Repetitive firing behavior in nerve cell models based upon a simplified form of the Hodgkin-Huxley equations

Repetitive Firing Behavior in Nerve Cell Models Based upon a Simplified Form of the Hodgkin-Huxley Equations ERIK SKAUGEN Departments of Physics and Physiology, Communicated

Universi(y of Oslo, Oslo, Norway.

by A. Moller

ABSTRACT An investigation

of a nerve cell model based upon a simplified

form of the Hodgkin-

Huxley equations has been carried through. Several different cases have been treated, and analytical solutions for the frequency-current relationship have been obtained for each case. The basic model is a nerve cell with a sharp threshold for firing, and where the steady state values of the membrane conductances are independent of the potential below threshold. The form of the frequency-current relationship was for this model given by where f is the firing frequency, i is the injected current, and to is f=[ln(l+l/i)+tJ’, approximately the duration of the action potential. This relationship can be reasonably well fitted by a power law. The largest discrepancy is found around the current threshold for firing, where the above relationship has a vertical tangent. The effect upon this basic model of an additional dependence of the steady state values of the membrane conductances upon the membrane potential was investigated. It was found that an increase of the sodium conductance with the potential made the frequency-current relationship more linear, while an increase of the potassium conductance with the potential made the relationship less linear. This last case resembled more the frequency-current relationship for the squid giant axon investigated by Hodgkin and Huxley. The effects of a slow component of the potassium conductance and the effects of an electrogenic pump were also investigated. Both of these mechanisms resulted in adaptation. From the onset of a current step increase, the firing frequency rapidly increased with time and thereafter gradually decreased toward a steady state, non-zero value. Both of these adapting mechanisms made the frequency-current relationship more linear, and the forms of the steady state relationships were almost the same in the two cases. It will thus be difficult to decide between the two mechanisms possibly responsible for adaptation from the observed steady state frequency-current relationship alone.

INTRODUCTION The most important property of a nerve cell as an information processing and transmitting element in the nervous system is perhaps the output firing frequency as a function of the input current. The input current may represent synaptic activity, or chemical or mechanical stimuli in sense cells. MATHEMATICAL Q American

Elsevier

BIOSCIENCES Publishing

26, 119-155

Company,

(1975)

Inc., 1975

119

120

E. SKAUGEN

Approximate empirical relationships between current and frequency are easily found for many nerve cells. Some of these relationships are generally applicable to a large number of nerve cells by adjusting the parameters in the general formula. One example is the “power law”, where the frequencyf is given as a function of the input current i byf= k(i- Q. Here k. i, and n are parameters which depend upon the particular cell chosen to be represented. The parameters in these empirical formulas have no apparent connection with measurable quantities of the nerve cell, such as membrane resistance and capacitance, the dimensions of the cell, etc. In order to find a frequency-current relationship where these quantities are parameters, a model of the nerve cell must be assumed. One famous example of such a model is Hodgkin and Huxley’s equations for the giant axon of a squid [I]. But in this model about sixteen parameters are necessary, and some of them are very difficult to measure. Only easily accessible parts of a few nerve cells have been analyzed so completely that the complete equations could be found [l-3]. I will try to follow a middle way between the two extremes mentioned above. A simplified model of a nerve cell, based upon Hodgkin and Huxley’s equations, is used to derive an analytical expression for the frequency-current relationship. This expression contains the most easily measurable parameters of the nerve cell membrane. It should thus be fairly easy to test whether the model gives a good representation of a particular cell or not. It is believed that for a large number of the different types of nerve cells, the approximations of the calculated frequency-current relationship actual one.

model are realistic, and is a good approximation

First a very simple model is considered, added to it in order to make it more realistic.

and

MODEL

BELOW

1: CONSTANT

CONDUCTANCES

then

new

that the to the

features

are

THRESHOLD

The first model presented here has been discussed before by Stein [4]. It will be used as a starting point in the calculation of the frequency-current relationship. Let us consider a nerve cell where there is a sharp threshold E, for firing, and where between two action potentials the membrane conductance g is independent of the membrane potential E. If the cell is sufficiently stimulated, for instance by an injected current i, it will fire repeatedly. During an action potential, and maybe a short while after it, the current will have no effect upon the membrane potential. This time interval is here called the total refractory period t,. It is equal to the minimum time interval

between

two firings

of the cell. We also assume

that the membrane

FIRING

BEHAVIOR

IN NERVE

CELL

121

MODELS

potential E and the conductance g are reset to the values Ep and g,, at the end of each total refractory period. From then on the membrane potential will increase toward the firing threshold, while the conductance is constant and equal to g,. If the capacitance C of the membrane is known, the firing frequency as a function of the current can be calculated.

FIG. 1 Membrane potential as a function of time in a nerve cell. Potential increases exponentially (unbroken line) from E, and towards the firing threshold E, after an action potential.

Firing

frequency

is (to+ 1,)-‘.

Symbols

are explained

in the text.

Due to the constant conductance, the membrane potential E will increase exponentially from E, towards the threshold Et. This is illustrated in Fig. 1, where some of the parameters used are also shown. The rise of E is governed by the simple differential equation dE -=_ dt

go C

(1)

where E, is the resting potential. If E, + i/g, is larger than the threshold, the cell will fire again. From Eq. (1) we can find the potential as a function of time. If it takes a time t, to rise from Ep to E,, the firing frequency of the cell is given by the reciprocal of the time interval to+ t,: -1

E. SKAUGEN

122

Figure 2 shows f/( go/ C) as a function of the normalized current i/[(E, E,)g,,] for different values of 1,. Equation (2) gives a threshold current i, equal to (E, - E,) go as indicated by the figure. From this normalized plot the actual frequency-current relationship for a certain nerve cell can be found in more conventional units. The membrane time constant C/g,, and the membrane parameters Et, E,, E, and g, must then be known. It is of interest to note that the form of the frequency-current relationship depends only upon the value of b,, relative to C/g,. The frequency scale is determined by the membrane time constant C/g,,, while the current scale is determined by the product (E, - E,) g,. The only effect of a change in the resting potential E, is to shift the curve along the current axis.

I

1

05 t,=lE,

-E,

lgo

1

0

/

CURRENT [(E,!Ep,g,]

FIG. 2 Firing frequency as a function of injected current for different values of the total refractory period tw Normalized units are used.

1 This simple model gives the frequency-current relationship from relatively easily measured parameters. There is, however, a certain ambiguity in determining the total refractory period t,. And at the end of this period we may not find the minimum potential. These difficulties may be expected, since the crudest approximations of this model are its assumptions about the events just after an action potential has occurred. MODEL 2: CONSTANT STEADY STATE CONDUCTANCES BELOW THRESHOLD The second model, which is a refinement accurate description of the known processes

of the first, will give a more in the nerve membrane. We

FIRING BEHAVIOR IN NERVE CELL MODELS

123

must then look at the different ionic components of the total membrane conductance g. It is here assumed that the sodium conductance g,, and the potassium conductance g, are the ones responsible for the generation of the action potentials, while the conductance g, of the rest of the ions is constant. It is convenient to define the duration t, of an action potential as the time interval when the membrane potential E is larger than the threshold E,. During an action potential the injected current is so negligible compared to the large ionic currents, that during this time, and at the end of it, the potential and the conductances are practically independent of the current. Immediately after an action potential the potassium conductance is very large, but drops rapidly towards its resting value. During the same time the sodium conductance changes much less, and this change is therefore neglected. If we assume the action potential to end at time zero, the total membrane conductance can now be written as

where g,, is the additional potassium conductance at t = 0, and where all the parameters are independent of the injected current. The parameter t, is the time constant for deactivation of potassium. We have here assumed the potassium conductance to consist of a single component, as in Hodgkin and Huxley’s equations. The more general case is treated in Appendix I. If we chose the potassium equilibrium potential as the zero point on the potential scale, the differential equation for the membrane potential is

l+%e-‘/k The general solution of this equation, the threshold at t=O, is

E(t)=e

-m[ E,+

E )I

with the condition

$(E,+

. that E is equal to

&)/,6G(f)dt],

t/q).

