BULLETIN
OF
BIAT]~EI%IATICAL B I O L O G Y
VOLUM]S 38, 1976
COMPARATIVE EVALUATION NERVE EXCITATION
OF
QUANTUM
THEORY
OF
• C. HODSON AND L. Y. WEI~ Biophysical Research Laboratory, Electrical E n g i n e e r i n g D e p a r t m e n t , U n i v e r s i t y of Waterloo, Waterloo, Ont., Canada
Deliberate evaluation of the q u a n t u m theory of nerve excitation is made b y comparing it with Hill's theory in fitting the experimental data on threshold-frequency relation, optimum frequency (v0) for nerve excitation and strength-durati0n relation. Decrease of v0 and increase of all the time constants (Hill's A and k, Wei's T~ and spike duration w) with decreasing temperature are interpreted on the basis of the dipole relaxation time T~, but inexplicable from Hill's theory or any other existing theory. The closeness of k, T2 and w values is explained. A variety of experimental results obtained by others is discussed. Finally, a comparison is made between the t t o d g k i n - H u x l e y equations and the q u a n t u m theory. Most of the facts (electrical and non-electrical) tend to support the thesis that nerve excitation is a macroscopic expression of q u a n t u m transitions of dipoles between energy states.
In the years of 1932-1936, there were several phenomenological theories developed on nerve excitation (Blair, (1932, 1934, 1936); Rashevsky, (1933, 1960); Hill, (1936a, 1936b); Hill, Katz, Solandt, (1936; Monnier, (1934)). Formally, the equations of Blair, of Rashevsky and of Hill are equivalent and so are their results. Their equations are of the following types, d u / d t = K I - a ( u - uo)
(A)
dv/dt = M(u-
(B)
uo) - b ( v - vo)
where u and v are excitatory and inhibitory factors. Blair used one equation of type (A). Rashevsky had two equations, each of type (A). Hill started from two equations of type (A) and (B). The relations derived from Hill's equations ~To whom requests for reprints should be directed. 277
278
c. HODSON AND L. Y. WEI
have been under extensive experimental tests (Hill, 1936b; Hill, Katz, Solandt, 1936; Solandt, 1936) and evaluated in detail by Katz (1939). With properly chosen constants, Hill relations as well as Blair's and Rashevsky's are able to fit a number of experimental data and "in m a n y cases with an almost embarassing accuracy" (in Katz's word). However, the equations (A) and (B) and the constants therein do not suggest or imply the physical nature or mechanism of the processes described. This was so because in the years before 1936 only a small number of physiological data were available and they were insufficient to provide any clue on a physical mechanism of nerve excitation. The year of 1968 was a big turning point. In t h a t year were made known three important physical facts on nerve membrane. The first was a direct evidence for negative surface charge on the membranes of giant axons (Segal, 1968). The second was the observation of birefringence changes coinciding with the time course of action potential during nerve excitation (Cohen, Keynes, Hille, 1968). The third was the more refined measurements of heat production and absorption showing their time correlation with the action potential (Howarth, Keynes, Ritchie, 1968). Based upon these and other physical facts and upon physical principles, Wei has proposed a dipole flip-flop mechanism for nerve excitation which has been shown to be able to interpret a wide variety of phenomena in nerve excitation (Wei, 1968, 1969a, 1969b, 1971a, 1971b, 1972, 1973, 1974). However, it is desirable to perform an experiment to test the theory deliberately. I n the test to be described, basic excitation phenomena were observed and the data were compared with Hill's theory and Wei's theory. Evaluation is done not only upon the agreement between theory and experiment but also upon the relative merits of the two theories in facing the experimental data. The quantum theory is also evaluated on its capability to interpret a variety of intricate results from the experiments done by other workers. Finally, a summarial comparison is made between the HodgkinHuxley equations and the quantum theory on their scopes of usefulness and limitations.
Theory. Nerve excitation is manifest by a sudden inward flow of Na ions. The basic question in physics is, what forces are acting on Na ions in the resting state and in the excited state ? To answer this question, we need to know the physical environment about the Na ions. This is illustrated in Figure 1. The Na ions in the external solution are facing the dipole barrier at the outer surface of the membrane. The dipole field (F0) is to drive or keep the Na ions out. Since the Na concentration is higher outside t h a n inside, there is a diffusional force arising from chemical potential which tends to drive Na ions inward. Besides these two forces, there could be some inter-molecular forces acting on Na ions in the solution. Hence the total force (f) on Na ions is, f = f l +f2 +f3
(1)
QUANTUM THEORY OF :NERVE EXCITATION
279
where f l is the electric force from the dipole field, f2, the diffusional force from chemical potential and f3, the intermolecular force. In order to drive Na ions inward, the "force condition" must be satisfied, i.e., f 1> 0
(2)
where the "greater than" sign means inward. This is simply a statement of Newton's l~w of motion, a first principle. Suppose now a stimulus is applied to the nerve membrane. Which force in (1) would be changed ? The stimulus is not likely to change ion concentrations and hence f~ will be unaffected. Change in fs, if any, would be insignificant because the external solution is conducting and has a high dielectric constant both of which will dilute the stimulating effect
Figure 1.
Dipole barriers and potential profile of nerve membrane
0
Figure 2.
-
Dipole energy bands
in the solution. Most probably, the dipoles would take quantum transitions under excitation. Consider that the dipoles have two stable orientations corresponding to two energy bands (see Figure 2). The dipoles in the lower band are the ones with negative poles facing out as shown in Figure 1, while those in the upper band have opposite orientation, The dipole field is given by F o = E o / d = ,7/~ = q ( N 1 - N 2 ) / ~
(3)
where E0 and d are the barrier potential and thickness of the dipole layer; a, its surface charge density; ~, the permittivity; q, the pole charge; ~V1 and N2, the dipole populations in the lower and upper bands. The electric force (fl) on Na ions b y the dipole field is then f l = Q~o
(4)
2so
c. ~ODSON A_~ L. Y. WEI
where Q is the charge of Na ion. It is clear from (3) and (4) that i f N i is decreased and 2V2 increased b y stimulation, F0 and hence f i will be reduced. When f i is reduced to such an extent that the "force condition" (2) is satisfied, then Na ions will be driven inward and nerve impulse results. The reduction of N1 (microscopic) and so of Fo (macroscopic) is called depolarizarion while the reverse, hyperpolarization. A little physics will show that a cathodic stimulus exceeding a threshold can flip dipoles from the lower states to the upper states and hence cause depolarization. When the outer barrier (E0) is sufficiently reduced, nerve excitation results according to the "force condition". This is a simple way of depolarizing nerve membrane as is usually done in neurophysiology. Let us consider the energy change in the upper state of electric dipoles, d U 2 / d t = - (l/T2) ( U 2 - ~2) + P
(5)
where T2 is the dipole relaxation time in a spontaneous process regardless of stimulation. This is the time taken from a perturbed condition to equilibrium. On the R.H.S. of (5), the first term gives the rate of energy change during the spontaneous process and P, the power received from stimulation, and given by P = CI~FN~
(6)
where C is the rate constant for stimulated transition;/o, the dipole moment; F, the strength of the stimulus and -Y2, the dipole population in the upper state. Since U2 -- N2E2, U2 -- -/Y2E2 where the quantities with "bars" are equilibrium values, then (5) becomes, dNz/dt = - (1/T2)(Nz--Yz) + (Cp/E2)FN2 = - a(N2 - R2) + bFN2
(7) (8)
Here a = l / T u is the spontaneous downward transition rate b y relaxation and bF, the stimulated transition rate. Equation (7) or (8) is Wefts equation for nerve excitation. This equation is unlike (A) and (B) in that: (a) the term b F N ~ contains both the stimulating strength F and the dependent variable N2, (b) every quantity has a well-defined physical meaning and most important, (c) the equation is derived from a physical mechanism (dipole flip-flop) rather than from physiological experience. Equation (7) or (8) can be solved for various forms of _F(t). Under a constant cathodic stimulus, the strength-duration relation is given by (Wei, 1971a), ( F / F ~ - i)t0 = T 2 In [(1 - F J F ) N . d R 2+ F o / F ]
(9)
Q U A N T U M T H E O R Y OF N E R V E E X C I T A T I O N
281
where Fc = the threshold for excitation (rheobase) = a/b = E 2 / C p T 2
(10)
Here we m a y estimate Fc. Taking E2 = 60 meV (Wei, 1971b), p = 10 eA and C T2 -- 10 (i.e. a stimulating transition rate C ten times the relaxation rate 1/T2), we obtain Fc = 6x 104 mV/cm or a stimulating voltage of 36 mV across a 60/~ membrane, which is in the right order of magnitude of what is often used in physiology (10-100 mV). to = the minimum time required for excitation at a given F, the strength of the stimulus and N u = the minimum population N2 as determined by the "force condition". From (9), one finds t h a t when F ~ Fe, Fto = FcT2 In N m / N 2 = const.
