Download Magnetic Stimulation Of Curved Nerves Assaf Rotem, Elisha Moses

Magnetic Stimulation Of Curved Nerves
Assaf Rotem, Elisha Moses
Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot, Israel.
Abstract
The significance of curvature in magnetic stimulation
Methods
Magnetic stimulation of nerves is attracting increased attention recently, as it has been found
to be useful in therapy of neural disorders in humans. In an effort to explain the mechanisms
of magnetic stimulation we focus on the dependence of magnetic stimulation on neuronal
morphology and in particular on the importance of curvature of axonal bundles. Using the
theory of passive membrane dynamics, we predict the Threshold Power (the minimum
stimulation power required to initiate an action potential) of specific axonal morphologies. In
the experimental section we show that magnetic stimulation of the frog sciatic nerve follows
our theoretical predictions. Furthermore, the voltage length constant of the nerve can be
measured based on these results alone.
The figure below demonstrates that the effect of curvature gradients (section C in the bottom
left frame) is more than 100 times greater than the effect of field gradients (section B in the
bottom left frame) when considering the magnetic stimulation of a single axon with a Magstim
Rapid Double 70mm Coil.
Frog sciatic nerves were removed with the Gastrocnemius muscle intact (approved by the
Weizmann IACUC). We used the Magstim Rapid with 70mm Double Coil. The variable
measured in the experiment was the Threshold Power, i.e. the minimum Magstim power
setting required in order to initiate an action potential in the nerve, indicated by muscle
twitching*.
Background – Transcranial Magnetic Stimulation
Experimental Setup. The double coil is placed below the plate, to which the nerve and muscle are fixed by glass pins.
Inset shows a half loop configuration (N=1) with the nerve wrapped around the center glass pin. The table presents
parameters used in the experiment and model (length and time constants are taken from [3])
The induced electric field
* To verify that muscle twitching is indeed a good indicator of nerve excitation, we also measured the nerve’s electrical response, and found that the
threshold potential as determined visually coincides precisely with the one determined electrically. Electric recording was performed with an extracellular pipette filled with frog ringer, using an Axon AM 3000 (no filtering) at X100-X1000 amplification.
Taken from Mark George’s Article on Brain Stimulation in Scientific American 09/03
Modeling the fields created by a Magstim Rapid Double 70mm Coil.
Two magnetic coils (9 windings in each coil) in a figure eight
configuration carry opposite electric currents. (right coil - clockwise
current). Blue vector map indicates the maximum electric field induced
by an abrupt discharge of current through the coils which are
positioned 5mm below the plane of the figure. The central circle
indicates the boundaries of the dish which was used in the experiment.
According to the physical phenomena of magnetic induction, rapidly varying magnetic fields induce electric
fields. A designated magnetic coil positioned over a skull can induce electric fields which are strong enough
to stimulate cortical neurons. The mechanism of stimulation is described below.
rm
A
ε
x
λ
τ
External field
14 nerves were curved from one half loop (N=1) through single loops up to 3.5 loops (N=7).
Threshold Powers of these configuration agree with the proposed model. We can use these results
to estimate the length constant, by plotting the absolute difference between Thresholds of
consecutive N's. It can be shown that these differences decay exponentially with a decay constant
which is proportional to the ratio between the length constant and the radius of curvature.
Model - curving nerves
  1.52  0.14mm
A) Threshold Power of 14 nerves was measured for
curving of N half loops. B) Average Threshold Power of
(A). C) Average Threshold Power of (A) inverted and
normalized. The red line is the model prediction.
Vm
Vm
Membrane potential
Results
In the figure below we predict the membrane potential induced by magnetic stimulation of
axons curved into half a loop and a complete loop. Our model predicts a lower excitation level
for complete loops compared to half loops as a result of a canceling effect between the maxima
and the minima of the complete loop.
