This is the talk I gave at the 21st meeting of Midwest Neurobiologists,  May 22, 1999
M. Robert Showalter

Inductance in Neural Line Models

                     Standard circuit modeling of a neural line with only R and C shows an effective inductance (proportionality                      between di/dt and dv/dx) of more than 1011 times the inductance now being assumed.                                                                                                                                                           

                     Analytical results generate a new neural line equation that shows similarly high effective inductances.                                                                                                            

                     The new equation fits oscillatory and resonance-like data that the presently used neural conduction equation                      does not.   The new equation has two modes, a high g mode that resembles the behavior of the presently used                      equation, and a low g mode that is similar to an ordinary EE transmission line with a huge effective inductance.                      The new equation predicts more stable action potentials, particularly when channel populations are spatially                      uneven. Channels and synapses are predicted to be more physiologically effective. A dendrite conduction                      velocity versus frequency test, with channels closed, should discriminate between these equations.


Here are points I hope you'll remember:

Neurons and systems of neurons sometimes LOOK like they have large inductance.

New math suggests that neurons COULD have large inductance.

IF that large neural inductance in assumed, a lot fits.

          Inductance would produce the long range neural oscillations that we see.

          Inductance would make action potentials stable under the uneven           conditions many axons and dendritic lines have.

We can TEST the question of neural line inductance directly with present technique.

People have been suggesting neural inductance for sixty years.

But it's been thought that neurons couldn't have any significant inductance, even though almost everybody's agreed that neurons LOOK like they do have inductance. The "no inductance" decision was based on MATHEMATICAL THEORY.

Mathematical theory is based on assumptions called "axioms," and in science, assumptions can be tested. The axioms involved here had caused many troubles. My friend and partner, Professor Stephen J. Kline, of Stanford worked with me for over a thousand hours on this set of problems. We found a longstanding error that matters in neurophysiology and some other places, and have fixed it. Steve died 18 months ago. I miss him a lot.

               (here is what I said at Steve's memorial service at Stanford Chapel .)



In a conducting line, inductance is the PROPORTION between voltage gradient and the rate of current change. Any physical effect that produces such a proportionate relation is an inductance in the ways that matter for neural conduction. Magnetism doesn't have to be involved.

Inductance is "electrical inertia". Everybody understands how inertia matters in mechanics.

                         (I show big bell on long string - swing bell as pendulum.)

When I take this bell, and swing it, it keeps moving after the lowest point on its arc. Without inertia it would stop the instant it got to its lowest point. Without inductance, currents cease when the voltage forces driving them stop, in a similar way.

The RATIO of inertia to damping counts. If I swung this bell in water instead of air, it would stop much more quickly. If this bell were immersed in honey, and I tried to swing it, it woud act as if it had no inertia at all.

In neural lines, resistance is a damping force, analogous to the honey. Inductance is thought to be negligible in comparison.

IF inductance were much larger than now thought, two things would happen.

                    Currents and waveforms would be more stable.

                   Neural structures might sometimes show PASSIVE RESONANCE.

Fig 10. Regan, David HUMAN BRAIN ELECTROPHYSIOLOGY: Evoked Potentials and Evoked Magnetic Fields in Science and Medicine Elsevior, 1989.

Here's data that shows resonance-like behavior measured in brain. This is David Regan's zoom FFT EEG and MEG data. The stimulus is a TV screen flashing at two frequencies. There are frequency power peaks at sums and differences of those frequencies. If there were an ensemble of very sharply tuned passive resonators in brain, you'd see curves like this.

For energy reasons, every peak must have represented at least tens of thousand of neurons or subcellular elements.

Regan got results like this for both MEG and EEG, and for many modalities of of evoked stimulation. A repetitive shock to your skin and a TV screen flash produced similar zoom FFT patterns.

Regan measured bandwidths of these kinds of pulses as less than 2 thousandths of a hertz. Or maybe tighter than the .002 Hz he could measure. That's tight.

Recently, a Japanese television cartoon broadcast bright and repetitive T.V. screen flashes, much like Regan's stimulus. . Hundreds of children and adults who saw the cartoon had epileptic seizures that resulted in hospital visits. Thousands of kids seem to have been effected.

(Sheryl WuDunn "TV Cartoon's Flashes send 700 Japanese into Seizures" December 18, 1997, and "Japan TV to Act Against Seizure-Causing Cartoon Flashes" December 20, 1997, THE NEW YORK TIMES.)

What's passive resonance? Here's an example: (RING ! ! !)

I deflect the bell. It vibrates, exchanging energy between elastic and inertial forms. Note that there isn't any logic switching involved. There's no "reverberatory circuit" with propagation of a signal around a loop. The resonance is inherent in the simple, unitary structure of the bell.

