Strong symmetry breaking rhythms created by folded nodes in a pair of symmetrically coupled, identical Koper oscillators.

The Koper model is a prototype system with two slow variables and one fast variable that possesses small-amplitude oscillations (SAOs), large-amplitude oscillations (LAOs), and mixed-mode oscillations (MMOs). In this article, we study a pair of identical Koper oscillators that are symmetrically coupled. Strong symmetry breaking rhythms are presented of the types SAO-LAO, SAO-MMO, LAO-MMO, and MMO-MMO, in which the oscillators simultaneously exhibit rhythms of different types. We identify the key folded nodes that serve as the primary mechanisms responsible for the strong nature of the symmetry breaking. The maximal canards of these folded nodes guide the orbits through the neighborhoods of these key points. For all of the strong symmetry breaking rhythms we present, the rhythms exhibited by the two oscillators are separated by maximal canards in the phase space of the oscillator.

Symmetry-breaking rhythms in coupled, identical fast-slow oscillators.

Symmetry-breaking in coupled, identical, fast-slow systems produces a rich, dramatic variety of dynamical behavior-such as amplitudes and frequencies differing by an order of magnitude or more and qualitatively different rhythms between oscillators, corresponding to different functional states. We present a novel method for analyzing these systems. It identifies the key geometric structures responsible for this new symmetry-breaking, and it shows that many different types of symmetry-breaking rhythms arise robustly. We find symmetry-breaking rhythms in which one oscillator exhibits small-amplitude oscillations, while the other exhibits phase-shifted small-amplitude oscillations, large-amplitude oscillations, mixed-mode oscillations, or even undergoes an explosion of limit cycle canards. Two prototypical fast-slow systems illustrate the method: the van der Pol equation that describes electrical circuits and the Lengyel-Epstein model of chemical oscillators.

A new class of chimeras in locally coupled oscillators with small-amplitude, high-frequency asynchrony and large-amplitude, low-frequency synchrony.

Chimeras are surprising yet important states in which domains of decoherent (asynchronous) and coherent (synchronous) oscillations co-exist. In this article, we report on the discovery of a new class of chimeras, called mixed-amplitude chimera states, in which the structures, amplitudes, and frequencies of the oscillations differ substantially in the decoherent and coherent regions. These mixed-amplitude chimeras exhibit domains of decoherent small-amplitude oscillations (phase waves) coexisting with domains of stable and coherent large-amplitude or mixed-mode oscillations (MMOs). They are observed in a prototypical bistable partial differential equation with oscillatory dynamics, spatially homogeneous kinetics, and purely local, isotropic diffusion. They are observed in parameter regimes immediately adjacent to regimes in which common large-amplitude solutions exist, such as trigger waves, spatially homogeneous MMOs, and sharp-interface solutions. Also, key singularities, folded nodes, and folded saddles arising commonly in multi-scale, bistable systems play important roles, and these have not previously been studied in systems with chimeras. The discovery of these mixed-amplitude chimeras is an important advance for understanding some processes in neuroscience, pattern formation, and physics, which involve both small-amplitude and large-amplitude oscillations. It may also be of use for understanding some aspects of electroencephalogram recordings from animals that exhibit unihemispheric slow-wave sleep.

Delayed loss of stability due to the slow passage through Hopf bifurcations in reaction-diffusion equations.

This article presents the delayed loss of stability due to slow passage through Hopf bifurcations in reaction-diffusion equations with slowly-varying parameters, generalizing a well-known result about delayed Hopf bifurcations in analytic ordinary differential equations to spatially-extended systems. We focus on the Hodgkin-Huxley partial differential equation (PDE), the cubic Complex Ginzburg-Landau PDE as an equation in its own right, the Brusselator PDE, and a spatially-extended model of a pituitary clonal cell line. Solutions which are attracted to quasi-stationary states (QSS) sufficiently before the Hopf bifurcations remain near the QSS for long times after the states have become repelling, resulting in a significant delay in the loss of stability and the onset of oscillations. Moreover, the oscillations have large amplitude at onset, and may be spatially homogeneous or inhomogeneous. Space-time boundaries are identified that act as buffer curves beyond which solutions cannot remain near the repelling QSS, and hence before which the delayed onset of oscillations must occur, irrespective of initial conditions. In addition, a method is developed to derive the asymptotic formulas for the buffer curves, and the asymptotics agree well with the numerically observed onset in the Complex Ginzburg-Landau (CGL) equation. We also find that the first-onset sites act as a novel pulse generation mechanism for spatio-temporal oscillations.