It is not possible to evaluate this integral in terms of finite sums of elementary functions. But E(t) can be expressed as an infinite series. We

E. SKAUGEN

124 then set ew

The solution

E(t)=

gKDtK _ Ce

-,,,

K)= g

~(+)&%

(6)

then becomes

exp(ae-‘1”)

E,e-“e-ko/c” ( _

(e-~~/Le-ko/c)~)

, (7) I

where a=

&D

P=

c/cgot,)

k/

cT

This equation is a much more accurate description of the membrane potential between two action potentials than the solution to Eq. (1). Two new quantities have been introduced, t, and g,,, while Ep has been dropped and to replaced with the more precisely defined t,. Figures 3 and 4 show the membrane potential E as a function of time after an action potential for different values of the parameter b;xD, from and the time constant t, g,, = 5g0 to g,, = 8OOg,. The time unit is C/g,, for deactivation of potassium is set equal to O.OlC/g,. The potential is in

TIME [C/go]

FIG. 3 Membrane potential as a function of time from the end of an action potential for different values of the additional potassium conductance g,, at i = 0. The values of the parameters are E, = 25 mV (relative to EK) and tK = 0.01 C/g,. I?, + i/g,= 20 mV.

FIRING

BEHAVIOR

IN NERVE

CELL

125

MODELS

NUMBERS GIVE

I =ZSmV.g,

TIME

FIG. 4

Membrane

potential

as a function

[C/g,,]

of time from the end of an action

for different values of the additional potassium conductance the parameters are E, =25 mV (relative to EK) and r,=O.Ol

g,, C/g,.

potential

at t=O. The values E,+ i/g,=45 mV.

of

mV and relative to the potassium equilibrium potential. The threshold E, is set equal to 25 mV. In Fig. 3 the current is adjusted so that E,+ i/g, is equal to 20 mV. This value is smaller than the threshold, and the cell will not fire again. In Fig. 4 E,+ i/g, is equal to 45 mV, and the cell will fire repeatedly. Several interesting things may be noted here. When g,,>g,, and the membrane time constant C/g,,> t,, as is the case in perhaps the majority of nerve cells, the events after an action potential may be divided into two stages. First comes a time interval where the increased potassium conducIn this interval the potential falls rapidly tance g,,e -‘/‘K dominates. towards the potassium equilibrium potential. This fall is approximately described by (4) where only the term with g,, is retained: dE -=--_e dt

gKD -l/IKE c ’

(8)

with the solution E=E,exp[-a(l-e-“‘K)].

(9)

As t increases, the potential E falls rapidly towards the asymptotic value E,e-“. This is approximately equal to the minimum value of the potential after an action potential. When the additional potassium conductance becomes smaller than g,, the equation for E is approximately given by Eq. (l), which is obtained from (4) by neglecting the term with g,,. The potential then increases exponentially towards the value E,+ i/g,. The

126

E. SKAUGEN

minimum value of the potential is fairly insensitive to the injected current i, as shown by Figs. 3 and 4. In order to describe the membrane potential between two action potentials, the higher order terms in Eq. (7) are necessary. But when only the firing frequency f is of interest, this equation is only used to find the time t, when the potential crosses the threshold E,. At this time all terms with the factor e-j’/‘,, j+O, would have become small compared to the other terms and may be dropped. (This assumes again a cell where C/g,> tK.) Equation (7) then becomes (10) where

,=1-T

&$&>l_ /=I

and where we have set ae-‘/*K = 0, e pjr/r~ = 0 for j f 0. It is here assumed that the time interval t, is large compared to the time constant t, for deactivation of potassium. Since in general t, must be somewhat smaller than the duration t, of an action potential, (because the potassium conductance must have time to change very greatly during an action potential), it is sufficient to assume that the firing frequency f [ = 1/(fr+ t,)] is small compared to the inverse of the duration t, of an action potential. For physiological firing frequencies this is in most cases true. The firing ; frequency of the nerve cell model can now be obtained from Eq. (10):

f=

g&,(1 -e-“/S) i--go(E,-K)

This equation has exactly the same form as Eq. (2) the frequency given by the simpler model we started with. By comparing these two equations we see that t,=t,+

ClnS, go

(12)

Ep=E& All parameters

in the equation

for the frequency-current

relationship

given

FIRING

BEHAVIOR

IN NERVE

CELL

127

MODELS

by (1 l), or by (2) and (12) combined,

are now clearly defined. They are:

C, the membrane capacitance, g,, the membrane conductance when E,, the threshold for firing, t,,, the time interval during an action t,, the time constant for deactivation g,,, the excess K conductance when

E < E,, potential where E > E,, of the K conductance, E < E,, E = E, after an action potential.

The parameters a and S used in Eq. (11) are given by (7) and (10). The last two parameters are not as easily measured as the others. An estimate of t, and g,, can be obtained by fitting Eq. (9) to the observed fall of the membrane potential after an action potential. The injected current should be close to threshold. A better result could be obtained if the membrane were hyperpolarized to approximately the value E,e-” [the steady state value of E in Eq. (9)] and the action potential were started by an additional, short, depolarizing current pulse. The fall of the potential would then be very closely described by (9). Figure 5 shows as an example the frequency-current relationships for a nerve cell where all parameters in (11) have been specified, except for t,, which has been given several different values. The parameters are: C= 5 nF, g,=O.2 PMho, E, = 25 mV, E,= 20 mV, t,=0.25 ms, and g,, = 20 PMho. These parameters are close to the mean values of the corresponding parameters for the crayfish stretch receptor (Nakajima et al., [5]). The curves in Figs. 3 and 4, where g,,/g,,= 100, show the membrane potential

I

I

I

I

1

2

3 CURRENT

/

1

L I nAl

FIG. 5 Firing frequency as a function of injected current for different values of the duration r, of the action potential. The values of the parameters used are: C=5 nF, g,=O.2 pMho, E,=25 mV, E,=20 mV, t,=0.25 ms, and g,,=20 BMho.

