A REDERIVATION OF THE ELECTRICAL
TRANSMISSION LINE EQUATIONS USING NETWORK THEORY
SHOWS NEW TERMS THAT MATTER IN NEURAL TRANSMISSION.

M. Robert Showalter

ABSTRACT:

Some of the logic of neurophysiology depends on the mathematical form of the passive electrical transmission equations that apply to nerves. The electrical transmission line is modelled here by network theory. The network model is inconsistent with the accepted transmission equations,

dv/dx = -Ri -L di/dt and di/dx = -Gv -C dv/dt.

The network model shows an effective inductance that depends on R and C. Results suggest that we have been underestimating effective inductance in many neural passages by large factors.

It would be hard to pick equations in engineering physics that are more useful, and that have been more extensively tested, than the electrical line transmission equations.

dv/dx = -Ri -L di/dt         (1)

di/dx = -Gv -C dv/dt       (2)

where v = voltage          i = current           x = position along the line
R = resistance/length    L=electromagnetic inductance/length
G= membrane conductance/length              C=capacitance/length

R, L, G, and C are modelled as uniform along the line. These equations are applied to different conductors. For example, R may vary by a factor of 1015 from a wire to a neural dendrite, with equations 1 and 2 applied in the same way to both the wire and the dendrite. Totally satisfactory performance of equations 1 and 2 over the entirety of this huge range has been assumed.

Equations 1 and 2 were first applied to submerged telegraph cables by Kelvin in 1855(1), and were combined by Heaviside in 1887(2) to form the "Heaviside equations" or "telegrapher's equations(3)" that are fundamental to telephony and much of electrical and computer engineering. Although Equations 1 and 2 both refer to flow of the same charge carriers on the same line, dv/dx depends only on R and L, and di/dx depends only on G and C. This paper shows that both dv/dx and di/dx are functions of all of the line properties R, L, G and C. Some new terms are shown that are negligible for the usual transmission lines of electrical engineering, but that can be dominant in the fine scale dendrites and unmyelinated axons of neurons.

NEUROLOGICAL BACKGROUND:

The dendrites and axons of nerves are conducting lines. They are thought to be without significant inductance. Many waveforms observed in dendrites and axons look like reflection traces that occur on electrical transmission lines, where inductance is important(4). Three decades ago Lieberstein suggested that inductance was important in neural transmission as well(5). However, when the matter was investigated(6) (7), no plausible source of the inductance required was found. Wilfred Rall expressed the conclusion drawn as follows(8). The "cable equation" Rall refers to is equation 1 with L=0, combined with equation 2.

"Comment: Cable Versus Wave Equations

" The cable equation, heat conduction equation, and diffusion equation are all partial differential equations of the same parabolic type. Such equations, and their solutions, differ significantly from those of the elliptical type (e.g. Laplace's equation or potential theory) and from those of hyperbolic type (e.g. electromagnetic wave equation.) When electromagnetic inductance is added to a telegraph cable or transmission line, the partial differential equation gains a term that is proportional to and this changes the equation from the parabolic type to the hyperbolic type; engineers have long known that this can enhance telegraph signal propagation. However important this is for engineering, a recent attempt to account for nerve propagation as a wave solution to such a differential equation can be dismissed for two important reasons: not only did this involve a misunderstanding of the nonlinear membrane properties elucidated by Hodgkin and Huxley, it tacitly assumes an amount of electromagnetic inductance that far exceeds the negligible amount present in nerve."

This paper shows an effective inductance that is inversely proportional to the cube of neural passage diameter. At the scale of most neurons, this effective inductance appears to be > 1011 times larger than the inductance that Rall considered.

Implications of this increased inductance for neural science and neural medicine, and evidence for it, are discussed in a companion paper(9). With the increased inductance, unresolved issues in neural transmission and neural time response can be explained. Ubiquitous but now unexplained effects, phenomenologically similar to resonance, can be explained in resonant terms.

