Nervous System Stimulation Using
Microwave
By Md Anas Boksh Mazady
Project report of ELCT 891B
This article is intended to investigate the possibilities of using microwave to stimulate the
human nervous system so that people with peripheral nervous system disorders could be
cured with non-invasive technique. We will start with first describing the human nervous
system, and then we will try having some insight of how the signal propagates through
this system. Later in this report we will review two papers, one of which will be dealing
with the modeling of the stimulation of a nerve fiber, and the other will be dealing with
the considerations to design a magnetic coil to serve as a stimulator. I will finish this
article by describing my own findings in designing a magnetic coil.
1. HUMAN NERVOUS SYSTEM
The human nervous system can broadly be categorized into two components, namely- the
central nervous system and the peripheral nervous system. Brain, spinal cord, and nerve
roots constitute the central nervous system. On the other hand, thirty one pairs of nerve
roots when branches off the spinal cord to enable the body to move and feel are called the
peripheral nerves.
As shown in figure 1, spinal cord is extended from the lower part of the brain through the
lower part of the body. It is responsible for carrying signal to and from the brain and
different organs. Information about movement of our different limbs and organs or the
information about how we feel in different environments, all of these communications
between the body and brain are conveyed by the cable like structure of the spinal cord.
This spinal cord is about an inch in diameter and 18 inches in length at its widest point.
The peripheral nerves can be categorized into five sub-sections according to their
function in human body.
1)
2)
3)
4)
5)
Cervical Nerves (8 pairs of nerves)
Thoracic Nerves (12 pairs of nerves)
Lumbar Nerves (5 pairs of nerves)
Sacral Nerves (5 pairs of nerves)
Coccyx Nerves (1 pair of nerve)
1
Figure 1: Human nervous system structure
Cervical nerves are responsible for the proper functioning of head, neck, breathing, upper
arms, wrists, hands etc. Thoracic nerves, on the other hand, are responsible for the
functioning of chest, abdominal muscle, internal organs etc. Similarly, lumbar nerves are
for leg muscles, sacral nerves for bathroom capabilities, and coccyx nerves for parental
capabilities
The spinal cord is protected by a hard shell called vertebrae.
Figure 2: Vertebrae
2
2. SIGNAL PROPAGATION IN HUMAN NERVOUS SYSTEM
Neuron is the unit particle of the nervous system. When a neuron receives a stimulus,
depending on the strength of the stimulus, it creates electrical pulses. These pulses then
propagate through its cable like structure.
Figure 3: A neuron cell
The pulses always have constant magnitudes and duration irrespective of the strength of
the stimulus. The intensity of the stimulus is conveyed by the number of pulses produced.
The cell has input ends named dendrites and a long tail – axon. The axons differ in
lengths for different parts of the body. The longest axons might even be more than 1
meter in length. Some of these axons are covered with fatty materials called myelin.
These myelin sheaths are about 2 mm long, separated by gaps named Node of Ranvier.
Human axons are very narrow in diameter. The largest axon might have a diameter of 20
µm. Squid have comparatively thicker axons. This is the reason why most of the
experiments are done on squid axons.
Neurons are of three types – sensory neurons, motor neurons, and interneurons. In a
simple reflex action a stimulus produced by a muscle is transported to the spine, where it
is carried to a motor neuron, the motor neurons then send impulses to control the muscle.
3
Figure 4: Reflex action
Body fluids are dissociated into ions. As a result they can carry electrical signal. Fluids
inside and outside the axons are separated by a thin membrane, axon membrane, which
has very high resistivity. Resistivities of the fluids inside and outside the axon are about
the same but they substantially differ in chemical compositions. Inside body fluid
consists of potassium and negatively charged large organic ions. On the other hand,
outside body fluid mostly consists of sodium and chloride ions.
Figure 5: Axon membrane and body fluids
In the resting condition the axon membrane is highly permeable to potassium than to
sodium, as a result, the inside potential of axon is -70 mV at resting condition with
respect to the outside. A mechanism called “sodium pump” hinders the sodium ions
leaking into the axon through the membrane.
