Andrew Fielding Huxley(1917–2012)Michael C. Mackey and Moisés Santillán, Coordinating Editors

In A Mathematician’s Apology, G. H. Hardywrote: “A mathematician, like a painter or apoet, is a maker of patterns. The mathemati-cian’s patterns, like the painter’s or the poet’s,must be beautiful; the ideas, like the colours

or the words, must fit together in a harmoniousway.” In our opinion these sentiments also applyto all other areas of science. The work of AndrewF. Huxley (who passed away on May 30, 2012) isone of the finest examples not only because ofits importance to applied mathematics, biomath-ematics, and physiology but also because of thebeauty of the underlying conceptual framework.A. F. Huxley has justifiably become one of thebest-known scientists of the twentieth century.

A. F. Huxley was the youngest son of thewriter and editor Leonard Huxley and his secondwife, Rosalind Bruce, half-brother of the writerAldous Huxley and biologist Julian Huxley, and thegrandson of biologist Thomas H. Huxley. In 1947 hemarried Jocelyn Richenda Gammell (Chenda) Pease(1925–2003), the daughter of the geneticist MichaelPease and his wife, Helen Bowen Wedgwood.

From 1935 to 1938 A. F. Huxley studied physics,mathematics, and physiology at Trinity College,Cambridge, where he met Alan L. Hodgkin. Atthe time, high table included a glittering arrayof scientific talent, including J. J. Thomson, LordRutherford, F. W. Aston, A. S. Eddington, F. G.Hopkins, G. H. Hardy, F. J. W. Roughton, W. A. H.Rushton, A. V. Hill, and E. D. Adrian.

In August 1939 Huxley joined Hodgkin at theMarine Biological Laboratory at Plymouth for hisfirst introduction to research, and they succeeded

Michael C. Mackey is Joseph Morley Drake Professor of Phys-iology at McGill University. His email address is michael.mackey@mcgill.ca.

Moisés Santillán is professor of biomedical engineering andphysics at the Center for Research and Advanced Studies ofThe National Polytechnic Institute, Monterrey, Mexico. Hisemail address is msantillan@cinvestav.mx.

DOI: http://dx.doi.org/10.1090/noti998

in recording electrically from the inside of the squidgiant axon. However, this work was interruptedby the outbreak of World War II. For the firstyear of the war, Huxley was a clinical student, butwhen medical teaching in London was stopped byair attacks, he changed to work on operationalresearch in gunnery, first for the Anti-AircraftCommand and later for the Admiralty.

In 1941 Huxley was elected to a research fellow-ship at Trinity College, Cambridge, and he took upthis position at the beginning of 1946, togetherwith a teaching appointment in the Departmentof Physiology. On his return to England after aresearch stay in the laboratory of Kenneth S. Cole,Hodgkin teamed up with Huxley to measure theelectrophysiological phenomena associated withthe generation of an action potential in the squidgiant axon. This work was published in a brilliantseries of five papers in the Journal of Physiology in1952. The final one [Hodgkin and Huxley, 1952d] isan intellectual tour de force, combining both exper-imental data analysis and mathematical modeling(the Hodgkin-Huxley equations). This work earnedHodgkin and Huxley the Nobel Prize in 1963, alongwith J. C. Eccles, “for their discoveries concerningthe ionic mechanisms involved in excitation andinhibition in the peripheral and central portions ofthe nerve cell membrane.”

Huxley, the mathematician/physiologist, wasnot content to stop at the Hodgkin-Huxley model,however, and went on to publish in 1957 hiscelebrated review of muscle contraction data andits synthesis into the mathematically formulatedcross-bridge theory, a theory that still stands in itsessential ingredients today.

Huxley held college and university posts inCambridge until 1960, when he became head ofthe Department of Physiology at University CollegeLondon. In 1969 he was appointed to a RoyalSociety Research Professorship, which he helduntil 1983, when he became Master of CambridgeUniversity’s Trinity College. Named a fellow of the

576 Notices of the AMS Volume 60, Number 5

Royal Society in 1955, he served as its presidentfrom 1980 to 1985. In 1974 he was knighted.

We present below a brief account of the twomajor scientific contributions by Huxley: theHodgkin-Huxley equations and the sliding filamenttheory, followed by brief personal pieces writtenby some of Andrew Huxley’s close collaborators,friends, and students.

