3
Subthreshold membrane phenomena
3.1 INTRODUCTION
In the previous chapter the subthreshold behavior of the nerve cell was discussed qualitatively. This chapter describes the physiological basis of the resting voltage and the subthreshold response of an axon to electric stimuli from a quantitativeperspective.
The membrane plays an important role in establishing the resting and active electric properties of an excitable cell, through its regulation of the movement of ions between the extracellular and intracellular spaces. The word ion (Greek for “that which goes”) was introduced by Faraday (1834). The ease with which an ion crosses the membrane, namely the membrane permeability, differs among ion species; this selective permeability will be seen to have important physiological consequences. Activation of a cell affects its behavior by altering these permeabilities. Another important consideration for transmembrane ion movement is the fact that the ionic composition inside the cell differs greatly from that outside the cell. Consequently, concentration gradients exist for all permeable ions that contribute to the net ion movement or flux. The principle whereby ions flow from regions of high to low concentration is called diffusion.One consequence of this ion flow is the tendency for ions to accumulate at the inner and outer membrane surfaces, a process by which an electric field is established within the membrane. This field exerts forces on the ions crossing the membrane since the latter carry an electric charge. Thus to describe membrane ion movements, electricfield forces as well as diffusional forces should be considered. Equilibrium is attained when the diffusional force balances the electric field force for all permeable ions.For a membrane that is permeable to only one type of ion, equilibrium requires that the force due to the electric field be equal and opposite to the force due to diffusion. In the next section we shall explore the Nernst equation, which expresses the equilibrium voltage associated with a given concentration ratio. Equilibrium can also be defined by equating the electrochemical potential on both sides of the membrane.The Nernst equation is derived from two basic concepts involving ionic flow – those resulting from an electric field force and those resulting from a diffusional force. A more rigorous thermodynamic treatment is available, and the interested reader should consult references such as van Rysselberghe (1963) and Katchalsky and Curran (1965).We shall also derive the GoldmanHodgkinKatz equation, which gives the steadystate value of the membrane voltage when there are several types of ions in the intracellular and extracellular media, and when the membrane is permeable to all of them. As will be seen, the GoldmanHodgkinKatz equation is a straightforward extension of the Nernst equation.A more detailed discussion of physical chemistry, which contributes to many topics in this chapter, can be found in standard textbooks such as Edsall and Wyman (1958) and Moore (1972).
3.2 NERNST EQUATION
3.2.1 Electric Potential and Electric Field
In electrostatics the electric potential Φ at point P is defined as the work required to move a unit positive charge from a reference position O to position P. If the reference potential is Φ_{O} and the potential at point P designated Φ_{P}, then the work W_{e}, required to move a quantity of charge Qfrom point O to point P is simply
W_{e} = Q(Φ_{P} – Φ_{O})  (3.1) 
where  W_{e}  = work [J] 
Q  = charge [C] (coulombs)  
Φ  = potential [V] 
In electrophysiological problems the quantity of ions is usually expressed in moles. (One mole equals the molecular weight in gramshence 6.0225 × 10²³, Avogadro’s number of molecules.) If one mole of an ion is transferred from a reference point O at potential Φ_{O} to an arbitrary point P at potential Φ_{P}, then from Equation 3.1 the required work is
W_{e} = zF(Φ_{P} – Φ_{O})  (3.2) 
where  W_{e}  = work [J/mol] 
z  = valence of the ions  
F  = Faraday’s constant [9.649 × 10^{4} C/mol]  
Φ  = potential [V] 
Faraday’s constant converts quantity of moles to quantity of charge for a univalent ion. The factor z, called valence, takes into account multivalent ions and also introduces the sign. Note that if Φ_{P} – Φ_{O} and z are both positive (i.e., the case where a positive charge is moved from a lower to higher potential), then work must be done, and W_{e} is positive as expected.
The electric field is defined by the force that it exerts on a unit charge. If a unit positive charge is moved from reference point O to a nearby point P, where the corresponding vector displacement is d, then the work done against the electric field force , according to the basic laws of mechanics, is the work dW given by
(3.3) 
Applying Equation 3.1 to Equation 3.3 (replacing Q by unity) gives:
(3.4) 
The Taylor series expansion of the scalar field about the point O and along the path s is:
(3.5) 
Since P is very close to O, the remaining higher terms may be neglected in Equation 3.5. The second term on the righthand side of Equation 3.5 is known as the directional derivative of Φ in the direction s. The latter, by the vectoranalytic properties of the gradient, is given by . Consequently, Equation 3.5 may be written as
(3.6) 
From Equations 3.4 and 3.6 we deduce that
(3.7) 
This relationship is valid not only for electrostatics but also for electrophysiological problems since quasistatic conditions are known to apply to the latter (see Section 8.2.2).
According to Ohm’s law, current density and electric field are related by
(3.8) 
where σ is the conductivity of the medium. This current, for obvious reasons, is called a conduction current.
We are interested mainly in those charged particles that arise from ionization in an electrolyte and, in particular, in those ions present in the intracellular and extracellular spaces in electrically excitable tissues. Because of their charges, these ions are subject to the electric field forces summarized above. The flux (i.e., flow per unit area per unit time) that results from the presence of an electric field depends on the electric resistance, which, in turn, is a function of the ionic mobility of the ionic species. The latter is defined by uk, the velocity that would be achieved by the kth ion in a unit electric field. Then the ionic flux is given by
(3.9) 
where  _{ke}  = ionic flux (due to electric field) [mol/(cm²·s)] 
u_{k}  = ionic mobility [cm²/(V·s)]  
z_{k}  = valence of the ion  
c_{k}  = ionic concentration [mol/cm³] 
and further:
=  the sign of the force (positive for cations and negative for anions)  
=  the mean velocity achieved by these ions in a unit electric field (according to the definition of u_{k})  
the subscript k denotes the k^{th} ion. 
Multiplying ionic concentration c_{k} by velocity gives the ionic flux. A comparison of Equation 3.8 with Equation 3.9 shows that the mobility is proportional to the conductivity of the k^{th} ion in the electrolyte. The ionic mobility depends on the viscosity of the solvent and the size and charge of the ion.
