Units Analysis

Overview

We will be using SI units for this model: meters, volts, siemens, ohms, amperes, farads, seconds. Significant issues are:

Due to numerical precision, we may have to scale the voltage up by 3 or 6 orders of magnitude. The problem is that when voltage gradients get too small (roughly on the order of 1e-6), SCIRun and Matlab are unable to distinguish them numerically from 0.
We may have problems with very small distance values in SCIRun. Specifically, objects on the order of 1e-6 length are rounded to zero and therefore not visible. It may be best to use microns for distance measures in SCIRun.
Conductivity, resistivity and capacitance values are often normalized differently depending on the type of tissue.

Cell membrane capacitance is usually expressed in Farads/meter^2. It is not clear how to scale this value for myelin.
Cell membrane resistance is usually expressed in Ohms * meter^2. Again, it is not clear how to scale this for myelin.
Volume conductivity values are normally expressed in Ohm * meters or Siemens / meter.

Note: due to numerical precision problems, it may be necessary to rescale variables. Specifically: microns, micro Volts, micro Amperes, Siemens / micron.
In a nutshell:

Conductivities are in S/m
Model dimensions are in microns
Therefore conductance is in microS
Current injection is in microA
Therefore, model output is in volts
Gradient is in Volt/micron

Our Experimental Model Parameters

Axoplasm resistivity: .909 S / m
Extracellular resistivity: .333 S / m
Membrane conductance per unit area (for thin cell membranes):

.4 mS / cm^2 = 4 S / m^2

Membrane capacitance per unit area:

2 micro F / cm^2 = .02 F / m^2
Membrane capacitance must be evaluated based on surface area

Membrane conductivity (for artificially thick cell membrane):

Membrane conductance per unit length = ( 4 S / m^2 ) * 7e-9 m = 28 e-9 S / m
Membrane conductance needs to be corrected for artificial thickness: 28e-9 * .31 micron / .007 micron = 1.24e-6 S / m

Myelin conductivity:

Assuming myelin is many cell membranes stacked in series, myelin properties could be estimated from membrane properties.
Myeling volume conductivity is assumed to be the same as membrane volume conductivity, 28 e-9 S / m

Fiber diameter: 12-20 microns for large, myelinated fibers (from here )

Large myelinated fibers in our model are approximately 12-20 pixels in diameter
This implies that one pixel is approximately one micron

Whole blood conductivity [Foster]: .60 S/m
On 2/5/03 Ian measured the pixel distance on the slides. Conversion factor is 0 .29 microns/pixel. Note that the scale bar on the digitized pictures is incorrect.

Conductivity values from Table 7.3, Bioelectromagnetism, Malmivuo & Plonsey, http://butler.cc.tut.fi/~malmivuo/bem/bembook/

Table 7.3. Resistivity values for various tissues

Tissue	r[Wm]	Remarks	Reference

Brain Cerebrospinal fluid Blood Plasma Heart muscle Skeletal muscle Liver Lung Fat Bone	2.2 6.8 5.8 0.7 1.6 0.7 2.5 5.6 1.9 13.2 7 11.2 21.7 25 177 15 158 215	gray matter white matter average Hct = 45 longitudinal transverse longitudinal transverse longitudinal circumferential radial (at 100 kHz)	Rush and Driscoll, 1969 Barber and Brown, 1984 " Barber and Brown, 1984 Geddes and Sadler, 1973 Barber and Brown, 1984 Rush, Abildskov, and McFee, 1963 Epstein and Foster, 1982 Rush, Abildskov, and McFee, 1963 Schwan and Kay, 1956 Rush, Abildskov, and McFee, 1963 Geddes and Baker, 1967 Rush and Driscoll, 1969 Saha and Williams, 1992

Model Volume Conductivity Summary

Tissue	Conductivity (SI units: S / m)
Connective Tissue
Endothelium	.6
Perineurium	.01
Intravascular (blood)	.6
Endoneurium	.333
Cell membrane	28 e-9
Axoplasm	.909
Myelin	28 e-9
Epineurium	.1

Information on Utah Electrode Array

From Danny:

there are 400 um between each electrode, so a 10 x 10 is a 4mm x 4mm.

since we are using a slant array, depth will vary. the long electrodes are
1.5mm long, the short electrodes are .5 mm.

we're (i asked dusty) not entirely sure about thickness, but it's probably
about 20 or 30 um, tops.

chris, the waveforms are really pretty simple. the stimuli are biphasic
pulses, negative first, with a width of 100 µs per phase with a 50 µs
interphase interval. current amplitude varies from 5-50 µA. i stimulate at
15 Hz.

The square wave is 100 us negative, then 50
us of interphase (maybe i'm not using that word correctly?), then 100 us of
positive phase. this is repeated at 15 Hz, so 16.7 ms later it goes through
this cycle again.