The Hodgkin-Huxley EquationsThe work for which Huxley is best known is hisformulation with Alan Hodgkin of the “Hodgkin-Huxley equations” in the final paper [Hodgkin andHuxley, 1952d] of a series of five papers publishedin 1952 [Hodgkin et al., 1952], [Hodgkin and Huxley,1952a], [Hodgkin and Huxley, 1952b], [Hodgkinand Huxley, 1952c], [Hodgkin and Huxley, 1952d].The first of the Hodgkin-Huxley equations, whichaccounts for the dynamics of the membrane poten-tial of the axon (measured relative to extracellularfluid), V , is

(1) Cm∂V∂t+ Ii = Ia +D

∂2V∂x2

,

where x is the distance along the axon, Ii isthe current carried by ions moving through themembrane, and Ia is the current applied to theaxon from an external source. The brilliance ofthe work of Huxley with Hodgkin came in theirmeticulous characterization of the ionic current Ii ,which from their experimental work they deducedconsisted of the sum of three independent currentscarried by sodium (Na+), potassium (K+), and a“leakage” current so Ii = INa + IK + IL. Using thevoltage clamp technique, they were then able tocharacterize the way in which these three currentsdepended on both membrane potential V andtime t . Introducing the notion of an equilibriumpotential for these three ionic species (e.g., theequilibrium potential VNa for sodium is that valueof the membrane potential V for which there is nonet sodium current), they then further assumedthat the currents were given by an “ohmic” relationsuch that each current was proportional to thedriving force for each ion and the proportionalityfactor for each was a conductance:

INa = gNa(V − VNa),IK = gK(V − VK),IL = gL(V − VL).

From these three relations they were then ableto deduce the dependence of the conductancesgNa, gK , gL on both potential V and time t . Thefactor gL was found to be independent of (V , t),but gNa, gK were highly nonlinear functions of(V , t) and the problem they then faced was how tocharacterize this dependence.

The problem was solved for potassium byassuming gK = gK[n(V, t)]4, where gK > 0 is amaximal conductance and n is a so-called gatingvariable satisfying the simple differential equation

(2)dndt= αn(1− n)− βnn.

This was used to fit the experimental data, andfrom these fits they were then able to deduce thedependence of αn, βn on V . Since the temporalevolution of the conductance gK(V , t) at constantVhad a distinctly sigmoidal nature, they concludedthat n had to be raised to the fourth powerto reproduce this property. In their final paper,Hodgkin and Huxley speculated that the formfor the potassium conductance might representthe movement of what we now call a molecularsubunit that regulated the opening and closingof the potassium gate. The fourth power wouldarise if it was assumed that four subunits hadto be simultaneously in the correct position forpotassium to move across the membrane.

The description of the behavior of the sodiumconductance was conceptually the same, thoughmore complicated in detail, since they had toassume that the equation for the sodium currentwas given by INa = gNa(V − VNa) with gNa =gNa[m(V, t)]3h(V, t), where m was an activationvariable and h was an inactivation variable. Thusthree m subunits had to be open to allow Na+

to cross the membrane, but one h subunit couldstop that movement. The kinetics ofm and h weredescribed by

(3)dmdt= αm(1−m)− βmm

and

(4)dhdt= αh(1− h)− βhh.

From the behavior of gNa as a function of t atconstant V , the functional dependence of theactivation (αm, βm) and inactivation (αh, βh) rateconstants on V was deduced.

Equations (1)–(4), along with the expressions forthe α’s and β’s, are now known as the Hodgkin-Huxley equations and were the first proposedto explain the nature of the action potential inan excitable cell based firmly on experimentalevidence. The explanation that they offered wasimpressive in its breadth, because the solutionsconformed to the “form, duration and amplitudeof (the action potential), both ‘membrane’ andpropagated; the conduction velocity, the impedancechanges during the (action potential); the refractoryperiod; ionic exchanges; subthreshold responses;and oscillations.” Though mention was not madein [Hodgkin and Huxley, 1952d] about difficultiesin the solution of the equations, Hodgkin in hisautobiography [Hodgkin, 1992, p. 291] notes that it

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Figure 1.

took Huxley three weeks of work using a Brunsvigamechanical calculator to compute one solution toequations (1)–(4).

Setting aside the difficulties that Hodgkin andHuxley had in solving their proposed equations,there is something buried in their structure that iseven more astonishing. The kinetic equation forthe temporal evolution of gK(V , t), as pointed out,could be interpreted as requiring the simultaneousopening of four n subunits. We now know thatpotassium channels are tetrameric structures withfour subunits that must all be in the correctposition before ion movement is allowed.