3.2.2 Diffusion
If a particular ionic concentration is not uniform in a compartment, redistribution occurs that ultimately results in a uniform concentration. To accomplish this, flow must necessarily take place from high to lowdensity regions. This process is called diffusion, and its quantitative description is expressed by Fick’s law (Fick, 1855). For the k^{th} ion species, this is expressed as
(3.10) 
where  _{kD}  = ionic flux (due to diffusion) [mol/(cm²·s)] 
D_{k}  = Fick’s constant (diffusion constant) [cm²/s]  
c_{k}  = ion concentration [mol/cm³] 
This equation describes flux in the direction of decreasing concentration (accounting for the minus sign), as expected.
Fick’s constant relates the “force” due to diffusion (i.e., –c_{k }) to the consequent flux of the kth substance. In a similar way the mobility couples the electric field force (Φ) to the resulting ionic flux. Since in each case the flux is limited by the same factors (collision with solvent molecules), a connection between u_{k} and D_{k} should exist. This relationship was worked out by Nernst (1889) and Einstein (1905) and is
(3.11) 
where  T  = absolute temperature [K] 
R  = gas constant [8.314 J/(mol·K)] 
3.2.3 NernstPlanck Equation
The total ionic flux for the k^{th} ion, _{k }, is given by the sum of ionic fluxes due to diffusion and electric field of Equations 3.10 and 3.9. Using the Einstein relationship of Equation 3.11, it can be expressed as
(3.12) 
Equation 3.12 is known as the NernstPlanck equation (after Nernst, 1888, 1889; Planck, 1890ab). It describes the flux of the k^{th} ion under the influence of both a concentration gradient and an electric field. Its dimension depends on those used to express the ionic concentration and the velocity. Normally the units are expressed as [mol/(cm²·s)].
The ionic flux can be converted into an electric current density by multiplying the former by zF, the number of charges carried by each mole (expressed in coulombs, [C]). The result is, for the k^{th} ion,
(3.13) 
where  _{k}  = electric current density due to the k^{th} ion [C/(s·cm²)] = [A/cm²] 
Using Equation 3.11, Equation 3.13 may be rewritten as
(3.14) 
3.2.4 Nernst Potential
Figure 3.1 depicts a small portion of a cell membrane of an excitable cell (i.e., a nerve or muscle cell). The membrane element shown is described as a patch. The significant ions are potassium (K^{+}), sodium (Na^{+}), and chloride (Cl^{–}), but we shall assume that the membrane is permeable only to one of them (potassium) which we denote as the k^{th} ion, to allow later generalization. The ion concentrations on each side of the membrane are also illustrated schematically in Figure 3.1. At the sides of the figure, the sizes of the symbols are given in proportion to the corresponding ion concentrations. The ions are shown to cross the membrane through channels, as noted above. The number of ions flowing through an open channel may be more than 10^{6} per second.
It turns out that this is a reasonable approximation to actual conditions at rest. The concentration of potassium is normally around 30 – 50 times greater in the intracellular space compared to the extracellular. As a consequence, potassium ions diffuse outward across the cell membrane, leaving behind an equal number of negative ions (mainly chloride). Because of the strong electrostatic attraction, as the potassium efflux takes place, the potassium ions accumulate on the outside of the membrane. Simultaneously, (an equal number of) chloride ions (left behind from the KCl) accumulate on the inside of the membrane. In effect, the membrane capacitance is in the process of charging, and an electric field directed inward increasingly develops in proportion to the net potassium efflux.The process described above does not continue indefinitely because the increasing electric field forms a force on the permeable potassium ion that is directed inward and, hence, opposite to the diffusional force. An equilibrium is reached when the two forces are equal in magnitude. The number of the potassium ions required to cross the membrane to bring this about is ordinarily extremely small compared to the number available. Therefore, in the above process for all practical purposes we may consider the intracellular and extracellular concentrations of the potassium ion as unchanging throughout the transient. The transmembrane potential achieved at equilibrium is simply the equilibrium potential.A quantitative relationship between the potassium ion concentrations and the aforementioned equilibrium potential can be derived from the NernstPlanck equation. To generalize the result, we denote the potassium ion as the k^{th} ion. Applying Equation 3.13 to the membrane at equilibrium we must satisfy a condition of zero current so that
(3.15) 
where the subscript k refers to an arbitrary k^{th} ion. Transposing terms in Equation 3.15 gives
(3.16) 
Since the membrane is extremely thin, we can consider any small patch as planar and describe variations across it as onedimensional (along a normal to the membrane). If we call this direction x, we may write out Equation 3.16 as
(3.17) 
Equation 3.17 can be rearranged to give
(3.18) 
Equation 3.18 may now be integrated from the intracellular space (i) to the extracellular space (o); that is:
(3.19) 
Carrying out the integrations in Equation 3.19 gives
(3.20) 
where c_{i,k} and c_{o,k} denote the intracellular and extracellular concentrations of the k^{th} ion, respectively. The equilibrium voltage across the membrane for the k^{th} ion is, by convention, the intracellular minus the extracellular potential (V_{k} = Φ_{i} – Φ_{o}), hence:
(3.21) 
where  V_{k}  = equilibrium voltage for the k^{th} ion across the membrane Φ_{i} – Φ_{o} i.e., the Nernst voltage [V] 
R  = gas constant [8.314 J/(mol·K)]  
T  = absolute temperature [K]  
z_{k}  = valence of the k^{th} ion  
F  = Faraday’s constant [9.649 × 104 C/mol]  
c_{i,k}  = intracellular concentration of the kth ion  
c_{o,k}  = extracellular concentration of the kth ion 
Equation 3.21 is the famous Nernst equation derived by Walther Hermann Nernst in 1888 (Nernst, 1888). By Substituting 37 °C which gives T = 273 + 37 and +1 for the valence, and by replacing the natural logarithm (the Napier logarithm) with the decadic logarithm (the Briggs logarithm), one may write the Nernst equation for a monovalent cation as:
(3.22) 
At room temperature (20 °C), the coefficient in Equation 3.22 has the value of 58; at the temperature of seawater (6 °C), it is 55. The latter is important when considering the squid axon.
