# GHK Equation and Action potential

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Can GHK equation be used to predict the membrane potential even if the cell is not at resting state?

To say it again, can we use GHK equation at every moment during Action potential?

I'm confused because GHK equation is derived at the situation with total ionic current is zero, but is that true also during AP?

The GHK voltage equation also known as just "Goldman equation" is always valid for determining the voltage at which the net current is zero, given internal/external ion concentrations and their permeabilities. This includes times during the action potential, though the result you get from GHK will be changing faster than the actual membrane potential itself, since the change in membrane potential brings about changes in permeabilities.

It's still useful to think about this equation to determine in which direction the membrane potential is changing instantaneously: it will always be changing towards the equilibrium potential. You can also qualitatively estimate the relative rate of change: if the current voltage is very far away from the equilibrium potential, the voltage will change more rapidly than if it is close to equilibrium.

However, this equation is just a steady-state equation. It doesn't tell you how quickly you will get to the new voltage or exactly which ions will move. For this, you need some other equations and other parameters, namely the actual permeabilities (rather than relative permeabilities) of the ions and the membrane capacitance.

The GHK flux equation can help: this will give you the flux of each ion at a given membrane potential (Wikipedia shows how this equation is related algebraically to the Nernst equation, which in turn is equivalent to the Goldman/GHK voltage equation for a single ion). You can also think in terms of the Hodgkin-Huxley model and the simple differential equation for current.

In summary, with respect to your question:

Can GHK equation be used to predict the membrane potential even if the cell is not at resting state?

It can predict where the membrane potential is headed, but not what it actually is or how fast it's changing, and the voltage you get from GHK will itself change as membrane potential changes affect the parameters that go into the GHK equation itself.

## GHK Equation and Action potential - Biology

A neuron can receive input from other neurons and, if this input is strong enough, send the signal to downstream neurons. Transmission of a signal between neurons is generally carried by a chemical called a neurotransmitter. Transmission of a signal within a neuron (from dendrite to axon terminal) is carried by a brief reversal of the resting membrane potential called an action potential. When neurotransmitter molecules bind to receptors located on a neuron’s dendrites, ion channels open. At excitatory synapses, this opening allows positive ions to enter the neuron and results in depolarization of the membrane—a decrease in the difference in voltage between the inside and outside of the neuron. A stimulus from a sensory cell or another neuron depolarizes the target neuron to its threshold potential (−55 mV). Na + channels in the axon hillock open, allowing positive ions to enter the cell (Figure 1).

Once the sodium channels open, the neuron completely depolarizes to a membrane potential of about +40 mV. Action potentials are considered an “all-or nothing” event, in that, once the threshold potential is reached, the neuron always completely depolarizes. Once depolarization is complete, the cell must now “reset” its membrane voltage back to the resting potential. To accomplish this, the Na + channels close and cannot be opened. This begins the neuron’s refractory period, in which it cannot produce another action potential because its sodium channels will not open. At the same time, voltage-gated K + channels open, allowing K + to leave the cell. As K + ions leave the cell, the membrane potential once again becomes negative. The diffusion of K + out of the cell actually hyperpolarizes the cell, in that the membrane potential becomes more negative than the cell’s normal resting potential. At this point, the sodium channels will return to their resting state, meaning they are ready to open again if the membrane potential again exceeds the threshold potential. Eventually the extra K + ions diffuse out of the cell through the potassium leakage channels, bringing the cell from its hyperpolarized state, back to its resting membrane potential.

### Practice Question

The formation of an action potential can be divided into five steps, which can be seen in Figure 1.

Figure 1. Action Potential

1. A stimulus from a sensory cell or another neuron causes the target cell to depolarize toward the threshold potential.
2. If the threshold of excitation is reached, all Na + channels open and the membrane depolarizes.
3. At the peak action potential, K + channels open and K + begins to leave the cell. At the same time, Na + channels close.
4. The membrane becomes hyperpolarized as K + ions continue to leave the cell. The hyperpolarized membrane is in a refractory period and cannot fire.
5. The K + channels close and the Na + /K + transporter restores the resting potential.

Potassium channel blockers, such as amiodarone and procainamide, which are used to treat abnormal electrical activity in the heart, called cardiac dysrhythmia, impede the movement of K + through voltage-gated K + channels. Which part of the action potential would you expect potassium channels to affect?

Figure 2. The action potential is conducted down the axon as the axon membrane depolarizes, then repolarizes.

## Membrane Potential across the Cell Membrane: 3 Types | Biology

Membrane potential is classified as: 1. Resting Membrane Potential 2. Action Potential 3. Graded Potential.

All cells in animal body tissue are electrically polarized, in other words they maintain a voltage difference across the plasma membrane known as membrane potential. The cell membrane acts as a barrier that prevents intracellular fluid from mixing with the extracellular fluid. Therefore, the electrical potential difference results from a complex interplay between protein structures embedded in the membrane called ion pumps and ion channels.

#### Type # 1. Genesis of Resting Membrane Potential:

The membrane potential across the cell membrane when the cell is at rest is called resting membrane potential (RMP). We always express RMP by comparing ICF potential to ECF potential keeping ECF potential to be zero. For example: RMP for large nerve fibre is –90 mV. That is the potential inside the fiber is 90 millivolts more negative than the ECF potential. The nerve cell at this state is said to be in polarized state.

RMP in a cell is generated due to two reasons:

i. Contribution of Simple Diffusion to the Genesis of RMP:

Simple diffusion through protein channels like sodium and potassium channels, which allows movement down the concentration gradient, is influenced by factors such as size, charge on surface of protein, hydration of the ion, etc.

Biophysical basis for membrane potential is caused by simple diffusion alone.

The Gibbs-Donnan effect (also known as the Donnan effect, Donnan law, Donnan equilibrium, or Gibbs-Donnan equilibrium) is a name for the behavior of charged particles near a semipermeable membrane which sometimes fail to distribute evenly across the two sides of the membrane. The usual cause is the presence of a different charged substance that is unable to pass through the membrane and thus creates an uneven electrical charge.

In the body, it is the Gibbs-Donnan effect of intracellular negatively charged protein forms the basis of the negative resting membrane potential. If it was not for the electrogenic activity of Na/K ATPases the resting membrane potential would be even more negative. This forms the basis for equilibrium potential of ions.

b. Equilibrium Potential and Nernst Equation:

Particular ion will flow across a membrane from the higher concentration to the lower concentration (down a concentration gradient), causing a current. However, this creates a voltage difference across the membrane that opposes the movement of ions. When this voltage reaches the equilibrium value, the two balances (concentration gradient and the voltage) and the flow of ion stops.

The voltage at which the flow of the ion stops is called the equilibrium potential of that ion. The equilibrium potential for any ion can be calculated by an equation called Nernst equation. Equilibrium potential for potassium is –94 mV equilibrium potential for sodium is +61 mV.

Equation for calculating the equilibrium potential for potassium is as follows:

Eeq K + = RT In [K + ]o/zF [K + ]i

i. Eeq K + is the equilibrium potential for potassium in volts

iii. T is the absolute temperature

iv. Z is the number of elementary charge of the ion

vi. [K + ] o is ECF concentration of potassium

vii. [K + ] i is ICF concentration of potassium

viii. RT is a constant and the value is calculated as 61 and formula can be written as-

c. Goldman-Hodgkin-Katz Equation:

If membrane is permeable to only one ion, the membrane potential is the equilibrium potential of that ion. But in reality, animal cell is permeable to many ions. So the membrane potential has to be calculated taking equilibrium potential of all ions into consideration.

Hence, membrane potential depends on:

i. Polarity of electric charge of each ion

ii. Permeability of membrane

iii. Concentration of ion in ICF and ECF.

The major ions involved in generating membrane potential are sodium, potassium and chloride. Membrane potential can be calculated from Goldman-Hodgkin-Katz equation ―

Where P is permeability of the cell membrane to the ion. Due to the negative charge of chloride ion, the concentration of chloride in ECF is written in the numerator.

