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.     

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.     

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.     

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.     

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.     

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.     

The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales

In recent work [J. Rubin and M. Wechselberger, Biol. Cybern. 97, 5 (2007)], we explained the appearance of remarkably slow oscillations in the classical Hodgkin-Huxley (HH) equations, modified by scaling a time constant, using recently developed theory about mixed-mode oscillations (MMOs). This theory is only rigorously valid, however, for epsilon sufficiently small, where epsilon is a parameter that arises from nondimensionalization of the HH system. Here, we illustrate how the parameter regime over which MMOs exist, and the features of the MMO patterns within this regime, vary with respect to several key parameters in the nondimensionalized HH equations, including epsilon. Moreover, we explain our findings in terms of the effects that these parameters are expected to have on certain organizing structures within the corresponding flow, generalized from analysis done previously in the singular limit.     

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.     

Transition metal-metal oxide hybrids as versatile materials for hydrogen storage

Metal atom located on metal oxide (MMO) is a promising material with various applications such as hydrogen storage. As one of the metal oxides, niobium oxide (NbO) presents fascinating properties that make it a possibly applicable in MMOs. Here, we investigated the feasibility of transition metal-NbO hybrids as MMO materials for application in the hydrogen storage technology. In this respect, the hydrogen adsorption of transition metals (Fe, Ni, Cu, Pd, Ag, and Pt) decorated on the NbO nanocluster has been explored using density functional theory calculations. We found that the adsorption energy of the H₂ molecule on the NbO adsorbent is remarkably increased by locating the transition metals on the NbO metal oxide. Our results reveal that the transition metals decorated on the NbO nanocluster can act as active sites for hydrogen adsorption. Among the studied transition metals, Pt shows the highest hydrogen capacity up to 6.52 wt%.     

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.     

Molecular,metabolic, and genetic control: An introduction

The living cell is a miniature, self-reproducing, biochemical machine. Like all machines, it has a power supply, a set of working components that carry out its necessary tasks, and control systems that ensure the proper coordination of these tasks. In this Special Issue, we focus on the molecular regulatory systems that control cell metabolism, gene expression, environmental responses, development, and reproduction. As for the control systems in human-engineered machines, these regulatory networks can be described by nonlinear dynamical equations, for example, ordinary differential equations, reaction-diffusion equations, stochastic differential equations, or cellular automata. The articles collected here illustrate (i) a range of theoretical problems presented by modern concepts of cellular regulation, (ii) some strategies for converting molecular mechanisms into dynamical systems, (iii) some useful mathematical tools for analyzing and simulating these systems, and (iv) the sort of results that derive from serious interplay between theory and experiment. (c) 2001 American Institute of Physics.     

Fractional order chaotic systems: history,achievements, applications,and future challenges

Motivated by the importance of study on the complex behaviors, which may be exhibited by fractional order differential equations, this review paper focuses on dynamical fractional order systems exhibiting chaotic behaviors. The review begins with a brief history on the first publications on the above-mentioned subject. Then, the review is continued by investigating the recent progresses relevant to fractional order chaotic systems. Furthermore, a summary on some applications for such systems, which have been reported in the literature, is presented. Finally, the paper is closed by discussing some open problems on the aforementioned research subject. These open problems, as future challenges for further study on fractional order chaotic systems, can specify some direction lines for continuing the research on that subject.

     

Resonances,Unstable Systems,and Irreversibility: Matter Meets Mind

The fundamental time-reversal invariance of dynamical systems can be broken in various ways. One way is based on the presence of resonances and their interactions giving rise to unstable dynamical systems leading to well-defined time arrows. Associated with these time arrows are semigroups bearing time orientations. Usually, when time symmetry is broken, two time-oriented semigroups result, one directed toward the future and one directed toward the past. If time-reversed states and evolutions are excluded due to resonances, then the status of these states and their associated backwards-in-time oriented semigroups is open to question. One possible role for these latter states and semigroups is as an abstract representation of mental systems as opposed to material systems. The beginnings of this interpretation will be sketched.     