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2.1 Action Potential
If the stimulus exceeds a certain threshold, typically 15 mV higher than the resting value,
an impulse is then produced which propagates down the axon. This propagating impulse
is known as action potential.
Figure 6: Action Potential
In figure 5, at resting condition the potential inside the axon is -70 mV. When a stimulus
is encountered of higher than 15 mV, a sudden rise in the inside potential is observed.
This typically goes up to +30 mV and then start to decrease. While decreasing, the
potential goes below the resting condition for some time, -90 mV, and then returns to the
resting condition. This impulse then propagates through the axon with a maximum speed
of 100 m/sec. Propagation of this action potential sees only a little attenuation as the axon
is insulated by the axon membrane. Moreover, the signal is reproduced at the nodes of
Ranvier. Nerve impulses are produced at a rate proportional to strength of the stimulus,
the upper limit however being limited by the maximum propagation speed of action
potential.
2.2 Propagation of Action Potential
When the potential difference between the inside and outside of an axon falls below some
threshold, less than about 55 mV, the membrane becomes more permeable to the outside
sodium ions. The rapid inflow of sodium ions quickly increases the inside potential to a
value near 30 mV. This increase in the potential then increases the permeability of axon
membrane to the sodium ions immediately ahead of it, which results in a spike in that
region, and the action potential propagates. At the peak of the action potential the
membrane closes its gate to sodium and opens it wide to potassium ions. Thus the inside
potential becomes negative again.
5
Figure 7: Propagation of action potential. (a) Membrane becoming permeable to
sodium, (b) Membrane closing its gate to sodium and opening to potassium making
the inside potential negative again.
Although the signal propagates by the transportation of the ions, the net flow of ions is
much lower than number of ions at rest. Let’s illustrate with some calculations.
We know,
∆Q = C ∆V
But the potential difference inside the axon, ∆V = 70 mV - (-30) mV = 100 mV
And the capacitance per unit length of a non-myelinated axon is C = 3 × 10 −7 F/m (see
table 1)
So the difference in charge appears to be ∆Q = 3 × 10 −8 C. Which gives the indication
that the number of sodium ions entering per meter of axon length is ∆Q/q = 1.875 × 10 11 ,
this is much lower a number than the number of ions at rest ( 7 × 1015 ).
6
During the propagation of the action potential, the above capacitance is continuously
discharged and recharged. The energy required to charge 1 m length of non-myelinated
axon can be calculated as
E=
1
1
CV 2 = × 3 × 10 −7 × (0.1) 2 = 1.5 × 10 −9 J/m
2
2
Since, the duration of the pulse is different for different part of the body. In the muscle
fiber the duration is usually longer, about 20 msec whereas in heart muscle it may last a
quarter of a second. Let’s assume a duration of 1/100 sec for this case, then the power
required to charge the capacitor is 1.5 × 10 −7 W/m.
At the far end the axon branches off to nerve endings which extend to the cells that are to
be activated. The nerve endings are not in direct contact with the cells. This gap is about
1 nm and known as “synapse”.
Figure 8: Synapse
When signal reaches the nerve ending, a chemical substance is released in the synapse
which activates the muscle cell. The deposition of this chemical substance occurs in
discrete packets.
3. AXON AS AN ELECTRIC CABLE
For a simplistic model of the axon, we can imagine it as an insulated electric cable. The
cable is submerged in a conducting fluid having some electrical characteristics as shown
in table 1. The axon membrane may be thought of as the insulation of the cable. The
membrane is not a perfect electric insulator as there is some leakage through it. This lossy
dielectric characteristic may be characterized by capacitance and resistance.
7
Figure 7 (a): An axon showing the flow of electrical signal
Figure 7 (b): Equivalent circuit of an axon
8
Table 1: Electrical properties of an axon
However, this model has some limitations as it fails to explain some key characteristics
of an axon. First, in the above circuit electrical signal flows at a speed comparable to the
speed of light ( 3 × 10 8 m/sec) but as we know the maximum speed of propagation of an
electrical signal through an axon is limited to 100 m/sec. Second, in the above circuit
electrical signals decay very quickly but in an axon action potentials propagate with very
low attenuation. These shortcomings of the above model will be accounted for in a
modified circuit model, Hodgkin-Huxley model, the description of which follows later in
this report.