At rest the cell membrane is 100 times more permeable to potassium diffusion than sodium because the hydrated form of potassium is smaller in size compared to that of hydrated form of sodium ion. This is reason why RMP will be close to equilibrium potential of potassium.

ii. Contribution of Sodium Potassium Pump for the Genesis of RMP:

It is an electrogenic pump which pumps three Na + to ECF and two K + to ICF against concentration gradient leaving a net deficient of one positive ion on the inside, which causes a negative voltage inside the cell membrane.

Let us calculate the RMP of the large nerve fiber:

i. The resting membrane potential of large nerve fiber, if potassium alone is considered as permeable is –94 mV which is calculated from Nernst equation.

ii. The RMP if sodium alone is considered will be +61 mV.

iii. But actually RMP is due to both sodium and potassium which can be calculated from Goldman equation to be –86 mV, which is nearest to equilibrium potential of potassium.

iv. But still we have –4 mV left to get –90 mV as RMP in large nerve fiber. What it is due to? It is due to the sodium potassium pump which leaves negativity inside the cell contributing to –4 mV by pumping an extra positive charge outside the cell.

v. So totally –86 mV and –4 mV negativity inside the cell is due to simple diffusion and active transport respectively which contributes to –90 mV RMP in large nerve fiber.

For calculating RMP in a muscle, calcium ion should also be considered for calculating Goldman equation (Fig. 2.18).

#### Type # 2. Action Potential (AP):

An action potential is a short lasting event in which the electrical membrane potential of cell rapidly rises and fall after sufficient strength of a stimulus is applied (during action). This occurs in excitable cell like neu­rons and muscle cell. In neurons, they play a central role in cell to cell communication (Fig. 2.19). In muscle cell, AP is the first step in chain of events leading to contraction.

The stages of action potential:

I. Resting (Polarized) Stage:

This is the membrane potential before the stimulus, i.e. RMP and the membrane is said to be polarized.

II. Depolarization Stage:

After the sufficient stimulus to an excitable cell, there occurs a change in voltage which opens the voltage gated sodium channel, (Sodium permeability is more than potassium during action in contrast to what it is at rest) allow­ing tremendous flow of positively charged sodium ions through voltage gated sodium channels by simple diffusion to the interior of cell.

The membrane potential which was negative compared with outside, now rapidly shifts to positive side due to flow of positive charge sodium ion. This shift from negative to positive potential is called as depolariza­tion (i.e. polarized to depolarized state).

III. Repolarization Stage:

The gated channel is open only 1/10000th of a second. Within a few 10000th of a second the sodium channels begin to close. But the voltage for opening potassium channel is attained, and therefore, voltage gated potassium channel opens. Since K + is more inside, the potassium flows tremendously to the outside of the cell (Fig. 2.20). This rapid diffusion of K + which is also a positive charge to outside of the cell, re-establishes the negative RMP. This stage is called repolarization stage. After a fraction of second the gated potassium channel closes.

The ionic basis with an example of action potential in a large nerve fiber is explained as:

RMP nerve fibre is –90 mV (resting stage) → Sufficient strength of a stimulus is applied → Opening of voltage-gated sodium channel → Sodium flows slowly from outside to inside through these channels thereby slowly decreasing the negativity inside the cell to –50 mV to –70 mV (threshold potential) → Once firing potential or threshold potential is reached, more voltage gated sodium channels is recruited which causes rapid inflow of sodium (sodium permeability increases by 500 to 5000 times) → The voltage inside the cell shoots up to +35 mV. This is depolarization stage → Inactivation of sodium gate at +35 mV occurs, but this is voltage required for opening the potassium gates → Potassium pours from inside to outside through these channel because ICF potassium is more than ECF concentration of potassium. There is rapid fall of potential. This event brings back the potential towards negativity. This is repolarization stage (sharp rise and fall of potential is called spike potential) → At the end there is slow fall of potential towards RMP called after depolarization → Potassium channel is slow to close, so there is extra outflow of the potassium ion to the ECF → More negativity than RMP is created called after hyperpolarization → Here comes Na + K + pump which is re-establish RMP

There will be some extra flow of K + ions outside the cell because they are very slow to close. This causes after hyperpolarization. The gates will not reopen until the membrane potential returns to the original RMP. This is possible only with the help of Na + K + pump which helps to re-establish the RMP.

Propagation of Action Potential:

An action potential elicited at any one point on an excitable membrane usually excites the adjacent portion of the membrane, resulting in propagation of action potential over the membrane. The local change in potential is carried inwards to several millimeter of adjacent membrane in both the directions, which slowly opens more Na + channel. This newly depolarized area in the same manner propagates the action potential. This transmission of depolarization process is called an impulse.

Saltatory Conduction:

The need for fast transmission of electrical signals in nervous system results in myelination of neuronal axons. Myelin sheath around axons is separated by intervals known as nodes of Ranvier. In saltatory conduction, an action potential at one node of Ranvier caused inward current that depolarize the membrane at the next node, provoking a new action potential, the AP hops from node to node because myelin offers resistance in internodal intervals.

The significance of this is that:

i. The conduction of AP is faster.

ii. The energy is conserved, which otherwise requires lot of energy if it travels through the entire neuron.

Properties of Action Potential:

It states that a threshold or sub-thre­shold stimulus is capable of producing action potential will produce the maximum possible amplitude of action potential or will not produce an action potential if the stimulus is sub-threshold. In other words, large strength do not create large action potential, therefore, action potential are said to be all-or-none.

It is the period during which excitability of excitable tissue to second stimulus is decreased.

i. Absolute Refractory Period (ARP):

It is the period during which even a second strongest stimulus cannot produce an action potential. The period extends from firing level to one-third of repolarization stage of first action potential. This is responsible for unidirectional conduction of action potential in axon.

Reason ― Inactivation of sodium gates.

ii. Relative Refractory Period (RRP):

It is the period during which a stronger than normal stimulus can produce action potential. It extends from one-third of repolarisation stage to after depolari­zation.

Reason ― The excitability is increased but threshold is decreased, so you need stronger stimulus.

III. Strength-Duration Curve:

This is a curve plotted to show the relationship of action potential with strength of a stimulus and duration of stimulus. A stimulus must be of adequate intensity and dura­tion to evoke a response. If it is too short, even a strong pulse will not be effective. A long pulse below certain strength will evoke only a local non- propagated response.

It is the minimum threshold strength which can pro­duce an action potential.

It is duration for which twice the rheobase strength has to be applied to produce an action potential. It is the measure of excitability of the tissue. Chronaxie is inversely proportional to excitability.

Variations in Action Potential in Other Tissues:

I. Plateau in Action Potential:

For example, in cardiac muscle. This type action potential happens in cardiac muscle.

b. Depolarization Phase:

Due to rapid sodium influx through voltage-gated sodium channel which is same as nerve action potential, but it is followed by plateau phase.

The membrane is held at high voltage for a few milliseconds prior to being repolarized.

This is due to two reasons:

(i) Voltage-gated calcium sodium channel, which is slow to open causing slow inflow of calcium and sodium,

(ii) Voltage-gated potassium channel are slow to open which causes slow outflow of potassium, and therefore, potential remains in positivity for some time. This delays the return of membrane potential to resting level.

d. Repolarization Phase:

Due to rapid outflow of potassium through K + channels and closing of slow calcium sodium channel which returns the action potential to resting level.

II. Pacemaker Potential:

For example ― cardiac pace­maker and smooth muscle (Fig. 2.25).

The cardiac pacemaker cells of the sinoatrial node in the heart provide a good example. They have self-induced rhythmical action potential without any stimulus. This is due to spontaneous excit­ability.