自由曲面成像光学系统设计:理论、发展与应用

传统球面以及非球面可供光学系统设计使用的自由度较少.自由曲面打破了旋转对称以及平移对称的几何约束,特别适用于校正非旋转对称系统的像差,同时可以减少系统中元件的数量,减小系统的体积与质量,实现传统光学系统难以实现的系统参数、结构与功能.自由曲面为光学设计的发展注入了巨大潜力,但同时也带来了全新的困难与挑战.概括性地总结了...     

ACTIVE CONTROL OF ENVIRONMENTAL NOISE, VII: PERFORMANCE OF MULTI-FREQUENCY, MULTI-CHANNEL, FREEFIELD SOUND CANCELLING SYSTEMS

The paper considers the performance of multi-frequency, multi-channel, free field sound cancelling systems for the reduction of discrete frequency and periodic noise. The approach uses a method of directional active noise control. Large sound reductions from these systems have been made possible through: (a) synthetically generating the cancelling sound and synchronizing with the primary source; (b) automatic alignment of all stability regions of the control system and; (c) avoiding instability produced by these multi-channel systems.     

Symbolic dynamics of one-dimensional maps: Entropies,finite precision,and noise

In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical systems theory: the topological and metric entropies. For the quadratic logistic equation, we find that the metric entropy converges very slowly in comparison to maps which are strictly hyperbolic. The effects of finite precision arithmetric and external noise on chaotic behavior are characterized with the symbolic dynamics entropies. Finally, we discuss the relationship of these measures of chaos to algorithmic complexity, and use algorithmic information theory as a framework to discuss the construction of models for chaotic dynamics.     

The Hannay angles: Geometry,adiabaticity, and an example

The Hannay angles were introduced by Hannay as a means of measuring a holonomy effect in classical mechanics closely corresponding to the Berry phase in quantum mechanics. Using parameter-dependent momentum mappings we show that the Hannay angles are the holonomy of a natural connection. We generalize this effect to non-Abelian group actions and discuss non-integrable Hamiltonian systems. We prove an averaging theorem for phase space functions in the case of general multi-frequency dynamical systems which allows us to establish the almost adiabatic invariance of the Hannay angles. We conclude by giving an application to celestial mechanics.Supported by the Deutsche ForschungsgemeinschaftSupported by the Akademie der Wissenschaften zu Berlin     

Time-delayed chameleon: Analysis,synchronization and FPGA implementation

In this paper we report a time-delayed chameleon-like chaotic system which can belong to different families of chaotic attractors depending on the choices of parameters. Such a characteristic of self-excited and hidden chaotic flows in a simple 3D system with time delay has not been reported earlier. Dynamic analysis of the proposed time-delayed systems are analysed in time-delay space and parameter space. A novel adaptive modified functional projective lag synchronization algorithm is derived for synchronizing identical time-delayed chameleon systems with uncertain parameters. The proposed time-delayed systems and the synchronization algorithm with controllers and parameter estimates are then implemented in FPGA using hardware–software co-simulation and the results are presented.     

Mesoscopic spin clusters, phase separation, and induced order in spin-gap magnets: A review

In many low-dimensional systems with antiferromagnetic interactions, a magnetic order of the Néel type is absent. These systems remain in a quantum-disordered (spin-liquid) state down to zero temperature. The disordered state appears to be stable to weak perturbations when magnetic excitations are separated from the ground state by an energy gap. The stability of the spin-liquid ground state is destroyed upon introduction of impurities or in a sufficiently strong magnetic field. This paper presents a review of the main results of the experimental investigations performed in order to reveal and identify mesoscopic spin clusters formed in the vicinity of impurity ions, to determine the spatial structure of an impurity-induced magnetic order in spin-gap systems, and to examine the low-frequency excitation spectra of field-induced antiferromagnetic phases.