Definitions

• (Current/Charge/Current Source) Density
• Kernel Density

The pykCSD project requires good knowledge of linear algebra, calculus, statistics and (bio)electromagnetism. Trying to fully understand the Kernel Current Source Density method I found few misleading definitions more or less connected with the topic. I would like to keep them here for the future reference.

(Current/Charge/Current Source) Density

Current Density ¹ $\vec{J}$ is the electric current $\Delta I$ (moving charge) per unit area S

$$\vec{J} = \lim_{S \rightarrow 0} \frac{\Delta I(S) \widehat{a}_{max}}{S} \left[ \frac{A}{m^2}\right], \text{ where } \Delta I = \lim_{\Delta t \rightarrow 0}\frac{\Delta Q}{\Delta t}$$

which is a vector.

Charge Density is electric charge per unit volume (or area or line)

$$\rho_q = \lim_{V \rightarrow 0} \frac{Q}{V} \left[ \frac{C}{m^3}\right]$$

which is a scalar.

Current Source Density ² has the meaning of volume density of current entering or leaving the medium at some point in space.

$$C(r,t) = \sum_{n=1}^N I_n(t)\delta ^3(r-r_n) \left[\frac{A}{m^3} \right]$$

and

$$\phi(r, t) = \frac{1}{4 \pi \sigma} \iiint \frac{ C(r', t) }{ \lvert r-r' \rvert } d^3 r'$$

Kernel Density

Despite its name, this is not directly related to the kCSD method.

• http://www.mathworks.com/help/stats/kernel-distribution.html
• http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/AV0405/MISHRA/kde.html
• http://scikit-learn.org/stable/modules/density.html
• http://en.wikipedia.org/wiki/Kernel_density_estimation