(11)
This is the "constant quantity" relation under strong stimulation. When F --~Fc. (9) gives, T u = T 2 ( N m / N 2 - 1) (12) T u Js called the utilization time or the minimum time required for excitation at rheobase strength. The chronaxie is obtained by setting F = 2Fc,
Tc~ = T2 In ( N m / N 2 + 1)/2
(13)
If the stimulus is harmonic, the threshold-frequency relation is given by (Wei, 1973), fc/F~ = a/o) = 1/coTu (14) at low frequencies and (fc/Fc)2/2 = (o~/a)2+ 1 = (0~T2)2+ 1
(15)
at high frequencies. In the middle range, the relation is much more complex and cannot be put in a closed form. In (14) and (15), fc is the threshold (A.C.) and oJ, the angular frequency. The optimum frequency for excitation is given by (Wei, 1973). o~o/a = 0.387 or
v0 = 0.0615a = 0.0615/T2
(16)
Equations (9)-(16) are to be tested from experiment. Among the constants, T~ is the most important because it appears in every and all equations. In the
282
C. H O D S O N A N D L. Y. W E I
previous papers (Wei, 1971a, 1971b, 1973), the value of T~ was assumed to be in the msec range. This is now to be confirmed from experiment. Since the relations derived from Hill's equations are to be used as references for comparison, we list t h e m as follows. The s t r e n g t h - d u r a t i o n relation takes the form of (Hill, 1936a), I = I0(1 - k/A)/(exp ( - t/A) - exp ( - t/k)
(17)
where I0 is the " t r u e rheobase". For v e r y slow accommodation, i.e. A --+ 0% (17) reduces to I = Io/(1-exp (-t/k) (18) From (18), the chronaxie is obtained b y setting I = 210, tch = 0.693k
(19)
However, if we let I = I0, (18) gives t = ~ . T h a t is, the utilization time does not exist for v e r y slow accommodation. E q u a t i o n (17) does n o t give tu in a closed form. This is to be examined from the plot of (17) with k and A determined from experiment. Under A.C. stimulation, the threshold-frequency relation derived by Hill (1936a) is, I / I o = (1 + 4nAken2)l/2 (1 + l/4x2A2n2) 1/2 (20) where n m the frequency. At low frequencies, (I/Io) 2 = 1 + 1/(2u;tn) 2 = 1 + 1/AAoJ9'
(21)
while at high frequencies, (I/Io) 2 = 1 + (2nkn) 2 = 1 +k2o~ 2
(22)
The optimum frequency for excitation is given by, re
=
l/2r:(kA) 1/2
(23)
Experimental method. Myxicola was obtained from Marine Research Associates (Lord's Cove, Deer Island, New Brunswick). Axon was cut out following the procedures used b y Binstock and Goldman (1969): I t was placed upon an assembly of p l a t i n u m wire (in groove) electrodes which was housed in a moist chamber. To control the temperature, the moist chamber was placed inside the Tenny J u n i o r environment chamber. Temperature was monitored with a copper-constantan thermo-couple located n e x t to the axon. Grass model SD9 stimulator was used to provide D.C. stimulation and Exact Model 126 function generator for A.C. stimulation. The recording electrodes were connected directly to the differential input of a grass model P15 preamplifier located close to the preparation. The o u t p u t of the amplifier was fed to the input of a Tektronix model 502A oscilloscope.
QUANTUH THEORY OF ~TERVE EXCITATION
283
Result. F r o m A.C. m e a s u r e m e n t s , we h a v e o b t a i n e d threshold versus f r e q u e n c y relations a t 22.1, 14.3 a n d 6.3°C. These are shown in Figures 3-5. On the low frequency side, we plot ] 2 vs 1/~ 2 while on t h e high f r e q u e n c y side, 12 vs v2. One sees t h a t the curves are good s t r a i g h t lines which are in general a g r e e m e n t O
'R,,
~2o
•~
Figure 3.
I
[
I
12
I0
8
I
I
6 4 [03Iv 2
(b)
f
I
I
[
2.
0
4
8
I
I
12 16 7,,,2/i0 4
I
I
20 24
The threshold-frequency curve and experimental points at 22°C
2O
I 6
I 5
'. 4
[ 3
I0~
Figure 4.
I ?-
I
i, I
el
0 y 2/104
.!/2
The threshold-frequency curve and experimental points at 14.3°C
15--
o
~10 O x
5 q
Figure 5.
4
3 102/Y
2
0
I
2
3
4
Y21103
5
6
7
8
P
The threshold-frequency curve and experimental points a~ 6.3°C
284
C. H O D S O N AND L. Y. WE1
with Equations (14) and (15) of Wei's theory and with Equations (21) and (22) of Hill's theory. Hill's 2 and k can be determined from the slopes of the curves on the low and high frequency sides respectively. The The spike duration w is directly measured. To determine the optimum frequency (vo) for excitation, we have replotted (not shown here) Figures 3-5 in log frequency scale so that the crowded points are spaced well apart. The frequency at which the threshold is the lowest gives v0. Then from (16), the dipole relaxation time T2 is calculated. Table 1 gives the constants measured and calculated. In this table, the last column gives the spike duration measured. The most interesting result from Table 1 is that v0 decreases and all the time constants (2, k, T2 and w) increase with TABLE I Constants obtained from A.C. Measurements Temp. (°C)
vo ( g z )
22.1 14.3 6.3
60 40 18.5
Io (ltA) 15.8 7.74 12.9
~, (ms)
]c (ms)
Tu (ms)
w (ms)
5.93 9.0 22.7
0.97 1.84 3.7
1.03 1.54 3.33
1.02 1.69 3.22
decreasing temperature. This cannot be understood from Hill's theory. In the quantum theory, all time and frequency constants are related to the dipole relaxation time T2. When the t e m p e r a t u r e is lowered, the dipoles will have smaller kinetic energy and thus it will take longer time for them to reorient themselves and hence to relax to the lower energy states. The above trend had already been seen in the classical experiment of Hill et al. (1936). For example, the optimum frequencies for excitations of frog's nerve (after soaked in normal Ringer's solution) they observed were: 21 Hz (6°C); 62 Hz (18.5°C) and 151 Hz (28°C). The similarity of the results from Hill et al. and from ours lends even greater support to our interpretation. Another interesting result from Table 1 is the closeness of k, T2 and w values at each temperature though they are determined b y different methods. That k should be close to T2 can be seen b y comparison of (15) and (22). The near equality of T2 to w in Table 1 is inherent with the dipole mechanism. Under stimulation, the dipole populations are perturbed and so is the dipole barrier potential (E0) according to (3). This barrier potential change is transmitted by the Na ion flow and is thus amplified. It is manifested as the "action potential". Hence the duration (w) of the action potential should be equal to that of the barrier potential change which is produced b y the change of dipole populations. The dipole relaxation time T2 is the time taken from t h e perturbed dipole
QUANTUM T H E O R Y OF N E R V E EXCITATION
285
population to the equilibrium population. That w is found nearly equal to T~ can thus be understood. The observed value of T~ in the msec range gives a great support to the quantum theory in two ways. Quantitatively, the calculated time constants from the theory are all in good agreement with the physiological data. On the other hand, T2 in the order of msec seems too large for free dipoles in water solution. This value would be conceivable for dipoles coupled together like electrons spins in a ferromagnet.
~
o
0
~--
EXPERIMENTAL
I
Io (I-K/X}
I= Ii / (l-e -I/K) !
o .5~
156
I
,65
I i,I],,I
I
io-4 STIMULUS PULSE
F i g u r e 6.
I TI11111
I
,63
I L r111il
162
DURATION |=ec|
S t r e n g t h - d u r a t i o n curves b a s e d on H i l l ' s t h e o r y a n d e x p e r i m e n t a l p o i n t s a t 22°C
From the D.C. measurements, we have obtained the strength-duration relation. Using the ~ and /~ values from Table 1, Equations (17) and (18) of Hill's theory are plotted along with the experimental data in Figures 6-8. Though the theoretical curves follow the general trend of the experimental points, the fit is not good particularly for the solid curves at long durations in Figures 6 and 7. The solid curves bend upwards after meeting 11 in Figures 6 and 7. The Hill theory predicts either an infinite utilization time (dashed curves in Figures 6-8) or higher stimulating strengths beyond the utilization time (solid curves in Figures 6 and 7), both contrary to the established facts. However from the experimental data we can obtain the utilization times at three temperatures and then from (12) and Table 1 (taking T2), we can calculate N~n/572 in Wefts theory. These are listed in Table 2. With T2 and Nm/IV2available, Equation (9) of Wei's theory can be calculated and plotted as shown in Figures 9-11. Agreement between theory and experiment is rather satisfactory.
286
C. EODSON AND L. Y. W E I -3 EXPERIMENTAL I Io ( t - K / X )
-
I0 s
note: I o = I z
I
]
10-5
Figure 7.
I TIIlII
I
1 T Ilftl]
I
(--~) ~--~K'ST-i
r III1111
10-4, tO-'~ STIMULUS PULSE DURATJON (see}
]O-Z
S t r e n g t h - d u r a t i o n curves based on Hill's t h e o r y and experimental points at 14.3°C \
i6 3
"tl~ ~"\
D --
\\
EXPERIMENTAL T
- =Io~( I - K / X )
D
ee~~
ul
q 155 b-
I0 6
1 i(~ 5
T t ItttIT
t i6 4
t t t11111 t5 3
I
1 t tilt11 16 2
STIMULUS PULSE DURATION (sec)
Figure 8.
