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.

Bifurcation and Chaos of Spontaneous Oscillations of Hair Bundles in Auditory Hair Cells

Various spontaneous oscillations and Hopf bifurcation have been observed in hair bundles of auditory hair cells, which play very important roles in the auditory function. In the present paper, the bifurcations and chaos of spontaneous oscillations of hair bundles are investigated in a theoretical model to explain the experimental observations. Firstly, the equivalent negative stiffness and symmetrical characteristic of the model are acquired. The model exhibits coexisting attractors symmetrical to each other or an attractor with symmetry by itself. The attractors include stable focus, stable periodic oscillations, and chaotic oscillations. Secondly, except for the well-known subcritical and supercritical Hopf bifurcations from the stable focus to period-1 limit cycle, the complex bifurcations of spontaneous oscillation patterns such as period-doubling bifurcation cascade to chaos and intermittency between periodic limit cycles and chaos, are observed. Various chaotic oscillations are distinguished. Lastly, a complex bifurcation process containing multiple modes of oscillations and bifurcations mentioned above is obtained, which provides the relationships between different spontaneous oscillation patterns. The results present not only the well-known Hopf bifurcation, but also the various spontaneous oscillations including periodic and chaotic patterns, which are consistent with the recent experimental results. The complex bifurcation process presents a global view of the nonlinear dynamics of complex spontaneous oscillations of hair bundles, which is very important for the auditory function.

Phase dependence of response curves to stimulation and their relationship: from a Wilson-Cowan model to essential tremor patient data

Essential tremor manifests predominantly as a tremor of the upper limbs. One therapy option is high-frequency deep brain stimulation, which continuously delivers electrical stimulation to the ventral intermediate nucleus of the thalamus at about 130 Hz. Investigators have been looking at stimulating less, chiefly to reduce side effects. One strategy, phase-locked deep brain stimulation, consists of stimulating according to the phase of the tremor, once per period. In this study, we aim to reproduce the phase dependent effects of stimulation seen in patient data with a biologically inspired Wilson-Cowan model. To this end, we first analyse patient data, and conclude that half of the datasets have response curves that are better described by sinusoidal curves than by straight lines, while an effect of phase cannot be consistently identified in the remaining half. Using the Hilbert phase we derive analytical expressions for phase and amplitude responses to phase-dependent stimulation and study their relationship in the linearisation of a stable focus model, a simplification of the Wilson-Cowan model in the stable focus regime. Analytical results provide a good approximation for response curves observed in patients with consistent significance. Additionally, we fitted the full non-linear Wilson-Cowan model to these patients, and we show that the model can fit in each case to the dynamics of patient tremor as well as the phase response curve, and the best fits are found to be stable foci for each patients (tied best fit in one instance). The model provides satisfactory prediction of how patient tremor will react to phase-locked stimulation by predicting patient amplitude response curves although they were not explicitly fitted. This can be partially explained by the relationship between the response curves in the model being compatible with what is found in the data. We also note that the non-linear Wilson-Cowan model is able to describe response to stimulation more precisely than the linearisation.

Bristol Classics Hub – reflections on the first year

The Bristol Classics Hub was set up in September 2016 to promote and support the teaching of classical subjects in state schools in the South West of England. Funded by Classics for All and the Institute of Greece, Rome and the Classical Tradition and delivered in partnership with the University of Bristol, the hub aims to widen access to Classics by offering a powerful and stable focus for regional development.

CHAOTIC FEATURE OF MARTIN PROCESS IMPOSED ON THE COSINE FUNCTION

By analyzing the phase diagram of Martin process on the cosine function, it is shown that with the change of system parameters the system will eventually converge to a chaotic attractor. The process is repeated and stable focus, period doubling bifurcation occurs during this process. Further computation gives the maximum Lyapunov exponent of the system and meanwhile, the bifurcation diagram is drawn. Thus it is proved from theory that the system exhibits strong chaotic properties.

GWN-INDUCED BURSTING, SPIKING, AND RANDOM SUBTHRESHOLD IMPULSING OSCILLATION BEFORE HOPF BIFURCATIONS IN THE CHAY MODEL

Gaussian white noise (GWN), as an intrinsic noise source, can give rise to various firing activities at the rest state before a supercritical or subcritical Hopf bifurcation (supH or subH) in the Chay system without or with external current input, when VK, VC, λn and I are considered as changeable control parameters. These firing activities are closely related to the global bifurcation mechanism of the whole system and the fast/slow dynamical subsystems, and can be tackled by means of bifurcation analysis. GWN can induce some typical bursting phenomena in the stochastic Chay system. Firstly, integer multiple "fold/homoclinic" or "circle/homoclinic" bursting due to GWN, with only one spike per burst, can arise from rest states before both subH and supH (with respect to the parameter VK), and their respective trajectories have the same shape and property. However, less spikes appear and their peaks are lower before supH, comparing with those before subH. Secondly, a "fold/fold" point–point hysteresis loop bursting due to GWN is generated before supH (with respect to the parameter VC) on the upper branch of a "Z"-shaped bifurcation curve between two fold bifurcations of the fast system. Thirdly, at a rest state before subH (with respect to the additional current I) situated on the lower branch of a "S"-shaped bifurcation curve between two fold bifurcations of the fast system, a GWN-induced firing pattern appears and is classified as "Hopf/homoclinic" bursting via "fold/homoclinic" point–point hysteresis loop. GWN-induced firing activities other than bursting can also be observed in the stochastic Chay system. For example, sometimes GWN-induced continuous spiking without any particular shape may arise at a rest state before supH (with respect to the parameter VK) for certain values of parameters. Moreover, under the situation that a stable node and a stable focus coexist before subH (with respect to the parameter I) and the attractive region of the stable node is larger than that of the stable focus, GWN only provoke random subthreshold impulsing oscillation near the stable node.

Soil nutrient cycles as a nonlinear dynamical system

An analytical model for the soil carbon and nitrogen cycles is studied from the dynamical system point of view. Its main nonlinearities and feedbacks are analyzed by considering the steady state solution under deterministic hydro-climatic conditions. It is shown that, changing hydro-climatic conditions, the system undergoes dynamical bifurcations, shifting from a stable focus to a stable node and back to a stable focus when going from dry, to well-watered, and then to saturated conditions, respectively. An alternative degenerate solution is also found in cases when the system can not sustain decomposition under steady external conditions. Different basins of attraction for "normal" and "degenerate" solutions are investigated as a function of the system initial conditions. Although preliminary and limited to the specific form of the model, the present analysis points out the importance of nonlinear dynamics in the soil nutrient cycles and their possible complex response to hydro-climatic forcing.