Modeling the interactions between stimulation and physiologically induced APs in a mammalian nerve fiber: dependence on frequency and fiber diameter

Abstract

Electrical stimulation of nerve fibers is used as a therapeutic tool to treat neurophysiological disorders. Despite efforts to model the effects of stimulation, its underlying mechanisms remain unclear. Current mechanistic models quantify the effects that the electrical field produces near the fiber but do not capture interactions between action potentials (APs) initiated by stimulus and APs initiated by underlying physiological activity. In this study, we aim to quantify the effects of stimulation frequency and fiber diameter on AP interactions involving collisions and loss of excitability. We constructed a mechanistic model of a myelinated nerve fiber receiving two inputs: the underlying physiological activity at the terminal end of the fiber, and an external stimulus applied to the middle of the fiber. We define conduction reliability as the percentage of physiological APs that make it to the somatic end of the nerve fiber. At low input frequencies, conduction reliability is greater than 95% and decreases with increasing frequency due to an increase in AP interactions. Conduction reliability is less sensitive to fiber diameter and only decreases slightly with increasing fiber diameter. Finally, both the number and type of AP interactions significantly vary with both input frequencies and fiber diameter. Modeling the interactions between APs initiated by stimulus and APs initiated by underlying physiological activity in a nerve fiber opens opportunities towards understanding mechanisms of electrical stimulation therapies.

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Acknowledgments

Work supported by NIH R01 AT009401 to S.V.S, Y.G., and W.S.A., and by NPRI postdoctoral fellowship awarded to P.S. We would like to thank Dr. Michael Caterina, Neurosurgery Pain Research Institute, The Johns Hopkins University School of Medicine for valuable and insightful discussions.

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Correspondence to Vijay Sadashivaiah.

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Appendix

Appendix

The fiber model and its parameters at 37°C Fiber geometry

ᅟ:

axon diameter, d = 6–12 μm (step size of 3 μm)

ᅟ:

ratio of axon to fiber diameter, rdD = 1

ᅟ:

fiber diameter, D = d × rdD

ᅟ:

ratio of li to fiber diameter, \(r_{\mathrm {l_{i}D}} = 100\)

ᅟ:

internodal length, \(l_{i} = D \times r_{\mathrm {l_{i}D}}\)

ᅟ:

fiber length, L = 10 cm

ᅟ:

nodal length, l = 2.5 μm

ᅟ:

number of nodes, \(n = \lceil 1 + \frac {L}{l + l_{i}}\rceil \)

Parameters

ᅟ:

αmA = 1.86,αmB = 65.6,αmC = 10.3

ᅟ:

αhA = 0.0336,αhB = − 27,αhC = 11.0

ᅟ:

αnA = 0.00789,αnB = − 9.2,αnC = 1.10

ᅟ:

αsA = 0.00122,αsB = 71.5,αsC = 23.6

ᅟ:

βmA = 0.0860,βmB = 61.3,βmC = 9.16

ᅟ:

βhA = 2.30,βhB = 55.2,βhC = 13.4

ᅟ:

βnA = 0.0142,βnB = 8,βnC = 10.5

ᅟ:

βsA = 0.000739,βsB = 3.9,βsC = 21.8

ᅟ:

gating coefficients α∗A, β∗A in ms− 1

ᅟ:

gating coefficients α∗B, β∗B, α∗C, β∗C in mV

ᅟ:

sodium permeability, PNa = 7.04 × 10− 3 cm/s

ᅟ:

potassium conductance (fast), gKf = 0.015 S/cm2

ᅟ:

potassium conductance (slow), gKf = 0.030 S/cm2

ᅟ:

leakage conductance, gL = 60 × 10− 3 S/cm2

ᅟ:

sodium concentration outside, [Na]o = 154 mM

ᅟ:

sodium concentration inside, [Na]i = 35 mM

ᅟ:

potassium concentration outside, [K]o = 5.6 mM

ᅟ:

potassium concentration inside, [K]i = 155 mM

ᅟ:

sodium equilibrium potential, VNa = − 84 mV

ᅟ:

potassium equilibrium potential, VK = + 60 mV

ᅟ:

resting membrane potential, Vr = − 84 mV

ᅟ:

Faraday constant, F = 96485 C/mol

ᅟ:

gas constant, R = 8.3144 J/Kmol

ᅟ:

absolute temperature, T = 310.15 K

ᅟ:

membrane potential, V mV

Membrane currents

ᅟ:

sodium current, INa mA/cm2

ᅟ:

potassium current (fast), IKf mA/cm2

ᅟ:

potassium current (slow), IKs mA/cm2

ᅟ:

leakage current, IL mA/cm2

ᅟ:

\(I_{Na} = m^{3}hP_{\text {Na}}\frac {VF^{2}}{RT}\frac {([\text {Na}]_{\mathrm {o}} - [\text {Na}]_{\mathrm {i}}e^{VF/RT})}{(1 - e^{VF/RT})}\)

ᅟ:

IKf = n4gKf(VVK)

ᅟ:

IKs = sgKs(VVK)

ᅟ:

IL = gL(VVL)

Simulation parameters

ᅟ:

simulation duration, tstop = 30 s

ᅟ:

simulation step size, tstep = 0.001 ms

ᅟ:

simulation repeats, nrep = 50

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Sadashivaiah, V., Sacré, P., Guan, Y. et al. Modeling the interactions between stimulation and physiologically induced APs in a mammalian nerve fiber: dependence on frequency and fiber diameter. J Comput Neurosci 45, 193–206 (2018). https://doi.org/10.1007/s10827-018-0703-y

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Keywords

  • Action potential interactions
  • Conduction reliability
  • Nerve fiber
  • Mechanistic model
  • Electrical stimulation