1
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Panahi S, Do Y, Hastings A, Lai YC. Rate-induced tipping in complex high-dimensional ecological networks. Proc Natl Acad Sci U S A 2023; 120:e2308820120. [PMID: 38091288 PMCID: PMC10743502 DOI: 10.1073/pnas.2308820120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2023] [Accepted: 11/15/2023] [Indexed: 12/24/2023] Open
Abstract
In an ecosystem, environmental changes as a result of natural and human processes can cause some key parameters of the system to change with time. Depending on how fast such a parameter changes, a tipping point can occur. Existing works on rate-induced tipping, or R-tipping, offered a theoretical way to study this phenomenon but from a local dynamical point of view, revealing, e.g., the existence of a critical rate for some specific initial condition above which a tipping point will occur. As ecosystems are subject to constant disturbances and can drift away from their equilibrium point, it is necessary to study R-tipping from a global perspective in terms of the initial conditions in the entire relevant phase space region. In particular, we introduce the notion of the probability of R-tipping defined for initial conditions taken from the whole relevant phase space. Using a number of real-world, complex mutualistic networks as a paradigm, we find a scaling law between this probability and the rate of parameter change and provide a geometric theory to explain the law. The real-world implication is that even a slow parameter change can lead to a system collapse with catastrophic consequences. In fact, to mitigate the environmental changes by merely slowing down the parameter drift may not always be effective: Only when the rate of parameter change is reduced to practically zero would the tipping be avoided. Our global dynamics approach offers a more complete and physically meaningful way to understand the important phenomenon of R-tipping.
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Affiliation(s)
- Shirin Panahi
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ85287
| | - Younghae Do
- Department of Mathematics, Nonlinear Dynamics Mathematical Application Center, Kyungpook National University, Daegu41566, Republic of Korea
| | - Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, CA95616
- Santa Fe Institute, Santa Fe, NM87501
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ85287
- Department of Physics, Arizona State University, Tempe, AZ85287
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2
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Penalva J, Desroches M, Teruel AE, Vich C. Slow passage through a Hopf-like bifurcation in piecewise linear systems: Application to elliptic bursting. CHAOS (WOODBURY, N.Y.) 2022; 32:123109. [PMID: 36587327 DOI: 10.1063/5.0101778] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/2022] [Accepted: 11/09/2022] [Indexed: 06/17/2023]
Abstract
The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly varying quantity replacing a static parameter induces the solutions of the resulting slow-fast system to feel the effect of the Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow-fast dynamical systems; in the present work, we study it for the first time in piecewise linear (PWL) slow-fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular, through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slow-fast system to exhibit a slow passage through a Hopf-like bifurcation, in link with possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the so-called way-in/way-out function. Finally, we investigate this slow passage effect in the Doi-Kumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.
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Affiliation(s)
- J Penalva
- Departament de Matemàtiques i Informàtica & IAC3, Universitat de les Illes Balears, Palma 07122, Spain
| | - M Desroches
- MathNeuro Project-Team, Inria at Université Côte d'Azur, Sophia Antipolis 06902, France
| | - A E Teruel
- Departament de Matemàtiques i Informàtica & IAC3, Universitat de les Illes Balears, Palma 07122, Spain
| | - C Vich
- Departament de Matemàtiques i Informàtica & IAC3, Universitat de les Illes Balears, Palma 07122, Spain
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3
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Cross-scale excitability in networks of quadratic integrate-and-fire neurons. PLoS Comput Biol 2022; 18:e1010569. [PMID: 36191049 PMCID: PMC9560555 DOI: 10.1371/journal.pcbi.1010569] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2022] [Revised: 10/13/2022] [Accepted: 09/13/2022] [Indexed: 11/05/2022] Open
Abstract
From the action potentials of neurons and cardiac cells to the amplification of calcium signals in oocytes, excitability is a hallmark of many biological signalling processes. In recent years, excitability in single cells has been related to multiple-timescale dynamics through canards, special solutions which determine the effective thresholds of the all-or-none responses. However, the emergence of excitability in large populations remains an open problem. Here, we show that the mechanism of excitability in large networks and mean-field descriptions of coupled quadratic integrate-and-fire (QIF) cells mirrors that of the individual components. We initially exploit the Ott-Antonsen ansatz to derive low-dimensional dynamics for the coupled network and use it to describe the structure of canards via slow periodic forcing. We demonstrate that the thresholds for onset and offset of population firing can be found in the same way as those of the single cell. We combine theoretical analysis and numerical computations to develop a novel and comprehensive framework for excitability in large populations, applicable not only to models amenable to Ott-Antonsen reduction, but also to networks without a closed-form mean-field limit, in particular sparse networks.
