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Dipole theory of interactions of nerve signals
Bulletin of Mathematical B~ology, Vol_ 42, pp. 79 94 Pergamon Press Ltd. 1980. Printed in Great Britain © Society for Mathematical Biology
0007-4985/80/0101-0079 $02.00/0
DIPOLE THEORY OF INTERACTIONS OF N E R V E SIGNALS
ruLING Y. WEI B i o p h y s i c a l Research L a b o r a t o r y , Electrical Engineering D e p a r t m e n t , U n i v e r s i t y of W a t e r l o o , W a t e r l o o , Ont., C a n a d a , N 2 L 3G1
Based upon the transition rate equation of dipoles in the membrane, we deal with two important aspects of interaction of nerve signals: (1) conditions for nerve excitation and (2) frequency spectrum analysis of nerve impulse.- Interrelations between signal amplitudes and frequencies are formulated in detail. There are several important conclusions which can be drawn from our calculations. First, to excite the nerve, low frequencies are generally more effective than high frequencies. Second, to sedate the nerve (i.e. to suppress undesired activities), high frequencies would suit better. Third, harmonics produced through interactions of nerve signals are not necessarily weaker than the fundamental frequencies. The great significance of our theory is that it indicates in principle the feasibility to alter or rewrite the information contents of a nerve message in our body by applying stimulations of appropriate strengths and frequencies. Thus, the theory provides a physical basis and hence some understanding for a new branch of m e d i c i n ~ n e u r o therapy such as Nogier's auriculotherapy, Lamy's phonophoresis, Voll's electroacupuncture and the fast rising TENS (transcutaneous electro-neuro stimulation).
1. Introduction. In the central nervous system (CNS) there .are myriad spontaneous activities going on all the time. Beside these, both the CNS and the peripheral nerves are subject t o external stimulations, either naturally or forcefully on occasions. How do these nerve signals interact with each other? Do they mutually reinforce, suppress or. even create new signals? W h a t governs their interactions? The major difficulty in dealing with the complications of nerve activities is that one could hardly grasp a physical entity upon which one could develop a rich theory. Most nerve theories are built around or on the surface of the phenomena. This phenomenological approach is adequate and perhaps easy to take .for simple problems, but would become intractable when encountering complex situations. Most nerve problems are intimately related to the fundamental question: 79
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L I N G Y. WEI
what is the physical origin of the nerve impulse? We have suggested that nerve impulse could be a macroscopic manifestation of the flip-flop (quantum transitions) of the dipoles on the nerve membrane surface (Wei, 1969a, b; 1971a; 1973). The other possible physical manifestations of dipole flip-flop are heat and light (Wei, 1971b; 1972). The dipole theory is simple and can provide a coherent understanding of electrical, thermal and optical phenomena in nerve (Wei, 1971a, b; 1972; 1973; 1976). It has been " experimentally tested and found in better agreement with experiment than some other theories (Hodson and Wei, 1976). The gist of the dipole theory is as follows: the surface dipoles of nerve membrane form a potential barrier which under resting condition blocks the entry of Na ions into the membrane. U p o n stimulation, some dipoles in the ground state flip up to the upper state and thus reduce the barrier. When the barrier is sufficiently lowered, there is a net inward driving force of Na ions and thereby a nerve impulse is produced. The falling phase of the impulse follows the flop down of the dipoles to the ground state. The energy of the dipoles in the upper state can go two ways: decreased by relaxation and increased by stimulation. The energy balance equation can be reduced to the transition rate equation (Wei, 1971a; 1973),
dN 2 - --
dt
2 (N2 --/~2 ) +
bFN2
(1)
where N2 is the dipole population in the upper state and N2, its equilibrium value, 2 is the relaxation rate, F is the stimulating strength, positive for cathodal (or depolarizing) and negative for anodal (hyperpolarizing) stimulation.
b =- C p / E 2 ,
(2)
where C is the stimulated upward transition rate, p is the dipole moment and E2 is the energy spacing between the upper state and the ground state. When equation (1) is multiplied by E2, one gets the original energy-balance equation. We may define bF as the effective transition rate by stimulation, upward or downward depending on the polarity ~of the stimulus. In this paper we shall treat interactions of nerve signals based mainly on equation (1).
D I P O L E T H E O R Y OF I N T E R A C T I O N S O F NERVE SIGNALS
81
We shall first quote the simple results obtained previously (Wei, 1971a; 1973). The reasoning of these results will considerably simplify the mathematics for the treatment of multiple signals and their interactions. For a single DC stimulus (F), the condition for nerve excitation is, 2. C o n d i t i o n s f o r N e r v e E x c i t a t i o n .
2 -= b F
(3)
F = Fc = 2 / b = ) L E 2 / C p .
(4)
or
Equation (3) is a condition for transition r a t e s the relaxation rate has to be equal to the "effective" transition rate by stimulation. Equation (4) is a condition for threshold strength. If, say, C ~ 2, then (4) becomes Fc = E 2 / p or
pFc = E 2
which is the well-known criterion for q u a n t u m transition. The solution of equation (1) can be put in a general form, u2 = a/g
(5)
where G is a polynomial function which does not concern us hereafter and H, for the d.c. stimulus, is
N2
H = 2 - bF
(6)
H = ,t - b F = 0
(7)
will be unstable when
which is the same as (3) or (4). We call H = 0 as the condition for mathematical instability. Physically, N2 does n o t go to infinity but rather follows equation (1) with 2 = bF, i.e. d N 2 - J~N 2 .
(8)
dt Since N 2 is an ever-increasing function of time, the nerve is surely to be excited according to the dipole theory.
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L I N G Y. W E I
We see in the above that for the simple case of d.c. stimulus, all three conditions, equations (3), (4) and (7) are equivalent. But each condition is associated with a unique concept. U n d e r certain circumstances, one concept may be more apparent than the others and thus the condition for excitation based on that concept could be written down easily without going through long mathematics. If the stimulus is a.c., the solution of (1) can be exact. But that exact solution involves an improper integral which makes interpretation difficult. By taking a Fourier series solution, one can obtain N 2 in the form of G/H. For the low frequency case, H = 0 leads to (Wei, 1973),
2 = bfco/2
(9)
where f is the amplitude and co, the angular frequency of the stimulus. For the high-frequency case, H = 0 becomes
1-2(bf/22) 2
1+~
=0.
(10)
With the above simple results at hand, we can proceed easily with multiple signals. Case 1. For multiple d.c. stimuli with strengths el, ~2,...,~,, the condition for nerve excitation, from (4) is
~%=2/b
(11)
1
where ei is taken positive for cathode (depolarizing) and negative for anodal (hyperpolarizing) stimulus, It is clear from (11) that a cathodal subthreshold stimulus is to lower the threshold and a n anodal subthreshold stimulus, to raise the threshold. This is well k n o w n in neurophysiology. Case 2. For multiple a.c. stimuli of low frequencies, //1(col), //2(co2),...,//,(co,), the condition for nerve excitation, according to (9), is b
I1
2=~//ico i
(12)
This equation can be derived from H - - 0 but m a y be taken as a transition rate condition. Case 3. For multiple a.c. stimuli of high frequencies, 61(col), 62(co2),-..,
D I P O L E T H E O R Y O F I N T E R A C T I O N S O F N E R V E SIGNALS
83
3,,(co.), the condition of mathematical instability H-~ 0 leads to 1 - ~ 2a~/(1 + c~) = 0
(13)
1
where a~ = b6J22
(14)
c~=co,/2.
