hodgkin and huxley's work (5) revealing the origins of cellular excitability is one of the great triumphs of physiology. In an extraordinarily deft series of papers, they were able to measure the essential electrical characteristics of neurons and synthesize them into a quantitative model that accounts for the excitability of neurons and other cells. The Hodgkin-Huxley equations, a set of four differential equations, predict many of the electrical characteristics of neurons and muscle cells; however, these equations are somewhat beyond the ken of most undergraduate biology students. I will show that if one wants to truly understand the origins of excitability, one cannot avoid its mathematical underpinnings. In this article, I also will demonstrate how it is possible, resorting only to elementary mathematics, to show why the combination of certain ion channel makes cells excitable.

What does it mean to understand a biological process? Is a description of what happens during some biological event enough? Some might claim that science simply describes how things happen, but this misses the essential power of science, which is to go beyond what has been observed into predicting what might happen, which can only be done through the agency of mathematics. To guide student's understanding of the process in the classroom, we need to take them beyond rote recitation to a visceral understanding of the forces at play in the process.

Most neuroscience textbook accounts of cellular excitability do not divulge the essence of what makes cells excitable because they seem reluctant to address the underlying mathematics and physics. This leads to students recounting what happens during an action potential but gives them little insight into the mechanism.

### Approach

To understand why certain cells are excitable, we will need to look at some elementary cellular phenomena and review the physics of electrical current flow in simple linear circuits. In what follows, we will set up models that represent how cells behave electrically. In particular, we will use two circuit elements, namely resistors and batteries, to build up equivalent circuits. What is meant by “equivalent circuits” is that as far as the electrical properties of the cells are concerned, these models capture, with a fair degree of accuracy, the electrical properties of the cell. Using these models, we can predict how the voltage changes within a cell when current is injected. This exercise of stripping down a system to a barebones representation is a useful one because at the end we will have a mental “device” for understanding the system. The model is not a crutch but an attempt to get at the essentials of what is occurring in the cell and make predictions. In the course of this article, we will also show that the passive leak of a neuron plays an important role in determining its excitability. Our overall approach to understanding cellular excitability has been adapted from the works of FitzHugh (3) and Noble (11).

Biology students often have difficulty in thinking about electrical circuits, and I find that it helps to use the widely used analogy of water flow in tubes, which exhibits the same characteristics (10). Because everyone has experience with water flow (rivers, faucets, straws, garden hoses, etc.), almost everyone has some intuition about what might change the rate of water flow. In what follows, we will assume an understanding of the basic principles of electrical current flow such as presented in the textbooks by Nicholls et al. (10) or Kandel et al. (7).

Most cells have a resting potential that is negative relative to the outside, usually somewhere between −80 and −50 mV. Cells rest at a negative potential because they are typically more permeable to K^{+} than Na^{+} and because cells maintain a high concentration of K^{+} inside and Na^{+} outside through the action of Na^{+}-K^{+}-ATPase, which operates continuously to maintain this asymmetric and nonequilibrium ion concentration.

The resting potential arises from the competition between diffusion of an ion down its concentration gradient and the countervailing potential it sets up. To see how this occurs, let us imagine that we have a membrane that is only permeable to K^{+} and that we have a KCl solution with a concentration 100 mM inside and 10 mM outside. Furthermore, let us assume initially that there is no potential difference between inside and outside. K^{+} can move across the membrane and will tend to move from inside to outside because there is a higher concentration on the inside. Let's say that 1 K^{+} moves from inside to outside. Now, because Cl^{−} cannot move, there is a charge of +1 on the outside and −1 on the inside. Because like charges repel, this makes it more difficult for a subsequent K^{+} to move from inside to outside. So, what happens is that K^{+} moves across the membrane until voltage matches the force that impels K^{+} to move down its gradient. This is essentially the scenario that Nernst set up and solved in 1888, deriving the following expression:
(1)

where *V* is membrane potential, *R* is the gas constant, T is absolute temperature, *z* is the charge on the ion, *F* is Faraday's constant, and [K^{+}]_{o} and [K^{+}]_{i} are the K^{+} concentrations outside and inside of the cell, respectively.

The most elementary electrical characteristics of a cell can be represented by a capacitance (equivalent to the dielectric of the lipid bilayer) in parallel with a conductance (equivalent to K^{+} channels that are open) and a battery [the Nernst potential associated with the K^{+} conductance (*E*_{ℓ})] (Fig. 1). We will refer to this conductance as the “leak” [*g*_{ℓ} = 1/*R*_{ℓ}, where *R*_{ℓ} is the leak resistance). For the time being, we will ignore the capacitance and focus on the battery-conductance branch. The voltage across this branch is the membrane potential and is given by Ohm's law, which, in this case, is as follows:
(2)

where *I* is current. Solving for *I* gives the following:
(3)

By convention, the voltage outside the cell is set to zero.

Now we can use this circuit to make a prediction of the resting potential. When a circuit is at equilibrium or rest, no current flows, and we can then use *Eq. 3* to predict the resting potential, which is equal to *E*_{ℓ}. Note that the resting membrane potential is typically higher than the K^{+} Nernst potential because *g*_{ℓ} is also somewhat permeable to Na^{+} and other ions.

