This document will provide specific details of 2D-Gaussian equations
used by the different method options within
gaussplotR::fit_gaussian_2D().
method = "elliptical"Using method = "elliptical" fits a two-dimensional,
elliptical Gaussian equation to gridded data.
G(x, y) = Ao + A * e−U/2
where G is the value of the 2D-Gaussian at each (x, y) point, Ao is a constant term, and A is the amplitude (i.e. scale factor).
The elliptical function, U, is:
U = (x′/a)2 + (y′/b)2
where a is the spread of Gaussian along the x-axis and b is the spread of Gaussian along the y-axis.
x′ and y′ are defined as:
x′ = (x − x0)cos(θ) − (y − y0)sin(θ) y′ = (x − x0)sin(θ) + (y − y0)cos(θ) where x0 is the center (peak) of the Gaussian along the x-axis, y0 is the center (peak) of the Gaussian along the y-axis, and θ is the rotation of the ellipse from the x-axis in radians, counter-clockwise.
Therefore, all together:
G(x, y) = Ao + A * e−((((x − x0)cos(θ) − (y − y0)sin(θ))/a)2 + (((x − x0)sin(θ) + (y − y0)cos(θ))/b)2)/2
Setting the constrain_orientation argument to a numeric
will optionally constrain the value of θ to a user-specified value. If a
numeric is supplied here, please note that the value will be interpreted
as a value in radians. Constraining θ to a user-supplied value can lead
to considerably poorer-fitting Gaussians and/or trouble with converging
on a stable solution; in most cases constrain_orientation
should remain its default: "unconstrained".
method = "elliptical_log"The formula used in method = "elliptical_log" uses the
modification of a 2D Gaussian fit used by Priebe et al. 20031.
G(x, y) = A * e(−(x − x0)2)/σx2 * e(−(y − y′(x)))/σy2
and
y′(x) = 2(Q + 1) * (x − x0) + y0 where A is the amplitude (i.e. scale factor), x0 is the center (peak) of the Gaussian along the x-axis, y0 is the center (peak) of the Gaussian along the y-axis, σx is the spread along the x-axis, σy is the spread along the y-axis and Q is an orientation parameter.
Therefore, all together:
G(x, y) = A * e(−(x − x0)2)/σx2 * e(−(y − (2(Q + 1) * (x − x0) + y0)))/σy2
This formula is intended for use with log2-transformed data.
Setting the constrain_orientation argument to a numeric
will optionally constrain the value of Q to a user-specified value, which
can be useful for certain kinds of analyses (see Priebe et al. 2003 for
more). Keep in mind that constraining Q to a user-supplied value can lead
to considerably poorer-fitting Gaussians and/or trouble with converging
on a stable solution; in most cases constrain_orientation
should remain its default: "unconstrained".
method = "circular"This method uses a relatively simple formula:
G(x, y) = A * e(−(((x − x0)2/2σx2) + ((y − y0)2/2σy2)))
where A is the amplitude (i.e. scale factor), x0 is the center (peak) of the Gaussian along the x-axis, y0 is the center (peak) of the Gaussian along the y-axis, σx is the spread along the x-axis, and σy is the spread along the y-axis.
That’s all!
🐢
Priebe NJ, Cassanello CR, Lisberger SG. The neural representation of speed in macaque area MT/V5. J Neurosci. 2003 Jul 2;23(13):5650-61. doi: 10.1523/JNEUROSCI.23-13-05650.2003.↩︎