The Hodgkin-Huxley equations were not onlyextremely influential in the further developmentof physiology, but they also had a great impact onmathematics. With respect to the full equations,the study of waves in nonlinear parabolic partialdifferential equations (reaction diffusion equationswith excitable dynamics) was initiated by Hodgkinand Huxley when they calculated a traveling wavesolution to their equations. Since this initial work,the study of these types of systems has become aflourishing area of research, with applications inbiology as well as many other applied areas. Further-more, various reductions of the Hodgkin-Huxleyequations (e.g., the Fitzhugh-Nagumo equations)have provided an extremely active and fertile areaof development in dynamical systems theory overthe past sixty years.

The Sliding Filament TheoryHuxley was also instrumental in clarifying themechanisms underlying muscle contraction. Hebuilt the first working interference microscope tofollow changes in the striation patterns in musclewhen it changes length, and this led him (withRolf Niedergerke) to lay the foundation of what

has become known as the sliding filament theory[Huxley and Niedergerke, 1954].

Myofibrils are one of the most abundant con-stituents of a muscle fiber. They are long proteinicfilaments aligned along the cell longitudinal axis.Under the microscope it becomes apparent thatthey are composed of a repeated series of so-calledsarcomeres, with sarcomeres appearing as darkand light bands and lines (see Figure 1).

Among other structural elements, sarcomeresconsist of thin and thick filaments that overlap, asdepicted in Figure 1, thus explaining the sarcomerearrangement of bands and lines. Thin filaments aremainly made of the protein actin, while the majorcomponent of thick filaments is the protein myosin.The I-band is the zone of thin filaments that is notoverlapped by thick filaments, the A-band containsthe full length of thick filaments, and the H-zone isthe zone of thick filaments that is not overlappedby thin filaments.

According to the sliding filament theory, sar-comeres are the muscle’s basic contracting unit.Contraction occurs because the two ends of thethick filaments slide along the thin filaments inopposite directions toward the Z-lines. As thishappens, the A-bands do not change their length,whereas the I-bands and the H-zone shorten. Allthis causes the Z-lines to come closer, thereforecausing sarcomere contraction.

The sliding filament theory has been fullycorroborated since it was proposed, but it wasnecessary to develop experimental techniques thatwere not available at the time this theory wasintroduced. Since a direct experimental verificationwas impossible when the theory was proposed,[Huxley, 1957] developed a mathematical model toexamine testable consequences of his theory.

A central assumption in the model was thatmyosin heads, which can attach to (by formingcross-bridges) and detach from specific sites onthe actin protein, are responsible for moving themyosin along actin filaments. This assumptionpredates the notion of rotation of the myosinhead. As a matter of fact, it was debatable atthat time whether the cross-bridges played animportant role in generating the relative sliding ofthe two filament types. Huxley considered a half-sarcomere (as depicted in Figure 2) and pictured

Figure 2.

578 Notices of the AMS Volume 60, Number 5

the myosin heads as attached to the parent myosinfilament by elastic tails. He further assumed thatwhen the muscle is stimulated, those heads inthe neighborhood of a binding site in the actinfilament can be expected to attach to it. Forcewould then be applied to the actin filament bythe stretched elastic tail of the cross-bridge, acontracting force if the elastic tail is in a stateof extension. Since a contraction velocity wouldtend to shorten the elastic tail, in order for sucha force to be created, one must assume that thetail is already extended when the cross-bridgeis formed. Huxley proposed that this extensioncould originate from thermal agitation. However,this agitation would likely make the tail extend orcontract with the same probability. Hence, Huxleysuggested that attachment would, by means ofan unspecified chemical-mechanical mechanism,be facilitated for cross-bridges that are displacedpositively and be more difficult for those whosetails are in a resting position or contracted.

This conceptual model can be translated intoequations by letting n(x, t) denote the density ofheads attached to the actin filament at position xand time t and v(t) represent the actin-filamentcontraction velocity. A balance equation can thenbe written as

∂n(x, t)∂t

− v(t)∂n(x, t)∂x

= F(x, t)−G(x, t),

where F(x, t) and G(x, t) respectively representthe myosin head binding and detachment rates.Huxley further assumed that these functionsare given by F(x, t) = (U − n(x, t))f (x),G(x, t) =n(x, t)g(x). In these equations U represents thetotal density of possible cross-bridges, which isassumed to be independent of both x and t .Furthermore, in agreement with the nonuniformityof the attachment rate, the functions f (x) and g(x)are assumed to be

f (x) ={ax if 0 ≤ x ≤ h,0 otherwise,

g(x) ={b if 0 ≤ x ≤ h,cx otherwise,

where a, b, c, h are all constant. In particular b� 1,while h is chosen to be smaller than the maximummyosin-tail extension length, X. Finally, the totalforce exerted by the attached filament tails can becomputed as

F(t) =∫ X−XE(x)n(x, t)dx,

in which E(x) represents the elastic force dueto a myosin tail elongated a length x. In hispaper, Huxley assumed this function to be givenby E(x) = Emx, with Em the tail elastic modulus.Solving these equations by employing the methodof characteristics, Huxley was able to reproduce the

celebrated force-velocity curves (experimentallyobtained by A. V. Hill) with a proper choiceof parameter values. Another important successof this model was that it correctly predictedthe dependence of longitudinal stiffness on thespeed of shortening, which had not previously beenmeasured. Huxley himself was aware of his model’sincompleteness; however, despite all criticisms, itsinfluence in the evolution of cross-bridge theorieshas been paramount: practically all current theoriesqualitatively share the central features of Huxley’smodel.