Amplitude-Modulated Bursting: A Novel Class of Bursting Rhythms.

We report on the discovery of a novel class of bursting rhythms, called amplitude-modulated bursting (AMB), in a model for intracellular calcium dynamics. We find that these rhythms are robust and exist on open parameter sets. We develop a new mathematical framework with broad applicability to detect, classify, and rigorously analyze AMB. Here we illustrate this framework in the context of AMB in a model of intracellular calcium dynamics. In the process, we discover a novel family of singularities, called toral folded singularities, which are the organizing centers for the amplitude modulation and exist generically in slow-fast systems with two or more slow variables.

A computational model for BMP movement in sea urchin embryos.

Bone morphogen proteins (BMPs) are distributed along a dorsal-ventral (DV) gradient in many developing embryos. The spatial distribution of this signaling ligand is critical for correct DV axis specification. In various species, BMP expression is spatially localized, and BMP gradient formation relies on BMP transport, which in turn requires interactions with the extracellular proteins Short gastrulation/Chordin (Chd) and Twisted gastrulation (Tsg). These binding interactions promote BMP movement and concomitantly inhibit BMP signaling. The protease Tolloid (Tld) cleaves Chd, which releases BMP from the complex and permits it to bind the BMP receptor and signal. In sea urchin embryos, BMP is produced in the ventral ectoderm, but signals in the dorsal ectoderm. The transport of BMP from the ventral ectoderm to the dorsal ectoderm in sea urchin embryos is not understood. Therefore, using information from a series of experiments, we adapt the mathematical model of Mizutani et al. (2005) and embed it as the reaction part of a one-dimensional reaction-diffusion model. We use it to study aspects of this transport process in sea urchin embryos. We demonstrate that the receptor-bound BMP concentration exhibits dorsally centered peaks of the same type as those observed experimentally when the ternary transport complex (Chd-Tsg-BMP) forms relatively quickly and BMP receptor binding is relatively slow. Similarly, dorsally centered peaks are created when the diffusivities of BMP, Chd, and Chd-Tsg are relatively low and that of Chd-Tsg-BMP is relatively high, and the model dynamics also suggest that Tld is a principal regulator of the system. At the end of this paper, we briefly compare the observed dynamics in the sea urchin model to a version that applies to the fly embryo, and we find that the same conditions can account for BMP transport in the two types of embryos only if Tld levels are reduced in sea urchin compared to fly.

The dynamics of hybrid metabolic-genetic oscillators.

The synthetic construction of intracellular circuits is frequently hindered by a poor knowledge of appropriate kinetics and precise rate parameters. Here, we use generalized modeling (GM) to study the dynamical behavior of topological models of a family of hybrid metabolic-genetic circuits known as "metabolators." Under mild assumptions on the kinetics, we use GM to analytically prove that all explicit kinetic models which are topologically analogous to one such circuit, the "core metabolator," cannot undergo Hopf bifurcations. Then, we examine more detailed models of the metabolator. Inspired by the experimental observation of a Hopf bifurcation in a synthetically constructed circuit related to the core metabolator, we apply GM to identify the critical components of the synthetically constructed metabolator which must be reintroduced in order to recover the Hopf bifurcation. Next, we study the dynamics of a re-wired version of the core metabolator, dubbed the "reverse" metabolator, and show that it exhibits a substantially richer set of dynamical behaviors, including both local and global oscillations. Prompted by the observation of relaxation oscillations in the reverse metabolator, we study the role that a separation of genetic and metabolic time scales may play in its dynamics, and find that widely separated time scales promote stability in the circuit. Our results illustrate a generic pipeline for vetting the potential success of a circuit design, simply by studying the dynamics of the corresponding generalized model.

