General functions - NRVM CaMKII model

Hill equation.

H(x, k) := \frac{x}{x + k}

H(x, k, n) := \frac{x^n}{x^n + k^n}

Relative exponential function. The definition here is reciprocal to the Python one.

exprel(x) := \frac{x}{ \exp(x) - 1}

Logistic function

expit(x) := \frac{1}{1 + \exp(-x)}

GHK flux equation.

GHK(p, z, V, S_i, S_o) := pzF\delta\frac{\exp(\delta)S_i - S_o}{\exp(\delta) - 1} ; \quad \delta = \frac{zVF}{RT}

Ion concentrations:

\begin{align} na_i &= \mathrm{[Na^+]_i} \\ k_i &= \mathrm{[K^+]_i} \\ ca_i &= \mathrm{[Ca^{2+}]_i} \\ na_o &= \mathrm{[Na^+]_o} \\ k_o &= \mathrm{[K^+]_o} \\ ca_o &= \mathrm{[Ca^{2+}]_o} \\ \end{align}

Reversal potentials:

\begin{align} E_{Na} &= V_T \ln \frac{na_o}{na_i} \\ E_{K } &= V_T \ln\frac{k_o}{k_i} \\ E_{Ca} &= 0.5V_T \ln \frac{ca_o}{ca_{sl}} \\ E_{Kr} &= V_T \ln\left( \frac{0.98 k_o + 0.02na_o}{0.98 k_i + 0.02a_i} \right) \\ \end{align}