On decomposing mixed-mode oscillations and their return maps

Alternating patterns of small and large amplitude oscillations occur in a wide variety of physical, chemical, biological, and engineering systems. These mixed-mode oscillations (MMOs) are often found in systems with multiple time scales. Previous differential equation modeling and analysis of MMOs have mainly focused on local mechanisms to explain the small oscillations. Numerical continuation studies reported different MMO patterns based on parameter variation. This paper aims at improving the link between local analysis and numerical simulation. Our starting point is a numerical study of a singular return map for the Koper model which is a prototypical example for MMOs, which also relates to local normal form theory. We demonstrate that many MMO patterns can be understood geometrically by approximating the singular maps with affine and quadratic maps. Motivated by our numerical analysis we use abstract affine and quadratic return map models in combination with two local normal forms that generate small oscillations. Using this decomposition approach we can reproduce many classical MMO patterns and effectively decouple bifurcation parameters for local and global parts of the flow. The overall strategy we employ provides an alternative technique for understanding MMOs.     

Mixed-mode oscillations in a stochastic,piecewise-linear system

We analyse a piecewise-linear FitzHugh–Nagumo model. The system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist. The addition of small noise induces mixed-mode oscillations (MMOs) in the vicinity of the canard point. We determine the effect of each model parameter on the stochastically driven MMOs. In particular we show that any parameter variation (such as a modification of the piecewise-linear function in the model) that leaves the ratio of noise amplitude to time-scale separation unchanged typically has little effect on the width of the interval of the primary bifurcation parameter over which MMOs occur. In that sense, the MMOs are robust. Furthermore, we show that the piecewise-linear model exhibits MMOs more readily than the classical FitzHugh–Nagumo model for which a cubic polynomial is the only nonlinearity. By studying a piecewise-linear model, we are able to explain results using analytical expressions and compare these with numerical investigations.     

Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network

Recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators [R. Curtu, A. Shpiro, N. Rubin, J. Rinzel, Mechanisms for frequency control in neuronal competition models, SIAM J. Appl. Dyn. Syst. 7 (2) (2008) 609-649] showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics. Several types of behavior were identified such as: fusion, equivalent to a steady state of identical activity levels for both neural units; oscillations due to either an escape or a release mechanism; and a winner-take-all state of bistability. The model consists of two neural populations interacting through reciprocal inhibition, each endowed with a slow negative-feedback process in the form of spike frequency adaptation. In this paper we report the occurrence of another complex oscillatory pattern, the mixed-mode oscillations (MMOs). They exist in the model at the transition between the relaxation oscillator dynamical regime and the winner-take-all regime. The system distinguishes itself from other neuronal models where MMOs were found by the following interesting feature: there is no autocatalysis involved (as in the examples of voltage-gated persistent inward currents and/or intrapopulation recurrent excitation) and therefore the two cells in the network are not intrinsic oscillators; the oscillations are instead a combined result of the mutual inhibition and the adaptation. We prove that the MMOs are due to a singular Hopf bifurcation point situated in close distance to the transition point to the winner-take-all case. We also show that in the vicinity of the singular Hopf other types of bifurcations exist and we construct numerically the corresponding diagrams.     

Mixed-mode oscillations and chaos from a simple second-order oscillator under weak periodic perturbation

In this study, we propose a remarkably simple oscillator that exhibits extremely complicated behaviors. The second-order nonautonomous differential equation discussed in this Letter is considered to be one of the simplest dynamics that can produce mixed-mode oscillations (MMOs) and chaos. Our model uses a Bonhoeffer-van der Pol (BVP) oscillator under weak periodic perturbation. The parameter set of the BVP equation is chosen such that a focus and a relaxation oscillation coexist when no perturbation is applied. Under weak periodic perturbation, various types of MMOs and chaos with remarkably complicated waveforms are observed.     