S t r e n g t h - d u r a t i o n curves based on HilFs t h e o r y and experimental points at 6.3°C TABLE
II
Nm/N2 V a l u e s T (°C) 22 14.3 6.3
Tu (msec.) (obs.) 3 3.9 7
N'm/_N2(calc.) 3.85 3.4 3.1
QUANTUM THEORY OF NERVE EXCITATION
287
In contrast with the Hill theory, the q u a n t u m t h e o r y gives a finite utilization time and agrees r a t h e r well with the e x p e r i m e n t a l points below the utilization time. I n Figures 9-11, the theoretical curves would drop slowly below the
EXPE%::?
o
E Ii__±__D~--O---o----. ~E I[ o-155 F-
-E
I0
I i0-5
I ? I IrIl[ I I I IllIiI I T I IItIII 1o-4 10-3 lO-Z STIMULUS PULSE DURATION ( s e c }
Figure 9. Strength-duration curves based on Wei's theory and experimental points at 22°C 163~ A
EXPERIMENTAL
:
l~
155
I
I IlllEIT
164
STIMULUS
l
1 T]11III
PULSE
io-3
DURATION
1
T I]1111I
id2
(see)
Figure 10. Stren~h-duration curve based on Wei's theory and e~oer~enta! points at ]4.3°(3
dotted line and F = (0.68-0.74) Fc as t - ~ oo. I n e x p e r i m e n t , it is e x t r e m e l y difficult to observe excitation as F goes below F c because of m e m b r a n e noise and external interference. Thus a comparison between t h e o r y and e x p e r i m e n t
288
c. HODSON AND L. Y. WEI
below Fc is meaningless. Table I I indicates that the required minimum number (Nm) of dipoles in the upper state for excitation of the nerve need only be a few times the equilibrium value N~. That Nm/-~ decreases with decreasing temperature is perhaps due to the fact that T2 is longer and hence the threshold Fc is lower (see (10)) at lower temperatures. As far as the strength-duration relation is concerned, Wei's theory predicts the existence and finiteness of the utilization time (tu) while this is not apparent from Hill's theory, as has been pointed out previously b y Johnson et al. (1954). Equation (12) suggests that the utilization time arises from the time of the excess dipole population (Nm- ~V2) spending in the upper state. 15"
165
[(~
o I
1~5
? IIIIII1
1
Figure 1l.
._o__ ..~_
] I~IIITI
1(~4 ST]MULUS
EX.ER,ME=2
~
I T IIIill
i(~ =' PULSE
DURATION
[(~Z (sec)
Strength-duration relation curve based on Wei's theory and experimental points at 6.3°C
Discussion. The quantum theory of nerve excitation is based upon the well known quantum mechanisms : stimulated and spontaneous quantum transitions of molecules between energy states. The important quantities involved are molecular quantities (such as Nz, N2, Tz and E2) whose behaviour is to obey physical principles. To illustrate the capability of the quantum theory, we shall attempt here to interpret a variety of experimental results obtained b y others. It is well known that Ca~+ tends to reduce the size of action potential (Frankenhaeuser and Hodgkia, 1957). This is thought to result from Ca ~+ binding with the membrane surface on which there is fixed negative charge. In the quantum theory, such a binding would lower the energy of the dipoles in the lower band (Figure 2) with their negative ends bound to Ca 2+. This is effectively to increase the separation energy ( E 2 - Ez) between the two bands. I f a dipole flips from the lower band to the upper band b y excitation, its polarity will be
QUANTUM THEORY OF, NERVE EXCITATIQN
289
reversed and its positive pole will then face the Ca ~+ ion. Because of mutual repulsion, this dipole in the upper state would flop down in shorter time (statistically speaking) than it would in the absence o f Ca 2+ ions. That is to say, Ca 2+ ions in good amount would tend to reduce the dipole relaxation time T2 from the upper state to the lower state. According to (10), increasing E~ and decreasing T2 b y Ca 2+ has the aggravated effect of raising the threshold F c for nerve excitation. Then reduction of action potential b y Ca 2+ becomes readily understood. The above arguments may also be applicable to a host of drugs or agents which have anaesthetic or tranquilizing actions upon nerve. Hill et al. (1936) found that the optimum frequency for excitation of frog's nerve after soaked in Ringer's solution containing high Ca (4-10 times normal) was increased to 94 Hz. at room temperature from 62 Hz. (after soaked in normal Ringer solution). Based on (16), this increase of v0 results from a decrease of T2 in high Ca as reasoned in the above. Next, we shall consider the temperature effects on nerve excitation. The classical experiment of Hodgkin and Katz (1949) showed that, (a) the peak amplitude of the action potential decreases with increasing temperature and (b) the duration particularly the repolarization phase of the action potential "becomes prolonged with decreasing temperature. Recently, Inoue et al. (1973) observed that the squid axon was abruptly depolarized when the temperature was lowered to about 5°C at or below a critical ratio of Ca : Na in the external solution. Results of similar nature were obtained b y Adey (1973) in cat cortex. He found that transient cooling b y 5-6°C in cat cortex for 1-2 min depolarized neurons b y 1-2 mV/C. Raising topical Ca levels to 20 mM blocked depolarization by transient cooling, but 7.5 mM solutions were without effect. All these temperature effects may be explained on the basis of dipole relaxation time T2. As the temperature is decreased, T~ increases and according to (10), the threshold F¢ will be lowered. As a result, the peak amplitude of the action potential will be increased or the membrane becomes depolarized more readily. The increase of T2, however, will prolong the action potential, particularly the repolarization phase which in the view of the quantum theory, is a macroscopic manifestation of dipole relaxation. As discussed in t h e above, the presence of Ca at high concentration would have the opposite effect, that is, raising the threshold. Thus depolarization b y cooling should be abolished by raising in Ca concentration above a certain level. This has indeed been verified in the experiments b y Inoue et al. and b y Adey. That the lower conduction velocity in nerve at lower temperatures as observed in our experiment and b y others (Hodgkin, Katz, 1949; Gasser, 1931; Turner, 1955; Wright, 1958) may be taken as a simple consequence of increasing relaxation time T2. Students of biology often wonder why the time constants in nerve excitation are in the msec range and why nerve impulse frequency seldom goes much higher than kHz. From the quantum theory, these questions can now be
290
0. HODSON AND L. Y. WEI
answered. The time and frequency constants in nerve excitation are all determined by the dipole relaxation time T2. I f T~ were in the msec range, the quantum theory would yield values of the said constants as observed. Our experiment has indeed produced T2 in the msec range. Here we see mutual benefits between theory and experiment. The quantum theory suggests a very simple way of determining T 2 by observing v0, the optimum frequency for nerve excitation. The experimental value of T2 in turn lends a strong support to the quantum theory in the interpretation of the time and frequency constants both qualitatively and quantitatively. Since T2 is one of the most important quantities in the quantum transitions of molecules, the quantum theory and the simple method of determining T2 will provide us an elegant way of studying macroscopic phenomena from a microscopic approach. In nerve excitation, the fundamental question remains to be the "all-ornone" response. W h a t is the underlying mechanism ? This can be understood from quantum mechanics. In a molecular system, with quantum energies Em and E n , if the incident energy (E~) is such t h a t Ei = E r a - E n
the probability for stimulated upward transition becomes infinitely great in principle and one gets "all" response. I f the above condition is not met, nothing spectacular is to be expected. Maser and laser work on this principle. Why is the nerve response pulsed rather than continuous ? Again from quantum mechanics, a system consisting of only two active levels or bands could produce an impulse wave but not a continuous wave under excitation. Ammonia maser is a well-known example (Singer, 1958). However, nerve excitation differs from maser or laser action in population requirement and in manifestation. To excite the nerve N2 need only reach 2 ~ m a S determined by the "force condition" while population inversion (N2 > N1) must be achieved for maser or laser action. In the nerve membrane, when N2 = #Vm, then according to the theory, the barrier potential would be reduced to such an extent as to allow sudden rush of Na ions inward, thus producing a nerve impulse. I n maser or laser, when N2 > / V z , a coherent emission of light is resulted. Since both systems make use of quantum transitions of molecules between energy states, one could gain much understanding of nerve action from maser or laser principle. A simple illustration is the temperature effect. I f an ammonia maser were placed in a hot bath, it would not produce coherent pulsed emission. This is because at high temperatures, T2 is so short t h a t LV~ cannot be accumulated at E2 long enough to exceed N1. The same should happen to a nerve membrane. Cole et al. (1970) have indeed observed t h a t at temperatures above 15°C, the response of squid giant axon is not "all-or-none" but a gradual function of the stimulating strength. I t is also interesting to note t h a t at lower temperatures, the peak amplitude of the action potential is greater (Hodgkin, Katz, 1949) and so is the
QUANTUMTHEORY OF NERVE EXCITATION
291
coherent emission from maser or laser (Singer, 1959). This should not be taken as a mere phenomenal coincidence b u t is derived from profound physical principles. What makes the quantum theory more appealing is that quantum transitions of molecules between energy states can give rise not only to electrical effects but also to thermal and optical effects. In nerve excitation, thermal and optical events have been observed to occur along with action potential (Cohen, Keynes, Hil]e, 1968; Howarth, Keynes, l~itchie, 1968; Fraser, Frey, 1968; Sherebrin, 1972; Abbott, Howarth, 1973). That the electrical, optical and thermal effects in nerve and their time correlations during excitation can be well accounted for by quantum transitions of dipoles has been shown previously (Wei, 1971b, 1972, 1974). From all the above discussions and from our experiment reported here we cannot escape the conclusion that the quantum theory is capable of not only accounting for a wide variety of excitation phenomena in nerve but also providing a deep insight into the mechanism underlying the processes. Finally, we wish to make a comparison of the quantum theory with the I,Hodgkin-Huxley equations which have been widely used since 1952. For this purpose, it is better to quote directly from one of H-H's classical papers. "The agreement must not be taken as evidence that our equations are anything more than an empirical description of the time course of the change in permeability to sodium and potassium". (Hodgkin, Huxley, 1952). Based on this statement of H - H and its implications and on the previous passage related to the quantum theory, we can make a list as below: Empirical description Physical mechanism Electrical effects Thermal effects Optical effects
H - H equations + + -
Quantum theory + + + +
The first two rows indicate the different approaches of the two theories and the next three rows show the consequences therefrom. The H - H equations have been most useful in dealing with voltage clamp data. However, this is also its inherent limitation. Beyond voltage clamp data, the empirical description of the H - H equations finds it hard to extend. For example, the problem of heat production and absorption is rather old and the failure of the H - H theory in this effect is well-known (Hodgkin, 1951 ; Cole, 1968). There is no way, from the H - I t equations, to account for any of the optical effects such as birefringence change (Cohen, Keynes, Hille, 1968), infrared emission (Fraser, Frey, 1968) and absorption (Sherebrin, 1972) in nerve. Such is not the case for the quantum theory which is limited only b y what forbids dipole transitions to occur. Even in the area of voltage clamp data, several authors have produced results that are
292
c. HODSON A ~ D T.. y . WEI
i n e x c e l l e n t a g r e e m e n t w i t h e x p e r i m e n t b y t a k i n g d i p o l e flip-flop t r a n s i t i o n s as a p h y s i c a l b a s i s ( A r n d t a n d R o p e r , 1972; A r n d t , B o n d a n d R o p e r , 1972; V a n L a m s w e e r d e - G a l l e z a n d Messen, 1974). I t is fair t o s a y t h a t e a c h t h e o r y has its m e r i t s a n d serves a m e a n i n g f u l p u r p o s e i n t h e p r o p e r d o m a i n of p r o b l e m s u n d e r s t u d y . T h e q u a n t u m t h e o r y is d i f f e r e n t f r o m t h e p r e v i o u s t h e o r i e s i n t h a t it suggests a p h y s i c a l m e c h a n i s m t h a t m a k e s i t p o s s i b l e for a n u n d e r s t a n d i n g of t h e v a r i o u s processes a s s o c i a t e d w i t h n e r v e e x c i t a t i o n a n d c o n d u c t i o n . T h e w o r k w a s s u p p o r t e d b y t h e N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a u n d e r G r a n t No. A1252.
