Emine Topcu

# The Math (and the Electricity) Behind Neuroscience

Neuroscience is a very fast-evolving branch of science. What is known to be true today can be invalidated tomorrow. It is important to follow the new research and new studies to be aware of the current state of knowledge. On this site, we will tap mostly into publications that are recently published. But I will do something different today and talk about a publication from 1952, by Hodgkin and Huxley (1). Why? Because it is still highly relevant. According to the Web of Science database, the article I am going to elaborate on today has close to 15,000 citations. It is a pioneering article.

Before dwelling on the mathematical equations known as Hodgkin-Huxley equations, I would like to give a brief introduction to the electrical behaviour of neurons for the readers who are unfamiliar with it. Neurons have positive and negative ions both inside and outside the cell at different concentrations. Similar to how a drop of colouring in a cup of water diffuses out, the ions also diffuse from where they are highly concentrated towards where they are less concentrated. So, there is a concentration gradient that affects the ionic flow. There is also the attraction of opposite charges and repelling of the like charges. The electrical gradient has an impact on the ionic flow as well. Neurons also have a cell membrane that the ions do not pass through freely. There are ion channels on the cell membranes that open and close and allow the flow of specific ions in or out.

Neurons have a resting membrane potential (or voltage), around -70 mV (these numbers can vary from one cellular environment to another, but the numbers I use here are representative enough to describe the phenomenon). The ionic flows mentioned above can cause a change in this membrane potential. It is when this potential reaches a threshold voltage of around -55 mV that we see an event called the action potential. Some voltage-gated ion channels open and cause a sudden increase up to 40 mV. And this spike is further carried through the axons of neurons, the long extensions that reach out to other neurons. Different ion channels continue to open and close based on the voltage changes, and the membrane potential shows a sudden fall, even further down the resting potential. This is called the refractory period, during which it is harder for the neuron to fire another action potential. Then everything stabilizes again. The changes in membrane potential during an action potential and cartoons of ion channels are displayed in the figure below:

The major players in these series of events are sodium (Na) and potassium (K) channels. Sodium channels, as the name suggests, allow the passage of sodium ions in, whereas potassium channels allow potassium ions out of the cell. The sodium channels also have another property called the “gate,” which locks the channel so the ions cannot pass through, even when it is open.

Now we are coming to the math side of things. Scientists wanted to create a mathematical model of how neurons behave. Three names are pioneers in this area: Alan Hodgkin, Andrew Huxley, and Bernard Katz. Their work with squid giant axons allowed us to understand the electrical behaviour of neurons (1,2). They published a series of articles, and the last one contained the formulation below: (1)

Equation [1] shows how the membrane potential (V) changes based on the currents of different ions (the ion flows). As sodium (Na) and potassium (K) are the primary players, we see their currents as IK and INa. They are not the only ions, however. There are other ions as well, and Il (the leak current) represents them as a whole.

The EK, ENa, and El are called the equilibrium potentials, and they are the voltage values the ions want to reach if they can freely pass through the cell membrane. Each ion is trying to reach a different equilibrium potential, and the difference between the actual potential and their equilibrium potential dictates the drive. So, the further the V is from the E value, the higher the value of the current. The g values in these formulas are the conductance of their corresponding channels: how well they conduct the current.

There are also some other parameters we see in equations [2] and [3]. These are the most interesting in my mind. The n, m, and h values take the value between 0 and 1, and they can be considered as the probability of channels to be open or close. The sodium current has two parameters, m and h. Why? That’s because of the gating property of the sodium channels I mentioned before. The Na channels not only open and close, but they also have a gate that closes.

But why is the fourth power of n used for K, the third power of m is used for Na? What Hodgkin and Huxley did was to use the experimental data and figure out the simplest equation that would produce similar results. There is a biological reality to these equations, like the presence of gates in Na channels. But there is also an educated guess component and trial and error. Hodgkin and Huxley mention that an equation with 5 or 6 exponential may provide more accurate results, but the cost of the complexity it creates is not worth the level of accuracy it adds (1). Also, these formulations are valid within a certain range of voltages, which is an acceptable assumption as the goal is to model neuronal behaviour in the biological ranges (1).

The list of equations they list in their article is not limited by the ones I listed here. The n, m, and h values also change by the membrane potential. Our neurons seem to do a lot of math in just a few milliseconds. For my master’s project, I am using a model similar to this, but it is called a “simplified model” (3). You can appreciate why. The cost-benefit calculations sometimes make us to chose simpler models to reach the results, which can later be expanded with more layers of complexity for higher precision.

All I have talked about are part of Neuronal Dynamics. If you would like to learn more about it, Coursera and edX platforms offer free online courses that are full of valuable information. Both sites have been my frequently visited websites for the past decade; I highly recommend these platforms in general.

Despite the title of the blog entry, I can in no way mention all the math used in neuroscience. Just a couple of days ago, I was learning about short-time Fourier transform to understand an article I was reading. If you like math and want to do something in Neuroscience, you are in for a treat!

**References:
**1. Hodgkin AL, Huxley AF. A Quantitative Description of Membrane Current and its Applicatinos to Conduction And Excitation in Nerve. J Physiol. 1952;117:500–44.

2. Hodgkin AL, Huxley AF, Katz B. Measurement of Current-Voltage Relations in the Membrane of Giant Axon of Loligo. J Physiol. 1952;116:424–48.

3. Izhikevich EM. Dynamical Systems in Neuroscience. Sejnowski TJ, Poggio TA, editors. Dynamical Systems in Neuroscience. The MIT Press; 2007.

Blog by __Emine Topcu__