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Table 1 Mathematical expressions and labels of the pulse shapes applied in the experiment. The pulse shapes originating from the provided equations are illustrated

From: Preferential activation of small cutaneous fibers through small pin electrode also depends on the shape of a long duration electrical current

Pulse shape

Equations

Label

Illustration

Exponential increase

\(i\left( t \right) = \left\{ {\begin{array}{*{20}l} {\frac{{I_{s} }}{{e^{{\frac{{T_{S} }}{\tau }}} - 1}}\left( {e^{{\frac{t}{\tau }}} - 1} \right),\quad 0 \le t < Ts } \\ {I_{s} \cdot e^{{\frac{ - t}{{\tau_{tr} }}}} , \qquad \qquad T_{s} \le t \le T_{tr} } \\ \end{array} } \right.\)

Exp

Linear increase

\(i\left( t \right) = \left\{ {\begin{array}{*{20}l} { \frac{{I_{s} }}{{T_{s} }} \cdot t, \qquad 0 \le t < Ts } \\ {I_{s} \cdot e^{{\frac{ - t}{{\tau_{tr} }}}} , \quad T_{s} \le t \le T_{tr} } \\ \end{array} } \right.\)

Lin

Bounded exponential

\(i\left( t \right) = \left\{ {\begin{array}{*{20}l} { \frac{{I_{s} }}{{1 - e^{{\frac{{ - T_{S} }}{\tau }}} }}\left( {1 - e^{{\frac{ - t}{\tau }}} } \right), \quad 0 \le t < Ts} \\ {I_{s} \cdot e^{{\frac{ - t}{{\tau_{tr} }}}} , \,\,\,\, \qquad \qquad T_{s} \le t \le T_{tr} } \\ \end{array} } \right.\)

B.Exp

Rectangular

\(i\left( t \right) = I_{s} \left( t \right), \quad 0 \le t < Ts\)

Rec

  1. \(I_{s}\)  stimulation current, \(T_{S}\)  stimulation duration, \(\tau\)  (\(T_{S}\) /2)  time constant. Trailing phase: \(T_{tr}\)  \(T_{S} *\) 1.4 and \(\tau_{tr}\)  \(\tau\) /6.6