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Physical Principles and Formalisms of Electrical Excitability

Alan Finkelstein, Alexander Mauro

10.1002/cphy.cp010106

Source: Supplement 1: Handbook of Physiology, The Nervous System, Cellular Biology of Neurons

Originally published: 1977

Published online: January 2011

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Membrane Potentials and Ionic Fluxes
1.1 Fundamental Concepts
1.2 Two Important Examples of Equilibrium Situations
1.3 Electrodes — the Measurement of Potential Difference
1.4 Quasi‐equilibrium Systems
1.5 Ion Transport (the Nernst‐Planck Flux Equations)
2 Formal Consequences of Voltage‐Dependent Conductances
2.1 The Nature of Electrical Excitability
2.2 null
3 Voltage‐Dependent Conductance in Thin Lipid Membranes
3.1 The Unmodified Thin Lipid Membrane
3.2 Nonvoltage‐dependent Modifiers
3.3 Voltage‐dependent Modifiers
Figure 1. Figure 1.

The Donnan system.
Figure 2. Figure 2.

Qualitative plots of the Donnan ratio, r, as a function of the impermeant ion concentration, [N], for an impermeant ion of valence +1 or −1. (See Eq. 26.)
Figure 3. Figure 3.

Concentrations and potentials (both plotted on ordinate) in compartments 1 and 2 for a Donnan equilibrium.
Figure 4. Figure 4.

Sketch of concentration and potential profiles (both plotted on ordinate) for a Donnan equilibrium as determined from the Poisson‐Boltzmann analysis (cf. Fig. 3, which is the result of the thermodynamic analysis). Note that at any point x the electrical force (+F dψ/dx for cations and −F dψ/dx for anions) and the phenomenological diffusion “force” (−RT/[Na+] d[Na+]/dx and −RT/[C] l/dx) balance. The space‐charge density is shown in the lower part of the figure.
Figure 5. Figure 5.

The two‐phase system.
Figure 6. Figure 6.

Concentrations and potentials in the water and oil phases. The potential, ψ2, in the oil phase is positive, because we have assumed that the partition coefficient for sodium, , between oil and water is greater than that for chloride, .
Figure 7. Figure 7.

Sketch of concentration and potential profiles for a phase‐boundary equilibrium as determined from the Poisson‐Boltzmann analysis (cf. Fig. 6, which is the result of the thermodynamic analysis; as in Fig. 6, ). The space‐charge density is shown in the lower part of the figure.
Figure 8. Figure 8.

Method for measuring the potential difference across a membrane.
Figure 9. Figure 9.

Method of measuring membrane potential by making contact with the solutions through 3 M KCl junctions.
Figure 10. Figure 10.

An ion‐exchange membrane separating two solutions.
Figure 11. Figure 11.

A: concentration and potential profiles for an ionexchange membrane of large positive fixed‐charge density separating two solutions of equal concentration of NaCl. B: same as A, except that NaCl concentrations in compartments 1 and 2 are unequal. (The Na+ and Cl− profiles within the membrane have a small negative slope that is not clearly seen in the figure.)
Figure 12. Figure 12.

An oil membrane separating two NaCl aqueous solutions.
Figure 13. Figure 13.

Concentration and potential profiles for a “thick” oil membrane separating two NaCl aqueous solutions. The potential within the membrane (ψm) is positive, because we have assumed that Na+ partitions better into the membrane than Cl−.
Figure 14. Figure 14.

A: a redrawing of Fig. 13 for the concentration profiles of a “thick” oil membrane, with the space‐charge regions shown (exaggerated). B: concentration profiles for a “thin” oil membrane. Note that there is no region within the membrane where electroneutrality ([Na+] = [Cl−]) holds. The entire membrane thickness corresponds to the regions near the boundaries in Fig. 14A.
Figure 15. Figure 15.

A membrane separating two “infinite,” well‐stirred aqueous solutions. Stimulating and recording electrodes are shown.
Figure 16. Figure 16.

Equivalent circuit for a homogeneous, uncharged membrane separating two solutions of arbitrary ionic composition.
Figure 17. Figure 17.

Concentration profile within a thick, homogeneous, uncharged membrane separating two NaCl solutions at different concentrations.
Figure 18. Figure 18.

Equivalent circuit for the single‐salt case for a thick, homogeneous, uncharged membrane.
Figure 19. Figure 19.

Current‐voltage characteristics for the single‐salt case for a thick, homogeneous, uncharged membrane. These characteristics follow from the equivalent circuit in Fig. 18. (Note that these are drawn for c2 > c1; if c1 > c2, ψD (the potential at I = 0) would change sign. That is, the plots for u > v and u < v would be interchanged.
Figure 20. Figure 20.

How an electric field would develop, if at t = 0 there is a sharp boundary in salt concentration from c1 to c2 and the mobilities (i.e., diffusion coefficients) of the ions are different. The profiles shown are those that would exist at some short time later if the ions were uncharged. We see, however, that since the ions are charged, this creates a space‐charge region that will give rise to an electric field that tends to couple the motion of the ions together.
Figure 21. Figure 21.

Concentration profiles within the membrane for the bi‐ionic case of a membrane separating 0.1 M NaCl from 0.1 M KC1. A, I = 0; B, I is large and positive; C, I is large and negative.
Figure 22. Figure 22.

Sketch of the steady‐state current‐voltage characteristic for the bionic case of Fig. 21. Note the characteristics (dashed lines) for a membrane separating either symmetrical 0.1 M NaCl solutions or symmetrical 0.1 M KCl solutions, which the bi‐ionic characteristic approaches asymptotically for large positive and negative potentials, respectively. Also shown are the chord and slope resistance lines at point P.
Figure 23. Figure 23.

Nonlinear steady‐state current‐voltage characteristic with chords drawn at (I1, ψ1) and (I3, ψ3). If current is stepped from I1 to I2, the voltage originally attains the value at P and then increases with time along the vertical line from P until it reaches the value ψ2. Similarly, if current is stepped from I3 to I4, the voltage originally attains the value at Q and then moves along the vertical line from Q until it reaches the value ψ4.
Figure 24. Figure 24.