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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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5
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Köksal Ersöz E, Modolo J, Bartolomei F, Wendling F. Neural mass modeling of slow-fast dynamics of seizure initiation and abortion. PLoS Comput Biol 2020; 16:e1008430. [PMID: 33166277 PMCID: PMC7676664 DOI: 10.1371/journal.pcbi.1008430] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2020] [Revised: 11/19/2020] [Accepted: 10/08/2020] [Indexed: 12/31/2022] Open
Abstract
Epilepsy is a dynamic and complex neurological disease affecting about 1% of the worldwide population, among which 30% of the patients are drug-resistant. Epilepsy is characterized by recurrent episodes of paroxysmal neural discharges (the so-called seizures), which manifest themselves through a large-amplitude rhythmic activity observed in depth-EEG recordings, in particular in local field potentials (LFPs). The signature characterizing the transition to seizures involves complex oscillatory patterns, which could serve as a marker to prevent seizure initiation by triggering appropriate therapeutic neurostimulation methods. To investigate such protocols, neurophysiological lumped-parameter models at the mesoscopic scale, namely neural mass models, are powerful tools that not only mimic the LFP signals but also give insights on the neural mechanisms related to different stages of seizures. Here, we analyze the multiple time-scale dynamics of a neural mass model and explain the underlying structure of the complex oscillations observed before seizure initiation. We investigate population-specific effects of the stimulation and the dependence of stimulation parameters on synaptic timescales. In particular, we show that intermediate stimulation frequencies (>20 Hz) can abort seizures if the timescale difference is pronounced. Those results have the potential in the design of therapeutic brain stimulation protocols based on the neurophysiological properties of tissue.
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Affiliation(s)
| | - Julien Modolo
- University of Rennes, Inserm-U1099, LTSI, Rennes, France
| | - Fabrice Bartolomei
- Aix Marseille University, Inserm, INS, Institut de Neurosciences des Systèmes, Marseille, France
- APHM, Timone Hospital, Clinical Neurophysiology, Marseille, France
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6
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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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7
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Velarde OM, Urdapilleta E, Mato G, Dellavale D. Bifurcation structure determines different phase-amplitude coupling patterns in the activity of biologically plausible neural networks. Neuroimage 2019; 202:116031. [PMID: 31330244 DOI: 10.1016/j.neuroimage.2019.116031] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2019] [Revised: 07/10/2019] [Accepted: 07/16/2019] [Indexed: 12/15/2022] Open
Abstract
Phase-amplitude cross frequency coupling (PAC) is a rather ubiquitous phenomenon that has been observed in a variety of physical domains; however, the mechanisms underlying the emergence of PAC and its functional significance in the context of neural processes are open issues under debate. In this work we analytically demonstrate that PAC phenomenon naturally emerges in mean-field models of biologically plausible networks, as a signature of specific bifurcation structures. The proposed analysis, based on bifurcation theory, allows the identification of the mechanisms underlying oscillatory dynamics that are essentially different in the context of PAC. Specifically, we found that two PAC classes can coexist in the complex dynamics of the analyzed networks: 1) harmonic PAC which is an epiphenomenon of the nonsinusoidal waveform shape characterized by the linear superposition of harmonically related spectral components, and 2) nonharmonic PAC associated with "true" coupled oscillatory dynamics with independent frequencies elicited by a secondary Hopf bifurcation and mechanisms involving periodic excitation/inhibition (PEI) of a network population. Importantly, these two PAC types have been experimentally observed in a variety of neural architectures confounding traditional parametric and nonparametric PAC metrics, like those based on linear filtering or the waveform shape analysis, due to the fact that these methods operate on a single one-dimensional projection of an intrinsically multidimensional system dynamics. We exploit the proposed tools to study the functional significance of the PAC phenomenon in the context of Parkinson's disease (PD). Our results show that pathological slow oscillations (e.g. β band) and nonharmonic PAC patterns emerge from dissimilar underlying mechanisms (bifurcations) and are associated to the competition of different BG-thalamocortical loops. Thus, this study provides theoretical arguments that demonstrate that nonharmonic PAC is not an epiphenomenon related to the pathological β band oscillations, thus supporting the experimental evidence about the relevance of PAC as a potential biomarker of PD.