(15)
Case 4. For the most general case of mixed multiple stimuli, ~s, flj(~oj)'s and 6k(OOk)'S. We define an effective relaxation rate
]
(16)
Then H = 0 gives 1 -- ~ a~,l(1 + c~)= 0
(17)
k
where ak = bak/22e
(18)
Ck=Ogk/2 e.
(19)
It is implicit from equations (16)-(19), there will be a frequency of O~o, (co~is
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LING Y. WEI
frequencies. The existence of optimum frequency for nerve excitation tends to suggest that the dipoles are coupled and their collective oscillation could produce a resonance. The resonance character of the dipole system could be the origin of the frequency-selective behavior of nerve excitation. We shall come to this point in the next section.
3. Frequency Spectrum Analysis.
We shall first treat a simple case.
Suppose F = f l sin o o , t + f 2 sin co2t
(20)
be the stimulation. Assume
N2 =Ao+A1 sin OJlt +B1
COS a ) l t
+ A2 sin eJ2t + B2 cos c02t
(21)
to be the response. Here in first-order approximation, we neglect all harmonics and mixed frequencies. Substituting the above into (1) and equating the coefficients of like terms, we obtain, A 1 = 2 a l A o / ( 1 + c 2)
(22)
A 2 = 2 a 2 A o / ( 1 + c22)
(23)
B1 = - c l A 1
(24)
B 2 = -- c 2 A 2
(25) (26)
Ao = N 2 / H
2a22 1 + c 22
(27)
a~ = bfl/2,,~ , a 2 = b f 2 / 2 2
(28)
H=I-
2a~ - l+c 2
where
c, =o),/~,c~=o~/~.
(29)
DIPOLE THEORY OF INTERACTIONS OF NERVE SIGNALS
85
The absolute amplitudes of the two frequency components of the response are:
R 1 = (A 2 +B2) I/z =2alAo(1 + c~)-l/z
(30)
R2 = (A~ + B 2 )1/2 = 2azAo (1 + c 2 )- 1/z.
(31)
Equations (30) and (31) show that the response amplitudes are proportional to the stimulating strengths and that at equal stimulating strength (fl =f2), the response amplitude at the lower frequency is greater. This is a first-order interaction. Next we consider (co1 +co2) and (0)2-col) components in addition to o~1 and 092 in the response. We assume. N2 = Ao + E (Ai sin colt + Bi cos COlt) 1,2
+ C 1 sin (co2 - col)t + D1 cos (co2 - 0) 1)t
+ C2 sin (COl-I- CO2)t-t- D 2 c o s (°91 -I- ~ 2 ) t
(32)
while F is still given by equation (20). Using the same procedure as before we obtain
+alAz)]
(33)
alB2)]
(34)
C2 -= gz[(a2 B 1 + a l B 2 ) - (cl + c 2 ) ( a 2 A t + a l A 2 ) ]
(35)
C1 = g l E ( a 2 B 1 - a l B 2 )
+ (c 2 -cl)(azA1
O l = g l [(azA1 + alA2 ) - (c2 -
D 2 -= _ g 2 [ ( a 2 A 1 +
c I )(a2B
1 -
alA2)+ (ca + c2)(a2B1 + alBz)]
(36)
where (c2 - q ) 2 ] - 1
(37)
g2 = [ 1 + (C 1 + C 2 ) 2 ] - 1
(38)
gl = [1 +
a l , a2, c 1 and
C 2 are given by (28) and (29). Since al, az, A1, B1, A2 and B2 are proportional to the respective field strength, then from equations (33)-(36), C1, D1, C2 and D 2 will be proportional to the cross product of field strengths (flfz)- It is predictable that at low field strengths, C's and D's may be small compared with A's and B's but at high field strengths, the former could be comparable with or even greater than the latter. This is a second-order interaction.
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L I N G Y. WEI
Now, we wish to find the h a r m o n i c c o m p o n e n t s in the response. To simplify m a t h e m a t i c s let us t a k e F = f s i n o)t
(39)
N 2 = Ao + ~. (Ai sin coit + Bi cos ogit )
(40)
and 3 i=1
and neglect all h a r m o n i c s higher t h a n third. Substituting (39) a n d (40) into e q u a t i o n (1), a n d following the same procedure as before, we o b t a i n
ak)A~ + (1 -ak)Ba]
(41)
B2 = - k'[(1 - a k ) A a + c(1 + ak)B~]
(42)
A3 = - kk'[(1 - c 2 - a 2)A1 + 2cB1]
(43)
B3 = -- k k ' [ - 2cAa + (1 - c z - a 2)B1]
(44)
A 2 = k'[-
c ( 1 nk
where a = bf/22,
c =- o9/2
k=a/g,
g=l+c
2
k' = a / g '
g' = (1 -- a k ) z + c 2(1 + a k ) z. W h e n a k , ~ 1 (low field case), g ' ~ g , k ' ~ a / g A2 = a g - l [ - - c A 1 + B 1 ]
(45)
B2 = - a g - l [ A 1
(46)
+cB~]
A3 = - a 2 g - 2[ (1 - c 2 - a 2)A 1 + 2cB1]
(47)
B3 - - a2g - 2[ _ 2cA1 + (1 - c 2 - a 2)B1].
(48)
If a k ~ 1 (high field case), k ' ~ g / a 3 A2 = - a -
lEcA1 + B 1 ]
B 2 = a - 1 [A1
-- cBx]
(49)
(50)
D I P O L E T H E O R Y O F I N T E R A C T I O N S O F NERVE SIGNALS a 3 -----
a - 2[a2A1 -- 2cB1] = Ai - 2ca- 2B 1
B3 = a - 2[2cA1 + a2B1] = 2ca- 2A 1-1-B
1.