LITERATURE Abort, B. C. and J. V. ttowarth, 1973. "Heat studies in excitable tissues." Physiol. Rev., 53, 120-158. Adey, W. 1%, 1973. "Biophysical and metabolic bases of cooling effects on cat cortical membrane potentials." UCLA Brain Res. Inst. Tech. Report. Arndt, 1%. and L. D. Roper. 1972. "Quantitative comparison of dipole models for steadystate currents in excitable membranes." Bull. Math. Biophys., 34, 305-324. ----, J. D. Bond and L. D. 1%oper. 1972. "A fit to nerve membrane rectification curves with a double-dipole layer membrane model." Bull. _Math. Biophys., 34, 151-172. Binstock, L. and L. Goldman• 1969. "Current and voltage clamped studies in Myxicola giant axons." J. Gen. Physiol., 54, 730-740. Blair, H. A. 1932. "On the intensity-time relation for stimulation by electric currents, I, II." J. Gen. Physiol., 15, 709-755. ----. 1934. "Conduction in nerve fibres." Ibid., 18, 125-142. ----. 1936. "Kinetics of the excitatory process." Cold Spr. Hath. Syrnp. quant. Biol., 4, 63-72. Cohen, L. B., 1%.D. Keynes and B. Hille. 1968. "Light scattering and birefringence changes during nerve activity." Nature, 218, 438-441. Cole, K. S. 1968. "Membranes, Ions and Impulses•" pp. 318-319. Berkeley- Univ• Calif• Press. , 1%. G u t t m a n and F. Bezanilla. 1970. "Nerve membrane excitation without threshold." Proc. 1Vatn. Acad. Sci., 65, 884-891. Frankcnhaeuser, B. and A. L. Hodgkin, 1957. "The action of calcium on the electrical properties of squid axons." J. Physiol., Lond., 137, 218-244. ~ Fraser, A. and A. H. Frey. 1968• "Electromagnetic emission at micro wavelengths from active nerves. '° Biophys. J., 8, 731-734. Gasser, I-I. S. 1931. "Nerve activity as modified b y temperature." A m . J. Physiol., 97, 254-270. Hill, A. V. 1936a. Excitation and accommodation in nerve. Proe. R. Soc. Lond., 119, 305-355. • 1936b. "The strength-duration relation for electric excitation ofmedullated nerve." Ibid., 119, 440-453. , B. Katz and D. Y. Solandt. 1936• "Nerve excitation by alternating current." Ibid., BI21, 74-133. I-Iodgkin, A. L. 1951. "The ionic basis of electrical activity in nerve and muscle." Biol. Rev., 26, 339-409. --~ and A. F. Huxley. 1952. "A quantitative description of membrane current and its application to conduction and excitation in nerve?' J. Physiol., Lond., 117, 500-544.
QUANTLrM T B _ E O R Y O F N E R V E ----
EXCITATION
293
a n d B. Katz. 1949• " T h e effect of temperature on the electrical activity of the
g i a n t a x o n of t h e s q u i d . " Ibid., 109, 240-249. H o w a x t h , J . V., R. D. K e y n e s a n d J . M. l~itchie. 1968. " T h e origin of t h e i n i t i a l h e a t associated w i t h a single i m p u l s e i n m a m m a l i a n n o n - m y e l i n a t c d n e r v e fibres." J . Physiol., Loved., 194, 745-793. Inoue, I., Y. K o b a t a k e a n d I. Tasaki. 1973. " E x c i t a b i l i t y , i n s t a b i l i t y a n d p h a s e t r a n s i t i o n s in s q u i d a x o n m e m b r a n e u n d e r i n t e r n a l p e r f u s i o n w i t h d i l u t e salt s o l u t i o n s . " Biochim. Biophys. Acts. 307, 471-477. J o h n s o n , F. H., H . E y r i n g a n d M. g. Polissar, 1954. The Kinetic Basis of Molecular Biology, p. 656. N e w Y o r k , Wiley. K a t z , B. 1939. Electric Excitation of Nerve, Oxford. U n i v . Press. Mormier, A. M. 1934. L'Excitation Eleetrique des Tissues. P a r i s : H e r m a n n . l~ashevsky, N. 1933. " O u t l i n e of a p h y s i c o - m a t h e m a t i c a l t h e o r y of e x c i t a t i o n a n d i n h i b i t i o n . " Protoplasma, 20, 42-56. - - - - - - 1960. Mathematical Biophysics. Vol. 1, 3rd. rev. N e w Y o r k : D o v e r . Segal, J. 1%. 1968. " S u r f a c e c h a r g e of g i a n t a x o n s of s q u i d a n d l o b s t e r . " Biophys. J., 8, 470-489. Sherebrin, M. H . 1972. " C h a n g e s in i n f r a r e d s p e c t r u m of n e r v e d u r i n g e x c i t a t i o n . " Nature, New Biol., 235, 122-124. Singer, J. R . 1959. Masers. N e w Y o r k : Wiley. Solandt, D. Y. 1936. " T h e m e a s u r e m e n t of ' a c c o m m o d a t i o n ' i n n e r v e . " Proc. B. Soc. Lond., B l 1 9 , 355-379. Turner, 1~. S. 1955. " R e l a t i o n b e t w e e n t e m p e r a t u r e a n d c o n d u c t i o n i n n e r v e fibres of different sizes." Physiol. Zool., 28, 55-61. V a n L a m s w e e r d e - G a l l e z , D. a n d A. Mcssen. 1974. " S u r f a c e dipoles, surface charges a n d n e g a t i v e s t e a d y - s t a t e r e s i s t a n c e i n biological m e m b r a n e s . " J. Biol. Phys., 2, 75-102. Wei, L. Y. 1968. " E l e c t r i c a l dipole t h e o r y of c h e m i c a l s y n a p t i c t r a n s m i s s i o n . " Biophys. J., 8, 396-414. - - . 1969a. " R o l e of surface dipoles on a x o n m e m b r a n e . " Science, 163, 280-282. 1969b. " M o l e c u l a r m e c h a n i s m s of n e r v e e x c i t a t i o n a n d c o n d u c t i o n . " Bull. Math. Biophys., 31, 39-58. - - . 1971a. " Q u a n t u m T h e o r y of n e r v e e x c i t a t i o n . " Ibid., 33, 187-194. . 1971b. " P o s s i b l e origin of a c t i o n p o t e n t i a l a n d b i r e f r i n g e n c e c h a n g e i n n e r v e a x o n . " Ibid., 33, 521-537. • 1972. " D i p o l e t h e o r y of h e a t p r o d u c t i o n a n d a b s o r p t i o n in n e r v e a x o n . " Biophys. J . , 12, 1159-1170. - - . 1973. " Q u a n t u m t h e o r y of t i m e - v a r y i n g s t i m u l a t i o n i n n e r v e . " Bull. Math. Biol., 35, 359-374. - - . 1974. " D i p o l e m e c h a n i s m s of electrical, optical a n d t h e r m a l e n e r g y t r a n d u c tions i n n e r v e m e m b r a n e . " Ann. N . Y . Acad. Sci., 227, 285-293. W r i g h t , E . B. 1958. " T h e effect of low t e m p e r a t u r e s o n single c r u s t a c e a n m o t o r n e r v e fibres." J. Cell. Comp. Physiol., 51, 29-65.