Lincoln E. Ford

Personal RemembrancesI was a somewhat disinterested student until myfirst course in physiology in medical school. WhatI learned was that physiologists were positingtheories of how things worked and conductingexperiments to test their theories. I became espe-cially fascinated with the nerve work of Hodgkinand Huxley. In my day it was almost a certaintythat all medical graduates would be drafted, so,to indulge my interest in research, I applied to goto the NIH as a commissioned officer to fulfill myNational Service obligation. While there I workedwith Richard Podolsky, and from the NIH I went tothe Peter Bent Brigham Hospital, where I learnedthat there was a great deal of useless researchbeing done. Wanting to be a better researcher, Iasked Podolsky if he would recommend me toAndrew Huxley, which he very kindly did. It wasa terrific experience. Andrew had peculiar workhabits that meshed very well with my own. Helived in Cambridge and came up to London forthe middle three days of every week. We usuallyate supper together two nights per week and thenworked until 11 o’clock or sometimes later.

At a personal level, Andrew Huxley was kindand sympathetic, particularly to younger scientists,but he had a reputation for being ferocious. Thisarose, in large part, from his overcommitment. Heonly rarely turned down a request for help or to beon a committee, with the result of being chronicallyshort of time. There is a class of person who joinscommittees for the pleasure of dominating thediscussion, frequently with specious arguments.He almost always spoiled their fun with irrefutablelogic that decimated their arguments and greatlyshortened the discussion.

Of all the very intelligent things he did in hislife, by far the smartest was his asking Richendato marry him. She once told me that what she

Lincoln E. Ford is professor emeritus of medicine and of cellu-lar and integrative physiology at Indiana University Schoolof Medicine. His email address is lieford@iupui.edu.

May 2013 Notices of the AMS 579

regarded as her primary role was to provide asecure home life for Andrew and their six childrenso that he could get on with his work without otherworries, and she did this admirably.

Clara Franzini-ArmstrongClay M. Armstrong and I were postdocs withSir Andrew at University College London in thesixties. The imposing atmosphere of UniversityCollege’s physiology and biophysics departments,with their tradition of muscle/nerve research(A. V. Hill, B. Katz), the gloomy, old-fashionedlaboratories, and the ceremony of afternoon teain the library were a part of the experience. Mostimportant was the interaction with Sir Andrew.Despite his encouraging gentleness, I was awed.His accomplishments had established him as aninnovative and rigorous scientist, and it was quiteclear that his mind was always immersed inmeaningful thoughts. We claimed that we couldhear the gears moving in his mind while he wasperforming mental computations.

Clay and I have been deeply influenced by A. F.Huxley’s work. Two cherished lessons from SirAndrew were honesty in publication and the factthat motherhood and science are not inconsistent.Even though he had suggested my postdoc projectand guided me through it, Sir Andrew did notcoauthor the final paper, because he had notdirectly contributed to the actual performanceof the work. I have tried to follow that preceptthroughout my life. The arrival of my first child inthe middle of my postdoc period did not disturbhim; he just gave me a generous period for recoveryand kept my position open. I went on to have threemore children and to enjoy a life in science. I owethat to his encouragement.

Saul WinegradI first met Sir Andrew Huxley when I was apostdoctoral fellow working at NIH. I becamerather enthusiastic during my presentation to him,when he listened attentively for several minutesand then said, “You needn’t take credit becausenature is so clever.” I thought, “Boy, did I blowthis one.” At the end of the discussion ProfessorHuxley said that it was important to “treasure yourexceptions,” and he then invited me to pursue myresearch in London where I also met and marriedmy wife, Dilys.

Clara Franzini-Armstrong is emeritus professor of celland developmental biology at the Perelman School ofMedicine, University of Pennsylvania. Her email address isarmstroc@mail.med.upenn.edu.

Saul Winegrad is emeritus professor of physiology at thePerelman School of Medicine, University of Pennsylvania.His email address is bsg@mail.med.upenn.edu.