128

E. SKAUGEN

as a function of time after an action potential for two values of the input current. The unit of time is the membrane time constant C/g,=25 ms. In the calculation the parameters a and S were found to have the values 1 and 1.09, respectively. This gives Ep = 7.3 mV and (C/g,)lnS = 2.14 ms. We can now compare Model 1 and Model 2. According to the results presented above, the minimum value of the potential is in Model 2 somewhat larger than the asymptotic value E,e-” derived from (9). In Model 1 the starting value Ep of the potential is set equal to the minimum value of the potential. It will therefore be larger than the value E,e-“/S used for E, in Model 2. By expanding 1nS in the expression for t, in (12) it is seen that to a first approximation the value of t,, is equal to t, + at,. As for Ep, this value will in general be different from the value chosen if Model 1 is used. These differences will always give a somewhat higher firing frequency in Model 1 than in Model 2 for the same value of the injected current. Even if Model 1 gives an expression for the firing frequency which is of the same form as that given by Model 2, the two models will give different frequency-current relationships for the same nerve cell. The threshold current will, however, always be the same in the two models. It is worth noting that the frequency-current curves obtained from these models can be closely approximated by a “power law,” i.e., a frequency current relationship of the form f= k(i- i,,)n. The largest discrepancy is found around the threshold current, where the curves given by the models presented here are almost vertical over a considerable frequency range. The models presented here are approximations to real nerve cells. A number of assumptions about the electrical behavior of the nerve cell membrane is made in order to calculate the frequency-current relationship. These assumptions are more or less good approximations to the observed behavior. By following the errors introduced by the assumptions through the calculation, it is possible to find in what direction the calculated frequency-current relationship must be changed in order to give a better representation of the frequency-current relationship in the real cell. We will briefly discuss some of the most important errors introduced by the assumptions made here. In Appendix II, where the concept of the threshold potential is investigated, it is shown that the value of the threshold potential is most sensitive to any dependence of the conductances upon the membrane potential at very low frequencies. Depending upon the particular cell in question, the threshold may either increase or decrease slightly as the frequency decreases. In the latter case the frequency will increase more gradually with an increase of the current than indicated by Eq. (11). In the first case the height of the “almost vertical” part of the current-frequency curve will

FIRING

BEHAVIOR

IN NERVE

CELL

129

MODELS

increase, and the curve may even become discontinuous at the threshold current. At high firing frequencies neither the sodium nor the potassium conductance will have time to reach its steady state value in the time interval between two action potentials (in the time interval when E < Et). When the threshold for firing is approached, the sodium conductance will be more inactivated, and the potassium conductance more activated, than at lower firing frequencies. This will increase the threshold and change the form of the action potential, in general making it smaller and of shorter duration. This will again change the value of the potassium conductance (which is approximately equal to g,,) at the end of an action potential. At medium frequencies the increase of the threshold will have the largest effect upon the firing frequency, making it somewhat smaller than predicted by Eq. (11). At high frequencies the effect of the decrease of the duration of the action potentials will dominate, and the frequency becomes larger than given by Eq. (11). At very high frequencies the firing mechanism will break down in real nerve cells, but this is not accounted for at all in the nerve cell models presented here. It is expected that these models give the most accurate description of the current-frequency relationship at medium and low frequencies, except possibly the very lowest frequencies. These results were found by analyzing the simplifications and assumptions made in deriving the current-frequency relationships presented here, and they were also confirmed by computer solutions of a complete set of equations of the Hodgkin-Huxley type, but adjusted to represent a nerve cell which can be approximated by our models. MODEL 3: STEADY STATE CONDUCTANCES AS LINEAR FUNCTIONS OF MEMBRANE POTENTIAL BELOW THRESHOLD The membrane conductances have so far been assumed to be independent of the membrane potential below threshold. To a first approximation we can include changes in the membrane conductances by setting g,, =

k,(E - E,) + gNO7

g,=ME-&)+&co,

g, =

gN0

+ gK0

(13) +

&O,

where gNa, g, and g,, are the conductances of the sodium, potassium and . remamng ions. gLOis a constant. gN0 and g, are the steady state conductances of the sodium and potassium ions at the membrane resting potential

130

E. SKAUGEN

E,. The coefficients k, and k, are constants. It is here assumed that the conductances depend upon the potential only, and that they are linear functions of the potential. In the general form of the Hodgkin-Huxley equations the conductances are non-linear functions both of the potential and of time. The non-linear relationship between the conductances and the potential can be approximated as closely as wanted by assuming the conductances to be piecewise linear functions of the potential. The solutions shown here can then still be used. The errors introduced by assuming the conductances to be independent of time will decrease as the firing frequency decreases. From the general form of the Hodgkin-Huxley equations it can be shown (Appendix III) that, when the rate dE/dt of increase of E is constant, the sodium and potassium conductances are given by gNa= g,,(E) + AgN,, and gK = gK(E) - AgK. The “lag” in g, relative to gK(E) can be approximately expressed by

‘h

tK

d&(E) dE

-gpnKdE-

dt ’

(14)

A corresponding, but more complicated, expression can be obtained for Ag,,. As a numerical example we may assume g,(E) to double its value when the potential E is increased 10 mV from the resting potential. If the other data are taken from the crayfish stretch receptor (t,mO.25 ms, E, - E,=20 mV), the relative error Ag,/g, in g, is less than 10% at a firing frequency of about 50 Hz. In accordance with the results of the preceding sections, it is assumed that the membrane potential has a value E, at a time t, after the start of an action potential (when E has increased to the threshold E,), and that these values are independent of the injected current. As before, these values may be “virtual,” only to be used as a convenient starting point for the calculation. This is discussed at the end of this section. The problem is now specified, the following differential equation for the nerve membrane must be solved: gm+g,+g,o)E-(g,aEm+g,E,+g,oE~++)}~

(15)

where ENa, E, and EL are the equilibrium potentials for the sodium ions, the potassium ions and the rest of the ions, respectively; and where at time t = 0, E = Ep. If the time t, at which E = E, can be found from the solution, the firing frequency is given by f = (to+ tT)-I.

FIRING

BEHAVIOR

IN NERVE

CELL

In order to shorten the notation

A =

131

MODELS

the following

symbols are used:

;(k,+k,),

B=AEr+

&.An-po). (16)

D=

$(E,(go-kNEN,)+i),

As before, the potentials are taken relative to the potassium potential E,; E, is then zero. The differential equation (15) can then be written as

e=-AE’+BE+D. dt

equilibrium

(17)

Figure 6 shows the general form of the solution of this equation for different values of the “starting” potential Ep. The lower, diverging curves correspond to cases where Ep is so small that one or both of the sodium and potassium conductances are negative. These cases must of course be ex-

FIG. 6 General form of the membrane potential as a function of time in a nerve cell where the membrane conductances are linear functions of the potential. Time courses for different starting values of the potential are shown.

132

E. SKAUGEN

chided, that is, the potential E, must be larger than 4 B/A - v/c If this is not the case, the approximation made by assuming the membrane conductances to be linear functions of the potential only, (13) is too coarse to be of any use. The solutions of interest are then the ones where the potential increases with time toward the stable value 4 B/A the differential equation (17) is then given by

+ v?? . The solution

of

(18) where t, is an integration constant, which is found by requiring that E = Ep at t = 0. The time t, when E has increased to the threshold E, can be found from this equation, and the firing frequency is obtained from (t,+ to)-‘: -1

2(E,-

E,)\/G

(VG -E,+:B/A)(\IG

+E,-:B/A) (19)

This expression for the current-frequency relationship resembles the expression found earlier [Eq. (1 l)]. Moreover, it can be shown that the equation above reduces to (11) if the coefficients k, and k, approach zero. The threshold current for firing can be found from (18). At the threshold the steady state value + B/A + L@ must be equal to the threshold E,. This requirement together with Eq. (16) gives i,=(E,-E,)[go+E,kK-(ENa-EI)kNl

potential

(20)