Published Data values

The following values were culled from [Malmivuo], Table 21.1, page 366:

Table 21.1: Electrical properties of myelinated nerve fibers
Parameter	Value	Rescaled Value
axoplasm resistivity	0.11 kohm * cm	.909 Siemens / meter
extracellular resistivity	0.3 kohm * cm	.333 Siemens / meter
nodal membrane capacitance/unit area	2.0 micro F / cm^2
nodal membrane conductance/unit area	0.4 mS / cm ^2	4 S / m^2
nodal gap width	0.5 micrometer
ratio of internode spacing to fiber diameter	100
ratio of axon diameter (internal myelin) to fiber diameter	0.7
nodal membrane resistance	29.9 Mohm
nodal membrane capacitance	2.2 pF
internodal resistance	14.3 Mohm

The following are culled from [Ranck]:

For myelinated axons, internodal length is 200 times axis cylinder diameter. Internal length is 120 times outside diameter, since the ratio of axis cylinder diameter to outside diameter is usually about .6 (p.15)
The nodal width is about 2 microns in all axons (p. 16)
One length constant is analagous to about two internodal lengths (p. 16)
The resistivity of grey matter is about 400 ohm * cm (p. 21)
When two electrode tips are separated by more than about two length constants of the fibers being stimulated then we can consider the two electrodes to be independent. When they are closer than about two length constants their effects interact (p. 36)

The following are culled from [Nicholson]:

The extracellular environment consists of the narrow gaps between cellular processes, probably not more than 200 Angstroms wide on average (p. 102).
For brain tissue, the extracellular resistance is fairly constant (about 400 ohm * cm) (p. 103).
Note that there is a table on p. 132 that is not reproduced here but contains bulk conductivities parallal and perpendicular to the fiber system for various CNS preparations.

Cell membrane thickness is about 7nm. One online reference is here.

Table 22-2 from Kandel: Afferent fiber groups in peripheral nerve

	Muscle Nerve	Cutaneous Nerve	Fiber Diameter (microns)	Conduction Velocity (m/s)
Myelinated
Large	I	A alpha	12-20	72-120
Medium	II	A beta	6-12	36-72
Small	III	A delta	1-6	4-36
Unmyelinated	IV	C	.2-1.5	.4-2.0

Table 36-1 from Kandel: Classification of Sensory Fibers from Muscle

Type	Receptor	Axon (microns)	Sensitive to
Ia	Primary spindle endings	12-20 myelinated	Muscle length and rate of change of length
Ib	Golgi tendon organs	12-20 myelinated	Muscle tension
II	Secondary spindle endings	6-12 myelinated	Muscle length (little rate sensitivity)
II	Nonspindle endings	6-12 myelinated	Deep pressure
III	Free nerve endings	2-6 myelinated	Pain, chemical stimuli and temp
IV	Free nerve endings	.5-2 unmyelinated	Pain, chemical stimuli and temp

Tasaki found the resistance and capacitance of the myelin sheath to be 290Mohms*mm and 1.6 microF/mm respectively in isolated frog nerve fiber.

In Vivo Experimental Data

Values from Danny, threshold current values for first detectable response (using step function for 100us, negative initial polarity, and 50us gap):


Date	Minimum Stimulus to evoke measurable muscle contraction (uA)	Maximum stimulus needed to elicit response (uA)	Minimum stimulus of electrode used in experiment	Maximum stimulus of electrode used in experiment
2-2-2	5	83	9	9
2-14-2	9	98	12	26
3-4-2	4	88	4	9
3-11-2	7	88	7	17
3-19-2	7	90	2	15
4-19-2	5	83
4-23-2	6	92	7	10
6-4-2	chronic implant
6-25-2	4	58	4	15
7-10-2	9	63	6	19

References

Malmivuo J, Plonsey R, Bioelectromagnetism, Oxford University Press, New York: 1995.
Available online at: http://butler.cc.tut.fi/~malmivuo/bem/bembook/

Ranck JB, Extracellular Stimulation, Electrophysiological Techniques, Nov. 1-2, 1979, Society for Neuroscience, Atlanta GA.

Nicholson C, Generation and Analysis of Extracellular Field Potentials, Electrophysiological Techniques, Nov. 1-2, 1979, Society for Neuroscience, Atlanta GA.

Foster KR, Schwan HP, Dielectric Properties of Tissues and Biological Materials: A Critical Review, Critical Reviews in Biomedical Engineering, Vol 17, Issue 1, 1989, pp:25-104.

Kandel ER, Schwartz JH, Jessell TM, Principles of Neural Science, 4^th edition, McGraw-Hill, New York: 2000.

Tasaki I, New measurements of the capacity and the resistance of the myelin sheath and the nodel membrane of the isolated frog nerve fiber, Amer J Physiol, vol. 181, p.639, 1955.