87
(51) (52)
Equations (45)-(52) show that as the stimulating strength ( f ) increases, the second harmonic components will first increase with f z (note: A~ and B~ are proportional to f ) and finally reach a plateau (i.e., independent of f ) , while the third harmonic components will first increase with f 3 and finally reach values almost equal to the fundamental components as evident from (51) and (52). The above analysis clearly indicates that the nerve response could contain significant components of sum and difference frequencies, and second and third harmonics when the stimulating strengths are not small. However, because of the resonance character of the dipole system aforementioned, those frequency components near COo, the resonance frequency would be enhanced and those far remote from ~oo would be damped. Hence the frequency components calculated in the above should be each multiplied by a frequency-selective function S(CO). In the language of Fourier transform, S(CO) is called a frequency window. One m a y assume for S(CO) a simple form such as
S(CO)=cosp(CO-COo)
(53)
where p is a shaping factor, or a more general form, ?l
S(CO)= l + ~ Ak(CO--COo)~
(54)
1
where COois the optimum frequency for nerve excitation, and some Ak's are negative. Besides the frequency window effect, there is an inhibitory effect in the nerve. In q u a n t u m mechanical concept, the inhibition is viewed as a manifestation of stimulated downward transition. In two-level solid state masers making use of the flip-flop of electron spins, the oscillation is in the form of pulses (Singer, 1959). During the dark period between pulses, the stimulation produces no effect and hence is said to be "inhibited." The reason for the darkness is that emission wipes out the population inversion in the upper state by stimulated downward transition and that it will take a long time (mostly the time required for thermal relaxation of electron spins with the lattice) before population inversion can be re-established in the upper state. The thermal relaxation time, as the name implies, can be shortened by raising the temperature or some other means. In modern
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LING Y. WEI
electrical engineering language, we say that both nerve membranes and a two-level solid state maser can do analog-to-digital (A/D) conversion and produce F M pulses. The digitized output (pulses) is an inevitable result of a two-level system unless the upper state molecules can be sorted out as in an ammonia maser (Singer, 1959). Since electric dipoles and electron spins (like magnetic dipoles) behave in similar ways, the inhibitory effect in the nerve could be reasonably understood as a manifestation of stimulated downward transition of membrane dipoles. Mathematically, we only need to add to the right-hand side of equation (1) a term to this effect. This has been done in a previous paper (Wei, 1971a) and the solution of the modified equation gives refractory period. When there are multiple stimulations, each of them will affect the stimulated downward transition rate and hence their interactions will manifest in the nerve response. One could in principle do calculations following similar processes as in the above with the modified equation. The mathematics, however, is too complex to be easily interpretable. To circumvent this difficulty, let us focus on two pulse trains in the response (Figure 1). The two pulse trains are represented by
tl
= ~ A exp -- kl ( t - nT1)
(55)
n
U2=~, Y2(t-mT2) ?n
= ~ Bexp - k 2 ( t - mT2).
(56)
m
-q,
_
U2
b i b2b3
Figure 1.
~
t AMP.
Freq.
u i Pulse train :
A
uI
Period Tj
u2 Pulse frain :
B
uz
Tz
Inhibitory interaction of two pulse trains.
D I P O L E T H E O R Y O F I N T E R A C T I O N S O F N E R V E SIGNALS
89
We presume that the rising phase of each pulse is due to stimulated upward transition and the falling phase (exponential decay), due to stimulated downward transition of dipoles. As shown in Figure 1, pulses ba, b2 and ba are under the canopy of pulse ax and they will be inhibited because the stimulated upward transition rates leading to bl, bz and ba are overpowered by the stimulated downward transition rates leading to the falling phase of pulse aa at the corresponding instants. The relative refractory period of the Ua-response in this case is given by B = A e x p - klz 1
z=~lnA/B.
(57)
The b-pulses will appear in the period (Ta--c). The apparent frequency of the over-all pulse train is v = v2( rl
= v 2 ( 1 - k~T~ In A )
(58)
where v2 = l/T2. If A < B , then from (57) and Figure 1, z = 0 and v=v 2 from (58). If z = T 1, all b-pulses will be inhibited, and we only have a-pulses, hence v = v j . This occurs when -c= k1 In A/B > T1 or
A =>Bexp k i T 1.
(59)
The above result shows that if there are two pulse trains at frequencies Vx and v2 in the nerve, and if the amplitude of the low frequency (Vl) train is increased from low to high values, the apparent frequency (v) of the overall pulse train should change from v 1 to v2 in a way like that given by (58). This change in v is the consequence of the inhibitory action which is conceived to originate from the stimulated downward transitions of membrane dipoles. Since the above result is so interesting and could have great implications in neurotherapy, we have designed a simple experiment to give it a test. We applied two stimuli at different frequencies to the giant axon of Myxicola and recorded the response at the end. The result is
90
L I N G Y. WEI
shown in Figs. 2 and 3 (Wei and Hodson, 1977). One sees that as the amplitude of the low frequency stimulus is increased, the apparent frequency of the overall response decreases from v2 to vl, as predicted. Here we need not be too concerned with the exact way by which v varies with A (amplitude) as long as the end points (v2 and vl) and the trend (downward) in between are correct. In this paper we have treated interactions of nerve signals from the viewpoint of dipole quantum processes: (1) spontaneous relaxation, (2) stimulated upward transition, (3) stimulated downward transition and (4) collective oscillation. As far as nerve excitation is concerned, the first two processes are predominant and are contained in equation (1). This equation of dipole excitation is quite similar to that for excitation of ferromagnetic magnons (quanta of spin waves) (Kittel, 1963) based upon the same principle of energy balance. Since many results derived from equation (1) have been confirmed in experiment (Hodson and Wei, 1976) and expressions such as (9) and (10) are essentially the same as those derived from Hill's theory (1936) and Rashevsky's theory (1933), then all the calculations done in this paper based on (1) should be correct to first approximation. Improvement could be made by adding a term on stimulated downward transition to equation (1), or better by writing three equations on N1, Na, and P (phonon population) as proposed in the works of Lee and Chiang (1976), and of Chiang (1978). Chiang in his recent paper (1978) had derived expressions for Na and K currents, and for action potential. However, in both of these papers, the field factor is contained in the exponential expressions for the transition probabilities. It is very difficult, if not impossible, in their formulations to treat interactions of multiple signals and to arrive at relations in closed forms. In our view, it appears better to start with a simple equation such as (1) and to obtain results that can be easily comprehensible. We think we have achieved this purpose in this paper. There are several important conclusions which can be drawn from our calculations. First, to excite the nerve, low frequencies are generally more effective than high frequencies. This is evident from (17), (22) and (23). Second, to sedate the nerve, i.e. to suppress many undesired nerve activities, high frequencies would suit better, as implied by equation (58). Third, by an ingenious combination of multiple stimuli or by applying sufficient strength of a single stimulus at a proper frequency, it is possible to regulate some body functions dictated by nerve impulses (such as in autonomic nerves). Our calculated results suggest many of these possibilities and it is up to biomedical engineers to use these results for practical benefit. For example, electro-sleep, electro-anesthesia, TENS (transcuDiscussion.
DIPOLE THEORY OF INTERACTIONS OF NERVE SIGNALS
Figure 2. Recordings taken with control stimulus (100Hz) applied alone (upper traces); test stimulus (30 Hz) applied alone (middle traces) and both stimuli applied simultaneously (lower traces), The strength of the control stimulus was fixed at 400mV, while that of the test stimulus was (a) 900mV, (b) 1.5V and (c) 2V (Wei and Hodson, 1977).
91
92
LING Y. WEI 24
Ax,onI~IRT~R'2 zo
o
.~ o_ o.;
]8
\
~
o-~
\ T h i s level limited by the control
1
arrangement
o ~
stimulus frequency, SI
I6 --
JElectrode
o\ ~
(200 Hz)
4"4-
SIo
-12 --
14
pe = _ J
O ~ o
-
I0 8 -O 4-
6
E
o-. These action potentials /~' orrived outside the 70 ms interval occupied by the test pulse
4 --
2 -o
I
I
I
I
I
I
I
I
I
2
5
4
5
6
7
8
Figure 3. The effect of test stimulus strength on the frequency of nerve impulses passing through the test stimulus site (Wei and Hodson, 1977)_
taneous electro-neuro stimulation), acupuncture, and various electronic aids for the blind and for deaf mutes are effective means to take advantage of interaction of nerve signals to achieve a specific purpose. Our theory clearly indicates that multiple stimuli, be they spontaneous or external, do not simply add up as some people tend to believe, but rather could produce many other frequency components which under some circumstances could be as strong as the fundamental components. To lull brain activities, say for the sake of mind concentration, both theory and practice dictate lessening inner and outer stimulations as much as possible. The dipole theory, howeyer, suggests more can be done_ Since all-interactions depend on the transition rate constant b, equation (2), one may reduce b by decreasing C, the stimulated upward transition rate and p, the dipole moment and/or by increasing E2, the spacing between the two energy states. C is temperature-dependent and can be reduced by lowering temperature. The effect is amply demonstrated in our daily experience. When keeping the head cool, one can think better. In very hot summer the head may be ringing and often feels topsy-turvy. To reduce p or to increase E2, one may introduce some polyvalent ions or chemical agents which either annul part of the polar charges or bind th,e polar heads: so that they find it difficult to turn from their ground state. Ca ions seem t o have these effects (Frankenhaeuser and Hodgkin, 1957; Gilbert and Ehrenstein, 1969). Very likely, anaesthetics and tranquillizers may work on t h e same principle (Narahashi, 1971; Blaustein and Goldman, 1966; Agin et al., i965).
