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Published equation contexts

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}

Why this formula appears here

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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π\pi

Symbol pi

pi occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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σ\sigma

Symbol σ

σ occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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nn

Symbol n

n appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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nn

Starting index or lower bound: n

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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∥r−rn∥\lVert \mathbf{r}-\mathbf{r}_n \rVert

Denominator: lVert r-r_n rVert

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

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}

Equation 2 · Neuroscience

What a Neural Recording Actually Records

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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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