4. DESIGN CONSIDERATIONS
Here we will review two papers which will give us some insight about the design
consideration of stimulation circuitry. The first paper describes an improved model of the
nerve fiber. The authors then research the location of maximum stimulation along the
nerve fiber. The second paper studies different magnetic coils to give an insight of which
configuration will be better suited for a particular application of nerve stimulation.
4.1.1 Model of a Nerve fiber
To continue from where we have stopped in Article 3, let’s review the paper by B. J.
Roth et al (Roth & Basser, June 1990).
9
Figure 8: An electrical circuit representing the passive cable
The passive electrical cable model of the axon shown in figure 8 is very much similar to
what we have discussed in figure 7 (b). The extracellular potential is neglected here,
marks the only difference than what we have previously discussed.
Let’s first assume the axon to be at resting condition. Applying Ohm’s law we can then
write
߲ܸ
‫ݎ‬௜ ‫ܫ‬௜ = −
߲‫ݔ‬
Using KCL, membrane current can be written as,
݅௠ = −
Again,
݅௠ = ܿ௠
߲‫ܫ‬௜
߲‫ݔ‬
߲ܸ ܸ
+
߲‫ݎ ݐ‬௠
Combining the above three equations we can write,
Where, length constant λ =
λ2
rm
ri
∂ 2V
∂V
−V = τ
2
∂x
∂t
And time constant, τ = cm rm
Axial component of electric field along the fiber, Ei = −
∂V
∂x
With electromagnetic induction by a time varying magnetic field,
Ei = −
∂V
+ ε x ( x, t )
∂x
10
Where ߝ௫ (‫ݔ‬, ‫ )ݐ‬is the induced electric field parallel to the the axis of the axon.
But the induced electric field equals the negative rate of change of the vector magnetic
r
potential
r
∂A
ε =−
∂t
Now,
߲ܸ
‫ݎ‬௜ ‫ܫ‬௜ = −
+ ߝ௫ (‫ݔ‬, ‫)ݐ‬
߲‫ݔ‬
So,
λ2
∂ 2V
∂V
∂ε
−V = τ
+ λ2 x
2
∂x
∂t
∂x
……………………. (1)
Let’s note that the derivative of the electric field, not the electric field itself, appears in
the equation. This is because the membrane current ݅௠ , not the axial current along the
fiber, depolarizes the axon.
The membrane current is a maximum when the spatial gradient of the axial component of
the current is maximum and the induced electric field is at its highest.
From equation (1) we can draw an interesting conclusion, stimulation is not maximum at
the locations where the induced electric field is maximum rather it would be zero there.
The locations where the gradient of the electric field is maximum will have maximum
stimulation.
This model suffers from the limitations that we have previously discussed in Article 3.
So, in the next article a modified approach has been discussed.
4.1.2 Hodgkin-Huxley Model
Figure 9: Hodgkin-Huxley model of nerve fiber
11
The Hodgkin-Huxley model of the nerve fiber takes care of the membrane permeability
to the sodium and potassium ions. The three conductances in each branch represent the
membrane leakage, the sodium channel, and the potassium channel. The value of this
parameters are summarized in table 2.
Table 2: Parameter values for Hodgkin-Huxley model
With this modification the cable equation becomes
ܽ ߲ ଶܸ
− ൫݃ே௔ ݉ଷ ℎ(ܸ − ‫ܧ‬ே௔ ) + ݃௄ ݊ସ (ܸ − ‫ܧ‬௄ ) + ݃௅ (ܸ − ‫ܧ‬௅ )൯
2ܴ௜ ߲‫ ݔ‬ଶ
ܽ ߲ߝ௫
߲ܸ
+
(‫ݔ‬, ‫)ݐ‬
= ‫ܥ‬௠
߲‫ ݐ‬2ܴ௜ ߲‫ݔ‬
Where, m, h, n can be found solving the below three differential equations,
And, ߙ௠ , ߚ௠ , ߙ௛ , ߚ௛ , ߙ௡ , ߚ௡ can be calculated from the below four differential equations,
12
Here the resting potential is assumed as -65 mV.
