Neuroscience Equations

The membrane potential ($V$) in a single branch of a neuron is described by the one dimensional cable equation:

$$\frac{1}{2\pi a}\frac{\partial}{\partial x} \left( \frac{\pi a^2}... ...al V}{\partial x} \right) = C_m \frac{\partial V}{\partial t} + I_{\mathrm{HH}}$$ (1.7)

If we consider the axial resistance and dendritic diameter constant, this is simplified to a cylindrical cable equation:

$$\frac{a}{2R_a}\frac{\partial^2V}{\partial x^2} = C_m \frac{\partial V}{\partial t} + I_{\mathrm{HH}}$$ (1.8)

In the the Hodgkin-Huxley formalism, the current is defined as:

$$I_{\mathrm{HH}} = \overline{g_{\mathrm{Na}}}m^3h(V-E_{\mathrm{Na}... ...verline{g_{\mathrm{K}}}n^4(V-E_{\mathrm{K}}) + g_{\mathrm{L}}(V-E_{\mathrm{L}})$$ (1.9)

$m$, $h$ and $n$ are voltage and time dependent variables between 0 and 1, each satisfying a simple exponential curve, described by:

$$\frac{\mathrm{d}h}{\mathrm{d}t} = \alpha_h(V) - (\alpha_h(V) + \beta_h(V)) \cdot h$$ (1.10)