Complex Mixed-mode Oscillations Research Articles

Periodicity, mixed-mode oscillations, and multiple timescales in a phosphoinositide-Rho GTPase network

While rhythmic contractile behavior is commonly observed at the cellular cortex, the primary focus has been on excitable or periodic events described by simple activator-delayed inhibitor mechanisms. We show that Rho GTPase activation in nocodazole-treated mitotic cells exhibits both simple oscillations and complex mixed-mode oscillations. Rho oscillations with a 20- to 30-s period are regulated by phosphatidylinositol (3,4,5)-trisphosphate (PIP3) via an activator-delayed inhibitor mechanism, while a slow reaction with period of minutes is regulated by phosphatidylinositol 4-kinase via an activator-substrate depletion mechanism. Conversion from simple to complex oscillations can be induced by modulating PIP3 metabolism or altering membrane contact site protein E-Syt1. PTEN depletion results in a period-doubling intermediate, which, like mixed-mode oscillations, is an intermediate state toward chaos. In sum, this system operates at the edge of chaos. Small changes in phosphoinositide metabolism can confer cells with the flexibility to rapidly enter ordered states with different periodicities.

Periodically kicked feedforward chains of simple excitable FitzHugh–Nagumo neurons

This article communicates results on regular depolarization cascades in periodically kicked feedforward chains of excitable two-dimensional FitzHugh–Nagumo systems driven by sufficiently strong excitatory forcing at the front node. The study documents a parameter exploration by way of changes to the forcing period, upon which the dynamics undergoes a transition from simple depolarization to more complex behavior, including the emergence of mixed-mode oscillations. Both rigorous studies and careful numerical observations are presented. In particular, we provide rigorous proofs for existence and stability of periodic traveling waves of depolarization, as well as the existence and propagation of a simple mixed-mode oscillation that features depolarization and refraction in alternating fashion. Detailed numerical investigation reveals a mechanism for the emergence of complex mixed-mode oscillations featuring a potentially high number of large amplitude voltage spikes interspersed by an occasional small amplitude reset that fails to cross threshold. Further careful numerical investigation provides insights into the propagation of this complex phenomenology in the downstream, where we see an effective filtration property of the network; the latter amounts to a successive reduction in the complexity of mixed-mode oscillations down the chain.

Complex mixed-mode oscillations based on a modified Rayleigh-Duffing oscillator driven by low-frequency excitations

The complex mixed-mode oscillation patterns are proposed and analyzed in a modified Rayleigh-Duffing oscillator based on the bifurcation theory in this paper. Four mixed-mode oscillations, namely “Homoclinic/Homoclinic-Homoclinic/Homoclinic” intermittent type, “fold/Homoclinic-Homoclinic/supHopf” intermittent type, “fold/Homoclinic-supHopf/supHopf” intermittent type and “fold/Homoclinic” type, are discussed in detail. Considering the low frequency excitations as slow-changing state variables, a modified autonomous system is obtained. The bifurcation characteristics of the fast subsystem are presented briefly by using the bifurcation theory. Then, we investigate the generation principle of the bifurcation delay phenomenon that performs a critical role in the production of two mixed-mode oscillations. This paper presents a fact that the dynamical behaviors are sensitive to the values of the system parameters and the parameters determine different forms of the repetitive spiking states that leads to different patterns of the mixed-mode oscillations. In addition, the theoretical analysis and numerical simulations are compared to illustrate the correctness of this paper.

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.

Complex mixed-mode oscillations in oscillators sharing nonlinearity

Complex mixed-mode oscillations in oscillators sharing nonlinearity

Fast autaptic feedback induced-paradoxical changes of mixed-mode bursting and bifurcation mechanism

Bursting is a complex multiple-time-scale nonlinear phenomenon in a nervous system and exhibits diverse patterns, which is modulated by the excitatory or inhibitory effect to achieve the physiological functions. According to the bifurcations of the fast subsystem, bursting is classified as different patterns including the mixed-mode bursting. Recently, many studies have found the paradoxical phenomena contrary to the common concept that the inhibitory effect induces the electrical activity to decrease or the excitatory effect induces the discharge activity to increase, which enriches the connotation of the nonlinear dynamics. To identify more examples of paradoxical phenomena of different bursting patterns and the underlying nonlinear mechanism, in the present study the paradoxical phenomena for the complex mixed-mode oscillations of the bursting pattern induced by the self-feedback mediated by both the inhibitory autapse and excitatory autapse and the bifurcation mechanism are acquired in the modified Morris-Lecar model. By using the fast/slow variable dissection method, the complex dynamics of the bursting is acquired, which is that the depolarization block behavior appears after the burst and before the quiescent state. The burst begins from a saddle-node bifurcation in an invariant cycle (SNIC) and terminates at a fold limit cycle (FLC) bifurcation. Furthermore, the FLC bifurcation is identified to play a key role in generating the paradoxical phenomenon. The inhibitory autapse induces the FLC bifurcation to shift leftward. However, the SNIC point remains unchanged. The change of FLC bifurcation point leads the parameter range of the burst to widen, the number of spikes per burst to become larger, and the average firing frequency to turn higher. Unlike the inhibitory autapse, the excitatory autapse induces the FLC bifurcation to shift rightward, and SNIC to be unchanged, thus reducing the bursting activity. Such results are different from those of the paradoxical phenomenon induced by the inhibitory autapse instead of excitatory autapse for the other bursting pattern and by the slow auatpse, which present a novel example and regulation mechanism of the paradoxical phenomena of the bursting patterns and show the diversity of the paradoxical phenomena, thus helping understand the potential functions of the bursting and self-feedback modulations of the brain neurons.