TRANSMISSION LINES AND KIRCHHOFF'S LAWS:

Fig 1 shows a neural conductor (transmission line). A tubular membrane is filled with and surrounded by an ionic fluid. The fluid inside the tube carries current (and signal) and has resistance R and electromagnetic inductance L per unit length. The outer fluid is grounded. In convenient units, the membrane separating the inner and outer ionic fluids has capacitance c in farads/cm2 and conductance g in mhos/cm2. The fluid has a Fig 1 shows an arbitrarily chosen length beta, picked from other indistinguishable length sectors illustrated. The circumferential lines that separate the sectors shown are analytic constructs, like grid lines. There is nothing special or physically causal about the beginning and endpoints for the length beta that happens to be chosen for analysis. The tube of Fig 1 is modelled as a pair of mathematical lines that are homogeneous, continuous, differentiable, and indefinitely long. The grounded external fluid is modelled as a 0 resistance, 0 inductance line. The inner, signal carrying line has uniformly distributed resistance R and inductance L, uniformly distributed capacitance C to the ground line, and uniformly distributed current leakage G to the ground line. This is a continuum MATHEMATICAL model. The scale beta could be 10-5, 10-50, or 10-500 meters long, and the logic and geometry of the model would stay the same.

This homogeneous and differentiable line pair is typically visualized as a series of arbitrarily small 4-pole elements connected end-for-end as shown in Fig 2, with a single 4-pole element set out in enlarged detail in Fig 2a. Fig 2a shows symbols for the discrete elements of R, L, G and C. Fig 2 is shown as one "length increment" long. That "length increment" might be a millimeter, or 10-50 meters long - the representation that is Fig 2 would look the same. We will show that we CANNOT derive equations 1 and 2 from an "almost differentially short" Fig 2. We will also show that an "almost differentially short" model like Fig 2, but consisting of 20, 80, 400, or n RLGC elements rather than the 10 that Fig 2, will yield results that depend on choice of n.

Let's call incremental voltage and current as functions of position x and time t KIRCHHOFF'S LAWS AND THE FUNCTIONAL DEPENDENCIES OF dv/dx AND di/dx:

Fig 2 has the form of a ladder filter, a common kind of bandpass filter(10). Let's consider the RLGC network of Fig 2 according to network theory, following Skilling's encyclopedia article(11). The elements of the network are the resistors, inductors and capacitors, corresponding to the symbols R, L, G, and C. (R corresponds to series resistance; G is the reciprocal of shunt resistance.) Nodes connect three or more elements together. Branches connect two nodes, and may include one or more elements in series. A loop, or circuit, is a single closed path for current. A mesh is a loop with no interior branch.

Charge is conserved. This fact applied to nodes is Kirchhoff's current law: the sum of currents entering a node is 0. Current equations of this form can be written for each of the N nodes in a circuit. N-1 of these equations will be independent. Voltage is a property, and hence path independent. This fact applied to loops is Kirchhoff's voltage law. The sum of voltage changes around a loop (from a point back to that point) is zero. Voltage equations of this form can be written for each of the loops in a circuit. Not all of these equations are independent.

The number of independent loops, Llp can be computed from

E = N + Llp

where N and E are the number of nodes and elements. For a planar, fully connected network like Fig 2, the number of independent loops is the same as the number of meshes.

Fig 2 has 10 RLGC elements, 20 nodes and 20 meshes. For m RLGC elements placed end-for-end, there will be 2m nodes and 2m meshes. E, the number of elements, is 4m. B, the number of branches, is 6m.

Details of setup and solution of the equations corresponding to loops and nodes are set out in Skilling(12). The essential point here is that solving for the currents and voltages of a network such as the one of Fig 2 can be reduced to either

solving Llp linear loop equations, solving currents from which voltages follow directly or

solving N linear node equations, solving voltages from which currents follow directly.

We can see some things about the network of Fig. 2 by inspection. In Fig 2, every loop has both a series and a shunt element, and every node has both a series and a shunt element. Whether circuit analysis were done using node equations or loop equations, voltage and current would be determined by systems of equations consisting only of equations with both series and shunt elements. This means that, in contrast to equation 1, the full dv/dx relation must have some dependence on G and C. In contrast to equation 2, the full di/dx relation must have some dependence on R and L.