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Affiliation(s)
- Osvaldo Matías Velarde
- Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Energía Atómica (CNEA), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad Nacional de Cuyo (UNCUYO), Av. E. Bustillo 9500, R8402AGP, San Carlos de Bariloche, Río Negro, Argentina
| | - Eugenio Urdapilleta
- Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Energía Atómica (CNEA), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad Nacional de Cuyo (UNCUYO), Av. E. Bustillo 9500, R8402AGP, San Carlos de Bariloche, Río Negro, Argentina
| | - Germán Mato
- Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Energía Atómica (CNEA), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad Nacional de Cuyo (UNCUYO), Av. E. Bustillo 9500, R8402AGP, San Carlos de Bariloche, Río Negro, Argentina.
| | - Damián Dellavale
- Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Energía Atómica (CNEA), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad Nacional de Cuyo (UNCUYO), Av. E. Bustillo 9500, R8402AGP, San Carlos de Bariloche, Río Negro, Argentina.
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8
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Köksal Ersöz E, Desroches M, Mirasso CR, Rodrigues S. Anticipation via canards in excitable systems. CHAOS (WOODBURY, N.Y.) 2019; 29:013111. [PMID: 30709107 DOI: 10.1063/1.5050018] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2018] [Accepted: 12/19/2018] [Indexed: 06/09/2023]
Abstract
Neurons can anticipate incoming signals by exploiting a physiological mechanism that is not well understood. This article offers a novel explanation on how a receiver neuron can predict the sender's dynamics in a unidirectionally-coupled configuration, in which both sender and receiver follow the evolution of a multi-scale excitable system. We present a novel theoretical viewpoint based on a mathematical object, called canard, to explain anticipation in excitable systems. We provide a numerical approach, which allows to determine the transient effects of canards. To demonstrate the general validity of canard-mediated anticipation in the context of excitable systems, we illustrate our framework in two examples, a multi-scale radio-wave circuit (the van der Pol model) that inspired a caricature neuronal model (the FitzHugh-Nagumo model) and a biophysical neuronal model (a 2-dimensional reduction of the Hodgkin-Huxley model), where canards act as messengers to the senders' prediction. We also propose an experimental paradigm that would enable experimental neuroscientists to validate our predictions. We conclude with an outlook to possible fascinating research avenues to further unfold the mechanisms underpinning anticipation. We envisage that our approach can be employed by a wider class of excitable systems with appropriate theoretical extensions.
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Affiliation(s)
- Elif Köksal Ersöz
- MathNeuro Team, Inria Sophia Antipolis Méditerranée, 06902 Sophia Antipolis, France
| | - Mathieu Desroches
- MathNeuro Team, Inria Sophia Antipolis Méditerranée, 06902 Sophia Antipolis, France
| | - Claudio R Mirasso
- Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Universitat de les Illes Baleares, Campus UIB, E-07122 Palma de Mallorca, Spain
| | - Serafim Rodrigues
- IKERBASQUE: The Basque Foundation for Science 48013 Bilbao, Basque Country, Spain
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9
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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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10
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Shoffner S, Schnell S. Approaches for the estimation of timescales in nonlinear dynamical systems: Timescale separation in enzyme kinetics as a case study. Math Biosci 2017; 287:122-129. [DOI: 10.1016/j.mbs.2016.09.001] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2016] [Revised: 08/30/2016] [Accepted: 09/01/2016] [Indexed: 10/21/2022]
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11
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Avitabile D, Desroches M, Knobloch E. Spatiotemporal canards in neural field equations. Phys Rev E 2017; 95:042205. [PMID: 28505875 DOI: 10.1103/physreve.95.042205] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2016] [Indexed: 06/07/2023]
Abstract
Canards are special solutions to ordinary differential equations that follow invariant repelling slow manifolds for long time intervals. In realistic biophysical single-cell models, canards are responsible for several complex neural rhythms observed experimentally, but their existence and role in spatially extended systems is largely unexplored. We identify and describe a type of coherent structure in which a spatial pattern displays temporal canard behavior. Using interfacial dynamics and geometric singular perturbation theory, we classify spatiotemporal canards and give conditions for the existence of folded-saddle and folded-node canards. We find that spatiotemporal canards are robust to changes in the synaptic connectivity and firing rate. The theory correctly predicts the existence of spatiotemporal canards with octahedral symmetry in a neural field model posed on the unit sphere.
