Understanding bursting oscillations as periodic slow passages through bifurcation and limit points

Slow passage through a Hopf-like bifurcation in piecewise linear systems: Application to elliptic bursting

The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly varying quantity replacing a static parameter induces the solutions of the resulting slow-fast system to feel the effect of the Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow-fast dynamical systems; in the present work, we study it for the first time in piecewise linear (PWL) slow-fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular, through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slow-fast system to exhibit a slow passage through a Hopf-like bifurcation, in link with possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the so-called way-in/way-out function. Finally, we investigate this slow passage effect in the Doi-Kumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.

Slow passage through resonance

The slow passage problem through a resonance is considered. As a model problem, we consider a damped harmonically forced oscillator whose forcing frequency is slowly ramped linearly in time. The setup is similar to the familiar slow passage through a Hopf bifurcation problem, where for slow variations of the control parameter, oscillations are delayed until the parameter has exceeded the critical value of the static-parameter problem by an amount that is the difference between the Hopf value and the initial value of the parameter. In sharp contrast, in the resonance problem there is an early onset of resonance, setting in when the ramped forcing frequency is midway between its initial value and the natural frequency for resonance in the unforced problem; we term this value the jump frequency. Numerically, we find that the jump frequency is independent of the system's damping coefficient, and so we also consider the undamped problem, which is analytically tractable. The analysis of the undamped problem confirms the numerical results found in the damped problem that the maximal amplitude obtained at the jump frequency scales as A~ε(-1/2), ε being the ramp rate, and that the jump frequency is midway between the initial frequency at the start of the ramp and the natural frequency of the unforced problem.

Amplification of information transfer in excitable systems that reside in a steady state near a bifurcation point to complex oscillatory behavior

We study the amplification of information transfer in excitable systems. We show that excitable systems residing in a steady state near a bifurcation point to complex oscillatory behavior incorporate several frequencies that can be exploited for a resonant amplification of information transfer. In particular, for excitable neurons that reside in a steady state near a bifurcation point to elliptic bursting oscillations, we show that in addition to the resonant frequency of damped oscillations around the stable focus, another frequency exists that resonantly enhances large amplitude bursts and thus amplifies the information transfer in the system. This additional frequency cannot be found by the local stability analysis and has never been used for amplifying the information transfer in a system. The results obtained for elliptic bursting oscillations can be generalized also to other complex oscillators, such as parabolic or square-wave bursters. Additionally, the biological importance of presented results in the field of neuroscience is outlined.

Detecting and controlling unstable periodic orbits that are not part of a chaotic attractor

A method for controlling unstable periodic orbits (UPOs) that have not been controllable before is presented. The method is based on detecting UPOs that are situated outside the skeleton of a chaotic attractor. The main idea is to exploit flexible parts of the attractor, which under weak external perturbations allow variable excursions of the trajectory away from its originally determined path. After the perturbation, the trajectory of the autonomous system seeks its path back to the chaotic attractor and reveals additional UPOs that are otherwise not used by the system. It is shown that these UPOs can be controlled as easily as the UPOs that form the basic chaotic attractor. The effectiveness of the proposed method is demonstrated on two different chaotic systems with very distinct response abilities to external perturbations. Additionally, some applications of the method in the fields of laser technology, information encoding, and biomedical engineering are discussed.

Periodically-modulated inhibition of living pacemaker neurons--III. The heterogeneity of the postsynaptic spike trains, and how control parameters affect it

Codings involving spike trains at synapses with inhibitory postsynaptic potentials on pacemakers were examined in crayfish stretch receptor organs by modulating presynaptic instantaneous rates periodically (triangles or sines; frequencies, slopes and depths under, respectively, 5.0 Hz, 40.0/s/s and 25.0/s). Timings were described by interspike and cross-intervals ("phases"); patterns (dispersions, sequences) and forms (timing classes) were identified using pooled graphs (instant along the cycle when a spike occurs vs preceding interval) and return maps (plots of successive intervals). A remarkable heterogeneity of postsynaptic intervals and phases characterizes each modulation. All cycles separate into the same portions: each contains a particular form and switches abruptly to the next. Forms differ in irregularity and predictability: they are (see text) "p:q alternations", "intermittent", "phase walk-throughs", "messy erratic" and "messy stammering". Postsynaptic cycles are asymmetric (hysteresis). This contrasts with the presynaptic homogeneity, smoothness and symmetry. All control parameters are, individually and jointly, strongly influential. Presynaptic slopes, say, act through a postsynaptic sensitivity to their magnitude and sign; when increasing, hysteresis augments and forms change or disappear. Appropriate noise attenuates between-train contrasts, providing modulations are under 0.5 Hz. Postsynaptic natural intervals impose critical time bases, separating presynaptic intervals (around, above or below them) with dissimilar consequences. Coding rules are numerous and have restricted domains; generalizations are misleading. Modulation-driven forms are trendy pacemaker-driven forms. However, dissimilarities, slight when patterns are almost pacemaker, increase as inhibition departs from pacemaker and incorporate unpredictable features. Physiological significance-(1) Pacemaker-driven forms, simple and ubiquitous, appear to be elementary building blocks of synaptic codings, present always but in each case distorted typically. (2) Synapses are prototype: similar behaviours should be widespread, and networks simulations benefit by nonlinear units generating all forms. (3) Relevant to periodic functions are that few variables need be involved in form selection, that distortions are susceptible to noise levels and, if periods are heterogeneous, that simple input cycles impose heterogeneous outputs. (4) Slow Na inactivations are necessary for obtaining complex forms and hysteresis. Formal significance--(1) Pacemaker-driven forms and presumably their modulation-driven counterparts, pertain to universal periodic, intermittent, quasiperiodic and chaotic categories whose formal properties carry physiological connotations. (2) Only relatively elaborate, nonlinear geometric models show all forms; simpler ones, show only alternations and walk-throughs. (3) Bifurcations resemble those of simple maps that can provide useful guidelines. (4) Heterogeneity poses the unanswered question of whether or not the entire cycle and all portions have the same behaviours: therefore, whether trajectories are continuous or have discontinuities and/or singular points.