Inflection, Canards and Folded Singularities in Excitable Systems: Application to a 3D FitzHugh–Nagumo Model

Specific kinds of physical and biological systems exhibit complex Mixed-Mode Oscillations mediated by folded-singularity canards in the context of slow-fast models. The present manuscript revisits these systems, specifically by analysing the dynamics near a folded singularity from the viewpoint of inflection sets of the flow. Originally, the inflection set method was developed for planar systems [Brons and Bar-Eli in Proc R Soc A 445(1924):305–322, 1994; Okuda in Prog Theor Phys 68(6):1827–1840, 1982; Peng et al. in Philos Trans R Soc A 337(1646):275–289, 1991] and then extended to N-dimensional systems [Ginoux et al. in Int J Bifurc Chaos 18(11):3409–3430, 2008], although not tailored to specific dynamics (e.g. folded singularities). In our previous study, we identified components of the inflection sets that classify several canard-type behaviours in 2D systems [Desroches et al. in J Math Biol 67(4):989–1017, 2013]. Herein, we first survey the planar approach and show how to adapt it for 3D systems with an isolated folded singularity by considering a suitable reduction of such 3D systems to planar non-autonomous slow-fast systems. This leads us to the computation of parametrized families of inflection sets of one component of that planar (non-autonomous) system, in the vicinity of a folded node or of a folded saddle. We then show that a novel component of the inflection set emerges, which approximates and follows the axis of rotation of canards associated to folded-node and folded-saddle singularities. Finally, we show that a similar inflection-set component occurs in the vicinity of a delayed Hopf bifurcation, a scenario that can arise at the transition between folded node and folded saddle. These results are obtained in the context of a canonical model for folded-singularity canards and subsequently we show it is also applicable to complex slow-fast models. Specifically, we focus the application towards the self-coupled 3D FitzHugh–Nagumo model, but the method is generically applicable to higher-dimensional models with isolated folded singularities, for instance in conductance-based models and other physical-chemical systems.

EXCITABILITY AND COMPLEX MIXED-MODE OSCILLATIONS IN STOCHASTIC BUSINESS CYCLE MODEL

Stochastically forced business cycle model is considered. We study a phenomenon of the noise-induced generation of complex mixed-mode oscillations in zones where the deterministic model exhibits a single attractor (equilibrium or limit cycle). A probabilistic mechanism of the transition from unimodal oscillations to the intermittency of small- and large-amplitude stochastic oscillations is investigated on the base of the stochastic sensitivity functions technique. We show that such intermittency, connected with the nonuniformity of phase portrait, can be parametrically analyzed by the confidence domains method.

Experimental study of complex mixed-mode oscillations generated in a Bonhoeffer-van der Pol oscillator under weak periodic perturbation.

Bifurcations of complex mixed-mode oscillations denoted as mixed-mode oscillation-incrementing bifurcations (MMOIBs) have frequently been observed in chemical experiments. In a previous study [K. Shimizu et al., Physica D 241, 1518 (2012)], we discovered an extremely simple dynamical circuit that exhibits MMOIBs. Our model was represented by a slow/fast Bonhoeffer-van der Pol circuit under weak periodic perturbation near a subcritical Andronov-Hopf bifurcation point. In this study, we experimentally and numerically verify that our dynamical circuit captures the essence of the underlying mechanism causing MMOIBs, and we observe MMOIBs and chaos with distinctive waveforms in real circuit experiments.

Complex mixed-mode oscillations in a Bonhoeffer–van der Pol oscillator under weak periodic perturbation

In this paper, we elucidate the extremely complicated bifurcation structure of a weakly driven relaxation oscillator by focusing on chaos, and notably, on complex mixed-mode oscillations (MMOs) generated in a simple dynamical model. Our model uses the Bonhoeffer–van der Pol (BVP) oscillator subjected to a weak periodic perturbation near a subcritical Andronov–Hopf bifurcation (AHB). The mechanisms underlying the chaotic dynamics can be explained using an approximate one-dimensional map. The MMOs that appear in this forced dynamical model may be more sophisticated than those appearing in three-variable slow–fast autonomous dynamics because the approximate one-dimensional mapping of the dynamics used herein is a circle map, whereas the one-dimensional first-return map that is derived from the three-variable slow–fast autonomous dynamics is usually a unimodal map. In this study, we generate novel bifurcations such as an MMO-incrementing bifurcation (MMOIB) and intermittently chaotic MMOs. MMOIBs trigger an MMO sequence that, upon varying a parameter, is followed by another type of MMO sequence. By constructing a two-parameter bifurcation diagram, we confirmed that MMOIBs occur successively. According to our numerical results, MMOIBs are often observed between two neighboring MMOs. Numerically, MMOIBs may occur as many times as desired. We also derive the universal constant of the associated successive MMOIBs. The existence of the universal constant suggests that MMOIBs could occur infinitely many times. Furthermore, intermittently chaotic MMOs appear in this dynamical circuit. The intermittently chaotic MMOs relate to a type of intermittent chaos that resembles MMOs at first glance, but includes rare bursts over a long time interval. Complex intermittently chaotic MMOs of various types are observed, and we clarify that the intermittently chaotic MMOs are generated by crisis-induced intermittency.