Calculation shows what these dependencies are. Software for these calculations is widely available. The de facto standard for analog circuit simulation is SPICE from the U. of C. at Berkeley, and its commercial embodiments(13) (14) (15). Typical design of complicated circuits is now a dialog between SPICE-derived simulation and experimental checking, and over time this dialog has checked and validated SPICE. Although SPICE does much else, the heart of SPICE consists of routines that solve circuit network equations consisting of passive elements like those of Fig 2. The network equations are derived from Kirchhoff's laws and the laws for R, L, and C. The equations are then solved by linear algebra, according to the standard patterns set out above. A powerful demo program of PSpice, a commercial version of SPICE, is free. The results below are were derived using this demo. The demo PSpice program, the input programs that generated that graphs shown below, and the computer programs that yielded the input instructions that generated the graphs, are made available by ftp(16). We are using PSpice here to do Kirchhoff network models of a transmission line. PSpice contains separate transmission line subroutines, based on equations 1 and 2, that are computationally much faster than the Kirchhoff models set out here. These subroutines embody assumptions that the Kirchhoff model does make, that we are investigating here.

Let Fig 1 refer to a section of neuronal line. We can model a set length of that line using a RGLC-RGLC-RGLC ... network such as Fig 2. For the same length, we can use a modelling network with different numbers of evenly spaced and matched RGLC sections, adjusting the values of the individual R, G, L and C elements so that they add up to the R, G, L and C over the modelled length . Sometimes we may simplify the model, and deal with RC sections with G and L set to 0, or model a series of RG sections with C and L set to 0. With the simpler RC and RG models we can compute more sections in series than we can with RGLC networks.

Consider Fig 1 as a neural dendrite of .1 micron diameter. We set membrane capacitance at 10-6 farad/cm2. The passage is filled with an ionic fluid with a resistivity, , of 110 ohm-cm. We choose the electromagnetic inductance of a single line, which is no function of diameter.) Membrane resistance, which occurs by opening and closing of populations of membrane channels, can vary by many orders of magnitude from time to time. Line properties are

R =1.40x1012 ohm/cm                   L =5x10-9 henries/cm

C =3.16x10-5 microfarad/cm                    G = variable

We'll set L = 0, judging it to be negligible in comparison with R. The case where G is also 0, so that the dendrite is an R-C system, will be treated first.

Graphs 31-34 show calculations for this R-C dendrite, corresponding to Fig 1, for a length of 10-5 cm (.1 micron.) The dendrite is modelled as the "ladder filter" of Fig 2, with 20, 40, and 80 RC loops in series. AC input current is 1 nanoamp swept from 1 to 40,000 Hz. Output impedance in the models is negligible (the end of the line dumps to ground through .0001 line resistance) so that voltage varies about linearly from one end of these lines to the other. Voltage drops across resistors are plotted as resistor voltage drop divided by resistor modelling length, x (x = .05µ, .025µ, and .0125µ for the 20, 40, and 80 loop cases.) v/x in these models depends on capacitance as well as resistance. Voltage drop across the resistors changes because current storage in membrane capacitance causes current to change along the line.

Graph 3-1 is for 20 RC groups. Curves are plotted for every 2d resistor.

Graph 3-2 models 40 RC groups. Curves are plotted for every 4th resistor.

Graph 3-3 models 80 RC groups. Curves are plotted for every 8th resistor.

Graph 3-4 is also for 80 groups, with C = 0. Graphs 31, 32, and 33 look much alike. Graph 34 shows that

gradients corresponding to dv/dx do not change from one resistor to the next, when capacitance is 0. Although these models are for a line length of 10-5 cm, similar effects are calculated for shorter lengths also. These PSpice calculations on this ladder model imply that no matter how short a delta x we choose, there will be a finite capacitance effect on values of [delta v(x,t)]/(delta x). We may investigate this dependence in more detail. We've set L=0, so Equation 1 reduces to dv/dx = -Ri. In finite increment form, this integrates over an interval delta x as The residual of voltage not explained by Ohm's law is the calculated voltage difference over an interval minus the voltage predicted by Ohm's law over that interval, using the average current over the interval. An "effective inductance" is calculated, as shown in graphs 4a, 4b, 4c, and 4d. The "effective inductance" per unit length shown in Figs 4A-4D works out to be proportional to R2C/4 (length2). The relation of the models of Figs 4B, 4C, and 4D to the model of Fig 4A illustrate this. Each of these models changes one value of the Fig 4A model, leaving others the same. Comparing output voltages, 4B/4A shows an effective inductance per unit length proportional to length2; 4C/4A shows proportionality to capacitance; and 4D/4A shows proportionality to the square of resistance. All the points on all the lines of Figs 4A-4C seem to be the same within 1 part per 10,000, except for the factors of 10 shown. The lines shown are straight, showing proportionality to di/dt (the operational definition of inductance.)