The change of voltage with time in response to steps of current. These responses follow from the analysis of Fig. 23 as given in the legend and text.
Figure 25. Figure 25.

RC (A) and RL (B) networks that would give responses similar to those shown in Fig. 24.
Figure 26. Figure 26.

Concentration profiles (A) and potential profile (B) for a positive fixed‐charge membrane separating NaCl solutions at different concentrations. Note the Donnan “jumps” in concentrations and potential at the interfaces.
Figure 27. Figure 27.

Concentration profile for an oil membrane separating a single salt at two different concentrations.
Figure 28. Figure 28.

Equivalent circuit for a mosaic membrane. The individual elements are shown to be ideally selective for a given ion, but this need not necessarily be the case.
Figure 29. Figure 29.

The Hodgkin‐Huxley equivalent circuit for the squid giant axon membrane in normal seawater.
Figure 30. Figure 30.

A circuit consisting of only a voltage‐dependent conductance element.
Figure 31. Figure 31.

Possible steady‐state g‐V characteristics and their corresponding I–V characteristics for the circuit of Fig. 30.
Figure 32. Figure 32.

A steady‐state I‐V characteristic with a region of negative slope. This is an enlargement of Fig. 31C″, illustrating why any point P in the region of negative slope is unstable under current‐clamp conditions (see text).
Figure 33. Figure 33.

Two circuits (A and C) with the same steady‐state g‐V characteristic (B).
Figure 34. Figure 34.

A: the steady‐state I‐V characteristic for the circuit of Fig. 33A, given the steady‐state g‐V characteristic of Fig. 33B. B: the steady‐state I‐V characteristic for the circuit of Fig. 33C, given the same steady‐state g‐V characteristic (i.e., Fig. 33B).
Figure 35. Figure 35.

The arrangement for forming and studying bilayers on a planar plastic partition separating two aqueous phases.
Figure 36. Figure 36.

Schematic drawing of a phospholipid bilayer. Filled circles, polar ends of the phospholipids; wavy lines, their fatty acid chains. The latter make up the hydrocarbon interior of the membrane, whereas the former anchor this region to the two aqueous phases.
Figure 37. Figure 37.

Diagrams of the response of a parallel RC circuit (B) to a step of current (A) or a step of voltage (C). (The voltageclamp response, C, is actually that which occurs if there is a small resistance in series with the circuit in B; without that resistance, the current response is a δ function.) These are the responses of an unmodified lipid bilayer.
Figure 38. Figure 38.

Schematic diagram of a cation carrier, such as valinomycin. The carbonyl oxygens provide a polar environment for the cation, and in fact substitute for the first hydration shell normally surrounding the cation in water. The exterior aspect of the molecule is nonpolar, thus making it compatible with the hydrocarbon interior of the membrane.
Figure 39. Figure 39.

Structural formulas for the cation carriers nonactin 11 and enniatin B 64.
Figure 40. Figure 40.

Structural formula of amphotericin B 47. Nystatin differs from amphotericin B in that it contains a tetraene and a diene in place of the heptaene chromophore; in other respects the molecules are very similar.
Figure 41. Figure 41.

Molecular model (CPK) of amphotericin B. In A the completely hydrophobic face of amphotericin B is seen; in B the molecule has been rotated 180° about its long axis to reveal the opposite face with its many hydroxyl groups. Note that the molecule consists of two chains: a polyene chain (seen on the right in A and on the left in B) and an amphipathic chain. The hydrophobic and hydrophilic faces of the amphipathic chain are seen in A and B, respectively. At the bottom of the figures are the polar amino sugar and carboxyl groups; at the top is a single hydroxyl group (seen most clearly in B). In C, a CPK model of lecithin is shown for comparison with amphotericin B.

From Finkelstein & Holz 19, reprinted by courtesy of Marcel Dekker, Inc
Figure 42. Figure 42.

Half of an amphotericin B‐created pore. Each amphotericin B molecule is schematized as a plane with a protuberance and a solid dot. The shaded portion of each plane represents the hydroxyl face of the amphipathic chain, the protuberance represents the amino sugar, and • represents the single hydroxyl group at the nonpolar end of the molecule. The aqueous phase is at the bottom of the figure and the middle of the membrane is at the top. We see that the interior of the pore is polar, whereas the exterior is completely nonpolar; there is also a wedge in the exterior of the pore, between each pair of amphotericin B molecules, that can accommodate a sterol molecule. Note the ring of hydroxyl groups in the middle of the membrane that can hydrogen bond with an identical structure from the other side to form a complete pore.

From Finkelstein & Holz 19
Figure 43. Figure 43.

Structural formula of valine‐gramicidin A 68. (Isoleucine gramicidin A has l‐Ile in place of the first l‐Val.)
Figure 44. Figure 44.

Conductance fluctuations for a membrane treated with a small amount of gramicidin A. (Actually one is seeing the current fluctuations in the face of a constant voltage across the membrane.) The membrane was formed from glyceryl monooleate dissolved in hexadecane and separates 0.5 M NaCl solutions at 23°C. The events marked α and β on the left‐hand side of the record occurred very infrequently.

From Hladky & Haydon 28
Figure 45. Figure 45.

Proposed structure of the gramicidin A channel. A: side view of a CPK molecular model of two molecules of gramicidin A in the π6(L,D) helix conformation, linked by three hydrogen bonds through their formyl ends. B: end view of the same model showing the central channel 2 Å in radius.

From Urry 75
Figure 46. Figure 46.

A: equivalent circuit for a membrane treated with both nystatin and valinomycin and separating KCl solutions at different concentrations. (The membrane capacitance is shown for completeness.) B: plot of IK and ICl as a function of V for the elements of the circuit in A. The slopes of these lines are gK and gCl, respectively. At point A, IK = −ICl; the value of V at this point is the membrane potential (in the absence of an external current source). Note that A must lie between EK and ECl; its exact position is determined by the slopes of IK and ICl (i.e., gK and gCl).
Figure 47. Figure 47.

