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See Neuron action potential

Sodium, Potassium, and other Ions.

$C \frac{dV}{dt}+I_{ions}=0$

$I_{ions}=\hat{g}_{K}(V-V_K)+\hat{g}_{Na}(V-V_{Na})+\hat{g_{L}}(V-V_L)$

$\hat{g}_{Na}=g_{Na} m^3 h (V-V_{Na})$. $3m$ gates, $1h$ gates

$\hat{g}_{K}=g_{Na} n^4 (V-V_{K})$. $4n$ gates. Fraction of the channels in which 4 gates are open, that is why we have the $n^4$.

Equations for gates m,h, n too (see Neuron action potential).

There is a simplified model: FitzHugh–Nagumo model

Can explore the model with no spatial variation using the patch clamp technique (no?), where we insert a conductive wire which equilibrates the potential along the axon