. Federico Faraci The 60th anniversary of the Hodgkin-Huxley model: a critical assessment from a historical and modeller’s viewpoint Master thesis, defended on May 16th 2013 Specialization: Applied Mathematics Thesis Advisors: Dr Sander Hille (Leiden University) Prof Dr Marc Timme (Max Planck Institute for Dynamics and Self-Organization, Göttingen) 1 . 2 Preface Mathematics is a powerful creation of the human mind. Like English, Dutch, Italian, German, mathematics is a language. It is the language that is univer- sally recognized in the scientific community as the most adequate to describe, predict, and eventually explain the phenomena we observe. There is one very fundamental reason for this: mathematics is, at the same time, flexible and precise as no other known language is. These two properties (flexibility and precision) make mathematics an exceptionally powerful tool. Looking back to the history of Science, we may find however cases in which the mathematical appearance of certain theories masked underlying unjustified hypotheses and distracted from critical thinking. In this thesis we provide a concrete example of one of these cases: the Hodgkin-Huxley model, nowadays the most popular mathematical model in the neurosciences. It will be shown here that although this model has a remarkable descriptive power, it has some major flaws from the explanatory perspective of the phenomena it describes. My interest in critically studying the Hodgkin-Huxley model arose from nu- merousdiscussionswithProf. Dr. KonradKaufmann,duringmystayingatthe Max Planck Institute for Dynamics and Self-Organization in Göttingen. I de- cided then to choose this specific topic for my thesis for two reasons: (a) I have been studying neuroscience since more than three years now, actively working inthefieldattheMPIsincemoreorlesstwo;and(b)ithappenedthatthisyear the 60th anniversary of the publication by Hodgkin and Huxley of their theory has been celebrated with a World conference on computational neuroscience, where it happened that no substantial criticism to the model was advanced. It goes without saying that, except from extremely rare exceptions, the same cel- ebrative attitude towards the model is observed through all the neuroscientific community, while the diverse experimental evidences against some of its most important hypotheses are still ignored. It appeared then to me that a com- prehensive, critical treatment of the mathematical theory advanced by the two Nobel-awarded physiologists would have been important, both for reconstruct- ing the historical development and derivation of the model, and for providing a critical assessment of the experimental evidence used to support the claim of the existence of an explanatory power of the model. Theresultsofmyefforts-necessarilyhighlyinterdisciplinary,attheinterface 3 between mathematics, biophysics and electrophysiology - are reported here in thisthesisforobtainingtheMasterdegreeinAppliedmathematics. Themathe- maticalcontentmainlyfocusesonthetechniquesusedinderivingthemodels,in particulartheonefromHodgkinandHuxleyinitsstaticandpropagatingform, and the identification of the relationships among them. A detailed analysis of the behaviour of possible solutions of the Hodgkin-Huxley model was beyond the scope of this thesis (and mathematically still a topic of advanced research). More importantly, it seems reasonable to say that one should first establish (or disprove)thevalueoftheHodgkin-Huxleymodelasanexplanatorymodelwith properhypothesessupportedbyexperimentalobservations,beforestartingsuch an analysis. For similar reasons, we did not discuss in depth simplifications of theHodgkin-HuxleymodellikeforexampletheFitzHugh-Nagumomodel,which is only briefly mentioned here. These have even less explanatory power than the detailed model they somehow approximate. I hope to have managed in such an intent to provide a valuable reading. Göttingen, April 2013 4 . Acknowledgements FirstofallIwouldliketothankmythesissupervisorsProfDrMarcTimmeand Dr Sander Hille for trusting in my work and giving me the possibility to pursue my way in Science. Without their open minds this work would have never been possible. I am deeply grateful also to the librarians of the Otto Hahn Library (Max Planck Institute for Biophysical Chemistry) for their kindness and valu- able help in providing me hardly-accessible articles and books. I can’t imagine my work if their help would be missing. I thank Prof Dr Konrad Kaufmann for the seeds of doubt, and Dr Ahmed El Hady for numerous discussions on exper- imental neurophysiology; I thank Marta, for trusting and feeding our Future. I dedicate this thesis to my parents, as a filtered drop of a whole thing they still can’t see, but always sustained. 5 . 