This is attributed to two reasons:

a. The resting membrane potential of pacemaker cells is between -55 mV to -60 mV. It is very close to the threshold potential which makes the cells to depolarize easily.

b. The resting sinoatrial nodal cells have sodium leaking channel called funny channels that are already open to sodium. Without any stimulus, the sodium leak to the inside of the pacemaker cell causes a slow rising RMP between heart beats, this slow rising membrane potential when reaches -40 mV attains threshold for firing due to rapid entry of sodium and calcium ions at -40 mV. After repolarization the cycle continues. Thus the inherent leaking of sodium ion causes self-excitation.

#### Type # 3. Graded Potential:

It is the localized change (depolarization or hyper-polarization) in the potential difference across a cell surface membrane. Strength of the graded potential varies with the intensity of the stimulus and causes local flows of current which decrease with distance from the stimulus point (Fig. 2.26).

Graded potentials are given different names according to their function (Fig. 2.27). The membrane potential at any point in a cell’s membrane is determined by the ion concentration differences between the intracellular and extracellular areas and by the permeability of the membrane to each type of ion.

The ion concentrations do not normally change very quickly (with the exception of calcium, where the baseline intracellular concentration is so low that even a small inflow may increase it by orders of magnitude), but the permeability can change in a fraction of a millisecond, as a result of activation of ligand-gated or voltage-gated ion channels.

The change in membrane potential can be large or small, depending on how many ion channels are activated and what type they are. Changes of this type are referred to as graded potentials, in contrast to action potentials, which have a fixed amplitude and time course. Graded membrane potentials are particularly important in neurons, where they are produced by synapses ― a temporary rise or fall in membrane potential produced by activation of a synapse is called a postsynaptic potential.

Comparison between Graded Potentials and Action Potentials:

1. Origin ― Arise mainly in dendrites and cell bodies

2. Types of channels ― Chemical, Mechanical or light

3. Conduction ― Not propagated, localized, thus permit communication over few mm

4. Amplitude ― Depends on strength of stimulus varies from less than 1 mV to more than 50 mV

5. Duration ― Longer, ranging from msec to several minutes

6. Polarity ― May be hyperpolarizing, depolarizing

1. Origin ― Arise at trigger zones and propagate along axon

2. Types of channels ― Voltage-gated ion channels

3. Conduction ― Propagated, thus permit communication over long distance

4. Amplitude ― All-or-none, typically 100mV

5. Duration ― Shorter, ranging from 0.5-2 msec

6. Polarity ― Always consist of depolarizing phase followed by repolarizing phase and then return to resting membrane potential

## A GUIDE TO INTERPRETING THE FIGURES

PK and PNa can be calculated during an action potential using the Microsoft SOLVER function (17) applied to Eq. 1, where the individual components of the action potential profile are identified as 1) the resting membrane potential, 2) the action potential peak, and 3) the afterhyperpolarization (AHP) (Fig. 1A). When PNa is calculated relative to a PK of 1, its value is low at rest (0.025). Application of a stimulus depolarizes the membrane potential, which approaches threshold (–50 mV) but PNa remains low, consistent with the critical point made by Hodgkin that it is local circuit currents that depolarize the membrane potential toward threshold (18, 19). As the membrane potential depolarizes, PNa increases and Na + becomes the dominant ion in determining membrane potential. At the peak of the action potential, the membrane potential approaches ENa and PNa increases to 13. As a result of Na + channel inactivation and the delayed activation of the K + current, the membrane potential rapidly repolarizes, reflected in the falling value of PNa and an increasing value of PK, which peaks during the AHP at a value of 400, where the membrane potential approaches EK. The membrane potential subsequently relaxes toward rest where PK = 40 (Fig. 1B).

Where the membrane potential equals 0 mV, the value of the numerator and denominator in the GHK equation must equal 1. If [K]o = [Na]i and [K]i = [Na]o, then PNa = PK = 1 at 0 mV, but the disparity of the transmembrane concentrations of K + and Na + in our model ensures that PNaPK at 0 mV.

While these relative permeabilities provide important information when viewed in isolation, it is more instructive to compare the membrane potential with the reversal potentials for Na + and K + to gain a more complete picture of the permeability changes that underlie the action potential. We have used a simulation of the Hodgkin Huxley squid axon (20) to reproduce a classic experiment carried out by Hodgkin and Katz in which action potentials were recorded in the presence of various [K + ]o (Fig. 2A). Hodgkin and Katz sought to identify to which component of the action potential Na + and K + contributed by independently reducing the concentration of the ions bathing the squid axon and then assessing which component(s) of the action potential were affected. They deduced that Na + contributes to the upstroke of the action potential but the K + contribution was subtler and affected the resting membrane potential, the rate of repolarization and the magnitude of the AHP. The resting [K + ]o was either 5 mM, 10 mM, 15 mM, or 20 mM with equimolar substitution of Na + such that [K + ]o + [Na + ]o = 460 mM. The main effects of this experiment were 1) as [K + ]o decreased, the resting potential hyperpolarized slightly 2) as [K + ]o decreased, the action potential peak was unchanged and 3) as [K + ]o decreased, the AHP amplitude significantly increased. These three effects can be satisfactorily explained by referring to each action potential component relative to Em, ENa, or EK. When plotted against [K + ]o on a log base 10 scale, EK [calculated as 55 log10([K + ]o/345)] is a straight line with a slope of 55 mV, whereas ENa 10[(460 – [K + ]o)/72]> commences as a horizontal line at low [K + ]o and then hyperpolarizes dramatically as [K + ]o increases. The membrane potential (Em) is calculated according to Eq. 1 with a PNa value of 0.04. The resting membrane potential of neurons and astrocytes can be modeled as Em and EK, respectively, and illustrates the key point that the rate at which the membrane potential depolarizes with increased [K + ]o within the physiological range (3–12 mM) is greater in astrocytes than neurons, making them ideal sensors of increased neuronal activity (21). If a vertical line is drawn from each of the four values of [K + ]o in the simulation (Fig. 2B), the intercept with Em, ENa, and EK provides a good estimate of 1) the resting membrane potential, 2) the action potential peak, and 3) the magnitude of the AHP, respectively. This suggests the membrane is predominantly permeable to K + , but has a small finite Na + permeability, at rest, but is predominantly permeable to Na + at the peak of the action potential and to K + at the AHP peak. In conclusion the membrane potential approaches the reversal potential of the ion to which it is most permeable at that instant.

## The enduring legacy of the “constant-field equation” in membrane ion transport

Osvaldo Alvarez, Ramon Latorre The enduring legacy of the “constant-field equation” in membrane ion transport. J Gen Physiol 2 October 2017 149 (10): 911–920. doi: https://doi.org/10.1085/jgp.201711839

In 1943, David Goldman published a seminal paper in The Journal of General Physiology that reported a concise expression for the membrane current as a function of ion concentrations and voltage. This body of work was, and still is, the theoretical pillar used to interpret the relationship between a cell’s membrane potential and its external and/or internal ionic composition. Here, we describe from an historical perspective the theory underlying the constant-field equation and its application to membrane ion transport.

### Introduction

The Goldman (1943) paper is, perhaps, one of the most well-known articles published in The Journal of General Physiology. Because of the great impact that this article still has on the study of transport of ions across membranes, the JGP editorial board thought that it deserved some discussion and comments.

