A POSSIBLE IIIECHANISII INVOLVED I N THE
COND'C'CTION PROCXSS OF THIN
SHEATHED XERVE FIBERS
WILLIAM J. F R Y AND RUT11 BAUMhN-1- F R Y
r?lLVe?Sity of Ilhnois, rrbana, Itlrnoks
FOUR FIGURES
TNTROnUCTIOS
The relati onship betu7ecn the speed of the conduction
process and the diameter of an invcrtehrate nerve is well
known. However, n o satisfactory theory yet exists which correlates this experimentally observed relation (Yrosser, '46).
To this end a theoretical investigation of localized propagated
mechanical disturbances in the fibers appears worthwhile. I n
part, this consideration is prompted by the order of magnitude
of the velocities in the nerves. The velocities of propagation
of the nerve impulses compare in magnitude to the velocities
of propagation of localized mechanical disturbances in various structures. Geometrically, the nerve fiber approximates
a cylinder consisting of a medium enclosed by a membrane.
We consider, therefore, the propagation of small ampltitude
transverse disturhances on a cylindrical membrane embedded
in a fluid. A possible role of shear waves in the nerve conduction process has been discussed by Rutherland ( '05, '06-'07,
'08- '09).
Results reported in the literature indicate the possibility of
mechanical rnotiori in nerve. For example, investigators (Curtis and Cole, '42; Hodgkin and Huxley, '45; Hodgkin and
Rushton, '46) have postulated the existence of a membrane
inductance in order to correlate certain aspects of the observed
'Work supported in p a r t by the Physiologg Branch of the Office of Naval Rcsearph under Contract K'F-ori-'il-Tauk XXI.
299
230
WILTJIAM J. F R Y AXiu LtLlTII BAUMANN F R Y
electrical characteristics of nerve on a passive electrical network basis. The magnitude of the required inductance, of the
order of 0.2Hcm2, is extremely large to accept as a pure
electrical element manifested in iierve tissue. However, if
inechaiiical motion coexists with electrical activity i n nerve,
this movement would manifest itself in the equivalent electrical circuit of the tissue as a n inductive element. If this is the
case, then ~ a l u e sfor thc inductance of 0.2 H em2 a r e not unre ason abl e.
Tobias and Solomon ('XI), Solomon aiid Tobias ('50) and
Tobias ( ' N a ) have observed dimensional aiid opacity changes
which occur in the region of clectrodes supplying polarizing
current. Tobias ( '5011) has discussed possible implications of
these observations. Their results when considered in conjuiiction with the work of Hill and Keynes ('49) on the opacity
changes accompanying propagated electrcial activity suggest
the possibili t y of mechanical motion during such activity.
The presence of a mechanical disturbance cluring activity is
also suggesicd hy the ease of stiniulatioii of a nerve by mechanical meaiis (Elair, '36). The extreme sensitivity of
mechano-receptors may be considered a s additional support
for such a picture.
Any theory of nerve conduction which involves dimensional
changes must iiiclude a mechanism €or coupling the electrical
aiid mechanical activity. Suitable coupling incclianisins might
iiivolve Iiezoelectric or electrostrietive materials. The theory
discussed herein does not depend critically on a n y one particular coupling scheme.
I n this and another paper (Fry and F r y , '50), which includes the detailed mathematical development, an expression
is presented ivhich relates the velocity of propagation of a
transverse disturhsnce along a tubular membrane to the diameter of the tube. Thc identification of this velocity with the
velocity of propagation of the spike potential of thin sheathed
nerves requires not unreasonahle ralues of the associated
parameters. The experimentally observed relation between
conduction velocity and fiber size for such iiwves can then be
3TIWHANISM FOR C O X D U C T I O N I X NERVE
231
realized theoretically. The picture developed can also be
reconciled with other aspects of the nerve conduction process.
I n the following discussion no attempt has been inade to
coiisider critically all available experimental data since coilcomiiant electrical a i d chcniical phenomena have not yet been
incorporated into this theory. Therefore, at this stage, the
clevelopnimt is to he regarded as suggestive.
