**This is the talk I gave at the
21st meeting of Midwest Neurobiologists, May 22, 1999
M.
Robert Showalter
Abstract:
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 10^{11} times the inductance now
being assumed.
http://www.wisc.edu/rshowalt/kirch1

Analytical results generate a new neural line equation that shows similarly high effective inductances. http://xxx.lanl.gov/html/math-ph/9807015/

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.

TALK:

**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 http://www.wisc.edu/rshowalt/klineul
.)

WHAT'S INDUCTANCE?

WHAT DOES INDUCTANCE DO?

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 :

THE BASIC LINE CALCULATIONS HAVE
BEEN ESTABLISHED FOR A CENTURY.

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

CONDUCTION.

ON

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:

THREE OF THE FOUR CROSSTERM PARTS DEPEND ON G - THEY DEPEND
ON THE MEAN VALUE OF CHANNEL OPENING THAT DETERMINES G. That means conduction
is very channel dependent in the new model.

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 http://xxx.lanl.gov/html/math-ph/9807015/
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. http://www.wisc.edu/rshowalt/kirch1

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 10^{11} times electromagnetic inductance.
It was proportional to the R^{2}C 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 showalte@macc.wisc.edu