Goldman was interested in explaining the phenomenon of electrical rectification found at the time in several different biological preparations such as Valonia, a species of algae found in the oceans, the squid giant axon, and the frog muscle (Blinks, 1930a,b Cole and Curtis, 1941 Katz, 1942). This phenomenon, in which the current is large when flowing in one direction and small in the reverse direction, was hard to understand. Thermodynamic equations are not applicable, because ion transport in cells is out of equilibrium. Goldman’s experimental work was an attempt to reproduce some of the electrical properties of biological membranes in artificial systems. However, his work is mostly remembered (and cited) for the section in which he considered that ions move according to their concentration (diffusion) and voltage gradients and that the electric field through the membrane is constant, hence the name “constant-field equation.” Under these assumptions, and considering that the ion concentrations at the internal and external membrane boundaries are directly proportional to those in the aqueous solutions, he was able to arrive at an explicit solution of the Nernst–Planck (NP) equation (Nernst, 1888 Planck, 1890). The NP equation describes the contribution of concentration and voltage gradients to the current density carried by single ionic species. Today, we know it as the constant-field current equation (Eq. 1):

where Ii is the current density carried by ion i, Pi is the permeability coefficient, zi the valence of ion I, V is the membrane voltage, Ci,in and Ci,out are the ion concentrations in the solutions bathing either side of the membrane, and R, T, and F have their usual meaning. In this equation, a positive current carries charge from the inside to the outside and the membrane voltage is the internal minus the external electric potential. Goldman concluded that the theory could account reasonably well for the variation of the resting potential of the giant axon of the squid with external K + concentration as determined by Curtis and Cole (1942).

The Second World War interrupted the work of most scientists, and it was not until 1949 that Goldman’s analysis of the movement of ions across membranes reappeared in the literature. This time, it was invoked to show that Bernstein’s membrane hypothesis (Bernstein, 1902), although correct in the identification of the origin of the resting potential, was mistaken regarding the ionic basis of the action potential. Bernstein’s theory proposed that membrane excitation consisted of a large increase in membrane permeability to all ions, in which case the membrane potential would fall nearly to zero. Although in the late 1930s there were already squid axon experiments indicating that there was a transient reversal of the membrane potential during nerve activity (Hodgkin and Huxley, 1939), there was no clear theoretical basis to explain this result. Hodgkin and Katz (1949) took advantage of the Goldman current equation and obtained an expression for the membrane potential, Vr, in terms of the internal and external ion composition and the ion permeabilities: the Goldman–Hodgkin–Katz (GHK) voltage equation (Eq. 2, which is formally equivalent to Eq. 18 in Goldman’s 1943 paper).

The use of this equation allowed Hodgkin and Katz to show that under resting conditions, the K + permeability is 20 times larger than Na + permeability. During nerve activity, the situation is reversed, and Na + permeability becomes 20 times greater than the K + permeability at the peak of the action potential (Hodgkin and Katz, 1949). Notably, the experimental strategy used by Hodgkin and Katz (1949) to determine the changes in active membrane potential with external Na + is essentially the same as the one used by Hille almost 20 yr later to determine the ion selectivity of different ion channels (see Eq. 2 in Hodgkin and Katz, 1949).

The sodium hypothesis could not explain, however, the fact that crustacean muscle fibers elicit action potentials after replacement of the external sodium by tetraethylammonium (Fatt and Katz, 1953). Some years later, Fatt and Ginsborg (1958) found that an isotonic solution of Sr 2+ or Ba 2+ could sustain large crustacean muscle action potentials. To interpret the data obtained in crustacean muscle fibers, Fatt and Ginsborg (1958) expanded the constant-field theory so as to include divalent cations, arriving at the conclusion that divalent cations determined the peak of the action potential in these muscle fibers. The GHK voltage equation that incorporates a mixture of monovalent and divalent cations predicted that in this type of preparation the permeability of Ba 2+ at the peak of the action potential was 50 times larger than that of K + . This was the first indication of the existence of Ca 2+ action potentials and hence voltage-dependent Ca 2+ channels.

Hodgkin and Huxley’s dissection of the ionic currents that underlie the action potential (Hodgkin and Huxley, 1952) identified voltage-dependent Na + and K + currents in the giant axon of the squid. On the other hand, in the early 1960s, Baker et al. (1962) were able to internally perfuse giant squid axons. This made it possible to learn about the selectivity of the Na + conductance as the ionic composition of the internal medium could now be controlled. By having only K + in the internal medium and mixtures of Rb + and Cs + , or Li + as a replacement for external Na + . Chandler and Meves (1965) were able to determine the relative selectivity of the sodium channel for alkali-metal ions using the GHK equation. They found that Li + (1.1) > Na + (1) > K + (0.08) > Rb + (0.025) > Cs + (0.016). Importantly, preference for Na + of the Na + system allowed Chandler and Meves (1965) to make inferences about the molecular nature of the structures defining ion selectivity, which, when considered in light of the theory developed by Eisenman (1962), were not far from the mark of what we know now about these proteins. Having only Na + outside and K + inside of the axon reduced the GHK to a simpler expression known as the bi-ionic potential, which give us a direct measurement of the PNa+/PK+ permeability ratio when the concentrations on both sides of the membrane are equal. In fact, when reduced to the bi-ionic case, the GHK equation is the most convenient way of determining the ratio between the permeabilities of the permeant ions. We should mention here that in the case of a membrane that is selective to either positive or negative ions, the constant-field assumption is not required and the voltage equation for the bi-ionic case can be obtained through integration, using the NP equation and the appropriate boundary conditions (see Appendix B e.g., Sten-Knudsen, 1978).

The group of Hille (Hille, 1971, 1972, 1973 Dwyer et al., 1980 Wollmuth and Hille, 1992) determined the ion selectivity of several ion channels, including Na + , K + , HCN (Ih), and ACh receptor channels. Using a series of inorganic and organic ions of different size and by determining the permeability ratios relative to the reference ions (Na + , K + ) by applying the GHK equation, they were able to obtain the approximate dimensions of the selectivity filter of these channels. Because the internal concentration of the reference ions was unknown in the different preparations, the experimental strategy used was to determine first the reversal potential in the presence of the reference ion and then in the presence of the test cation. They obtained the permeability ratio between the test ion and the reference ion from the difference of these potentials because the GHK equation contains only the permeability coefficients and concentrations of the test and reference ions.

The GHK voltage equation can also be used to calculate the membrane voltage when there is a contribution to the total current from electrogenic pumps, as is the case, for example, of the Na + /K + pump or of electrogenic exchangers such as the Na + /Ca 2+ exchanger (e.g., Sjodin, 1980 Mullins, 1981 Armstrong, 2003).

A quick review of the JGP archives revealed that the GHK equation has also been used to determine the relative ion permeability in mechanotransducer channels (Beurg et al., 2015), in Ca 2+ -activated Cl − channels (Jeng et al., 2016), and in T-type Ca 2+ channels (Smith et al., 2017), to name just a few. In other words, after 74 yr, the GHK equation is still the theoretical framework to interpret the reversal potentials obtained in different ionic media. Despite the fact that several reviews and paper chapters dealt with the derivation of the constant-field equation, we thought it was of interest to discuss the theory underlying it and its approximations from an historical perspective.

### A historical note about diffusion and the NP equation

Michael Faraday, the founder of electrochemistry, studied the process of electrolysis and introduced the terms: electrode, ions, cations, and anions (Faraday, 1834). Adolf Fick stated the empirical laws of diffusion and Svante Arrhenius established that salts dissociate in solution resulting in positively charged cations and negatively charged anions (Fick, 1855 Arrhenius, 1887). Jacobus van ’t Hoff discovered that dilute solutions behaved very much like ideal gases and introduced the concept of osmotic pressure (van ’t Hoff, 1887). All these concepts and experimental data were necessary to set up the scenario that led to the so-called NP equation, which when integrated assuming a constant field, results in the Goldman current equation.

