Mixed-mode Oscillations Research Articles

All Possible Bursting Attractors in the Neighborhood of Hopf Bifurcation Point Under Periodic Excitation

For a vector field with Hopf bifurcation at the origin, when periodic excitation is introduced in which the frequency of the excitation is far less than the natural frequency, bursting oscillations, characterized by the alternations between the large-amplitude and the small-amplitude oscillations, can often be observed. The paper tries to answer the common question of how many types of bursting attractors may appear in such a vector field. Based on the normal form of Hopf bifurcation up to the third order, by regarding the whole exciting term as a slow-varying bifurcation parameter, all possible equilibrium branches of the generalized autonomous system are derived, the bifurcation sets of which divide the parameter space into regions associated with different types of dynamical behaviors. It is found that, there exist totally four possible combinations of the regions that the trajectory of the full system may visit as the slow-varying parameter varies. Accordingly, four qualitatively different behaviors, i.e. periodic symmetric oscillations, periodic symmetric mixed-mode oscillations, fold/Hopf/Hopf/fold and fold-Hopf/fold-Hopf bursting attractors, are presented, the mechanism of which is obtained by overlapping the transformed phase portrait and the equilibrium branches in the generalized autonomous system. Furthermore, the dynamics of the generalized autonomous system in different regions in the parameter space may influence the structures of the bursting oscillations when the trajectory enters the corresponding regions, while the bifurcations at the boundaries of the regions may lead to changes between the quiescent states and spiking states in the bursting attractors.

Complex Dynamic Behavior During the Electro-Reduction of Bromate Ions

This article describes the electrochemical reduction of bromate ions in strongly alkaline media on platinum electrodes. Bromates’ importance stems from its wide use in a variety of industrial processes, its incorporation into energy storage devices, as well as its known toxicity, which has led to it being designated as a human carcinogen. Interestingly, during the electro-reduction of bromate in alkaline solutions sustained oscillatory behavior was observed in both current and potential under both potentiostatic and galvanostatic control. Mechanistic characterization using electrochemical impedance spectroscopy (EIS) lead to the discovery that the oscillatory behavior is driven by the presence of a negative differential resistance (NDR). The EIS measurements allowed for the classification of the bromate oscillator into the HN-NDR class, which was supported by potentiostatic oscillations emerging once sufficient external resistance was included into the system as well as oscillating under galvanostatic conditions. Through a systematic exploration of experimental conditions, the electrochemical bromate oscillator also displayed complexity in the form of mixed-mode oscillations as well as complex bursting patterns.

Mixed-mode Oscillations in Filippov System

The mechanism of the mixed mode oscillations of a class of non-smooth Filippov systems under multistable coexistence was studied in this paper. Based on a Lorenz-type chaotic model with multi-attractor coexistence, the Filippov system was established by introducing non-smooth terms as well as an external excitation. With multiple stable attractors in the discontinuous vector field, the parameter changes have led to complex transition patterns between the attractors and the non-smooth interface, or between the attractors. When an order gap exists between the exciting frequency and the natural frequency, implying the mixed-mode oscillations. Here we have taken several excitation amplitudes to cover different coexistence regions, a set of mixed mode oscillation patterns were obtained. Besides, the bifurcation set of two generalized autonomous subsystems and the coexistence region of attractors were discussed. Combined with the transformed phase diagram method, the bifurcation mechanism of bursting oscillation and the sliding dynamical behaviors of the system at the discontinuous interface has revealed with slow varying parameters access in different regions of multistable attractors coexistence. The alternations between quiescent and spiking states become more frequent and complex, leading to the change of the structure of the bursting oscillation modes. Moreover, the non-smooth partition interface of the system yields multiple non-smooth bifurcations, which will also affect the oscillation modes of the generalized autonomous system.

Complex Periodic Mixed-Mode Oscillation Patterns in a Filippov System

The main task of this article is to study the patterns of mixed-mode oscillations and non-smooth behaviors in a Filippov system with external excitation. Different types of periodic spiral crossing mixed-mode oscillation patterns, i.e., “cusp-F−/fold-F−” oscillation, “cusp-F−/two-fold/two-fold/fold-F−” oscillation and “two-fold/fold-F−” oscillation, are explored. Based on the analysis of the equilibrium and tangential singularities of the fast subsystem, spiral crossing oscillation around the tangential singularities is investigated. Meanwhile, by combining the fast and slow analysis methods, we can observe that the cusp, two-fold and fold-cusp singularities play an important role in generating all kinds of complex mixed-mode oscillations.