DIPOLE THEORY OF INTERACTIONS OF NERVE SIGNALS
93
T h e great significance of this theoretical w o r k is t h a t it indicates in principle the feasibility of altering or rewriting the i n f o r m a t i o n c o n t e n t of a nerve message in o u r b o d y b y a p p l y i n g stimulations of a p p r o p r i a t e strengths a n d frequencies. If some b o d y functions (such as secretion of gastric juice) are c o n t r o l l e d b y nerve message (say in a u t o n o m i c n e r v o u s systems), t h e n disorders of these functions c o u l d be n o r m a l i z e d b y external stimulations a c c o r d i n g to o u r theory. W e t h i n k t h a t this is highly p r o b a b l e as a physical principle for some modalities of physical t h e r a p y . A m o n g notable examples are Nogier's (1972) a u r i c u l o t h e r a p y and L a m y ' s (1967, 1969) p h o n o p h o r e s i s in F r a n c e , Voll's (1974, 1975) e l e c t r o a c u p u n c t u r e in G e r m a n y a n d the very recent rising t r e n d of p r o m o t i n g T E N S ( t r a n s c u t a n e o u s e l e c t r o - n e u r o s t i m u l a t i o n ) as a t h e r a p y in the U.S.A. (Burton a n d M a u r e r , 1974; L o e s e r et al., 1975; Veale, 1976). T h e w o r k s of T a n y a n d S a w a t s u g a (1975) a n d of F r a z e e (1975) b y using a wide electromagnetic spectrum, h a r m o n i c s a n d d o u b l e frequencies for stimulations are p a r t i c u l a r l y d e m o n s t r a t i v e for a n d justified f r o m the t h e o r y presented here. It is h o p e d t h a t further d e v e l o p m e n t along this line will open a new era for physical medicine t h a t c o u l d c o m p l e m e n t the conventional chemical (drug) medicine for the benefit of m a n k i n d . This w o r k is s u p p o r t e d in p a r t b y the N a t i o n a l C a n a d a u n d e r G r a n t N o . A-1252.
R e s e a r c h Council of
LITERATURE Agin, D., L. Hersh and D. Holtzman. 1965. "The action of anesthetics on excitable membranes: A quantum chemical analysis." Proc. Natn Acad. Sci., U.S A., 53, 952-958. Blaustein, M. P_ and D. E. Goldman. 1966. "Action of anionic and cationic nerve-blocking agents: Experimental and interpretation." Science, 153, 429-432. Burton, C. and D. D. Maurer. 1974. "Pain suppression by transcutaneous electronics stimulation_" 1EEE Trans. BME, 21, 81 88. Chiang, C. 1978. "On the nerve impulse equation: The dynamical responses of nerve impulse. Bull. Math. Biol., 40, 247 256. Frankenhaeuser, B. and A. L_ Hodgkin. 1957. "The action of calcium on the electrical properties of squid axon." J_ Physiol., Lond., 137, 218-244. Frazee, J. S. 1975. "Experimental harmonic progression and simultaneous dual frequency stimulation." Am. J. Acup., 3, 315-324. Gilbert, D. L. and G_ Ehrenstein. 1969. "Effect of divalent cations on potassium conductance of squid axons: Determination of surface charge." Biophys. J., 9, 447-463. Hill, A. V. 1936. "Excitation and accommodation in nerve." Proc. R. Soc. Lond., Bl19, 305355. Hodson, C. and L. Y_ Wei. 1976. "Comparative evaluation of quantum theory of nerve excitation." Bull. Math. Biol., 38, 277 293. Kittel, C. 1963. Quantum Theory of Solids, pp. 69-70. New York, John Wiley. Lamy, J. 1967, 1969. Acupuncture Phonophor~se-Technique, Clinique. Tome I; Tome II. Librairie Maloine. S.A_ Paris. Lee, C. Y. and (2. Chiang. 1976. "Nerve excitations by the coupling of dipoles and the membrane matrix." Bull_ Math. Biol., 38, 59-70.
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Loeser, J. D., et al. 1975. "Relief of pain by transcutaneous stimulation." J. Neurosurg., 42, 308-314. Narahashi, T. 1971. "Neurophysiological basis for drug action: Ionic mechanism, site of action and active form in nerve fibres." In Biophysics and Physiology of Excitable Membranes, Ed. W. J. Adelman, Jr., pp. 423-462_ New York: Van Nostrand Reinhold. Nogier, P. F. M. 1972. Treatise of Auriculo-therapy. Maisonneuve, Moulins-Les-Metz, France. Rashevsky, N. 1933. "Outline of a physico-mathematical theory of excitation and inhibition." Protoplasma, 20, 4~56. Singer, J. R. 1959. Masers. New York: John Wiley• Solandt, D. Y. 1936. "The measurement of accommodation in nerve•" Proc. R. Soc. Lond_, Bl19, 355-379. Tany, M. and S. Sawatsuga. 1975. "New development: electromagnetic acupuncture•" Am. J. Acup., 3, 58-66. Veale, J. R. 1976. Report on Transcutaneous electrical nerve stimulation for pain relief Food and Drug Adrn.0 Washington, D.C. Voll, R. 1974. Kopfherde-Diagnostik und Therapie Mittels Elektroakupunktur. Uelzen. Germany: ML. Verlag. • 1975. "Twenty years of electroacupuncture therapy using low-frequency current pulses•" Am. J. Acup., 3, 291 314. Wei, L. Y. 1969a. "Role of surface dipoles on axon membrane." Science, 163, 280-282. . 1969b. "Molecular mechanisms of nerve excitation and conduction." Bull. Math. Biophys., 31, 39 58. . 1971a. "Quantum theory of nerve excitation." Bull. Math. Biophys., 33, 187-194. • 1971b. "Possible origin of action potential and birefringence change in nerve axon." Bull. Math. Biophys. 33, 521 537. 1972. "Dipole theory of heat production and absorption in nerve axon." Biophys. J., 12, 1159 1170. • 1973. Quantum theory of time-varying stimulation in nerve• Bull. Math. Biol., 35, 359 374. 1974. "'Dipole mechanisms of electrical, optical and thermal energy transductions in nerve membrane•" Ann. N.Y. Acad. Sci., 277, 285-293• -, and C. Hodson. 1977. "Nerve transmission and acupuncture mechanism." Am. J_ Acup., 5, 69-83. RECEIVED 10-30-78 REVISED 2-2-79
0007-4985/80/0101-0079 $02.00/0
DIPOLE THEORY OF INTERACTIONS OF N E R V E SIGNALS
ruLING Y. WEI B i o p h y s i c a l Research L a b o r a t o r y , Electrical Engineering D e p a r t m e n t , U n i v e r s i t y of W a t e r l o o , W a t e r l o o , Ont., C a n a d a , N 2 L 3G1
Based upon the transition rate equation of dipoles in the membrane, we deal with two important aspects of interaction of nerve signals: (1) conditions for nerve excitation and (2) frequency spectrum analysis of nerve impulse.- Interrelations between signal amplitudes and frequencies are formulated in detail. There are several important conclusions which can be drawn from our calculations. First, to excite the nerve, low frequencies are generally more effective than high frequencies. Second, to sedate the nerve (i.e. to suppress undesired activities), high frequencies would suit better. Third, harmonics produced through interactions of nerve signals are not necessarily weaker than the fundamental frequencies. The great significance of our theory is that it indicates in principle the feasibility to alter or rewrite the information contents of a nerve message in our body by applying stimulations of appropriate strengths and frequencies. Thus, the theory provides a physical basis and hence some understanding for a new branch of m e d i c i n ~ n e u r o therapy such as Nogier's auriculotherapy, Lamy's phonophoresis, Voll's electroacupuncture and the fast rising TENS (transcutaneous electro-neuro stimulation).