#### Nonelectrolyte diffusion

Walther Nernst (Nernst, 1888) compared the diffusion of gasses and ions in solution. Gas diffusion is much faster than ion diffusion, because ions undergo considerable friction when moving through a viscous medium. Therefore, the velocity of ion movement must be proportional to the force driving the diffusion. Nernst reformulated the Fick law of diffusion in physical terms. Nernst (1888) considered a slice in a diffusion cylinder of cross section q and height dx (Fig. 1). The volume of the slice is qdx. If there is an osmotic pressure difference dp across the slice, the force acting on the molecules in the slice is qdp. The number of moles contained in the slice is the concentration c times the volume qdx. Therefore, the force acting upon each mole, in N/mol, is The number of molecules crossing the area q is the number of molecules contained in a 1-cm-high cylinder times the velocity of the molecules. Nernst (1888) defined K as the force required to drive the molecules at a velocity of 1 cm −1 . Therefore, the velocity of the molecules will be (1/K)(1/c)dp/dx. On the other hand, the number of molecules, S, crossing area q in time interval Δt is the number of molecules contained in a cylinder of volume q times the distance they traveled in Δt at a velocity (1/K)(1/c)dp/dx:

## Contents

### What is a RMP? [ edit | edit source ]

Resting membrane potential is:

• the unequal distribution of ions on the both sides of the cell membrane
• the voltage difference of quiescent cells
• the membrane potential that would be maintained if there weren’t any stimuli or conducting impulses across it
• determined by the concentrations of ions on both sides of the membrane
• a negative value, which means that there is an excess of negative charge inside of the cell, compared to the outside.
• much depended on intracellular potassium level as the membrane permeability to potassium is about 100 times higher than that to sodium.

### Producing and maintaining RMP [ edit | edit source ]

RMP is produced and maintained by:

• Primary active transport – if it spends energy. This is how the Na + /K + ATPase pump functions.
• Secondary active transport – if it involves an electrochemical gradient. This is not involved in maintaining RMP.

### Ion affection of resting membrane potential [ edit | edit source ]

RMP is created by the distribution of ions and its diffusion across the membrane. Potassium ions are important for RMP because of its active transport, which increase more its concentration inside the cell. However, the potassium-selective ion channels are always open, producing an accumulation of negative charge inside the cell. Its outward movement is due to random molecular motion and continues until enough excess negative charge accumulates inside the cell to form a membrane potential.

### Na + /K + ATPase pump affection of the RMP [ edit | edit source ]

The Na + /K + ATPase pump creates a concentration gradient by moving 3Na + out of the cell and 2K + into the cell. Na + is being pumped out and K + pumped in against their concentration gradients. Because this pump is moving ions against their concentration gradients, it requires energy.

### Ion channels affection of resting membrane potential [ edit | edit source ]

The cell membrane contains protein channels that allow ions to diffuse passively without direct expenditure of metabolic energy. These channels allow Na + and K + to move across the cell membrane from a higher concentration toward a lower. As these channels have selectivity for certain ions, there are potassium- and sodium- selective ion channels. All cell membranes are more permeable to K + than to Na + because they have more K + channels than Na + .

### The Nernst Equation [ edit | edit source ]

Ihhs a mathematical equation applied in physiology, to calculate equilibrium potentials for certain ions.

• R = Gas Constant
• T = Absolute temperature (K)
• E = The potential difference across the membrane
• F = Faradays Constant (96,500 coulombs/mole)
• z = Valency of ion

### The Goldman-Hodgkin-Katz Equation [ edit | edit source ]

Is a mathematical equation applied in Physiology, to determine the potential across a cell's membrane, taking in account all the ions that are permeable through it.

• E = The potential difference across the membrane
• P = Permeability of the membrane to sodium or potassium
• [ ] = Concentration of sodium or potassium inside or outside

### Measuring resting potentials [ edit | edit source ]

In some cells, the RPM is always changing. For such, there is never any resting potential, which is only a theoretical concept. Other cells with membrane transport functions that change potential with time, have a resting potential. This can be measured by inserting an electrode into the cell. Transmembrane potentials can also be measured optically with dyes that change their optical properties according to the membrane potential.

## 4.4 Nernst equilibrium potential

The electrical activity generated in a neuron is a result of ions flowing across the neuron’s membrane which is caused by the following two principles: opposite charges attract, and concentration gradients seek to equalize. This potential difference is referred to as the membrane potential. In order for ions to flow, a concentration gradient must be established because the difference in concentration across the membrane leads it to pass either into the neuron or out of the neuron. This is accomplished by the sodium-potassium pump, which uses just below 10% of your body’s daily energy to pump three sodium ions out of the neuron for every two potassium ions pumped in, thus forming two respective concentration gradients.

With the concentration gradient established, the sodium and potassium ions will flow down the concentration gradients when their respective channels open, generating an electrical current that propagates down the axon. We must also take into account the fact that each ion possesses a charge–or charges in the case of Ca++ and Mg++–and that as this charge is built up on one side of the cell, this will generate an electrical force that will begin to repel ions with similar charge as they try to flow down their concentration gradient. When the force of the concentration gradient matches the electrical force attracting or repelling the ion, this is known as the Nernst potential for that ion, also referred to as the reversal potential. This means that means that both sodium and potassium possess their own respective Nernst potentials. Nernst potentials are especially important because they allow us to calculate the membrane voltage when a particular ion is in equilibrium, which helps to define the role it plays in an action potential.

The Nernst potential for an ion can be derived from the following equation: [E_ = fracln(frac<[out]><[in]>) ]

Expression Meaning
(E_) Nernst potential
R Gas constant: 8.314 (J/mol cdot K)
ln() Natural log
z Valence
T Temperature in Kelvin
[out] Extracellular ion concentration
[in] Intracellular ion concentration

While the Nernst potential will give the equilibrium point for a single ion, it also has a relation to the equilibrium potential or the resting potential the membrane, which is potential at which there is no net flow of ions, leading to a halt in the flow of electric current. The equilibrium potential is really a weighted average of all of the Nernst potentials and is modeled by the Goldman-Hodgkin-Katz equation which is shown below: [V_ = fracln(frac<>[K+]_+P_[Na+]_+P_[Cl-]_><>[K+]_+P_[Na+]_+P_[Cl-]_>) ]

This equation utilizes the membrane permeability, P, in conjunction with the concentration of each ion inside and outside of the cell to produce the equilibrium potential of a membrane. Using this equation alongside the Nernst potential, the driving force, which is a representation of the pressure for an ion to move in or out of the cell, can be calculated using the following equation: [DF = V_-E_ ]

The Nernst Potential, the Goldman-Hodgkin-Katz equation, and the driving force present necessary calculations that allow for better understanding of the flow of ions in relation to an action potential.

Worked Example:
Consider the following table of ion concentrations and relative permeabilities:

Ion Intracellular concentration (mM) Extracellular concentration (mM) Permeability
K+ 150 4 1
Na+ 15 145 0.05
Cl- 10 110 0.45

If the extracellular concentration of Na+ was increased by a factor of ten:
a. What is the new Nernst potential for sodium?
b. What is the resting potential of the neuron?

• R is the ideal gas constant: 8.314 (kg cdot m^2 cdot K^ <1>cdot mol^<-1>s^<-2>)
• T is the temperature in Kelvin: (310 K at human temperature)
• z is the valence. Here, we use +1 because the Na+ ion has a charge of +1.
• F is Faraday’s constant: 96.49 (kJ cdot V^ <-1>cdot mol^<-1>)
• Extracellular sodium is increased by a factor of 10: ([Na+]_) = 10 * 145 mM = 1450 mM.

From these values:
[E_ = frac<8.314 cdot 310> <1 cdot 96.49>cdot ln frac<1450><15>] [E_ = frac<2577.34> <96.49>cdot ln(96.67)] [E_ = 26.7109545 * (4.5713031) = 122.1 mV] Therefore, if the extracellular concentration of sodium was increased tenfold, the new Nernst potential for sodium would be 122.1 mV.

Solution, part b:
We will use the Goldman Hodgkin Katz (GHK) equation to find the resting potential of the neuron:
[V_ = fracln(frac<>[K+]_+P_[Na+]_+P_[Cl-]_><>[K+]_+P_[Na+]_+P_[Cl-]_>) ]

From the values given above:
[V_ = frac<8.314 cdot 310> <1 cdot 96.49>cdot ln frac<(.05*1450)+(1*4)+(.45*10)><(.05*15)+(1*150)+(.45*110)>]
[V_ = frac<2577.34> <96.49>cdot ln frac<72.5+4+4.5><75+150+49.5>] [V_ = 26.71099545 cdot ln frac<81> <200.25>= -24.18 mV]
If the extracellular concentration of sodium was increased by a factor of 10, the neuron’s resting potential would be -24.2 mV instead of -70 mV.

