Mixed-Mode Oscillations in a Stochastic, Piecewise-Linear System

We analyze a piecewise-linear FitzHugh-Nagumo model. The system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist. The addition of small noise induces mixed-mode oscillations (MMOs) in the vicinity of the canard point. We determine the effect of each model parameter on the stochastically driven MMOs. In particular we show that any parameter variation (such as a modification of the piecewise-linear function in the model) that leaves the ratio of noise amplitude to time-scale separation unchanged typically has little effect on the width of the interval of the primary bifurcation parameter over which MMOs occur. In that sense, the MMOs are robust. Furthermore we show that the piecewise-linear model exhibits MMOs more readily than the classical FitzHugh-Nagumo model for which a cubic polynomial is the only nonlinearity. By studying a piecewise-linear model we are able to explain results using analytical expressions and compare these with numerical investigations.Comment: 25 pages, 10 figure

Mini-Workshop: Dynamics of Stochastic Systems and their Approximation

The aim of this workshop was to bring together specialists in the area of stochastic dynamical systems and stochastic numerical analysis to exchange their ideas about the state of the art of approximations of stochastic dynamics. Here approximations are considered in the analytical sense in terms of deriving reduced dynamical systems, which are less complex, as well as in the numerical sense via appropriate simulation methods. The main theme is concerned with the efficient treatment of stochastic dynamical systems via both approaches assuming that ideas and methods from one ansatz may prove beneficial for the other. A particular goal was to systematically identify open problems and challenges in this area

A mathematical framework for critical transitions: normal forms, variance and applications

Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find early-warning signs. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Based on this elementary classification, we analyze stochastic fluctuations and calculate scaling laws of the variance of stochastic sample paths near critical transitions for fast subsystem bifurcations up to codimension two. The theory is applied to several models: the Stommel-Cessi box model for the thermohaline circulation from geoscience, an epidemic-spreading model on an adaptive network, an activator-inhibitor switch from systems biology, a predator-prey system from ecology and to the Euler buckling problem from classical mechanics. For the Stommel-Cessi model we compare different detrending techniques to calculate early-warning signs. In the epidemics model we show that link densities could be better variables for prediction than population densities. The activator-inhibitor switch demonstrates effects in three time-scale systems and points out that excitable cells and molecular units have information for subthreshold prediction. In the predator-prey model explosive population growth near a codimension two bifurcation is investigated and we show that early-warnings from normal forms can be misleading in this context. In the biomechanical model we demonstrate that early-warning signs for buckling depend crucially on the control strategy near the instability which illustrates the effect of multiplicative noise.Comment: minor corrections to previous versio

Scale-free avalanches in arrays of FitzHugh-Nagumo oscillators

The activity in the brain cortex remarkably shows a simultaneous presence of robust collective oscillations and neuronal avalanches, where intermittent bursts of pseudo-synchronous spiking are interspersed with long periods of quiescence. The mechanisms allowing for such a coexistence are still a matter of an intensive debate. Here, we demonstrate that avalanche activity patterns can emerge in a rather simple model of an array of diffusively coupled neural oscillators with multiple timescale local dynamics in vicinity of a canard transition. The avalanches coexist with the fully synchronous state where the units perform relaxation oscillations. We show that the mechanism behind the avalanches is based on an inhibitory effect of interactions, which may quench the spiking of units due to an interplay with the maximal canard. The avalanche activity bears certain heralds of criticality, including scale-invariant distributions of event sizes. Furthermore, the system shows an increased sensitivity to perturbations, manifested as critical slowing down and a reduced resilience.Comment: 9 figure

Samoorganizacija u spregnutim ekscitabilnim sistemima: sadejstvo višestrukih vremenskih skala i šuma