1. Introduction. In the central nervous system (CNS) there .are myriad spontaneous activities going on all the time. Beside these, both the CNS and the peripheral nerves are subject t o external stimulations, either naturally or forcefully on occasions. How do these nerve signals interact with each other? Do they mutually reinforce, suppress or. even create new signals? W h a t governs their interactions? The major difficulty in dealing with the complications of nerve activities is that one could hardly grasp a physical entity upon which one could develop a rich theory. Most nerve theories are built around or on the surface of the phenomena. This phenomenological approach is adequate and perhaps easy to take .for simple problems, but would become intractable when encountering complex situations. Most nerve problems are intimately related to the fundamental question: 79
80
L I N G Y. WEI
what is the physical origin of the nerve impulse? We have suggested that nerve impulse could be a macroscopic manifestation of the flip-flop (quantum transitions) of the dipoles on the nerve membrane surface (Wei, 1969a, b; 1971a; 1973). The other possible physical manifestations of dipole flip-flop are heat and light (Wei, 1971b; 1972). The dipole theory is simple and can provide a coherent understanding of electrical, thermal and optical phenomena in nerve (Wei, 1971a, b; 1972; 1973; 1976). It has been " experimentally tested and found in better agreement with experiment than some other theories (Hodson and Wei, 1976). The gist of the dipole theory is as follows: the surface dipoles of nerve membrane form a potential barrier which under resting condition blocks the entry of Na ions into the membrane. U p o n stimulation, some dipoles in the ground state flip up to the upper state and thus reduce the barrier. When the barrier is sufficiently lowered, there is a net inward driving force of Na ions and thereby a nerve impulse is produced. The falling phase of the impulse follows the flop down of the dipoles to the ground state. The energy of the dipoles in the upper state can go two ways: decreased by relaxation and increased by stimulation. The energy balance equation can be reduced to the transition rate equation (Wei, 1971a; 1973),
dN 2 - --
dt
2 (N2 --/~2 ) +
bFN2
(1)
where N2 is the dipole population in the upper state and N2, its equilibrium value, 2 is the relaxation rate, F is the stimulating strength, positive for cathodal (or depolarizing) and negative for anodal (hyperpolarizing) stimulation.
b =- C p / E 2 ,
(2)
where C is the stimulated upward transition rate, p is the dipole moment and E2 is the energy spacing between the upper state and the ground state. When equation (1) is multiplied by E2, one gets the original energy-balance equation. We may define bF as the effective transition rate by stimulation, upward or downward depending on the polarity ~of the stimulus. In this paper we shall treat interactions of nerve signals based mainly on equation (1).
D I P O L E T H E O R Y OF I N T E R A C T I O N S O F NERVE SIGNALS
81
We shall first quote the simple results obtained previously (Wei, 1971a; 1973). The reasoning of these results will considerably simplify the mathematics for the treatment of multiple signals and their interactions. For a single DC stimulus (F), the condition for nerve excitation is, 2. C o n d i t i o n s f o r N e r v e E x c i t a t i o n .
2 -= b F
(3)
F = Fc = 2 / b = ) L E 2 / C p .
(4)
or
Equation (3) is a condition for transition r a t e s the relaxation rate has to be equal to the "effective" transition rate by stimulation. Equation (4) is a condition for threshold strength. If, say, C ~ 2, then (4) becomes Fc = E 2 / p or
pFc = E 2
which is the well-known criterion for q u a n t u m transition. The solution of equation (1) can be put in a general form, u2 = a/g
(5)
where G is a polynomial function which does not concern us hereafter and H, for the d.c. stimulus, is
N2
H = 2 - bF
(6)
H = ,t - b F = 0
(7)
will be unstable when
which is the same as (3) or (4). We call H = 0 as the condition for mathematical instability. Physically, N2 does n o t go to infinity but rather follows equation (1) with 2 = bF, i.e. d N 2 - J~N 2 .
(8)
dt Since N 2 is an ever-increasing function of time, the nerve is surely to be excited according to the dipole theory.
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L I N G Y. W E I
We see in the above that for the simple case of d.c. stimulus, all three conditions, equations (3), (4) and (7) are equivalent. But each condition is associated with a unique concept. U n d e r certain circumstances, one concept may be more apparent than the others and thus the condition for excitation based on that concept could be written down easily without going through long mathematics. If the stimulus is a.c., the solution of (1) can be exact. But that exact solution involves an improper integral which makes interpretation difficult. By taking a Fourier series solution, one can obtain N 2 in the form of G/H. For the low frequency case, H = 0 leads to (Wei, 1973),
2 = bfco/2
(9)
where f is the amplitude and co, the angular frequency of the stimulus. For the high-frequency case, H = 0 becomes
1-2(bf/22) 2
1+~
=0.
(10)
With the above simple results at hand, we can proceed easily with multiple signals. Case 1. For multiple d.c. stimuli with strengths el, ~2,...,~,, the condition for nerve excitation, from (4) is
~%=2/b
(11)
1
where ei is taken positive for cathode (depolarizing) and negative for anodal (hyperpolarizing) stimulus, It is clear from (11) that a cathodal subthreshold stimulus is to lower the threshold and a n anodal subthreshold stimulus, to raise the threshold. This is well k n o w n in neurophysiology. Case 2. For multiple a.c. stimuli of low frequencies, //1(col), //2(co2),...,//,(co,), the condition for nerve excitation, according to (9), is b
I1
2=~//ico i
(12)
This equation can be derived from H - - 0 but m a y be taken as a transition rate condition. Case 3. For multiple a.c. stimuli of high frequencies, 61(col), 62(co2),-..,
D I P O L E T H E O R Y O F I N T E R A C T I O N S O F N E R V E SIGNALS
83
3,,(co.), the condition of mathematical instability H-~ 0 leads to 1 - ~ 2a~/(1 + c~) = 0
(13)
1
where a~ = b6J22
(14)
c~=co,/2.