RECEIVED 6 - 2 5 - 7 5 l~EVISED 1 0 - 1 7 - 7 5
OF
BIAT]~EI%IATICAL B I O L O G Y
VOLUM]S 38, 1976
COMPARATIVE EVALUATION NERVE EXCITATION
OF
QUANTUM
THEORY
OF
• C. HODSON AND L. Y. WEI~ Biophysical Research Laboratory, Electrical E n g i n e e r i n g D e p a r t m e n t , U n i v e r s i t y of Waterloo, Waterloo, Ont., Canada
Deliberate evaluation of the q u a n t u m theory of nerve excitation is made b y comparing it with Hill's theory in fitting the experimental data on threshold-frequency relation, optimum frequency (v0) for nerve excitation and strength-durati0n relation. Decrease of v0 and increase of all the time constants (Hill's A and k, Wei's T~ and spike duration w) with decreasing temperature are interpreted on the basis of the dipole relaxation time T~, but inexplicable from Hill's theory or any other existing theory. The closeness of k, T2 and w values is explained. A variety of experimental results obtained by others is discussed. Finally, a comparison is made between the t t o d g k i n - H u x l e y equations and the q u a n t u m theory. Most of the facts (electrical and non-electrical) tend to support the thesis that nerve excitation is a macroscopic expression of q u a n t u m transitions of dipoles between energy states.
In the years of 1932-1936, there were several phenomenological theories developed on nerve excitation (Blair, (1932, 1934, 1936); Rashevsky, (1933, 1960); Hill, (1936a, 1936b); Hill, Katz, Solandt, (1936; Monnier, (1934)). Formally, the equations of Blair, of Rashevsky and of Hill are equivalent and so are their results. Their equations are of the following types, d u / d t = K I - a ( u - uo)
(A)
dv/dt = M(u-
(B)
uo) - b ( v - vo)
where u and v are excitatory and inhibitory factors. Blair used one equation of type (A). Rashevsky had two equations, each of type (A). Hill started from two equations of type (A) and (B). The relations derived from Hill's equations ~To whom requests for reprints should be directed. 277
278
c. HODSON AND L. Y. WEI
have been under extensive experimental tests (Hill, 1936b; Hill, Katz, Solandt, 1936; Solandt, 1936) and evaluated in detail by Katz (1939). With properly chosen constants, Hill relations as well as Blair's and Rashevsky's are able to fit a number of experimental data and "in m a n y cases with an almost embarassing accuracy" (in Katz's word). However, the equations (A) and (B) and the constants therein do not suggest or imply the physical nature or mechanism of the processes described. This was so because in the years before 1936 only a small number of physiological data were available and they were insufficient to provide any clue on a physical mechanism of nerve excitation. The year of 1968 was a big turning point. In t h a t year were made known three important physical facts on nerve membrane. The first was a direct evidence for negative surface charge on the membranes of giant axons (Segal, 1968). The second was the observation of birefringence changes coinciding with the time course of action potential during nerve excitation (Cohen, Keynes, Hille, 1968). The third was the more refined measurements of heat production and absorption showing their time correlation with the action potential (Howarth, Keynes, Ritchie, 1968). Based upon these and other physical facts and upon physical principles, Wei has proposed a dipole flip-flop mechanism for nerve excitation which has been shown to be able to interpret a wide variety of phenomena in nerve excitation (Wei, 1968, 1969a, 1969b, 1971a, 1971b, 1972, 1973, 1974). However, it is desirable to perform an experiment to test the theory deliberately. I n the test to be described, basic excitation phenomena were observed and the data were compared with Hill's theory and Wei's theory. Evaluation is done not only upon the agreement between theory and experiment but also upon the relative merits of the two theories in facing the experimental data. The quantum theory is also evaluated on its capability to interpret a variety of intricate results from the experiments done by other workers. Finally, a summarial comparison is made between the HodgkinHuxley equations and the quantum theory on their scopes of usefulness and limitations.
Theory. Nerve excitation is manifest by a sudden inward flow of Na ions. The basic question in physics is, what forces are acting on Na ions in the resting state and in the excited state ? To answer this question, we need to know the physical environment about the Na ions. This is illustrated in Figure 1. The Na ions in the external solution are facing the dipole barrier at the outer surface of the membrane. The dipole field (F0) is to drive or keep the Na ions out. Since the Na concentration is higher outside t h a n inside, there is a diffusional force arising from chemical potential which tends to drive Na ions inward. Besides these two forces, there could be some inter-molecular forces acting on Na ions in the solution. Hence the total force (f) on Na ions is, f = f l +f2 +f3
(1)
QUANTUM THEORY OF :NERVE EXCITATION
279
where f l is the electric force from the dipole field, f2, the diffusional force from chemical potential and f3, the intermolecular force. In order to drive Na ions inward, the "force condition" must be satisfied, i.e., f 1> 0
(2)
where the "greater than" sign means inward. This is simply a statement of Newton's l~w of motion, a first principle. Suppose now a stimulus is applied to the nerve membrane. Which force in (1) would be changed ? The stimulus is not likely to change ion concentrations and hence f~ will be unaffected. Change in fs, if any, would be insignificant because the external solution is conducting and has a high dielectric constant both of which will dilute the stimulating effect
Figure 1.
Dipole barriers and potential profile of nerve membrane
0
Figure 2.
-
Dipole energy bands
in the solution. Most probably, the dipoles would take quantum transitions under excitation. Consider that the dipoles have two stable orientations corresponding to two energy bands (see Figure 2). The dipoles in the lower band are the ones with negative poles facing out as shown in Figure 1, while those in the upper band have opposite orientation, The dipole field is given by F o = E o / d = ,7/~ = q ( N 1 - N 2 ) / ~
(3)
where E0 and d are the barrier potential and thickness of the dipole layer; a, its surface charge density; ~, the permittivity; q, the pole charge; ~V1 and N2, the dipole populations in the lower and upper bands. The electric force (fl) on Na ions b y the dipole field is then f l = Q~o
(4)
2so
c. ~ODSON A_~ L. Y. WEI
where Q is the charge of Na ion. It is clear from (3) and (4) that i f N i is decreased and 2V2 increased b y stimulation, F0 and hence f i will be reduced. When f i is reduced to such an extent that the "force condition" (2) is satisfied, then Na ions will be driven inward and nerve impulse results. The reduction of N1 (microscopic) and so of Fo (macroscopic) is called depolarizarion while the reverse, hyperpolarization. A little physics will show that a cathodic stimulus exceeding a threshold can flip dipoles from the lower states to the upper states and hence cause depolarization. When the outer barrier (E0) is sufficiently reduced, nerve excitation results according to the "force condition". This is a simple way of depolarizing nerve membrane as is usually done in neurophysiology. Let us consider the energy change in the upper state of electric dipoles, d U 2 / d t = - (l/T2) ( U 2 - ~2) + P
(5)
where T2 is the dipole relaxation time in a spontaneous process regardless of stimulation. This is the time taken from a perturbed condition to equilibrium. On the R.H.S. of (5), the first term gives the rate of energy change during the spontaneous process and P, the power received from stimulation, and given by P = CI~FN~
(6)
where C is the rate constant for stimulated transition;/o, the dipole moment; F, the strength of the stimulus and -Y2, the dipole population in the upper state. Since U2 -- N2E2, U2 -- -/Y2E2 where the quantities with "bars" are equilibrium values, then (5) becomes, dNz/dt = - (1/T2)(Nz--Yz) + (Cp/E2)FN2 = - a(N2 - R2) + bFN2
(7) (8)
Here a = l / T u is the spontaneous downward transition rate b y relaxation and bF, the stimulated transition rate. Equation (7) or (8) is Wefts equation for nerve excitation. This equation is unlike (A) and (B) in that: (a) the term b F N ~ contains both the stimulating strength F and the dependent variable N2, (b) every quantity has a well-defined physical meaning and most important, (c) the equation is derived from a physical mechanism (dipole flip-flop) rather than from physiological experience. Equation (7) or (8) can be solved for various forms of _F(t). Under a constant cathodic stimulus, the strength-duration relation is given by (Wei, 1971a), ( F / F ~ - i)t0 = T 2 In [(1 - F J F ) N . d R 2+ F o / F ]
(9)
Q U A N T U M T H E O R Y OF N E R V E E X C I T A T I O N
281
where Fc = the threshold for excitation (rheobase) = a/b = E 2 / C p T 2
(10)
Here we m a y estimate Fc. Taking E2 = 60 meV (Wei, 1971b), p = 10 eA and C T2 -- 10 (i.e. a stimulating transition rate C ten times the relaxation rate 1/T2), we obtain Fc = 6x 104 mV/cm or a stimulating voltage of 36 mV across a 60/~ membrane, which is in the right order of magnitude of what is often used in physiology (10-100 mV). to = the minimum time required for excitation at a given F, the strength of the stimulus and N u = the minimum population N2 as determined by the "force condition". From (9), one finds t h a t when F ~ Fe, Fto = FcT2 In N m / N 2 = const.