Structural formula of alamethicin 61. Aib, α‐aminoisobutyric acid.
Figure 48. Figure 48.

Voltage‐clamp responses of monazomycin‐treated phosphatidylethanolamine membranes. Membranes (area = 1 mm2) were formed at room temperature in 0.1 M KCl; monazomycin was then added to the inside chamber (33 μg/ml in A and 17 μg/ml in B, C, and D). The arrows designate the I = 0 level, and the vertical blips in A, B, and D mark the onset of stimulation. A: successive current responses to positive rectangular voltage pulses of 25, 29, 33, 37, and 41 mV. B: record illustrating the difference in kinetics for turning on and turning off conductance. The initial stimulus is +60 mV; after the current reaches a steady state, the stimulus is switched to −60 mV. C: higher time resolution of the conductance turnoff. The current has already reached a steady‐state value for a voltage of +65 mV, and the stimulus is then switched to −65 mV. D: further illustration of difference in kinetics for conductance turnon and turnoff. The initial stimulus is +55 mV; after the current reaches a steady state, the voltage is reduced to +50 mV. Note that the initial drop in current when the voltage is reduced from +55 to +50 mV is much greater than the initial rise in current (hardly visible) when the voltage is increased from 0 to +55 mV. This reflects the much larger steady‐state conductance at +55 mV than at 0 mV.

From Muller & Finkelstein 57
Figure 49. Figure 49.

Schematic drawing of the nonlinear steady‐state I‐V characteristic of a monazomycin‐treated membrane separating identical solutions. At V1, the value of the current is that corresponding to P. Any change in potential (to V2 for example) leads to an “instantaneous” value of the current at t = 0 corresponding to a point on the chord drawn between P and the origin. The current subsequently relaxes (along the dashed vertical line) to its final value (at t =∞) on the steady‐state characteristic.
Figure 50. Figure 50.

The effect of divalent cation added to the outside solution on the steady‐state g‐V characteristic of a membrane with monazomycin in the inside solution. The phosphatidylglycerol membrane was formed at room temperature in 0.01 M KCl. Monazomycin was then added to the inside chamber, and after approximately 20 min, the g‐V characteristic labeled [Mg2+] = 0.0 mM was taken. MgS04 was then added to the outside chamber to the concentrations indicated on the curves. It required only 1 min (the time to stir in MgSO4) for the new g‐V characteristic to be established. Note the large shift of the g‐V characteristic to the right along the voltage axis. Similar results are obtained with Ca2+. Slope of lines = e‐fold conductance change per 4.0 mV; monazomycin concentration = 0.5 μg/ml; membrane area = 1 mm2.

From Muller & Finkelstein 58
Figure 51. Figure 51.

Potential profiles for a phosphatidylglycerol membrane separating the salt solutions indicated. At large (greater than 100 Å) distances from the membrane interfaces, the potential on both sides is zero (in the open‐circuited condition). The quantity of interest is ψ0, the value of the potential at the two interfaces. A: symmetrical salt solutions. ψ0 is the same at each interface, and there is no electric field within the membrane. B: divalent cation is present only in the REAR (outside) solution. Because of the asymmetry in surface potentials, there exists a potential difference across the membrane proper (though none measurable between the two solutions) and hence a field within the membrane. The sign of the potential difference is such as to turn off the monazomycin‐induced conductance. As far as the field that monazomycin “sees,” the situation in B is identical to having no Ca2+ present but instead passing sufficient current across the membranes to make the inside compartment −60 mV with respect to the outside. This is the basis for the shifts of the g‐V characteristics to the right in Fig. 50.

From Muller & Finkelstein 58
Figure 52. Figure 52.

A: steady‐state I‐V characteristics of a monazomycintreated phosphatidylethanolamine membrane: development of a negative‐slope conductance region by the addition of a salt gradient. The membrane was formed at room temperature in 0.02 M KCl; monazomycin was then added to the FRONT (inside) chamber and approximately 15 min later the I‐V characteristic labeled was obtained. A KCl gradient was established by raising [KCl] to 0.153 M in the REAR (outside) chamber. This gave rise to a diffusion emf of 42 mV. The I‐V characteristic labeled was taken approximately 1 min after the gradient was established. The gradient (along with the emf) was then abolished by raising [KCl] in the front (inside) chamber to 0.153 M. The I‐V characteristic labeled was taken approximately 1 min after the gradient was abolished. B: the steady‐state g‐V characteristics of a monazomycin‐treated membrane in the absence ( and ) and presence of a diffusion emf created by a salt gradient. The g‐V characteristics have been calculated from the corresponding I‐V characteristics in A by Equation 95 (for curves and ) and by Equation 99 (for curve ) with emf = 42 mV.

From Muller & Finkelstein 57
Figure 53. Figure 53.

Voltage‐clamp responses of a monazomycin‐treated membrane with a positive emf created by a salt gradient. The inside compartment, which has monazomycin (5 μg/ml), contains 0.01 M NaCl, and the outside compartment contains 0.077 M NaCl. The resulting emf is approximately 40 mV. Membrane area = 0.01 mm2. The membrane was formed from a mixture of ox‐brain lipid and tocopherol. See text and Fig. 54 for a discussion of these records.
Figure 54. Figure 54.

Schematic drawings (lower two) of the records in Fig. 53 along with the steady‐state I‐V characteristic of the membrane (upper drawing). See text for a discussion of these drawings.
Figure 55. Figure 55.

Equivalent circuit for a membrane treated with both nystatin and monazomycin and separating NaCl solutions with [NaCl]outside > [NaCl]inside. (The membrane capacitance is shown for completeness.)
Figure 56. Figure 56.