Bifurcation analysis of mixed-mode oscillations and Farey trees in an extended Bonhoeffer–van der Pol oscillator

The two-variable Bonhoeffer–van der Pol oscillator, which is equivalent to the FitzHugh–Nagumo model, can be represented by a natural circuit, i.e., a circuit consisting only of simple two-terminal elements. An extended Bonhoeffer–van der Pol (BVP) oscillator is a circuit extended to a three-variable system from a two-variable BVP oscillator; this is obtained by adding an inductor–resistor branch to the original BVP oscillator. The extended BVP oscillator is known to generate mixed-mode oscillations. In this study, we classify the bifurcations of the simple sequences and their daughters, which constitute asymmetric Farey trees. Because the extended BVP oscillator is an extremely simple circuit, the parents–daughter processes can be quite precisely explained. We confirm that simple sequences 1s (s≥0) are born via saddle–node bifurcations and can be basic parents, which correspond to each stable fixed point of a Poincaré return map, where 1s represents one large excursion followed by a number s of small peaks. The two basic parents 1s and 1s+1 generate daughters [1s+1,1s×n] sequentially for the successive values of n via mixed-mode oscillation-incrementing bifurcations. The terms [1s+1,1s×n] indicate that 1s+1 is followed by 1sn-times. The daughters are born in a similar fashion to period-adding bifurcations generated by the circle map, and the parents–daughter processes satisfy Farey arithmetic and are terminated by a saddle–node bifurcation through which 1s appears. We create multiple two-parameter bifurcation diagrams and confirm that these phenomena are stably observed in these diagrams over broad ranges, i.e., complex bifurcations, such as codimension-two bifurcations or cusps, do not appear in the diagrams. Furthermore, the theoretical results are confirmed using laboratory measurements and experiments.

Reciprocally inhibitory circuits operating with distinct mechanisms are differently robust to perturbation and modulation.

Reciprocal inhibition is a building block in many sensory and motor circuits. We studied the features that underly robustness in reciprocally inhibitory two neuron circuits. We used the dynamic clamp to create reciprocally inhibitory circuits from pharmacologically isolated neurons of the crab stomatogastric ganglion by injecting artificial graded synaptic (ISyn) and hyperpolarization-activated inward (IH) currents. There is a continuum of mechanisms in circuits that generate antiphase oscillations, with 'release' and 'escape' mechanisms at the extremes, and mixed mode oscillations between these extremes. In release, the active neuron primarily controls the off/on transitions. In escape, the inhibited neuron controls the transitions. We characterized the robustness of escape and release circuits to alterations in circuit parameters, temperature, and neuromodulation. We found that escape circuits rely on tight correlations between synaptic and H conductances to generate bursting but are resilient to temperature increase. Release circuits are robust to variations in synaptic and H conductances but fragile to temperature increase. The modulatory current (IMI) restores oscillations in release circuits but has little effect in escape circuits. Perturbations can alter the balance of escape and release mechanisms and can create mixed mode oscillations. We conclude that the same perturbation can have dramatically different effects depending on the circuits' mechanism of operation that may not be observable from basal circuit activity.

Complex mixed-mode oscillations in oscillators sharing nonlinearity

Complex mixed-mode oscillations in oscillators sharing nonlinearity

Variability of complex oscillatory regimes in the stochastic model of cold-flame combustion of a hydrocarbon mixture.