(11)
This is the "constant quantity" relation under strong stimulation. When F --~Fc. (9) gives, T u = T 2 ( N m / N 2 - 1) (12) T u Js called the utilization time or the minimum time required for excitation at rheobase strength. The chronaxie is obtained by setting F = 2Fc,
Tc~ = T2 In ( N m / N 2 + 1)/2
(13)
If the stimulus is harmonic, the threshold-frequency relation is given by (Wei, 1973), fc/F~ = a/o) = 1/coTu (14) at low frequencies and (fc/Fc)2/2 = (o~/a)2+ 1 = (0~T2)2+ 1
(15)
at high frequencies. In the middle range, the relation is much more complex and cannot be put in a closed form. In (14) and (15), fc is the threshold (A.C.) and oJ, the angular frequency. The optimum frequency for excitation is given by (Wei, 1973). o~o/a = 0.387 or
v0 = 0.0615a = 0.0615/T2
(16)
Equations (9)-(16) are to be tested from experiment. Among the constants, T~ is the most important because it appears in every and all equations. In the
282
C. H O D S O N A N D L. Y. W E I
previous papers (Wei, 1971a, 1971b, 1973), the value of T~ was assumed to be in the msec range. This is now to be confirmed from experiment. Since the relations derived from Hill's equations are to be used as references for comparison, we list t h e m as follows. The s t r e n g t h - d u r a t i o n relation takes the form of (Hill, 1936a), I = I0(1 - k/A)/(exp ( - t/A) - exp ( - t/k)
(17)
where I0 is the " t r u e rheobase". For v e r y slow accommodation, i.e. A --+ 0% (17) reduces to I = Io/(1-exp (-t/k) (18) From (18), the chronaxie is obtained b y setting I = 210, tch = 0.693k
(19)
However, if we let I = I0, (18) gives t = ~ . T h a t is, the utilization time does not exist for v e r y slow accommodation. E q u a t i o n (17) does n o t give tu in a closed form. This is to be examined from the plot of (17) with k and A determined from experiment. Under A.C. stimulation, the threshold-frequency relation derived by Hill (1936a) is, I / I o = (1 + 4nAken2)l/2 (1 + l/4x2A2n2) 1/2 (20) where n m the frequency. At low frequencies, (I/Io) 2 = 1 + 1/(2u;tn) 2 = 1 + 1/AAoJ9'
(21)
while at high frequencies, (I/Io) 2 = 1 + (2nkn) 2 = 1 +k2o~ 2
(22)
The optimum frequency for excitation is given by, re
=
l/2r:(kA) 1/2
(23)
Experimental method. Myxicola was obtained from Marine Research Associates (Lord's Cove, Deer Island, New Brunswick). Axon was cut out following the procedures used b y Binstock and Goldman (1969): I t was placed upon an assembly of p l a t i n u m wire (in groove) electrodes which was housed in a moist chamber. To control the temperature, the moist chamber was placed inside the Tenny J u n i o r environment chamber. Temperature was monitored with a copper-constantan thermo-couple located n e x t to the axon. Grass model SD9 stimulator was used to provide D.C. stimulation and Exact Model 126 function generator for A.C. stimulation. The recording electrodes were connected directly to the differential input of a grass model P15 preamplifier located close to the preparation. The o u t p u t of the amplifier was fed to the input of a Tektronix model 502A oscilloscope.
QUANTUH THEORY OF ~TERVE EXCITATION
283
Result. F r o m A.C. m e a s u r e m e n t s , we h a v e o b t a i n e d threshold versus f r e q u e n c y relations a t 22.1, 14.3 a n d 6.3°C. These are shown in Figures 3-5. On the low frequency side, we plot ] 2 vs 1/~ 2 while on t h e high f r e q u e n c y side, 12 vs v2. One sees t h a t the curves are good s t r a i g h t lines which are in general a g r e e m e n t O
'R,,
~2o
•~
Figure 3.
I
[
I
12
I0
8
I
I
6 4 [03Iv 2
(b)
f
I
I
[
2.
0
4
8
I
I
12 16 7,,,2/i0 4
I
I
20 24
The threshold-frequency curve and experimental points at 22°C
2O
I 6
I 5
'. 4
[ 3
I0~
Figure 4.
I ?-
I
i, I
el
0 y 2/104
.!/2
The threshold-frequency curve and experimental points at 14.3°C
15--
o
~10 O x
5 q
Figure 5.
4
3 102/Y
2
0
I
2
3
4
Y21103
5
6
7
8
P
The threshold-frequency curve and experimental points a~ 6.3°C
284
C. H O D S O N AND L. Y. WE1
with Equations (14) and (15) of Wei's theory and with Equations (21) and (22) of Hill's theory. Hill's 2 and k can be determined from the slopes of the curves on the low and high frequency sides respectively. The The spike duration w is directly measured. To determine the optimum frequency (vo) for excitation, we have replotted (not shown here) Figures 3-5 in log frequency scale so that the crowded points are spaced well apart. The frequency at which the threshold is the lowest gives v0. Then from (16), the dipole relaxation time T2 is calculated. Table 1 gives the constants measured and calculated. In this table, the last column gives the spike duration measured. The most interesting result from Table 1 is that v0 decreases and all the time constants (2, k, T2 and w) increase with TABLE I Constants obtained from A.C. Measurements Temp. (°C)
vo ( g z )
22.1 14.3 6.3
60 40 18.5
Io (ltA) 15.8 7.74 12.9
~, (ms)
]c (ms)
Tu (ms)
w (ms)
5.93 9.0 22.7
0.97 1.84 3.7
1.03 1.54 3.33
1.02 1.69 3.22
decreasing temperature. This cannot be understood from Hill's theory. In the quantum theory, all time and frequency constants are related to the dipole relaxation time T2. When the t e m p e r a t u r e is lowered, the dipoles will have smaller kinetic energy and thus it will take longer time for them to reorient themselves and hence to relax to the lower energy states. The above trend had already been seen in the classical experiment of Hill et al. (1936). For example, the optimum frequencies for excitations of frog's nerve (after soaked in normal Ringer's solution) they observed were: 21 Hz (6°C); 62 Hz (18.5°C) and 151 Hz (28°C). The similarity of the results from Hill et al. and from ours lends even greater support to our interpretation. Another interesting result from Table 1 is the closeness of k, T2 and w values at each temperature though they are determined b y different methods. That k should be close to T2 can be seen b y comparison of (15) and (22). The near equality of T2 to w in Table 1 is inherent with the dipole mechanism. Under stimulation, the dipole populations are perturbed and so is the dipole barrier potential (E0) according to (3). This barrier potential change is transmitted by the Na ion flow and is thus amplified. It is manifested as the "action potential". Hence the duration (w) of the action potential should be equal to that of the barrier potential change which is produced b y the change of dipole populations. The dipole relaxation time T2 is the time taken from t h e perturbed dipole
QUANTUM T H E O R Y OF N E R V E EXCITATION
285
population to the equilibrium population. That w is found nearly equal to T~ can thus be understood. The observed value of T~ in the msec range gives a great support to the quantum theory in two ways. Quantitatively, the calculated time constants from the theory are all in good agreement with the physiological data. On the other hand, T2 in the order of msec seems too large for free dipoles in water solution. This value would be conceivable for dipoles coupled together like electrons spins in a ferromagnet.
~
o
0
~--
EXPERIMENTAL
I
Io (I-K/X}
I= Ii / (l-e -I/K) !
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DURATION |=ec|
S t r e n g t h - d u r a t i o n curves b a s e d on H i l l ' s t h e o r y a n d e x p e r i m e n t a l p o i n t s a t 22°C
From the D.C. measurements, we have obtained the strength-duration relation. Using the ~ and /~ values from Table 1, Equations (17) and (18) of Hill's theory are plotted along with the experimental data in Figures 6-8. Though the theoretical curves follow the general trend of the experimental points, the fit is not good particularly for the solid curves at long durations in Figures 6 and 7. The solid curves bend upwards after meeting 11 in Figures 6 and 7. The Hill theory predicts either an infinite utilization time (dashed curves in Figures 6-8) or higher stimulating strengths beyond the utilization time (solid curves in Figures 6 and 7), both contrary to the established facts. However from the experimental data we can obtain the utilization times at three temperatures and then from (12) and Table 1 (taking T2), we can calculate N~n/572 in Wefts theory. These are listed in Table 2. With T2 and Nm/IV2available, Equation (9) of Wei's theory can be calculated and plotted as shown in Figures 9-11. Agreement between theory and experiment is rather satisfactory.
286
C. EODSON AND L. Y. W E I -3 EXPERIMENTAL I Io ( t - K / X )
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S t r e n g t h - d u r a t i o n curves based on Hill's t h e o r y and experimental points at 14.3°C \
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STIMULUS PULSE DURATION (sec)
Figure 8.
