Proc. Nat. Acad. Sci. USA
Vol. 73, No. 1, pp. 10O-103, January 1976 Biochemistry
Spike-forming model of neural membrane: Computer simulation of additional perfused axon experiments* (membrane conductance/gates and channels/gate conversion/action potentials/voltage clamps)
DEAN E. WOOLDRIDGE 4545 Via Esperanza, Santa Barbara, California 93110
Contributed by Dean E. Wooldridge, October 30, 1975
Here v is the membrane potential divided by 25 mV, the value of the Maxwell-Boltzmann coefficient at room temperature. The first term of Eq. 1 represents the electronic conductance of the membrane, assumed independent of the nature of the internal and external fluids. The second and third terms are the ionic currents generated by the entry of Cland Ca++ ions into the axon through the system of "fixed" pores. These pores are assumed "fixed" in the sense that their relatively simple gate molecules are not affected by the adjacent fluid. Thus, for the first three components of the membrane current equation, the values of the constants are the same as those found to be appropriate for the squid giant axon in ref. 2; these terms have been directly copied from Eq. 4 of that paper. The remainder of Eq. 1 specifies the contribution to the total membrane current made by Ca++ and Cl- ions that enter the axon through the "convertible" pores. In ref. 1 it was established that the inner gates of these pores change their properties as a function of v; in ref. 2 it was specified that these properties are also affected by entry of ions from the internal fluid into the molecular configuration of the inner gates, with the magnitudes and details of the effects different for different types of ion. For this reason the only constants for the convertible pores whose numerical values can be carried over from ref. 2 are those that do not involve characteristics of the inner gates. These are the constants 61 AA/cm2 and 1.8 ,A/cm2 shown in the above equation. The four remaining constants have been given the symbolic designations of Eq. 3, ref. 2, from which the numerical Eq. 4 was derived. The value of each a depends on the relative ease of escape of its type of ion from a convertible pore, by way of the inner and outer gates. The notation for these as will be simplified to aca and aci henceforth. The values of the two ys depend on the relative ease of entry into and loss from the inner gate configuration by Ca++ ions supplied to the channel by the external fluid. P designates the fraction of the convertible gates in a "poisoned" state, in the sense that the gates are unable to pass any ions and are prevented from changing into a conducting configuration for at least a few milliseconds. The meanings of all the symbols used in Eq. 1 are explained more fully in ref. 2. Values of constants that lead to action potential spikes For the sodium-free axons now under consideration, it is
ABSTRACT In this paper a physical model of the neural membrane that I developed is tested for its ability to account for the details of the action potential spikes observed in squid giant axons, when perfused with sodium- and potassium-free fluids and surrounded by a calcium chloride solution. The near-zero resting potential of these perfused axons is accounted for by the model. The sizes, shapes, and conductances of the spikes observed with a number of different perfusates are also accounted for, when suitable values are assigned to the so-far unmeasured membrane constants that control the model properties. The three papers comprising the series are summarized at the end of this paper. In the first of this series of papers, a model of the neural membrane was described; its action potential spiking characteristics were shown to be similar to those of the squid giant neuron in the normal chemical environment of potassium-rich internal fluid, sodium- and calcium-rich external fluid (1). In the second paper, some features were added to the membrane model; the result was the appearance of properties similar to those observed in perfusion experiments in which the normal axoplasm of a squid giant neuron is replaced by a potassium-free, sodium-rich fluid, and the axon is immersed in a sodium- and potassium-free, calcium-rich bath (2). This third paper is essentially a continuation of the second, in that it uses the same model for computer simulation of axon perfusion experiments involving internal fluids in which the cation is not sodium, but one of several types of polyatomic molecules (3-5). The current equation Since it seems unlikely that the neural membrane should be permeable to the cations of all of the dozen or so types of internal fluid found to permit action potential spiking, I have left out of the model dependence on any substantial membrane permeability to ions other than calcium, chlorine, and sodium. And sodium is absent from the external as well as the internal fluid in most of the experimental work to be dealt with in this paper. Hence we can start the present considerations by writing down a modified form of Eq. 4, ref. 2, to give the current through the membrane as a function of the potential across it, under the anticipated sodium-free conditions. The resulting expression for membrane current, in gA/cm2, is Im = 1.3v + ev/(l + 0.5e) - 3.7/(e2v + 0.4) - (1 -
