Andrew Fielding Huxley (1917–2012) Michael C. Mackey and Moisés Santillán, Coordinating Editors n A Mathematician’s Apology, G. H. Hardy in recording electrically from the inside of the squid wrote: “A mathematician, like a painter or a giant axon. However, this work was interrupted poet, is a maker of patterns. The mathemati- by the outbreak of World War II. For the first cian’s patterns, like the painter’s or the poet’s, year of the war, Huxley was a clinical student, but Imust be beautiful; the ideas, like the colours when medical teaching in London was stopped by or the words, must fit together in a harmonious air attacks, he changed to work on operational way.” In our opinion these sentiments also apply research in gunnery, first for the Anti-Aircraft to all other areas of science. The work of Andrew Command and later for the Admiralty. F. Huxley (who passed away on May 30, 2012) is In 1941 Huxley was elected to a research fellow- one of the finest examples not only because of ship at Trinity College, Cambridge, and he took up its importance to applied mathematics, biomath- this position at the beginning of 1946, together ematics, and physiology but also because of the with a teaching appointment in the Department beauty of the underlying conceptual framework. of Physiology. On his return to England after a A. F. Huxley has justifiably become one of the research stay in the laboratory of Kenneth S. Cole, best-known scientists of the twentieth century. Hodgkin teamed up with Huxley to measure the A. F. Huxley was the youngest son of the electrophysiological phenomena associated with writer and editor Leonard Huxley and his second the generation of an action potential in the squid wife, Rosalind Bruce, half-brother of the writer giant axon. This work was published in a brilliant Aldous Huxley and biologist Julian Huxley, and the series of five papers in the Journal of Physiology in grandson of biologist Thomas H. Huxley. In 1947 he 1952. The final one [Hodgkin and Huxley, 1952d] is married Jocelyn Richenda Gammell (Chenda) Pease an intellectual tour de force, combining both exper- (1925–2003), the daughter of the geneticist Michael imental data analysis and mathematical modeling Pease and his wife, Helen Bowen Wedgwood. (the Hodgkin-Huxley equations). This work earned From 1935 to 1938 A. F. Huxley studied physics, Hodgkin and Huxley the Nobel Prize in 1963, along mathematics, and physiology at Trinity College, with J. C. Eccles, “for their discoveries concerning Cambridge, where he met Alan L. Hodgkin. At the ionic mechanisms involved in excitation and the time, high table included a glittering array inhibition in the peripheral and central portions of of scientific talent, including J. J. Thomson, Lord the nerve cell membrane.” Rutherford, F. W. Aston, A. S. Eddington, F. G. Huxley, the mathematician/physiologist, was Hopkins, G. H. Hardy, F. J. W. Roughton, W. A. H. not content to stop at the Hodgkin-Huxley model, Rushton, A. V. Hill, and E. D. Adrian. however, and went on to publish in 1957 his In August 1939 Huxley joined Hodgkin at the celebrated review of muscle contraction data and Marine Biological Laboratory at Plymouth for his its synthesis into the mathematically formulated first introduction to research, and they succeeded cross-bridge theory, a theory that still stands in its essential ingredients today. Michael C. Mackey is Joseph Morley Drake Professor of Phys- Huxley held college and university posts in iology at McGill University. His email address is michael. [email protected]. Cambridge until 1960, when he became head of the Department of Physiology at University College Moisés Santillán is professor of biomedical engineering and physics at the Center for Research and Advanced Studies of London. In 1969 he was appointed to a Royal The National Polytechnic Institute, Monterrey, Mexico. His Society Research Professorship, which he held email address is [email protected]. until 1983, when he became Master of Cambridge DOI: http://dx.doi.org/10.1090/noti998 University’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 president The problem was solved for potassium by 4 from 1980 to 1985. In 1974 he was knighted. assuming gK = g¯K [n(V ; t)] , where g¯K > 0 is a We present below a brief account of the two maximal conductance and n is a so-called gating major scientific contributions by Huxley: the variable satisfying the simple differential equation Hodgkin-Huxley equations and the sliding filament dn (2) = α (1 − n) − β n: theory, followed by brief personal pieces written dt n n by some of Andrew Huxley’s