Picture of Andrew Huxley taken by SaulWinegrad on the occasion of Huxley’s 90th

birthday celebration. Courtesy of Saul Winegrad.

The years that followed were filled with deepen-ing friendship and memorable experiences, someof which stand out. During one of our early visits tothe Huxleys at Cambridge, Andrew was particularlyinterested in showing Dilys and me the FellowsRoom at Trinity College, Cambridge University.Restrictions would not allow Dilys to enter, butAndrew wanted both of us to see it. He went aheadto reconnoiter, looking around corners and alongcorridors. He then signaled for us to come, andso we did. Andrew was more chuffed than weat having successfully broken tradition. We thenfollowed tradition and dined with him at high tableat Trinity, followed by Stilton cheese, claret, andsuperb port at the Masters Lodge.

Andrew once said to me, as he was reviewinga thesis in which the author had praised themultidisciplinary approach, that multidisciplineis fine, especially when all of the disciplines arepresent in the same person. I shall miss him forhis warm friendship, his ability and willingness tobring his powerful intellect to problems outsidehis own major interest, and his intellectual honestyand integrity. I shall miss him and all of theseadmirable characteristics present in one person.

J. Walter WoodburyIn late July 1961 I spent an afternoon at UniversityCollege London with Andrew Huxley. KnowingHuxley’s penchant for mathematics, I summarizedsome research from my lab showing that currentinjected into a single cell in a planar sheet ofrat atrium spreads to adjacent cells and thatthe steady state current spread was accuratelydescribed by the 2-D equivalent of the 1-D cableequation. The solution is a Bessel function of the

J. Walter Woodbury is a retired professor of physiologyfrom the University of Utah. His email address is walt.woodbury@utah.edu.

580 Notices of the AMS Volume 60, Number 5

Picture of Andrew Huxley taken by WalterWoodbury in July 1961. Courtesy of J. WalterWoodbury.

second kind with imaginary argument. I mentionedin passing that I hadn’t worked out the solution ofthe 3-D cable equation. He was obviously intrigued,because when I saw him a day or two later at thePhysiological Society meetings at Oxford, he hadworked out the 3-D solution!

As I prepared to leave, I asked Professor Huxleyfor permission to take a photograph of him. Hereadily agreed and carefully composed himselfin his office chair. As you can see, the phototurned out very well. He was quiet, unfailinglypolite and cordial, the embodiment of the reservedBritish gentleman. I knew from his papers thathe was an enormously talented, insightful, andcreative scientist. My overall impression was thatbehind the reserve there was a highly capable andgenuinely caring human being. There is a pleasantglow attached to my memory of the afternoon Ispent with Andrew Huxley.

Albert M. GordonI was privileged to work with and study underAndrew Huxley as a postdoctoral fellow in hislaboratory at University College London (UCL)from December 1960 until June 1962. I had justcompleted my Ph.D. in solid state physics fromCornell University and was making the transitionto physiology and biophysics. The Prof (as wecalled him) had just moved to UCL as the JodrellProfessor in Physiology and department head. Hewas now working on muscle, having developed thesliding filament theory of contraction, along withothers.

Albert M. Gordon is professor emeritus of physiology andbiophysics at the University of Washington School of Medicine.His email address is amg@u.washington.edu.

The Prof provided the complete model of whatit meant to be a scientist, combining excellence inexperimentation, in instrumentation, and in math-ematical theory. His experiments were carefullycontrolled and designed to clearly test importanthypotheses. The hypotheses came out of his the-oretical studies. The experiments were possiblebecause of his dexterity in dissection combinedwith his excellence in instrumentation. He showedthat it was not sufficient to just have a hypothesisto explain qualitatively the experimental data, butthat one needed mathematical models that couldexplain the experimental observations.

Andrew Huxley was an outstanding scientist,mentor, and friend. I will miss him and the annualChristmas card with his photographs of scenesfrom his favorite vacations in Scotland and theLake District.

Reinhardt RüdelTo begin with, I must say that I am suffering froma progressive paralysis of the legs, which wasnot yet evident in the years 1968 to 1970, whenI was working as a postdoc in the lab of A. F.Huxley. Prof, as we used to call him confidentiallyyet respectfully, had a general predilection forcompetitions. This was well known to us from theannual departmental outings when he was eager towin all games, in particular the tennis tournament,the “yard-of-ale” (his record: 23 s!), and even theharmless musical chairs!