P)[61acwga/(e
aCBCa)
easy to show that values of the ys and as similar to those of the sodium-containing axon of ref. 2 do not lead to action potential spiking. Curve A of Fig. 1 is a plot of Eq. 1 for P = 0, using the ref. 2 values of 0.1 and 5 for YA and 'YKI, and values of 300 for aci and aCa to conform with the previous
1.8A-yAKe K4 a(CK (/(e' + aCK CI)] + (1
*
+
+ YAe
-t +
YAYKe)
[1]
Third paper of a series. See refs. 1 and 2. 100
Biochemistry: Wooldridge requirement that these terms >>1. For this current/potential relation, the only stable value for the membrane potential is +3.85 (+96 mV). Quite a different result was obtained when the corresponding relation for the sodium-containing axon was plotted in Fig. 1, ref. 2, for this curve displayed the triple crossover of the current axis that leads to spiking. It had a resting potential of about -1.5 v units instead of +3.85, and a negative conductance region associated with the triple crossover whereby the potential would suddenly jump to about +0.7 when a stimulus current of a little more than 1 gA/cm2 was injected into the axon. It is, of course, absence of the 8.9 MA/cm2 "sodiumpump" current that prevents the essential triple crossover in the sodium-free case. In the sodium-containing case, this current component lifted the curve enough to put part of it above the axis and thereby make possible a stable low-v resting potential and a spiking capability. If such a capability is to be secured for the model when operating under sodiumfree conditions, some other way must be found to endow the current/potential curve with a triple crossover. Analysis of Eq. 1 shows that this result cannot be achieved by any combination of values of aca, adc, and P. as long as YA and YK1 are kept small as in the sodium-containing situation. The only way a triple crossover can be obtained is by a large increase in 'YA, UKp, or both. This is illustrated by Curve B of Fig. 1. For plotting this curve, aca and acl were each kept at 300 and the poisoning factor P was held at 0 as before, to approximate the expected conditions in the resting neuron. But 'YA and 'YK1 were each multiplied by a factor of VI'T0, giving "YA = 3.2, 'YAYK1 = 500. The result is a spike-forming characteristic, with a resting potential very slightly above 0, and a jump of potential to +4 when the stimulus current exceeds about 2 MA/cm2. The return of the spike to a low value of v then depends on the gradual build-up of P. as described in connection with Fig. 1 in ref. 2. It is interesting to note that there is no combination of values for the as and ys that permits spiking-that is, yields a triple crossover of the current axis-and at the same time gives a resting potential higher than 0.8 (+20 mV) or lower than 0. This is generally compatible with some experimental observations. In one set of experiments (3) the resting potentials were measured in spiking axons perfused with several different types of sodium-free internal fluid. The values reported were all close to 0-never higher than +15 mV. To be sure, some small negative values were also reporteddown to -10 mV (0.4 v unit)-which does not appear consistent with predictions from the model. However, a small negative resting potential can be displayed by the model if the membrane is not completely denied permeability to ions of the internal perfusate. A current of 2 MLA/cm2 could lower the resting potential to -7 or -8 mV, with only a small effect on the other characteristics of the action potential spikes we are about to consider. Such an additional component has not been put into the current equation, however, and the resting potentials will be slightly positive for all the calculations of this paper. For these calculations two simplifying assumptions will be installed in Eq. 1. One is that e-v 20, say, this condition is adequately satisfied. The other simplifying assumption is that, for any of the perfusates to be considered, the ratio of "VA to its previous value of 0.1 for sodium-containing perfusate is the same as the ratio of YK1 to its previous value of 5. Physically, this would imply that the replacement of Na+ by the perfusate
Proc. Nat. Acad. Sci. USA 73 (1976) +10 0
-2
-I
0
+1
+2
101
+3
N
E -10
-20
;:-30_30 -40 wZ cr
-0
0 -60
FIG. 1. Computed membrane current/potential curves (potential in v units). For both curves, ac. = aCl = 300, P = 0. For A, YA = 0.1, 'YK1 = 5. For B, 'YA = 3.2, YA'YK1 = 500. Only curve B permits spiking.