To measure the voltage within a cell, we use a very fine glass microelectrode (or patch electrode) filled with a conductive electrolyte solution, which is inserted into the cell and is in electrical contact with the cytoplasm. The electrode is, in turn, connected to an amplifier called a “current clamp,” which measures the voltage and allows one to inject current into the cell (Fig. 1).

When cells are depolarized by the injection of a small positive charge, the membrane charges up exponentially and then declines back exponentially to rest when the charge flow ceases (Fig. 1). This charging and discharging is what is termed “passive,” depending as it does on the passive properties of the cell, its resistance and capacitance. Similar events occur for hyperpolarizing currents. Without the capacitance, membrane potential would change with no delay when current is injected, so it is the capacitance that slows the rise of membrane potential. We will not linger on the properties of capacitors but refer the reader to Refs. 7 or 10.

For a cell to be excitable, it needs voltage-gated ion channels, pathways whose conductivity changes in response to changes in transmembrane potential. To demonstrate this, we will use the example of a voltage-gated Na^{+} channel that does not exhibit inactivation, i.e., it is persistent. For this channel, current is as follows:
(4)

where *g*_{Na} is Na^{+} conductance and *E*_{Na} is the Nernst potential associated with Na^{+} conductance. This is not much different from the leak channel discussed above; the only new term here is *p*(*V*), which we will call the “probability of activation” and give it the form of the following Boltzman equation (4, 12) (Fig. 2*A*):
(5)

where *V*_{0.5} is the voltage at which *p* is half maximal and *z* is the charge on the channel. As the voltage increases, *p* goes from zero to one, from being shut to open.

Students might wonder at this point whether a channel can be half open, and it might be worthwhile noting that channels exhibit quantal characteristics, i.e., they are either open or closed. Moreover, they gate in a stochastic fashion, so that a *p* of 0.5 represents half of the channels being open.

To study what happens in an excitable cell, we will use a different kind of amplifier, called a “voltage clamp”, which allows us to control the voltage across the membrane while measuring the current. Let's look at what currents flow as we change the voltage across our voltage-gated channel. We can do the following experiment: we hold the cell at a potential (the holding potential) and abruptly increase (i.e., step) it for a time to a new potential (the step potential). We then plot the current measured as a function of the step current, yielding a current-voltage (*I–V*) relationship, as shown in Fig. 2*B*. This characteristic nose-like profile is very important in the generation of action potentials. Notice that it is simply the product of the gating function *p* and the *I–V* relationship of the open Na^{+}channel (*Eq. 6*).

It may not seem like much to go from measuring the voltage to measuring current; however, this switch made an enormous difference to unscrambling the process of action potential generation. It greatly simplifies measuring the properties of ion channels, since currents simply add up. If the measured current is the sum of two currents (*I*_{x} and *I*_{y}), all we need to do to measure all currents is to measure the total current (*I*_{x} + *I*_{y}) and then block one of the currents, say *I*_{y}, and subtract *I*_{x} from the total current to get *I*_{y}. Moreover, it is easier to comprehend the sources of currents rather than changes in voltage.

An action potential is a bit like a little explosion, that is, it is an unstable event that, once triggered, proceeds to its conclusion, making it difficult to determine what occurs during the event. The voltage clamp prevents the occurrence of an action potential, by fixing the voltage and measuring the currents that flow. Voltage-clamp experiments allow one to build a systematic model of how ion channels respond to changes in potential, and it is this model that is our lens into understanding how the action potential occurs. Solving the equations numerically allow us to reproduce the phenomena; however, understanding can only be achieved by diving into the mathematics, which is what we will do here.

The voltage-gated Na^{+} channel is by itself insufficient to make the cell excitable. We need one other essential but rather unprepossessing channel, a simple leak. If we add the leak and persistent Na^{+} channel, we then get an *I–V* relationship, as shown in Fig. 3*A*, where the whole cell *I–V* relationship is given by the following:
(6)

where *I*_{a} is the applied current (vide infra).

It is this “N”-shaped, kinky curve that is the very source of the instability that characterizes an excitable system. To see why that is so, we need to look at the points where the current is zero, which are potential equilibria. Notice that this curve simply allows one to predict what current will flow if the system is at a particular voltage, but we can use our knowledge of the direction of current flow to predict how the system changes with time.

Positive charge flowing into a cell, or negative charge out, will increase membrane potential (i.e., depolarize it). By convention, this direction of current flow is termed “inward” current and is given a negative sign. “Outward” current corresponds to positive charge flowing out of the cell (or negative charge in) and is given a positive sign and decrease the membrane potential (i.e., hyperpolarize it).

Let's look at *point a* (membrane potential = −60 mV, current = 0; Fig. 3*A*). It is an equilibrium point so the system could potentially be “content” there; however, let's say that the voltage is perturbed so that it becomes more positive. Now the current flow is outward, i.e., positive charge flows out of the cell, making it more negative. Thus, the system will be driven back to *point a* (Fig. 3*A*). If the voltage is changed to a little below *point a*, it generates inward current, which drives the voltage back to *point a*. So, we can see that *point a* is what is called a “stable point,” i.e., any small perturbation around the point will send it back to the point. The system behaves like a ball at the bottom of a valley; if you push it left or right, it rolls back to the same location (Fig. 3*B*).

