86 



HANDBOOK OF PHYSIOLOGV 



NEUROPHYSIOLOGY 



further to discuss the spread of electricity alons a 

 uniform resting axon. 



In figure qD, the electric properties of an axon im- 

 mersed in a large volume of sea water are represented 

 by a network of resistances and capacities. Since we 

 are interested only in the change of potentials, the 

 batteries are omitted in the figure. The resistance of 

 the axoplasm of a unit length is represented by r-,; it 

 is related to the specific resistance of the axoplasm 

 Ri by the expression 



"■i = 





(4-3) 



where D is the diameter of the fiber. 



Symbols fn, and („, denote, respectively, the resist- 

 ance and the capacity of the memljrane covering the 

 axoplasm of a unit length. They are related to the 

 corresponding figures for a unit area, R„. and Cm, by 

 the formulae 



':« = -^ . (4-4) 



c„, = ttDC,,,. (4-5) 



Let V(^x, t) denote the potential of the axoplasm, 

 referred to the potential of the surrounding fluid 

 medium, at position x and time /. Then the ssmi^ols 

 f'(.v — A.v, and ['(v + ^x, can be used to denote 

 the potentials at position (,v — A.v) and at (.v -|- A.v), 

 respectively. The axon is now imaginati\elv divided 

 into a series of .segments of a length lA.v. The a.xoplasm 

 resistance (to a longitudinal current) of such a seg- 

 ment is then rjA.v. Similarly, the membrane capacity 

 and the resistance of one segment are given by 

 c„Ax and rn,/A,v, respectively. By applying Ohm's 

 law, it is found that the longitudinal current in the 

 section labelled i is equal to [("(.v, /) — '(•* — ^x, 0]/ 

 (riAAr). Similarly, the longitudinal current through 

 ■section 2 is equal to [F(.v + A.v, /) — r'(.v, 0]/C''iA.v). 

 The difference between the current through i and 

 that through 2 is equal to the membrane current, 

 which has the form given by equation (4-1). This 

 us to the equation 



I^Cv + \x. - r(.v, /) F(.v, - F(.v - Sx.O 



r,Ax 



r\Ax 



dVix, Vix, 



= f„,A.v 1 . 



dl r,^/Ax 



By taking the limit A.v to zcio, we obtain the well 

 known cable equation: 



r, a.v2 



dl'Cx, 



It is obvious that the spread of currents in other non- 

 myelinated nerve fibers and in a uniform muscle fiber 

 can be described by the same equation. 



In the steady state the potential is a function of 

 position X alone. Equation (4-6) is then reduced to 



dnXx) 



dx^ 



= VixX 



(4-7) 



in which \\x) represents ['(a:, k). The general solu- 

 tion of this equation is 



r(.v) = Aft-^l^- -|-Be+''\ 



C4-8) 



where X, the 'space constant', is related to the mem- 

 brane resistance and the axoplasm resistance by the 

 expression 



Constants A and B in equation (4-8) depend on the 

 boundary conditions. 



In a special case where a constant current of in- 

 tensity /o is .sent into the axon at .v = o, constant B 

 has to be equal to zero; otherwise, \\x') approaches 

 infinity as .v increases. At .v = o where the current is 

 sent into the axon, dr(A)'d.v is equal to — '2 ''i^» > the 

 factor '2 being introduced to meet the situation where 

 the current spreads on both sides of the point .v = o. 

 From these boundary conditions, it is found that 

 A = ^-n^lfi and B = o. The solution of equation 

 (4-7) for this special case is, therefore. 



K*) = }i nX-Zoe- 



C4-10) 



at 



-h Vix, 0. 



(4-6) 



The 'effective' resistance }-2^i^ can be expressed by 

 virtue of equation (4-9) as 3^2 ^/^^n^i ■ The space 

 constant, X, is a measure of the spread of electricity 

 along the axon; the greater the value of X, the more 

 extensive is the spread. In the squid giant axon, X is 

 of the order of 0.6 cm (20). Solutions of the general 

 cable equation for several special cases have been 

 achieved (30, 63, 130). 



C'SiBLE PROPERTIES OF THE MYELINATED NERVE FIBER 



Large nerve fibers in the vertebrate nerve have a 

 thick layer of fatty substance, the myelin sheath, be- 

 tween the cylinder of the axoplasm and the outermost 

 layer of connective tissue, the neurilemma or the 

 sheath of Schwann. The myelin sheath is broken at 

 so-called nodes of Ranvier where the surface of the 

 axis cvlindcr is covered dircctlv bv the neurilemma. 