Mixed-mode oscillations and cluster patterns in an electrochemical relaxation oscillator under galvanostatic control

We studied the dynamics of a prototypical electrochemical model, the electro-oxidation of hydrogen in the presence of poisons, under galvanostatic conditions. The lumped system exhibits relaxation oscillations, which develop mixed-mode oscillations (MMOs) for low preset currents. A fast-slow analysis of the homogeneous dynamics reveals that the MMOs arise from a fast oscillating subsystem and a one-dimensional slow manifold. In the spatially extended system, the galvanostatic constraint imposes a synchronizing global coupling that drives the system into cluster patterns. The properties of the cluster patterns (CPs) result from an intricate interplay of the nature of the local oscillators, the global constraint, and a nonlocal coupling through the electrolyte. In particular, we find that the global constraint suppresses small-amplitude oscillations of MMOs and prevents domains oscillating out of phase from occupying equal regions in phase space. The nonlocal coupling causes each individual clustered region to oscillate on a different limit cycle. Typically multistability of CPs is found. Coexisting patterns possess different oscillation periods and a different total fraction in space that occupies the in-phase or out-of-phase state, respectively.     

Bifurcations of mixed-mode oscillations in a stellate cell model

Experimental recordings of the membrane potential of stellate cells within the entorhinal cortex show a transition from subthreshold oscillations (STOs) via mixed-mode oscillations (MMOs) to relaxation oscillations under increased injection of depolarizing current. Acker et al. introduced a 7D conductance based model which reproduces many features of the oscillatory patterns observed in these experiments. For the first time, we present a comprehensive bifurcation analysis of this model by using the software package AUTO. In particular, we calculate the stable MMO branches within the bifurcation diagram of this model, as well as other MMO patterns which are unstable. We then use geometric singular perturbation theory to demonstrate how the bifurcations are governed by a 3D reduced model introduced by Rotstein et al. We extend their analysis to explain all observed MMO patterns within the bifurcation diagram. A key role in this bifurcation analysis is played by a novel homoclinic bifurcation structure connecting to a saddle equilibrium on the unstable branch of the corresponding critical manifold. This type of homoclinic connection is possible due to canards of folded node (folded saddle-node) type.     

Characterizing mixed mode oscillations shaped by noise and bifurcation structure

Many neuronal systems and models display a certain class of mixed mode oscillations (MMOs) consisting of periods of small amplitude oscillations interspersed with spikes. Various models with different underlying mechanisms have been proposed to generate this type of behavior. Stochastic versions of these models can produce similarly looking time series, often with noise-driven mechanisms different from those of the deterministic models. We present a suite of measures which, when applied to the time series, serves to distinguish models and classify routes to producing MMOs, such as noise-induced oscillations or delay bifurcation. By focusing on the subthreshold oscillations, we analyze the interspike interval density, trends in the amplitude, and a coherence measure. We develop these measures on a biophysical model for stellate cells and a phenomenological FitzHugh-Nagumo-type model and apply them on related models. The analysis highlights the influence of model parameters and resets and return mechanisms in the context of a novel approach using noise level to distinguish model types and MMO mechanisms. Ultimately, we indicate how the suite of measures can be applied to experimental time series to reveal the underlying dynamical structure, while exploiting either the intrinsic noise of the system or tunable extrinsic noise.     

Rhythms of the brain: an examination of mixed mode oscillation approaches to the analysis of neurophysiological data

In the nervous system many behaviorally relevant dynamical processes are characterized by episodes of complex oscillatory states, whose periodicity may be expressed over multiple temporal and spatial scales. In at least some of these instances the variability in oscillatory amplitude and frequency can be explained in terms of deterministic dynamics, rather than being purely noise-driven. Recently interest has increased in studying the application of mixed-mode oscillations (MMOs) to neurophysiological data. MMOs are complex periodic waveforms where each period is comprised of several maxima and minima of different amplitudes. While MMOs might be expected to occur in brain kinetics, only a few examples have been identified thus far. In this article, we review recent theoretical and experimental findings on brain oscillatory rhythms in relation to MMOs, focusing on examples at the single neuron level but also briefly touching on possible instances of the phenomenon across local and global brain networks.     