(15)
Case 4. For the most general case of mixed multiple stimuli, ~s, flj(~oj)'s and 6k(OOk)'S. We define an effective relaxation rate
]
(16)
Then H = 0 gives 1 -- ~ a~,l(1 + c~)= 0
(17)
k
where ak = bak/22e
(18)
Ck=Ogk/2 e.
(19)
It is implicit from equations (16)-(19), there will be a frequency of O~o, (co~is
84
LING Y. WEI
frequencies. The existence of optimum frequency for nerve excitation tends to suggest that the dipoles are coupled and their collective oscillation could produce a resonance. The resonance character of the dipole system could be the origin of the frequency-selective behavior of nerve excitation. We shall come to this point in the next section.
3. Frequency Spectrum Analysis.
We shall first treat a simple case.
Suppose F = f l sin o o , t + f 2 sin co2t
(20)
be the stimulation. Assume
N2 =Ao+A1 sin OJlt +B1
COS a ) l t
+ A2 sin eJ2t + B2 cos c02t
(21)
to be the response. Here in first-order approximation, we neglect all harmonics and mixed frequencies. Substituting the above into (1) and equating the coefficients of like terms, we obtain, A 1 = 2 a l A o / ( 1 + c 2)
(22)
A 2 = 2 a 2 A o / ( 1 + c22)
(23)
B1 = - c l A 1
(24)
B 2 = -- c 2 A 2
(25) (26)
Ao = N 2 / H
2a22 1 + c 22
(27)
a~ = bfl/2,,~ , a 2 = b f 2 / 2 2
(28)
H=I-
2a~ - l+c 2
where
c, =o),/~,c~=o~/~.
(29)
DIPOLE THEORY OF INTERACTIONS OF NERVE SIGNALS
85
The absolute amplitudes of the two frequency components of the response are:
R 1 = (A 2 +B2) I/z =2alAo(1 + c~)-l/z
(30)
R2 = (A~ + B 2 )1/2 = 2azAo (1 + c 2 )- 1/z.
(31)
Equations (30) and (31) show that the response amplitudes are proportional to the stimulating strengths and that at equal stimulating strength (fl =f2), the response amplitude at the lower frequency is greater. This is a first-order interaction. Next we consider (co1 +co2) and (0)2-col) components in addition to o~1 and 092 in the response. We assume. N2 = Ao + E (Ai sin colt + Bi cos COlt) 1,2
+ C 1 sin (co2 - col)t + D1 cos (co2 - 0) 1)t
+ C2 sin (COl-I- CO2)t-t- D 2 c o s (°91 -I- ~ 2 ) t
(32)
while F is still given by equation (20). Using the same procedure as before we obtain
+alAz)]
(33)
alB2)]
(34)
C2 -= gz[(a2 B 1 + a l B 2 ) - (cl + c 2 ) ( a 2 A t + a l A 2 ) ]
(35)
C1 = g l E ( a 2 B 1 - a l B 2 )
+ (c 2 -cl)(azA1
O l = g l [(azA1 + alA2 ) - (c2 -
D 2 -= _ g 2 [ ( a 2 A 1 +
c I )(a2B
1 -
alA2)+ (ca + c2)(a2B1 + alBz)]
(36)
where (c2 - q ) 2 ] - 1
(37)
g2 = [ 1 + (C 1 + C 2 ) 2 ] - 1
(38)
gl = [1 +
a l , a2, c 1 and
C 2 are given by (28) and (29). Since al, az, A1, B1, A2 and B2 are proportional to the respective field strength, then from equations (33)-(36), C1, D1, C2 and D 2 will be proportional to the cross product of field strengths (flfz)- It is predictable that at low field strengths, C's and D's may be small compared with A's and B's but at high field strengths, the former could be comparable with or even greater than the latter. This is a second-order interaction.
86
L I N G Y. WEI
Now, we wish to find the h a r m o n i c c o m p o n e n t s in the response. To simplify m a t h e m a t i c s let us t a k e F = f s i n o)t
(39)
N 2 = Ao + ~. (Ai sin coit + Bi cos ogit )
(40)
and 3 i=1
and neglect all h a r m o n i c s higher t h a n third. Substituting (39) a n d (40) into e q u a t i o n (1), a n d following the same procedure as before, we o b t a i n
ak)A~ + (1 -ak)Ba]
(41)
B2 = - k'[(1 - a k ) A a + c(1 + ak)B~]
(42)
A3 = - kk'[(1 - c 2 - a 2)A1 + 2cB1]
(43)
B3 = -- k k ' [ - 2cAa + (1 - c z - a 2)B1]
(44)
A 2 = k'[-
c ( 1 nk
where a = bf/22,
c =- o9/2
k=a/g,
g=l+c
2
k' = a / g '
g' = (1 -- a k ) z + c 2(1 + a k ) z. W h e n a k , ~ 1 (low field case), g ' ~ g , k ' ~ a / g A2 = a g - l [ - - c A 1 + B 1 ]
(45)
B2 = - a g - l [ A 1
(46)
+cB~]
A3 = - a 2 g - 2[ (1 - c 2 - a 2)A 1 + 2cB1]
(47)
B3 - - a2g - 2[ _ 2cA1 + (1 - c 2 - a 2)B1].
(48)
If a k ~ 1 (high field case), k ' ~ g / a 3 A2 = - a -
lEcA1 + B 1 ]
B 2 = a - 1 [A1
-- cBx]
(49)
(50)
D I P O L E T H E O R Y O F I N T E R A C T I O N S O F NERVE SIGNALS a 3 -----
a - 2[a2A1 -- 2cB1] = Ai - 2ca- 2B 1
B3 = a - 2[2cA1 + a2B1] = 2ca- 2A 1-1-B
1.
87
(51) (52)
Equations (45)-(52) show that as the stimulating strength ( f ) increases, the second harmonic components will first increase with f z (note: A~ and B~ are proportional to f ) and finally reach a plateau (i.e., independent of f ) , while the third harmonic components will first increase with f 3 and finally reach values almost equal to the fundamental components as evident from (51) and (52). The above analysis clearly indicates that the nerve response could contain significant components of sum and difference frequencies, and second and third harmonics when the stimulating strengths are not small. However, because of the resonance character of the dipole system aforementioned, those frequency components near COo, the resonance frequency would be enhanced and those far remote from ~oo would be damped. Hence the frequency components calculated in the above should be each multiplied by a frequency-selective function S(CO). In the language of Fourier transform, S(CO) is called a frequency window. One m a y assume for S(CO) a simple form such as
S(CO)=cosp(CO-COo)
(53)
where p is a shaping factor, or a more general form, ?l
S(CO)= l + ~ Ak(CO--COo)~
(54)
1
where COois the optimum frequency for nerve excitation, and some Ak's are negative. Besides the frequency window effect, there is an inhibitory effect in the nerve. In q u a n t u m mechanical concept, the inhibition is viewed as a manifestation of stimulated downward transition. In two-level solid state masers making use of the flip-flop of electron spins, the oscillation is in the form of pulses (Singer, 1959). During the dark period between pulses, the stimulation produces no effect and hence is said to be "inhibited." The reason for the darkness is that emission wipes out the population inversion in the upper state by stimulated downward transition and that it will take a long time (mostly the time required for thermal relaxation of electron spins with the lattice) before population inversion can be re-established in the upper state. The thermal relaxation time, as the name implies, can be shortened by raising the temperature or some other means. In modern
88
LING Y. WEI
electrical engineering language, we say that both nerve membranes and a two-level solid state maser can do analog-to-digital (A/D) conversion and produce F M pulses. The digitized output (pulses) is an inevitable result of a two-level system unless the upper state molecules can be sorted out as in an ammonia maser (Singer, 1959). Since electric dipoles and electron spins (like magnetic dipoles) behave in similar ways, the inhibitory effect in the nerve could be reasonably understood as a manifestation of stimulated downward transition of membrane dipoles. Mathematically, we only need to add to the right-hand side of equation (1) a term to this effect. This has been done in a previous paper (Wei, 1971a) and the solution of the modified equation gives refractory period. When there are multiple stimulations, each of them will affect the stimulated downward transition rate and hence their interactions will manifest in the nerve response. One could in principle do calculations following similar processes as in the above with the modified equation. The mathematics, however, is too complex to be easily interpretable. To circumvent this difficulty, let us focus on two pulse trains in the response (Figure 1). The two pulse trains are represented by
tl
= ~ A exp -- kl ( t - nT1)
(55)
n
U2=~, Y2(t-mT2) ?n
= ~ Bexp - k 2 ( t - mT2).