The induced electric field is related to the coil current and geometry by the following
equation
ε (r , t ) =
dI (t ) µ 0 N
(−
dt
4π
∫
dl ′
)
r − r′
Where, I(t) is the current through the coil, r is the location where the electric field is
calculated, r ′ is the position of the differential element of the coil dl ′ .
4.1.3 Numerical Computation of Stimulation
For this computation, the coil was placed at a distance of 1 cm from the nerve. The coil
had 30 turns and a radius of 2.5 cm. The nerve was parallel to x-axis and was tangent to
the coil at y = rc . The coil was assumed to be a 64-sided polygon.
13
Figure 10: Coil arrangement for stimulation
4.1.4 Electric Field and its gradient
ࢊࡵ
Figure 11 (a): Electric field profile (ࢊ࢚ = 1 ‫ܣ‬/ߤ‫)ݏ‬
14
Figure 11 (a) shows the electric field profile for the arrangement of figure 10. The
corresponding electric field gradient is shown in figure 11 (b).
Figure 11 (b): Electric field gradient
It is interesting to note that the maximum of electric field gradient is not below the center
of the coil nor even at the edges.
Figure 11 (c) shows the contour plot of the spatial profile of electric field gradient. The
bold circle represents the position of the magnetic coil with the arrow indicating the
direction of rate of change of current. The dotted line indicates the location of the axon.
The minus signs indicate the location of depolarization, i.e. stimulation, whereas the plus
signs indicate the location of hyper polarization.
15
Figure 11 (c): Contour plot of electric field gradient
4.1.5 The current in the magnetic coil
The simplest way to produce a current pulse is to use a capacitor and inductor in parallel
with a switch continually going back and forth. The capacitor is initially charged to V0 by
Figure 12: Current pulse generator
16
a dc voltage source. Then the switch S is closed and the capacitor is discharged through
the inductance L and resistance R of the magnetic coil (MC).
The inductance L of a N turn coil with radius rc, and wire radius rw is given by
8rc
) − 1.75)
rw
Thus, a coil of radius 2.5 cm, wire radius 1 mm and having 30 turns has an inductance of
0.165 mH.
L = µ 0 rc N 2 (ln(
With R = 3 Ω, L = 0.165 mH, C = 200 µF
ܴଶ
1
−
= 5.2342 × 10଻ ≫ 1
ଶ
‫ܥܮ‬
4‫ܮ‬
So, the coil current I(t) would have an overdamped response.
‫ܸ = )ݐ(ܫ‬଴ ‫߱ܥ‬ଶ ݁
Where,
ିఠభ ௧
߱ଵ =
߱ଵ ଶ
ቆ൬ ൰ − 1ቇ sinh (߱ଶ ‫)ݐ‬
߱ଶ
ܴ
= 9.07 ݉‫ି ݏ‬ଵ
2‫ܮ‬
ܴ ଶ
1
ඨ
߱ଶ = ൬ ൰ −
= 7.21 ݉‫ି ݏ‬ଵ
2‫ܮ‬
‫ܥܮ‬
V0 = 200 V
So, the resulting I(t) would be as shown in figure 13 (a).
17
Figure 13 (a): Current in a magnetic coil
The time derivative of I(t) would then be as shown in figure 13 (b).
Figure 13 (b): Time derivative of current in a magnetic coil (MC)
Taking this time derivative of current into consideration the electric field gradient in
figure 11 (b) will be modified to figure 14.
18
Figure 14: Electric field gradient taking MC current into consideration
4.1.6 Conclusion from (Roth & Basser, June 1990)
From this paper we can draw several important conclusions.
1. The nerve fiber is stimulated by the gradient of the electric field that is parallel to
the axis of the nerve fiber.
2. The electric field component hyperpolarizes or depolarizes the axon and may
invoke an action potential depending on the strength of the electric field.
