Nonlinear Oscillations and Chaos in Chemical Cardiorespiratory Control

We report progress made on an analytic investigation of low-frequency cardiorespiratory variability in humans. The work is based on an existing physiological model of chemically-mediated blood-gas control via the central and peripheral chemoreceptors, that of Grodins, Buell & Bart (1967). Scaling and simplification of the Grodins model yields a rich variety of dynamical subsets; the thesis focusses on the dynamics obtained under the normoxic assumption (i.e., when oxygen is decoupled from the system). In general, the method of asymptotic reduction yields submodels that validate or invalidate numerous (and more heuristic) extant efforts in the literature. Some of the physiologically-relevant behaviour obtained here has therefore been reported before, but a large number of features are reported for the first time. A particular novelty is the explicit demonstration of cardiorespiratory coupling via chemosensory control. The physiology and literature reviewed in Chapters 1 and 2 set the stage for the investigation. Chapter 3 scales and simplifies the Grodins model; Chapters 4, 5, 6 consider carbon dioxide dynamics at the central chemoreceptor. Chapter 7 begins analysis of the dynamics mediated by the peripheral receptor. Essentially all of the dynamical behaviour is due to the effect of time delays occurring within the conservation relations (which are ordinary differential equations). The pathophysiology highlighted by the analysis is considerable, and includes central nervous system disorders, heart failure, metabolic diseases, lung disorders, vascular pathologies, physiological changes during sleep, and ascent to high altitude. Chapter 8 concludes the thesis with a summary of achievements and directions for further work. ACKNOWLEDGEMENTS I owe a great debt of thanks to my supervisor, Dr Andrew Fowler, for his guidance of this work. I have found Andrew always available and approachable (if not in his office, at the Royal Oak!); his feedback and comments on my work were prompt, thorough, and educative. I especially acknowledge his great forbearance, sometimes for months on end, as I endeavoured to come to terms with applied mathematics in general, and differential-delay equations in particular. I would like to thank Dr Guy Kember for many useful discussions (though he may deny their ‘use’ !) and for first producing some of the numerical results presented in Chapter 4. My grateful thanks to Professor Michael Mackey, who read drafts of the Chapters 1 and 2, and generally provided much encouragement. OCIAM has provided a happy working environment (and a mean cricket team during the summers!). My grateful thanks to the people who have shared office-space with me (and taken so many phone messages) during this period: Paul Emms, Stuart Doole, Clare Johnson, Taryn Malcolm and Timo Taskinen. Thanks also to Abbey Perumpanani, an always-encouraging ‘fellow-traveller’ ! I should like to particularly record my gratitude to Dr Hilary Ockendon for furnishing me with the ‘executiveoffice’ of DH31 during my last year. I would like to thank the Mathematical Institute’s Computer Officer, Chris Cooper, for breathing life into kaveri.maths and giving me virtually unlimited disc space. Thanks also to the administrative staff of the Institute, notably Sheila Robinson, Kathy Hunt and Val Willoughby, for being helpful in so many ways. I gratefully acknowledge the support of a Rhodes scholarship during the early stages of my graduate career in Oxford, and a Wellcome Prize studentship subsequently. I thankfully mention the benefit of a Centenary senior scholarship at St Hugh’s College; also the ever-helpful attitude of the administration there. I acknowledge the cheer and support of my personal tutor in College, Dr Mary Lunn. I have relied on the support, cheer and friendship of many over these past few years; I cannot name them here individually for the fear of leaving someone out. In general, I acknowledge with pride the friendship of many wonderful people that I met and got to know in Oxford as a newly-arrived student from India. Before that, I was fortunate to belong to a remarkable batch of medical students who graduated from JIPMER, Pondicherry in 1989. That I have seen the completion of this thesis is tribute to their idealism, enthusiasm and maturity. Long live the Too Much! Taking a step further back, I recall with affection my schoolmates in Bishop Cotton Boys’ School, Bangalore, and our (with hindsight) superb teachers. This thesis has relied as much on what they taught us, as anything I gained in ‘higher’ education. I owe a great deal to Mitta Uncle. He has had more confidence in me than I have had in myself; his advice, time and again, has proved crucial in difficult situations. I can only say, with deep gratitude, ‘thank you for everything’. I thank Amma, Papa, Nanda and Poornashri for providing me a wonderful family and being fountains of love and support. Thanks also to my in-laws, for their advice, love and cheer. Finally, I acknowledge with love the companionship and understanding of Sujata.