ion in the complex inner gate molecule increases by the same amount the binding energy of both of the Ca++ ions that can attach to the under side of the gate. While this seems more or less reasonable, the main justification for the assumption, as for the first simplifying assumption, is that it leads to spike shapes and sizes similar to those experimental*ly observed. Incidentally, it should be noted that the large. increases in the value of the ys on which spiking of the model depends do not call for unreasonably large increases in the tightness of binding of the calcium ions: the 32:1 increase in each used for Fig. lB corresponds to a change of less than 0.1 electronvolt in the binding energy. With the simplifications just described, Eq. 1 becomes Im = L3v + eU/(l + 0V) - 3.7/(e2v + 0.4) - (1 -
P)[61/(1
+
e2V/aca) - 0.9r2e-4v]
(1 + 0.lre-2U + 0.5r2e-v) [2] where r has been set in to represent the ratio of each y to its counterpart in the internal-sodium situation, and may be expected to have a value >>1 for any sodium-free spike-forming axon. Spike-determining differential equations The reaction rate equations for the fast and slow poisoning processes that comprise P are given in Eqs. 5 of ref. 2. When equipped with the new expressions for the 'ys in terms of r, these equations become dPf/dt = 1- Pf-PsXC + C2e-'2) tf(1 + 0.lre-2 + 0.5r2e4U)-Pf + tf [3a] = dP./dt (1-Pf-PP) t(1++ 0.1r 0.5r2e4)-S + kse c [3b] In Eq. 3a the value of 1.5 from ref. 2 will be used for C2 in all cases. Actually the term C2e-2v could be neglected. It turns out that, for all the calculations to be made, its effect is negligible as long as C2 is less than about 10. tf, tf + C1, ts, and kse2v are time constants of the reactions that determine whether the inner gate of a convertible pore has a "poisoned" or "nonpoisoned" molecular configuration (2). The third differential equation of the set that jointly determines Pf, Ps, and v is obtained by taking explicit account of the membrane capacitance and the stimulus current, as explained in connection with Eqs. 6 and 7, ref. 2. The resulting equation is dv/dt = 26 sec-1'ext - L3v - e/(l + 0.5et) + 3.7/(e2' + 0.4) +(1 - Pf - P)[61/(l + e2 /aca) - 0.9r2e-4] *. (1 + 0.1re-2U + 0.5r2e4,)t [4]
Here Iext is the probe-injected stimulus current in it is positive when in a direction to increase v.
liA/cm2;
102
Biochemistry: Wooldridge
Proc. Nat. Acad. Sci. USA 73 (1976) 4 2
t'J II 0
3
0
~~A
ol-
'ext o
Vol. 73, No. 1, pp. 10O-103, January 1976 Biochemistry
Spike-forming model of neural membrane: Computer simulation of additional perfused axon experiments* (membrane conductance/gates and channels/gate conversion/action potentials/voltage clamps)
DEAN E. WOOLDRIDGE 4545 Via Esperanza, Santa Barbara, California 93110
Contributed by Dean E. Wooldridge, October 30, 1975
Here v is the membrane potential divided by 25 mV, the value of the Maxwell-Boltzmann coefficient at room temperature. The first term of Eq. 1 represents the electronic conductance of the membrane, assumed independent of the nature of the internal and external fluids. The second and third terms are the ionic currents generated by the entry of Cland Ca++ ions into the axon through the system of "fixed" pores. These pores are assumed "fixed" in the sense that their relatively simple gate molecules are not affected by the adjacent fluid. Thus, for the first three components of the membrane current equation, the values of the constants are the same as those found to be appropriate for the squid giant axon in ref. 2; these terms have been directly copied from Eq. 4 of that paper. The remainder of Eq. 1 specifies the contribution to the total membrane current made by Ca++ and Cl- ions that enter the axon through the "convertible" pores. In ref. 1 it was established that the inner gates of these pores change their properties as a function of v; in ref. 2 it was specified that these properties are also affected by entry of ions from the internal fluid into the molecular configuration of the inner gates, with the magnitudes and details of the effects different for different types of ion. For this reason the only constants for the convertible pores whose numerical values can be carried over from ref. 2 are those that do not involve characteristics of the inner gates. These are the constants 61 AA/cm2 and 1.8 ,A/cm2 shown in the above equation. The four remaining constants have been given the symbolic designations of Eq. 3, ref. 2, from which the numerical Eq. 4 was derived. The value of each a depends on the relative ease of escape of its type of ion from a convertible pore, by way of the inner and outer gates. The notation for these as will be simplified to aca and aci henceforth. The values of the two ys depend on the relative ease of entry into and loss from the inner gate configuration by Ca++ ions supplied to the channel by the external fluid. P designates the fraction of the convertible gates in a "poisoned" state, in the sense that the gates are unable to pass any ions and are prevented from changing into a conducting configuration for at least a few milliseconds. The meanings of all the symbols used in Eq. 1 are explained more fully in ref. 2. Values of constants that lead to action potential spikes For the sodium-free axons now under consideration, it is