Example
We discuss the subject of equilibrium further by means of the example described in Figure 3.2, depicting an axon lying in a cylindrical experimental chamber. The potential inside the axon may be changed with three interchangeable batteries (A, B, and C) which may be placed between the intracellular and extracellular spaces. We assume that the intracellular and the extracellular spaces can be considered isopotential so that the transmembrane voltage V_{m} (difference of potential across the membrane) is the same everywhere. (This technique is called voltage clamp, and explained in more detail in Section 4.2.) Furthermore, the membrane is assumed to be permeable only to potassium ions. The intracellular and extracellular concentrations of potassium are c_{i,K} and c_{o,K}, respectively. In the resting state, the membrane voltage V_{m} (= Φ_{i} – Φ_{o}) equals V_{K}, the Nernst voltage for K^{+} ions according to Equation 3.21.In Figure 3.2 the vertical axis indicates the potential Φ, and the horizontal axis the radial distance r measured from the center of the axon. The membrane is located between the radial distance values r_{i} and r_{o}. The length of the arrows indicates the magnitude of the voltage (inside potential minus outside potential). Their direction indicates the polarity so that upward arrows represent negative, and downward arrows positive voltages (because all the potential differences in this example are measured from negative potentials). Therefore, when ΔV is positive (downward), the transmembrane current (for a positive ion) is also positive (i.e., outward).A. Suppose that the electromotive force emf of the battery A equals V_{K}. In this case V_{m} = V_{K} and the condition corresponds precisely to the one where equilibrium between diffusion and electric field forces is achieved. Under this condition no net flow of potassium ions exists through the membrane (see Figure 3.2A). (The flow through the membrane consists only of diffusional flow in both directions.)B. Suppose, now, that the voltage of battery B is smaller than V_{K} (V_{m} < V_{K}). Then the potential inside the membrane becomes less negative, a condition known as depolarization of the membrane. Now the electric field is no longer adequate to equilibrate the diffusional forces. This imbalance is ΔV = V_{m} – V_{K} and an outflow of potassium (from a higher electrochemical potential to a lower one) results. This condition is illustrated in Figure 3.2B.C. If, on the other hand, battery C is selected so that the potential inside the membrane becomes more negative than in the resting state (V_{m} > V_{K}), then the membrane is said to be hyperpolarized. In this case ions will flow inward (again from the higher electrochemical potential to the lower one). This condition is described in Figure 3.2C.
 (A) equilibrium at rest,(B) depolarized membrane, and(C) hyperpolarized membrane.The diffusional force arising from the concentration gradient is equal and opposite to the equilibrium electric field
V_{K}
 which, in turn, is calculated from the Nernst potential (see Equation 3.21). The Nernst electric field force
V_{K}
 is described by the open arrow. The thin arrow describes the actual electric field
V_{m}
 across the membrane that is imposed when the battery performs a voltage clamp (see Section 4.2 for the description of voltage clamp). The bold arrow is the net electric field driving force Δ
V
 in the membrane resulting from the difference between the actual electric field (thin arrow) and the equilibrium electric field (open arrow).
3.3 ORIGIN OF THE RESTING VOLTAGE
The resting voltage of a nerve cell denotes the value of the membrane voltage (difference between the potential inside and outside the membrane) when the neuron is in the resting state in its natural, physiological environment. It should be emphasized that the resting state is not a passive state but a stable active state that needs metabolic energy to be maintained. Julius Bernstein, the founder of membrane theory, proposed a very simple hypothesis on the origin of the resting voltage, depicted in Figure 3.3 (Bernstein, 1902; 1912). His hypothesis is based on experiments performed on the axon of a squid, in which the intracellular ion concentrations are, for potassium, c_{i,K} = 400 mol/m³; and, for sodium, c_{i,Na} = 50 mol/m³. It is presumed that the membrane is permeable to potassium ions but fully impermeable to sodium ions.The axon is first placed in a solution whose ion concentrations are the same as inside the axon. In such a case the presence of the membrane does not lead to the development of a difference of potential between the inside and outside of the cell, and thus the membrane voltage is zero.The axon is then moved to seawater, where the potassium ion concentration is c_{o,K} = 20 mol/m³ and the sodium ion concentration is c_{o,Na} = 440 mol/m³. Now a concentration gradient exists for both types of ions, causing them to move from the region of higher concentration to the region of lower concentration. However, because the membrane is assumed to be impermeable to sodium ions, despite the concentration gradient, they cannot move through the membrane. The potassium ions, on the other hand, flow from inside to outside. Since they carry a positive charge, the inside becomes more negative relative to the outside. The flow continues until the membrane voltage reaches the corresponding potassium Nernst voltage – that is, when the electric and diffusion gradients are equal (and opposite) and equilibrium is achieved. At equilibrium the membrane voltage is calculated from the Nernst equation (Equation 3.21).The hypothesis of Bernstein is, however, incomplete, because the membrane is not fully impermeable to sodium ions. Instead, particularly as a result of the high electrochemical gradient, some sodium ions flow to the inside of the membrane. Correspondingly, potassium ions flow, as described previously, to the outside of the membrane. Because the potassium and sodium Nernst voltages are unequal, there is no membrane voltage that will equilibrate both ion fluxes. Consequently, the membrane voltage at rest is merely the value for which a steadystate is achieved (i.e.,where the sodium influx and potassium efflux are equal). The steady resting sodium influx and potassium efflux would eventually modify the resting intracellular concentrations and affect the homeostatic conditions; however, the NaK pump, mentioned before, transfers the sodium ions back outside the membrane and potassium ions back inside the membrane, thus keeping the ionic concentrations stable. The pump obtains its energy from the metabolism of the cell..
 The origin of the resting voltage according to Julius Bernstein.