Worked Example: Consider the unicorn neuron’s resting potential. Due to their mythical identity and magic, the resting potential for a unicorn neuron is different than that of humans. Unlike humans, unicorns have a magic ion that alters their resting potentials. The relevant ion concentrations and permeabilities are found in the table below:

1. What is the Nernst potential for the magic ion of the unicorn neuron, assuming magic+ moves just as other ions do?
2. What is the resting potential of the unicorn neuron?
• R is the ideal gas constant: 8.314 (kg cdot m^2 cdot K^ <1>cdot mol^<-1>s^<-2>)
• T is the temperature in Kelvin: (310 K at human temperature)
• z is the valence. Here, we use +1 because the Magic+ ion has a charge of +1.
• F is Faraday’s constant: 96.49 (kJ cdot V^ <-1>cdot mol^<-1>)
• Extracellular sodium is increased by a factor of 10: ([Na+]_) = 10 * 145 mM = 1450 mM.

From these values:
[E_ = frac<8.314 cdot 310> <1 cdot 96.49>cdot ln frac<30><200>] [E_ = frac<2577.34> <96.49>cdot -1.90 = -50.8 mV] The Nernst potential of the magic+ ion is -50.8 mV. Thus the ionic and concentration gradients are at equilibrium for the magic+ ion when the potential difference of the neuron is -50.8 mV.

[V_ = 26.71099545 cdot ln frac<48.75> <420.25>= -57.3 mV] The resting potential of the unicorn neuron is -57.3 mV. This happens as a result of the permeabilities and concentrations of all four ions in this example, especially as a result of the magic+ ion’s high permeability. This makes the resting potential most similat to its Nernst potential.

The GHK voltage equation for /> positive ionic species and /> negative:

This results in the following if we consider a membrane separating two -solutions:

It is "Nernst-like" but has a term for each permeant ion. The Nernst equation can be considered a special case of the Goldman equation for only one ion:

• = The membrane potential
• = the permeability for that ion
• = the extracellular concentration of that ion
• = the intracellular concentration of that ion
• = The ideal gas constant
• = The temperature in kelvins

The first term, before the parenthesis, can be reduced to 61.5 log for calculations at human body temperature (37 C)

Note that the ionic charge determines the sign of the membrane potential contribution.

## [Physiology] Action Potential and the Nernst Equation

(1) Why isn't the membrane potential of sodium +60 mV as would be calculated using the Nernst equation? Looking at the AP graph in my book, I see it peaks out at +40 mV instead.

(2) Does the Nernst equation describes the membrane potential acquired when the electro-gradient is sufficient enough in strength to stop the concentration gradient?

(3) From what I understand, the Nernst equation describes the potential voltage attained when the cell is permeable to only one ion, right?

The Nernst equation is used to calculate the voltage, or charge differential, across a membrane, given that the membrane is permeable to one ion. This charge differential is similar to potential energy in its capacity to do "work". When we plug in the concentration values of sodium across the cell membrane into the Nernst equation, we find that the equilibrium voltage potential is +60 mV. This number is the membrane potential inside of the cell. Furthermore, I use the word “equilibrium”, given that the initial concentrations that we plug are derived from “normal” physiological conditions. Additionally, this is the voltage inside of the cell when the concentration gradient cancels out the opposing electro-gradient.

Now, in one of my books, it writes that the concentration gradient is actually stronger, in effect. That’s why we have ion leakage and require the sodium-potassium pump, to restore normal physiological conditions. That’s a bit confusing to me, because I thought this is the point where the electro cancels out the chemical gradient. If we have further leakage, then the electro isn’t powerful enough, just yet.

To be honest, this is frustrating. Everybody seems to have a conflicting idea of what the hell is going on.

I know this seems confusing - I'll try to explain as best as I can, but if anything is unclear please let me know!

The Nernst equation only gives you an accurate picture of the membrane potential if the system is in complete thermodynamic equilibrium as you've alluded to above (although in biology this usually means death), and only calculates potential based on one ion that is moving along its electrochemical gradient. In a real biological situation, it's often the case that neither of these assumptions are accurate. For example, your focus may be on sodium ions, but there are significant charge contributions of potassium and chlorine ions that influence sodium's electrochemical gradient, which are not accounted for by the Nernst equation. Moreover, there is a constant flux of ions across membranes against their electrochemical gradients (e.g. due to sodium/potassium pumps), which actively work to prevent equilibrium from being established - this too is not accounted for by the Nernst equation. The only time the Nernst equation might be applicable is if a particular type of channel dominates a membrane (or a section of membrane), effectively permitting the free flow of the ions it accommodates and enabling membrane potential at those areas to approximate the potential you would calculate from the Nernst equation (i.e. the Nernst potential).

In most cases, however, the measured resting membrane potential is closest to the Nernst potential of whichever ion the membrane is most permeable to. In neurons at rest, for instance, the membrane is most permeable to potassium (due to a predominance of open potassium leakage channels), so you would expect the resting membrane potential of such a neuron (usually about -70 mV) to be close to the Nernst potential of potassium (about -80 mV), which it is. The discrepancy comes from the fact that the membrane is still permeable to other ions, which influence each other's electrochemical gradients as I mentioned above.

Instead of the Nernst equation, the Goldman-Hodgkin-Katz equation is used to calculate resting membrane potential in biological systems. It's almost like a bulked up version of the Nernst equation but it takes into account the concentrations of as many different ions as youɽ like as well as their relative permeabilities through the membrane. For these reasons, the GHK equation provides a superior approximation of the resting membrane potential than the Nernst equation does. If you plugged in the proper values for sodium, potassium, and chlorine ions into the GHK equation, you would calculate the resting membrane potential to be about -70 mV.

To address your final point regarding the strength of the electrochemical (electrical+concentration) gradient (of potassium, I'm guessing), you need to picture what the flux of these ions across the membrane looks like. You're absolutely right that the electrochemical gradient dominates however, if nothing else was happening, the electrochemical gradient would eventually lead to equal concentrations of potassium ions on both sides of the membrane due to movement through the potassium leakage channels. However, this never happens - why? Because the sodium/potassium pumps are constantly pumping potassium ions back into the cell, against their electrochemical gradient, in order to prevent that gradient from disappearing and avoid a literally deadly thermodynamic equilibrium.

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Despite the enormous complexity of the brain, it is possible to obtain an understanding of its function by paying attention to two major details:

• First, the ways in which individual neurons, the components of the nervous system, are wired together to generate behavior.
• Second, the biophysical, biochemical, and electrophysiological properties of the individual neurons.

A good place to begin is with the components of the nervous system and how the electrical properties of the neurons endow nerve cells with the ability to process and transmit information.

1.1 Introduction to the Action Potential

Figure 1.1
Tap the colored circles (light stimulus) to activate.

Theories of the encoding and transmission of information in the nervous system go back to the Greek physician Galen (129-210 AD), who suggested a hydraulic mechanism by which muscles contract because fluid flowing into them from hollow nerves. The basic theory held for centuries and was further elaborated by René Descartes (1596 – 1650) who suggested that animal spirits flowed from the brain through nerves and then to muscles to produce movements (See this animation for modern interpretation of such a hydraulic theory for nerve function). A major paradigm shift occurred with the pioneering work of Luigi Galvani who found in 1794 that nerve and muscle could be activated by charged electrodes and suggested that the nervous system functions via electrical signaling. However, there was debate among scholars whether the electricity was within nerves and muscle or whether the nerves and muscles were simply responding to the harmful electric shock via some intrinsic nonelectric mechanism. The issue was not resolved until the 1930s with the development of modern electronic amplifiers and recording devices that allowed the electrical signals to be recorded. One example is the pioneering work of H.K. Hartline 80 years ago on electrical signaling in the horseshoe crab Limulus . Electrodes were placed on the surface of an optic nerve. (By placing electrodes on the surface of a nerve, it is possible to obtain an indication of the changes in membrane potential that are occurring between the outside and inside of the nerve cell.) Then 1-s duration flashes of light of varied intensities were presented to the eye first dim light, then brighter lights. Very dim lights produced no changes in the activity, but brighter lights produced small repetitive spike-like events. These spike-like events are called action potentials, nerve impulses, or sometimes simply spikes. Action potentials are the basic events the nerve cells use to transmit information from one place to another.

