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Equation 2 · Part 14 · What a Neural Recording Actually Records

Numerator: I_n(t)

ϕ(r,t)=14πσ∑nIn(t)∥r−rn∥\phi(\mathbf{r},t) = \frac{1}{4\pi\sigma}\sum_{n}\frac{I_n(t)}{\lVert \mathbf{r}-\mathbf{r}_n \rVert}
In(t)I_n(t)

What this part means

The complete quantity above the fraction bar.

Its job in the formula

In(t)I_n(t) occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

Start with the most direct measurement available. A metal or silicon electrode placed in the extracellular space senses the potential produced by transmembrane currents in the surrounding tissue. Under the standard volume-conductor treatment, with the medium approximated as homogeneous, isotropic and purely resistive, and each current source treated as a point, the potential at position r\mathbf{r} is a superposition: ϕ(r,t)=14πσ∑nIn(t)∥r−rn∥\phi(\mathbf{r},t) = \frac{1}{4\pi\sigma}\sum_{n}\frac{I_n(t)}{\lVert \mathbf{r}-\mathbf{r}_n \rVert}. Three assumptions are visible in that expression, and all three are approximations rather than facts. The medium is not homogeneous. Conductivity may be frequency-dependent. And the sum runs over every current source in range, not over the neuron of…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the article section

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