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A model for compound action potentials and currents in a nerve bundle I: The forward calculation

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Abstract

We describe a model for the Compound Action Currents (CACs) and Compound Action Potentials (CAPs) produced by a peripheral nerve bundlein vitro. The Single Fiber Action Currents (SFACs) and the extracellular Single Fiber Action Potentials (SFAPs) are calculated using a generalized volume conduction model. Frequency-dependent conductivities, variations in the intracellular action potentials with recording temperature and axon conduction velocity, and the effects of axonal myelination are incorporated into the volume conduction calculation. We demonstrate how the propagation distance and the recording radius affect the simulated Compound Action Signals (CASs) of various nerve bundles. We also demonstrate how the frequency-dependent and-independent conductivities affect the CASs simulated by our model. For this simulation, some of the parameters for the nerve bundles and Conduction Velocity Distributions (CVDs) were obtained from the literature. In accompanying papers, we use the simulated CASs to investigate the effects of variations in the model parameters on the CVDs predicted by our inverse model.

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Abbreviations

φ i (ρ,z):

intracellular potential (mV)

φ e (ρ,z):

potential in the saline bath (mV)

φ s (ρ,θ,z):

sheath potential (mV)

φ b (ρ,θ,z):

bundle potential (mV)

φ m (z):

transmembrane potential (mV)

φ i (ρ,k):

Fourier transform of the intracellular potential (mV·k)

φ e (k):

Fourier transform of the potential in the saline bath (mV·k)

φ m (k):

Fourier transform of the transmembrane potential (mV·k)

k :

spatial frequency (1/mm)

t :

time (ms)

a :

axon radius (mm)

p :

volume fraction

ρ:

recording radius (mm)

ν:

conduction velocity of the action potential (m/sec)

σ i :

intracellular conductivity of the axon in the nerve bunde (Ω−1m−1)

σ e :

extracellular conductivity of the saline bath (Ω−1m−1)

σ s :

conductivity of the sheath around the nerve bundle (Ω−1m−1)

σ z :

axial conductivity of the nerve bundle (Ω−1m−1)

σ ρ :

radial conductivity of the nerve bundle (Ω−1m−1)

b :

radius of the nerve bundle (mm)

δ:

thickness of the sheath around the nerve bundle (mm)

s :

distance of the axon from the center of the nerve bundle (mm)

\(\bar s\) :

mean distance between center of bundle and center of axon (mm)

A :

cross sectional area of the bundle (mm2)

d * :

internal diameter of the nerve fiber (μm)

τ j :

time delay between the stimulus time and the arrival time at the recording site of thejth nerve fiber (msec)

D :

external diameter of the nerve fiber (μm)

Z m :

membrane impedance times unit area (Ωm2)

λ:

length constant (mm)

B :

magnetic field (pT)

c :

inner radius of the toroid (mm)

d :

outer radius of the toroid (mm)

e :

width of the toroid (mm)

ρ eff :

effective toroid radius (mm)

Γ:

circumference of the circle with radius ρ eff

ω:

temporal angular frequency (1/s)

T :

temperature (°C)

l :

the propagation distance of the action signal (mm)

N :

number of active fibers in the nerve bundle

Υ:

delay of SFAS due to activation time and virtual cathode effect (msec)

l n :

internodal distance (mm)

Ј:

current density (A/m2)

Iin :

total current passing through the toroid (A)

L n :

length of the node of Ranvier (μm)

C N :

capacitance of the node of Ranvier (F)

C m :

effective membrane capacitance per unit area (F/m2)

R N :

resistance of the node of Ranvier (Ω)

R m :

effective resistance times unit area (Ωm2)

R sh :

resistance of the sheath (Ω)

C sh :

capacitance of the sheath (F)

Q 10 :

temperature coefficient

ATD:

arrival time distribution

CAC:

compound action current (μA)

CAP:

compound action potential (μV)

CAS:

compound action signal (μA or μV)

CVD:

conduction velocity distribution

FDH:

fiber diameter histogram

FFT:

fast Fourier transform

NNLS:

nonnegative least squares

SFAC:

single fiber action current (μA)

SFAP:

single fiber action potential (μV)

SFAS:

single fiber action signal (μA or μV)

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Authors and Affiliations

  1. Living State Physics Group Department of Physics and Astronomy, Vanderbilt University, Station B, Box 1807, 37235, Nashville, TN

    Ranjith S. Wijesinghe, Frans L. H. Gielen & John P. Wikswo Jr.

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  1. Ranjith S. Wijesinghe
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  2. Frans L. H. Gielen
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  3. John P. Wikswo Jr.
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Wijesinghe, R.S., Gielen, F.L.H. & Wikswo, J.P. A model for compound action potentials and currents in a nerve bundle I: The forward calculation. Ann Biomed Eng 19, 43–72 (1991). https://doi.org/10.1007/BF02368460

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  • DOI: https://doi.org/10.1007/BF02368460

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