If the conductances are independent of the potential E, both the coefficients k, and k, are zero, and (20) reduces to the simpler expression found in the preceding models. If the threshold and the resting potentials are kept constant, and the rate of increase of the potassium (kK) or sodium (kN) conductance is increased, the threshold current is increased or decreased, respectively. The threshold current is in most cases more sensitive to a change in k, than a change in k,, as the factor EN,- E, is in general larger than E,. It should be noted that in Eqs. (18) and (19) only the parameter G depends upon the injected current i. Both the parameters A and B are independent of the current, while G increases linearly with the current. Figure 7 shows the current-frequency relationships obtained from (19) for different values of the coefficients k, and k,. The parameter to is kept

FIRING

BEHAVIOR

IN NERVE

CELL

133

MODELS

FIG. 7 Firing frequency as a function of injected current for different values of k, and k,, which are the rates of increase with the potential of the sodium and the potassium conductances, respectively. In the pair of numbers denoting each curve the first number gives the value of k, in Mho/mV, the second the value of k,. to= t, + (C/gc)ln S = 0.2C/ go. Membrane conductances are linear functions of the membrane potential. Normalized units are used.

constant at to=0.2C/go. As in Fig. 2, the frequency and current units are g,/ C and (E, - E’) g,, respectively. At the zero point on the current scale, the injected current is equal to (E, - E,)g,, the threshold current when

k,=

k,=O.

All the curves shown are of the same general form, but the height of the “almost vertical” part of the curves around the threshold current varies greatly. An approximate expression can be found for this “height” relative to the “height” when k, = k, = 0 (Appendix IV):

H=l+;{(ZE,-E,)k,-(E,,+E,-ZE,)k,}.

(21)

From this expression it is seen that the height of of the current-frequency curves increases when creases when k, is increased, in accordance with In Appendix IV it is also shown that relative =O, the curves are compressed along the current

CF=

H2 H-(K-Ep)(k,+W

If this compression

the “almost vertical” part k, is increased, but deFig. 7. to the curve for k,= kN scale by a factor

is larger than unity,

(22)

’

the “almost

vertical”

part of the

134

E. SKAUGEN

current-frequency curve will appear somewhat larger in height than that given by Eq. (21) because there will be an overall increase of the steepness of the curve when the factor CF is increased. It is seen, both from the last equations and from Fig. 7, that the current-frequency relationship becomes more linear when k, is increased relative to k,. But the minimum potential Ep then approaches the unstable region shown in Fig. 6, and the approximations made here cannot be used. For the curve farthest to the left in Fig. 7, where k, = 0 and k, = 0.005, E, lies close to the boundary of the unstable region at the threshold current (k,=0.0058). (As the current is increased, the boundary of the unstable region in Fig. 6 moves downwards.) For larger values of k, the expression for the current-frequency relationship given here cannot be used. When the factor k, is increased relative to k,, the “almost vertical” part of the current-frequency curves increases in height. As the injected current is steadily increased, the frequency “jumps” from zero to a relatively high value as the current threshold is passed, and thereafter it increases slowly. This type of behavior was found in the giant squid axon investigated by Hodgkin and Huxley. This type of axon does not at all fit the assumptions used here: the duration of the action potential is large, the membrane time constant is small, and the firing frequency is relatively high (if it fires at all). So we cannot use the models presented here to describe this axon. From the data presented by Hodgkin and Huxley, however, it is seen that around the resting potential the potassium conductance increases much faster than the sodium conductance when the potential is increased (in terms of absolute values of the conductances). The form of the current-frequency relationship for the squid giant axon is thus qualitatively explained by the model presented here. MODEL 4: CONSTANT STEADY STATE CONDUCTANCES FOR SMALL VALUES OF POTENTIAL AND CONDUCTANCES LINEAR FUNCTIONS OF MEMBRANE POTENTIAL FOR LARGER VALUES So far the values of E, and t, in the last model have not been defined in terms of measurable membrane parameters. The parameters E, and t, are “virtual” and do not correspond to the minimum value of the potential and the duration of the action potential, although there is a connection. We could have taken into account the large potassium conductance after an action potential and solved the equations by an approximate method as in Model 2, but it is much simpler to use the results of the preceding sections directly. It must then be assumed that for a certain time after the action potential has ended, the steady state values of the membrane conductances

FIRING

BEHAVIOR

IN NERVE

CELL

MODELS

135

are constant. This condition is closely approximated membrane conductances to be defined by

gNa

=

by assuming

kN(E-Er)+gNO

if E > E,

bd&-E,)+&w=th,

ifE
~K(E-J%)+~KO

ifE>E,

gK= k,(E,

-

4)

+

gKO=

gKl

the

(23)

ifE
g, = g,, = constant, g0 =

gN0

+

gK0

+

&.O*

g,=gN,+gKl+&.,.

The membrane conductances are now piecewise linear functions of the potential, and in most cases these functions can be fitted closer to the actual conductance-potential relationships than simple linear functions as given by Eq. (13) in the preceding model. The conductances cannot now become negative, and there will be no unstable region for low values of the potential, as shown in Fig. 6. In Fig. 8 the general form of the conductancepotential relationships used in the models presented here is shown. These relationships may be compared with the steady state potassium conductance-

I

*

SQUIDGIANT AXON

I FOTASSIUMI

FIG. 8 General form of the membrane conductances below threshold as functions of membrane potential for the different models investigated. Model 1: constant conductances. Model 2: constant steady state conductances. Model 3: conductances are linear functions of the potential. Model 4: conductances are piecewise linear functions of the potential.

136

E. SKAUGEN

potential relationship for the squid giant axon (dotted line), which is taken as an example [ 11. We can now assume that after an action potential and until the time when the potential E has again increased to E,, the potassium conductance is given by g, = g,, + g,, exp( - t/ tK), where t = 0 when E has decreased to E, after an action potential. This will not be quite true in the short time interval while the potential falls from E, to E,. Due to the change of the steady state value of the potassium conductance in this time interval, the fall of the conductance will not be purely exponential. But this time interval is so short compared to the time between two action potentials that an approximation here will have little effect upon the firing frequency. The sodium conductance is in accordance with the assumptions used in Model 2 [Eq. (3)], set equal to g,, in this time interval. The time interval t,, from the start of an action potential to the time when the membrane potential E has again increased to E, after the action potential, can now be found from Model 2 with E,= E,, go=g, and [Eq. (1111:

E,=E,=(g,,E,,+g,,E,)/g,

,+ (El-Ete-“lS)gl i+(E,-E,)g,

+ ClnS+ gl

t,,

(24)

where a and S are given by (7) and (10). Note that the value of t, obtained here is an approximation, because higher order terms in the complete equation (7) has been neglected. The approximation is good if the time interval t, - t, is large compared to the time constant t, for deactivation of the potassium conductance. The time interval t, - t, increases when E, is increased; therefore E, should not be too small. The potential E, in Eq. (19) now corresponds to E, here, and from (19) the firing frequency as a function of the injected current is then obtained:

2(E,-E,)fl (G

- E, + $/A)(*

+ E, - &B/A)

(25)

where a and S are defined in Eqs. (7) and (10) and where g, is replaced by g, in Eq. (7). The threshold current ir is found from the condition that the arguments of both the natural logarithms in Eq. (25) must be positive

FIRING

BEHAVIOR

IN NERVE

CELL

137

MODELS

numbers. At the threshold one or both of them are positively infinite, which gives zero frequency. The condition for positively infinite arguments is given by Eq. (20) for the first argument, and by the condition i + (ES - E,) g, =0 for the second. Combining these conditions we obtain

ir=Max{(E,-E,)[go+E,k,-(E,,-E,)k,l;(E,-E,)g,}.