3.4 MEMBRANE WITH MULTIION PERMEABILITY
3.4.1 Donnan Equilibrium
The assumption that biological membranes are permeable to a single ion only is not valid, and even low permeabilities may have an important effect. We shall assume that when several permeable ions are present, the flux of each is independent of the others (an assumption known as the independence principle and formulated by Hodgkin and Huxley (1952a)). This assumption is supported by many experiments.The biological membrane patch can be represented by the model drawn in Figure 3.4, which takes into account the primary ions potassium, sodium, and chloride. If the membrane potential is V_{m}, and since V_{k} is the equilibrium potential for the k^{th} ion, then (V_{m} – V_{k}) evaluates the net driving force on the k^{th} ion. Considering potassium (K), for example, the net driving force is given by (V_{m} – V_{K}); here we can recognize that V_{m} represents the electric force and V_{K} the diffusional force (in electric terms) on potassium. When V_{m} = V_{K} ,the net force is zero and there is no flux since the potential is the same as the potassium equilibrium potential. The reader should recall, that V_{K} is negative; thus if V_{m} – V_{K} is positive, the electric field force is less than the diffusional force, and a potassium efflux (a positive transmembrane current) results, as explained in the example given in Section 3.2.4.The unequal intracellular and extracellular composition arises from active transport (NaK pump) which maintains this imbalance (and about which more will be said later). We shall see that despite the membrane ion flux, the pump will always act to restore normal ionic composition. Nevertheless, it is of some interest to consider the end result if the pump is disabled (a consequence of ischemia, perhaps). In this case, very large ion movements will ultimately take place, resulting in changed ionic concentrations. When equilibrium is reached, every ion is at its Nernst potential which, of course, is also the common transmembrane potential. In fact, in view of this common potential, the required equilibrium concentration ratios must satisfy Equation 3.23 (derived from Equation 3.21)
(3.23) 
Note that Equation 3.23 reflects the fact that all ions are univalent and that chloride is negative. The condition represented by Equation 3.23 is that all ions are in equilibrium; it is referred to as the Donnan equilibrium.
3.4.2 The Value of the Resting Voltage, GoldmanHodgkinKatz Equation
The relationship between membrane voltage and ionic flux is of great importance. Research on this relationship makes several assumptions: first, that the biological membrane is homogeneous and neutral (like very thin glass); and second, that the intracellular and extracellular regions are completely uniform and unchanging. Such a model is described as an electrodiffusion model. Among these models is that by GoldmanHodgkinKatz which is described in this section.In view of the very small thickness of a biological membrane as compared to its lateral extent, we may treat any element of membrane under consideration as planar. The GoldmanHodgkinKatz model assumes, in fact, that the membrane is uniform, planar, and infinite in its lateral extent. If the xaxis is chosen normal to the membrane with its origin at the interface of the membrane with the extracellular region, and if the membrane thickness is h, then x = h defines the interface of the membrane with the intracellular space. Because of the assumed lateral uniformity, variations of the potential field Φ and ionic concentration c within the membrane are functions of x only. The basic assumption underlying the GoldmanHodgkinKatz model is that the field within the membrane is constant; hence
(3.24) 
where  Φ_{0}  = potential at the outer membrane surface 
Φ_{h}  = potential at the inner membrane surface  
V_{m}  = transmembrane voltage  
h  = membrane thickness 
This approximation was originally introduced by David Goldman (1943).
The Nernst equation evaluates the equilibrium value of the membrane voltage when the membrane is permeable to only one kind of ion or when all permeable ions have reached a Donnan equilibrium. Under physiological conditions, such an equilibrium is not achieved as can be verified with examples such as Table 3.1. To determine the membrane voltage when there are several types of ions in the intra and extracellular media, to which the membrane may be permeable, an extended version of the Nernst equation must be used. This is the particular application of the GoldmanHodgkinKatz equation whose derivation we will now describe.For the membrane introduced above, in view of its one dimensionality, we have , , and, using Equation 3.12, we get
(3.25) 
for the k^{th} ion flux. If we now insert the constant field approximation of Equation 3.24 (dΦ/d_{x} = V_{m}/h) the result is
(3.26) 
(To differentiate ionic concentration within the membrane from that outside the membrane (i.e., inside versus outside the membrane), we use the symbol c^{m} in the following where intramembrane concentrations are indicated.) Rearranging Equation 3.26 gives the following differential equation:
(3.27) 
We now integrate Equation 3.27 within the membrane from the lefthand edge (x = 0) to the righthand edge (x = h). We assume the existence of resting conditions; hence each ion flux must be in steady state and therefore uniform with respect to x. Furthermore, for V_{m} to remain constant, the total transmembrane electric current must be zero. From the first condition we require that j_{k}(x) be a constant; hence on the lefthand side of Equation 3.27, only c_{k}^{m}(x) is a function of x. The result of the integration is then
(3.28) 
where  c_{k}^{h}  = concentration of the k^{th} ion at x = h 
c_{k}^{0}  = concentration of the k^{th} ion at x = 0 
Both variables are defined within the membrane.Equation 3.28 can be solved for j_{k}, giving
(3.29) 
The concentrations of the k^{th} ion in Equation 3.29 are those within the membrane. However, the known concentrations are those in the intracellular and extracellular (bulk) spaces. Now the concentration ratio from just outside to just inside the membrane is described by a partition coefficient, β. These are assumed to be the same at both the intracellular and extracellular interface. Consequently, since x = 0 is at the extracellular surface and x = h the intracellular interface, we have
(3.30) 
where  β  = partition coefficient 
c_{i}  = measurable intracellular ionic concentration  
c_{o}  = measurable extracellular ionic concentration 
The electric current density J_{k} can be obtained by multiplying the ionic flux j_{k} from Equation 3.29 by Faraday’s constant and valence. If, in addition, the permeability P_{k} is defined as
(3.31) 
then
(3.32) 
When considering the ion flux through the membrane at the resting state, the sum of all currents through the membrane is necessarily zero, as noted above. The main contributors to the electric current are potassium, sodium, and chloride ions. So we may write
(3.33) 
By substituting Equation 3.32 into Equation 3.33, appending the appropriate indices, and noting that for potassium and sodium the valence z = +1 whereas for chloride z = 1, and canceling the constant z_{k}²F²/RT, we obtain:
(3.34) 
In Equation 3.34 the expression for sodium ion current is seen to be similar to that for potassium (except for exchanging Na for K); however, the expression for chloride requires, in addition, a change in sign in the exponential term, a reflection of the negative valence.
