Mathematics have proven to be “unreasonably effective” in many branches of science. Physics, of course is math’s most fertile domain, but mathematics has also played a starring role in genetics, epidemiology and neuroscience.
In this post, I will share a famous example of the application of math to neuroscience in the Nobel Prize-winning work of British biologists, Alan Hodgkin and Andrew Huxley. Their work led to first quantitative model that explains the way the brain’s nerve cells or neurons conduct electrochemical impulses, impulses that are responsible for our perceptions of the world, our emotions, and our behavior.
The Road to Discovery
In the 1950s, Hodgkin and Huxley were investigating how nerve cells talked to one another. They wanted to know how electrical signals traveled along nerve fibers or axons, causing nerve excitations that are responsible for our perceptions, emotions and behavior.
Complex chemical changes take place at the cellular level of the brain that causes excitation to occur and dissipate. When Hodgkin and Huxley began their research journey, they had to conduct painstaking experiments to calculate the flow of sodium and potassium ions across the membranes of a very big nerve fiber – that of a squid. They worked out empirically how those electrical flows depended on the voltage across the nerve cell membrane, and how this voltage was altered by the flowing ions. For more complex calculations such as the speed and shape of a neural impulse as it traveled down an axon, they needed the help of a computer because the math was too hard. Using a hand-cranked mechanical calculator (remember his was in the 1950s), Huxley solved the problem over a period of three weeks, and together with Hodgkin, reported their findings, including their famous system of “excitation equations” in five seminal papers over the course of one year, 1952 in the Journal of Physiology . For their groundbreaking work in discovering the ionic basis of how nerve cells work, Hodgkin and Huxley shared the 1963 Nobel Prize for Physiology or Medicine.
The Hodgkin-Huxley Equations
Hodgkin and Huxley faced a hugely challenging task in coming up with their system of equations. The problem is something called “the curve of dimensionality”, which bedevils mathematicians, physicists as well as biologists studying highly nonlinear systems. The problem is this: when we apply nonlinear dynamics to biology, we find ourselves in a world with many dimensions. This is like the problem Newton faced when he studied the so-called Three-Body Problem in physics where one has a “state space” of 18 dimensions because at any instant, a body such as a planet is located somewhere in ordinary three-dimensional space specified by three numbers x, y, and z plus it also moves in any of three velocities corresponding to those coordinates, giving 18 dimensions in total for each body (no wonder Newton was reported as saying that this problem gave him a headache!)
In Hodgkin and Huxley’s case, the problem was how to mathematically model the changing concentrations of all the ions such as sodium, potassium, calcium, chloride, and others as they flow in and out of flowing in and out of the nerve membrane. A problem like this would involve hundreds of variables: one to represent their changing ionic concentrations in the cell, another to represent the changing voltage across the cell membrane, yet another to capture the membrane’s changing ability to conduct the various ions and let them pass from or into the cell and so on. Clearly, the problem that Hodgkin and Huxley was investigating would have given them a more severe headache than the three-body problem did for Newton.
Miraculously, Hodgkin and Huxley managed to pull off a feat – they came up with a system of equations that give the variables their “dance instructions” so to speak and tell them how to move on to their trajectories. In this way, their equations could be used to predict the paths electrical impulses move across the vast network of nerve fibers. Their work was a stroke of pure genius, an “eureka moment” for mathematical biology. See Box 1 for a glimpse of how their system of equations look like.
Even today, the conductance-based model of Hodgkin and Huxley is sometimes referred to as the “crown jewel” of the subject because of its remarkable descriptive and predictive power. It is found in every modern neuroscience textbook, and has served as a foundation of neuroscientific research, from molecular to circuit level . The brilliance of Hodgkin and Huxley is also a paean to the “unreasonable” ability of the human mind to probe nature in ways that are mysterious and deep, using symbols that are entirely man-made.
 For a listing of Hodgkin and Huxley’s seminal papers, all published in the Journal of Physiology in 1952, see https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3424715/
 W.A. Catterall, I.M. Raman, H.P.C. Robinson, T.J. Sejnowski, O.Paulsen, “The Hodgkin-Huxley heritage: from channels to circuits”, Journal of Neuroscience, 32 (2012), pp. 14064-14073. See also the review article, “Thinking about the nerve impulse: A critical analysis of the electricity-centered conception of nerve excitability”, by Benjamin Drukarch et al. in Progress in Neurobiology (2018), 69, pp. 172-85. A popular account of the HH model can be found in Steven Strogatz (2019), Infinite Powers, chapter 11.