Introduction to focus issue: rhythms and dynamic transitions in neurological disease: modeling, computation, and experiment.

Rhythmic neuronal oscillations across a broad range of frequencies, as well as spatiotemporal phenomena, such as waves and bumps, have been observed in various areas of the brain and proposed as critical to brain function. While there is a long and distinguished history of studying rhythms in nerve cells and neuronal networks in healthy organisms, the association and analysis of rhythms to diseases are more recent developments. Indeed, it is now thought that certain aspects of diseases of the nervous system, such as epilepsy, schizophrenia, Parkinson's, and sleep disorders, are associated with transitions or disruptions of neurological rhythms. This focus issue brings together articles presenting modeling, computational, analytical, and experimental perspectives about rhythms and dynamic transitions between them that are associated to various diseases.

Mixed-mode bursting oscillations: dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster.

This article concerns the phenomenon of Mixed-Mode Bursting Oscillations (MMBOs). These are solutions of fast-slow systems of ordinary differential equations that exhibit both small-amplitude oscillations (SAOs) and bursts consisting of one or multiple large-amplitude oscillations (LAOs). The name MMBO is given in analogy to Mixed-Mode Oscillations, which consist of alternating SAOs and LAOs, without the LAOs being organized into burst events. In this article, we show how MMBOs are created naturally in systems that have a spike-adding bifurcation or spike-adding mechanism, and in which the dynamics of one (or more) of the slow variables causes the system to pass slowly through that bifurcation. Canards are central to the dynamics of MMBOs, and their role in shaping the MMBOs is two-fold: saddle-type canards are involved in the spike-adding mechanism of the underlying burster and permit one to understand the number of LAOs in each burst event, and folded-node canards arise due to the slow passage effect and control the number of SAOs. The analysis is carried out for a prototypical fourth-order system of this type, which consists of the third-order Hindmarsh-Rose system, known to have the spike-adding mechanism, and in which one of the key bifurcation parameters also varies slowly. We also include a discussion of the MMBO phenomenon for the Morris-Lecar-Terman system. Finally, we discuss the role of the MMBOs to a biological modeling of secreting neurons.

An elementary model of torus canards.

We study the recently observed phenomena of torus canards. These are a higher-dimensional generalization of the classical canard orbits familiar from planar systems and arise in fast-slow systems of ordinary differential equations in which the fast subsystem contains a saddle-node bifurcation of limit cycles. Torus canards are trajectories that pass near the saddle-node and subsequently spend long times near a repelling branch of slowly varying limit cycles. In this article, we carry out a study of torus canards in an elementary third-order system that consists of a rotated planar system of van der Pol type in which the rotational symmetry is broken by including a phase-dependent term in the slow component of the vector field. In the regime of fast rotation, the torus canards behave much like their planar counterparts. In the regime of slow rotation, the phase dependence creates rich torus canard dynamics and dynamics of mixed mode type. The results of this elementary model provide insight into the torus canards observed in a higher-dimensional neuroscience model.

Coordination of cellular pattern-generating circuits that control limb movements: the sources of stable differences in intersegmental phases.

