Axons are responsible for the transmission of information between different points of the nervous system and their function is analogous to the wires that connect different points in an electric circuit. However, this analogy cannot be pushed very far. In an electrical circuit the wire maintains both ends at the same electrical potential when it is a perfect conductor or it allows the passage of an electron current when it has electrical resistance. As we will see in these lectures, the axon, as it is part of a cell, separates its internal medium from the external medium with the plasma membrane and the signal conducted along the axon is a transient potential difference that appears across this membrane. This potential difference, or membrane potential, is the result of ionic gradients due to ionic concentration differences across the membrane and it is modified by ionic flow that produces ionic currents perpendicular to the membrane. These ionic currents give rise in turn to longitudinal currents closing local ionic current circuits that allow the regeneration of the membrane potential changes in a different region of the axon. This process is a true propagation instead of the conduction phenomenon occurring in wires. To understand this propagation we will study the electrical properties of axons, which include a description of the electrical properties of the membrane and how this membrane works in the cylindrical geometry of the axon.


Much of our understanding of the ionic mechanisms responsible for the initiation and propagation of the action potential (AP) comes from studies on the squid giant axon by A. L. Hodgkin and A. F. Huxley in 1952. The giant axon has a diameter in excess of 0.5 mm, allowing the introduction of electrodes and change of solutions in the internal medium. These studies have general relevance because the properties of the squid axon are very similar to non-myelinated nerves in other invertebrates and vertebrates, including man.


The Capacitance and Resistance of the membrane.


The plasma membrane is made of a molecular lipid bilayer. Inserted in this bilayer, there are membrane proteins that have the important function of transporting materials across the membrane. The lipid bilayer acts like an insulator separating two conducting media: the external medium of the axon and the internal medium or axoplasm. This geometry constitutes an electric capacitor(2) where the two conducting plates are the ionic media and the membrane is the dielectric. The capacitance c of a capacitor increases with the area of the plates and decreases with the separation between the plates according to the the relation

where A is the membrane area, d is the separation between the plates or the membrane thickness and is the dielectric constant. In the case of the membrane, it is more convenient to define the capacitance as independent of the amount of area involved and call it the specific capacitance Cm which is defined as the capacitance per unit area or c/A. Replacing this definition in eq (1) we find

As the thickness d is only 25 A, the specific capacitance of the membrane is very high, close to 1 µF/cm2. Having the properties of a capacitor, the membrane is able to separate electric charge, achieved by a difference in the number of anions and cations on each side of the membrane; this charge separation, in turn produces a potential difference across the membrane. In a capacitor the potential difference V is related to the charge Q by

where C is the capacitance. It is important to notice that in the case of the plasma membrane a small amount of charge separation is able to generate a large potential difference. For example, to obtain a membrane potential of 100 mV it is necessary to separate the product of Cm =1(µF/cm2) times Vm = 0.1 (Volts), that is, Q=0.1 µCoulombs/cm2.To get an idea of the magnitude of this charge, we can compute the number p of monovalent ions that must be separated across the membrane to explain this charge

p = 0.1 x 10-6 (coul/sq cm)/1.6 x 10-19 (coul/ion) = 6.25 x 1011 (ion/cm2)


which corresponds to only 6250 ions per µm2 of membrane.


The energy to put an ion into the lipid bilayer is so large that we would expect the membrane to be practically impermeable to ions. However, experimentally, it has been found that the membrane presents a finite permeability to cations and anions. Today we know that this permeability is mediated through specialized proteins that can act as carriers or channels for the passage of charged species. The detail of the operation of channels will be described later. What is relevant here is the fact that ions can penetrate through specialized pathways which constitute the membrane electrical conductance (conductance is the reciprocal of resistance). This conductance will be another element of our electric circuit that will represent the electrical characteristics of the membrane.


Again we will define units that are independent of the membrane area. If we measure the electrical resistance of a membrane of 1 sq cm of surface area, it will be 10 times larger than the resistance of a membrane with the same characteristics but of 10 sq cm surface area. This is because the resistance decreases as the access area increases. For this reason, we can define the membrane specific resistance Rm in units of ohms sq cm or specific conductance Gm in units of Siemens/cm2 (S/cm2).


The Membrane Potential.


The Nernst Potential.