The dynamics of complex systems typically involves multiple spatial and temporal scales, while emergent phenomena are often associated with critical transitions in which a small parameter variation causes a sudden shift to a qualitatively different regime. In the vicinity of such transitions, complex systems are highly sensitive to external perturbations, potentially resulting in dynamical switching between different (meta)stable states. Such behavior is typical for many biological systems consisting of coupled excitable units. In neuronal systems, for instance, self-organization is influenced by the interplay between noise from diverse sources and a multi-timescale structure arising from both local and coupling dynamics. The present thesis is devoted to several types of self-organized dynamics in coupled stochastic excitable systems with multiple timescale dynamics. The excitable behavior of single units is well understood, in terms of both the nonlinear threshold-like response to external perturbations and the characteristic non-monotonous response to noise, embodied by different resonant phenomena. However, the excitable behavior of coupled systems, as a new paradigm of emergent dynamics, involves a number of fundamental open problems, including how interactions modify local dynamics resulting in excitable behavior at the level of the coupled system, and how the interplay of multiscale dynamics and noise gives rise to switching dynamics and resonant phenomena. This thesis comprises a systematic approach to addressing these issues, consisting of three complementary lines of research. In particular, within the first line of research, we have extended the notion of excitability to coupled systems, considering the examples of a small motif of locally excitable units and a population of stochastic neuronal maps. In the case of the motif, we have classified different types of excitable responses and, by applying elements of singular perturbation theory, identified what determines the motif’s threshold-like response. Regarding the neuronal population, we have established the concept of macroscopic excitability whereby an entire population of excitable units acts like an excitable element itself. To examine the stability viiand bifurcations of the macroscopic excitability state, as well as the associated stimulusresponse relationship, we have derived the first effective mean-field model for the collective dynamics of coupled stochastic maps.Dinamika kompleksnih sistema se tipiˇcno odigrava na nekoliko prostornih i vremenskih skala, pri ˇcemu su emergentni fenomeni ˇcesto povezani sa kritiˇcnim prelazima, pri kojima mala promena vrednosti parametra izaziva naglu i kvalitativnu promenu dinamiˇckog režima. U blizini takvih prelaza, kompleksni sistemi su vrlo osetljivi na eksterne peturbacije, što može izazvati dinamiku alterniranja (switching) izmedu razliˇcitih (meta)stabilnih stanja. ¯ Takvo ponašanje je tipiˇcno za mnoštvo bioloških sistema saˇcinjenih od spregnutih ekscitabilnih jedinica, medu kojima su i neuronski sistemi, kod kojih na samoorganizaciju utiˇcu koe- ¯ fekti šuma iz raznolikih izvora i višestrukosti vremenskih skala koja potiˇce od lokalne dinamike i dinamike interakcija. Ova disertacija je posve´cena prouˇcavanju nekoliko vrsta samoorganizuju´ce dinamike u spregnutim stohastiˇckim ekscitabilnim sistemima sa dinamikom koja se odvija na višestrukim vremenskim skalama (multiscale dinamika). Ekscitabilno ponašanje pojedinaˇcnih jedinica je detaljno istraženo, kako u pogledu nelinearnog pragovskog (threshold-like) odgovora na eksterne perturbacije, tako i u pogledu karakteristiˇcnog nemonotonog odgovora na šum, manifestovanog kroz razne rezonantne fenomene. Medutim, pri razmatranju ¯ ekscitabilnog ponašanja spregnutih sistema kao nove paradigme emergentne dinamike, na fundamentalnom nivou postoje brojna otvorena pitanja, ukljuˇcuju´ci kako interakcije modifikuju lokalnu dinamiku rezultuju´ci ekscitabilnoš´cu na nivou spregnutog sistema, kao i kako sadejstvo multiscale dinamike i šuma dovodi do switching-a i rezonantnih fenomena. U ovoj disertaciji, saˇcinjenoj od tri komplementarne linije istraživanja, sistematiˇcno pristupamo traženju odgovora na navedena pitanja. U sklopu prve linije istraživanja, proširili smo koncept ekscitabilnosti na spregnute sisteme, razmatraju´ci primere malog motiva saˇcinjenog od lokalno ekscitabilnih jedinica i populacije stohastiˇckih neuronskih mapa. U sluˇcaju motiva, klasifikovali smo razliˇcite vrste ekscitabilnih odgovora i pokazali šta odreduje pragovsko ponašanje, primenivši elemente ¯ teorije singularnih perturbacija. U sluˇcaju populacije, uveli smo koncept makroskopske ixekscitabilnosti pri kojoj se cela populacija ekscitabilnih jedinica ponaša kao ekscitabilni element. Kako bismo ispitali stabilnost i bifurkacije stanja makroskopske ekscitabilnosti, kao i odgovor sistema na perturbaciju, izveli smo prvi efektivni model srednjeg polja (mean-field) za kolektivnu dinamiku spregnutih stohastičkih mapa