Model - The axon as a passive cable
Vm
Modeling the field gradients of the coil described in the left frame A) The maximal absolute value of electric field
gradients along a straight axon oriented parallel to the y axis. Each point in the figure indicates the electric field
gradient that would have been induced if the axon was positioned at that point. The central circle indicates the
boundaries of the dish and the black line illustrates a straight axon above the coil center. B) The maximal
absolute value of electric field gradients along an axon which bends from the –y direction to +y direction in half
a loop of radius 0.4mm. Each point in the figure indicates the electric field gradient that would have been
induced if the axon bend was positioned at that point. The central circle indicates the boundaries of the dish and
the black line illustrates an axon bend over the center of the coil (not to scale). All distances are in units of the
length constant.
cm
rm
 x
x
Axon direction
Summary
cm
Stimulating mechanism of TMS on nerves is dominated by nerve morphology (curvature,
branching or endings), which induces effects that are more than 100 times greater than the
stimulating effect created by the field gradient of a typical TMS coil.
Length constant
Time constant
Passive cable equation
Exponential fit of Absolute difference between consecutive
N’s provides an estimate of λ/rt. Using the known value for rt
we can derive the average length constant of the nerve.
B
Straight axon
Non-uniform external field
C
The Power Threshold is a simple and reliable method to explore the passive properties of
nerves. The measured voltage length constant is consistent with the range of measured
values in the literature [2].
Curved axon
Uniform external field
A physical equation (A) is used to model the sub-threshold (passive) membrane potential of an axon in the
presence of an external electric field (this electric field can result from external electric stimulation or from
external magnetic stimulation). B) Any gradients in this electric field will induce an accumulation of ionic
charge which will in turn will build up membrane potential. C) Alternatively, if the axon is curved, ionic charge
will accumulate at the curve hillock even if the external electric field is uniform.
In the future, applying the method we have presented here on increasingly complex neural
networks (neural cultures, slices and living brains) will help us predict and understand
better the interaction of the stimulation with curved neural substrate, and possibly improve
our ability to affect activity in the cortex of humans.
A) Zooming in on the center of the stimulating coil, the passive effect of the induced Electric Field (blue vector
map) on an axon (thick curve) is shown (color coding represents maximal membrane potential). The axon was
curved to form half a loop of radius rt around the origin (see top right frame). B) Same as (A) with an axon
curved to form a complete loop of radius rt around the origin C) The maximal membrane potential along the two
axons curved at either half (solid line) or complete (dashed line) loop at the time of maximum induced field. All
distances are in units of the length constant λ. Notice the canceling effects in the complete loop curvature.
References
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Rotem A, Moses E. Magnetic stimulation of curved nerves, IEEE Trans Biomed Eng. Accepted Jun 2005.
B. Katz. Nerve, muscle, and synapse. New York, McGraw-Hill, 1966.
P. J. Basser and B. J. Roth. “Stimulation of a myelinated nerve axon by electromagnetic induction,” Med Biol Eng Comput, 29(3):261–268, May 1991.
Background - Motivation
SINCE the first Transcranial Magnetic Stimulation (TMS) was conducted by Barker
et al. [4] in 1985, it has become a remarkable tool for neuroscience research. As a
painless means to probe into human brains, TMS continuously gains diagnostic and
therapeutic applications [5] - [7]. Despite the impressive progress, there is a wide
agreement that the full clinical potential of TMS, mainly as an alternative to
Electroconvulsive Therapy (ECT - electric shocks) is still unrealized [8] - [10], with
one of the main drawbacks being the incomplete comprehension of the physical
mechanisms that underlie TMS. It is particularly not clear why certain regions in the
brain are excitable while others are not, but brain geometry and the anatomy of sulci
and gyri seem to play a role in this mechanism [11] - [15]. This fact motivates us to
further explore how curvature modulates the effect of magnetic stimulation.
References
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Rotem A, Moses E. Magnetic stimulation of curved nerves, IEEE Trans Biomed Eng. Accepted Jun 2005.
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