Self-organization in coupled excitable systems: interplay between multiple timescale dynamics and noise

Abstract

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.