Equation 3 · What a Neural Recording Actually Records
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol t
t is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol h
known, that it is the same across regions, subjects and states, and that the mapping from n to y is linear.
Symbol τ
τ is one of the signed contributions combined to compute the quantity on the left.
Symbol n
n is one of the signed contributions combined to compute the quantity on the left.
Symbol d
d is one of the signed contributions combined to compute the quantity on the left.
Symbol varepsilon
varepsilon is one of the signed contributions combined to compute the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subtraction
Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →Starting index or lower bound: 0
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.
Ending index or upper bound: infty
This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.
How to interpret it
Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
Nearly all standard analysis assumes that neural activity and the measured signal are related by a linear time-invariant system, so that the measured time course is a convolution of an underlying activity time course with a canonical response function plus noise: . Writing it this way exposes what is being assumed rather than measured: that h is known, that it is the same across regions, subjects and states, and that the mapping from n to y is linear. Each of those is an approximation with documented exceptions.
Sources cited in the article section
- [10] Neurophysiological investigation of the basis of the fMRI signal ↗
- [11] What we can do and what we cannot do with fMRI ↗
These citations give research context. Read each source to check which claims it supports.
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