How is it possible to separate charge across the membrane? Let us take a simple example. Assume that we have a membrane separating two compartments (Fig. 1) that has channels that are only permeable to potassium and no other ions can permeate. Initially the channels are closed and we add 100 mM of KCl to the lower compartment (say the interior of the cell) and 10 mM of KCl to the upper compartment (the outside). As we have added a neutral salt, there will be the same number of cations than anions in the lower compartment and the same will be true for the upper compartment (even though the total number of ions is 10 times lower in the upper compartment).The consequence of the electroneutrality in each side will be zero charge difference across the membrane and consequently the membrane potential difference will be zero. This is because from eq. (3) we can write that V=Q/C, where V is the potential difference and Q is the excess charge. The permanent thermal motion of the ions will make them move randomly but they will not be able to cross the membrane because they are poorly permeant through the bilayer and the channels are closed. Suppose that at one point we open the channels. Then, as there are 10 times more K+ ions in the bottom compartment than in the top compartment, there will be 10 times more chances of an ion crossing up than down. This initial situation is schematically pictured in Fig. 1A where a K+ ion (the black balls) is crossing the channel in the upward direction leaving a Cl- ion behind. This flow, which is proportional to the concentration gradient, increases the top compartment by one positive charge and the lower compartment by one negative charge, producing a charge separation (Fig. 1B). This charge separation introduces a non-random new electrostatic force acting on the ions, as pictured in the box insets of Fig. 1. This electrostatic force tends to drive the ions from the top compartment into the bottom compartment and at the same time it tends to brake the flow in the opposite direction. The final result is that the charge separation will build up a voltage across the membrane (V=Q/C) that will continue to increase until the flow in both directions becomes equal due to the increased electrostatic force that will tend to balance the flow produced by the concentration gradient. When that happens, any ion that crosses in one direction will be counterbalanced by another crossing in the opposite direction, maintaining an equilibrium situation. This potential difference is then called the equilibrium potential or Nernst potential.


The above discussion can be put in more quantitative terms by expressing the net ion flow j in terms of the chemical and electrical gradients:

where D is the diffusion coefficient, C is the concentration, R is the gas constant, V is the voltage, z is the valence, F is the Faraday constant, and T is the temperature. When j=0 (no net flow), eq (4) can be integrated and we get:

which is the Nernst equation that relates the voltage V across the membrane which is in equilibrium with the concentration gradient established by the concentrations Co, outside, and Ci, inside. It is customary to call this voltage the equilibrium potential of the ion "N" Ee, and by calling the external and internal concentrations of the ion "N" No and Ni, respectively we can rewrite eq. (5) as follows:

More than one specific channel.


In the nerve membrane there are several types of channels, each of which is selective to a specific ion, such as Na+ or K+. Therefore the situation of zero net flow across the membrane does not depend on one particular ion concentration gradient but it involves the concentration of the other permeant ions and their relative permeabilities. In this situation we have to consider the individual fluxes jNa, jK, etc. and the solution when the sum of all the flows is zero gives the Goldman-Hodgkin-Katz equation which can be written as:

where the permeabilities for the ion k is written as Pk and the concentrations of the ion are given by its chemical symbol followed by the subindex indicating the side of the membrane, with i for inside and o for outside. Thus, according to this equation, the voltage across the membrane is determined by the concentrations of all the ions and is most affected by the ion with the highest permeability. If E, as computed from eq. (7) is equal to the E of the Nernst equation (eq. 6) for one particular ion, we say that that ion is in equilibrium.


Real channels are not perfectly selective. The selectivity of ion channels is not perfect and, for example in K+ channels, for every 20 K+ ions that flow through the channel, one Na+ ion can get through. This means that we cannot apply the Nernst equation to compute the potential that produces zero flow across the channel because more than one ion is involved. Instead, we could use the Goldman-Hodgkin-Katz equation (with PK/PNa=20, in the case of the K channel) and the potential predicted by the equation would be called reversal potential, instead of equilibrium potential, at which the net flow of charge through the channel is zero.


The electric Model of the Membrane



We have now described the capacitance of the membrane, mainly given by the bilayer, the resistance of the membrane given by the ionic channels, and we must now include the membrane potential. As explained in the previous paragraphs, this membrane potential exists even in the absence of stimulus or external electric field due to the charge separation produced by the ion redistribution under the influence of chemical and electrical gradients. In resting conditions, this voltage is called the resting potential and it can be represented as a battery that must be in series with the membrane resistance (Fig. 2). This battery of electromotive force (3)Em and the membrane resistance Rm may be considered as the equivalent electric circuit of the membrane of the axon, which includes all the membrane resistances and batteries of the different systems of ionic channels each one with its own conductance gi (where gi =1/Ri) and reversal potential Ei. Notice that the battery is in series with the resistance which implies that the membrane potential is equal to Em only when there is no current flow through the resistance Rm, that is, in open circuit conditions, or when it is measured electrometrically (without draining current through the membrane).



The circuit in Fig. 2 is a minimal representation of the membrane. It is important to note that this representation refers to a membrane element of very small dimensions, such that the potential could be considered constant along each side of the membrane. This is only true if the element is made infinitesimally small because any finite size may not be isopotential due to the electrical resistance of the medium and the geometry of the axon. As we will see below, in the case of the axon we will have to locate this basic circuit of the membrane element in the cylindrical geometry of the axon which will make the final circuit quite complicated. But before incorporating the geometry, we can study the basic properties of our elemental circuit which will be extremely useful in understanding the electrical behavior of the elementary unit that constitutes the axonal membrane.


