ptp u Ottawa L'Université canadienne Canada's university FACULTÉ DES ÉTUDES SUPÉRIEURES I=J FACULTY OF GRADUATE AND ETPOSTOCTORALES u Ottawa posdoctoral studies L'Université canadienne Canada's university Brian Heffernan autëïïrîéIXtïïésF/MîorWthësïs M.Sc. (Systems Science) Department of Systems Science ~TÂCULTElÎCCtÎ7D^ Neuronal Models of Consonance and Dissonance TITRE DE LA THÈSE / TITLE OF THESIS André Longtin DIRECTEUR (DIRECTRICE) DE LA THESE / THESIS SUPERVISOR CO-DIRECTEUR (CO-DIRECTRICE) DE LA THESE / THESIS CO-SUPERVISOR ÇJiristianGiguCTe John Lewis Gary W. Slater Le Doyen de la Faculté des études supérieures et postdoctorales / Dean of the Faculty of Graduate and Postdoctoral Studies NEURONAL MODELS OF CONSONANCE AND DISSONANCE by Brian Heffernan Thesis submitted to the Faculty of Graduate and Postdoctoral Studies In partial fulfillment of the requirements For the M. Sc. degree in Systems Science Interdisciplinary Studies Faculty of Graduate and Postdoctoral Studies University of Ottawa © Brian Heffernan, Ottawa, Canada, 2010 Library and Archives Bibliothèque et ?F? Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l'édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre référence ISBN: 978-0-494-69075-8 Our file Notre référence ISBN: 978-0-494-69075-8 NOTICE: AVIS: The author has granted a non- L'auteur a accordé une licence non exclusive exclusive license allowing Library and permettant à la Bibliothèque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l'Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans le loan, distribute and sell theses monde, à des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, électronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriété du droit d'auteur ownership and moral rights in this et des droits moraux qui protège cette thèse. Ni thesis. Neither the thesis nor la thèse ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent être imprimés ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformément à la loi canadienne sur la Privacy Act some supporting forms protection de la vie privée, quelques may have been removed from this formulaires secondaires ont été enlevés de thesis. cette thèse. While these forms may be included Bien que ces formulaires aient Inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. 1+1 Canada Abstract One of the fundamental questions in music is why certain combinations of tones sound pleasant (or consonant) and others unpleasant (or dissonant). This question's impor- tance is highlighted by the cross-cultural overlap in preferred musical tone combinations found across the globe and throughout human history as well as the seemingly innate predisposition towards the processing of these same tone combinations by infants; the phenomenon indeed appears to be a universal one. In order to gain insight into the pos- sible neurophysiological mechanism(s) underlying this phenomenon, plausible neuronal models are constructed and investigated in order to assess their explanatory power. This thesis investigates two different models: one which explores a correspondence between synchrony in nonlinear dynamical systems and consonance assessments, and another which explores the relationship between the stochastic resonance - already a viable mech- anism for pitch extraction - and its consonance assessments. The results obtained using the first model indicate that the previously reported correspondence was the result of an error of analysis, and that any such correspondence is parameter dependent. A modified version of the model reflecting a higher degree of biophysical realism was constructed; initial results show an interesting modal structure in the relationship between intervals and their synchronized states. The second model further establishes the temporal coding of consonance via interspike interval statistics and shows a good correspondence with psychoacoustic consonance assessments, and a nearly perfect correspondence with mu- sical consonance assessments. The latter result is of particular interest as it not only provides a biologically plausible account of the musical consonance of intervals, but may also provide insight into the actual neural machinery behind the development of auditory perception more generally. 