S t r e n g t h - d u r a t i o n curves based on HilFs t h e o r y and experimental points at 6.3°C TABLE
II
Nm/N2 V a l u e s T (°C) 22 14.3 6.3
Tu (msec.) (obs.) 3 3.9 7
N'm/_N2(calc.) 3.85 3.4 3.1
QUANTUM THEORY OF NERVE EXCITATION
287
In contrast with the Hill theory, the q u a n t u m t h e o r y gives a finite utilization time and agrees r a t h e r well with the e x p e r i m e n t a l points below the utilization time. I n Figures 9-11, the theoretical curves would drop slowly below the
EXPE%::?
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Figure 9. Strength-duration curves based on Wei's theory and experimental points at 22°C 163~ A
EXPERIMENTAL
:
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Figure 10. Stren~h-duration curve based on Wei's theory and e~oer~enta! points at ]4.3°(3
dotted line and F = (0.68-0.74) Fc as t - ~ oo. I n e x p e r i m e n t , it is e x t r e m e l y difficult to observe excitation as F goes below F c because of m e m b r a n e noise and external interference. Thus a comparison between t h e o r y and e x p e r i m e n t
288
c. HODSON AND L. Y. WEI
below Fc is meaningless. Table I I indicates that the required minimum number (Nm) of dipoles in the upper state for excitation of the nerve need only be a few times the equilibrium value N~. That Nm/-~ decreases with decreasing temperature is perhaps due to the fact that T2 is longer and hence the threshold Fc is lower (see (10)) at lower temperatures. As far as the strength-duration relation is concerned, Wei's theory predicts the existence and finiteness of the utilization time (tu) while this is not apparent from Hill's theory, as has been pointed out previously b y Johnson et al. (1954). Equation (12) suggests that the utilization time arises from the time of the excess dipole population (Nm- ~V2) spending in the upper state. 15"
165
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Strength-duration relation curve based on Wei's theory and experimental points at 6.3°C
Discussion. The quantum theory of nerve excitation is based upon the well known quantum mechanisms : stimulated and spontaneous quantum transitions of molecules between energy states. The important quantities involved are molecular quantities (such as Nz, N2, Tz and E2) whose behaviour is to obey physical principles. To illustrate the capability of the quantum theory, we shall attempt here to interpret a variety of experimental results obtained b y others. It is well known that Ca~+ tends to reduce the size of action potential (Frankenhaeuser and Hodgkia, 1957). This is thought to result from Ca ~+ binding with the membrane surface on which there is fixed negative charge. In the quantum theory, such a binding would lower the energy of the dipoles in the lower band (Figure 2) with their negative ends bound to Ca 2+. This is effectively to increase the separation energy ( E 2 - Ez) between the two bands. I f a dipole flips from the lower band to the upper band b y excitation, its polarity will be
QUANTUM THEORY OF, NERVE EXCITATIQN
289
reversed and its positive pole will then face the Ca ~+ ion. Because of mutual repulsion, this dipole in the upper state would flop down in shorter time (statistically speaking) than it would in the absence o f Ca 2+ ions. That is to say, Ca 2+ ions in good amount would tend to reduce the dipole relaxation time T2 from the upper state to the lower state. According to (10), increasing E~ and decreasing T2 b y Ca 2+ has the aggravated effect of raising the threshold F c for nerve excitation. Then reduction of action potential b y Ca 2+ becomes readily understood. The above arguments may also be applicable to a host of drugs or agents which have anaesthetic or tranquilizing actions upon nerve. Hill et al. (1936) found that the optimum frequency for excitation of frog's nerve after soaked in Ringer's solution containing high Ca (4-10 times normal) was increased to 94 Hz. at room temperature from 62 Hz. (after soaked in normal Ringer solution). Based on (16), this increase of v0 results from a decrease of T2 in high Ca as reasoned in the above. Next, we shall consider the temperature effects on nerve excitation. The classical experiment of Hodgkin and Katz (1949) showed that, (a) the peak amplitude of the action potential decreases with increasing temperature and (b) the duration particularly the repolarization phase of the action potential "becomes prolonged with decreasing temperature. Recently, Inoue et al. (1973) observed that the squid axon was abruptly depolarized when the temperature was lowered to about 5°C at or below a critical ratio of Ca : Na in the external solution. Results of similar nature were obtained b y Adey (1973) in cat cortex. He found that transient cooling b y 5-6°C in cat cortex for 1-2 min depolarized neurons b y 1-2 mV/C. Raising topical Ca levels to 20 mM blocked depolarization by transient cooling, but 7.5 mM solutions were without effect. All these temperature effects may be explained on the basis of dipole relaxation time T2. As the temperature is decreased, T~ increases and according to (10), the threshold F¢ will be lowered. As a result, the peak amplitude of the action potential will be increased or the membrane becomes depolarized more readily. The increase of T2, however, will prolong the action potential, particularly the repolarization phase which in the view of the quantum theory, is a macroscopic manifestation of dipole relaxation. As discussed in t h e above, the presence of Ca at high concentration would have the opposite effect, that is, raising the threshold. Thus depolarization b y cooling should be abolished by raising in Ca concentration above a certain level. This has indeed been verified in the experiments b y Inoue et al. and b y Adey. That the lower conduction velocity in nerve at lower temperatures as observed in our experiment and b y others (Hodgkin, Katz, 1949; Gasser, 1931; Turner, 1955; Wright, 1958) may be taken as a simple consequence of increasing relaxation time T2. Students of biology often wonder why the time constants in nerve excitation are in the msec range and why nerve impulse frequency seldom goes much higher than kHz. From the quantum theory, these questions can now be
290
0. HODSON AND L. Y. WEI
answered. The time and frequency constants in nerve excitation are all determined by the dipole relaxation time T2. I f T~ were in the msec range, the quantum theory would yield values of the said constants as observed. Our experiment has indeed produced T2 in the msec range. Here we see mutual benefits between theory and experiment. The quantum theory suggests a very simple way of determining T 2 by observing v0, the optimum frequency for nerve excitation. The experimental value of T2 in turn lends a strong support to the quantum theory in the interpretation of the time and frequency constants both qualitatively and quantitatively. Since T2 is one of the most important quantities in the quantum transitions of molecules, the quantum theory and the simple method of determining T2 will provide us an elegant way of studying macroscopic phenomena from a microscopic approach. In nerve excitation, the fundamental question remains to be the "all-ornone" response. W h a t is the underlying mechanism ? This can be understood from quantum mechanics. In a molecular system, with quantum energies Em and E n , if the incident energy (E~) is such t h a t Ei = E r a - E n
the probability for stimulated upward transition becomes infinitely great in principle and one gets "all" response. I f the above condition is not met, nothing spectacular is to be expected. Maser and laser work on this principle. Why is the nerve response pulsed rather than continuous ? Again from quantum mechanics, a system consisting of only two active levels or bands could produce an impulse wave but not a continuous wave under excitation. Ammonia maser is a well-known example (Singer, 1958). However, nerve excitation differs from maser or laser action in population requirement and in manifestation. To excite the nerve N2 need only reach 2 ~ m a S determined by the "force condition" while population inversion (N2 > N1) must be achieved for maser or laser action. In the nerve membrane, when N2 = #Vm, then according to the theory, the barrier potential would be reduced to such an extent as to allow sudden rush of Na ions inward, thus producing a nerve impulse. I n maser or laser, when N2 > / V z , a coherent emission of light is resulted. Since both systems make use of quantum transitions of molecules between energy states, one could gain much understanding of nerve action from maser or laser principle. A simple illustration is the temperature effect. I f an ammonia maser were placed in a hot bath, it would not produce coherent pulsed emission. This is because at high temperatures, T2 is so short t h a t LV~ cannot be accumulated at E2 long enough to exceed N1. The same should happen to a nerve membrane. Cole et al. (1970) have indeed observed t h a t at temperatures above 15°C, the response of squid giant axon is not "all-or-none" but a gradual function of the stimulating strength. I t is also interesting to note t h a t at lower temperatures, the peak amplitude of the action potential is greater (Hodgkin, Katz, 1949) and so is the
QUANTUMTHEORY OF NERVE EXCITATION
291
coherent emission from maser or laser (Singer, 1959). This should not be taken as a mere phenomenal coincidence b u t is derived from profound physical principles. What makes the quantum theory more appealing is that quantum transitions of molecules between energy states can give rise not only to electrical effects but also to thermal and optical effects. In nerve excitation, thermal and optical events have been observed to occur along with action potential (Cohen, Keynes, Hil]e, 1968; Howarth, Keynes, l~itchie, 1968; Fraser, Frey, 1968; Sherebrin, 1972; Abbott, Howarth, 1973). That the electrical, optical and thermal effects in nerve and their time correlations during excitation can be well accounted for by quantum transitions of dipoles has been shown previously (Wei, 1971b, 1972, 1974). From all the above discussions and from our experiment reported here we cannot escape the conclusion that the quantum theory is capable of not only accounting for a wide variety of excitation phenomena in nerve but also providing a deep insight into the mechanism underlying the processes. Finally, we wish to make a comparison of the quantum theory with the I,Hodgkin-Huxley equations which have been widely used since 1952. For this purpose, it is better to quote directly from one of H-H's classical papers. "The agreement must not be taken as evidence that our equations are anything more than an empirical description of the time course of the change in permeability to sodium and potassium". (Hodgkin, Huxley, 1952). Based on this statement of H - H and its implications and on the previous passage related to the quantum theory, we can make a list as below: Empirical description Physical mechanism Electrical effects Thermal effects Optical effects
H - H equations + + -
Quantum theory + + + +
The first two rows indicate the different approaches of the two theories and the next three rows show the consequences therefrom. The H - H equations have been most useful in dealing with voltage clamp data. However, this is also its inherent limitation. Beyond voltage clamp data, the empirical description of the H - H equations finds it hard to extend. For example, the problem of heat production and absorption is rather old and the failure of the H - H theory in this effect is well-known (Hodgkin, 1951 ; Cole, 1968). There is no way, from the H - I t equations, to account for any of the optical effects such as birefringence change (Cohen, Keynes, Hille, 1968), infrared emission (Fraser, Frey, 1968) and absorption (Sherebrin, 1972) in nerve. Such is not the case for the quantum theory which is limited only b y what forbids dipole transitions to occur. Even in the area of voltage clamp data, several authors have produced results that are
292
c. HODSON A ~ D T.. y . WEI
i n e x c e l l e n t a g r e e m e n t w i t h e x p e r i m e n t b y t a k i n g d i p o l e flip-flop t r a n s i t i o n s as a p h y s i c a l b a s i s ( A r n d t a n d R o p e r , 1972; A r n d t , B o n d a n d R o p e r , 1972; V a n L a m s w e e r d e - G a l l e z a n d Messen, 1974). I t is fair t o s a y t h a t e a c h t h e o r y has its m e r i t s a n d serves a m e a n i n g f u l p u r p o s e i n t h e p r o p e r d o m a i n of p r o b l e m s u n d e r s t u d y . T h e q u a n t u m t h e o r y is d i f f e r e n t f r o m t h e p r e v i o u s t h e o r i e s i n t h a t it suggests a p h y s i c a l m e c h a n i s m t h a t m a k e s i t p o s s i b l e for a n u n d e r s t a n d i n g of t h e v a r i o u s processes a s s o c i a t e d w i t h n e r v e e x c i t a t i o n a n d c o n d u c t i o n . T h e w o r k w a s s u p p o r t e d b y t h e N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a u n d e r G r a n t No. A1252.