The denominator can be eliminated from Equation 3.34 by first multiplying the numerator and denominator of the last term by factor –e^{FVm/RT} and then multiplying term by term by 1 – e^{FVm/RT}. Thus we obtain
(3.35) 
Multiplying through by the permeabilities and collecting terms gives:
(3.36) 
From this equation, it is possible to solve for the potential difference V_{m}across the membrane, as follows:
(3.37) 
where V_{m} evaluates the intracellular minus extracellular potential (i.e., transmembrane voltage). This equation is called the GoldmanHodgkinKatz equation. Its derivation is based on the works of David Goldman (1943) and Hodgkin and Katz (1949). One notes in Equation 3.37 that the relative contribution of each ion species to the resting voltage is weighted by that ion’s permeability. For the squid axon, we noted (Section 3.5.2) that P_{Na}/P_{K} = 0.04, which explains why its resting voltage is relatively close to V_{K} and quite different from VNa.
By substituting 37 °C for the temperature and the Briggs logarithm (with base 10) for the Napier logarithm (to the base e), Equation 3.37 may be written as:
(3.38) 
Example
It is easy to demonstrate that the GoldmanHodgkinKatz equation (Equation 3.37) reduces to the Nernst equation (Equation 3.21). Suppose that the chloride concentration both inside and outside the membrane were zero (i.e., c_{o,Cl} = c_{i,Cl} = 0). Then the third terms in the numerator and denominator of Equation 3.37 would be absent. Suppose further that the permeability to sodium (normally very small) could be taken to be exactly zero (i.e., P_{Na} = 0). Under these conditions the GoldmanHodgkinKatz equation reduces to the form of the Nernst equation (note that the absolute value of the valence of the ions in question z = 1). This demonstrates again that the Nernst equation expresses the equilibrium potential difference across an ion permeable membrane for systems containing only a single permeable ion.
3.4.3 The Reversal Voltage
The membrane potential at which the (net) membrane current is zero is called the reversal voltage (V_{R}). This designation derives from the fact that when the membrane voltage is increased or decreased, it is at this potential that the membrane current reverses its sign. When the membrane is permeable for two types of ions, A^{+} and B^{+}, and the permeability ratio for these ions is P_{A}/P_{B}, the reversal voltage is defined by the equation:
(3.39) 
This equation resembles the Nernst equation (Equation 3.21), but it includes two types of ions. It is the simplest form of the GoldmanHodgkinKatz equation (Equation 3.37).
3.5 ION FLOW THROUGH THE MEMBRANE
3.5.1 Factors Affecting Ion Transport Through the Membrane
This section explores the flow of various ions through the membrane under normal resting conditions.The flow of ions through the cell membrane depends mainly on three factors:
 the ratio of ion concentrations on both sides of the membrane
 the voltage across the membrane,and
 the membrane permeability.
The effects of concentration differences and membrane voltages on the flow of ions may be made commensurable if, instead of the concentration ratio, the corresponding Nernst voltage is considered. The force affecting the ions is then proportional to the difference between the membrane voltage and the Nernst voltage.Regarding membrane permeability, we note that if the biological membrane consisted solely of a lipid bilayer, as described earlier, all ionic flow would be greatly impeded. However, specialized proteins are also present which cross the membrane and contain aqueous channels. Such channels are specific for certain ions; they also include gates which are sensitive to membrane voltage. The net result is that membrane permeability is different for different ions, and it may be affected by changes in the transmembrane voltage, and/or by certain ligands.As mentioned in Section 3.4.1, Hodgkin and Huxley (1952a) formulated a quantitative relation called the independence principle. According to this principle the flow of ions through the membrane does not depend on the presence of other ions. Thus, the flow of each type of ion through the membrane can be considered independent of other types of ions. The total membrane current is then, by superposition, the sum of the currents due to each type of ions.
3.5.2 Membrane Ion Flow in a Cat Motoneuron
We discuss the behavior of membrane ion flow with an example. For the cat motoneuron the following ion concentrations have been measured (see Table 3.1).


the membrane [mol/m^{3}] 
the membrane [mol/m^{3}] 



Na^{+}  
K^{+}  
Cl^{–}  

For each ion, the following equilibrium voltages may be calculated from the Nernst equation:
V_{Na} = 61 log_{10}(15/150) = +61 mVV_{K} = 61 log_{10}(150/5.5) = 88 mVV_{Cl} = +61 log_{10}(9/125) = 70 mVThe resting voltage of the cell was measured to be 70 mV.
When Hodgkin and Huxley described the electric properties of an axon in the beginning of the 1950s (see Chapter 4), they believed that two to three different types of ionic channels (Na^{+}, K^{+}, and Cl^{–}) were adequate for characterizing the excitable membrane behavior. The number of different channel types is, however, much larger. In 1984, Bertil Hille (Hille, 1984/1992) summarized what was known at that time about ion channels. He considered that about four to five different channel types were present in a cell and that the genome may code for a total number of 50 different channel types. Now it is believed that each cell has at least 50 different channel types and that the number of different channel proteins reaches one thousand.We now examine the behavior of the different constituent ions in more detail.
Chloride Ions
In this example the equilibrium potential of the chloride ion is the same as the resting potential of the cell. While this is not generally the case, it is true that the chloride Nernst potential does approach the resting potential. This condition arises because chloride ion permeability is relatively high, and even a small movement into or out of the cell will make large changes in the concentration ratios as a result of the very low intracellular concentration. Consequently the concentration ratio, hence the Nernst potential, tends to move toward equilibrium with the resting potential.
Potassium ions
In the example described by Table 3.1, the equilibrium voltage of potassium is 19 mV more negative than the resting voltage of the cell. In a subsequent section we shall explain that this is a typical result and that the resting potential always exceeds (algebraically) the potassium Nernst potential. Consequently, we must always expect a net flow of potassium ions from the inside to the outside of a cell under resting conditions. To compensate for this flux, and thereby maintain normal ionic composition, the potassium ion must also be transported into the cell. Such a movement, however, is in the direction of increasing potential and consequently requires the expenditure of energy. This is provided by the NaK pump,that functions to transport potassium at the expense of energy.