Bistable property of a membrane treated with both nystatin and monazomycin. The membrane was formed at room temperature in a solution containing 0.01 M KCl and 0.001 M MgS04. Nystatin was added to both chambers to a concentration of 5 μg/Ml. After approximately 20 min the membrane reached a steady conductance of about 10−7 Ω−1. At this point the KCl concentration was raised in the outside chamber to 0.077 M. This produced a membrane emf of −41 mV. Monazomycin was then added to the front chamber to a concentration of 3 μg/ml. This produced no significant change in the emf. After approximately 15 min, the above record was obtained. (In the record the horizontal line is V = 0; above the line are negative potentials, and below the line are positive potentials.) At a, a positive step of current = 3.75 nA was applied and at b it was removed. The potential returned toward the original −41 mV. At c, the current was again applied, and at d it was removed. This time the potential continued increasing and achieved a new stable state of +41 mV. At e, a negative step of current = −4.05 nA was applied, and at f it was removed. The potential returned to +41 mV. At g the current was again applied and at h it was removed. This time the potential continued to decrease and flipped back to the original state of −41 mV. Note the much smaller IR drops at e and g compared with that at a, thus demonstrating that the conductance is much higher at +41 mV (where the monazomycin‐induced conductance is turned on) than at −41 mV (where only the nystatin‐induced conductance is significant).

From Muller & Finkelstein 57
Figure 57. Figure 57.

Schematic plots of the steady‐state I‐V characteristics for the nystatin‐ and monazomycin‐induced conductance elements of the circuit in Fig. 55. The slope of the dashed line is the monazomycin‐induced conductance, gNa, in the resting state (at point A). (The slope of the line drawn between ENa and any point on the INa characteristic gives the value of gNa at the voltage corresponding to that point.)
Figure 58. Figure 58.

Inactivation of monazomycin‐induced conductance by dodecyltriethylammonium bromide (C12). A membrane formed from phosphatidylglycerol and cholesterol separates symmetrical 0.1 M KCl solutions and contains in the inside compartment 1 μg/ml monazomycin and 1.3 × 10−5 M dodecyltriethylammonium bromide. The current record (traced from the oscillograph recording) is in response to a voltage step (applied at the upward arrow and removed at the downward arrow) of 88 mV. Membrane area = 1 mm2. Vertical line, 200 nA; horizontal line, 5 s. Note the contrast of this record with Fig. 48 A, where C12 is absent.

From Heyer 27
Figure 59. Figure 59.

A: membrane treated with valinomycin on both sides, monazomycin and C12 on one side, and separating NaCl‐ and KCl‐containing solutions as shown. B: equivalent circuit for the situation depicted in A.
Figure 60. Figure 60.

Schematic plots of the I‐V characteristics for the valinomycin‐ and monazomycin‐induced conductance elements of the circuit in Fig. 59. (To stress the analogy both to the axonal membrane and to Fig. 57, we have called the monazomycin‐induced emf and current ENa and INa, respectively, rather than Ecat and Icat.) The solid INa characteristic is the sodium current that would develop in the absence of inactivation (i.e., in the absence of Ccat). The dashed INa characteristic is the steady‐state current that develops in the presence of C12. Note that as drawn there is only one point (A) in the steady state where Ik = −INa, whereas there are three points (A, M, and B) that can satisfy this condition in the absence of inactivation.
Figure 61. Figure 61.

Steady‐state I‐V characteristic for an alamethicintreated membrane. The membrane (formed from sphingomyelin and tocopherol) separates 0.1 M KCl solutions buffered to pH 7 with 5 mM histidine chloride. Alamethicin is present only in the inside aqueous compartment at a concentration of 10−7 g/ml.

From Mueller & Rudin 53
Figure 62. Figure 62.

Voltage‐clamp responses of alamethicin‐treated membrane. A lecithin‐cholesterol membrane separates 0.5 M NaCl solutions with alamethicin in the inside aqueous compartment. Note the exponentiallike rise in current here, in contrast to the marked S‐shaped rise seen with monazomycin in Fig. 48 A.

From Mauro et al. 44
Figure 63. Figure 63.

Schematic drawing of the steady‐state g‐V characteristic and the consequent I‐V characteristic for an EIM‐treated membrane separating symmetrical solutions.
Figure 64. Figure 64.

Equivalent circuit for a membrane treated with both EIM and protamine, separating KCl solutions at different concentrations.
Figure 65. Figure 65.

Hypothetical steady‐state g‐V characteristics of gK and gCl for the circuit of Fig. 64. We have assumed that the relative amounts of EIM and protamine are such that there are many more potential K+ ‐conducting channels than Cl− ‐conducting channels. Note that [KCl]inside > [KCl]outside, so that EK is negative and ECl is positive.
Figure 66. Figure 66.

Schematic plots of the I‐V characteristics for the elements of the circuit in Fig. 64 based on the g‐V characteristics of Fig. 65. Solid curves, steady‐state characteristics; dashed line, instantaneous K+ characteristic in the resting state (i.e., with the system sitting at point A).
Figure 67. Figure 67.

Action potential produced with EIM and protamine. The membrane (formed from a mixture of sphingomyelin, phosphatidylserine, cholesterol, and tocopherol) was formed in 3 mM KCl buffered to pH 7 with 5 mM histidine chloride; EIM was then added to the inside solution to bring the resistance down to about 105 Ωcm2. The KCl concentration in the inside compartment was raised to 100 mM, and this produced a resting potential of −50 mV. Protamine was then added to the inside compartment to a concentration appropriate to produce the record shown.

Reconstructed from a motion picture by P. Mueller and D. O. Rudin
Figure 68. Figure 68.

Current versus time record showing discrete current jumps for a single EIM channel in an oxidized cholesterol membrane. The membrane separates 0.1 M KCl solutions buffered to pH 7 with 5 mM histidine chloride. Upper trace, current record; lower trace, applied voltage.

From Ehrenstein et al. 12
Figure 69. Figure 69.

