Hodgkin Huxley Model

Famous and standard model for Neuronal Current prediction. While quite complicated for large population models, HH does a great job at modeling single neurons by directly capturing the biophysics of the Potassium and Sodium channels.

$$\{\begin{array}{l}{C}_m\frac{dV}{dt}={\bar{g}}_{Na}m{}^{3}h(E_{Na}-V)\\ +{\bar{g}}_{K}n{}^4(E_K-V)+g_L(E_L-V)\\ \frac{dm}{dt}={\alpha }_m(V)(1-m)-{\beta }_m(V)m\\ \frac{dh}{dt}={\alpha }_h(V)(1-h)-{\beta }_h(V)h\\ \frac{dn}{dt}={\alpha }_n(V)(1-n)-{\beta }_n(V)n\end{array}$$

Membrane Voltage

$$\{\begin{array}{l}\\ C_m\frac{dV}{dt}=\stackrel{I_{N{a}^{+}}}{\overbrace{{\bar{g}}_{Na}m{}^{3}h(E_{Na}-V)}}\\ +\underset{I_{K^{+}}}{\underbrace{{\bar{g}}_{K}n{}^4(E_K-V)}}+\underset{I_{leak}}{\underbrace{g_L(E_L-V)}}\\ \end{array}$$

Gates