3. Hodgkin-Huxley model was introduced to represent a nerve fiber.
4. Maximum stimulation is neither below the center of the coil nor below its edges.
4.2 Magnetic Coil Design Considerations
Here we will discuss another paper (Lin, Hsiao, & Dhaka, May 2000). In these paper
coils of various geometric orientations was studied varying radius, number of turns,
slinky, wire length etc. Different geometry of coils that were studied are shown in figure
15.
19
Figure 15: Different geometries of magnetic coils
4.2.1 Theoretical Computations
Electric field generated by a magnetic coil (MC) can be calculated by
∂A
∂t
∂I
dl ′
= −∇V − ( µ 0 / 4π ) ∫
∂t
r
E = −∇V −
A typical current wave form, generated by a capacitor initially charged to V0 volt and
discharging through a coil (RLC circuit) is governed by the following equation
I = (V 0 / ωL) sin(ωt ) exp(− at )
∂I / ∂t = (V 0 / L)[cos(ωt ) − (a / ω ) sin(ωt )] exp(− at )
E = −∇V − {(V 0 / L)[cos(ωt ) − (a / ω ) sin(ωt )] × exp(− at )}( µ 0 / 4π ) ∫ dl ′ / r
Where, a = R /(2 L) and ω = (1 / LC ) − a 2
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For uniform current distribution along the coil ∇V becomes insignificant compared to the
other factors. So,
1.
E∞ ∫
2.
E∞
dl ′
r
1
L
4.2.2 Experimental Measurements
Table 3:
21
Table 4:
Table 3 & 4 summarizes different arrangement of the coils used in this experiment.
Figure 16 shows the experimental set up.
A plastic container of 30 ܿ݉ × 30 ܿ݉ × 25 ܿ݉ was used in this experiment. The
container was filled with a saline solution of conductivity 0.002 S/cm (0.8% NaCl). The
electrodes of the electric field probe were 0.2 cm – 0.5 cm apart. The frequency of the
coil excitation current was 20 Hz.
22
Figure 16: Experimental setup
The electric field distribution for coils IA-IE at 3 mm above them was found as follows
Figure 17: Electric field distribution (IA-IE)
23
From figure 17 we can clearly see that the ratio of
ா೛ೝ೔೘ೌೝ೤
ாೞ೐೎೚೙೏ೌೝ
increases as number of
slinky increases. This happens because with increased number of slinky in other planes
than the horizontal, effective r increases as a result Esecondary decreases.
Electric field distribution of circular (IB) vs rectangular (IF) coil is shown below for same
resistance R.
Figure 18: Electric field profile coil IB and IF
We can observe that the circular coil has higher primary electric field but less focalization
as compared to the rectangular coil (2.6 vs. 2.2).
ா೛ೝ೔೘ೌೝ೤
Electric field profiles of coils IIB, IB, and IID with diameter 11.5, 7.62, and 5.08 cm
respectively are plotted in figure 19. This figure shows that the ratio
ாೞ೐೎೚೙೏ೌೝ
ଵ
decreases
as the coil diameter increases. Eprimary decreases because ‫ ܦߙܮ‬and ‫ ߙܧ‬. Esecondary is higher
௅
for larger diameter coil because of less interference from the side limb of the other coils.
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Figure 19: Electric field profile for different diameter coil
The field penetration by different diameter coils, IB (7.62 cm), IIB (11.5 cm), and IID
(5.08 cm), were investigated in the following plot. Here we can see that lower diameter
coil has higher initial field strength, but higher diameter coils penetrate deeper.
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Figure 20: Electric field penetration of coils IID, IB, and IIB
4.2.3 Analytic Computation
A computer program FARADAY was used to simulate the electric field in different coil
arrangements. This software uses boundary integral equation to solve for the fields. For
this simulation current through the coil conductor was 30,000 A at 4 KHz.
ா೛ೝ೔೘ೌೝ೤
The analytic computation results of electric fields for coils IA-IE is shown in figure 21.
The ratio
ாೞ೐೎೚೙೏ೌೝ
is observed to increase as the number of slinky increases. This is
because effective r increases and thus Esecondary decreases.