Processes of the cold-flame combustion of a mixture of two hydrocarbons are studied on the base of a three-dimensional nonlinear dynamical model. Bifurcation analysis of the deterministic model reveals mono- and bistability parameter zones with equilibrium and oscillatory attractors. For this model, effects of random disturbances in the bistability parameter zone are studied. We show that random forcing causes transitions between coexisting stable equilibria and limit cycles with the formation of complex stochastic mixed-mode oscillations. Properties of these oscillatory regimes are studied by means of statistics of interspike intervals. A phenomenon of anti-coherence resonance is discussed. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Inhibitory autapse with time delay induces mixed-mode oscillations related to unstable dynamical behaviors near subcritical Hopf bifurcation

<abstract><p>Mixed-mode oscillations (MMOs) consisting of spikes alternating with a series of subthreshold oscillations have been observed in various neurons related to some physiological functions. In the present paper, inhibitory-autapse-induced MMOs are simulated by using the Hodgkin-Huxley neuron model, and the underlying dynamical mechanism is identified to be related to dynamics of unstable behaviors near subcritical Hopf bifurcation. For the monostable spiking, a delayed inhibitory current pulse activated by a spike can suppress the phase trajectory corresponding to depolarization phase of the next spike to the unstable focus nearby or the neighborhood outside of unstable limit cycle, respectively. Then the trajectory rotates multiple cycles away and converges to the stable limit cycle, resulting in an evolution process of membrane potential from small-amplitude subthreshold oscillations to a large-amplitude spike, i.e., MMOs. For the spiking coexisting with the resting state, inhibitory autapse induces MMOs and resting state from the spiking. The difference in the MMOs from those induced by the excitatory autapse is identified. The result presents the underlying nonlinear mechanisms of inhibitory autapse to suppress the neuronal firing and reveals the potential role to control the neuronal firing patterns near subcritical Hopf bifurcation.</p></abstract>

Amplitude-modulated spiking as a novel route to bursting: Coupling-induced mixed-mode oscillations by symmetry breaking.

Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are increasingly well understood and are often caused by folded singularities, canard orbits, or singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for diffusively coupled FitzHugh-Nagumo oscillators with negative coupling constant, and we show that it is caused by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity. Inspired by earlier work on models of pancreatic beta-cells [Sherman, Bull. Math. Biol. 56, 811 (1994)], we then identify a new type of bursting dynamics due to diffusive coupling of cells firing action potentials when isolated. In the presence of coupling, small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. Confirming the hypothesis from the earlier work that this behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem, we find that it is caused by symmetry breaking. Moreover, we show that it is organized by a FSN in the averaged system, which causes a singular Hopf bifurcation. Such behavior is related to the recently studied dynamics caused by the so-called torus canards.

Bifurcations of mixed-mode oscillations in three-timescale systems: An extended prototypical example

We study a class of multi-parameter three-dimensional systems of ordinary differential equations that exhibit dynamics on three distinct timescales. We apply geometric singular perturbation theory to explore the dependence of the geometry of these systems on their parameters, with a focus on mixed-mode oscillations (MMOs) and their bifurcations. In particular, we uncover a novel geometric mechanism that encodes the transition from MMOs with single epochs of small-amplitude oscillations (SAOs) to those with double-epoch SAOs; the former feature SAOs or pseudo-plateau bursting either “below” or “above” in their time series, while in the latter, SAOs or pseudo-plateau bursting occur both “below” and “above.” We identify a relatively simple prototypical three-timescale system that realizes our mechanism, featuring a one-dimensional S-shaped 2-critical manifold that is embedded into a two-dimensional S-shaped critical manifold in a symmetric fashion. We show that the Koper model from chemical kinetics is merely a particular realization of that prototypical system for a specific choice of parameters; in particular, we explain the robust occurrence of mixed-mode dynamics with double epochs of SAOs therein. Finally, we argue that our geometric mechanism can elucidate the mixed-mode dynamics of more complicated systems with a similar underlying geometry, such as a three-dimensional, three-timescale reduction of the Hodgkin–Huxley equations from mathematical neuroscience.

Complex Oscillatory Dynamics in a Three-Timescale El Ni\~No Southern Oscillation Model

We study the three-timescale dynamics of a model that describes the El Ni\~no Southern Oscillation (ENSO) phenomenon, which was proposed in [A. Roberts, J. Guckenheimer, E. Widiasih, A. Timmermann, and C. K. Jones, Mixed-mode oscillations of El Ni\~no--Southern Oscillation, Journal of the Atmospheric Sciences, 73 (2016), pp. 1755--1766]. While ENSO phenomena are inherently characterised by the presence of multiple distinct timescales, the above model has previously been studied in a two-timescale context only. Here, we uncover the geometric mechanisms that are responsible for complex oscillatory dynamics in a three-timescale regime, and we demonstrate that the system exhibits a variety of qualitatively different behaviours, such as mixed-mode oscillation (MMO) with ``plateaus" -- trajectories where epochs of quiescence alternate with dramatic excursions -- and relaxation oscillation. The latter, although emergent also in the two-timescale context in appropriate parameter regimes, had not been documented previously. Moreover, we show that these mechanisms are relevant to models from other fields of ecological and population dynamics, as the underlying geometry is similar to the unfolding of Rosenzweig--MacArthur-type models in three dimensions.

