Some scaling facts (verbal from chuck, not double checked)
=~ means "proportional to" ~=~ means "almost proportional to" for example, if X =~ Y^.94, we say ~=~ #x means "number of neurons in structure X" vol(X) means the volume of structure X surf(X) means the surface area of structure X x^y/z means x^(y/z)
in the following, whatever we say about isocortex probably also applies to allocortex, if the thickness of allocortex turns out to be (roughly) constant, which we suspect it will be, but we haven't checked that yet
Q: which of these are base facts and which are derived from the others? for the base facts, citation?
#thalamus^3/2 =~ #isocortex vol(isocortex) ~=~ vol(thalamus) combined; also, V1 vs LGN vol(primary olfactory cortex) ~=~ vol(olfactory bulb) (mb equal, actually) vol(fish tectum) =~ #(fish tectum)^3/2 vol(8 brainstem nuclei) =~ #(8 brainstem nuclei)^3/2 (that this holds separately for each of those nuclei) vol(claustrum) =~ #claustrum^3/2 vol(magno of LGN) =~ #(magno of LGN)^3/2 vol(parvo of LGN) =~ #(parvo of LGN)^3/2 vol(thalamus) =~ #thalamus^3/2 (fact from este armstrong's data) (every nuclei separately) Q: vol(olfactory bulb) =~ #(olfactory bulb)^3/2? no we don't know, but we suspect. what we know is that glomeruli covers the surface of the bulb, but we haven't measured # of glomeruli. # of mitral cells is proportional to # of glomeruli, probably corpus callosum time increases enormously with scale (maybe this is not true for every single collosal fiber) in MT and 21a and probably other higher-order areas have angle sensitivity and pinwheels relations between map size and size of dendritic arbors: subcorticals which go like sunhwa's paper (.5): RGC (definitely), tectal cells (tectal dendrites our sample is too small but it seems to indicate this so far) also suspect thalamic nrns (each of thalamic nuclei individually), each of brainstem nuclei indiv, claustrum (but "map" is uncertain) but what we actually really know for those is that # of nrns and volume of the structure is a 3/2 power law (#nrns^3/2 = vol), so the above is just a guess, but it is consistent (and is inconsistent with 0 power law for dendritic arbors) in the cortex, we know that as cortex gets bigger, synapses per cortical nrn does not get (much) bigger (2/1000, and that's mostly from slightly increased cortical thickness) surf(isocortex) =~ #isocortex vol(isocortex) is only proportional to its surface area to within a factor of 3. therefore, vol(isocortex) is proportional to #isocortex within a factor of 3. factor of 3 is across areas and also across species
this is derived from the fact that the # of cells under a bit of cortical surface is almost constant, except in visual areas. the "almost" there is because some areas are a little thicker and some are a little thinner.
vol(each part of the amygdala except the central nucleus) ~=~ vol(isocortex) probably vol(central nucleus of amygdala) ~= vol(isocortex)^x (x is around .5 to .6; fact from nikoosh's thesis) vol(most of amygdala) ~= vol(hippocampus) ~= vol(medulla) (fact from nikoosh's thesis; is "most of amygdala" = "each part of the amygdala except the central nucleus and cortical nucleus"? yes) vol(most of amygdala) ~= vol(hippocampus) ~= vol(medulla) ~= vol(entire brain with some stuff subtracted, notably white matter) (maybe) vol(white matter) ~=~ vol(brain - white matter)^4/3 (maybe) retinal ganglion cells have flat axonal arbors into tectum
they "tile with overlap" to form a (bounded) plane in tectum. a 2D picture of the world is continuously mapped onto this plane. the tectum is a layered structure like cortex. this 2D map is one of the layers. other layers have intra-tectal axons. tectal cells have dendritic arbors which cover a column, that is, they spread out to cover some area tangential to the input plane, and the arbor also has a boundary orthogonal to the input plane (not sure about the orthogonality). the area of the plane which is covered by one tectal dendritic arbor is called its "pixel area". the intra-tectal axons in the non-input layers can travel tangentially further than the extent of the source cell's dendrites, i.e. they can transmit messages to neurons dealing with pixel areas different from the pixel area of the message source.
#tectum =~ #(retinal gangion)^2 (all stuff about the fish tectum: pretty sure but not positive)
size of each ret gang axonal arbor into tectum is a fixed proportion of the area of the tectal surface (which is ~= to area of input map)
surf(tectum) =~ #tectum
vol(tectum) =~ #tectum^3/2
thickness(tectum) =~ #tectum^1/2 (derived from the previous two)
tectal pixel area =~ (area of entire map provided by retinal ganglion (the "plane" referred to above))^1/2 (or was it .6 or .62 or .63? forgot)
note: the scaling law between tectal pixel area and the input map from retinal ganglion cells has the same scaling law as for retinal ganglion cell pixel areas vs. their input map size. the latter scaling law is covered in sunwha's paper.
Q: note: the vol vs. # scaling law is 3/2 for both the olfactory bulb and the tectum. In the olfactory bulb, this can be explained b/c the nrns are on the surface, the vol vs. surface scales by 3/2, but this isn't the case in the tectum, supporting the idea that the reason for 3/2 is computational rather than developmental (but i Q this)?
note: if you are going from 2d to 3d (by # of features), and you want the resolution to be the same for all features, then a 3/2 scaling law. the other tele derivation is sunhwa paper on the one hand vs. max on resolution on the other. also 3rd tele derivation: wavelet transform of 2D image is 4D; but one of those 4 dimensions is fixed, and used to compute an edge detector; so there are 3 features (3rd is like the first; 4th feature is spatial frequency)