(26)

The threshold current is a piecewise linear function of both k, and k,, and thus reflects the assumed conductance-potential relationships. When the threshold current is determined by the term (E, - ES) g,, the parameter k, is so large and/or the parameter k, is so small that the effective threshold for firing has been decreased from E, to E,; when the potential increases to E,, the cell will fire. Figure 9 shows the frequency-current relationships for different values of k, and k, when t, = 0.2C/g, and E, = E,. The other parameters are E, = 25 mV, E, = 15 mV, and E,e-“IS = 5 mV (Se 1). Normalized units are used. This figure should be compared with Fig. 7, where the same values of the parameters have been used. It is seen that the curves in Figs. 7 and 9 are very similar. The largest difference is observed for the largest values of k, (up to 0.005). This may be expected, since Model 3 cannot be used for

2 J?

-05

0

05 ~?lRRENT [IE,

15 -E,lg,]

FIG. 9 Firing frequency as a function of injected current for different values of k, and k,, which are the rates of increase with the potential of the sodium and the potassium conductances, respectively. In the pair of numbers denoting each curve the first number gives the value of k, in Mho/mV, the second the value of k,. t,,= to + (C/gc)lnS = 0.2C/ g,. Membrane conductances are piecewise linear functions of the membrane potential. Normalized units are used.

138

E. SKAUGEN

values of k, larger than 0.0058 when k, = 0. The more realistic model given by Eq. (25) (Fig. 9) gives slightly more curved curves for the currentfrequency relationship, and the curves for different values of k, and k, are a little farther apart. But the differences are so small that the simpler Eqs. (19), (20) and (21) of Model 3 give a good approximation when these equations can be used. The parameters Ep and t, in these equations are then given by Eq. (12), as they also are for Eq. (2). Figure 9 shows that as k, is increased beyond the critical value of 0.0058 in Model 3, a constant threshold current is approached [Eq. (26)], and the frequency-current relationships become less linear again. TWO POSSIBLE

MECHANISMS

FOR ADAPTATION

A certain degree of adaptation is very often observed in nerve cells, even in those classified as tonic nerve cells. By adaptation is here meant that at a constant injected current the firing frequency gradually decreases and eventually reaches a steady state level. The firing frequency is in this case a function both of the injected current and of time, or the previous history of the nerve cell. If the firing frequency falls to zero within a reasonably short time, the nerve cell should be classified as phasic rather than as tonic. A certain phasic behavior is found in the model defined by Hodgkin and Huxley’s original equations; if the injected current is raised rapidly to a value slightly below the threshold current for steady state firing, one or a few action potentials are triggered off, but then the firing ceases. If the injected current is larger than the threshold for steady state firing, no adaptation is found; the firing frequency does not change with time when the current is constant. But adaptation is easily introduced in this model by making some not unreasonable assumptions of slow changes of some of the parameters. I will now show how the effect of a long term change of the potassium conductance, and also the effect of an electrogenic Na/K pump, can be included in the models presented. Both of these processes lead to adaptation, and processes similar to either one or the other of these processes are believed to be responsible for the adaptation observed in the majority of nerve cells. Model 2, with constant subthreshold conductances [Eq. (1 I)], will be used as a basis for the calculations. Since the details of the mechanisms leading to adaptation in all probability differ from cell to cell, only the general effects of these mechanisms will be considered here. The adaptation is calculated in the case where the injected current is increased suddenly from zero to a certain value i, and thereafter kept constant at this value. At the onset of the current step, the firing frequency

FIRING BEHAVIOR IN NERVE CELL MODELS

139

as a function of i is given by Eq. (1 l), because the adapting mechanism has not yet had any effect. The firing frequency thereafter decreases until a steady state (if one exists) is reached. Numerical calculations showed that in the absence of saturation, the decrease of the frequency was to a first approximation exponential, with a time constant equal to the time constant of the adapting mechanism-either the time constant of the slow potassium conductance or that of the electrogenic Na/K pump. The frequencycurrent relationship in this steady state can be calculated from very general and simple assumptions about the adapting mechanism. In contrast to this, it was found that in order to calculate the decrease of the firing frequency as a function of time, the adapting mechanism had to be specified in much greater detail. This will not be done here, since too many arbitrary assumptions then have to be introduced. MODEL 5: CONSTANT STEADY STATE CONDUCTANCES BELOW THRESHOLD, AND A SLOWLY CHANGING COMPONENT OF THE POTASSIUM CONDUCTANCE The case where the adaptation is due to a slowly changing component of the potassium conductance will be investigated first. The complete potassium conductance can then be written as &c=g,,+glc,e-“‘~+gS.

(27)

The parameters are defined in (11) and in (13), except for the new term g,. This parameter is zero when the membrane potential E is kept below threshold, but increases to a steady state value g&(E) > 0 when the potential is kept at a constant value E which is larger than the threshold. We assume g, to be described by a first-order process, that is,

ds, df

-=

f[g,Wgsl, s

where t, is the time constant for change of gs. g&(E) is different from zero during an action potential only, i.e., in the time from t =0 to t = to. According to the earlier discussion the’time course of the potential during an action potential is a function of time only, and independent of all the other parameters. But then’the parameter g,,,(E) can also be considered as a function of time only, and we accordingly write it as g,e( t), where gJ t) = 0 for t, < t < l/f. The solution for the differential equation (28) is then gs=e-‘/r~(

g,,+ +~“‘g&t)e’/cdt),

(29)

140

E. SKAUGEN

where t=O at the onset of an action potential, and where g, is the value of g, at t = 0. The above integral is a constant for a particular nerve cell, since the function g&(t) and the parameters t, and t, are independent of the injected current. The value k, of this integral is a measure of the strength of the slowly changing component of the potassium conductance. The change in g, during the time interval 1/f from the onset of one action potential to the start of the next is then AgS=c+(gSO+ikS)-gSO. If a steady state is assumed,

(30)

then Ag, = 0, and g,, can be found from (30): (31)

The time constant t, is large compared to the other time constants. For frequencies f>> l/tS, the term exp[l /(ft,)] - 1 is to a good approximation equal to 1/(ftJ, and Eq. (3 1) reduces to g,, = k.J

(32)

In this case the value of g, does not change much during the time interval between two action potentials, because, according to (29) g, then decays with the time constant t,, which is large compared to l/f. If g, is set relationship in the steady constant and equal to g,,, the frequency-current state can now be found from (11). Instead of g, the complete membrane conductance g,+g, must be used. The resting potential E, is then changed to Ergo/( g,, + g,). The frequency-current relationship now obtained can only be solved explicitly for the injected current,

i=

In normalized

(33)

units this becomes

(34)

FIRING

BEHAVIOR

IN NERVE

CELL

141

MODELS

CURRENT

FIG. 10

Steady

state firing

frequency

as a function

[IE,

- Epl go1

of injected

current

for different

values of k,/ C in the case of a slowly changing component of the potassium conductance. k, denotes the strength of this component. &,= r, +(C/g,,)lnS=0,2C/g,. I-e-“/S=0.56. Normalized units are used.