Fig 5 corresponds to Fig 4A, and shows that the calculated effective inductance term is a small residual compared to system voltages in this calculation. Fig. 6 also corresponds to the case of Fig 4a, and shows the 400 loop model line of 4A divided by di/dt for frequencies from 15-40 kzh. [ di/dt=10-9 amps*(2 pi frequency) amps/sec]. The vertical axis plots 5.0585286 millivolts +/- .2 nanovolts. Fig 6 illustrates the good but still limited quality of the PSpice model numerics, the close proportionality of the curve to di/dt, and a value of effective inductance that is .9850 [(R2C)/12 (delta x)2], a value close to R2C/12 (length2). Figs 4A-4D were plotted for models with 40, 20 and 10 times more sections than the 10 section length illustrated in Fig 2 for a computational reason. Fig 7 illustrates calculated results analogous to those of Fig 4A, with the same .1 micron length, same capacitance, and same conductance, with the length divided into 400 and smaller numbers of sections. Calculated values depend strongly on the number of sections. Data bars in Fig. 7 are fit to a polynomial curve for interpolation. The behavior shown in Fig 7 may merit the term "insidious." The effect occurs in proportion to crossproducts (such as R2C) that may vary over many tens of orders of magnitude from case to case. For a conventional wire, this product might be of the order of 10-80 or less, too small to attend to (and too small for numerics to catch.) Even when the crossproduct magnitude is large, the effect may be missed if only a few sections are calculated. However, the effect is systematic and nonlinear, and it builds up. I have been limited to about 400 RC sections, and have only tested ten sets of values for R and C, but I now suspect that calculations with more sections must stabilize, subject to numerical build-up errors, near the limit suggested by Fig. 7. Fig 7 shows that the residual shown in Figs 4A-4C requires a fine grid scale of calculation to approach equilibrium. However, at courser scales, far from equilibrium, the functional dependence of the relation on the variables is the same. For instance, curves like those of Figs 4A-4c were calculated with 80, 60, 40, and 20 sections (rather than 400, 300, 200 and 100 sections of the Fig 4 series.) Values were different in the proportion shown by Fig 7, and each of the curves in a single calculated case departed noticeably from a 803:603:403:203 relation, with departures following the proportions shown in Fig 7. Even so, comparing these courser cases, at a length ratio of 10:1 produced a calculated voltage deviation from Ohm's law in an 1000:1 ratio. This ratio held to .1% or better for all the curves compared. Similar comparisons showed dependence on R2 and C to similar precision, according to the logic shown in the Fig 4 series.

Figs 4A and 4B show that the integrated inductive effect over a length goes as length3, so that "effective inductance" per unit length goes as the square of length. Real measurements measure over intervals, never at points. Here, the effective inductance depends radically on the scale at which it is "effectively measured," an important departure from the behavior of measurable quantities in the world.

Inductance generates a LAG between current and voltage. The RC ladder model shows such an i-v phase shift for the scale of small neural dendrites. Fig 8 plots voltage and current phase, and Fig 9 the phase difference between them. The value of effective inductance calculated from this phase shift matches the inductance calculated as a deviation from Ohm's law to four significant figures. For the .1 micron diameter, .1 micron length of Fig 4A, the phase shift and voltage effect corresponds to an inductance per unit length of 507 henries/cm from Fig 4a - a factor of 1011 larger than the electromagnetic inductance (5x10-9 henries/cm) calculated for the dendrite. Evaluated over a 1 micron length, the effective inductance per unit length is 100 times larger still.

