Dinamika FitzHugh-Nagumo modela

master's thesis

• Author: Ivona Jelić
• Mentor: Maja Resman (mentor)
• Committee member: Maja Resman (predsjednik povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-11-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu proučavamo FitzHugh-Nagumo model. Radi se o modelu širenja impulsa kroz neuron, danom sustavom parcijalnih diferencijalnih jednadžbi. Sustav se ne može eksplicitno riješiti pa se okrećemo kvalitativnoj analizi: pokušavamo predvidjeti ponašanje rješenja bez eksplicitnog rješavanja sustava. Diskretizacijom prelazimo s kontinuiranog modela na diskretni dinamički sustav u dvije dimenzije, ovisan o nekim parametrima. Fiksiramo sve parametre osim parametra koji kontrolira nelinearan član sustava, te u ovisnosti o tom parametru analiziramo ponašanje orbita sustava u vremenu te analiziramo kako se to ponašanje mijenja promjenom parametra. U radu uvodimo i osnovne pojmove iz teorije dinamičkih sustava i teorije bifurkacija.

Abstract (english)

In this thesis we consider the FitzHugh-Nagumo model, a model which describes the propagation of an impulse through a neuron. The model is given by a system of partial differential equations. It is not possible to solve the system explicitly, so we turn to qualitative analysis: we predict the behavior of solutions without explicitly solving the system. By discretization of the system, we get a discrete dynamical two-dimensional system, which depends on some parameters. We fix all parameters except for the one controlling the non-linear part of the system, which we refer to as the leading parameter. We analyze the behaviour of orbits of the system over time, depending on the leading parameter. We describe how the behavior changes as the leading parameter varies. We also introduce some basic notions from theory of dynamical systems, bifurcation theory and chaos theory.

• Keywords: FitzHugh-Nagumo model; sustav parcijalnih diferencijalnih jednadžbi; diskretizacija; dinamički sustavi
• Keywords (english): FitzHugh-Nagumo model; system of partial differential equations; discretization; dynamical systems
• Language: croatian
• URN:NBN: urn:nbn:hr:217:148140
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-05-14 12:08:46