Contains reports on one research project.Bell Telephone Laboratories, IncorporatedNational Institutes of HealthTeagle Foundation, IncorporatedU. S. Air Force under WADD Contract AF33(616)-778

Berendsen, H. J. C.

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XIV. NEUROPHYSIOLOGY*W. S. McCulloch J. D. Cowan Diane MajorJ. A. Aldrich R. C. Gesteland R. MelzackF. S. Axelrod B. Howlandt N. OnestoH. J. C. Berendsen K. Kornacker W. H. PittsM. Blum J. Y. Lettvin A. TaubJ. E. Brown P. D. WallA. INFLUENCE ON CONDUCTANCE BETWEEN TWO PARALLEL PLATES OF ACYLINDER RESTING ON ONE PLATEMeasurements carried out on the dielectric properties of squid axons at ultrahighfrequencies yielded results that are not sufficiently conclusive to report. However, aboundary-value problem that occurred during this research has been solved by approxi-mation; its solution may be of general interest and is therefore reported here.We have measured the complex admittance between 300 mc and 1000 mc of a parallel-plate condensor with a squid giant axon resting on one of the plates. The space betweenthe plates was filled with mineral oil of dielectric constant 2. 5. The axon, presumably,can be considered as a homogeneous cylinder, since the membrane is "invisible" at thefrequencies used.To our knowledge, there is no literature concerning the influence of a cylinder on thecurrent between two parallel plates, with the cylinder resting on one plate. We note thatthe low-frequency problem of a conducting cylinder in a conducting medium is equiv-alent to the high-frequency problem of a dielectric cylinder in a dielectric medium, andthat cr can be replaced by jo . Lord Rayleigh (1) calculated the influence of a cylinderon a regular array of cylinders placed symmetrically between two plates, and his resultshave been simplified and modified for application to practical systems (see, for example,Cole (2)), including random arrays of large numbers of cylinders (3).Vd00(a)---- SPECIFIC CONDUCTANCE I( r 0b)(b)Fig. XIV-1. (a) Boundary-value problem of cylinder betweentwo plates. (b) Reduced problem.This work was supported in part by Bell Telephone Laboratories, Incorporated;National Institutes of Health; Teagle Foundation, Incorporated; and in part by theU. S. Air Force under WADD Contract AF33(616)-7783.tStaff Member, Lincoln Laboratory, M. I. T.247(XIV. NEUROPHYSIOLOGY)The problem can be stated as follows (see Fig. XIV-la). Consider two parallel platesat a distance d, extending infinitely in the z-direction (perpendicular to the plane of thedrawing), and to a distance that is large with respect to d, in the x-direction. Let theplates have a potential difference V, let the material between the plates have a specificconductance a 1l , and let the current between the plates, per unit length in the z-direction,be i o . Now place a homogeneous cylinder with specific conductance 02 on one of theplates. If the current per unit length is i, what then is i - io as a function of a, d, a-2-, and V? The problem is simplified (Fig. XIV-lb) by expressing the potential 4 inunits of V, making the distance between plates 1, the radius of the cylinder ro = a/2d,and expressing the current per unit length in units of ao1 V. The reduced specific con-ductance of the medium is 1 and that of the cylinder is ao = - 2 / 1. We shall also use thereduced specific resistance p = 1/-2Z .Solution by analytical methods seemed infeasible, and we used a model measurementcombined with an analytical approximation method. Equipotential lines were plotted for- = 0 and cr = oo, and for a/d between 0. 5 and 0. 9, with the aid of Teledeltos paper. Fromthis plot, (i-i )/ -lV was determined and plotted as a function of a/d. (See Fig. XIV-2for a0 = 0, and Fig. XIV-3 for p = 0 or cr = oo.) It is worth noting that the approximation0.5 0.6 0.7 0.8 0.9Fig. XIV-2. Solutions for r-2 < -I.of a model; the otherThe cr = 0 values are measured by meansvalues are analytically interpolated.248cr=0.8~----a =0.6(XIV. NEUROPHYSIOLOGY)P=oP =0.2P=0.3P=0.40 P=0.5P=0.60P=07OF P=O.80.5 0.6 0.7 0.8 0.9dFig. XIV-3. Solutions for G-2 1> -method of solving boundary-value problems by manual construction of sets of equipoten-tial and current lines in patterns consisting of squares leads to results of comparableaccuracy.Solutions for intermediate values of a- were approximated by fitting linear combina-tions of the potentials for o- = 0 and ao = 1 into the boundary conditions. Let the potentialdistribution outside the cylinder be o for o = 0 and t1 for - = i. The potential distribu-tion for an arbitrary ar between 0 and 1 is now set as1 = kl + (1-\) 4o (1)inside the cylinder, and as22 z A + A r cos 0 + AZr cos 20 + ... (2)inside the cylinder. Equation 1 automatically satisfies the boundary conditions at theelectrodes. At the surface of the cylinder the conditions are:(a) P is continuous, or 1 (ro) = 2(ro).2490"2 >0 1P = 0",S20*2Tib(XIV. NEUROPHYSIOLOGY)(b) r r=r r =ro oCondition (b) yields the relation Al = -\/. Higher terms cannot be fitted. By Fourieranalysis of the experimentally determined 0 (r, 0) with respect to 0, another relationbetween Al and X is found. From the value of X thus determined the current per unitlength is calculated:(i-io) = (l-k)(i-io)o 0The results are given in Fig. XIV-2.For o- > 1 (0<p<l), an analogous procedure is followed. Since the constants Ai inEq. 2 vanish for -c = oo, another description has to be used at the cylinder surface:S= A + (B r+C 1 r cos 0 + .which is valid outside the cylinder. Conditions (a) and (b) yield2 p-1C r2  B1 o p+1 1Fourier analysis is now applied to( ) dOr=r owhich is experimentally determined from the current lines for c = oo. The best linearcombination of 4)1 and pcc is then determined. The results are given in Fig. XIV-3.The inaccuracy of these results is estimated at 5 per cent for a-Z < a-1 and at 10 percent for cr2 > -1.H. J. C. BerendsenReferences1. Lord Rayleigh, The influence of obstacles arranged in rectangular order uponthe properties of a medium, Phil. Mag. Part V, 34, 481 (1892).2. K. S. Cole and H. J. Curtis, a chapter in Medical Physics, edited byOtto Glasser, Vol. II (Year Book Publishers, Inc. , Chicago, 1950), p. 82.3. S. Velick and M. Gorin, Electrical conductance of suspensions of ellipsoids andits relation to the study of Avian Erythrocytes, J. Gen. Physiol. 23, 753 (1940).250

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