Voltage dependence of the fraction of time a channel is open. Dashed curve, relative conductance of a many‐channel membrane; ○, determined from one‐channel experiment; ▴, determined from two‐channel experiment; ▪, determined from fourchannel experiment.

From Ehrenstein et al. 12
Figure 70. Figure 70.

A: fluctuations in the current through a membrane, formed from glycerol monooleate, in the presence of a very small amount of alamethicin. The aqueous phase was 2 M KCl, and the applied potential 210 mV. The baseline at each end of the group of fluctuations corresponds to the conductance of the pure lipid membrane. B: current fluctuations for the same system as in A recorded over a considerable period on a storage oscilloscope (1 large division = 5 × 10−10 A). The intensities of the lines represent the probabilities of finding the various conductance levels. This sort of record makes clear that within a group of fluctuations such as in A, the current tends to take certain well‐defined values.

From Gordon & Haydon 24


Figure 1.

The Donnan system.



Figure 2.

Qualitative plots of the Donnan ratio, r, as a function of the impermeant ion concentration, [N], for an impermeant ion of valence +1 or −1. (See Eq. 26.)



Figure 3.

Concentrations and potentials (both plotted on ordinate) in compartments 1 and 2 for a Donnan equilibrium.



Figure 4.

Sketch of concentration and potential profiles (both plotted on ordinate) for a Donnan equilibrium as determined from the Poisson‐Boltzmann analysis (cf. Fig. 3, which is the result of the thermodynamic analysis). Note that at any point x the electrical force (+F dψ/dx for cations and −F dψ/dx for anions) and the phenomenological diffusion “force” (−RT/[Na+] d[Na+]/dx and −RT/[C] l/dx) balance. The space‐charge density is shown in the lower part of the figure.



Figure 5.

The two‐phase system.



Figure 6.

Concentrations and potentials in the water and oil phases. The potential, ψ2, in the oil phase is positive, because we have assumed that the partition coefficient for sodium, , between oil and water is greater than that for chloride, .



Figure 7.

Sketch of concentration and potential profiles for a phase‐boundary equilibrium as determined from the Poisson‐Boltzmann analysis (cf. Fig. 6, which is the result of the thermodynamic analysis; as in Fig. 6, ). The space‐charge density is shown in the lower part of the figure.



Figure 8.

Method for measuring the potential difference across a membrane.



Figure 9.

Method of measuring membrane potential by making contact with the solutions through 3 M KCl junctions.



Figure 10.

An ion‐exchange membrane separating two solutions.



Figure 11.

A: concentration and potential profiles for an ionexchange membrane of large positive fixed‐charge density separating two solutions of equal concentration of NaCl. B: same as A, except that NaCl concentrations in compartments 1 and 2 are unequal. (The Na+ and Cl− profiles within the membrane have a small negative slope that is not clearly seen in the figure.)



Figure 12.

An oil membrane separating two NaCl aqueous solutions.



Figure 13.

Concentration and potential profiles for a “thick” oil membrane separating two NaCl aqueous solutions. The potential within the membrane (ψm) is positive, because we have assumed that Na+ partitions better into the membrane than Cl−.



Figure 14.

A: a redrawing of Fig. 13 for the concentration profiles of a “thick” oil membrane, with the space‐charge regions shown (exaggerated). B: concentration profiles for a “thin” oil membrane. Note that there is no region within the membrane where electroneutrality ([Na+] = [Cl−]) holds. The entire membrane thickness corresponds to the regions near the boundaries in Fig. 14A.



Figure 15.

A membrane separating two “infinite,” well‐stirred aqueous solutions. Stimulating and recording electrodes are shown.



Figure 16.

Equivalent circuit for a homogeneous, uncharged membrane separating two solutions of arbitrary ionic composition.



Figure 17.

Concentration profile within a thick, homogeneous, uncharged membrane separating two NaCl solutions at different concentrations.



Figure 18.

Equivalent circuit for the single‐salt case for a thick, homogeneous, uncharged membrane.



Figure 19.

Current‐voltage characteristics for the single‐salt case for a thick, homogeneous, uncharged membrane. These characteristics follow from the equivalent circuit in Fig. 18. (Note that these are drawn for c2 > c1; if c1 > c2, ψD (the potential at I = 0) would change sign. That is, the plots for u > v and u < v would be interchanged.



Figure 20.

How an electric field would develop, if at t = 0 there is a sharp boundary in salt concentration from c1 to c2 and the mobilities (i.e., diffusion coefficients) of the ions are different. The profiles shown are those that would exist at some short time later if the ions were uncharged. We see, however, that since the ions are charged, this creates a space‐charge region that will give rise to an electric field that tends to couple the motion of the ions together.



Figure 21.

Concentration profiles within the membrane for the bi‐ionic case of a membrane separating 0.1 M NaCl from 0.1 M KC1. A, I = 0; B, I is large and positive; C, I is large and negative.



Figure 22.

Sketch of the steady‐state current‐voltage characteristic for the bionic case of Fig. 21. Note the characteristics (dashed lines) for a membrane separating either symmetrical 0.1 M NaCl solutions or symmetrical 0.1 M KCl solutions, which the bi‐ionic characteristic approaches asymptotically for large positive and negative potentials, respectively. Also shown are the chord and slope resistance lines at point P.



Figure 23.

Nonlinear steady‐state current‐voltage characteristic with chords drawn at (I1, ψ1) and (I3, ψ3). If current is stepped from I1 to I2, the voltage originally attains the value at P and then increases with time along the vertical line from P until it reaches the value ψ2. Similarly, if current is stepped from I3 to I4, the voltage originally attains the value at Q and then moves along the vertical line from Q until it reaches the value ψ4.



Figure 24.

The change of voltage with time in response to steps of current. These responses follow from the analysis of Fig. 23 as given in the legend and text.



Figure 25.

RC (A) and RL (B) networks that would give responses similar to those shown in Fig. 24.



Figure 26.

Concentration profiles (A) and potential profile (B) for a positive fixed‐charge membrane separating NaCl solutions at different concentrations. Note the Donnan “jumps” in concentrations and potential at the interfaces.