Frank Lehmann-Horn and I organized the 7thInternational Conference on Neuromuscular Dis-eases in Munich, 1990. One of our novel ideaswas to add the theoretical subjects of physiology,pharmacology, and genetics to the canon of clinicalsubjects treated at the conference. To convince theskeptical clinicians, we asked Professor Huxley toparticipate in the conference as a kind of figure-head. Of course we organized a press conferencefor our star guest. The journalists asked him allkinds of questions about squid neurons and hisNobel Prize, but he could not be diverted from ex-pressing his happiness about his theoretical workhaving so quickly assumed such importance forthe understanding of human diseases. He stressedthat this was another example of pure researchbeing a useful forerunner to applied research.

Prof also came to our social event, a Bavarianget-together in one of Munich’s most famousbeer halls, the Löwenbräukeller . Beer is servedthere only in steins of one liter. Participantsin the evening told me afterwards that theywere impressed by Professor Huxley’s purposeful

Reinhardt Rüdel is a professor at the Division of Neu-rophysiology of Ulm University. His email address isreinhardt.ruedel@uni-ulm.de.

May 2013 Notices of the AMS 581

method of mustering the jar before he put itto his mouth with one hand without spilling adrop. On the other hand, they reported that themuscle expert seemed a bit stranded when hehad to dissect the enormous grilled haxen (calf’sleg) that followed. After dinner we engaged in awheelchair ballet. The dancers wearing reflectingjackets under stroboscopic illumination operated inperfect synchrony, obtaining almost unnoticeablysupportive momentum from their partners.

When I drove him to the airport two days later Iasked him about his favorite impressions of theconference. Huxley pondered for a moment, andthen he replied with the little stutter that wascharacteristic for him when disclosing an opinion:“Well, I was most impressed by the wheelchairballet.” I considered this as highest appreciation.

Vincenzo LombardiMy collaboration with Andrew Huxley was funda-mental in my scientific education and the mostimportant driving force for my research achieve-ments. My first visit to his laboratory at theUniversity College London took place from Octo-ber 1979 to July 1980, when he developed thestriation follower, the optoelectronic apparatusfor investigating the function of molecular motorsinside a muscle cell, with still presently unequalledresolution. At that time Andrew was busy most ofthe day with his duties as president of the RoyalSociety and often came to the lab only after dinner.He usually spent quite a large part of the nighttesting and assembling the optical components,checking the performance of the electronic circuitsI had built during the day, and planning my workand tests for the following day. Back in Florence Iwas able to set up my lab thanks to the enthusiasmand generous perseverance of Andrew in involvinghis coworkers in every single step during therealization of his ideas.

Andrew is rightfully recognized as one of themost prominent scientists of the twentieth century.He was an extraordinarily gifted man of science, buthe also possessed other not-so-popular qualities,such as his natural curiosity for any aspect of life,his widespread knowledge of the natural sciences,and his dedication to the education of youngsters.

Andrew was an exceptionally interesting andpleasant person. For instance, during his visit to mycountry house in Fietri, Chianti, in the summer of1989, Andrew enjoyed in a special way discoveringthe variety of butterflies in the yard and astonishedeverybody by following them and telling the properscientific name of each species. From my first visit

Vincenzo Lombardi is professor of physiology at the Univer-sity of Florence. His email address is vincenzo.lombardi@unifi.it.

to his lab at University College London in 1980and until April 2011, we exchanged hundreds ofletters that accompanied and supported my life inany sort of event. These letters are my personallegacy and remind me of his role as Magister vitae.

Gabriella PiazzesiIn 1990 I started to collaborate with ProfessorHuxley after he and Vincenzo Lombardi had movedthe laboratory from University College London tothe Department of Physiology of the Universityof Cambridge. As a researcher at the Universityof Florence, I had the opportunity to spend onemonth a year in his lab.

During my collaborative visits, several timesI had the honor to be invited to join him fordinner at Trinity College or to have supper in hishouse in Grantchester, where I could enjoy thefamiliar and warm atmosphere he and his wife,Richenda, extended to me. On the occasions thatthe data analysis and discussion required us tomeet but I was unable to go to Cambridge, he andRichenda came to Florence. Beyond the scientificwork, this was a perfect opportunity to reciprocatetheir hospitality, exploiting the unique Florenceenvironment. A most remarkable quality of Andrewwas his ability to freely share with his youngercoworkers any aspect of life.

Makoto EndoI worked under Professor Andrew Huxley fromSeptember 1962 to April 1964 at the Department ofPhysiology, University College London. The resultsof our work were reported in my name only. Huxleydeclined to add his name to the report as beingthe person to suggest this work to me and givinghis valuable advice throughout the experimentsbecause, he pointed out, he himself had not carriedout the experiments. I adopted this attitude in mylater work with postdocs.