G DEPENDENCES:

So far we've shown that equation 1 depends on C. Equation 1 also depends on G. The dependence, as a deviation from Ohm's law over an interval delta x, depends on both voltage and current. Using the same procedures shown above for the RC line, and assuming that coefficients close to 1 represent 1, the voltage dependency of (delta v)/(delta x) is The current dependency is Kirchhoff's law models of equation 2 after the pattern of Fig 2 indicate that di/dx = -Gv -C dv/dt must also be incomplete. The i(x,t)/x relation changes with R and L. Because electromagnetic L seems negligible in neurons, we investigate only the R dependence. Using the same techniques illustrated in Figs 3 and Figs 4, we find the i/x relation below. SUMMARY OF RESULTS:

Compared to the finite increment form of equation 1 and the finite increment form of equation 2 our Kirchhoff network model estimates equations 1K and 2K: and CONCLUSION:

The standard equations 1 and 2 convert directly to differential equations, but the numerically derived Kirchhoff equations do not convert directly to differential equations. The subexpressions may appear infinitessimal by the usages of calculus. However, these numerically derived subexpressions are definite results of a linear algebraic process. We cannot dismiss these subexpressions by a limiting argument. Setting delta x at zero, the terms in delta x will be zero, too. But we know from our network model that if delta x is then evaluated at larger and larger values, the subexpression will not stay "infinitessimal" but will grow. What are we to make of these Kirchhoff model derived correllations, with these awkward subexpressions? For many purposes in transmission modelling, we'll need differential equations analogous to (1) and (2). We therefore wish to argue for a particular differential equation, based on our knowledge of the Kirchhoff correllations 1 and 2.

Let's take an approach that appeals to data. If we could use a specific scaling delta x, and algebraically simplify the subexpressions into expressions that did not depend on any particular value of delta x, we'd then have and These equations could be converted directly into differential equations useful for modelling neural transmission. More generally, reasoning from our Kirchhoff model results, we could assume equations of the form of these equations, and try to fit them to data.

I do this in a companion paper9. The ki's that match biological data appear to be about 1. There appears to be an effective inductance in neural transmission that can be, for neural diameters, between 1012 and 1021 times greater than the inductance that is now assumed.

NOTES:

1. William Thomson, Lord Kelvin Proceedings of the Royal Society May 1855. Reprinted as article LXXXIII in Mathematical and Physical Papers by Sir William Thomson, Cambridge Press, 1884.

2. O. Heaviside ELECTRICAL PAPERS V.II Section XXXII 76-81 Chelsea Publishing Company 1970.

3. S. Rosenstark TRANSMISSION LINES IN COMPUTER ENGINEERING p 5 McGraw Hill, 1994.

4. David T. Stephenson "Transmission Lines" Fig 2. McGraw-Hill Encyclopedia of Science and Technology, 7th ed 1992.

5. Lieberstein, H.M. Mathematical Biosciences 1, 45-69 (1967).

6. S. Kaplan and D. Trujillo Mathematical Biosciences, 7 , 379-404, 1970.

7. A.C. Scott Mathematical Biosciences 11: 277-290, 1971.

8. W. Rall HANDBOOK OF PHYSIOLOGY: THE NERVOUS SYSTEM ( Kandell, E.R., ed) p. 60. (American Physiological Society Bethesda, Md. 1977.)

9. M.R. Showalter "New terms in the transmission line equations derived using network theory fit biological data" simultaneous submission

10. DeVerl S. Humpherys THE ANALYSIS, DESIGN, AND SYNTHESIS OF ELECTRICAL FILTERS Prentice Hall, Englewood Cliffs, N.J. 1970.

11. Hugh H. Skilling "Network theory" in McGraw Hill Encyclopedia of Science and Technology 5th ed. New York, 1982.

12. Skilling, op. cit. and H.H. Skilling ELECTRIC NETWORKS Wiley, New York, 1974.

13. L.W. Nagel SPICE2: A computer program to simulate semiconductor circuits, Memorandum No. m530 (May 1975) The Regents of the University of California, Berkeley, Ca.

14. W.J. McCalla FUNDAMENTALS OF COMPUTER-AIDED CIRCUIT SIMULATION Kluwer Academic 1988.

15. Paul W. Tuinenga SPICE: A guide to circuit simulation and analysis using PSpice 2nd ed. Prentice Hall, Englewood Cliffs, N.J. 1992.

16. ftp at angus.macc.wisc.edu