Figure 27.

Concentration profile for an oil membrane separating a single salt at two different concentrations.



Figure 28.

Equivalent circuit for a mosaic membrane. The individual elements are shown to be ideally selective for a given ion, but this need not necessarily be the case.



Figure 29.

The Hodgkin‐Huxley equivalent circuit for the squid giant axon membrane in normal seawater.



Figure 30.

A circuit consisting of only a voltage‐dependent conductance element.



Figure 31.

Possible steady‐state g‐V characteristics and their corresponding I–V characteristics for the circuit of Fig. 30.



Figure 32.

A steady‐state I‐V characteristic with a region of negative slope. This is an enlargement of Fig. 31C″, illustrating why any point P in the region of negative slope is unstable under current‐clamp conditions (see text).



Figure 33.

Two circuits (A and C) with the same steady‐state g‐V characteristic (B).



Figure 34.

A: the steady‐state I‐V characteristic for the circuit of Fig. 33A, given the steady‐state g‐V characteristic of Fig. 33B. B: the steady‐state I‐V characteristic for the circuit of Fig. 33C, given the same steady‐state g‐V characteristic (i.e., Fig. 33B).



Figure 35.

The arrangement for forming and studying bilayers on a planar plastic partition separating two aqueous phases.



Figure 36.

Schematic drawing of a phospholipid bilayer. Filled circles, polar ends of the phospholipids; wavy lines, their fatty acid chains. The latter make up the hydrocarbon interior of the membrane, whereas the former anchor this region to the two aqueous phases.



Figure 37.

Diagrams of the response of a parallel RC circuit (B) to a step of current (A) or a step of voltage (C). (The voltageclamp response, C, is actually that which occurs if there is a small resistance in series with the circuit in B; without that resistance, the current response is a δ function.) These are the responses of an unmodified lipid bilayer.



Figure 38.

Schematic diagram of a cation carrier, such as valinomycin. The carbonyl oxygens provide a polar environment for the cation, and in fact substitute for the first hydration shell normally surrounding the cation in water. The exterior aspect of the molecule is nonpolar, thus making it compatible with the hydrocarbon interior of the membrane.



Figure 39.

Structural formulas for the cation carriers nonactin 11 and enniatin B 64.



Figure 40.

Structural formula of amphotericin B 47. Nystatin differs from amphotericin B in that it contains a tetraene and a diene in place of the heptaene chromophore; in other respects the molecules are very similar.



Figure 41.

Molecular model (CPK) of amphotericin B. In A the completely hydrophobic face of amphotericin B is seen; in B the molecule has been rotated 180° about its long axis to reveal the opposite face with its many hydroxyl groups. Note that the molecule consists of two chains: a polyene chain (seen on the right in A and on the left in B) and an amphipathic chain. The hydrophobic and hydrophilic faces of the amphipathic chain are seen in A and B, respectively. At the bottom of the figures are the polar amino sugar and carboxyl groups; at the top is a single hydroxyl group (seen most clearly in B). In C, a CPK model of lecithin is shown for comparison with amphotericin B.

From Finkelstein & Holz 19, reprinted by courtesy of Marcel Dekker, Inc


Figure 42.

Half of an amphotericin B‐created pore. Each amphotericin B molecule is schematized as a plane with a protuberance and a solid dot. The shaded portion of each plane represents the hydroxyl face of the amphipathic chain, the protuberance represents the amino sugar, and • represents the single hydroxyl group at the nonpolar end of the molecule. The aqueous phase is at the bottom of the figure and the middle of the membrane is at the top. We see that the interior of the pore is polar, whereas the exterior is completely nonpolar; there is also a wedge in the exterior of the pore, between each pair of amphotericin B molecules, that can accommodate a sterol molecule. Note the ring of hydroxyl groups in the middle of the membrane that can hydrogen bond with an identical structure from the other side to form a complete pore.

From Finkelstein & Holz 19


Figure 43.

Structural formula of valine‐gramicidin A 68. (Isoleucine gramicidin A has l‐Ile in place of the first l‐Val.)



Figure 44.

Conductance fluctuations for a membrane treated with a small amount of gramicidin A. (Actually one is seeing the current fluctuations in the face of a constant voltage across the membrane.) The membrane was formed from glyceryl monooleate dissolved in hexadecane and separates 0.5 M NaCl solutions at 23°C. The events marked α and β on the left‐hand side of the record occurred very infrequently.

From Hladky & Haydon 28


Figure 45.

Proposed structure of the gramicidin A channel. A: side view of a CPK molecular model of two molecules of gramicidin A in the π6(L,D) helix conformation, linked by three hydrogen bonds through their formyl ends. B: end view of the same model showing the central channel 2 Å in radius.

From Urry 75


Figure 46.

A: equivalent circuit for a membrane treated with both nystatin and valinomycin and separating KCl solutions at different concentrations. (The membrane capacitance is shown for completeness.) B: plot of IK and ICl as a function of V for the elements of the circuit in A. The slopes of these lines are gK and gCl, respectively. At point A, IK = −ICl; the value of V at this point is the membrane potential (in the absence of an external current source). Note that A must lie between EK and ECl; its exact position is determined by the slopes of IK and ICl (i.e., gK and gCl).



Figure 47.

Structural formula of alamethicin 61. Aib, α‐aminoisobutyric acid.



Figure 48.