Figure 10 shows this frequency-current relationship for t, = 0.2C/g, and for different values of k,. The parameters 1 - e-O/S and S are set equal to 0.56 and 1, respectively. It is seen that the relationship becomes more linear as k, is increased. Note that the curve for k, =0 also shows the frequency-current relationship at the onset of the current step, while the final steady state relationship (full adaptation) is given by the curve with the actual value of k,. Equations (33) and (34) do not hold true for very small frequencies, because l/f then no longer is small compared to the time constant t,, as was assumed in the derivation of these equations. In this case g, must be found from (31) which gives a smaller value than the linear relationship (32). The firing frequency will then be higher than shown by Fig. 10, but the frequency will be difficult to calculate, as the component g, of the conductance cannot now be considered as constant in the long time interval between two firings. It should also be noted that if g, is not described by a first order process as assumed in Eq. (28), there can be a non-linear relationship between g, and f, and also between the firing frequency and the injected current, even when k, is large and dominates the relationship. The decrease of the firing frequency towards the steady state value with increasing time cannot be expressed by a simple analytical formula. If a constant frequency is maintained, the value of g, increases exponentially with time constant t, towards the steady state value. But when the frequency

142

E. SKAUGEN

decreases as g, increases, as it will if the injected current is kept constant, the growth of g, will be more and more slowed relative to the case where the frequency is kept constant. In the case of a constant current, the value of g, will thus start to increase with a time “constant” smaller than t,, but as g, approaches the steady state, the time “constant” for the growth increases towards t,. MODEL 6: CONSTANT BELOW THRESHOLD,

STEADY STATE CONDUCTANCES AND AN ELECTROGENIC Na/K PUMP

The electrogenic Na/K pump restores the ionic concentrations inside a nerve cell when one or several action potentials have disturbed the resting state. Sodium ions are pumped out of the cell, and potassium ions are pumped in. These two processes are believed to be partly coupled, but for the electrogenic pump the inward flow I, of potassium ions is smaller by a factor 1 -b than the outward flow ZNa of sodium ions. This gives a net outward current lp = I,, - I, = hl,,. In the steady state the total inward flow i,, of sodium ions due to action potentials must be equal to the total outward flow I,, due to the pump. Since the inward flow i,, is proportional to the number of action potentials per time unit, that is, the firing frequency f, we have ip = bI,, = bi,, = kJ,

(35)

where k, is a constant. This is the mean value of the pump current, but if the time constant of the pump is large compared to l/f, the pump current can be assumed constant and equal to $, during the time interval between two firings of the cell. Seen from the cell, the total effective inward current is then constant and equal to i - $,, and by inserting this value instead of i in Eq. (1 l), and solving with respect to the injected current i, we obtain

i=

E,(l-G)go iexp[ h$_I,)]_l

+kJ+(Et-Er)go, (36)

where Eq. (35) has been used. In normalized

units this becomes (37)

where kp also is in normalized units. Figure 11 shows this frequency-current relationship for t, = 0.2C/go and for different values of the pump parameter

FIRING

BEHAVIOR

IN NERVE

CELL

143

MODELS

kp

05 CURRENT

[I+Epl

C]

10 [lE,-Ep)gol

FIG. 11 Steady state firing frequency as a function of injected current for different values of kp in the case of an electrogenic Na/K pump. kp denotes the effect of the pump. 1,,= t, + (C/g& S =0.2C/g,. Normalized units are used.

kP. Note that the curves are very similar to those shown for Model 4 in Fig. 10. The two quite different basic mechanisms for adaptation discussed here have almost identical effects upon the firing frequency in the steady state situation. As for Fig. 10, the curves shown in Fig. 11 are lower limits for the firing frequency. For frequencies which are not small compared to the inverse of the pump time constant, the firing frequency will be found between the curve for kp = 0 and the curve for the actual value of k,. The decrease of the firing frequency towards the steady state value depends upon many pump parameters which are not specified here. Numerical calculations have for instance shown that the effect of saturation can be large and may make the decrease of the frequency very slow compared to the decrease expected from the time constant of the pump alone. But the detailed mechanisms of the electrogenic pump must be specified in order to find the effects of the different pump parameters upon the decrease of the frequency. This demands, however, too many assumptions to be of much general interest, and it will not be done here. CONCLUSIONS

AND DISCUSSION

The expressions for the frequency-current relationship presented here are obtained from nerve cell models which take note of the most basic membrane properties, but the finer details of membrane changes during activity of the cell are not included. This gives the possibility of applying the expressions presented here to most nerve cells, as the parameters

144

E. SKAUGEN

required in the expressions are relatively few and can be rather easily measured. In the models developed here the time course of the membrane potential is calculated in the time interval between two action potentials. It is assumed that the cell fires when a certain value of the potential, the threshold potential, is reached. The changes of the membrane parameters during the following action potential are not accounted for directly in the models. An unspecified action potential with a constant duration is simply assumed. It is also assumed that the action potential leaves the cell in a certain state which is independent of the injected current. This state is then used as a starting point for the calculation of the following subthreshold events. A constant threshold Et for firing and a constant duration 1, of the action potential are in many cases useful approximations to the results of the complicated membrane events during an action potential. As these events probably are very different in detail from cell to cell, a more complicated model which had specified these events would also become less general. Of the different models presented here, the central one is a model of a cell where the steady state, sub-threshold membrane conductances are independent of the potential. This cell then has a constant membrane conductance below threshold if it is a long time since the last action potential. With this model as a starting point the effects of different changes in the membrane mechanisms are calculated. The effects of potential dependent, sub-threshold membrane conductances are considered first. Then the effects of a long term component of the potassium conductance and of an electrogenic Na/K pump are calculated. For all these models analytical expressions for the frequency-current relationship are obtained. Since the models include only a part of the processes in a nerve cell which may influence the firing frequency, the expressions obtained cannot be expected to account in detail for the firing behavior. These expressions are at best more or less accurate approximations to the measured frequency-current relationship. The accuracy of the expressions increases when we go from a nerve cell to another, where either: (a) the membrane time constant in the resting state is larger compared to the duration of the action potential, or (b) the threshold for firing is sharper (it does not change so much when the injected current is changed), or (c) the duration of the action potential is more constant (it does not change so much when the injected current is changed), or

FIRING

BEHAVIOR

IN NERVE

145

CELL MODELS

(d) the steady state, subthreshold membrane change so much with a change of the potential.

