Table 4. Mathematical empirical cat relationships between the motoneuron (MN) properties ${\mathit{S}}_{\mathit{n}\mathit{e}\mathit{u}\mathit{r}\mathit{o}\mathit{n}},\mathit{}{\mathit{D}}_{\mathit{s}\mathit{o}\mathit{m}\mathit{a}},\mathit{}\mathit{R},\mathit{}{\mathit{R}}_{\mathit{m}},\mathit{}\mathit{C},\mathit{}\mathit{\tau },\mathit{}{\mathit{I}}_{\mathit{t}\mathit{h}},$$\mathit{A}\mathit{H}\mathit{P}$, and $\mathit{A}\mathit{C}\mathit{V}$.

Each column provides the relationships between one and the eight other MN properties. If one property is known, the complete MN profile can be reconstructed by using the pertinent line in this table. All constants and properties are provided in SI base units (metres, seconds, ohms, farads, and amperes). The relationships involving $R_m$ were obtained from theoretical relationships involved in Rall’s cable theory (see ‘Discussion’).

	${\displaystyle S_{neuron}\left[{\mathrm{m}}^2\right]}$	${\displaystyle D_{soma}\left[\mathrm{m}\right]}$	${\displaystyle R\left[\mathrm{\Omega }\right]}$	${\displaystyle R_m\left[\mathrm{\Omega }\cdot {\mathrm{m}}^2\right]}$	${\displaystyle C\left[\mathrm{F}\right]}$	${\displaystyle \tau \left[\mathrm{s}\right]}$	${\displaystyle I_{th}\left[\mathrm{A}\right]}$	${\displaystyle AHP\left[\mathrm{s}\right]}$	${\displaystyle ACV\left[\mathrm{m}\cdot {\mathrm{s}}^{-1}\right]}$
${\displaystyle S_{neuron}\left[{\mathrm{m}}^2\right]}$		${\displaystyle D_{soma}=1.8\cdot {10}^2\cdot S_{neuron}}$	${\displaystyle R=\frac{1.7\cdot {10}^{-10}}{S_{neuron}^{2.43}}}$	${\displaystyle R_m=\frac{1.0\cdot {10}^{-10}}{S_{neuron}^{1.43}}}$	${\displaystyle C=1.3\cdot {10}^{-2}\cdot S_{neuron}}$	${\displaystyle \tau =\frac{1.0\cdot {10}^{-12}}{S_{neuron}^{1.48}}}$	${\displaystyle I_{th}=3.8\cdot {10}^8\cdot S_{neuron}^{2.52}}$	${\displaystyle AHP=\frac{1.0\cdot {10}^{-11}}{S_{neuron}^{1.51}}}$	${\displaystyle ACV=3.0\cdot {10}^6\cdot S_{neuron}^{0.69}}$
${\displaystyle D_{soma}\left[\mathrm{m}\right]}$	${\displaystyle S_{neuron}=5.5\cdot {10}^{-3}\cdot D_{soma}}$		${\displaystyle R=\frac{5.1\cdot {10}^{-5}}{D_{soma}^{2.43}}}$	${\displaystyle R_m=\frac{1.7\cdot {10}^{-7}}{D_{soma}^{1.43}}}$	${\displaystyle C=7.9\cdot {10}^{-5}\cdot D_{soma}}$	${\displaystyle \tau =\frac{2.3\cdot {10}^{-9}}{D_{soma}^{1.48}}}$	${\displaystyle I_{th}=7.8\cdot {10}^2\cdot D_{soma}^{2.52}}$	${\displaystyle AHP=\frac{2.7\cdot {10}^{-8}}{D_{soma}^{1.51}}}$	${\displaystyle ACV=8.1\cdot {10}^4\cdot D_{soma}^{0.69}}$
${\displaystyle {\displaystyle R\left[\mathrm{\Omega }\right]}}$	${\displaystyle S_{neuron}=\frac{9.5\cdot {10}^{-5}}{R^{0.41}}}$	${\displaystyle D_{soma}=\frac{1.7\cdot {10}^{-2}}{R^{0.41}}}$		${\displaystyle R_m=5.7\cdot {10}^{-5}\cdot R^{0.59}}$	${\displaystyle C=\frac{1.5\cdot {10}^{-6}}{R^{0.40}}}$	${\displaystyle \tau =9.6\cdot {10}^{-7}\cdot R^{0.61}}$	${\displaystyle I_{th}=\frac{2.7\cdot {10}^{-2}}{R^{1.04}}}$	${\displaystyle AHP=1.3\cdot {10}^{-5}\cdot R^{0.62}}$	${\displaystyle ACV=\frac{4.9\cdot {10}^3}{R^{0.29}}}$
