For my master's project (mathematical modelling of cardiac cells) I need to solve a system of highly nonlinear equations numerically.
I've tried MATLAB, MAPLE, C++, even WolframAlpha and I'm at a total loss and beginning to panic.
Here is a link to the equations. https://i.stack.imgur.com/40M8l.jpg I'm really desperate.... please help!
Please verify these - they are impossible to read with such a low quality image.
Solve for the initial values $V_0, n_0, m_0$ and $h_0$ given by
$$\begin{align} \tag{1} n_0 &= \dfrac{\alpha_{n_0}}{\alpha_{ n_0} + \beta_{n_0}} \\ \tag{2} m_0 &= \dfrac{\alpha_{m_0}}{\alpha_{ m_0} + \beta_{ m_0}} \\ \tag{3} h_0 &= \dfrac{\alpha_{ h_0}}{\alpha_{ h_0} + \beta_{ h_0}} \\ \tag{4} V_0 &= \dfrac{I + \overline{g_K}~ n_0^4~ V_k + \overline{g_{N_a}}~ m_0^3 ~h_0~ V_{N_a} + g_l~ V_l}{\overline{g_K}~ n_0^4 + \overline{g_{N_a}}~ m_0^3~ h_0 +~ g_l}\end{align} $$
where
$$\begin{align} \tag{5} \alpha_{n_0} &= \dfrac{0.01(10 - V_0)}{\large e^{\frac{10-V_0}{10}}-1 } \\ \tag{5} \beta_{n_0} &= \large \dfrac{0.125}{e^{\frac{V_0}{80}}} \\ \tag{6} \alpha_{m_0} &= \dfrac{0.01(25 - V_0)}{\large e^{\frac{25-V_0}{10}}-1 }\\ \tag{6} \beta_{m_0} &= \large \dfrac{4}{e^{\frac{V_0}{18}}} \\ \tag{7} \alpha_{h_0} &= \dfrac{0.07}{\large e^{\frac{V_0}{20}}}\\ \tag{7} \beta_{h_0} &= \large \dfrac{1}{e^{\frac{30-V_0}{10}}+1} \\ \end{align}$$
and $I, \overline{g_K}, \overline{g_{N_a}}, g_l, V_K, V_{N_a}, V_l$ are constants. Example values
$$\begin{align} \tag{8} I &= 0.0 \\ \tag{9} \overline{g_K} &= 36.0 \\ \tag{10} \overline{ g_{N_a}} &= 120.0 \\ \tag{11} g_l &= 0.3 \\ \tag{12} V_K &= 12.0 \\ \tag{13} V_{N_a} &= 115.0 \\ \tag{14} V_l &= 10.6 \end{align}$$