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Stochastic Oscillations Motivated by the Noisy and Rhythmic Firing

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Stochastic Oscillations Motivated by the Noisy and Rhythmic Firing Submitted by Irene Tubikanec Submitted at Department of Stochastics Supervisor Univ.-Prof.in Dr.in Stochastic Oscillations Evelyn Buckwar December 2016 Motivated by the Noisy and Rhythmic Firing Activity of Neurons Master Thesis to obtain the academic degree of Diplom-Ingenieurin in the Master’s Program Industriemathematik JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, Osterreich¨ www.jku.at DVR 0093696 „Don’t worry dreams aren’t real. They’re just neurons firing randomly in your brain.“ Katherine Applegate Eidesstattliche Erklärung Ich erkläre an Eides statt, dass ich die vorliegende Masterarbeit selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe. Die vorliegende Masterarbeit ist mit dem elektronisch übermittelten Textdokument identisch. Linz, 01. Dezember 2016 Irene Tubikanec, BSc BSc Abstract A human brain contains billions of neurons. These are extremely complex dynamical systems that are affected by intrinsic channel noise and extrinsic synaptic noise. Os- cillatory behaviour is a phenomena that arises in single neurons as well as in neuronal networks. Based on the noisy and rhythmic firing activity of nerve cells, the aimof this thesis is to provide an overview on the existing theory of stochastic oscillations. The main question is how oscillations can be defined mathematically in a stochastic setting. This work suggests two very different definitions of a stochastic oscillator according to certain well-known deterministic tools. The first one states that the solution of a specific two-dimensional stochastic differential equation is a stochastic oscillator if it has infinitely many simple zeros almost surely. The second definition of a stochastic oscillator corresponds to the concept of a random periodic solution that is based on the cocycle property. Two particular stochastic equations are introduced each satisfying one of these definitions. For this purpose, two detailed proofs onthe validity of the first and the second definition, respectively, are presented. A second topic addressed by this work is the stability of stochastic equations. The system energy carries information on how the trajectory propagates in the phase plane. Besides, Lyapunov exponents describe the asymptotic exponential growth of random dynamical systems. Finally, this work provides an application of the developed theory to specific standard oscillatory models as well as to the Van der Pol oscillator on which the FitzHugh- Nagumo neuron model is based. Sample path simulations of the stochastic equations are provided by the implementation of the Euler-Maruyama method in MATLAB. Furthermore, an exact simulation method applied to the stochastic harmonic oscillator equation is introduced. Zusammenfassung Das menschliche Gehirn umfasst Milliarden von Neuronen. Das sind hochkomplexe dynamische Systeme, die intrinsischem Kanal- und extrinsischem Synapsenrauschen unterliegen. Oszillierendes Verhalten tritt in einzelnen Zellen als auch in Netzwerken auf. Aufgrund der rhythmischen und zufälligen neuronalen Impulse, ist es Ziel dieser Arbeit einen Überblick der aktuellen Theorie stochastischer Oszillatoren zu geben. Das Hauptanliegen ist die Beantwortung der Frage wie Oszillatoren in einem stochastis- chen Rahmen mathematisch definiert werden können. In Anlehnung an determinis- tische Aussagen, betrachtet diese Arbeit zwei mögliche Definitionen. Bei der ersten Variante wird die Lösung einer zweidimensionalen stochastischen Differentialgleichung als Oszillator definiert, falls sie fast sicher unendlich viele einfache Nullstellen aufweist. Die zweite Definition entspricht dem Konzept einer periodischen Lösung und basiert auf der cocycle Eigenschaft. Es werden zwei Gleichungen vorgestellt, welche je eine der beiden stochastischen Definitionen erfüllen. Die entsprechenden Beweise werden detailliert präsentiert. Des Weiteren wird die Stabilität von stochastischen Systemen thematisiert. Die Sys- temenergie gibt Auskunft über die Entwicklung der Trajektorie im Phasenraum. Außer- dem beschreiben Lyapunov Exponenten das asymptotische exponentielle Wachstum von stochastischen dynamischen Systemen. Neben der Anwendung erarbeiteter Theorien auf Standardmodelle erfolgt eine Betrach- tung der Van der Pol Gleichung, die den Grundstein des bekannten FitzHugh-Nagumo Neuronenmodells bildet. Die Simulation von Beispielpfaden erfolgt mit der Imple- mentierung des Euler-Maruyama Verfahrens in MATLAB. Ergänzend wird eine exakte Simulationsmethode für den harmonischen Oszillator diskutiert. Acknowledgments I would like to express my deep gratitude to my supervisor Prof.in Evelyn Buckwar