety of fast-slow systems. In this article, we consider the Hindmarsh-Rose neuron model, where, as it is known in the literature, there are homoclinic bifurcations involved in the bursting dynamics. However, the global homoclinic structure is far from being fully understood. Working in a three-parameter space, the results of our numerical analysis show a complex atlas of bifurcations, which extends from the singular limit to regions where a fast-slow perspective no longer applies. Based on this information, we propose a global theoretical description. Surfaces of codimension-one homoclinic bifurcations are exponentially close to each other in the fast-slow regime. Remarkably, explained by the specific properties of these surfaces, we show how the Hindmarsh-Rose model exhibits isolas of homoclinic bifurcations when appropriate two-dimensional slices are considered in the three-parameter space. On the other hand, these homoclinic bifurcation surfaces contain curves corresponding to parameter values where additional degeneracies are exhibited. These codimension-two bifurcation curves organize the bifurcations associated with the spike-adding process and they behave like the "spines-of-a-book," gathering "pages" of bifurcations of periodic orbits. Depending on how the parameter space is explored, homoclinic phenomena may be absent or far away, but their organizing role in the bursting dynamics is beyond doubt, since the involved bifurcations are generated in them. This is shown in the global analysis and in the proposed theoretical scheme.
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Affiliation(s)
- Roberto Barrio
- Departamento de Matemática Aplicada and IUMA, University of Zaragoza, E-50009 Zaragoza, Spain
| | - Santiago Ibáñez
- Departamento de Matemáticas, University of Oviedo, E-33007 Oviedo, Spain
| | - Lucía Pérez
- Departamento de Matemáticas, University of Oviedo, E-33007 Oviedo, Spain
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8
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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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9
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Hasan CR, Krauskopf B, Osinga HM. Saddle Slow Manifolds and Canard Orbits in [Formula: see text] and Application to the Full Hodgkin-Huxley Model. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2018; 8:5. [PMID: 29675585 PMCID: PMC5908812 DOI: 10.1186/s13408-018-0060-1] [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: 11/19/2017] [Accepted: 04/05/2018] [Indexed: 05/22/2023]
Abstract
Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron models typically involve observable differences in time scales. The Hodgkin-Huxley model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5-32, 2007) nondimensionalized this model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio ε. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type.In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in [Formula: see text]. We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin-Huxley model in organizing MMOs and determining the firing rates of action potentials. For ε sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the model.
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Affiliation(s)
- Cris R. Hasan
- Department of Mathematics, The University of Auckland, Auckland, New Zealand
| | - Bernd Krauskopf
- Department of Mathematics, The University of Auckland, Auckland, New Zealand
| | - Hinke M. Osinga
- Department of Mathematics, The University of Auckland, Auckland, New Zealand
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Baier G, Taylor PN, Wang Y. Understanding Epileptiform After-Discharges as Rhythmic Oscillatory Transients. Front Comput Neurosci 2017; 11:25. [PMID: 28458634 PMCID: PMC5394159 DOI: 10.3389/fncom.2017.00025] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2017] [Accepted: 03/29/2017] [Indexed: 01/24/2023] Open
Abstract
Electro-cortical activity in patients with epilepsy may show abnormal rhythmic transients in response to stimulation. Even when using the same stimulation parameters in the same patient, wide variability in the duration of transient response has been reported. These transients have long been considered important for the mapping of the excitability levels in the epileptic brain but their dynamic mechanism is still not well understood. To investigate the occurrence of abnormal transients dynamically, we use a thalamo-cortical neural population model of epileptic spike-wave activity and study the interaction between slow and fast subsystems. In a reduced version of the thalamo-cortical model, slow wave oscillations arise from a fold of cycles (FoC) bifurcation. This marks the onset of a region of bistability between a high amplitude oscillatory rhythm and the background state. In vicinity of the bistability in parameter space, the model has excitable dynamics, showing prolonged rhythmic transients in response to suprathreshold pulse stimulation. We analyse the state space geometry of the bistable and excitable states, and find that the rhythmic transient arises when the impending FoC bifurcation deforms the state space and creates an area of locally reduced attraction to the fixed point. This area essentially allows trajectories to dwell there before escaping to the stable steady state, thus creating rhythmic transients. In the full thalamo-cortical model, we find a similar FoC bifurcation structure. Based on the analysis, we propose an explanation of why stimulation induced epileptiform activity may vary between trials, and predict how the variability could be related to ongoing oscillatory background activity. We compare our dynamic mechanism with other mechanisms (such as a slow parameter change) to generate excitable transients, and we discuss the proposed excitability mechanism in the context of stimulation responses in the epileptic cortex.
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Affiliation(s)
- Gerold Baier
- Cell and Developmental Biology, University College LondonLondon, UK
| | - Peter N Taylor
- Institute of Neuroscience, Newcastle UniversityNewcastle upon Tyne, UK.,Interdisciplinary Computing and Complex BioSystems (ICOS), School of Computing Science, Newcastle UniversityNewcastle, UK.,Institute of Neurology, University College LondonLondon, UK
| | - Yujiang Wang
- Institute of Neuroscience, Newcastle UniversityNewcastle upon Tyne, UK.,Interdisciplinary Computing and Complex BioSystems (ICOS), School of Computing Science, Newcastle UniversityNewcastle, UK.,Institute of Neurology, University College LondonLondon, UK
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11
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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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12
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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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13
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Abbasian AH, Fallah H, Razvan MR. Symmetric bursting behaviors in the generalized FitzHugh-Nagumo model. BIOLOGICAL CYBERNETICS 2013; 107:465-476. [PMID: 23801268 DOI: 10.1007/s00422-013-0559-1] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2012] [Accepted: 05/16/2013] [Indexed: 06/02/2023]
Abstract
In the current paper, we have investigated the generalized FitzHugh-Nagumo model. We have shown that symmetric bursting behaviors of different types could be observed in this model with an appropriate recovery term. A modified version of this system is used to construct bursting activities. Furthermore, we have shown some numerical examples of delayed Hopf bifurcation and canard phenomenon in the symmetric bursting of super-Hopf/homoclinic type near its super-Hopf and homoclinic bifurcations, respectively.
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Affiliation(s)
- A H Abbasian
- School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran
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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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