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Neuroscience Equations

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

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

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

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

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

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

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

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



Subsections
next up previous contents
Next: A Single compartment Up: From Numerical Theory to Previous: The Trapezoidal Rule   Contents
2002-11-15