LITERATURE Abort, B. C. and J. V. ttowarth, 1973. "Heat studies in excitable tissues." Physiol. Rev., 53, 120-158. Adey, W. 1%, 1973. "Biophysical and metabolic bases of cooling effects on cat cortical membrane potentials." UCLA Brain Res. Inst. Tech. Report. Arndt, 1%. and L. D. Roper. 1972. "Quantitative comparison of dipole models for steadystate currents in excitable membranes." Bull. Math. Biophys., 34, 305-324. ----, J. D. Bond and L. D. 1%oper. 1972. "A fit to nerve membrane rectification curves with a double-dipole layer membrane model." Bull. _Math. Biophys., 34, 151-172. Binstock, L. and L. Goldman• 1969. "Current and voltage clamped studies in Myxicola giant axons." J. Gen. Physiol., 54, 730-740. Blair, H. A. 1932. "On the intensity-time relation for stimulation by electric currents, I, II." J. Gen. Physiol., 15, 709-755. ----. 1934. "Conduction in nerve fibres." Ibid., 18, 125-142. ----. 1936. "Kinetics of the excitatory process." Cold Spr. Hath. Syrnp. quant. Biol., 4, 63-72. Cohen, L. B., 1%.D. Keynes and B. Hille. 1968. "Light scattering and birefringence changes during nerve activity." Nature, 218, 438-441. Cole, K. S. 1968. "Membranes, Ions and Impulses•" pp. 318-319. Berkeley- Univ• Calif• Press. , 1%. G u t t m a n and F. Bezanilla. 1970. "Nerve membrane excitation without threshold." Proc. 1Vatn. Acad. Sci., 65, 884-891. Frankcnhaeuser, B. and A. L. Hodgkin, 1957. "The action of calcium on the electrical properties of squid axons." J. Physiol., Lond., 137, 218-244. ~ Fraser, A. and A. H. Frey. 1968• "Electromagnetic emission at micro wavelengths from active nerves. '° Biophys. J., 8, 731-734. Gasser, I-I. S. 1931. "Nerve activity as modified b y temperature." A m . J. Physiol., 97, 254-270. Hill, A. V. 1936a. Excitation and accommodation in nerve. Proe. R. Soc. Lond., 119, 305-355. • 1936b. "The strength-duration relation for electric excitation ofmedullated nerve." Ibid., 119, 440-453. , B. Katz and D. Y. Solandt. 1936• "Nerve excitation by alternating current." Ibid., BI21, 74-133. I-Iodgkin, A. L. 1951. "The ionic basis of electrical activity in nerve and muscle." Biol. Rev., 26, 339-409. --~ and A. F. Huxley. 1952. "A quantitative description of membrane current and its application to conduction and excitation in nerve?' J. Physiol., Lond., 117, 500-544.
QUANTLrM T B _ E O R Y O F N E R V E ----
EXCITATION
293
a n d B. Katz. 1949• " T h e effect of temperature on the electrical activity of the
g i a n t a x o n of t h e s q u i d . " Ibid., 109, 240-249. H o w a x t h , J . V., R. D. K e y n e s a n d J . M. l~itchie. 1968. " T h e origin of t h e i n i t i a l h e a t associated w i t h a single i m p u l s e i n m a m m a l i a n n o n - m y e l i n a t c d n e r v e fibres." J . Physiol., Loved., 194, 745-793. Inoue, I., Y. K o b a t a k e a n d I. Tasaki. 1973. " E x c i t a b i l i t y , i n s t a b i l i t y a n d p h a s e t r a n s i t i o n s in s q u i d a x o n m e m b r a n e u n d e r i n t e r n a l p e r f u s i o n w i t h d i l u t e salt s o l u t i o n s . " Biochim. Biophys. Acts. 307, 471-477. J o h n s o n , F. H., H . E y r i n g a n d M. g. Polissar, 1954. The Kinetic Basis of Molecular Biology, p. 656. N e w Y o r k , Wiley. K a t z , B. 1939. Electric Excitation of Nerve, Oxford. U n i v . Press. Mormier, A. M. 1934. L'Excitation Eleetrique des Tissues. P a r i s : H e r m a n n . l~ashevsky, N. 1933. " O u t l i n e of a p h y s i c o - m a t h e m a t i c a l t h e o r y of e x c i t a t i o n a n d i n h i b i t i o n . " Protoplasma, 20, 42-56. - - - - - - 1960. Mathematical Biophysics. Vol. 1, 3rd. rev. N e w Y o r k : D o v e r . Segal, J. 1%. 1968. " S u r f a c e c h a r g e of g i a n t a x o n s of s q u i d a n d l o b s t e r . " Biophys. J., 8, 470-489. Sherebrin, M. H . 1972. " C h a n g e s in i n f r a r e d s p e c t r u m of n e r v e d u r i n g e x c i t a t i o n . " Nature, New Biol., 235, 122-124. Singer, J. R . 1959. Masers. N e w Y o r k : Wiley. Solandt, D. Y. 1936. " T h e m e a s u r e m e n t of ' a c c o m m o d a t i o n ' i n n e r v e . " Proc. B. Soc. Lond., B l 1 9 , 355-379. Turner, 1~. S. 1955. " R e l a t i o n b e t w e e n t e m p e r a t u r e a n d c o n d u c t i o n i n n e r v e fibres of different sizes." Physiol. Zool., 28, 55-61. V a n L a m s w e e r d e - G a l l e z , D. a n d A. Mcssen. 1974. " S u r f a c e dipoles, surface charges a n d n e g a t i v e s t e a d y - s t a t e r e s i s t a n c e i n biological m e m b r a n e s . " J. Biol. Phys., 2, 75-102. Wei, L. Y. 1968. " E l e c t r i c a l dipole t h e o r y of c h e m i c a l s y n a p t i c t r a n s m i s s i o n . " Biophys. J., 8, 396-414. - - . 1969a. " R o l e of surface dipoles on a x o n m e m b r a n e . " Science, 163, 280-282. 1969b. " M o l e c u l a r m e c h a n i s m s of n e r v e e x c i t a t i o n a n d c o n d u c t i o n . " Bull. Math. Biophys., 31, 39-58. - - . 1971a. " Q u a n t u m T h e o r y of n e r v e e x c i t a t i o n . " Ibid., 33, 187-194. . 1971b. " P o s s i b l e origin of a c t i o n p o t e n t i a l a n d b i r e f r i n g e n c e c h a n g e i n n e r v e a x o n . " Ibid., 33, 521-537. • 1972. " D i p o l e t h e o r y of h e a t p r o d u c t i o n a n d a b s o r p t i o n in n e r v e a x o n . " Biophys. J . , 12, 1159-1170. - - . 1973. " Q u a n t u m t h e o r y of t i m e - v a r y i n g s t i m u l a t i o n i n n e r v e . " Bull. Math. Biol., 35, 359-374. - - . 1974. " D i p o l e m e c h a n i s m s of electrical, optical a n d t h e r m a l e n e r g y t r a n d u c tions i n n e r v e m e m b r a n e . " Ann. N . Y . Acad. Sci., 227, 285-293. W r i g h t , E . B. 1958. " T h e effect of low t e m p e r a t u r e s o n single c r u s t a c e a n m o t o r n e r v e fibres." J. Cell. Comp. Physiol., 51, 29-65.
RECEIVED 6 - 2 5 - 7 5 l~EVISED 1 0 - 1 7 - 7 5