(56)
m
-q,
_
U2
b i b2b3
Figure 1.
~
t AMP.
Freq.
u i Pulse train :
A
uI
Period Tj
u2 Pulse frain :
B
uz
Tz
Inhibitory interaction of two pulse trains.
D I P O L E T H E O R Y O F I N T E R A C T I O N S O F N E R V E SIGNALS
89
We presume that the rising phase of each pulse is due to stimulated upward transition and the falling phase (exponential decay), due to stimulated downward transition of dipoles. As shown in Figure 1, pulses ba, b2 and ba are under the canopy of pulse ax and they will be inhibited because the stimulated upward transition rates leading to bl, bz and ba are overpowered by the stimulated downward transition rates leading to the falling phase of pulse aa at the corresponding instants. The relative refractory period of the Ua-response in this case is given by B = A e x p - klz 1
z=~lnA/B.
(57)
The b-pulses will appear in the period (Ta--c). The apparent frequency of the over-all pulse train is v = v2( rl
= v 2 ( 1 - k~T~ In A )
(58)
where v2 = l/T2. If A < B , then from (57) and Figure 1, z = 0 and v=v 2 from (58). If z = T 1, all b-pulses will be inhibited, and we only have a-pulses, hence v = v j . This occurs when -c= k1 In A/B > T1 or
A =>Bexp k i T 1.
(59)
The above result shows that if there are two pulse trains at frequencies Vx and v2 in the nerve, and if the amplitude of the low frequency (Vl) train is increased from low to high values, the apparent frequency (v) of the overall pulse train should change from v 1 to v2 in a way like that given by (58). This change in v is the consequence of the inhibitory action which is conceived to originate from the stimulated downward transitions of membrane dipoles. Since the above result is so interesting and could have great implications in neurotherapy, we have designed a simple experiment to give it a test. We applied two stimuli at different frequencies to the giant axon of Myxicola and recorded the response at the end. The result is
90
L I N G Y. WEI
shown in Figs. 2 and 3 (Wei and Hodson, 1977). One sees that as the amplitude of the low frequency stimulus is increased, the apparent frequency of the overall response decreases from v2 to vl, as predicted. Here we need not be too concerned with the exact way by which v varies with A (amplitude) as long as the end points (v2 and vl) and the trend (downward) in between are correct. In this paper we have treated interactions of nerve signals from the viewpoint of dipole quantum processes: (1) spontaneous relaxation, (2) stimulated upward transition, (3) stimulated downward transition and (4) collective oscillation. As far as nerve excitation is concerned, the first two processes are predominant and are contained in equation (1). This equation of dipole excitation is quite similar to that for excitation of ferromagnetic magnons (quanta of spin waves) (Kittel, 1963) based upon the same principle of energy balance. Since many results derived from equation (1) have been confirmed in experiment (Hodson and Wei, 1976) and expressions such as (9) and (10) are essentially the same as those derived from Hill's theory (1936) and Rashevsky's theory (1933), then all the calculations done in this paper based on (1) should be correct to first approximation. Improvement could be made by adding a term on stimulated downward transition to equation (1), or better by writing three equations on N1, Na, and P (phonon population) as proposed in the works of Lee and Chiang (1976), and of Chiang (1978). Chiang in his recent paper (1978) had derived expressions for Na and K currents, and for action potential. However, in both of these papers, the field factor is contained in the exponential expressions for the transition probabilities. It is very difficult, if not impossible, in their formulations to treat interactions of multiple signals and to arrive at relations in closed forms. In our view, it appears better to start with a simple equation such as (1) and to obtain results that can be easily comprehensible. We think we have achieved this purpose in this paper. There are several important conclusions which can be drawn from our calculations. First, to excite the nerve, low frequencies are generally more effective than high frequencies. This is evident from (17), (22) and (23). Second, to sedate the nerve, i.e. to suppress many undesired nerve activities, high frequencies would suit better, as implied by equation (58). Third, by an ingenious combination of multiple stimuli or by applying sufficient strength of a single stimulus at a proper frequency, it is possible to regulate some body functions dictated by nerve impulses (such as in autonomic nerves). Our calculated results suggest many of these possibilities and it is up to biomedical engineers to use these results for practical benefit. For example, electro-sleep, electro-anesthesia, TENS (transcuDiscussion.
DIPOLE THEORY OF INTERACTIONS OF NERVE SIGNALS
Figure 2. Recordings taken with control stimulus (100Hz) applied alone (upper traces); test stimulus (30 Hz) applied alone (middle traces) and both stimuli applied simultaneously (lower traces), The strength of the control stimulus was fixed at 400mV, while that of the test stimulus was (a) 900mV, (b) 1.5V and (c) 2V (Wei and Hodson, 1977).
91
92
LING Y. WEI 24
Ax,onI~IRT~R'2 zo
o
.~ o_ o.;
]8
\
~
o-~
\ T h i s level limited by the control
1
arrangement
o ~
stimulus frequency, SI
I6 --
JElectrode
o\ ~
(200 Hz)
4"4-
SIo
-12 --
14
pe = _ J
O ~ o
-
I0 8 -O 4-
6
E
o-. These action potentials /~' orrived outside the 70 ms interval occupied by the test pulse
4 --
2 -o
I
I
I
I
I
I
I
I
I
2
5
4
5
6
7
8
Figure 3. The effect of test stimulus strength on the frequency of nerve impulses passing through the test stimulus site (Wei and Hodson, 1977)_
taneous electro-neuro stimulation), acupuncture, and various electronic aids for the blind and for deaf mutes are effective means to take advantage of interaction of nerve signals to achieve a specific purpose. Our theory clearly indicates that multiple stimuli, be they spontaneous or external, do not simply add up as some people tend to believe, but rather could produce many other frequency components which under some circumstances could be as strong as the fundamental components. To lull brain activities, say for the sake of mind concentration, both theory and practice dictate lessening inner and outer stimulations as much as possible. The dipole theory, howeyer, suggests more can be done_ Since all-interactions depend on the transition rate constant b, equation (2), one may reduce b by decreasing C, the stimulated upward transition rate and p, the dipole moment and/or by increasing E2, the spacing between the two energy states. C is temperature-dependent and can be reduced by lowering temperature. The effect is amply demonstrated in our daily experience. When keeping the head cool, one can think better. In very hot summer the head may be ringing and often feels topsy-turvy. To reduce p or to increase E2, one may introduce some polyvalent ions or chemical agents which either annul part of the polar charges or bind th,e polar heads: so that they find it difficult to turn from their ground state. Ca ions seem t o have these effects (Frankenhaeuser and Hodgkin, 1957; Gilbert and Ehrenstein, 1969). Very likely, anaesthetics and tranquillizers may work on t h e same principle (Narahashi, 1971; Blaustein and Goldman, 1966; Agin et al., i965).