THEORY AiYD DISCU8810N
JVe consider the propagation of a ineehaiiical distnrhmcc
of the trailsverse type along a tubular membrane, as illustrated in figure 1. Thc interior of the tube is filled with a
liquid of density p, and the system is irmneiwd in a fluid of
density p?. The membrane is under teilsion, perhaps the result
of a diffcrenee in hydrostatic pressure bet\\-een the interior
43
4
h'ig. 1 A transverse disturbance on a tubular membrane. The amplitllde of
the clisplacenicnt is shown i n an exaggerated form.
liquict and the exterior medium. Let T denote this tension in
units of force per unit length, I f me assume that the disturbance is symmetrical, i,e., independcnt of 0, (fig. 1) and that its
amplitude is Small, the following approximate expression is
ohtwilled f o r the velocity of propagation of a sinnsoiclal disturbance guided along the iiieiiibrane
where w = 2nf, f j s thr f reyuency of the sinusoidsl disturbance,
V uis the velocity, T is the tension in the nieinhrnne in units of
232
WTLLIAN J. FRY A N D R U T H R A U M A F K FRY
force per unit length, r, is the radius of the tube and p1 is the
density of the internal fluid. This expression is applicable
The effect of viscosity
only if w is not larger than about
has been neglected in deriving expression (1).Ordinarily, the
\Tiscosity term affccts the expression f o r velocity only slightly
and introduces an amplitude damping factor. An indication of
the amount of damping introduced by viscosity for small
amplitude disturbances, such as those discussed h e w might
be giren by the damping factor of an acoustic disturbance of
the same frequency. On the basis of measurements made at
high acoustic frequencies, it appears that the damping factor
f o r the frequencies and path lengths involved herein would
be small.
An immediate consequence of (1) is that the velocity is independent of the density of the external fluid. An exact analysis shows that the velocity varies less than 10% f o r values of
the parameters appropriate fo r a giant squid axon if the
density of the external fluid is changed from that corresponding to water to a zero value f o r an w = ( 1 0 ) s (representing a
frequency of 16,000 cycles per see.). It is assumed that the
tension does not change.
If we postulatc that the tension T is determined soIeIy by a
difference in hydrostatic pressures, Po,we can derive the formula 'I
=
' roFu.The relation (1)then becomes
If we further assume that the pressure difference P, is independent of fiber size, i.e., the quantity Pain (2) is constant, we
obtain the result that the phase velocity varies as ras. I f the
tension arises through some mechanism other than a difference in hydrostatic pressure, it appcars necessary to take
T cc r, in order to obtain agreement with experiment.
The disturbance which is propagated along a nerve fiber is
not an infinite sine wave train of some iixcd frequency but is
rather one which is localized spatially. Howerer, further
MECHANISM FOB C O S D U C T I O N I N NEEVF,
233
analysis shows that the d o c i t y , V,, of propagation of a localized disturbance varies in exactly the same fashion, i.e.,
o r with the terisioii T as V, cc Ti.
If oiic plots experimentally determined conduction relocities as a function of fiber diameter f o r incertchrate nerves
with thin sheaths, for wliich data are tabulated lip Prosser
('46), the graph of figurc 2 is obtained. All of tlic data. available in Prosscr 's article for crustncea and cephalopod molluscs
are included on the graph. 'The straight line is plotted f o r a
DIAMETER ( P )
Fig. 2
diameter.
Logarithmic plot of experimental data : conduetion velocity versns dber
234
WILLIAM J. FRY A N D BTJTH BBUMANN FRY
power dependence relation of 0.5. Good agreement with the
experimental observations is obtained o w r the entire range of
fiber diameters from 1p to about 800 p. Pumphrey and Young
( ' 3 5 ) indicate that their data although not excluding a 0.5
relationship might be fitted somewhat better by about a 0.62
powcr dependence. If such is the case, agreement with the
theory developed herein can be obtained loy assuming that the
pressure difference P, in ( 3 ) is proportional t o the square root
of the radius, ro, of the fiber. However, for the subsequent
discussion, we will consider Poconstant since this appears to
us to be the most natural choice yith regard to existing esperimental observation. The valuc of P,may vary with the
animal classification and the type of fiber system.