1.2 Features of Action Potentials

The recordings in the figure above illustrate three very important features of nerve action potentials. First, the nerve action potential has a short duration (about 1 msec). Second, nerve action potentials are elicited in an all-or-nothing fashion. Third, nerve cells code the intensity of information by the frequency of action potentials. When the intensity of the stimulus is increased, the size of the action potential does not become larger. Rather, the frequency or the number of action potentials increases. In general, the greater the intensity of a stimulus, (whether it be a light stimulus to a photoreceptor, a mechanical stimulus to the skin, or a stretch to a muscle receptor) the greater the number of action potentials elicited. Similarly, for the motor system, the greater the number of action potentials in a motor neuron, the greater the intensity of the contraction of a muscle that is innervated by that motor neuron.

Action potentials are of great importance to the functioning of the brain since they propagate information in the nervous system to the central nervous system and propagate commands initiated in the central nervous system to the periphery. Consequently, it is necessary to understand thoroughly their properties. To answer the questions of how action potentials are initiated and propagated, we need to record the potential between the inside and outside of nerve cells using intracellular recording techniques.

1.3 Intracellular Recordings from Neurons

The potential difference across a nerve cell membrane can be measured with a microelectrode whose tip is so small (about a micron) that it can penetrate the cell without producing any damage. When the electrode is in the bath (the extracellular medium) there is no potential recorded because the bath is isopotential. If the microelectrode is carefully inserted into the cell, there is a sharp change in potential. The reading of the voltmeter instantaneously changes from 0 mV, to reading a potential difference of -60 mV inside the cell with respect to the outside. The potential that is recorded when a living cell is impaled with a microelectrode is called the resting potential, and varies from cell to cell. Here it is shown to be -60 mV, but can range between -80 mV and -40 mV, depending on the particular type of nerve cell. In the absence of any stimulation, the resting potential is generally constant.

It is also possible to record and study the action potential. Figure 1.3 illustrates an example in which a neuron has already been impaled with one microelectrode (the recording electrode), which is connected to a voltmeter. The electrode records a resting potential of -60 mV. The cell has also been impaled with a second electrode called the stimulating electrode. This electrode is connected to a battery and a device that can monitor the amount of current (I) that flows through the electrode. Changes in membrane potential are produced by closing the switch and by systematically changing both the size and polarity of the battery. If the negative pole of the battery is connected to the inside of the cell as in Figure 1.3A, an instantaneous change in the amount of current will flow through the stimulating electrode, and the membrane potential becomes transiently more negative. This result should not be surprising. The negative pole of the battery makes the inside of the cell more negative than it was before. A change in potential that increases the polarized state of a membrane is called a hyperpolarization. The cell is more polarized than it was normally. Use yet a larger battery and the potential becomes even larger. The resultant hyperpolarizations are graded functions of the magnitude of the stimuli used to produce them.

Now consider the case in which the positive pole of the battery is connected to the electrode (Figure 1.3B). When the positive pole of the battery is connected to the electrode, the potential of the cell becomes more positive when the switch is closed (Figure 1.3B). Such potentials are called depolarizations. The polarized state of the membrane is decreased. Larger batteries produce even larger depolarizations. Again, the magnitude of the responses are proportional to the magnitude of the stimuli. However, an unusual event occurs when the magnitude of the depolarization reaches a level of membrane potential called the threshold. A totally new type of signal is initiated the action potential. Note that if the size of the battery is increased even more, the amplitude of the action potential is the same as the previous one (Figure 1.3B). The process of eliciting an action potential in a nerve cell is analogous to igniting a fuse with a heat source. A certain minimum temperature (threshold) is necessary. Temperatures less than the threshold fail to ignite the fuse. Temperatures greater than the threshold ignite the fuse just as well as the threshold temperature and the fuse does not burn any brighter or hotter.

If the suprathreshold current stimulus is long enough, however, a train of action potentials will be elicited. In general, the action potentials will continue to fire as long as the stimulus continues, with the frequency of firing being proportional to the magnitude of the stimulus (Figure 1.4).

Action potentials are not only initiated in an all-or-nothing fashion, but they are also propagated in an all-or-nothing fashion. An action potential initiated in the cell body of a motor neuron in the spinal cord will propagate in an undecremented fashion all the way to the synaptic terminals of that motor neuron. Again, the situation is analogous to a burning fuse. Once the fuse is ignited, the flame will spread to its end.

1.4 Components of the Action Potentials

The action potential consists of several components (Figure 1.3B). The threshold is the value of the membrane potential which, if reached, leads to the all-or-nothing initiation of an action potential. The initial or rising phase of the action potential is called the depolarizing phase or the upstroke. The region of the action potential between the 0 mV level and the peak amplitude is the overshoot. The return of the membrane potential to the resting potential is called the repolarization phase. There is also a phase of the action potential during which time the membrane potential can be more negative than the resting potential. This phase of the action potential is called the undershoot or the hyperpolarizing afterpotential. In Figure 1.4, the undershoots of the action potentials do not become more negative than the resting potential because they are "riding" on the constant depolarizing stimulus.

1.5 Ionic Mechanisms of Resting Potentials

Before examining the ionic mechanisms of action potentials, it is first necessary to understand the ionic mechanisms of the resting potential. The two phenomena are intimately related. The story of the resting potential goes back to the early 1900's when Julius Bernstein suggested that the resting potential (Vm) was equal to the potassium equilibrium potential (EK). Where

The key to understanding the resting potential is the fact that ions are distributed unequally on the inside and outside of cells, and that cell membranes are selectively permeable to different ions. K + is particularly important for the resting potential. The membrane is highly permeable to K + . In addition, the inside of the cell has a high concentration of K + ([K + ]i) and the outside of the cell has a low concentration of K + ([K + ]o). Thus, K + will naturally move by diffusion from its region of high concentration to its region of low concentration. Consequently, the positive K + ions leaving the inner surface of the membrane leave behind some negatively charged ions. That negative charge attracts the positive charge of the K + ion that is leaving and tends to "pull it back". Thus, there will be an electrical force directed inward that will tend to counterbalance the diffusional force directed outward. Eventually, an equilibrium will be established the concentration force moving K + out will balance the electrical force holding it in. The potential at which that balance is achieved is called the Nernst Equilibrium Potential.

An experiment to test Bernstein's hypothesis that the membrane potential is equal to the Nernst Equilibrium Potential (i.e., Vm = EK) is illustrated to the left.

The K + concentration outside the cell was systematically varied while the membrane potential was measured. Also shown is the line that is predicted by the Nernst Equation. The experimentally measured points are very close to this line. Moreover, because of the logarithmic relationship in the Nernst equation, a change in concentration of K + by a factor of 10 results in a 60 mV change in potential.

Note, however, that there are some deviations in the figure at left from what is predicted by the Nernst equation. Thus, one cannot conclude that Vm = EK. Such deviations indicate that another ion is also involved in generating the resting potential. That ion is Na + . The high concentration of Na + outside the cell and relatively low concentration inside the cell results in a chemical (diffusional) driving force for Na + influx. There is also an electrical driving force because the inside of the cell is negative and this negativity attracts the positive sodium ions. Consequently, if the cell has a small permeability to sodium, Na + will move across the membrane and the membrane potential would be more depolarized than would be expected from the K + equilibrium potential.