Neuronal mechanisms in nervous systems that keep intersegmental phase lags the same at different frequencies are not well understood. We investigated biophysical mechanisms that permit local pattern-generating circuits in neighboring segments to maintain stable phase differences. We use a modified version of an existing model of the crayfish swimmeret system that is based on three known coordinating neurons and hypothesized intersegmental synaptic connections. Weakly coupled oscillator theory was used to derive coupling functions that predict phase differences between neurons in neighboring segments. We show how features controlling the size of the lag under simplified network configurations combine to create realistic lags in the full network. Using insights from the coupled oscillator theory analysis, we identify an alternative intersegmental connection pattern producing realistic stable phase differences. We show that the persistence of a stable phase lag to changes in frequency can arise from complementary effects on the network with ascending-only or descending-only intersegmental connections. To corroborate the numerical results, we experimentally constructed phase-response curves (PRCs) for two different coordinating interneurons in the swimmeret system by perturbing the firing of individual interneurons at different points in the cycle of swimmeret movement. These curves provide information about the contribution of individual intersegmental connections to the stable phase lag. We also numerically constructed PRCs for individual connections in the model. Similarities between the experimental and numerical PRCs confirm the plausibility of the network configuration that has been proposed and suggest that the same stabilizing balance present in the model underlies the normal phase-constant behavior of the swimmeret system.

A simple model of chaotic advection and scattering.

In this work, we study a blinking vortex-uniform stream map. This map arises as an idealized, but essential, model of time-dependent convection past concentrated vorticity in a number of fluid systems. The map exhibits a rich variety of phenomena, yet it is simple enough so as to yield to extensive analytical investigation. The map's dynamics is dominated by the chaotic scattering of fluid particles near the vortex core. Studying the paths of fluid particles, it is seen that quantities such as residence time distributions and exit-vs-entry positions scale in self-similar fashions. A bifurcation is identified in which a saddle fixed point is created upstream at infinity. The homoclinic tangle formed by the transversely intersecting stable and unstable manifolds of this saddle is principally responsible for the observed self-similarity. Also, since the model is simple enough, various other properties are quantified analytically in terms of the circulation strength, stream velocity, and blinking period. These properties include: entire hierarchies of fixed points and periodic points, the parameter values at which these points undergo conservative period-doubling bifurcations, the structure of the unstable manifolds of the saddle fixed and periodic points, and the detailed structure of the resonance zones inside the vortex core region. A connection is made between a weakly dissipative version of our map and the Ikeda map from nonlinear optics. Finally, we discuss the essential ingredients that our model contains for studying how chaotic scattering induced by time-dependent flow past vortical structures produces enhanced diffusivities. (c) 1995 American Institute of Physics.

A showcase of torus canards in neuronal bursters.

Rapid action potential generation - spiking - and alternating intervals of spiking and quiescence - bursting - are two dynamic patterns commonly observed in neuronal activity. In computational models of neuronal systems, the transition from spiking to bursting often exhibits complex bifurcation structure. One type of transition involves the torus canard, which we show arises in a broad array of well-known computational neuronal models with three different classes of bursting dynamics: sub-Hopf/fold cycle bursting, circle/fold cycle bursting, and fold/fold cycle bursting. The essential features that these models share are multiple time scales leading naturally to decomposition into slow and fast systems, a saddle-node of periodic orbits in the fast system, and a torus bifurcation in the full system. We show that the transition from spiking to bursting in each model system is given by an explosion of torus canards. Based on these examples, as well as on emerging theory, we propose that torus canards are a common dynamic phenomenon separating the regimes of spiking and bursting activity.

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.

Analysis of state-dependent transitions in frequency and long-distance coordination in a model oscillatory cortical circuit.

Changes in behavioral state are typically accompanied by changes in the frequency and spatial coordination of rhythmic activity in the neocortex. In this article, we analyze the effects of neuromodulation on ionic conductances in an oscillating cortical circuit model. The model consists of synaptically-coupled excitatory and inhibitory neurons and supports rhythmic activity in the alpha, beta, and gamma ranges. We find that the effects of neuromodulation on ionic conductances are, by themselves, sufficient to induce transitions between synchronous gamma and beta rhythms and asynchronous alpha rhythms. Moreover, these changes are consistent with changes in behavioral state, with the rhythm transitioning from the slower alpha to the faster gamma and beta as arousal increases. We also observe that it is the same set of underlying intrinsic and network mechanisms that appear to be simultaneously responsible for both the observed transitions between the rhythm types and between their synchronization properties. Spike time response curves (STRCs) are used to study the relationship between the transitions in rhythm and the underlying biophysics.