Basic current equation through the conductive pathways:


The current through the conductive part of the membrane can be expressed as a product of a conductance and a driving force:

where V-Ee is the driving force and g is the conductance (reciprocal value of resistance, R=1/g) expressed in Siemens( S=1/ohm). V is the membrane potential and Ee is the reversal potential for that pathway. If the pathway is selective to only one ion species, Ee corresponds to the equilibrium potential of that species e (the potential predicted by the Nernst equation).


There are several types of conductance and their classification is done according to the type of channel involved. Thus, we have: Sodium, selective to Na; Potassium, selective to K; Chloride, selective to Cl; Calcium, selective to Ca, etc. Currents through each conductance type: INa, IK,ICl, etc


The Resting Membrane Conductance


In the resting membrane, the resting potential V is constant and there is no net current crossing in or out of the cell, although each individual conductance may be carrying a net current, therefore

Replacing the current expression for each conductance, from eq (8) into eq (9):

Solving for V in eq (10) we get:


Equation 11 is similar to the Goldman-Hodgkin-Katz equation but it considers conductances and reversal potentials instead of permeabilities and concentrations. In several excitable membranes, such as the squid axon, gCl is small but there are other type of conductances that have been lumped together under the name of leakage or gL, which is then substituted for gCl.





In this section, we will study the currents through the different ionic pathways and the resulting voltage across the membrane. The conventions will be that the inside of the membrane faces down and all the voltages will be referred to the outside. For example, if the membrane potential is -80 mV it means that the inside is negative with respect to the outside. The variable conductances (or variable resistances) are represented by a box with a side cursor (or arm) that can slide up and down the side of the box (see Fig. 3). Thus, when the cursor is close to the top the resistance between the two ends is minimum and becomes maximum when the arm is close to the bottom (Fig. 3). The equilibrium (or reversal) potentials are represented by batteries with the value written near the battery symbol. In addition, the intensity of the current is represented by the degree of shading of the wires, being black the highest intensity and white zero current. The arrows indicate the direction of positive ions current flow.


With these conventions, we can study the influence of the conductances and reversal potentials in the final membrane potential in the steady state.


First Case. Only one conductance is present (Figs 4A and 4B). All the other conductances are zero (that is, resistances are infinite) represented by an interruption in the connection of the variable resistor. This means that there is no current circulation in any of the branches with infinite resistance. This is represented in Fig. 4A for the case of finite K conductance and all the others zero. Notice that the membrane potential will be equal to the K equilibrium potential (in this case -73.3 mV) because there is no current circulation through any other resistor and, since we are considering steady state, the capacitor does not drain any current (we will study later the origin of capacitive current). In Figure 4B there is a representation of the case when only the Na conductance is present and all the others are zero. As expected, in this case the membrane potential will be identical to the Na reversal potential, in this case +41.3 mV. The situation depicted in Fig. 4 is a rough approximation of what occurs during an action potential: at resting the membrane is selective to K and the membrane potential is near -73 mV and during the action potential the membrane becomes selective to Na and the membrane potential reverses and becomes positive.


Second Case (Figs. 5 and 6). Now we consider the more realistic situation where all the conductances have a finite value, although different among them. In the resting condition, the predominant conductance is potassium as shown in Fig. 5. In the resting condition all the conductances are small but the K conductance is larger than all the others. Notice that the relative values of the conductances (in mS/sq cm) are gK=0.37, gNa=0.02 and gL=0.3. The K battery will drive an outward current through the K conductance that will be drained by the leakage and Na branches to make the total current equal zero. The resulting membrane potential will be between the K reversal potential and Na reversal potential, but closer to the EK because gK is the largest conductance. Finally, let us consider the case when the Na conductance predominates as during the rising phase of the action potential. In this case, the actual sodium conductance increases several fold over the resting condition, becoming the dominant conductance. This produces a large flow of inward current through the Na conductance that is drained out by the K and leakage conductances (see Fig. 6).





The action potential is the result of changing the membrane from a K selective condition to a Na selective condition. We must explain now how this occurs. To understand how these conductances shifts occur, we must study their ability to change with the membrane voltage, or their "voltage dependence". This leads to a description of voltage dependent conductances. In voltage dependent conductances the value of the conductance depends  on the membrane potential.




1. A potential difference exists between two points whenever the introduction of a conducting path between the two points would result in spontaneous charge

transfer or electric current.


2. An electric capacitor is a device that allows the separation of electric charge on two conducting plates by a material, or dielectric, that is not able to sustain an

electric current.


3. The electromotive force of a battery is the voltage measured from the battery in the absence of current circulation