${\displaystyle {\displaystyle R_m\left[\mathrm{\Omega }\cdot {\mathrm{m}}^2\right]}}$	${\displaystyle S_{neuron}=\frac{1.5\cdot {10}^{-7}}{R_m^{0.70}}}$	${\displaystyle D_{soma}=\frac{2.6\cdot {10}^{-5}}{R_m^{0.70}}}$	${\displaystyle R=6.8\cdot {10}^6\cdot R_m^{1.70}}$		${\displaystyle C=\frac{1.8\cdot {10}^{-9}}{R_m^{0.69}}}$	${\displaystyle \tau =1.4\cdot {10}^{-2}\cdot R_m^{1.04}}$	${\displaystyle I_{th}=\frac{2.2\cdot {10}^{-9}}{R_m^{1.76}}}$	${\displaystyle AHP=2.2\cdot {10}^{-1}\cdot R_m^{1.06}}$	${\displaystyle ACV=\frac{5.4\cdot {10}^1}{R_m^{0.48}}}$
${\displaystyle {\displaystyle C\left[\mathrm{F}\right]}}$	${\displaystyle S_{neuron}=1.2\cdot {10}^2\cdot C}$	${\displaystyle D_{soma}=2.1\cdot {10}^4\cdot C}$	${\displaystyle R=\frac{1.5\cdot {10}^{-15}}{C^{2.48}}}$	${\displaystyle R_m=\frac{1.7\cdot {10}^{-13}}{C^{1.46}}}$		${\displaystyle \tau =\frac{8.6\cdot {10}^{-16}}{C^{1.51}}}$	${\displaystyle I_{th}=6.5\cdot {10}^{13}\cdot C^{2.57}}$	${\displaystyle AHP=\frac{7.7\cdot {10}^{-15}}{C^{1.54}}}$	${\displaystyle ACV=8.2\cdot {10}^7\cdot C^{0.71}}$
${\displaystyle \tau \left[\mathrm{s}\right]}$	${\displaystyle S_{neuron}=\frac{8.4\cdot {10}^{-9}}{{\tau }^{0.67}}}$	${\displaystyle D_{soma}=\frac{1.5\cdot {10}^{-6}}{{\tau }^{0.67}}}$	${\displaystyle R=7.1\cdot {10}^9\cdot {\tau }^{1.64}}$	${\displaystyle R_m=3.6\cdot {10}^1\cdot {\tau }^{0.96}}$	${\displaystyle C=\frac{1.6\cdot {10}^{-10}}{{\tau }^{0.66}}}$		${\displaystyle I_{th}=\frac{1.6\cdot {10}^{-12}}{{\tau }^{1.70}}}$	${\displaystyle AHP=1.7\cdot {10}^1\cdot {\tau }^{1.02}}$	${\displaystyle ACV=\frac{7.5}{{\tau }^{0.47}}}$
${\displaystyle {\displaystyle I_{th}\left[\mathrm{A}\right]}}$	${\displaystyle S_{neuron}=4.0\cdot {10}^{-4}\cdot I_{th}^{0.40}}$	${\displaystyle D_{soma}=7.1\cdot {10}^{-2}\cdot I_{th}^{0.40}}$	${\displaystyle R=\frac{3.1\cdot {10}^{-2}}{I_{th}^{0.96}}}$	${\displaystyle R_m=\frac{7.4\cdot {10}^{-6}}{I_{th}^{0.57}}}$	${\displaystyle C=5.9\cdot {10}^{-6}\cdot I_{th}^{0.39}}$	${\displaystyle \tau =\frac{1.2\cdot {10}^{-7}}{I_{th}^{0.59}}}$		${\displaystyle AHP=\frac{1.5\cdot {10}^{-6}}{I_{th}^{0.60}}}$	${\displaystyle ACV=1.3\cdot {10}^4\cdot I_{th}^{0.27}}$
${\displaystyle {\displaystyle AHP\left[\mathrm{s}\right]}}$	${\displaystyle S_{neuron}=\frac{5.5\cdot {10}^{-8}}{AH{P}^{0.66}}}$	${\displaystyle D_{soma}=\frac{9.8\cdot {10}^{-6}}{AH{P}^{0.66}}}$	${\displaystyle R=7.5\cdot {10}^7\cdot AH{P}^{1.61}}$	${\displaystyle R_m=2.5\cdot AH{P}^{0.95}}$	${\displaystyle C=\frac{9.7\cdot {10}^{-10}}{AH{P}^{0.65}}}$	${\displaystyle \tau =6.2\cdot {10}^{-2}\cdot AH{P}^{0.98}}$	${\displaystyle I_{th}=\frac{1.8\cdot {10}^{-10}}{AH{P}^{1.67}}}$		${\displaystyle ACV=\frac{2.7\cdot {10}^1}{AH{P}^{0.46}}}$
${\displaystyle {\displaystyle ACV\left[\mathrm{m}\cdot {\mathrm{s}}^{-1}\right]}}$	${\displaystyle S_{neuron}=4.6\cdot {10}^{-10}\cdot AC{V}^{1.44}}$	${\displaystyle D_{soma}=8.3\cdot {10}^{-8}\cdot AC{V}^{1.44}}$	${\displaystyle R=\frac{8.3\cdot {10}^{12}}{AC{V}^{3.50}}}$	${\displaystyle R_m=\frac{2.3\cdot {10}^3}{AC{V}^{2.06}}}$	${\displaystyle C=9.0\cdot {10}^{-12}\cdot AC{V}^{1.41}}$	${\displaystyle \tau =\frac{7.4\cdot {10}^1}{AC{V}^{2.14}}}$	${\displaystyle I_{th}=1.1\cdot {10}^{-15}\cdot AC{V}^{3.64}}$	${\displaystyle AHP=\frac{1.4\cdot {10}^3}{AC{V}^{2.18}}}$