6 Contents I Introduction 9 1.1 Objectives of the thesis 9 1.2 Fundamentals of single cell neurophysiology 11 II Theoretical foundations of the Hodgkin-Huxley model 17 2.1 Premises to the sodium hypothesis 17 2.2 The sodium hypothesis 22 2.3 Quantitative models before 1952 25 2.3.1 Nernst 1889-1908 26 2.3.2 Lapicque 1907-1926 30 2.3.3 Blair 1932 32 2.3.4 Rashevsky 1933 35 2.3.5 Hill 1936 39 2.3.6 Hodgkin and Huxley 1945 44 2.4 The legacy of the early quantitative models 47 III The Hodgkin-Huxley Model 51 3.1 The static model 51 3.1.1 The potassium conductance 53 3.1.2 The sodium conductance 57 3.2 The propagating Action Potential 63 IV Criticism on the sodium hypothesis 67 4.1 Sodium independence in non-squid systems 68 4.2 Sodium independence in the squid giant axon 71 4.3 Evidence of the sodium transmembrane flow 73 V Models of nerve excitation after 1952 75 5.1 The FitzHugh-Nagumo model 76 5.2 The Heimburg-Jackson model 78 7 VI Discussion 81 APPENDIX 85 A. Equivalence of Rashevsky’s and Hill’s theories 85 B. The Heimburg-Jackson model: analytical considerations 87 BIBLIOGRAPHY 91 8 CHAPTER I Introduction Inthehistoryofthebiologicalsciences,thereexistsnomathematicalmodelthat has been welcomed with such a broad consensus as the Hodgkin-Huxley model. Sinceitspublicationin1952,thetheorydevelopedbythetwophysiologistsfrom Cambridge University laid the foundations for the interpretation and planifica- tion of experiments, for the understanding of diseases and illness conditions as well as for the design of new drugs. Nobel prizes have been awarded for having conceived techniques which could be used to collect data interpretable as if in support of the theory (eg. Neher and Sakmann, Nobel Prizes for Medicine or Physiology1991“fortheirdiscoveriesconcerningthefunctionofsingleionchan- nelsincells")orforhavingelucidatedthefinestructureofmacromoleculesespe- ciallyrelevantwithintheframeworkofthemodel(eg. MacKinnon, NobelPrize for Chemistry 2003 "for structural and mechanistic studies of ion channels"). Nowadays in every university, every neuroscience course includes at least one lecture dedicated to the mathematical interpretation of nerve excitation given by Hodgkin and Huxley. 1.1 Objectives of the thesis This year, the 60th year since the publication of the model of the action po- tential,theOrganizationforComputationalNeurosciencescelebratedtherecur- rencybyheldingacongressattheAlmaMaterofthetwoscientists,namelythe Trinity College in Cambridge. On the webpage of the event one could read: “This publication [HodgkinandHuxley1952] and the mathematical model it describes is at the core of our modern understanding of how the action potential is generated, and has had profound effects on many fields of biological science in particular on computational studies of neural function” 9 The Journal of Physiology - the journal where the model was originally published as well as one of the most influent journals in physiology since more than one hundred years ago - dedicated the issue of June of the current year to the epoch-making achievements and legacy of the Hodgkin-Huxley model. In the articles published, there appear sentences such as: “It [theHHmodel]remainsoneofthebestexamplesofhowphenomenological description with mathematical modelling can reveal mechanisms long before they can directly be observed” (Schwiening 2012) or “The modern era of research on electrical signalling in nerve, muscle and otherexcitablecellsbeganin1952withaseriesoffourseminalpapersbyHodgkin and Huxley on analysis of the action potential of the squid giant axon using the voltage clamp technique” (Catterall 2012) and even “Looking forward, we expect that the Hodgkin-Huxley contribution will con- tinue to propel biomedical research, in areas as diverse as muscle physiology and pharmacology, autonomic physiology, neuroscience disease patophysiology and even clinical medicine” (Vandenberg and Waxman 2012) In this work we show that the model developed by Hodgkin and Huxley cannot be considred valid in its full generality. This not only because it has obvious discrepancies with what could in principle be defined fine details such as for example with some specific neuronal behaviours, but because the very funda- mental aspects of the theory do not conform with experimental evidence. The thesis is structured as follows: After an introductory section on the basic concepts of neuroscience (Section 1.2),anin-depthanalysiswillbeprovidedofthemajorscientificinfluencesofthe twophysiologists(ChapterII).Thepurelyqualitativeaswellasthequantitative ideas (models) that led to the development of the Hodgkin-Huxley theory will be analyzed. The third chapter is dedicated to the model as originally conceived in 1952. There the derivation of the equations for both the static membrane voltage variation and the propagated action potential will be treated in detail. 10