Page 1 |
Save page Remove page | Previous | 1 of 165 | Next |
|
small (250x250 max)
medium (500x500 max)
large ( > 500x500)
Full Resolution
All (PDF)
|
This page
All
Subset |
THE EFFECTS OF NOISE ON BIFURCATIONS IN CIRCLE MAPS WITH
APPLICATIONS TO INTEGRATE-AND-FIRE MODELS IN NEURAL BIOLOGY
by
John Mayberry
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(APPLIED MATHEMATICS)
May 2008
Copyright 2008 John Mayberry
Object Description
| Title | The effects of noise on bifurcations in circle maps with applications to integrate-and-fire models in neural biology |
| Author | Mayberry, John |
| Author email | jmayberr@usc.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Applied Mathematics |
| School | College of Letters, Arts and Sciences |
| Date defended/completed | 2008-03-10 |
| Date submitted | 2008 |
| Restricted until | Unrestricted |
| Date published | 2008-04-01 |
| Advisor (committee chair) | Baxendale, Peter |
| Advisor (committee member) |
Ziane, Mohammed Newton, Paul |
| Abstract | A stochastic bifurcation is generally defined as either a change in the number of stable invariant measures (dynamical or D-bifurcations) or a change in the qualitative shape of invariant measures (phenomenological or P-bifurcations) for a stochastic dynamical system. Some authors have observed that these definitions can fail to capture important information regarding the evolution of certain Markov Chains arising from first-passage-time distributions of stochastic differential equations since the definitions deal only with static information about the chain (i.e. information regarding invariant or stationary distributions). In this work we perform a more rigorous investigation of these observations by studying changes to the spectra of transition operators for two different classes of examples of Markov Chains obtained by taking small perturbations of deterministic dynamical systems. The first class deals with small Gaussian perturbations of discrete time dynamical systems on the circle while the second class arises naturally from the study of sequences of firing times in noisy integrate-and-fire models for chemical potential in neurons. We show that bifurcations in the deterministic system can often lead to changes in the number of eigenvalues of the transition operator for the corresponding perturbed process which approach the unit circle as the noise intensity goes to 0, a phenomenon we call a lambda-bifurcation. Although in both classes of examples, the perturbed process always has a unique stationary distribution, these changes in the number of eigenvalues with modulus close to 1 can have significant effects on both the shape of and the rate of convergence to the stationary distribution of the process. |
| Keyword | stochastic bifurcations; integrate-and-fire models; Markov chains; transition operators; first passage times; Gaussian perturbations; Ornstein-Uhlenbeck process |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Type | texts |
| Legacy record ID | usctheses-m1068 |
| Rights | Mayberry, John |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | http://www.usc.edu/isd/libraries/services/ask_a_librarian/email/ |
| Filename | etd-Mayberry-20080401 |
| Archival file | uscthesesreloadpub_Volume14/etd-Mayberry-20080401.pdf |
Description
| Title | Page 1 |
| Full text | THE EFFECTS OF NOISE ON BIFURCATIONS IN CIRCLE MAPS WITH APPLICATIONS TO INTEGRATE-AND-FIRE MODELS IN NEURAL BIOLOGY by John Mayberry A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2008 Copyright 2008 John Mayberry |
Comments
Post a Comment for Page 1