We would like to obtain some idea of the required pressure
difference if this is the mechanism which gives rise to the
tension, in order to realize velocities in agreement with esperiment. Therefore, we have carried out the following approximate analysis.
From an examination of the shape of the spike potential and
the shape of the membrane conductance change with time, it
are present with
is clear that frequencies of the order of
reasonable amplitudes (Cole arid Curtis, '39). This iinplics
If we assume that a steep
ralues of o of the order of
wavefront is necessary in order to insure propagation (possibly t o trigger an energy release process which is necessary
to compensate for both electrical and mechanical losses), n-e
may calculate a value f o r the required pressure by assuming
that the vclocity of propagation of a spike is of the order of
the phase velocity of the high frequency components of the
pulse. A graphical analysis f o r an exciting pulse of the form
indicated in figure 3a yields the results indicated in figure 3b.
The exciting function is repetitive in time with a period of
approximately 5 times the width of tlie individual pulses. This
was convenient f o r the purposes of the graphical analysis. It
should be noted that the exciting function is symmetrical.
W i e n this exciting function for displacement i s applied t o the
tube at the position z=O, a clisturbancc will spread out in
both directions. At points removed from z =0, the pulse will
arrive at some latey time. If we observe the displacement of
the tube at such points, the ciirves of figure 31) result. The
curve labeled (c) is obtained for a point twice a s far from
z = 0 as the curve labeled (b). These curves show that the
disturbance maintains a sharp wivefront at least initially and
that a relatively slow drop-off develops as evidenced by the
-t
I
-t
3a
l
l
3b
Fig. 3 ( a ) Periodic excitation function plotted against
t h e ; (b) the p r o p
gated disturbance; a, the exriting pulse, h, and r, the propagated piilse obscrred
at two points along the tube.
236
WlLLIAh1 J. FEY A F D R U T H BAU>lAE\'N FRY
iioii-symiiietrical shapes of the curves (b) and ( c ) . At points
further removed froin the source of disturbaiice, the sllalne is
drastically changed. This is partially a conseyuellce of the
choice of a periodic excitation function. Siiicc the phase v e h it>-is proportioiial to tlie square root of the Prequenc~7,it is not
possible f o r a pulse to retain its initial shape icle~ilicallyas it
progresses along the tube. Hornever, this does not appear t o
be a serious difficulty since in any case an energy release
mechanism must be included in a complete theory to account
f o r energy losses, 130th electrical and niechanical. Observed
electrical activity (noiipassive) could then correspond to an
energy release process triggered by the mechanical disturbance. The steepness of the wavefront of the mechanical motion
might, for example, be the important factor f o r such triggering action. The resultant electrical activity \vould then initiate
further mechanical activity because of the mutual coupling
action.
If we insert into equation ( 2 ) values appropriate for a giant
squid axon, a value of w = ( l 0 ) j sec.-l, ro= 2.5(10)-? CM, p, =
1g i ~ ~ / c mT7,
~ , =2 (
cm/sec., we obtain P, O.5( l0)7
dynes/cni2 o r about 5 atmospheres. The choice of a value for
L:, is somewhat arbitrary. From formula (a),we see that a
doubling of the value of o decreases the value of the required
pressure by a factor of 4. The calculated value of the pressure is very sensitive to the value of the velocity. A reduction
of 30% in the velocity changes the required pressure to about
1; atmospheres. Pressure differences of this order of magnitude appear not unrcasonable from the point of view of thc
strength of the materials involved.
I n addition to the above results, it is possible to obtain
correlation of other aspects of the nerve conduction process.
For example, thc effects of longitudinal tension and transverse pressure might be elucidated. Experimentally, the conduction velocity is observed to increase when a nerve is pulled
along its length (Bullock, '45). Such a pulling would have the
effect of increasing the tension, T, in formula, (1) and lvould
thus increase the conduction velocity. Quantitative results on
-
AlECHANISiIl FOR CONDTrCTIOX IN NERVE
237
single fibers might permit a deterinination of iiiteriial presbure. The experinieiital evideiice that lateral squeezing reduces
the conduction velocity (Gasser and Erlanger, '29) might also
he wecoiiciled with this theory since such lateral compression
Tvould 1-educc the tension, T, and thus reduce the conduction
wlocity as shown by relation (1).