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Affiliation(s)
- D Avitabile
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG9 7RD, United Kingdom
| | - M Desroches
- Inria Sophia Antipolis Méditerranée Research Centre, MathNeuro Team, 2004 route des Lucioles-Boîte Postale 93 06902 Sophia Antipolis, Cedex, France
| | - E Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
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12
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Wang L, Wang H, Yu L, Chen Y. Spike-Threshold Variability Originated from Separatrix-Crossing in Neuronal Dynamics. Sci Rep 2016; 6:31719. [PMID: 27546614 PMCID: PMC4992847 DOI: 10.1038/srep31719] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2016] [Accepted: 07/26/2016] [Indexed: 11/09/2022] Open
Abstract
The threshold voltage for action potential generation is a key regulator of neuronal signal processing, yet the mechanism of its dynamic variation is still not well described. In this paper, we propose that threshold phenomena can be classified as parameter thresholds and state thresholds. Voltage thresholds which belong to the state threshold are determined by the 'general separatrix' in state space. We demonstrate that the separatrix generally exists in the state space of neuron models. The general form of separatrix was assumed as the function of both states and stimuli and the previously assumed threshold evolving equation versus time is naturally deduced from the separatrix. In terms of neuronal dynamics, the threshold voltage variation, which is affected by different stimuli, is determined by crossing the separatrix at different points in state space. We suggest that the separatrix-crossing mechanism in state space is the intrinsic dynamic mechanism for threshold voltages and post-stimulus threshold phenomena. These proposals are also systematically verified in example models, three of which have analytic separatrices and one is the classic Hodgkin-Huxley model. The separatrix-crossing framework provides an overview of the neuronal threshold and will facilitate understanding of the nature of threshold variability.
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Affiliation(s)
- Longfei Wang
- Institute of Theoretical Physics, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Hengtong Wang
- College of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Lianchun Yu
- Institute of Theoretical Physics, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Yong Chen
- Center of Soft Matter Physics and its Application, Beihang University, Beijing 100191, China.,School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China
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13
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Perryman C, Wieczorek S. Adapting to a changing environment: non-obvious thresholds in multi-scale systems. Proc Math Phys Eng Sci 2014; 470:20140226. [PMID: 25294963 DOI: 10.1098/rspa.2014.0226] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2014] [Accepted: 07/15/2014] [Indexed: 11/12/2022] Open
Abstract
Many natural and technological systems fail to adapt to changing external conditions and move to a different state if the conditions vary too fast. Such 'non-adiabatic' processes are ubiquitous, but little understood. We identify these processes with a new nonlinear phenomenon-an intricate threshold where a forced system fails to adiabatically follow a changing stable state. In systems with multiple time scales, we derive existence conditions that show such thresholds to be generic, but non-obvious, meaning they cannot be captured by traditional stability theory. Rather, the phenomenon can be analysed using concepts from modern singular perturbation theory: folded singularities and canard trajectories, including composite canards. Thus, non-obvious thresholds should explain the failure to adapt to a changing environment in a wide range of multi-scale systems including: tipping points in the climate system, regime shifts in ecosystems, excitability in nerve cells, adaptation failure in regulatory genes and adiabatic switching in technology.
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Affiliation(s)
- Clare Perryman
- Mathematics Research Institute, University of Exeter , Exeter EX4 4QF, UK
| | - Sebastian Wieczorek
- Mathematics Research Institute, University of Exeter , Exeter EX4 4QF, UK ; Department of Applied Mathematics , Western Gateway Building, University College Cork , Ireland
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14
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Tonnelier A. Threshold curve for the excitability of bidimensional spiking neurons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022701. [PMID: 25215752 DOI: 10.1103/physreve.90.022701] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2014] [Indexed: 06/03/2023]
Abstract
We shed light on the threshold for spike initiation in two-dimensional neuron models. A threshold criterion that depends on both the membrane voltage and the recovery variable is proposed. This approach provides a simple and unified framework that accounts for numerous voltage threshold properties including adaptation, variability, and time-dependent dynamics. In addition, neural features such as accommodation, inhibition-induced spike, and postinhibitory (-excitatory) facilitation are the direct consequences of the existence of a threshold curve. Implications for neural modeling are also discussed.