1.6 Goldman-Hodgkin and Katz (GHK) Equation

When a membrane is permeable to two different ions, the Nernst equation can no longer be used to precisely determine the membrane potential. It is possible, however, to apply the GHK equation. This equation describes the potential across a membrane that is permeable to both Na + and K + .

Note that α is the ratio of Na + permeability (PNa) to K + permeability (PK). Note also that if the permeability of the membrane to Na + is 0, then alpha in the GHK is 0, and the Goldman-Hodgkin-Katz equation reduces to the Nernst equilibrium potential for K + . If the permeability of the membrane to Na + is very high and the potassium permeability is very low, the [Na + ] terms become very large, dominating the equation compared to the [K + ] terms, and the GHK equation reduces to the Nernst equilibrium potential for Na + .

If the GHK equation is applied to the same data in Figure 1.5, there is a much better fit. The value of alpha needed to obtain this good fit was 0.01. This means that the potassium K + permeability is 100 times the Na + permeability. In summary, the resting potential is due not only to the fact that there is a high permeability to K + . There is also a slight permeability to Na + , which tends to make the membrane potential slightly more positive than it would have been if the membrane were permeable to K + alone.

1.7 Membrane Potential Laboratory

Click here to go to the interactive Membrane Potential Laboratory to experiment with the effects of altering external or internal potassium ion concentration and membrane permeability to sodium and potassium ions. Predictions are made using the Nernst and the Goldman, Hodgkin, Katz equations.

 Membrane Potential Laboratory

If a nerve membrane suddenly became equally permeable to both Na + and K + , the membrane potential would:

A. Not change

B. Approach the new K + equilibrium potential

C. Approach the new Na + equilibrium potential

D. Approach a value of about 0 mV

E. Approach a constant value of about +55 mV

If a nerve membrane suddenly became equally permeable to both Na + and K + , the membrane potential would:

A. Not change This answer is INCORRECT.

A change in permeability would depolarize the membrane potential since alpha in the GHK equation would equal one. Initially, alpha was 0.01. Try substituting different values of alpha into the GHK equation and calculate the resultant membrane potential.

B. Approach the new K + equilibrium potential

C. Approach the new Na + equilibrium potential

D. Approach a value of about 0 mV

E. Approach a constant value of about +55 mV

If a nerve membrane suddenly became equally permeable to both Na + and K + , the membrane potential would:

A. Not change

B. Approach the new K + equilibrium potential This answer is INCORRECT.

The membrane potential would approach the K+ equilibrium potential only if the Na + permeability was decreased or the K + permeability was increased. Also there would be no "new" equilibrium potential. Changing the permeability does not change the equilibrium potential.

C. Approach the new Na + equilibrium potential

D. Approach a value of about 0 mV

E. Approach a constant value of about +55 mV

If a nerve membrane suddenly became equally permeable to both Na + and K + , the membrane potential would:

A. Not change

B. Approach the new K + equilibrium potential

C. Approach the new Na + equilibrium potential This answer is INCORRECT.

The membrane potential would approach the Na + equilibrium potential only if alpha in the GHK equation became very large (e.g., decrease PK or increase PNa). Also, there would be no "new" Na + equilibrium potential. Changing the permeability does not change the equilibrium potential it changes the membrane potential.

D. Approach a value of about 0 mV

E. Approach a constant value of about +55 mV

If a nerve membrane suddenly became equally permeable to both Na + and K + , the membrane potential would:

A. Not change

B. Approach the new K + equilibrium potential

C. Approach the new Na + equilibrium potential

D. Approach a value of about 0 mV This answer is CORRECT!

Roughly speaking, the membrane potential would move to a value half way between EK and ENa. The GHK equation could be used to determine the precise value.

E. Approach a constant value of about +55 mV

If a nerve membrane suddenly became equally permeable to both Na + and K + , the membrane potential would:

A. Not change

B. Approach the new K + equilibrium potential

C. Approach the new Na + equilibrium potential

D. Approach a value of about 0 mV

E. Approach a constant value of about +55 mV This answer is INCORRECT.

The membrane potential would not approach a value of about +55 mV (the approximate value of ENa) unless there was a large increase in the sodium permeability without a corresponding change in the potassium permeability. Alpha in the Goldman equation would need to approach a very high value.

If the concentration of K + in the cytoplasm of an invertebrate axon is changed to a new value of 200 mM (Note: for this axon normal [K]o = 20 mM and normal [K]i = 400 mM):

A. The membrane potential would become more negative

B. The K + equilibrium potential would change by 60 mV

C. The K + equilibrium potential would be about -60 mV

D. The K + equilibrium potential would be about -18 mV

E. An action potential would be initiated

If the concentration of K + in the cytoplasm of an invertebrate axon is changed to a new value of 200 mM (Note: for this axon normal [K]o = 20 mM and normal [K]i = 400 mM):

A. The membrane potential would become more negative This answer is INCORRECT.

The normal value of extracellular potassium is 20 mM and the normal value of intracellular potassium is 400 mM, yielding a normal equilibrium potential for potassium of about -75 mV. If the intracellular concentration is changed from 400 mM to 200 mM, then the potassium equilibrium potential as determined by the Nernst equation, will equal about -60 mV. Since the membrane potential is normally -60 mV and is dependent, to a large extent, on EK, the change in the potassium concentrationand hence EK would make the membrane potential more positive, not more negative.

B. The K + equilibrium potential would change by 60 mV

C. The K + equilibrium potential would be about -60 mV

D. The K + equilibrium potential would be about -18 mV

E. An action potential would be initiated

If the concentration of K + in the cytoplasm of an invertebrate axon is changed to a new value of 200 mM (Note: for this axon normal [K]o = 20 mM and normal [K]i = 400 mM):

A. The membrane potential would become more negative

B. The K + equilibrium potential would change by 60 mV This answer is INCORRECT.

The potassium equilibrium potential would not change by 60 mV. The potassium concentration was changed just from 400 mM to 200 mM. One can use the Nernst equation to determine the exact value that the equilibrium potential would change by. It was initially about -75 mV and as a result of the change in concentration, the equilibrium potential becomes -60 mV. Thus, the equilibrium potential does not change by 60 mV, it changes by about 15 mV.

C. The K + equilibrium potential would be about -60 mV

D. The K + equilibrium potential would be about -18 mV

E. An action potential would be initiated

If the concentration of K + in the cytoplasm of an invertebrate axon is changed to a new value of 200 mM (Note: for this axon normal [K]o = 20 mM and normal [K]i = 400 mM):

A. The membrane potential would become more negative

B. The K + equilibrium potential would change by 60 mV

C. The K + equilibrium potential would be about -60 mV This answer is CORRECT!

This is the correct answer. See the logic described in responses A and B.

D. The K + equilibrium potential would be about -18 mV

E. An action potential would be initiated

If the concentration of K + in the cytoplasm of an invertebrate axon is changed to a new value of 200 mM (Note: for this axon normal [K]o = 20 mM and normal [K]i = 400 mM):

A. The membrane potential would become more negative

B. The K + equilibrium potential would change by 60 mV

C. The K + equilibrium potential would be about -60 mV

D. The K + equilibrium potential would be about -18 mV This answer is INCORRECT.

Using the Nernst equation, the new potassium equilibrium potential can be calculated to be -60 mV. A value of -18 mV would be calculated if you substituted [K]o = 200 and [K]i= 400 into the Nernst equation.

E. An action potential would be initiated

If the concentration of K + in the cytoplasm of an invertebrate axon is changed to a new value of 200 mM (Note: for this axon normal [K]o = 20 mM and normal [K]i = 400 mM):

A. The membrane potential would become more negative

B. The K + equilibrium potential would change by 60 mV

C. The K + equilibrium potential would be about -60 mV

D. The K + equilibrium potential would be about -18 mV

E. An action potential would be initiated This answer is INCORRECT.

The membrane potential would not depolarize sufficiently to reach threshold (about -45 mV).

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