Application of hydrostatic pressure to the ~17holesystem
would have no direct effect on the conduction velocity if it is
not applied too rapidly, that is, if equilibrium were maintained,
the pressure difference hetween inside and outside could reiliain constant. It is assumed, of course, that no changes in
structure or shifts of equilibria occur under application of the
pressure. This coriclusion is consistent with the fact that experimentally rather high hydrostatic pressures are required
to affect the conduction process appreciably (Grundfest, '36).
However, sudden changes in pressure or rnechanical motion
might give rise to a disturbance.
As one moves radially away from the kurface of the tube,
the amplitude of the disturbaiice in the external medium falls
off rapidly f o r the higher frequency components of the p r o p '3
.sated pulse. For example, if we consider thc numerical case
discussed above, which is appropriate for the giant squid axon,
me obtain f o r the frequency coiiiponent corresponding to a n
(J of
tlic result that a t one diameter distance f r o m the
surface of the tube, the amplitude of tlie disturbance is less
than 1/10 the value at the membrane.
Electrical activity can he associated with meclianical moveirient in a variety of ways. F o r the present, perhaps the most
natural posbulate is that a piezoelectric o r electrostrictive
irieclianism is operative as indicated above. The dependence
of the velocity of propagation oil the resistivity of the external
nzediuln rrliglit then follov- from the influence which the electrical characteristics exert 011 the mechanical quantities because of strong niutnal coupling. The detailed de~Telopment
of
such a picture awaits further consideration.
238
WILLIAM J. FRY A N D R U T H BdCMANN FRY
STJMhfARY
The possibility of mechanical motion as a contributing fact o r in the coriduction process of thin sheathed invertebrate
nerve fibers is discussed. However, it is riot implied that such
activity in nerve is the only important factor in conduction.
It is shown that the experimentally observed relation between the velocity of conduction and the fiber diameter can be
explained on such a basis. The effect of lateral squeezing and
longitudinal stretching can also be correlated.
ACKNOWLEDGMEST
We wish to thank Dr. V. J. Wulff for his interest and for
helpful discussions.
LITERATURE CITED
BLAIR,H. A. 1936 The time-intensity curre and latent addition ill the niechanieal stimulation of nerve. Am. J. Physiol.. 114: 586-593.
BULLOCK,
T. H. 1945 Functional organization of the giant fiber system of
lumbricus. J. Seurophysiol., 8 : 55-73.
COLE, K. S., AUD H. J . CURTIS 1939 Electrical impedance of the squid giant
axon during activity. J. Gen. Physiol., 22: 649-670.
CURTIS, H. J., ANT) K. S. COLE 1942 Membrane resting and action potentials
from the squid giant axon. ,T. Cell. and Comp. Physiol., 19 : 135-144.
FRY,W. J., AND R. 13. FRY 1950 Theoretical consideration of a possible mechanism in the conduction process of thin sheathed nerve fihers. Bull. hfath.
Eiophys., l?. (In press.)
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1929 The role of fiber size in the estahlislment
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GRUNDFEST,H. 193G Effrets of hydrostatic pressure on the excitability, the rccovcry and the potential sequence of f r o g nerve. Cold Spring Harbor
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I€ILL,
D. K., A N D R. D. KEYNES1949 Opacity changes in stiniulated nerve. J.
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A . L., AND W. A. € I . RUSIITOF 1946 The electrical constants of a
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C. L. 1946 The phvsiology of nervons systcnis of inrertphrate animals.
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R. J., AND J. 2. YOTJEG 1938 Tlic rates o f conduction of ncrve fibers
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S., AND J. hf. TOBIAS 1950 Furtlier obsrraations on electrically induced transparency changes in nerve. Anodal rigidity, rathoclal softening and the effects of sevcral ions. (111 preparation.)
MECHANISICI FOR CONDCCTIO3- TN SERVE
239
SUTHERLAND,
W. 1905 The nature of the propagation of nerve impulse. Arri. J.
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