Introduction to focus issue: mixed mode oscillations: experiment, computation, and analysis

Mixed mode oscillations (MMOs) occur when a dynamical system switches between fast and slow motion and small and large amplitude. MMOs appear in a variety of systems in nature, and may be simple or complex. This focus issue presents a series of articles on theoretical, numerical, and experimental aspects of MMOs. The applications cover physical, chemical, and biological systems.     

Flavoured leptogenesis in the CTP formalism

Within the Closed Time Path (CTP) framework, we derive kinetic equations for particle distribution functions that describe leptogenesis in the presence of several lepton flavours. These flavours have different Standard-Model Yukawa couplings, which induce flavour-sensitive scattering processes and thermal dispersion relations. Kinetic equilibrium, which is rapidly established and maintained via gauge interactions, allows to simplify these equations to kinetic equations for the matrix of lepton charge densities. In performing this simplification, we notice that the rapid flavour-blind gauge interactions damp the flavour oscillations of the leptons. Leptogenesis turns out to be in the parametric regime where the flavour oscillations are overdamped and flavour decoherence is mainly induced by flavour sensitive scatterings. We solve the kinetic equations for the lepton number densities numerically and show that they interpolate between the unflavoured and the fully flavoured regimes within the intermediate parametric region, where neither of these limits is applicable.     

Excitable elements controlled by noise and network structure

We study collective dynamics of complex networks of stochastic excitable elements, active rotators. In the thermodynamic limit of infinite number of elements, we apply a mean-field theory for the network and then use a Gaussian approximation to obtain a closed set of deterministic differential equations. These equations govern the order parameters of the network. We find that a uniform decrease in the number of connections per element in a homogeneous network merely shifts the bifurcation thresholds without producing qualitative changes in the network dynamics. In contrast, heterogeneity in the number of connections leads to bifurcations in the excitable regime. In particular we show that a critical value of noise intensity for the saddle-node bifurcation decreases with growing connectivity variance. The corresponding critical values for the onset of global oscillations (Hopf bifurcation) show a non-monotone dependency on the structural heterogeneity, displaying a minimum at moderate connectivity variances.     

Synchronisationseffekte in einer linearen Anordnung aus N Josephson-Verbindungen

Synchronization Effects in a Linear Array of N Josephson Junctions Within the frame of the RSJ model we investigate the synchronization of the oscillations in a linear array of N identical Josephson junctions shunted by an electromagnetic resonator. Using an adiabatic approximation to the first order of the parameter I_cI the reduced equations of the slowly varying phases are derived. These equations allow the detailed investigation of all the stationary states of the system. Only the coherent state in the inductive regime and the radiationless state in the capacitive regime are found to be stable. Including noise effects we discuss the order parameter concept for the resonator current in the case N ? 1.     

Wave Theory of a Traveling-Wave Tube Operated Near the Cutoff

We outline the main principles of the wave theory of interaction between an electron beam and waves in a slow-wave structure near the edge of the transmission band. Formulation of the basic equations and the boundary conditions is considered taking consistently into account that the interaction parameter is small. A comparison of the results with a discrete version of the theory is discussed. We also consider the starting conditions for the oscillation regime, the linear amplification regimes, and some effects found within the framework of the nonstationary nonlinear theory, e.g., parasitic self-excitation in the amplification regime and hard excitation of the oscillation regime.     