Voltage‐clamp responses of monazomycin‐treated phosphatidylethanolamine membranes. Membranes (area = 1 mm2) were formed at room temperature in 0.1 M KCl; monazomycin was then added to the inside chamber (33 μg/ml in A and 17 μg/ml in B, C, and D). The arrows designate the I = 0 level, and the vertical blips in A, B, and D mark the onset of stimulation. A: successive current responses to positive rectangular voltage pulses of 25, 29, 33, 37, and 41 mV. B: record illustrating the difference in kinetics for turning on and turning off conductance. The initial stimulus is +60 mV; after the current reaches a steady state, the stimulus is switched to −60 mV. C: higher time resolution of the conductance turnoff. The current has already reached a steady‐state value for a voltage of +65 mV, and the stimulus is then switched to −65 mV. D: further illustration of difference in kinetics for conductance turnon and turnoff. The initial stimulus is +55 mV; after the current reaches a steady state, the voltage is reduced to +50 mV. Note that the initial drop in current when the voltage is reduced from +55 to +50 mV is much greater than the initial rise in current (hardly visible) when the voltage is increased from 0 to +55 mV. This reflects the much larger steady‐state conductance at +55 mV than at 0 mV.

From Muller & Finkelstein 57


Figure 49.

Schematic drawing of the nonlinear steady‐state I‐V characteristic of a monazomycin‐treated membrane separating identical solutions. At V1, the value of the current is that corresponding to P. Any change in potential (to V2 for example) leads to an “instantaneous” value of the current at t = 0 corresponding to a point on the chord drawn between P and the origin. The current subsequently relaxes (along the dashed vertical line) to its final value (at t =∞) on the steady‐state characteristic.



Figure 50.

The effect of divalent cation added to the outside solution on the steady‐state g‐V characteristic of a membrane with monazomycin in the inside solution. The phosphatidylglycerol membrane was formed at room temperature in 0.01 M KCl. Monazomycin was then added to the inside chamber, and after approximately 20 min, the g‐V characteristic labeled [Mg2+] = 0.0 mM was taken. MgS04 was then added to the outside chamber to the concentrations indicated on the curves. It required only 1 min (the time to stir in MgSO4) for the new g‐V characteristic to be established. Note the large shift of the g‐V characteristic to the right along the voltage axis. Similar results are obtained with Ca2+. Slope of lines = e‐fold conductance change per 4.0 mV; monazomycin concentration = 0.5 μg/ml; membrane area = 1 mm2.

From Muller & Finkelstein 58


Figure 51.

Potential profiles for a phosphatidylglycerol membrane separating the salt solutions indicated. At large (greater than 100 Å) distances from the membrane interfaces, the potential on both sides is zero (in the open‐circuited condition). The quantity of interest is ψ0, the value of the potential at the two interfaces. A: symmetrical salt solutions. ψ0 is the same at each interface, and there is no electric field within the membrane. B: divalent cation is present only in the REAR (outside) solution. Because of the asymmetry in surface potentials, there exists a potential difference across the membrane proper (though none measurable between the two solutions) and hence a field within the membrane. The sign of the potential difference is such as to turn off the monazomycin‐induced conductance. As far as the field that monazomycin “sees,” the situation in B is identical to having no Ca2+ present but instead passing sufficient current across the membranes to make the inside compartment −60 mV with respect to the outside. This is the basis for the shifts of the g‐V characteristics to the right in Fig. 50.

From Muller & Finkelstein 58


Figure 52.

A: steady‐state I‐V characteristics of a monazomycintreated phosphatidylethanolamine membrane: development of a negative‐slope conductance region by the addition of a salt gradient. The membrane was formed at room temperature in 0.02 M KCl; monazomycin was then added to the FRONT (inside) chamber and approximately 15 min later the I‐V characteristic labeled was obtained. A KCl gradient was established by raising [KCl] to 0.153 M in the REAR (outside) chamber. This gave rise to a diffusion emf of 42 mV. The I‐V characteristic labeled was taken approximately 1 min after the gradient was established. The gradient (along with the emf) was then abolished by raising [KCl] in the front (inside) chamber to 0.153 M. The I‐V characteristic labeled was taken approximately 1 min after the gradient was abolished. B: the steady‐state g‐V characteristics of a monazomycin‐treated membrane in the absence ( and ) and presence of a diffusion emf created by a salt gradient. The g‐V characteristics have been calculated from the corresponding I‐V characteristics in A by Equation 95 (for curves and ) and by Equation 99 (for curve ) with emf = 42 mV.

From Muller & Finkelstein 57


Figure 53.

Voltage‐clamp responses of a monazomycin‐treated membrane with a positive emf created by a salt gradient. The inside compartment, which has monazomycin (5 μg/ml), contains 0.01 M NaCl, and the outside compartment contains 0.077 M NaCl. The resulting emf is approximately 40 mV. Membrane area = 0.01 mm2. The membrane was formed from a mixture of ox‐brain lipid and tocopherol. See text and Fig. 54 for a discussion of these records.



Figure 54.

Schematic drawings (lower two) of the records in Fig. 53 along with the steady‐state I‐V characteristic of the membrane (upper drawing). See text for a discussion of these drawings.



Figure 55.

Equivalent circuit for a membrane treated with both nystatin and monazomycin and separating NaCl solutions with [NaCl]outside > [NaCl]inside. (The membrane capacitance is shown for completeness.)



Figure 56.

Bistable property of a membrane treated with both nystatin and monazomycin. The membrane was formed at room temperature in a solution containing 0.01 M KCl and 0.001 M MgS04. Nystatin was added to both chambers to a concentration of 5 μg/Ml. After approximately 20 min the membrane reached a steady conductance of about 10−7 Ω−1. At this point the KCl concentration was raised in the outside chamber to 0.077 M. This produced a membrane emf of −41 mV. Monazomycin was then added to the front chamber to a concentration of 3 μg/ml. This produced no significant change in the emf. After approximately 15 min, the above record was obtained. (In the record the horizontal line is V = 0; above the line are negative potentials, and below the line are positive potentials.) At a, a positive step of current = 3.75 nA was applied and at b it was removed. The potential returned toward the original −41 mV. At c, the current was again applied, and at d it was removed. This time the potential continued increasing and achieved a new stable state of +41 mV. At e, a negative step of current = −4.05 nA was applied, and at f it was removed. The potential returned to +41 mV. At g the current was again applied and at h it was removed. This time the potential continued to decrease and flipped back to the original state of −41 mV. Note the much smaller IR drops at e and g compared with that at a, thus demonstrating that the conductance is much higher at +41 mV (where the monazomycin‐induced conductance is turned on) than at −41 mV (where only the nystatin‐induced conductance is significant).