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Affiliation(s)
- Arnaud Tonnelier
- INRIA-Grenoble, 655 avenue de l'Europe, Montbonnot, 38334 Saint Ismier, France
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15
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Dahlem MA, Schumacher J, Hübel N. Linking a genetic defect in migraine to spreading depression in a computational model. PeerJ 2014; 2:e379. [PMID: 24860703 PMCID: PMC4017887 DOI: 10.7717/peerj.379] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2014] [Accepted: 04/23/2014] [Indexed: 11/20/2022] Open
Abstract
Familial hemiplegic migraine (FHM) is a rare subtype of migraine with aura. A mutation causing FHM type 3 (FHM3) has been identified in SCN1A encoding the Nav1.1 Na+ channel. This genetic defect affects the inactivation gate. While the Na+ tail currents following voltage steps are consistent with both hyperexcitability and hypoexcitability, in this computational study, we investigate functional consequences beyond these isolated events. Our extended Hodgkin–Huxley framework establishes a connection between genotype and cellular phenotype, i.e., the pathophysiological dynamics that spans over multiple time scales and is relevant to migraine with aura. In particular, we investigate the dynamical repertoire from normal spiking (milliseconds) to spreading depression and anoxic depolarization (tens of seconds) and show that FHM3 mutations render gray matter tissue more vulnerable to spreading depression despite opposing effects associated with action potential generation. We conclude that the classification in terms of hypoexcitability vs. hyperexcitability is too simple a scheme. Our mathematical analysis provides further basic insight into also previously discussed criticisms against this scheme based on psychophysical and clinical data.
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Affiliation(s)
- Markus A Dahlem
- Department of Physics, Humboldt Universität zu Berlin , Berlin , Germany
| | - Julia Schumacher
- Bernstein Center for Computational Neuroscience, Technische Universität Berlin , Berlin , Germany
| | - Niklas Hübel
- Department of Theoretical Physics, Technische Universität Berlin , Berlin , Germany
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16
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Cannon J, McCarthy MM, Lee S, Lee J, Börgers C, Whittington MA, Kopell N. Neurosystems: brain rhythms and cognitive processing. Eur J Neurosci 2014; 39:705-19. [PMID: 24329933 PMCID: PMC4916881 DOI: 10.1111/ejn.12453] [Citation(s) in RCA: 130] [Impact Index Per Article: 13.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2013] [Revised: 10/29/2013] [Accepted: 11/11/2013] [Indexed: 11/30/2022]
Abstract
Neuronal rhythms are ubiquitous features of brain dynamics, and are highly correlated with cognitive processing. However, the relationship between the physiological mechanisms producing these rhythms and the functions associated with the rhythms remains mysterious. This article investigates the contributions of rhythms to basic cognitive computations (such as filtering signals by coherence and/or frequency) and to major cognitive functions (such as attention and multi-modal coordination). We offer support to the premise that the physiology underlying brain rhythms plays an essential role in how these rhythms facilitate some cognitive operations.
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Affiliation(s)
- Jonathan Cannon
- Department of Mathematics and StatisticsBoston University111 Cummington MallBostonMA02215USA
| | - Michelle M. McCarthy
- Department of Mathematics and StatisticsBoston University111 Cummington MallBostonMA02215USA
| | - Shane Lee
- Department of NeuroscienceBrown UniversityProvidenceRIUSA
| | - Jung Lee
- Department of Mathematics and StatisticsBoston University111 Cummington MallBostonMA02215USA
| | | | | | - Nancy Kopell
- Department of Mathematics and StatisticsBoston University111 Cummington MallBostonMA02215USA
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17
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Weinberg SH. High-frequency stimulation of excitable cells and networks. PLoS One 2013; 8:e81402. [PMID: 24278435 PMCID: PMC3835437 DOI: 10.1371/journal.pone.0081402] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2013] [Accepted: 10/22/2013] [Indexed: 11/27/2022] Open
Abstract
High-frequency (HF) stimulation has been shown to block conduction in excitable cells including neurons and cardiac myocytes. However, the precise mechanisms underlying conduction block are unclear. Using a multi-scale method, the influence of HF stimulation is investigated in the simplified FitzhHugh-Nagumo and biophysically-detailed Hodgkin-Huxley models. In both models, HF stimulation alters the amplitude and frequency of repetitive firing in response to a constant applied current and increases the threshold to evoke a single action potential in response to a brief applied current pulse. Further, the excitable cells cannot evoke a single action potential or fire repetitively above critical values for the HF stimulation amplitude. Analytical expressions for the critical values and thresholds are determined in the FitzHugh-Nagumo model. In the Hodgkin-Huxley model, it is shown that HF stimulation alters the dynamics of ionic current gating, shifting the steady-state activation, inactivation, and time constant curves, suggesting several possible mechanisms for conduction block. Finally, we demonstrate that HF stimulation of a network of neurons reduces the electrical activity firing rate, increases network synchronization, and for a sufficiently large HF stimulation, leads to complete electrical quiescence. In this study, we demonstrate a novel approach to investigate HF stimulation in biophysically-detailed ionic models of excitable cells, demonstrate possible mechanisms for HF stimulation conduction block in neurons, and provide insight into the influence of HF stimulation on neural networks.
