It is assumed that the ions will behave statistically, such that it diffuses and electrically attracted. The ion gates on the membrane will also follow the random statistical process in which it will open and closes at a rate \( \alpha \), \( \beta \) that are dependent on the membrane voltage.

\begin{align*} \frac{1}{2\pi a}\frac{\partial}{\partial x}\left(\frac{\pi a^2}{R_a} \frac{\partial V}{\partial x}\right) &= C_m \frac{\partial V}{\partial t} + \bar{g}_\mathrm{Na} m^3 h (V - E_\mathrm{Na}) + \bar{g}_\mathrm{K} n^4 h(V-E_\mathrm{K}) + g_\text{leak}(V-E_\text{leak}) \\ \frac{\mathrm{d}n}{\mathrm{d}t} &= \alpha_n(V)(1-n) - \beta_n(V)n\\ \frac{\mathrm{d}m}{\mathrm{d}t} &= \alpha_m(V)(1-m) - \beta_m(V)m\\ \frac{\mathrm{d}h}{\mathrm{d}t} &= \alpha_h(V)(1-h) - \beta_h(V)h\\ \end{align*}

• \( a \) is the radius of the neural path and \( R_a \) is the axial resistance of it.
• The \( \alpha \) and \( \beta \) are measured experimentally.
• \( n \), \( m \), \( h \) are the fraction of gates of the type that are open.
• \( \bar{g}_\mathrm{Na} \), \( \bar{g}_\mathrm{K} \) and, \( g_\text{leak} \) are constants conductance measured when every gates are open.
• The equilibrium potentials \( E_\mathrm{Na} \), \( E_\mathrm{K} \) are given by the Goldman equation.

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