for the opportunity to write this thesis, for giving me the possibility to decide on topics and contents, for enabling me to participate on specific conferences and for her engagement throughout my whole work. I would also like to thank my colleague and best friend Bernadett Stadler for construc- tive discussions and for proofreading. Special thanks to my life partner Michaela Fehringer for her patience and for her mental assistance. Finally, I owe my deepest gratitude to my parents who supported me during my whole life and, in particular, during my studies. Especially, I would like to acknowledge my mother for respecting all of my decisions and for her long lasting blind trust. Contents 1 Introduction1 1.1 Problem Description . .1 1.2 Research Questions . .2 1.3 Structure of the Thesis . .2 2 Introduction to Neuroscience4 2.1 Single Neurons . .4 2.1.1 Components and Functionality . .4 2.1.2 Noise and Rhythmicity . .5 2.2 Single Neuron Models . .6 2.2.1 Hodgkin-Huxley Model . .6 2.2.1.1 Equivalent Electrical Circuit . .7 2.2.1.2 Types of Ion Channels . .8 2.2.2 FitzHugh-Nagumo Model . .9 2.2.3 Van der Pol Model . 10 3 Deterministic Oscillation Theory 12 3.1 Definition of a Deterministic Oscillator 1 . 12 3.2 Definition of a Deterministic Oscillator 2 . 13 3.2.1 Deterministic Flow Property . 13 3.2.2 Fixed Points and Periodic Solutions . 15 3.3 System Stability . 16 3.3.1 Stability of Fixed Points . 16 3.3.2 Stability of Limit Cycles . 19 4 Stochastic Oscillation Theory 21 4.1 Definition of a Stochastic Oscillator 1 . 23 4.1.1 Stochastic Harmonic Oscillator . 23 Contents viii 4.1.2 Theorem of Girsanov . 24 4.2 Definition of a Stochastic Oscillator 2 . 29 4.2.1 Stochastic Cocycle Property . 30 4.2.2 Random Fixed Points and Periodic Solutions . 34 4.3 System Stability . 38 4.3.1 System Energy . 38 4.3.2 Lyapunov Exponents . 39 5 Theory Application on Specific Models 43 5.1 Deterministic Applications . 43 5.1.1 Harmonic Oscillator . 43 5.1.2 Damped Harmonic Oscillator . 45 5.1.3 Van der Pol Oscillator . 47 5.1.4 Nonlinear Oscillator . 49 5.2 Stochastic Applications . 50 5.2.1 Stochastic Harmonic Oscillator . 51 5.2.2 Stochastic Damped Harmonic Oscillator . 57 5.2.3 Stochastic Van der Pol Oscillator . 59 5.2.4 Stochastic Nonlinear Oscillator . 62 6 Conclusion 71 6.1 Experiences . 71 6.2 Future Work . 72 Bibliography 76 Chapter 1 Introduction A human brain contains billions of extremely complex dynamical systems called neu- rons. By using mathematical models that capture their main dynamics, one tries to reconstruct the firing of single neurons as well as neuronal interactions. Understanding how the human brain is working, and especially how neurons manage their interactive communication, could contribute to the comprehension of specific mechanisms that lead to diseases such as Alzheimer’s, Parkinson’s or Epilepsy. 1.1 Problem Description Oscillatory activity arises at each level of neuronal systems. Many neurons fire action potentials repetitively. The mechanisms, inducing rhythmic firing, differ across each level of neural systems. Since oscillatory behaviour has been realised in diseases as well, understanding those mechanisms and being able to control them is an important medical problem [11]. Mathematics and especially stochastics offer central possibilities to make a contribution to this crucial topic. Neurons communicate with the use of electrical impulses that are called spikes or ac- tion potentials. These signals propagate along the filamentary extensions of neural cells by passing through ion channels that fluctuate rapidly between the open and closed state. For that reason, neurons are affected by intrinsic channel noise. Since nerve cells are surrounded by amounts of other neurons, extrinsic synaptic noise arises in Chapter 1 Introduction 2 this dynamical systems as well. To capture neuronal noise the application of stochastic differential equations (SDEs), which model time dependent phenomena that underlie random effects, is of essential importance. SDEs came into being in course of the twen- tieth century and have been build on a more solid ground by Itô [20] and Stratonovich [32] in the middle of that century. Many crucial developments, based on their work, have been achieved in recent decades. Therefore, this special type of equations is a very innovative mathematical tool that provides still many open possibilities for research. 1.2 Research Questions Based on the noisy and rhythmic firing activity of neurons, the aim of this thesis isto provide an overview on stochastic oscillation theory. This work mainly addresses the following questions: 1. How can oscillations be defined mathematically in a stochastic setting according to the well-known deterministic tools? 2. Which tools concerning the treatment of system stability
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