Influence of frequency and temperature on the mechanisms of nerve conduction block induced by high-frequency biphasic electrical current

Abstract

The influences of stimulation frequency and temperature on mechanisms of nerve conduction block induced by high-frequency biphasic electrical current were investigated using a lumped circuit model of the myelinated axon based on Schwarz and Eikhof (SE) equations. The simulation analysis showed that a temperature–frequency relationship was determined by the axonal membrane dynamics (i.e. how fast the ion channels can open or close.). At a certain temperature, the axonal conduction block always occurred when the period of biphasic stimulation was smaller than the action potential duration (APD). When the temperature decreased from 37 to 15°C, the membrane dynamics slowed down resulting in an APD increase from 0.4 to 2.4 ms accompanied by a decrease in the minimal blocking frequency from 4 to 0.5 kHz. The simulation results also indicated that as the stimulation frequency increased the mechanism of conduction block changed from a cathodal/anodal block to a block dependent upon continuous activation of potassium channels. Understanding the interaction between the minimal blocking frequency and temperature could promote a better understanding of the mechanisms of high frequency induced axonal conduction block and the clinical application of this method for blocking nerve conduction.

Appendix

The ionic current I _i,,n at nth node is described as:

$$ \begin{array}{*{20}c} {I_{{i,n}} = i_{{Na}} + i_{K} + i_{L} } \\ {i_{{Na}} = P_{{Na}} m^{3} h\frac{{EF^{2} }} {{RT}}\frac{{[{\text{Na}}]_{0} - [{\text{Na}}]_{i} e^{{{EF} \mathord{\left/ {\vphantom {{EF} {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} }} {{1 - e^{{{EF} \mathord{\left/ {\vphantom {{EF} {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} }}} \\ {i_{K} = P_{K} n^{2} \frac{{EF^{2} }} {{RT}}\frac{{[K]_{0} - [K]_{i} e^{{{EF} \mathord{\left/ {\vphantom {{EF} {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} }} {{1 - e^{{{EF} \mathord{\left/ {\vphantom {{EF} {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} }}} \\ {i_{L} = g_{L} (V_{n} - V_{L} )} \\ {E = V_{n} + V_{{{\text{rest}}}} } \\ \end{array} $$

where P _Na (0.00328 cm/s) and P _K (0.000134 cm/s) are the ionic permeabilities for sodium and potassium currents, respectively; g _L (86 kΩ⁻¹ cm⁻²) is the maximum conductance for leakage current. V_L (0 mV) is the reduced equilibrium membrane potential for leakage ions, in which the resting membrane potential V _rest (−78 mV) has been subtracted. [Na] _i (8.71 mmol/l) and [Na] _o (154 mmol/l) are sodium concentrations inside and outside the axon membrane. [K] _i (155 mmol/l) and [K] _o (5.9 mmol/l) are potassium concentrations inside and outside the axon membrane. F (96,485 C/mole) is Faraday constant. R (8,314.4 mJ K⁻¹ mol⁻¹) is gas constant. m, h and n are dimensionless variables, whose values always change between 0 and 1. m and h represent activation and inactivation of sodium channels, whereas n represents activation of potassium channels.

The evolution equations for m, h and n are the following:

$$ \begin{array}{*{20}c} {{dm} \mathord{\left/ {\vphantom {{dm} {dt}}} \right. \kern-\nulldelimiterspace} {dt} = {\left[ {\alpha _{m} {\left( {1 - m} \right)} - \beta _{m} m} \right]}k_{m} } \\ {{dh} \mathord{\left/ {\vphantom {{dh} {dt}}} \right. \kern-\nulldelimiterspace} {dt} = {\left[ {\alpha _{h} {\left( {1 - h} \right)} - \beta _{h} h} \right]}k} \\ {{dn} \mathord{\left/ {\vphantom {{dn} {dt}}} \right. \kern-\nulldelimiterspace} {dt} = {\left[ {\alpha _{n} {\left( {1 - n} \right)} - \beta _{n} n} \right]}k} \\ \end{array} $$

and

$$ \begin{array}{*{20}c} {\alpha _{m} = \frac{{1.87{\left( {V_{n} - 25.41} \right)}}} {{1 - \exp {\left( {\frac{{25.41 - V_{n} }} {{6.06}}} \right)}}}} \\ {\beta _{m} = \frac{{3.97{\left( {21 - V_{n} } \right)}}} {{1 - \exp {\left( {\frac{{V_{n} - 21}} {{9.41}}} \right)}}}} \\ {\alpha _{h} = - \frac{{0.55{\left( {V_{n} + 27.74} \right)}}} {{1 - \exp {\left( {\frac{{V_{n} + 27.74}} {{9.06}}} \right)}}}} \\ {\beta _{h} = \frac{{22.6}} {{1 + \exp {\left( {\frac{{56 - V_{n} }} {{12.5}}} \right)}}}} \\ {\alpha _{n} = \frac{{0.13{\left( {V_{n} - 35} \right)}}} {{1 - \exp {\left( {\frac{{35 - V_{n} }} {{10}}} \right)}}}} \\ {\beta _{n} = \frac{{0.32{\left( {10 - V_{n} } \right)}}} {{1 - \exp {\left( {\frac{{V_{n} - 10}} {{10}}} \right)}}}} \\ {k_{m} = 2.2^{{{{\left( {T - 310} \right)}} \mathord{\left/ {\vphantom {{{\left( {T - 310} \right)}} {10}}} \right. \kern-\nulldelimiterspace} {10}}} } \\ {k = 3^{{{{\left( {T - 310} \right)}} \mathord{\left/ {\vphantom {{{\left( {T - 310} \right)}} {10}}} \right. \kern-\nulldelimiterspace} {10}}} } \\ \end{array} $$

where T is the temperature used in the simulation study (in °K). The initial values for m, h and n (when V _n  = 0 mV) are 0.0077, 0.76 and 0.0267, respectively.