Buckling of spherical shells adhering onto a rigid substrate

Deformation of a spherical shell adhering onto a rigid substrate due to van der Waals attractive interaction is investigated by means of numerical minimization (conjugate gradient method) of the sum of the elastic and adhesion energies. The conformation of the deformed shell is governed by two dimensionless parameters, i.e., C_s/epsilon and C_b/epsilon where C_s and C_b are respectively the stretching and the bending constants, and epsilon is the depth of the van der Waals potential between the shell and substrate. Four different regimes of deformation are characterized as these parameters are systematically varied: (i) small deformation regime, (ii) disk formation regime, (iii) isotropic buckling regime, and (iv) anisotropic buckling regime. By measuring the various quantities of the deformed shells, we find that both discontinuous and continuous bucking transitions occur for large and small C_s/epsilon, respectively. This behavior of the buckling transition is analogous to van der Waals liquids or gels, and we have numerically determined the associated critical point. Scaling arguments are employed to explain the adhesion induced buckling transition, i.e., from the disk formation regime to the isotropic buckling regime. We show that the buckling transition takes place when the indentation length exceeds the effective shell thickness which is determined from the elastic constants. This prediction is in good agreement with our numerical results. Moreover, the ratio between the indentation length and its thickness at the transition point provides a constant number (2–3) independent of the shell size. This universal number is observed in various experimental systems ranging from nanoscale to macroscale. In particular, our results agree well with the recent compression experiment using microcapsules.     

Strongly interacting Fermi gases with density imbalance

We consider density-imbalanced Fermi gases of atoms in the strongly interacting, i.e., unitarity, regime. The Bogoliubov-de Gennes equations for a trapped superfluid are solved. They take into account the finite size of the system, as well as give rise to both phase separation and Fulde-Ferrel-Larkin-Ovchinnikov-type oscillations in the order parameter. We show how radio-frequency spectroscopy reflects the phase separation, and can provide direct evidence of the FFLO-type oscillations via observing the nodes of the order parameter.     

二维映射神经元模型中频率依赖的随机共振

通过数值模拟方法, 研究了在具有稳定次阈值振荡特性的二维映射神经元体系中, 噪声对体系非线性动力学的调控作用. 通过计算发现了噪声诱导的动作电位和随机共振现象. 另外,还研究了体系的控制参数及输入信号的频率对体系动力学的影响, 发现了该体系中频率依赖的随机共振现象. 关键词： 二维映射神经元模型 次阈值振荡 高斯白噪声 随机共振     

Fano versus Kondo resonances in a multilevel "semiopen" quantum dot

Linear conductance across a large quantum dot via a single level epsilon(0) with large hybridization to the contacts is strongly sensitive to quasibound states localized in the dot and weakly coupled to epsilon(0). The conductance oscillates with the gate voltage due to interference of the Fano type. At low temperature and Coulomb blockade, Kondo correlations damp the oscillations on an extended range of gate voltage values, by freezing the occupancy of the epsilon(0) level itself. As a consequence, the antiresonances of Fano origin are washed out. The results are in good correspondence with experimental data for a large quantum dot in the semiopen regime.     

Response of a rotorcraft model with damping non-linearities

The linearized equations of motion of a helicopter in contact with the ground have solutions which can be linearly stable or unstable, depending on the system parameters. The present study includes physical non-linearities in the helicopter model. This allows one to determine if a steady-state response exists and, if so, what the frequency and amplitude of the oscillations will be. In this way, one can determine how serious the linearly unstable operating regime is and whether destructive oscillations are possible when the system is in the linearly stable regime. The present analysis applies to helicopters having fully articulated rotors.     

Cosmological Post-Newtonian Expansions to Arbitrary Order

We prove the existence of a large class of one parameter families of solutions to the Einstein-Euler equations that depend on the singular parameter e = v_T/c{\epsilon=v_T/c} (0 < e < e₀){(0< \epsilon < \epsilon_0)}, where c is the speed of light, and v _T is a typical speed of the gravitating fluid. These solutions are shown to exist on a common spacetime slab M @ [0,T)×\mathbb T³{M\cong [0,T)\times \mathbb {T}^3}, and converge as e\searrow 0{\epsilon \searrow 0} to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter e{\epsilon} to any specified order with expansion coefficients that satisfy e{\epsilon}-independent (nonlocal) symmetric hyperbolic equations.