close collaborators, This was used to fit the experimental data, and friends, and students. from these fits they were then able to deduce the dependence of αn; βn on V. Since the temporal The Hodgkin-Huxley Equations evolution of the conductance gK (V ; t) at constant V The work for which Huxley is best known is his had a distinctly sigmoidal nature, they concluded formulation with Alan Hodgkin of the “Hodgkin- that n had to be raised to the fourth power Huxley equations” in the final paper [Hodgkin and to reproduce this property. In their final paper, Huxley, 1952d] of a series of five papers published Hodgkin and Huxley speculated that the form in 1952 [Hodgkin et al., 1952], [Hodgkin and Huxley, for the potassium conductance might represent 1952a], [Hodgkin and Huxley, 1952b], [Hodgkin the movement of what we now call a molecular and Huxley, 1952c], [Hodgkin and Huxley, 1952d]. subunit that regulated the opening and closing The first of the Hodgkin-Huxley equations, which of the potassium gate. The fourth power would accounts for the dynamics of the membrane poten- arise if it was assumed that four subunits had tial of the axon (measured relative to extracellular to be simultaneously in the correct position for fluid), V, is potassium to move across the membrane. The description of the behavior of the sodium @V @2V (1) C + I = I + D ; conductance was conceptually the same, though m @t i a @x2 more complicated in detail, since they had to where x is the distance along the axon, Ii is assume that the equation for the sodium current the current carried by ions moving through the was given by INa = gNa(V − VNa) with gNa = membrane, and Ia is the current applied to the 3 g¯Na[m(V ; t)] h(V ; t), where m was an activation axon from an external source. The brilliance of variable and h was an inactivation variable. Thus the work of Huxley with Hodgkin came in their three m subunits had to be open to allow Na+ meticulous characterization of the ionic current Ii, to cross the membrane, but one h subunit could which from their experimental work they deduced stop that movement. The kinetics of m and h were consisted of the sum of three independent currents described by carried by sodium (Na+), potassium (K+), and a dm “leakage” current so I = I + I + I . Using the (3) = α (1 − m) − β m i Na K L dt m m voltage clamp technique, they were then able to and characterize the way in which these three currents dh depended on both membrane potential V and (4) = αh(1 − h) − βhh: time t. Introducing the notion of an equilibrium dt potential for these three ionic species (e.g., the From the behavior of gNa as a function of t at constant V , the functional dependence of the equilibrium potential VNa for sodium is that value of the membrane potential V for which there is no activation (αm; βm) and inactivation (αh; βh) rate net sodium current), they then further assumed constants on V was deduced. that the currents were given by an “ohmic” relation Equations (1)–(4), along with the expressions for such that each current was proportional to the the α’s and β’s, are now known as the Hodgkin- driving force for each ion and the proportionality Huxley equations and were the first proposed factor for each was a conductance: to explain the nature of the action potential in an excitable cell based firmly on experimental INa = gNa(V − VNa), evidence. The explanation that they offered was IK = gK (V − VK ), impressive in its breadth, because the solutions conformed to the “form, duration and amplitude I = g (V − V ). L L L of (the action potential), both ‘membrane’ and From these three relations they were then able propagated; the conduction velocity, the impedance to deduce the dependence of the conductances changes during the (action potential); the refractory gNa; gK ; gL on both potential V and time t. The period; ionic exchanges; subthreshold responses; factor gL was found to be independent of (V ; t), and oscillations.” Though mention was not made but gNa; gK were highly nonlinear functions of in [Hodgkin and Huxley, 1952d] about difficulties (V ; t) and the problem they then faced was how to in the solution of the equations, Hodgkin in his characterize this dependence. autobiography [Hodgkin, 1992, p. 291] notes that it May 2013 Notices of the AMS 577 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 proteinic filaments aligned along the cell longitudinal axis. Under the microscope it becomes apparent that they are composed of a repeated series of so-called sarcomeres, with sarcomeres appearing as dark and light bands and lines (see Figure 1).