26
Figure 21: Electric field profile of coils IA-IE
Coils IIB, IB, and IID were studied to learn the behavior of different diameter coils. The
diameters were 11.5, 7.62, and 5.08 cm respectively. The initial field strengths were 454,
517, and 789 V/m respectively. So, the smallest diameter coil had the highest field
strength. This happens for the same reason as stated in Art 4.2.2. Eprimary increases
ଵ
because ‫ ܦߙܮ‬and ‫ ߙܧ‬. Esecondary is higher for larger diameter coil because of less
௅
interference from the side limb of the other coils. The ratio primary to secondary field
was found to be 1.5, 2.0, and 2.3 respectively. So the ratio decreases as the coil diameter
increases, agreeing to the experimental results. But the larger diameter coil maintained a
higher field strength after 30 mm distance.
4.2.4 Findings from (Lin, Hsiao, & Dhaka, May 2000)
Slinky arrangement of coils produces more focalized electric field than the planar circular
coils. Peak primary to secondary electric field ratios were found to be 1, 2.2, 2.85, 2.62,
and 3.54 for coils with slinky 1 to 5 respectively.
Coils with larger diameters have better penetration depth than those with smaller
diameter coils. Coils with less number of turns have higher initial field strength.
5. MY FINDINGS
27
I used XFDTD 6.5 to simulate a homogeneous rectangular phantom of 225 ݉݉ ×
150 ݉݉ × 150 ݉݉. 2 mm shell was used in the phantom that was facing the magnetic
coil. 30,000 A electric current was flowing through the coil conductor at 915 MHz.
Muscle simulating tissue had conductivity 0.97 S/m, relative permittivity of 41.5 and
density 1000 Kg/m3. The shell had conductivity of 0 S/m, relative permittivity of 3.7 and
density 0 Kg/m3. Two coils were placed such that the wires in one edge of a coil were at
the middle of the two wires of the other coil. In this way, we can facilitate maximum
electric field near the intersection of the two coils. The other edges would try to decrease
the effect of the other coil. So, we will be having maximum focused electric field. Each
of the coils had 4 turns and radius of 38.1 mm. The wire had a length of 80 mm and
radius of 0.25 mm. The wire material was selected to perfect electrical conductor (PEC).
The coils were placed such that the nearest coil element was 3 mm away from the tissue.
Base cell size of 1 mm was chosen in all directions. Liao absorbing boundary was used to
simulate.
Figure 22: The phantom and coil arrangement for simulation
Results
After the simulation was performed the electric field profile at the underneath of the
intersection of the two coils (mid points) and the side edges (side limb) were plotted in
figure 23.
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Figure 23: Electric field profile of midpoint and side limb with respect to time
From the above figure, Eprimary is 2.2 × 10ହ V/m and Esecondary is 1.1 × 10ହ V/m. So the
ratio is 2.
6. CONCLUSION
How signal propagates through human nervous system has been described in this report
in terms of electrical point of view. Some electrochemical changes has also been
discussed. A simplified model of the nerve fiber was then presented which was later
modified with Hodgkin-Huxley model. An analytic model of stimulation in the nerve
fiber was then developed based on this model. It was found that maximum stimulation
was neither beneath the center of the coil nor at the edges. Then different magnetic coils
were studied to find a suitable one for a particular application of nerve stimulation.
Stimulation of peripheral nerves at different organs demands different magnetic coils.
Some may require higher penetration and some may require a focalized beam. At last a
simple simulation model was developed in XFDTD. Future scope of this project lies in
studying the behavior of magnetic coil at different frequencies and study of magnetic coil
built with different magnetic materials. Some in vitro experiments are also left for future
work.
29
Bibliography
Davidovits, P. (2007). Physics in Biology and Medicine.
Lin, V. W.-H., Hsiao, I. N., & Dhaka, a. V. (May 2000). Magnetic Coil Design
Considerations for Functional Magnetic Stimulation. IEEE Trans. on Biomed. Eng. , Vol.
47, No. 5.
Roth, B. J., & Basser, P. J. (June 1990). A model of the stimulation of a nerve fibre by
electromagnetic induction. IEEE Transactions on Biomedical Engineering , vol. 37, No.
6.
Spine Universe. (n.d.). Retrieved April 2009, from www.spineuniverse.com
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