# Doubt about current density units: pA.m or pA/m2?

Hi, in a previous thread (http://neuroimage.usc.edu/forums/showthread.php?1132-Sources-localisation-interpretation&highlight=units) it’s been posted that ‘pA.m are physical units of current source density on the cortical surface’.

I’m a little confused because I’d have expected current density units to be pA/m2 (current per cross-sectional area); also, I’d have expected pA.m to be units corresponding to the strength of the dipoles. Does this mean that the sources yielded by Brainstorm are in terms of dipole strengths and not current densities? Am I understanding something wrong?

I’d really appreciate if someone could clarify this to me.

Best
Jose

Hi Jose,

Short answer: Brainstorm currently presents each grid point as a current dipole, in units of A-m. We may infer [B]surface[/B] density by dividing the dipole strength by the implied area around each dipole, and we may infer [B]volume[/B] density by additionally dividing out the implied cortical thickness.

So if we run the Brainstorm Median Nerve Sample as a minimum norm, for 15,000 vertices, we see peaks of about 500 pA-m around 35ms post-stimulus. If we reducepatch the cortex to 1,500 vertices, we see peaks of 5,000 pA-m. Under the Scouts, if we select all clusters, Brainstorm reports in both cases a cortical surface area of about 2000 cm^2. Therefore each dipole in the 1500 dipole model represents ten times the area of the 15,000 dipole model, hence each dipolar moment is 10x greater.

[I]As a rough approximation, for min norm models employing 15,000 vertices in an adult model, then the [B]rough[/B] conversion would be 500 pA-m => 50 nA/mm [B]surface[/B] density (see below calculations).[/I]

While in principle I would agree it would be nice to present as as density (we have discussed this among the research team in the past), please read the more detailed answer below.

[B]Very detailed answer below. This seemed like a good opportunity to discuss the source strength assumptions used in source modeling.[/B]

The terminology is a bit loose in the community. The minimum norm image is often stated as a “current density,” so you are correct in properly anticipating units of A/m^2, a proper [B]volume[/B] density. However, since we don’t know the true [B]effective[/B] cortical thickness, then many researchers speak of [B]surface[/B] densities, in units of A/m, effectively integrating out the thickness measure (A/m^2 * the presumed thickness of the cortex in m = A/m). Finally, dipole models simply integrate the full volume and use units of A-m of dipolar current (A/m^2 times the presumed thickness of the cortex times the presumed area of the cortex = A/m^2 * m^3 = A-m).

The units in the literature also can be confusing. For reference in understanding units, read carefully the changing units presented here (A:amps, m:meters, with prefixes of m-milli, micro, n-nano, p-pico, f-femto).

A dipole strength of 10,000 pA-m can be equated as 10 nA-m = 10,000 nA-mm = 10 microA-mm = 2 microA along a wire 5 mm in length. In invasive brain studies in our Epilepsy Center, we often stimulate the brain starting at 2 mA into a bipolar pair of strip electrodes 5 mm apart, thus neural stimulation is an effective 10 mA-mm = 10,000 nA-m dipolar current. Evoked dipolar strength is nominally of order 10 nA-m, and a good epileptic spike is of order 100 nA-m. Thus neural stimulation can be modeled as a dipolar strength at least 100 to 1,000 times stronger than what neural activity may be modeled as.

A single pyramidal cell is believed to possibly generate a 20 fA-m signal along its presumed 2 mm [B]effective[/B] length (Hamalainen 1993), but more recent modeling suggests it may be as high as 200 fA-m (Okada 2006). So a 20 nA-m dipole strength represents from 100,000 to 1,000,000 pyramidal cells. MEG source modeling is therefore about macro models of millions of cells, i.e. nothing close to microcellar modeling.

If the [B]surface[/B] density is 100 nA/mm, then a 10 mm by 10 mm patch of cortex would create a net dipole strength of 100 nA/mm * 100 mm^2 = 10,000 nA-mm strength = 10 nA-m strength per square cm of cortex.

This [B]surface[/B] density of 100 nA/mm is a good working [B]ORDER OF MAGNITUDE[/B] (literally, we could easily be off 1/10 to 10x) concept of neural strength in evoked activity. If the [B]effective[/B] cortical thickness is 2 mm, then the [B]volume[/B] density would be 50 nA/mm^2 to a working order of magnitude, but the net calculation would remain the same if we consider the same square cm patch of cortex.

Similarly, the evoked dipolar strength of 10 nA-m and the interictal spike dipolar strength of 100 nA-m are good working Order of Magnitude dipolar strengths. Note, the interictal spike is believed to possibly represent a higher [B]density[/B] of synchronous response among the neuronal population, hence it does [B]NOT[/B] necessarily follow that the interictal spike has ten times the cortical area of the evoked source.

As an experimental example cross-check, in the above Median Nerve example, we have 2000 cm^2 cortex divided into 15,000 dipoles (in the normal denser case of tesselation), so (crudely) each dipole may represent 0.1 cm^2 of cortical area (keeping the math simple). Thus a dipole strength of 500 pA-m divided by an average area of 0.1 cm^2 yields a [B]surface[/B] density of 50,000,000 pA/m = 50 nA/mm, in excellent agreement with our nominal surface density of 100 nA/mm for evoked activity.

Back to Brainstorm: Render a cortical surface and look at the tessellation (turn on Edges in the Surface Panel). The actual area of each triangle is highly variable, driven by the segmentation programs and the reducepatch programs. In theory, each vertex point (where we calculate the dipole) has a region of support equal to 1/3 of the total area of the triangles attached to that vertex. Thus the tesselated area attached to each vertex is itself quite variable.

Thus if we divide each dipolar point by its presumed area (to yield a [B]surface[/B] density in A/m), we introduce another level of variability that may be misleading or confusing. And the effective cortical thickness is a another guess (to yield a [B]volume[/B] density A/m^2). In reality, we don’t know the effective cortical length of the pyramidal cells, nor the density of neurons per unit volume, nor the density per unit surface area of cortex.

[I]Hence for now we keep it simple and report units of A-m, since the conversion to A/m^2 would only be another layer of modeling complexity best left to the user’s own interpretations. The summary rough calculation given above will put you in the right order of magnitude.
[/I]

– John

PS, I welcome cross-checks of my math above.

[I]Edit 2013-Sep-11:[/I] Typical invasive electrode separations, center-on-center, are 5, 6, 9, and 10 mm, depending on the style of intracranial electrode used. Altered stimulation example to be 5 mm in the paragraph above and simplified the example.

Dear Dr. Mosher,

Thank you very much for your thorough reply. The issue is perfectly clear now.

Best regards
Jose

Dear John,

I have a very basic follow-up question: does the dot stand for a "multiplication (*)" or a "division (/)" or something else?

Many thanks!

You mean as in A.m ?
That's a multiplication.