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.

Appendix

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

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

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ratio of axon to fiber diameter, r_dD = 1

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fiber diameter, D = d × r_dD

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ratio of l_i to fiber diameter, $r_{\mathrm {l_{i}D}} = 100$

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

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fiber length, L = 10 cm

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nodal length, l = 2.5 μm

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

Parameters

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α_mA = 1.86,α_mB = 65.6,α_mC = 10.3

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α_hA = 0.0336,α_hB = − 27,α_hC = 11.0

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α_nA = 0.00789,α_nB = − 9.2,α_nC = 1.10

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α_sA = 0.00122,α_sB = 71.5,α_sC = 23.6

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β_mA = 0.0860,β_mB = 61.3,β_mC = 9.16

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β_hA = 2.30,β_hB = 55.2,β_hC = 13.4

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β_nA = 0.0142,β_nB = 8,β_nC = 10.5

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β_sA = 0.000739,β_sB = 3.9,β_sC = 21.8

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gating coefficients α_∗A, β_∗A in ms^− 1

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gating coefficients α_∗B, β_∗B, α_∗C, β_∗C in mV

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

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

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

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

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sodium concentration outside, [Na]_o = 154 mM

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sodium concentration inside, [Na]_i = 35 mM

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potassium concentration outside, [K]_o = 5.6 mM

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potassium concentration inside, [K]_i = 155 mM

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sodium equilibrium potential, V_Na = − 84 mV

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potassium equilibrium potential, V_K = + 60 mV

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resting membrane potential, V_r = − 84 mV

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Faraday constant, F = 96485 C/mol

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gas constant, R = 8.3144 J/Kmol

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absolute temperature, T = 310.15 K

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membrane potential, V mV

Membrane currents

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sodium current, I_Na mA/cm²

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potassium current (fast), I_Kf mA/cm²

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potassium current (slow), I_Ks mA/cm²

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leakage current, I_L mA/cm²

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

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I_Kf = n⁴g_Kf(V − V_K)

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I_Ks = sg_Ks(V − V_K)

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I_L = g_L(V − V_L)

Simulation parameters

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simulation duration, t_stop = 30 s

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simulation step size, t_step = 0.001 ms

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simulation repeats, n_rep = 50