ABSTRACT In this paper a physical model of the neural membrane that I developed is tested for its ability to account for the details of the action potential spikes observed in squid giant axons, when perfused with sodium- and potassium-free fluids and surrounded by a calcium chloride solution. The near-zero resting potential of these perfused axons is accounted for by the model. The sizes, shapes, and conductances of the spikes observed with a number of different perfusates are also accounted for, when suitable values are assigned to the so-far unmeasured membrane constants that control the model properties. The three papers comprising the series are summarized at the end of this paper. In the first of this series of papers, a model of the neural membrane was described; its action potential spiking characteristics were shown to be similar to those of the squid giant neuron in the normal chemical environment of potassium-rich internal fluid, sodium- and calcium-rich external fluid (1). In the second paper, some features were added to the membrane model; the result was the appearance of properties similar to those observed in perfusion experiments in which the normal axoplasm of a squid giant neuron is replaced by a potassium-free, sodium-rich fluid, and the axon is immersed in a sodium- and potassium-free, calcium-rich bath (2). This third paper is essentially a continuation of the second, in that it uses the same model for computer simulation of axon perfusion experiments involving internal fluids in which the cation is not sodium, but one of several types of polyatomic molecules (3-5). The current equation Since it seems unlikely that the neural membrane should be permeable to the cations of all of the dozen or so types of internal fluid found to permit action potential spiking, I have left out of the model dependence on any substantial membrane permeability to ions other than calcium, chlorine, and sodium. And sodium is absent from the external as well as the internal fluid in most of the experimental work to be dealt with in this paper. Hence we can start the present considerations by writing down a modified form of Eq. 4, ref. 2, to give the current through the membrane as a function of the potential across it, under the anticipated sodium-free conditions. The resulting expression for membrane current, in gA/cm2, is Im = 1.3v + ev/(l + 0.5e) - 3.7/(e2v + 0.4) - (1 -
P)[61acwga/(e
aCBCa)
easy to show that values of the ys and as similar to those of the sodium-containing axon of ref. 2 do not lead to action potential spiking. Curve A of Fig. 1 is a plot of Eq. 1 for P = 0, using the ref. 2 values of 0.1 and 5 for YA and 'YKI, and values of 300 for aci and aCa to conform with the previous
1.8A-yAKe K4 a(CK (/(e' + aCK CI)] + (1
*
+
+ YAe
-t +
YAYKe)
[1]
Third paper of a series. See refs. 1 and 2. 100
Biochemistry: Wooldridge requirement that these terms >>1. For this current/potential relation, the only stable value for the membrane potential is +3.85 (+96 mV). Quite a different result was obtained when the corresponding relation for the sodium-containing axon was plotted in Fig. 1, ref. 2, for this curve displayed the triple crossover of the current axis that leads to spiking. It had a resting potential of about -1.5 v units instead of +3.85, and a negative conductance region associated with the triple crossover whereby the potential would suddenly jump to about +0.7 when a stimulus current of a little more than 1 gA/cm2 was injected into the axon. It is, of course, absence of the 8.9 MA/cm2 "sodiumpump" current that prevents the essential triple crossover in the sodium-free case. In the sodium-containing case, this current component lifted the curve enough to put part of it above the axis and thereby make possible a stable low-v resting potential and a spiking capability. If such a capability is to be secured for the model when operating under sodiumfree conditions, some other way must be found to endow the current/potential curve with a triple crossover. Analysis of Eq. 1 shows that this result cannot be achieved by any combination of values of aca, adc, and P. as long as YA and YK1 are kept small as in the sodium-containing situation. The only way a triple crossover can be obtained is by a large increase in 'YA, UKp, or both. This is illustrated by Curve B of Fig. 1. For plotting this curve, aca and acl were each kept at 300 and the poisoning factor P was held at 