Suppression effects of excitatory and inhibitory self-feedbacks on neuronal spiking near Hopf bifurcation

Neural activities driven by synaptic inputs are important for neural coding. In general, excitatory synaptic inputs facilitate the firing activities of neurons, and inhibitory synaptic inputs suppress the firing activities. In the present work, the autapse model with adjustable current decay speed is considered, and the suppression effects of excitatory autapse on neuronal firing responses are simulated near subcritical Hopf bifurcation in the Morris-Lecar (ML) neuron model, which are compared with the suppression effects of the inhibitory autapse. Furthermore, the dynamical mechanisms of the suppression effects are acquired with the help of the phase response curve and phase trajectories perturbed by excitatory autaptic current. Near the subcritical Hopf bifurcation, the ML neuron exhibits monostable periodic spiking and coexistence of periodic spiking and resting state. For monostable periodic spiking, excitatory autaptic current with fast and middle decay speeds can induce the periodic spiking with reduced firing frequency and the mixed-mode oscillations (MMOs) that are alternations between subthreshold oscillations and a spike, respectively, and inhibitory autaptic current with middle and slow decay speeds can also induce these two behaviors, respectively. For the periodic spiking in the coexistence region, besides the above two behaviors, excitatory autaptic current with middle decay speed and inhibitory autaptic current with slow decay speed can induce the change from spiking to resting state. The results enrich the paradoxical cases that excitatory inputs suppress the neuronal firing responses, and present the different nonlinear mechanisms in the suppression effects of excitatory and inhibitory self-feedbacks on the neuronal spiking, which provide novel measures to modulate neuronal firing activity.

The relationship between mixed-mode oscillation and the kinetics of potassium conductances in a mathematical model of vibrissa motoneurons

The relationship between mixed-mode oscillation and the kinetics of potassium conductances in a mathematical model of vibrissa motoneurons

Non-chaos-mediated mixed-mode oscillations in an extended Hindmarsh-Rose neuronal oscillator with time delay

This paper proposes an extended neuron model with time delay. It aims to investigate the effect of time delay on the dynamical behavior of the system under different conditions. The existence of the Hopf bifurcation of the system and the stability of its periodic solution are proved by the central manifold theorem. Numerical results show that the system has abundant dynamical performance, including chaos, period-adding, and intermittent chaos.

Bifurcation mechanism of doubly nested mixed-mode oscillations

Bifurcation mechanism of doubly nested mixed-mode oscillations

A multiple timescale network model of intracellular calcium concentrations in coupled neurons: Insights from ROM simulations

In Fernández-García and Vidal [Physica D 401 (2020) 132129], the authors have analyzed the synchronization features between two identical 3D slow-fast oscillators, symmetrically coupled, and built as an extension of the FitzHugh—Nagumo dynamics generating Mixed-Mode Oscillations. The third variable in each oscillator aims at representing the time-varying intracellular calcium concentration in neurons. The global model is therefore six-dimensional with two fast variables and four slow variables with strong symmetry properties. In the present article, we consider an extension of this model in two different directions. First, we consider heterogeneity among cells and analyze the coupling of two oscillators with different values for one parameter which tunes the intrinsic frequency of the output. We therefore identify new patterns of antiphasic synchronization, with non trivial signatures and that exhibit a Devil’s Staircase phenomenon in signature transitions when varying the coupling gain parameter value. Second, we introduce a network of N cells divided into two clusters: the coupling between neurons in each cluster is excitatory, while the coupling between the two clusters is inhibitory. Such system aims at modelling the interactions between neurons tending to synchronization in each of two subpopulations that inhibit each other, like ipsi- and contra-lateral motoneurons assemblies. To perform the numerical simulations in this case when N is large, as an initial step towards the network analysis, we consider Reduced Order Models in order to save computational costs. We present the numerical reduction results in a network of 100 cells. For the sake of validation of the numerical reduction method, we both compare the outputs and CPU times obtained with the original and the reduced models in different cases of network coupling structures.

