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)’.

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

Purpose This 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/approach The 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. Findings Periodic 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/value By 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.

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.

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.

Transition from Anti-coherence resonance to coherence resonance for mixed-mode oscillations and period-1 firing of nervous system

Identification of dynamics of the mixed-mode oscillations (MMOs), which exhibit transition between oscillations with large and small amplitudes, is very important for nonlinear physics. In this paper, the MMOs with transition between subthreshold oscillations and spikes are investigated in a neuron model. In the absence of noise, the MMOs appear between the resting state and period-1 firing with increasing depolarization current. After introducing white noise, coherence resonance (CR) is evoked from the resting state and non-CR is induced from period-1 firing far from the MMOs, which is consistent with the traditional viewpoint. However, an interesting result that a transition from anti-CR to CR is evoked by noise from both the MMOs and the period-1 firing near the MMOs is acquired, which is characterized by the increase, decrease and increase again of the coefficient of variations of interspike intervals (ISIs) with increasing noise intensity. At small noise intensity, more subthreshold oscillations are evoked by noise to reduce the firing frequency, resulting in faster increase of standard deviation (SD) of ISIs than that of mean value of ISIs, which is the cause for the anti-CR. The decrease of SD is faster for middle noise intensity and is lower for strong noise intensity, which is the cause for the CR. The different stochastic responses of MMOs and period-1 firing nearby at different levels of noise insanity are the dynamical mechanism for the transition from anti-CR to CR. Such results present potential functions of the MMOs and period-1 firing on information processing in the nervous system with noise and extend the conditions for the CR and anti-CR phenomena, which enriches the contents of nonlinear dynamics.

Mixed-mode oscillations for slow-fast perturbed systems

This article concerns the dynamics of mixed-mode oscillations (MMOs) emerging from the calcium-based inner hair cells (IHCs) model in the auditory cortex. The paper captures the MMOs generation mechanism based on the geometric singular perturbation theory (GSPT) after exploiting the average analysis for reducing the full model. Our analysis also finds that the critical manifold and folded surface are central to the mechanism of the existence of MMOs at the folded saddle for the perturbed system. The system parameters, such like the maximal calcium channels conductance, controls the firing patterns, and many new oscillations occur for the IHCs model. Tentatively, we conduct dynamic analysis combined with dynamic method based on GSPT by giving slow-fast analysis for the singular perturbed models and bifurcation analysis. In particular, we explore the two-slow-two-fast and three-slow-one-fast IHCs perturbed systems with layer and reduced problems so that differential-algebraic equations are obtained. This paper reveals the underlying dynamic properties of perturbed systems under singular perturbation theory.

Synchronization and Pattern Formation in a Memristive Diffusive Neuron Model

In this article, we construct an excitable memristive diffusive neuron model by considering a biophysical slow–fast bursting oscillator and study the effects of electromagnetic induction on the dynamics of the single model as well as the coupled systems. We explore various firing regimes such as tonic spiking, bursting, and mixed-mode oscillations depending on the bifurcation structure with different injected current stimuli, then perform a comparative analysis on the synchronization of the coupled oscillators by setting the model into two different network architectures. First, a diffusively coupled network is considered, and later a global network is constructed. The results suggest that the diffusively connected neurons show complete synchronization at higher couplings for bursting and tonic spiking regimes. Furthermore, we show that the extended spatial system can generate spiral-like patterns in the vicinity of a Hopf bifurcation point and observe the impact of Gaussian white noise to study its effects on pattern formation. These types of patterns are robust in the excitable model. Our results might contribute significantly to the dynamical studies of irregular neural computation.