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The Effects of Spike Frequency Adaptation and Negative Feedback on the Synchronization of Neural Oscillators
"... this article, we de#ne synchrony to mean a zero phase lag when two identical neurons are identically coupled. This is a mathematical idealization but allows us to be precise when we describe the locking behavior of coupled oscillators ..."
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Cited by 68 (6 self)
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this article, we de#ne synchrony to mean a zero phase lag when two identical neurons are identically coupled. This is a mathematical idealization but allows us to be precise when we describe the locking behavior of coupled oscillators
A universal model for spikefrequency adaptation
 Neural Comput
, 2003
"... Spikefrequency adaptation is a prominent feature of neural dynamics. Among other mechanisms various ionic currents modulating spike generation cause this type of neural adaptation. Prominent examples are voltagegated potassium currents (Mtype currents), the interplay of calcium currents and int ..."
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Cited by 57 (9 self)
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Spikefrequency adaptation is a prominent feature of neural dynamics. Among other mechanisms various ionic currents modulating spike generation cause this type of neural adaptation. Prominent examples are voltagegated potassium currents (Mtype currents), the interplay of calcium currents and intracellular calcium dynamics with calciumgated potassium channels (AHPtype currents), and the slow recovery from inactivation of the fast sodium current. While recent modeling studies have focused on the effects of specic adaptation currents, we derive a universal model for the ringfrequency dynamics of an adapting neuron which is independent of the specic adaptation process and spike generator. The model is completely dened by the neuron's onset fIcurve, steadystate fIcurve, and the time constant of adaptation. For a specic neuron these parameters can be easily determined from electrophysiological measurements without any pharmacological manipulations. At the same time, the simplicity of the model allows one to analyze mathematically how adaptation inuences signal processing on the singleneuron level. In particular, we elucidate the specic nature of highpass lter properties caused by spikefrequency adaptation. The model is limited to ring frequencies higher than the reciprocal adaptation time constant and to moderate uctuations of the adaptation and the input current. As an extension of the model, we introduce a framework for combining an arbitrary spike generator with a generalized adaptation current. 1 J. Benda & A. V. M. Herz: A Universal Model for SpikeFrequency Adaptation 2
A mathematical framework for critical transitions: bifurcations, fastslow systems and stochastic dynamics
 Physica D
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Mixedmode oscillations with multiple time scales
 SIAM REV
, 2012
"... Mixedmode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made t ..."
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Cited by 34 (17 self)
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Mixedmode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose smallamplitude oscillations are produced by a local, multipletimescale “mechanism. ” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.
Effects of noisy drive on rhythms in networks of excitatory and inhibitory neurons
 Neural Comp
, 2005
"... Abstract. Synchronous rhythmic spiking in neuronal networks can be brought about by the interaction between Ecells and Icells (excitatory and inhibitory cells): The Icells gate and synchronize the Ecells, and the Ecells drive and synchronize the Icells. We refer to rhythms generated in this wa ..."
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Cited by 33 (4 self)
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Abstract. Synchronous rhythmic spiking in neuronal networks can be brought about by the interaction between Ecells and Icells (excitatory and inhibitory cells): The Icells gate and synchronize the Ecells, and the Ecells drive and synchronize the Icells. We refer to rhythms generated in this way as “PING ” (PyramidalInterneuronal Gamma) rhythms. The PING mechanism requires that the drive II to the Icells be sufficiently low; the rhythm is lost when II gets too large. This can happen in (at least) two different ways. In the first mechanism, the Icells spike in synchrony, but get ahead of the Ecells, spiking without being prompted by the Ecells. We call this phase walkthrough of the Icells. In the second mechanism, the Icells fail to synchronize, and their activity leads to complete suppression of the Ecells. Noisy spiking in the Ecells, generated by noisy external drive, adds excitatory drive to the Icells and may lead to phase walkthrough. Noisy spiking in the Icells adds inhibition to the Ecells, and may lead to suppression of the Ecells. An analysis of the conditions under which noise leads to phase walkthrough of the Icells or suppression of the Ecells shows that PING rhythms at frequencies far below the gamma range are robust to noise only if network parameter values are tuned very carefully. Together with an argument explaining why the PING mechanism
Geometric Singular Perturbation Analysis of Neuronal Dynamics
 in Handbook of Dynamical Systems
, 2000
"... : In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which d ..."
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: In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which display bursting oscillations. There are, in fact, several dierent classes of bursting solutions; these have been classied by the geometric properties of how solutions evolve in phase space. We describe several of the bursting classes and then review related rigorous mathematical analysis. We then discuss the dynamics of small networks of neurons. We are primarily interested in whether excitatory or inhibitory synaptic coupling leads to either synchronous or desynchronous rhythms. We demonstrate that all four combinations are possible, depending on the details of the intrinsic and synaptic properties of the cells. Finally, we discuss larger networks of neuronal oscillators involving two dis...
Origin of chaos in a twodimensional map modeling spikebursting neural activity
 Int. J. Bif. and Chaos
, 2003
"... Origin of chaos in a simple twodimensional map model replicating the spiking and spikingbursting activity of real biological neurons is studied. The map contains one fast and one slow variable. Individual dynamics of a fast subsystem of the map is characterized by two types of possible attractors: ..."
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Cited by 16 (4 self)
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Origin of chaos in a simple twodimensional map model replicating the spiking and spikingbursting activity of real biological neurons is studied. The map contains one fast and one slow variable. Individual dynamics of a fast subsystem of the map is characterized by two types of possible attractors: stable fixed point (replicating silence) and superstable limit cycle (replicating spikes). Coupling this subsystem with the slow subsystem leads to the generation of periodic or chaotic spikingbursting behavior. We study the bifurcation scenarios which reveal the dynamical mechanisms that lead to chaos at alternating silence and spiking phases.
Canards and curvature: nonsmooth approximation by pinching, Nonlinearity
, 2011
"... Abstract. In multiple timescale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: ..."
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Abstract. In multiple timescale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and ecology. Canards were initially studied using nonstandard analysis, and later the same results were proved by standard techniques such as matched asymptotics, invariant manifold theory and parameter blowup. More recently, canardlike behaviour has been linked to surfaces of discontinuity in piecewisesmooth dynamical systems. This paper provides a new perspective on the canard phenomenon by showing that the nonstandard analysis of canard explosions can be recast into the framework of piecewisesmooth dynamical systems. An exponential coordinate scaling is applied to a singularly perturbed system of ordinary differential equations. The scaling acts as a lens that resolves dynamics across all timescales. The changes of local curvature that are responsible for canard explosions are then analyzed. Regions where different timescales dominate are separated by hypersurfaces, and these are pinched together to obtain a piecewisesmooth system, in which curvature changes manifest as discontinuityinduced bifurcations. The method is used to classify canards in arbitrary dimensions, and to derive the parameter values over which canards form either small cycles (canards without head) or large cycles (canards with head). Canards and curvature: nonsmooth approximation by pinching 2 1.