Canard-induced mixed mode oscillations as a mechanism for the Bonhoeffer-van der Pol circuit under parametric perturbation

PurposeThis paper aims to investigate the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. There is a singular periodic orbit constructed by the switching between the stable focus and large amplitude relaxation cycles. Using a generalized fast/slow analysis, the authors show the generation mechanism of two distinct kinds of MMOs.Design/methodology/approachThe parametric modulation can be used to generate complicated dynamics. The BVP circuit is constructed as an example for second-order differential equation with periodic perturbation. Then the authors draw the bifurcation parameter diagram in terms of a containing two attractive regions, i.e. the stable relaxation cycle and the stable focus. The transition mechanism and characteristic features are investigated intensively by one-fast/two-slow analysis combined with bifurcation theory.FindingsPeriodic perturbation can suppress nonlinear circuit dynamic to a singular periodic orbit. The combination of these small oscillations with the large amplitude oscillations that occur due to canard cycles yields such MMOs. The results connect the theory of the singular Hopf bifurcation enabling easier calculations of where the oscillations occur.Originality/valueBy treating the perturbation as the second slow variable, the authors obtain that the MMOs are due to the canards in a supercritical case or in a subcritical case. This study can reveal the transition mechanism for multi-time scale characteristics in perturbed circuit. The information gained from such results can be extended to periodically perturbed circuits.

Emergent Dynamics and Spatio Temporal Patterns on Multiplex Neuronal Networks.

We present a study on the emergence of a variety of spatio temporal patterns among neurons that are connected in a multiplex framework, with neurons on two layers with different functional couplings. With the Hindmarsh-Rose model for the dynamics of single neurons, we analyze the possible patterns of dynamics in each layer separately and report emergent patterns of activity like in-phase synchronized oscillations and amplitude death (AD) for excitatory coupling and anti-phase mixed-mode oscillations (MMO) in multi-clusters with phase regularities when the connections are inhibitory. When they are multiplexed, with neurons of one layer coupled with excitatory synaptic coupling and neurons of the other layer coupled with inhibitory synaptic coupling, we observe the transfer or selection of interesting patterns of collective behavior between the layers. While the revival of oscillations occurs in the layer with excitatory coupling, the transition from anti-phase to in-phase and vice versa is observed in the other layer with inhibitory synaptic coupling. We also discuss how the selection of these spatio temporal patterns can be controlled by tuning the intralayer or interlayer coupling strengths or increasing the range of non-local coupling. With one layer having electrical coupling while the other synaptic coupling of excitatory(inhibitory)type, we find in-phase(anti-phase) synchronized patterns of activity among neurons in both layers.

Electrochemical oscillations during the oxidation of 2-(dimethylamino)ethanethiol

• The electrochemical oxidation of DMAET led to oscillations in current and potential. • Electrochemical impedance spectroscopy was used to confirm the presence of negative differential resistance. • High frequency oscillations were observed under potentiostatic conditions. • The system supports complexity in the form of bursting and mixed-mode oscillations. This paper describes the electrochemical oxidation of 2-(dimethylamino)ethanethiol (DMAET), a known hydrolysis product of V-type nerve agents, which can be used to mimic the behavior of chemical warfare agents. Interestingly, during its oxidation, two distinct regions of dynamic (i.e., oscillatory) behavior were uncovered. Mechanistic investigation using electrochemical impedance spectroscopy (EIS) lead to the discovery that one of the oscillatory regions was driven by the presence of negative differential resistance (NDR) and that the occurrence of the other was dependent on the concentration of supporting electrolyte. EIS measurements indicated that the DMAET electrochemical oscillator belongs to the HN-NDR class of oscillators, which was supported by the presence of sustained oscillations in both current (under potentiostatic conditions) and potential (under galvanostatic conditions). Furthermore, the oxidation of DMAET on platinum electrodes resulted in rich dynamical behavior in the form of complex bursting phenomena.