Effects of delayed linear electrical perturbation of the Belousov-Zhabotinsky reaction: a case of complex mixed mode oscillations in a batch reactor

Delayed feedbacks are quite common in many physical and biological systems and in particular many physiological systems. Delay can cause a stable system to become unstable and vice versa. One of the well-studied non-biological chemical oscillators is the Belousov-Zhabotinskii (BZ) reaction. This gives relaxation oscillations for a considerable period of time under batch conditions. This paper deals with the effect of perturbing the limit cycle oscillation of BZ reaction by employing a delayed electrical feedback to the system under batch conditions. The parameters chosen to study are external resistance and delay. For various resistances and delays the system was electrically perturbed and found to exhibit various complex mixed mode oscillations. The dynamic features are accounted for by the Oregonator model, with time delay incorporated in one of the variables.

Investigation of Dynamic Behavior of the Bray−Liebhafsky Reaction in the CSTR. Determination of Bifurcation Points

Experimental results obtained by operating the Bray−Liebhafsky (BL) reaction in the CSTR are presented. The dynamic behavior of the BL reaction is examined at several operation points in the concentration phase space, by varying different parameters, the specific flow rate, temperature, and mixed inflow concentrations of the feed species, one at a time. Different types of bifurcation leading to simple periodic orbits, supercritical and subcritical Hopf bifurcations, saddle node infinite period bifurcation (SNIPER), saddle loop infinite period bifurcation, and jug handle bifurcation, are observed. Moreover, complex dynamic behavior, including transition from simple periodic oscillations to complex mixed-mode oscillations and chaos, and bistability are also discovered.

Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram

Abstract The bifurcation structure of a three-variable Van der Pol-Duffing-type model is studied in some detail, with special attention to the mixed-mode solutions, a type of complex periodic behavior frequently encountered in oscillating chemical reactions. The mixed-mode oscillations in the model occur close to two Hopf bifurcations, which are arranged with the saddle-node bifurcations in a so-called cross-shaped phase diagram, a bifurcation diagram also typical for chemical reactions. The mixed-mode oscillations are shown to lie on isolated bifurcation curves, which are all born in a single codimension-two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch. With the introduction of an additional slow time scale, the same model can exhibit more complex mixed-mode oscillations and torus bifurcations.

Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram

The bifurcation structure of a three-variable Van der Pol-Duffing-type model is studied in some detail, with special attention to the mixed-mode solutions, a type of complex periodic behavior frequently encountered in oscillating chemical reactions. The mixed-mode oscillations in the model occur close to two Hopf bifurcations, which are arranged with the saddle-node bifurcations in a so-called cross-shaped phase diagram, a bifurcation diagram also typical for chemical reactions. The mixed-mode oscillations are shown to lie on isolated bifurcation curves, which are all born in a single codimension-two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch. With the introduction of an additional slow time scale, the same model can exhibit more complex mixed-mode oscillations and torus bifurcations.

Organizing centres and symbolic dynamic in the study of mixed-mode oscillations generated by models of biological autocatalytic processes

The organization of the complex mixed-mode oscillations generated, in a three-dimensional variable space, by an autocatalytic process formalized as a cubic monomial is analyzed. The generation of the temporal patterns is elucidated by complementary approaches dealing with the three-variable differential continuous system itself and with successive discrete applications modelling its first return map. The extent to which the underlying bifurcation structures could constitute a fingerprint of autocatalytic processes is discussed in connection with the modelling of biological systems.

Characterisation of oscillations in the H2+ O2 reaction in a continuous-flow reactor

The H2+ O2 reaction is one of the simplest chemical oscillators and of great practical significance. In a well stirred flow system, equimolar mixtures support oscillatory ignitions, with a simple periodic waveform, over a range of ambient temperature and operating pressure. Here we report a series of experiments in which the outputs from the monitoring thermocouple and photomultiplier are data captured onto computer. These ‘time series’ are then treated via a number of techniques from dynamical systems theory, including fast Fourier transforms and attractor reconstructions. This allows the routes by which oscillations set in and are extinguished to be recognised. New responses have then been predicted in the light of these theories and observed experimentally. With a stoichiometric 2H2+ O2 mixture, complex mixed-mode oscillations and birhythmicity are also observed over some range of experimental conditions.