Sodium Ions
The equilibrium potential of sodium is +61 mV, which is given by the concentration ratio (see Table 3.1). Consequently, the sodium ion is 131 mV from equilibrium, and a sodium influx (due to both diffusion and electric field forces) will take place at rest. Clearly neither sodium nor potassium is in equilibrium, but the resting condition requires only a steadystate. In particular, the total membrane current has to be zero. For sodium and potassium, this also means that the total efflux and total influx must be equal in magnitude. Since the driving force for sodium is 6.5 times greater than for potassium, the potassium permeability must be 6.5 times greater than for sodium. Because of its low resting permeability, the contribution of the sodium ion to the resting transmembrane potential is sometimes ignored, as an approximation.In the above example, the ionic concentrations and permeabilities were selected for a cat motoneuron. In the squid axon, the ratio of the resting permeabilities of potassium, sodium and chloride ions has been found to be P_{K}:P_{Na}:P_{Cl} = 1:0.04:0.45.
3.5.3 NaK Pump
The longterm ionic composition of the intracellular and extracellular space is maintained by the NaK pump. As noted above, in the steady state, the total passive flow of electric current is zero, and the potassium efflux and sodium influx are equal and opposite (when these are the only contributing ions). When the NaK pump was believed to exchange 1 mol potassium for 1 mol sodium, no net electric current was expected. However recent evidence is that for 2 mol potassium pumped in, 3 mol sodium is pumped out. Such a pump is said to be electrogenic and must be taken into account in any quantitative model of the membrane currents (Junge, 1981).
3.5.4 Graphical Illustration of the Membrane Ion Flow
The flow of potassium and sodium ions through the cell membrane (shaded) and the electrochemical gradient causing this flow are illustrated in Figure 3.5. For each ion the clear stripe represents the ion flux; the width of the stripe, the amount of the flux; and the inclination (i.e., the slope), the strength of the electrochemical gradient.As in Figure 3.2, the vertical axis indicates the potential, and the horizontal axis distance normal to the membrane. Again, when ΔV is positive (downward), the transmembrane current (for a positive ion) is also positive (i.e., outward). For a negative ion (Cl^{–}), it would be inward.

 A model illustrating the transmembrane ion flux. (After Eccles, 1968.) (Note that for K
^{+}

 and Cl
^{–}
 passive flux due to diffusion and electric field are shown separately)
3.6 CABLE EQUATION OF THE AXON
Ludvig Hermann (1905b) was the first to suggest that under subthreshold conditions the cell membrane can be described by a uniformly distributed leakage resistance and parallel capacitance. Consequently, the response to an arbitrary current stimulus can be evaluated from an elaboration of circuit theory. In this section, we describe this approach in a cell that is circularly cylindrical in shape and in which the length greatly exceeds the radius. (Such a model applies to an unmyelinated nerve axon.)
3.6.1 Cable Model of the Axon
Suppose that an axon is immersed in an electrolyte of finite extent (representing its extracellular medium) and an excitatory electric impulse is introduced via two electrodes – one located just outside the axon in the extracellular medium and the other inside the axon, as illustrated in Figure 3.6. The total stimulus current (I_{i}), which flows axially inside the axon, diminishes with distance since part of it continually crosses the membrane to return as a current (I_{o}) outside the axon. Note that the definition of the direction of positive current is to the right for both I_{i} and I_{o}, in which case conservation of current requires that I_{o} = –I_{i}. Suppose also that both inside and outside of the axon, the potential is uniform within any crossection (i.e., independent of the radial direction) and the system exhibits axial symmetry. These approximations are based on the crosssectional dimensions being very small compared to the length of the active region of the axon. Suppose also that the length of the axon is so great that it can be assumed to be infinite.Under these assumptions the equivalent circuit of Figure 3.7 is a valid description for the axon. One should particularly note that the limited extracellular space in Figure 3.6 confines current to the axial direction and thus serves to justify assigning an axial resistance Ro to represent the interstitial fluid. In the model, each section, representing an axial element of the axon along with its bounding extracellular fluid,is chosen to be short in relation to the total axon length. Note, in particular, that the subthreshold membrane is modeled as a distributed resistance and capacitance in parallel. The resistive component takes into account the ionic membrane current imI; the capacitance reflects the fact that the membrane is a poor conductor but a good dielectric, and consequently, a membrane capacitive current imC must be included as a component of the total membrane current. The axial intracellular and extracellular paths are entirely resistive, reflecting experimental evidence regarding nerve axons..
 The experimental arrangement for deriving the cable equation of the axon.
 The equivalent circuit model of an axon. An explanation of the component elements is given in the text.
The components of the equivalent circuit described in Figure 3.7 include the following: Note that instead of the MKS units, the dimensions are given in units traditionally used in this connection. Note also that quantities that denote “per unit length” are written with lowercase symbols.
r_{i}  =  intracellular axial resistance of the axoplasm per unit length of axon [kΩ/cm axon length] 
r_{o}  =  extracellular axial resistance of the (bounding) extracellular medium per unit length of axon [kΩ/cm axon length] 
r_{m}  =  membrane resistance times unit length of axon [kΩ·cm axon length] (note that this is in the radial direction, which accounts for its dimensions) 
c_{m}  =  membrane capacitance per unit length of axon [µF/cm axon length] 
We further define the currents and voltages of the circuit as follows (see Figures 3.6 and 3.7):
I_{i}  =  total longitudinal intracellular current [µA] 
I_{o}  =  total longitudinal extracellular current [µA] 
i_{m}  =  total transmembrane current per unit length of axon [µA/cm axon length] (in radial direction) 
i_{mC}  =  capacitive component of the transmembrane current per unit length of axon [µA/cm axon length] 
i_{mI}  =  ionic component of the transmembrane current per unit length of axon [µA/cm axon length] 
Φ_{i}  =  potential inside the membrane [mV] 
Φ_{o}  =  potential outside the membrane [mV] 
V_{m}  =  Φ_{i} – Φ_{o} membrane voltage [mV] 
V_{r}  =  membrane voltage in the resting state [mV] 
V’  =  V_{m} – V_{r} = deviation of the membrane voltage from the resting state [mV] 
A graphical sketch defining of various potentials and voltages in the axon is given in Figure 3.8.