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Affiliation(s)
- Seth H. Weinberg
- Department of Applied Science, The College of William and Mary, Williamsburg, Virginia, United States of America
- * E-mail:
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18
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Yu N, Li YX, Kuske R. A computational study of spike time reliability in two types of threshold dynamics. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2013; 3:11. [PMID: 23945258 PMCID: PMC3849148 DOI: 10.1186/2190-8567-3-11] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/20/2012] [Accepted: 04/23/2013] [Indexed: 06/02/2023]
Abstract
Spike time reliability (STR) refers to the phenomenon in which repetitive applications of a frozen copy of one stochastic signal to a neuron trigger spikes with reliable timing while a constant signal fails to do so. Observed and explored in numerous experimental and theoretical studies, STR is a complex dynamic phenomenon depending on the nature of external inputs as well as intrinsic properties of a neuron. The neuron under consideration could be either quiescent or spontaneously spiking in the absence of the external stimulus. Focusing on the situation in which the unstimulated neuron is quiescent but close to a switching point to oscillations, we numerically analyze STR treating each spike occurrence as a time localized event in a model neuron. We study both the averaged properties as well as individual features of spike-evoking epochs (SEEs). The effects of interactions between spikes is minimized by selecting signals that generate spikes with relatively long interspike intervals (ISIs). Under these conditions, the frequency content of the input signal has little impact on STR. We study two distinct cases, Type I in which the f-I relation (f for frequency, I for applied current) is continuous and Type II where the f-I relation exhibits a jump. STR in the two types shows a number of similar features and differ in some others. SEEs that are capable of triggering spikes show great variety in amplitude and time profile. On average, reliable spike timing is associated with an accelerated increase in the "action" of the signal as a threshold for spike generation is approached. Here, "action" is defined as the average amount of current delivered during a fixed time interval. When individual SEEs are studied, however, their time profiles are found important for triggering more precisely timed spikes. The SEEs that have a more favorable time profile are capable of triggering spikes with higher precision even at lower action levels.
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Affiliation(s)
- Na Yu
- Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2
- Department of Cell Biology and Anatomy, Louisiana State University Health Sciences Center, New Orleans, LA, 70112, USA
| | - Yue-Xian Li
- Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2
| | - Rachel Kuske
- Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2
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Coombes S, Timofeeva Y. Editorial for special issue on neurodynamics. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2013; 3:10. [PMID: 23945204 PMCID: PMC3849104 DOI: 10.1186/2190-8567-3-10] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2013] [Accepted: 06/18/2013] [Indexed: 06/02/2023]
Abstract
"Neurodynamics" is an interdisciplinary area of mathematics where dynamical systems theory (deterministic and stochastic) is the primary tool for elucidating the fundamental mechanisms responsible for the behaviour of neural systems (whether biological or synthetic). A meeting on this topic was held at the International Centre for Mathematical Sciences in Edinburgh from March 5-7 in 2012. In this special issue, we have invited seven of the main contributors to this event to expand on their presentations and highlight the use of mathematics in understanding the dynamics of neural systems.
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Affiliation(s)
- Stephen Coombes
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
| | - Yulia Timofeeva
- Department of Computer Science and Centre for Complexity Science, University of Warwick, Coventry, CV4 7AL, UK
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