From Muller & Finkelstein 57


Figure 57.

Schematic plots of the steady‐state I‐V characteristics for the nystatin‐ and monazomycin‐induced conductance elements of the circuit in Fig. 55. The slope of the dashed line is the monazomycin‐induced conductance, gNa, in the resting state (at point A). (The slope of the line drawn between ENa and any point on the INa characteristic gives the value of gNa at the voltage corresponding to that point.)



Figure 58.

Inactivation of monazomycin‐induced conductance by dodecyltriethylammonium bromide (C12). A membrane formed from phosphatidylglycerol and cholesterol separates symmetrical 0.1 M KCl solutions and contains in the inside compartment 1 μg/ml monazomycin and 1.3 × 10−5 M dodecyltriethylammonium bromide. The current record (traced from the oscillograph recording) is in response to a voltage step (applied at the upward arrow and removed at the downward arrow) of 88 mV. Membrane area = 1 mm2. Vertical line, 200 nA; horizontal line, 5 s. Note the contrast of this record with Fig. 48 A, where C12 is absent.

From Heyer 27


Figure 59.

A: membrane treated with valinomycin on both sides, monazomycin and C12 on one side, and separating NaCl‐ and KCl‐containing solutions as shown. B: equivalent circuit for the situation depicted in A.



Figure 60.

Schematic plots of the I‐V characteristics for the valinomycin‐ and monazomycin‐induced conductance elements of the circuit in Fig. 59. (To stress the analogy both to the axonal membrane and to Fig. 57, we have called the monazomycin‐induced emf and current ENa and INa, respectively, rather than Ecat and Icat.) The solid INa characteristic is the sodium current that would develop in the absence of inactivation (i.e., in the absence of Ccat). The dashed INa characteristic is the steady‐state current that develops in the presence of C12. Note that as drawn there is only one point (A) in the steady state where Ik = −INa, whereas there are three points (A, M, and B) that can satisfy this condition in the absence of inactivation.



Figure 61.

Steady‐state I‐V characteristic for an alamethicintreated membrane. The membrane (formed from sphingomyelin and tocopherol) separates 0.1 M KCl solutions buffered to pH 7 with 5 mM histidine chloride. Alamethicin is present only in the inside aqueous compartment at a concentration of 10−7 g/ml.

From Mueller & Rudin 53


Figure 62.

Voltage‐clamp responses of alamethicin‐treated membrane. A lecithin‐cholesterol membrane separates 0.5 M NaCl solutions with alamethicin in the inside aqueous compartment. Note the exponentiallike rise in current here, in contrast to the marked S‐shaped rise seen with monazomycin in Fig. 48 A.

From Mauro et al. 44


Figure 63.

Schematic drawing of the steady‐state g‐V characteristic and the consequent I‐V characteristic for an EIM‐treated membrane separating symmetrical solutions.



Figure 64.

Equivalent circuit for a membrane treated with both EIM and protamine, separating KCl solutions at different concentrations.



Figure 65.

Hypothetical steady‐state g‐V characteristics of gK and gCl for the circuit of Fig. 64. We have assumed that the relative amounts of EIM and protamine are such that there are many more potential K+ ‐conducting channels than Cl− ‐conducting channels. Note that [KCl]inside > [KCl]outside, so that EK is negative and ECl is positive.



Figure 66.

Schematic plots of the I‐V characteristics for the elements of the circuit in Fig. 64 based on the g‐V characteristics of Fig. 65. Solid curves, steady‐state characteristics; dashed line, instantaneous K+ characteristic in the resting state (i.e., with the system sitting at point A).



Figure 67.

Action potential produced with EIM and protamine. The membrane (formed from a mixture of sphingomyelin, phosphatidylserine, cholesterol, and tocopherol) was formed in 3 mM KCl buffered to pH 7 with 5 mM histidine chloride; EIM was then added to the inside solution to bring the resistance down to about 105 Ωcm2. The KCl concentration in the inside compartment was raised to 100 mM, and this produced a resting potential of −50 mV. Protamine was then added to the inside compartment to a concentration appropriate to produce the record shown.

Reconstructed from a motion picture by P. Mueller and D. O. Rudin


Figure 68.

Current versus time record showing discrete current jumps for a single EIM channel in an oxidized cholesterol membrane. The membrane separates 0.1 M KCl solutions buffered to pH 7 with 5 mM histidine chloride. Upper trace, current record; lower trace, applied voltage.

From Ehrenstein et al. 12


Figure 69.

Voltage dependence of the fraction of time a channel is open. Dashed curve, relative conductance of a many‐channel membrane; ○, determined from one‐channel experiment; ▴, determined from two‐channel experiment; ▪, determined from fourchannel experiment.

From Ehrenstein et al. 12


Figure 70.

A: fluctuations in the current through a membrane, formed from glycerol monooleate, in the presence of a very small amount of alamethicin. The aqueous phase was 2 M KCl, and the applied potential 210 mV. The baseline at each end of the group of fluctuations corresponds to the conductance of the pure lipid membrane. B: current fluctuations for the same system as in A recorded over a considerable period on a storage oscilloscope (1 large division = 5 × 10−10 A). The intensities of the lines represent the probabilities of finding the various conductance levels. This sort of record makes clear that within a group of fluctuations such as in A, the current tends to take certain well‐defined values.

From Gordon & Haydon 24
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Article Title: Physical Principles and Formalisms of Electrical Excitability
 

How to Cite

Alan Finkelstein, Alexander Mauro. Physical Principles and Formalisms of Electrical Excitability. Compr Physiol 2011, Supplement 1: Handbook of Physiology, The Nervous System, Cellular Biology of Neurons: 161-213. First published in print 1977. doi: 10.1002/cphy.cp010106