Huxley visited Japan several times, the first timebeing in summer 1965 to attend the PhysiologicalCongress held in Tokyo. He stayed in Tokyo abouta month, and toward the end he gave a lecture onhis famous work on nerve excitation at WasedaUniversity. While in Japan he became interestedin Japanese, about which he knew nothing beforecoming. After only three weeks he could readboth types of Japanese alphabets and even someChinese characters. He then proposed to give hislecture in Japanese! He was confident enough,

Gabriella Piazzesi is professor of physiology at theUniversity of Florence. Her email address is gabriella.piazzesi@unifi.it.Makoto Endo is professor emeritus at the University of Tokyoand Tohoku University. His email address is qqf38h69@topaz.ocn.ne.jp.

582 Notices of the AMS Volume 60, Number 5

because he had successfully done a similar thingin Russia despite having no previous knowledge ofRussian. People at Waseda University were quiteapprehensive but were steamrollered by Huxley’sself-confidence.

Huxley gave me the English manuscript of hislecture two days before his lecture, and I translatedit into Japanese. With the aid of a Japanese-Englishdictionary, Huxley checked the Japanese version,commenting that unless he knew what he wastalking about, the lecture would be unsuccessful.Having rushed through the translation, I made amistake: I wrote “Osmotic pressure is low,” whereasit should have been “Osmotic pressure is high.”Huxley came to me and politely said that thissounded odd but was probably how it should besaid in Japanese, allowing me to find my mistake!His lecture at Waseda University was extremelysuccessful. He was even able to insert a joke andmake everybody laugh. We were even more amazed,because after returning to England, his Japanesevocabulary further increased!

Jan LännergrenIt is now thirty-nine years since I pushed open theheavy oak door marked “Anatomy” in Gower Street,entering University College London. High ceiling,a smell of wax polish, stairs leading upwards.At the second floor there was the entrance toAndrew Huxley’s lab (previously occupied by thelegendary muscle physiologist A. V. Hill). Insidethere were three people: Bob Simmons, LincolnFord, and Andrew Huxley (“Prof”—I soon learnedthat all called him Prof). He was very kind and verywelcoming and presented the other two.

Setting up for the experiments at King’s wasprobably the most exciting and instructive timeof my whole stay in London. Prof was presentduring the whole process, and it was absolutelymarvelous to realize how deep his knowledge ofoptics and other things was. During that time Ilearned a massive amount of basic optics. Prof wasvery patient and took ample time to explain thingsto his humble pupil. He also suggested a course inoptics at King’s, which I took.

In summary, my one and a half years with Prof,later Sir Andrew F. Huxley, was the most enjoyableand educating period of my scientific life. I amextremely grateful for having had the opportunityto work together with a true scientific giant.

Jan Lännergren is professor of physiology at KarolinskaInstitutet, Stockholm, Sweden. His email address is jan.lannergren@ki.se.

Top: Picture of the mechanical calculator AndrewHuxley used to do the calculations of thepropagated action potential. Bottom: AndrewHuxley and Lee D. Peachey. Pictures taken in2009. Courtesy of Lee Peachey.

Lee D. PeacheyI have many fond memories of Andrew Huxley,going back more than fifty years. I first met himin 1956 while I was a graduate student at theRockefeller Institute for Medical Research (nowRockefeller University) when he visited for a weekto lecture and meet with the students. In Juneof 1958, I joined Huxley in Cambridge. Thatfirst collaboration was the beginning of a longfriendship with Andrew and led to several morecollaborations. It seemed as if nothing was too bigfor Andrew to master or too small to be worthyof his full attention. Our relationship also had apersonal side. Andrew and Richenda were verygood friends to my wife, Helen, and me and openedmany doors for us.

During my last visit with Andrew in 2009, Iasked about the mechanical calculator that hehad used to do the calculations of the propagatedaction potential. He found it, sat down with it, and

Lee D. Peachey is professor of biology at the Universityof Pennsylvania. His email address is lpeachey@sas.upenn.edu.

May 2013 Notices of the AMS 583

Plaque mounted in July 2012 outside theDepartment of Physiology, Development and

Neuroscience, Cambridge University, tocommemorate the sixty-year anniversary of thefamous Hodgkin and Huxley 1952 publications.

Photograph courtesy of Professor Idan Segev.

became totally engrossed, reminding himself whento turn the crank, when to flip the lever to movethe carriage left or right, and so on.

Andrew Huxley’s intellectual brilliance and skillas an experimenter, his infectious enthusiasm in allthings, and his generosity in sharing his knowledgerubbed off on me and on many others who workedwith him, and through us to more generations ofphysiological scientists. It was very special to haveknown him for so long and to have gained so muchfrom our friendship and collaborations. He was avery important part of my life.