Passive resonance and active resonance aren't mutually exclusive. A system can have both. Many systems that work especially well DO have both.

Here's are facts :


With the right values of R, L, G, and C, and with the right treatment of current sources and sinks, the basic line equations and solutions APPLY TO NEURAL


Here are some other facts:

Conduction velocity of an RC line without inductance goes as the SQUARE ROOT of frequency.

When inductance predominates, velocity is CONSTANT as a function of frequency, above a threshold.

That's a big, testable difference.

With channels off, velocity without inductance is as the square root of frequency, velocity with the S-K theory is flat with frequency. Channel blocker technique is getting better and better. Optical observation techniques, which could look at pulse motion, are getting better and better. Patch clamp techniques for observing dendrite voltage at different points on a line are impressive, and getting better. The conduction difference shown here can be directly tested. I hope it will be. I'd be glad if someone here might get interested.

Based on what I've seen, I think we've been looking a inductance for a long time. We expect roughly constant conduction velocities in neural lines. For the complicated waveforms neural lines carry, that takes inductance, or a degree of perfection and complication in channel activity that seems unlikely to me.

Lines that have relatively large inductance (or inertia) can also show COLUMN RESONANCE, with electrical columns of charge vibrating like the air in an organ pipe.

Here's our main analytical result. The new equation looks just like the standard line equation, except that hatted values of R, L, G, and C are substituted for the usual ones.

The hatted value are as follows:


I'd like to discuss the math with anybody who asks. It is there for people to see on the web, and the citations are in my abstract.

Here's the new rule that generates our model:

When we derive a finite increment equation from a coupled finite increment physical model, that equation will include crossterms that represent several physical laws in interaction together over space. We must insist on algebraic simplification of these crossterms at UNIT SCALE.

This violates nobody's valid axioms, but does ask people to change some reflexes.

              (For much more detail, see and especially the first two appendices.)

Using this analysis, Steve Kline and I quickly derived a neural equation with enough inductance to fit Regan's resonance-like data. I modeled a number of spine shapes. Here's one I looked at.

Here's a fairly typical resonance curve we got, with resonant magnification shown on a linear and a log plot. Resonance magnifications for spines that acted like "organ pipes" in column resonance were similar. A steady state oscillation outside the spine, of the right frequency and stable phase, would be magnified between forty thousand to eightythousand times, if it went on long enough and the spine didn't break down electrically.

(I think epileptic spine damage, a characteristic of serious epilepsy, could happen when spines                 are destroyed in resonance in this way. My guess is that this is the way it does happen.                 According to this theory, voltage differences across membrane might get so big that the                 membrane breaks down electrically. After that, the spine or dendritic branch involved would                 shrivel.)

If you open ONE CHANNEL on the spine bag shown, the spine is so damped that it has almost no resonance. So the spine is calculated to be a CHANNEL SWITCHED passive resonator with great sensitivity. What could be better for an information processing brain?

Calculated resonant magnifications are high enough, and bandwidths are tight enough, to fit Regan's data. I think spines do act this way.

I looked at crossterms in another, independent way, with a discrete computer model. You can look at the model on the web.

I have a copy if somebody here wants to look. I calculated inductance from the phase shift, and the voltage loss. Agreement between the two calculations was good four significant figures or better. For a .1 micron model, .1 micron long, this model showed an effective (nonmagnetc) inductance that was more than 1011 times electromagnetic inductance. It was proportional to the R2C term of our analysis.

The S-K model behavior depends on the channel opening that determines G.

In this figure, the fraction of a wave that remains after one wave length of propagation is plotted. For the old theory, virtually all the wave damps out. No resonance is possible. For the new theory, with large G, that's also true. But for low G's, and small lines, most remains - enough that passive resonance in the low G state is possible.

So when channels are fairly open, the new theory isn't very different from the old. When g is very low, systems are more sensitive, and sharply tuned switched resonance is possible.

The new equation fits in with powerful ephatic coupling and tuning. There are something like a 1000 papers on neural oscillation in the last five years - it is not understood. I believe that this new transmission equation is correct, and hope that it might cast some new light on that subject, and related subjects in the neurosciences and neural medicine.

Let me close with the points I care most about.

Neurons and systems of neurons sometimes LOOK like they have large inductance.

New math suggests that neurons COULD have large inductance.

         IF that large neural inductance in assumed, a lot fits.

         Inductance would produce the long range neural oscillations that we see.

Inductance would make action potentials stable under the uneven conditions many axons and dendritic lines have.

We can TEST the question of neural line inductance directly with present technique.

Thank you.

                  At the meeting, questions were of a supportive kind, and comments thereafter were also.   Perhaps other readers,                   from other perspectives, may have other perspective to offer.    I'd be grateful for any comments, pro or con,                   anyone might want to communicate.          Bob Showalter