conductances

do not

For a fairly large class of simple nerve cells the accuracy of the calculated frequency-current relationships seems to be good. For these cells the models can be used directly to account for the cells’ firing properties. The crayfish stretch receptor is probably such a simple cell. This cell shows adaptation, and there is evidence that it is an electrogenic pump which causes it (see Sokolove [6]). As an example it was tried to fit Model 6 to this cell. This receptor shows an approximately exponential change of the membrane potential towards a new value when there is a subthreshold step change of the injected current. This means that the total membrane conductance must be approximately constant below threshold, and we can try to apply Model 2 (or Model 6 when we want to include adaptation). The assumptions in this model are well fitted by the crayfish stretch receptor. The duration of the action potential is always small compared to the membrane time constant C/g,, and to the inverse of the firing frequency, as tom:2 ms, while C/g,%25 ms, and f seldom exceeds 50 s-i (l/f>20 ms). I have chosen to use the experimental frequency-current relationship obtained by Sokolove [7] for the crayfish stretch receptor. He measured this relationship both at the onset of a current step increase from zero (no adaptation) and at the steady state (full adaptation). It should thus be possible to fit his data with Fig. 11 for reasonable parameter values. Sokolove did not present any data for the cell membrane parameters. But by using the data for the crayfish stretch receptor given by Nakajima [5], and changing these somewhat (within the limit of variability from cell to cell), a good fit between the experimental and the model frequency-current relationships can be obtained. The fitted membrane parameters are then: C= 5.7 nF, g,=O.23 PMho, (C/g,)lnS+ t, =2.5 ms, E,- Ep =20 mV, and E, - E, = 2.2 mV. These values were also chosen to correspond to values of t, and g,, which describe reasonably well the observed fall of the membrane potential at the end of the receptors’ action potentials (t,eO.25 ms, gKD=20 PMho). The upper curve in Fig. 12 shows the model frequency-current relationship at the onset of the current step increase from zero (no adaptation). The open circles are the corresponding measurements given by Sokolove. The filled circles show the measured frequency in the steady state; the current had then been kept constant for some time. This frequency-current relationship is represented by Eq. (36) for Model 6. The best fit was obtained for kp =0.09 nA/s-’ and is shown as the lower curve in Fig. 12. The membrane parameters are as before. It is seen that the model in this

146

E. SKAUGEN So

50 F

P

LO -

2% i y

30-

2 :IL 20

-

10 -

I

I

I

I

2

3 CURRENT

1

L

1

5

[nA]

FIG. 12 Firing frequency as a function of injected current in the case of an electrogenic Na/K pump. The current is increased suddenly from zero to the constant value shown by the current axis. Upper curve shows frequency at the onset of the current step (model), open circles from experiment. Lower curve shows frequency long after the onset of the current step (model, steady state), filled circles from experiment. Experimental results from Sokolove [7].

case gives frequency-current relationships which agree well with experiments, both for no adaptation and for full adaptation. The membrane parameters have been chosen to give the best fit, but all parameters are still close to the mean membrane parameters for the receptor as measured by Nakajima [5]. The lower curve for the steady state (full adaptation) is obtained from the upper curve by choosing the value of a single parameter k,. It should also be noted that the frequency-current relationships given by the models can in no way be fitted to any experimental frequency-current relationship. The curves must always be of the form shown here. The value of the parameter kp obtained here lies in the expected range. When kp =0.09 IA/S-’ it gives for the crayfish stretch receptor a net inward sodium current of 3(MO PA/cm’ during an action potential. This value is, for instance, not too far from the corresponding value obtained by direct measurements of the squid giant axon, where the net inward sodium current is 100-150 PA/cm* during an action potential (calculated from data taken from Hodgkin and Huxley [l]; they measured directly the charge transfer of sodium ions during an action potential). It is felt, however, that the greatest value of the models is that they in any case give relatively simple, analytical expressions which can be considered as first order approximations to the actual frequency-current relationship. These expressions show the main effect upon the firing fre-

FIRING

BEHAVIOR

IN NERVE

CELL

147

MODELS

quency of the most important membrane mechanisms which are found in nerve cells. Conversely, from the measured frequency-current relationship alone, much can be derived about the membrane properties of the actual nerve cell. This may be an easily obtained guide to further observations. APPENDIX

I. AN EXTENSION

OF MODEL

2

Assume that at the end of an action potential the membrane conductances have fixed values which decay towards their steady state values as long as E < E,. The conductances can be written as

gk = g,, + 5 gKje - ‘/IQ, j- 1

gN,=gNO+

(1.1)

5 gNje-f'fNJ3 j- I

where t = 0 at the end of the action potential. The case where g, also is a sum of decaying conductances is a straightforward extension. The differential equation for the membrane potential E is

p!{&[

1+B(t)]E+

C(t)},

(1.2)

where A = E, + i/go,

C(t)=

2

fj,g,e-‘/‘N,. 1’

As before, the potassium potential scale, i.e., E,=O.

equilibrium

potential

is taken

as zero on the

148

E. SKAUGEN

The solution

of (1.2) is

’ {A +

C(u)}e’go/C)D(“)du

,

(1.3)

1 where

D(t)=Sf[l+B(u)]du 0

( gNjtNj

+

glSJtKj) - + OJ

By expanding, become

the

time

dependent

.!1 ( gNjtN,e terms

-r’rNJ+ gKjtKje-‘l%)

in exp[( gO/C)D (t)]

in (1.3)

(1.4) Equation (1.3) can now be integrated term by term and E expressed as an infinite sum. In the case where the time constants tNj and tKj are all small compared to the membrane time constant C/g,, all terms with factors exp( - t/ tNj) and exp( - t/ tKj) will become insignificant compared to terms with factors exp( - tgo/C) as t increases. The expression for the potential then reduces to

E=[

E~e-YS,(E,+~)+E,,S2]e-“0/“‘+4+~.

(1.5)

where

a=

F

l_C,~gorN~) +q J

l_C,~gor,,) /

FIRING

BEHAVIOR

IN NERVE

CELL

149

MODELS

Only the first-order terms in S, and S2 are shown.’ Note that Si and S, are constants. If (1 S) is a good approximation at the time t, when the threshold E, is reached, we can set E = E, and solve with respect to the time t, in order to find the firing frequency (t, + to)-‘:

1 -1

E,(l-e-“lS,)-E,,S,IS,

g,

+ zlnS,+I,

i--(K-E,)g, ) which is of the same form as Eq. (11) obtained

APPENDIX

II. THE THRESHOLD

,

(1.6)

for Model 2.

POTENTIAL

After the end of an action potential the threshold E, decreases towards the steady state value Etw In the calculations I have so far considered E, as constant and used the value E,,. I will here develop an approximate expression for the threshold potential E, which is time dependent.

‘The terms in the sums S, and S, are rather following procedure: For S,: (a) Multiply

out

(b) Multiply

each term by the factor

complicated,

but can be found

1 1 -(C/g,) (c) Set t =O. For S,: (a) Multiply out

Then follow (b) and (c) for S,.

(total time constant

in term)-’

from the

150

E. SKAUGEN

The total inward current I==&+ Above a pulse E, and induced former

I through

the membrane

is given [Eq. (15)] by

-gNa(E-ENa)-gK(E-EK)-gL(E-&)+i.

(2.1)

the threshold E, an increase of E externally induced (for instance by in the injected current i) will increase I, giving a further increase of an action potential is started. Below the threshold an externally increase of E will decrease I, and E will decrease and return to its value. At the threshold E, we must then have dl _=dE

+(E,-E,,)-g,,-$$(E,-E,)-gK-g,=O.