DIPOLE THEORY OF INTERACTIONS OF NERVE SIGNALS
93
T h e great significance of this theoretical w o r k is t h a t it indicates in principle the feasibility of altering or rewriting the i n f o r m a t i o n c o n t e n t of a nerve message in o u r b o d y b y a p p l y i n g stimulations of a p p r o p r i a t e strengths a n d frequencies. If some b o d y functions (such as secretion of gastric juice) are c o n t r o l l e d b y nerve message (say in a u t o n o m i c n e r v o u s systems), t h e n disorders of these functions c o u l d be n o r m a l i z e d b y external stimulations a c c o r d i n g to o u r theory. W e t h i n k t h a t this is highly p r o b a b l e as a physical principle for some modalities of physical t h e r a p y . A m o n g notable examples are Nogier's (1972) a u r i c u l o t h e r a p y and L a m y ' s (1967, 1969) p h o n o p h o r e s i s in F r a n c e , Voll's (1974, 1975) e l e c t r o a c u p u n c t u r e in G e r m a n y a n d the very recent rising t r e n d of p r o m o t i n g T E N S ( t r a n s c u t a n e o u s e l e c t r o - n e u r o s t i m u l a t i o n ) as a t h e r a p y in the U.S.A. (Burton a n d M a u r e r , 1974; L o e s e r et al., 1975; Veale, 1976). T h e w o r k s of T a n y a n d S a w a t s u g a (1975) a n d of F r a z e e (1975) b y using a wide electromagnetic spectrum, h a r m o n i c s a n d d o u b l e frequencies for stimulations are p a r t i c u l a r l y d e m o n s t r a t i v e for a n d justified f r o m the t h e o r y presented here. It is h o p e d t h a t further d e v e l o p m e n t along this line will open a new era for physical medicine t h a t c o u l d c o m p l e m e n t the conventional chemical (drug) medicine for the benefit of m a n k i n d . This w o r k is s u p p o r t e d in p a r t b y the N a t i o n a l C a n a d a u n d e r G r a n t N o . A-1252.
R e s e a r c h Council of
LITERATURE Agin, D., L. Hersh and D. Holtzman. 1965. "The action of anesthetics on excitable membranes: A quantum chemical analysis." Proc. Natn Acad. Sci., U.S A., 53, 952-958. Blaustein, M. P_ and D. E. Goldman. 1966. "Action of anionic and cationic nerve-blocking agents: Experimental and interpretation." Science, 153, 429-432. Burton, C. and D. D. Maurer. 1974. "Pain suppression by transcutaneous electronics stimulation_" 1EEE Trans. BME, 21, 81 88. Chiang, C. 1978. "On the nerve impulse equation: The dynamical responses of nerve impulse. Bull. Math. Biol., 40, 247 256. Frankenhaeuser, B. and A. L_ Hodgkin. 1957. "The action of calcium on the electrical properties of squid axon." J_ Physiol., Lond., 137, 218-244. Frazee, J. S. 1975. "Experimental harmonic progression and simultaneous dual frequency stimulation." Am. J. Acup., 3, 315-324. Gilbert, D. L. and G_ Ehrenstein. 1969. "Effect of divalent cations on potassium conductance of squid axons: Determination of surface charge." Biophys. J., 9, 447-463. Hill, A. V. 1936. "Excitation and accommodation in nerve." Proc. R. Soc. Lond., Bl19, 305355. Hodson, C. and L. Y_ Wei. 1976. "Comparative evaluation of quantum theory of nerve excitation." Bull. Math. Biol., 38, 277 293. Kittel, C. 1963. Quantum Theory of Solids, pp. 69-70. New York, John Wiley. Lamy, J. 1967, 1969. Acupuncture Phonophor~se-Technique, Clinique. Tome I; Tome II. Librairie Maloine. S.A_ Paris. Lee, C. Y. and (2. Chiang. 1976. "Nerve excitations by the coupling of dipoles and the membrane matrix." Bull_ Math. Biol., 38, 59-70.
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Loeser, J. D., et al. 1975. "Relief of pain by transcutaneous stimulation." J. Neurosurg., 42, 308-314. Narahashi, T. 1971. "Neurophysiological basis for drug action: Ionic mechanism, site of action and active form in nerve fibres." In Biophysics and Physiology of Excitable Membranes, Ed. W. J. Adelman, Jr., pp. 423-462_ New York: Van Nostrand Reinhold. Nogier, P. F. M. 1972. Treatise of Auriculo-therapy. Maisonneuve, Moulins-Les-Metz, France. Rashevsky, N. 1933. "Outline of a physico-mathematical theory of excitation and inhibition." Protoplasma, 20, 4~56. Singer, J. R. 1959. Masers. New York: John Wiley• Solandt, D. Y. 1936. "The measurement of accommodation in nerve•" Proc. R. Soc. Lond_, Bl19, 355-379. Tany, M. and S. Sawatsuga. 1975. "New development: electromagnetic acupuncture•" Am. J. Acup., 3, 58-66. Veale, J. R. 1976. Report on Transcutaneous electrical nerve stimulation for pain relief Food and Drug Adrn.0 Washington, D.C. Voll, R. 1974. Kopfherde-Diagnostik und Therapie Mittels Elektroakupunktur. Uelzen. Germany: ML. Verlag. • 1975. "Twenty years of electroacupuncture therapy using low-frequency current pulses•" Am. J. Acup., 3, 291 314. Wei, L. Y. 1969a. "Role of surface dipoles on axon membrane." Science, 163, 280-282. . 1969b. "Molecular mechanisms of nerve excitation and conduction." Bull. Math. Biophys., 31, 39 58. . 1971a. "Quantum theory of nerve excitation." Bull. Math. Biophys., 33, 187-194. • 1971b. "Possible origin of action potential and birefringence change in nerve axon." Bull. Math. Biophys. 33, 521 537. 1972. "Dipole theory of heat production and absorption in nerve axon." Biophys. J., 12, 1159 1170. • 1973. Quantum theory of time-varying stimulation in nerve• Bull. Math. Biol., 35, 359 374. 1974. "'Dipole mechanisms of electrical, optical and thermal energy transductions in nerve membrane•" Ann. N.Y. Acad. Sci., 277, 285-293• -, and C. Hodson. 1977. "Nerve transmission and acupuncture mechanism." Am. J_ Acup., 5, 69-83. RECEIVED 10-30-78 REVISED 2-2-79