David R. TrenthamPerhaps of greatest interest to this readershipis the impact on mathematics of the Hodgkin-Huxley equations in that they describe an excitablenonlinear system that has a solution in the formof a nonlinear solitary wave. As these waves aredifferent from solitons in conservative nonlinearsystems, their discovery made a huge impact inthe emerging field of nonlinear dynamics andtheir applications in biology and engineering [Scott,1975]. Apart from providing the mathematical basisfor the entire fields of neural [Scott, 1975] andcardiac [Noble et al., 2012] modelling, the Hodgkin-Huxley equations became one of the cornerstonesin the theory of nonlinear waves (autowaves) inphysical, chemical, and biological excitable systems[Krinskiı, 1984], [Guckenheimer and Holmes, 1997],[Keener and Sneyd, 2009]. Interestingly, afterdecades of research, soliton-like behavior wasalso discovered in the Hodgkin-Huxley equations[Aslanidi and Mornev, 1999].

In July 2012 the Huxley family celebrated thelife of Andrew on the grounds of his lovely homein Grantchester, near Cambridge. It was a beautiful

David R. Trentham is honorary professor in the RandallDivision of Cell and Molecular Biophysics at King’s CollegeLondon. His email address is drtrentham@aol.com.

summer’s day and memorable for the gatheringof Huxley, Darwin, Pease, and Wedgwood familymembers. One could imagine a similar reunionof such names in the same idyllic surroundingsa century earlier. The dynastic nature of thesefamilies is well embedded in English culture, asis reflected in the ditty below, which appearedin Punch [Anon., 1964] following the engagementof Angela Darwin (née Huxley), one of the guestspresent.

A DARWIN is marrying a HUXLEYA fate which no Darwin escapes;

For the Huxleys speak only to Darwins,And the Darwins speak only to apes.

How Andrew would have enjoyed being at such agathering!

ReferencesAnonymous, —. Punch, 247(3):83, 1964.O. V. Aslanidi and O. A. Mornev, Soliton-like regimes

and excitation pulse reflection (echo) in homogeneouscardiac Purkinje fibers: Results of numerical simulations,J. Biol. Phys., 25(2-3):149–164, 1999.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations,Dynamical Systems, and Bifurcations of Vector Fields, vol-ume 42 of Applied Mathematical Sciences, Springer, NewYork, corr. 5th print edition, 1997. ISBN 0387908196.

A. L. Hodgkin, Chance and Design: Reminiscences of Sci-ence in Peace and War, Cambridge University Press,1992.

A. L. Hodgkin and A. F. Huxley, Currents carried bysodium and potassium ions through the membrane of thegiant axon of Loligo, J. Physiol., 116(4):449–472, 1952a.

, The components of membrane conductance in thegiant axon of Loligo, J. Physiol., 116(4):473–496, 1952b.

, The dual effect of membrane potential on sodiumconductance in the giant axon of Loligo, J. Physiol., 116(4):497–506, 1952c.

, A quantitative description of membrane currentand its application to conduction and excitation in nerve,J. Physiol., 117(4):500–544, 1952d.

A. L. Hodgkin, A. F. Huxley, and B. Katz, Measurement ofcurrent-voltage relations in the membrane of the giantaxon of Loligo, J. Physiol., 116(4):424–448, 1952.

A. F. Huxley, Muscle structure and theories of contraction,Prog. Biophys. Biophys. Chem., 7:255–318, 1957.

A. F. Huxley and R. Niedergerke, Structural changes inmuscle during contraction: Interference microscopy ofliving muscle fibers, Nature, 173:971–973, 1954.

J. P. Keener and J. Sneyd, Mathematical Physiology, vol-ume 8 of Interdisciplinary Applied Mathematics, Springer,New York, 2nd edition, 2009. ISBN 9780387094199 (set).

V. I. Krinskiı, Self-organization: Autowaves and StructuresFar from Equilibrium: Proceedings of an InternationalSymposium, Pushchino, USSR, July 18-23, 1983, volume 28of Springer Series in Synergetics, Springer-Verlag, Berlin,1984. ISBN 0387150803 (U.S.).

D. Noble, A. Garny, and P. J. Noble, How theHodgkin-Huxley equations inspired the Cardiac Phys-iome Project, J. Physiol., 590(Pt. 11):2613–28, June 2012.doi10.1113/jphysiol.2011.224238.

A. C. Scott, Electrophysics of a nerve-fiber, Rev. Mod.Phys., 47(2):487–533, 1975.

584 Notices of the AMS Volume 60, Number 5