We note once again that the direction of positive current is defined as the direction of the positive xaxis both inside and outside the axon. Therefore, for all values of x, conservation of current requires that I_{i} + I_{o} = 0 provided that x does not lie between stimulating electrodes. For a region lying between the stimulating electrodes, I_{i} + I_{o} must equal the net applied current..
 A graphical sketch depicting various potentials and voltages in the axon used in this book.
In the special case when there are no stimulating currents (i.e., when I_{i} = I_{o}= I_{m} = 0), then V_{m} = V_{r} and V‘ = 0. However, once activation has been initiated we shall see that it is possible for I_{i} + I_{o} = 0 everywhere and V’ 0 in certain regions.
Since V_{r} , the membrane resting voltage, is the same everywhere, it is clear that
(3.40) 
based on the definition of V’ given above.
3.6.2 The SteadyState Response
We first consider the stationary case (i.e., δ/δt = 0) which is the steadystate condition achieved following the application of current step. This corresponds to the limit t . The steadystate response is illustrated in Figure 3.9. It follows from Ohm’s law that
(3.41) 
From the current conservation laws, it follows also that the transmembrane current per unit length, i_{m}, must be related to the loss of I_{i} or to the gain of I_{o} as follows:
(3.42) 
Note that this expression is consistent with I_{i} + I_{o} = 0. The selection of the signs in Equation 3.42 is based on outwardflowing current being defined as positive. From these definitions and Equations 3.40 and 3.41 (and recalling that V’ = Φi – Φo – V_{r}), it follows that
(3.43) 
Furthermore, by differentiating with respect to x, we obtain:
(3.44) 
 (A) Stimulation of a nerve with current step.(B) Variation of the membrane voltage as a function of distance.
Substituting Equation 3.42 into Equation 3.44 gives:
(3.45) 
which is called the general cable equation.
Under stationary and subthreshold conditions the capacitive current c_{m}dV’/dt = 0; so that the membrane current per unit length is simply i_{m} = V’/r_{m}; according to Ohm’s law. Consequently, Equation 3.45 can be written in the form
(3.46) 
whose solution is
(3.47) 
The constant λ in Equation 3.47 has the dimension of length and is called the characteristic length or length constant of the axon. It is called also the space constant. The characteristic length λ is related to the parameters of the axon by Equation 3.46, and is given by:
(3.48) 
The latter form of Equation 3.48 may be written because the extracellular axial resistance r_{o} is frequently negligible when compared to the intracellular axial resistance r_{i}.
With the boundary conditions:
the constants A and B take on the values A = V'(0) and B = 0, and from Equation 3.47 we obtain the solution:
(3.49) 
This expression indicates that V’ decreases exponentially along the axon beginning at the point of stimulation (x = 0), as shown in Figure 3.9B. At x = λ the amplitude has diminished to 36.8% of the value at the origin. Thus λ is a measure of the distance from the site of stimulation over which a significant response is obtained. For example at x = 2λ the response has diminished to 13.5%, whereas at x = 5λ it is only 0.7% of the value at the origin.
3.6.3 Stimulation with a StepCurrent Impulse
In this section we consider the transient (rather than steadystate) response of the axon to a subthreshold currentstep input. In this case the membrane current is composed of both resistive and capacitive components reflecting the parallel RC nature of the membrane:
(3.50) 
where  i_{m}  = the total membrane current per unit length [µA/cm axon length] 
i_{mR}  = the resistive component of the membrane current per unit length [µA/cm axon length]  
i_{mC}  = the capacitive component of the membrane current per unit length [µA/cm axon length] 
Under transient conditions Equation 3.50 substituted into Equation 3.45 may be written:
(3.51) 
The left side of Equation 3.51 evaluates the total membrane current i_{m}, whereas on the right side the first term represents the resistive component (formed by the ionic currents), and the second term the capacitive current which must now be included since /t 0 . Equation 3.51 may also be written in the form:
(3.52) 
which can be easily expressed as
(3.53) 
where τ = r_{m}c_{m} is the time constant of the membrane and λ is the space constant as defined in Equation 3.48.
Here the time constant was derived for a long, thin axon corresponding to a onedimensional problem. The time constant may be derived with a similar method also for the surface of a membrane as a twodimensional problem. In such case instead of the variables defined “times unit length” and “per unit length”, variables defined “times unit area” and “per unit area” are used. Then we obtain for the time constant τ = R_{m}C_{m}.The temporal and spatial responses of the membrane voltage for several characteristic values of x and t are illustrated in Figure 3.10. One should note that the behavior of V’ as a function of x is nearly exponential for all values of t, but the response as a function of t for large values of x differs greatly from an exponential behavior (becoming Sshaped). These curves illustrate the interpretation of λ, the space constant, as a measure of the spatial extent of the response to the stimulating current. For values of x/λ less than around 2, τ is essentially a measure of the time to reach steady state. However, for large x/λ this interpretation becomes poor because the temporal curve deviates greatly from exponential. In Figure 3.10, where λ = 2.5 mm, the electrode at x = 5 mm is at 2λ, and the amplitude, after an interval τ, has reached only 37% of steady state. Were we to examine x = 25 mm (corresponding to 5λ), only 0.8% of steadystate would be reached after the interval τ.
 (A) The physical setup, including the waveform of the applied current and the placement of stimulating and recording electrodes.(B) The spatial response at τ = 13, 35, 100 ms; and
t
 =
 . The latter curve is the steadystate response and corresponds to Equation 3.49.(C) The temporal response of three axial sites at
x
 = 0, 2.5, 5 mm.
While a closedform solution to Equation 3.53 can be described, we have chosen to omit it from this text because of its complexity. One can find a derivation in Davis and Lorente de No (1947). Rather than include this analytical material, we have chosen instead to illustrate the temporal and spatial response of the transmembrane voltage to a current step for a range of values of λ and τ. This is provided in Figure 3.11.Specifically, Figure 3.11 describes the subthreshold transmembrane voltage response to a current step of very long duration introduced extracellularly at the center of a cable of infinite length. The response, when the current is turned on, is shown in the lefthand side of the figure, whereas the response, when the current is subsequently turned off, is on the right. The transmembrane voltage is described as a function of time for given positions of the fiber. The transmembrane voltage is also described as a function of position at given times following the application of the current or its termination. The figure is drawn from a recalculation of its quantities from the original publication of Hodgkin and Rushton (1946).Note that distance is shown normalized to the space constant , whereas time is normalized to the time constant . Normalization, such as this, results in “universal” curves that can be adapted to any actual value of and . Note also that the points on a particular voltage versus distance curve drawn at some values of t in the upper graph can also be found at the same values of t in the lower graph for the particular distance values, and vice versa. The fact that the upper and lower curves show the same phenomenon but in different dimensions is emphasized by the dotted vertical lines which indicate the corresponding location of points in the two sets of curves.Table 3.2 lists measured values of characteristic lengths and time constants for several axons for several different species. A significant variation from species to species is seen.