(2.2)

Note that dg,/dE=O, because g, =g,-, is constant. The sodium and potassium conductances must be specified. As an approximation to the form of Hodgkin and Huxley’s equations [l] we set (2.3) g,, = hm$?,,x(

1-

e-‘/‘h)e’E-E~)/EogNo,

where E,, = lim,,, E, (with no firings) and t,, is the time constant for the inactivation (h process) of sodium conductance. Since the time constant for m is very small, it is assumed that m is always close to its steady state value, which around E,, is proportional to exp[(E - E,,)/ E,]. For the squid giant axon E ax3 mV. The potassium conductance decays from its value g,,+ g,, at time t = 0 (the end of the action potential) and towards g,,. The sodium variable h increases from almost zero at t = 0 towards its steady state value. Both t, and th are large compared to the time constant of m. Neither g, nor h will change much before m has almost reached its steady state value if E is changed; h and g, are therefore assumed independent of E. This can be done because we are not interested in the actual time course of the potential. If (2.3) is used in (2.2) we obtain, by differentiating and solving with respect to the exponent,

E,=E,,+

[( In l-

EoE ENa_ f0

Eo,

(2.4)

FIRING

BEHAVIOR

IN NERVE

CELL

151

MODELS

where in order to obtain Em= lim,,,E*

we have set goEo

gNo=ENa-

(2.5)

Et0 ’

The factor E,/( E,, - E,,,) is in general small (0.03 for the squid giant axon) and can be neglected. If the last logarithm is expanded and only the first order term retained, we find

;(?;-T; Na

I

(2.6) 10

0

When the cell fires, t will be larger than both tK and t,, unless the firing frequency is very high. Equation (2.6) then reduces to

E = E,,+ f

Eo(EN,

gKD _e-~/lK+e-~/b

-

Co)

ENa- E&- E, ’

go

(2.7)

For reasonable firing frequencies, where l/f is considerably larger than t, and t,, (-t,), E, will be very close to E,, and can be considered as constant. The above expression for E, is fairly accurate if the steady state values of g, and h are constants up to the threshold (as assumed in Models 1 and 2). If, however, one or both of the steady state values of the parameters g, and h start to change before E has increased to the threshold, as m must do, the situation may be different. For high frequencies the potential E increases so fast that g, and h do not have time to change, and (2.7) is still fairly accurate. For low frequencies E may increase so slowly that g, and h have their steady state values when the threshold is reached. The threshold is then higher than given by (2.7) which can be considered a lower limit. In this case the threshold will increase as the injected current decreases. This will increase the height of the “almost vertical” part of the frequencycurrent curves, and a discontinuity around the threshold is possible. APPENDIX III. POTENTIAL AND TIME DEPENDENT CONDUCTANCES POTASSIUM

According

CONDUCTANCE

to the Hodgkin-Huxley g, =

formulation

n4&,

[ 11, we can set (3.1)

E. SKAUGEN

152

where gK is a constant

and n a dimensionless dn

-

variable

governed

by

f[n(E)-n].

=

dt

(3.2)

n

Here n(E) and t, are functions of E only. n(E) increases with E, and n must be smaller than n(E) to obtain a positive increase dn/dt of n. When E increases, the stable situation is that n lags behind n(E), and by such an amount that n increases as fast as n(E). This gives n=n(E)-tn4Ez.

The lag hg,

in g, relative to gK(E)

dE

(3.3)

is given by

{ [n(E)14-n4}&.

b,=g,(E)-gK= Combining

dn(E)

(3.3) and (3.4) gives dn(E)

dE z.

~g,<4[@)]3ikt,T

By using the fact that dg,(E)/dE=4[n(E)13g,dn(E)/dE, dg,(E)

(3.5) we obtain

dE

(3.6)

Ag,<4t,dEx. SODIUM

(3.4)

CONDUCTANCE

For the sodium conductance g,,

we have =

dh

-dt = dm

and m(E)

;[h(E)- h],

(3.7)

= ,‘[m(E)-m],

dt

where t,,, t,,,, h(E) described by

hm3i?Na>

m

are functions

h=h(E)-t,dEz,

m=m(E)-t,

of E only. The stable situation

dh(E)

dm(El -dEdt’

dE

dE

is

(3.8)

FIRING

BEHAVIOR

IN NERVE

CELL

153

MODELS

h(E) decreases while m(E) increases when E increases. h is thus larger than h(E) [dh(E)/dE
gNa

-

g,,(E)

= { hm3-h(E)[m(E)]3}&,. Combining

(3.9)

(3.8) and (3.9), we obtain

%h(E)[m(E)]’

$?Na I

(3.10) The relative change of g,,(E) with E is small compared to the relative change of h(E) with E, because the increase of m(E) almost balances the decrease of h(E) in gNa(E)= h(E)[m(E)]3gN,. In addition t,, is much larger than t,,,, Equation (3.10) therefore reduces to A&q,<----

th- h,,

dh(E) dE dE

h(E)

(3.11)

dt g,,(E),

where Ag,, is positive. APPENDIX IV. FREQUENCY-CURRENT RELATIONSHIP AROUND THRESHOLD

IN MODEL

3

When the injected current i is very close to the current threshold i, given by (20), Eq. (19) for the frequency can be approximated by a function which is of the same form as Eqs. (2) and (11) for the simpler Models 1 and 2. We then set i = i,+ Ai, and by inserting this into (16) and by using the threshold condition

3 B/2A

+c

= E,, we obtain

D= $(E,(g,-k,E,,)+i,)+

2Afl

=d4AD+

;Ai

(4.1)

(4.2)

B2 =$x

Since Ai is small, (4.2) can be expanded 2AL’77 x2AE,-

B+

and reduced 2A Ai C(2AE,B)

to (4.3)

154

E. SKAUGEN

When

Ai is very small, the last term can be neglected,

difference 2A fl reduces to

f

=

CUE,-

- (2AE, - B) is a factor. Equation

1+

1 B)C/g,

go

except where the

(19) in Model 3 then

WC&

(G-Ep)g, Ai

[A(E,+E,)-B]g,

(4.4) This equation is of the same form as Eq. (2) in Model i = i, + Ai and i, = (E, - E,)g, can be written as

f=

kin

1+ (E,-2)go

1

(

+t, 3

1, which by using

-I,

(4.5)

I

Since the contribution from to is small when Ai is small, we see that around the current threshold the frequency-current relationship in Model 3 is of the same form as in Model 1 (and 2), but that the frequency in (4.5) is multiplied by a factor

while the current

CF=

in (4.5) is multiplied

WK-

N2C

[A(&-Q-B]go Equation

by a factor HZ = H-(E,-E,)(k,+k,)

(16) has been used in the calculation

’

(4.7)

of (4.6) and (4.7).

It is a pleasure to thank Lars Wallue for many helpful suggestions and for valuable criticisms, and Jan Jansen, Michaef Brown and Arild Nj2 for helpful criticism of the manuscript. This work was supported by the Norwegian Research Council for Science and the Humanities. REFERENCES

1. A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application (1952).

to conduction

and excitation

in nerve, J. Physiol. (Land.) 117, 500-544

FIRING 2

3 4 5

6 7

BEHAVIOR

IN NERVE

CELL

155

MODELS

B. Frankenhaeuser and A. F. Huxley, The action potential in the myelinated nerve fibre of Xenopus Iuecis as computed on the basis of voltage clamp data, J. Physiol. (Lend) 171, 302-315 (1964). J. A. Connor and C. F. Stevens, Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma, J. Physiol. (Lend.) 213, 31-53 (1971). R. B. Stein, The frequency of nerve action potentials generated by applied currents, Proc. Roy. Sot. B 167, 6486 (1967). S. Nakajima and Kayoko Onodera, Membrane properties of the stretch receptor neurones of crayfish with particular reference to mechanisms of sensory adaptation, J. Physiol. (Lo&) 200, 161-185 (1969). P. G. Sokolove and I. M. Cooke, Inhibition of impulse activity in a sensory neuron by an electrogenic pump, J. Gen. Physiol. 57, 125-163 (1971). P. G. Sokolove, Computer simulation of after-inhibition in crayfish slowly adapting stretch

receptor

neuron,

Biophys. J. 12, 1429-1451

(1972).