Now, there is another equilibrium point, *point c*, on the right (Fig. 3*A*), and if you go through the same exercise described above, you will find that it is a stable point. But *point b*, as indicated by the open circle in Fig. 3*A*, is rather different. If you increase the voltage, an inward current is generated depolarizing the cell, which in turn increases the inward current. The net result of this is that the voltage increases until it reaches the rightmost equilibrium point. Again, starting at equilibrium *point b*, if the voltage decreases, this leads to outward current, which will sweep the current to the left until it reaches the resting potential, *point a*. Equilibrium *point b* behaves like a ball sitting on top of a hill, nudging it to the left or right will lead it to roll down the hill (Fig. 3*B*). In a neuron, it effectively serves as a threshold. If current is injected, this leads to depolarization, which does not get beyond *point b*; it subsides back to *point a*. However, if it exceeds *point b*, it gets swept to *point c*, and the voltage increases in a step-like fashion.

With the most elementary mathematics, we can get a sense of what it is like to be at a particular voltage. This little exercise of changing the voltage around an equilibrium point allows us to predict where the system will go.

So far in our analysis, we have perturbed the neuron by changing the voltage; however, in fact, the activation of synapses induces current flow, which changes the membrane potential. To see how current injection can alter membrane potential, we consider *Eq. 6*. Current injection simply shifts the *I–V* curve up (hyperpolarizing) or down (depolarizing), displacing *point a* to the left or right, respectively (Fig. 4). A current stimulus will excite the cell if it is large enough to make equilibria *points a* and *b* disappear, leaving only *point c*. This accounts for a cell's responses to steady current inputs. For rapid transient current inputs, we can approximate the cell's response by shifting the voltage to the right or left along the *I–V* curve in response to depolarizing our hyperpolarizing current pulses, respectively. Once the current is turned off, the system then moves to *point a* or *point c* depending on its position along the *I–V* curve.

Our system with only a leak and a voltage-gated persistent Na^{+} current is not much of a neuron, because if it is excited it transits to *point c* but gets stuck there and cannot repolarize back to *point a*. However, this simplified system allows us to see how such a set of channels creates a threshold. To get the voltage back home to rest, we need to introduce another process, namely, the activation of a K^{+} current. Let us assume that after the cell is depolarized, this then rapidly increases the leak current. Now the *I–V* relationship has the form shown in Fig. 5 with a single equilibrium point. The voltage hence transitions rapidly back to this equilibrium point since there is only one point. The leak channel then has to decline back to rest for the neuron to recover its excitability. What occurs typically in neurons is that a K^{+} current distinct from the leak activates with a slower time course than the voltage-gated Na^{+} channel, driving the repolarization process.

Another process that can return the voltage to the resting potential after excitation is Na^{+} channel inactivation. This is a process whereby after Na^{+} channels are gated open, they spontaneously move into a closed, inactivated state. Let's imagine that we have the same system as described above and that the system is excited and moves into the excited state, *point c*. Now if the Na^{+} channel starts shutting off, this will have the effect of making the excited state move progressively to the left, i.e., repolarizing the cell. This will occur until *point c* disappears as the curve detaches and then there is only one equilibrium point and the system then is back at rest (Fig. 6). In reality, inactivation only plays a small role in repolarization, but this example does show that it is possible for a neuron to be built with only a leak and an inactivating Na^{+} current.

### Discussion

The approach detailed here provides an introduction to systems biology thinking, where one has to synthesize all the essential elements into a model to account for a system's behavior. It also shows something that is glossed over in almost all accounts of excitability, namely, the importance of what, at first blush, seems like a factor of little consequence, the leak. As can be seen from our analysis, the system is not excitable when there is either no leak conductance or the leak is too high.

The following textbooks are recommended for more advanced mathematical approaches to cellular excitability and nonlinear dynamics: Refs. 2, 6, 8, and 13. Free programs for simulating action potentials and voltage-clamp responses are available on the web (1, 9). For a BBC television interview with Alan Hodgkin and Andrew Huxley discussing their work on the squid giant axon, see Ref. 14.

We have approached the question of cellular excitability in a pedestrian fashion, that is, we have used only the most basic concepts to get at the core mechanisms at play in cellular excitability. I believe that going through the process of putting this system together gives the student a much better sense of the mechanics of excitation than the usual verbal recitations.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: A.R.K. conception and design of research; A.R.K. performed experiments; A.R.K. analyzed data; A.R.K. interpreted results of experiments; A.R.K. prepared figures; A.R.K. drafted manuscript; A.R.K. edited and revised manuscript; A.R.K. approved final version of manuscript.

## ACKNOWLEDGMENTS

The author thanks Dr. Rodica Curtu for helpful comments on an earlier version of the paper.

- Copyright © 2014 The American Physiological Society