*) The specific resistance and specific capacitance of the membrane can be calculated from values ofresistance and capacitance per unit length by use of the following:

3.7 STRENGTHDURATION RELATION
When an excitable membrane is depolarized by a stimulating current whose magnitude is gradually increased, a current level will be reached, termed the threshold, when the membrane undergoes an action impulse. The latter is characterized by a rapid and phasic change in membrane permeabilities, and associated transmembrane voltage. An illustration of this process was given in Figure 2.8, where the response to stimulus level 2 is subthreshold, whereas stimulus 3 appears just at threshold (since sometimes an action potential (3B) results whereas at other times a passive response (3A) is observed). An action potential is also clearly elicited for the transthreshold stimulus of 4.Under active conditions the membrane can no longer be characterized as linear, and the RC model described in the previous section is not applicable. In the next chapter, we present a detailed study of the active membrane.A link between this chapter, which is limited to the passive membrane, and the next, which includes the nonlinear membrane, lies in the modeling of conditions that lead to excitation. Although it is only an approximation, one can consider the membrane just up to the point of activation as linear (i.e., passive). Consequently, membrane behavior within this limit can be analyzed using ordinary electric circuits. In particular, if threshold values are known, it then becomes possible to elucidate conditions under which activation will just be achieved. Since activation is affected not only by the strength of a stimulating current but also its duration, the result is the evaluation of strengthduration curves that describe the minimum combinations of strength and duration just needed to produce the activation (Arvanitaki, 1938), as was illustrated in Figure 2.10.A simple example of these ideas is furnished by a cell that is somewhat spherical in shape and in which one stimulating electrode is placed intracellularly and the other extracellularly. One can show that for cells of such shape, both the intracellular and extracellular space is isopotential at all times. Thus, if a current is passed between the electrodes, it passes uniformly across the membrane so that all membrane elements behave similarly. As a consequence, the corresponding electric circuit is a lumped R_{m} and C_{m} in parallel. The value of R_{m} is the membrane resistance times unit area, whereas C_{m} is the membrane capacitance per unit area.If I_{s} is the stimulus current per unit area, then from elementary circuit theory applied to this parallel RC circuit, we have
(3.56) 
where  V’  = change in the membrane voltage [mV] 
I_{s}  = stimulus current per unit area [µA/cm²]  
R_{m}  = membrane resistance times unit area [kΩ·cm²]  
t  = stimulus time [ms]  
τ  = membrane time constant = R_{m}C_{m} [ms]  
C_{m}  = membrane capacitance per unit surface [µF/cm²] 
Unfortunately, this simple analysis cannot be applied to cells with other shapes (e.g., the fiberlike shape of excitable cells), where the response to a stimulating current follows that governed by Equation 3.53 and described in Figure 3.11. However, Equation 3.56 could still be viewed as a firstorder approximation based on a lumpedparameter representation of what is actually a distributedparameter structure. Following this argument, in Figure 3.12 we have assumed that a long fiber can be approximated by just a single (lumped) section, hence leading to an equation of the type described in Equation 3.56. A characteristic response based on Equation 3.56 is also shown in Figure 3.12..
 (A) An approximate lumpedparameter RCnetwork which replaces the actual distributed parameter structure.(B) The response of the network to a current pulse of magnitude
I_{s}
 is exponential and is shown for a pulse of very long duration.
The membrane is assumed to be activated if its voltage reaches the threshold value. We consider this condition if we substitute V‘ = δV_{th} into Equation 3.56, where Vth is the change in the resting voltage needed just to reach the threshold voltage. Equation 3.56 may now be written in the form:
(3.57) 
The smallest current that is required for the transmembrane voltage to reach threshold is called the rheobasic current. With this stimulus current, the required stimulus duration is infinite. Because the rheobasic current is given by I_{rh} = δV_{th}/R_{m}, the strengthduration curve takes on the form:
(3.58) 
The strengthduration curve is illustrated in Figure 3.13. Here the stimulus current is normalized so that the rheobasic current has the strength of unity. (Note again, that this result is derived for a spaceclamp situation.)The time needed to reach the threshold voltage with twice the rheobasic stimulus current is called chronaxy. For the relation between chronaxy and the membrane time constant, Equation 3.57 can be written as:
(3.59) 
(3.60) 
(3.61) 
If the stimulus current is twice rheobasic current, then I_{s} = 2(δV_{th}/R_{m}), and we obtain for chronaxy:
(3.62) 
 (B) The subthreshold transient response prior to excitation.
The analytical results above are approximate for several reasons. First, the excitable tissue cannot normally be well approximated by a lumped R since such elements are actually distributed. (In a spaceclamp stimulation the membrane can be more accurately represented with a lumped model.) Also the use of a linear model is satisfactory up to perhaps 80% of the threshold, but beyond this the membrane behaves nonlinearly. Another approximation is the idea of a fixed threshold; in a subsequent chapter, we describe accommodation, which implies a threshold rising with time.
In a particular situation, a strengthduration curve can be found experimentally. In this case, rheobase and chronaxy are more realistic measures of the stimulusresponse behavior. This type of data for chronaxy is given in Table 3.3, which lists chronaxies measured for various nerve and muscle tissues. Note that, in general, the faster the expected response from the physiological system, the shorter the chronaxy value.

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FURTHER READING
Edsall JT, Wyman J (1958): Biophysical Chemistry, Vol. 1, 699 pp. Academic Press, New York.
Moore WJ (1972): Physical Chemistry, 4th ed., 977 pp. PrenticeHall, Englewood Cliffs, N.J.