0 as before, to approximate the expected conditions in the resting neuron. But 'YA and 'YK1 were each multiplied by a factor of VI'T0, giving "YA = 3.2, 'YAYK1 = 500. The result is a spike-forming characteristic, with a resting potential very slightly above 0, and a jump of potential to +4 when the stimulus current exceeds about 2 MA/cm2. The return of the spike to a low value of v then depends on the gradual build-up of P. as described in connection with Fig. 1 in ref. 2. It is interesting to note that there is no combination of values for the as and ys that permits spiking-that is, yields a triple crossover of the current axis-and at the same time gives a resting potential higher than 0.8 (+20 mV) or lower than 0. This is generally compatible with some experimental observations. In one set of experiments (3) the resting potentials were measured in spiking axons perfused with several different types of sodium-free internal fluid. The values reported were all close to 0-never higher than +15 mV. To be sure, some small negative values were also reporteddown to -10 mV (0.4 v unit)-which does not appear consistent with predictions from the model. However, a small negative resting potential can be displayed by the model if the membrane is not completely denied permeability to ions of the internal perfusate. A current of 2 MLA/cm2 could lower the resting potential to -7 or -8 mV, with only a small effect on the other characteristics of the action potential spikes we are about to consider. Such an additional component has not been put into the current equation, however, and the resting potentials will be slightly positive for all the calculations of this paper. For these calculations two simplifying assumptions will be installed in Eq. 1. One is that e-v 20, say, this condition is adequately satisfied. The other simplifying assumption is that, for any of the perfusates to be considered, the ratio of "VA to its previous value of 0.1 for sodium-containing perfusate is the same as the ratio of YK1 to its previous value of 5. Physically, this would imply that the replacement of Na+ by the perfusate
Proc. Nat. Acad. Sci. USA 73 (1976) +10 0
-2
-I
0
+1
+2
101
+3
N
E -10
-20
;:-30_30 -40 wZ cr
-0
0 -60
FIG. 1. Computed membrane current/potential curves (potential in v units). For both curves, ac. = aCl = 300, P = 0. For A, YA = 0.1, 'YK1 = 5. For B, 'YA = 3.2, YA'YK1 = 500. Only curve B permits spiking.
ion in the complex inner gate molecule increases by the same amount the binding energy of both of the Ca++ ions that can attach to the under side of the gate. While this seems more or less reasonable, the main justification for the assumption, as for the first simplifying assumption, is that it leads to spike shapes and sizes similar to those experimental*ly observed. Incidentally, it should be noted that the large. increases in the value of the ys on which spiking of the model depends do not call for unreasonably large increases in the tightness of binding of the calcium ions: the 32:1 increase in each used for Fig. lB corresponds to a change of less than 0.1 electronvolt in the binding energy. With the simplifications just described, Eq. 1 becomes Im = L3v + eU/(l + 0V) - 3.7/(e2v + 0.4) - (1 -
P)[61/(1
+
e2V/aca) - 0.9r2e-4v]
(1 + 0.lre-2U + 0.5r2e-v) [2] where r has been set in to represent the ratio of each y to its counterpart in the internal-sodium situation, and may be expected to have a value >>1 for any sodium-free spike-forming axon. Spike-determining differential equations The reaction rate equations for the fast and slow poisoning processes that comprise P are given in Eqs. 5 of ref. 2. When equipped with the new expressions for the 'ys in terms of r, these equations become dPf/dt = 1- Pf-PsXC + C2e-'2) tf(1 + 0.lre-2 + 0.5r2e4U)-Pf + tf [3a] = dP./dt (1-Pf-PP) t(1++ 0.1r 0.5r2e4)-S + kse c [3b] In Eq. 3a the value of 1.5 from ref. 2 will be used for C2 in all cases. Actually the term C2e-2v could be neglected. It turns out that, for all the calculations to be made, its effect is negligible as long as C2 is less than about 10. tf, tf + C1, ts, and kse2v are time constants of the reactions that determine whether the inner gate of a convertible pore has a "poisoned" or "nonpoisoned" molecular configuration (2). The third differential equation of the set that jointly determines Pf, Ps, and v is obtained by taking explicit account of the membrane capacitance and the stimulus current, as explained in connection with Eqs. 6 and 7, ref. 2. The resulting equation is dv/dt = 26 sec-1'ext - L3v - e/(l + 0.5et) + 3.7/(e2' + 0.4) +(1 - Pf - P)[61/(l + e2 /aca) - 0.9r2e-4] *. (1 + 0.1re-2U + 0.5r2e4,)t [4]
Here Iext is the probe-injected stimulus current in it is positive when in a direction to increase v.
liA/cm2;
102
Biochemistry: Wooldridge
Proc. Nat. Acad. Sci. USA 73 (1976) 4 2
t'J II 0
3
0
~~A
ol-
'ext o