11 Acknowledgements I would like thank André for providing me with the opportunity to study under his kind, enthusiastic and intellectually humbling advisement (as well as for purchasing a shiny new iMac upon which to run an endless sea of sims). Of all the areas of research I could have stumbled into, I don't believe that any of them could have been nearly so suiting to my passions and abilities. I would like to thank all of the longtinlab group for stimulating parts of my brain that I had presumed to be either dead or dormant and for being cool in general. In particular, a one future Dr. Jason Boulet who, aside from sharing a passion for good beer, shared with me his exceptional knowledge of matlab and all things technical, as well as his insight into the perplexing conceptual models I so often wrestled with. Without his help, this would have been an even more trying process. To my good friends who spared me from a complete immersion into geekdom and nerdery - I thank thee. To Mark and to the MSC: I am thankful for all the good people I met, and for learning so much about what I do and do not want to do with my life. To my family, for always supporting me, even if they don't so much know exactly what it is I'm studying. I truly could not be more blessed. Lastly, to mononucleosis, for showing me to the woman I love. If ever anything good could have come out of being so ill, this is it. Merci pour chaque moment de chaque jour que t'es dans ma vie mon chou. Bisous. m Dedication For my mom and dad. THIS PAGE IS INTENTIONALLY LEFT BLANK. ? Contents 1 Introduction 1 1.1 References 5 2 Background 7 2.1 Music: A Brief Introduction 8 2.1.1 What is music? 9 2.1.2 On Origins: Why is Music? 11 2.2 Systems Relevant to Music 13 2.2.1 Physical: The Basics of Sound & Musical Acoustics 13 2.2.2 Biological: Music and the Brain 18 2.2.3 Cognitive and Perceptive: Music and the Mind 24 2.3 The Basics of Auditory Neuroscience 38 2.3.1 The Leaky Integrate-and-Fire Neuron Model 39 2.3.2 Ordinary Measures of Neural Firing 41 2.3.3 Auditory Coding Schemes 42 2.4 Relevant Non-Linear Dynamics 49 2.4.1 Stochastic Resonance 49 2.4.2 The Sine-Circle Map & Mode-Locking 52 2.5 Summary 54 2.6 References 56 vi 3 Artide I 63 4 Artide II 76 4.1 Background 77 4.2 Model 80 4.3 Simulation Methods 80 4.4 Results 81 4.5 Analysis 82 4.6 Discussion 84 4.7 Conclusion 88 4.8 Figures 89 4.9 References 108 5 Conclusions 112 5.1 Summary 112 5.2 Final Thoughts 114 5.3 References 116 A Code 118 A.l Shapira Lots & Stone Model 118 A. 1.1 diadsweepForPublication . m 118 A. 1.2 modelockingForPublication .m 122 A.2 Cariani's Model 125 A.2.1 chialvoGSR2 .m 125 A.2. 2 chialvoGSRCaller.m 126 VIl List of Tables 2.1 The Systems Relevant to Music 14 4.1 The Wester Dyads: Consonance and Adjusted Amplitude Values 76 vin List of Figures 2.1 The Celestial Spheres of Pythagoras 13 2.2 Pure Tones 16 2.3 The Harmonic Series: Natural Modes of a String 17 2.4 The Human Ear 21 2.5 The Cochlea 21 2.6 The Auditory Pathway 23 2.7 Equal-loudness Contours 26 2.8 Virtual Pitch 27 2.9 The "Missing Fundamental" 27 2.10 Spectrogram of a Violin Playing 29 2.11 The Western Dyads: Equal Temperament vs. Just Intonation 31 2.12 The Other Pythagorean Theorem 33 2.13 Overlapping Partíais of a Perfect 5th 35 2.14 Beats & Roughness: The Basilar Membrane as a Series of Overlapping Band-pass Filters 36 2.15 Stumpf's Theory of Tonal Fusion 37 2.16 A Neuron 40 2.17 Schematic: Stimulus to Neural Response to Percept 43 2.18 Sensory Coding Schemes 43 2.19 Temporal Coding of the Auditory Nerve 47 2.20 Summary of Pitch Percepts Described by Temporal Coding 48 ix 2.21 Stochastic Resonance 49 2.22 Ghost Stochastic Resonance 51 2.23 The Devil's Staircase 53 2.24 Arnold Tongues 54 4.1 Cariani's all-order ISIH, Harmonic Sieve, & Pitch Salience 82 4.2 Cariani's Consonance 83 4.3 Schematic of Conceptual Model 84 4.4 The Stochastic Resonance of an LIF - Replicating Barbi et al 84 4.5 First-order ISIH's for Consonances Under Identical Parameterization . 85 4.6 First-order ISIH's for the Dissonances Under Identical Parameterization . 86 4.7 All-order ISIH for the Consonances Under Identical Parameterization . 87 4.8 All-order ISIH for the Consonances Under Identical Parameterization (Con- tinued) 88 4.9 All-order ISIH's for the Dissonances Under Identical Parameterization . 89 4.10 All-order ISIH's for the Dissonances Under Identical Parameterization (Continued) 90 4.11 First-order ISIH's for the Consonances with Adjusted Amplitudes ...