GP modelling a 2nd derivative?

[Mike Lawrence]

In neuroscience, we often collect EEG data to measure the activity of the brain. However, EEG voltage is a relative measure, and the specific choice of reference point can change the observed voltage topology quite dramatically. Apparently ([1]) there is a known-but-rarely-applied "solution" to this issue whereby you compute the 2nd derivative of the voltage topology using an arbitrary reference, resulting in a current density topology that provides an absolute measure of brain activity, and one that will be the same regardless of the reference chosen to generate the original voltage topology. 

Ignoring for the moment the issue of the topology being on the surface of a sphere (which is being discussed elsewhere, [2]), is there a recommended way to model this scenario in stan? I know that one method would be to simply model the voltage as a function of a spatiotemporal GP, then put computation of the 2nd derivative of the GP in the transformed parameters section, but possibly there's a more direct approach whereby the voltage is modelled as a transformation of a current density GP?

[1] https://books.google.ca/books?id=SzavAwAAQBAJ&lpg=PA165&ots=I9lMzlss5L&dq=Luck%20current%20density&pg=PA165#v=onepage&q&f=false

[2] https://groups.google.com/forum/?fromgroups#!topic/stan-users/z2hj0Zb1I6A

--
Mike Lawrence
Graduate Student
Department of Psychology & Neuroscience
Dalhousie University

~ Certainty is (possibly) folly ~

[Daniel Lee]

Hi Mike,

First, the book link [1] is dead.

More below.

On Monday, July 27, 2015 at 10:36:31 AM UTC-4, Mike Lawrence wrote:

In neuroscience, we often collect EEG data to measure the activity of the brain. However, EEG voltage is a relative measure, and the specific choice of reference point can change the observed voltage topology quite dramatically. Apparently ([1]) there is a known-but-rarely-applied "solution" to this issue whereby you compute the 2nd derivative of the voltage topology using an arbitrary reference, resulting in a current density topology that provides an absolute measure of brain activity, and one that will be the same regardless of the reference chosen to generate the original voltage topology. 

Ignoring for the moment the issue of the topology being on the surface of a sphere (which is being discussed elsewhere, [2]), is there a recommended way to model this scenario in stan?

Not sure. Do you have a way to calculate the 2nd derivative?
 

I know that one method would be to simply model the voltage as a function of a spatiotemporal GP, then put computation of the 2nd derivative of the GP in the transformed parameters section, but possibly there's a more direct approach whereby the voltage is modelled as a transformation of a current density GP?

If you've got a form of the voltage as a function of that estimated GP, then I don't see why not.
 

[Mike Lawrence]

On Wed, Aug 12, 2015 at 11:44 AM, Daniel Lee <bea...@alum.mit.edu> wrote:

First, the book link [1] is dead.

Darn, sorry, I didn't know a google books link would die like that. Searching books.google.com for "Luck current density" should bring up results with a first reference in Luck's "An introduction to the event-related potential technique" and clicking this hit should bring you to the section of the book describing current density as it relates to eeg. Just in case, I've taken a screen cap of the relevant section and attached it as a png.

 

Ignoring for the moment the issue of the topology being on the surface of a sphere (which is being discussed elsewhere, [2]), is there a recommended way to model this scenario in stan?

Not sure. Do you have a way to calculate the 2nd derivative?

I don't know; I actually think that maybe it might be better to think of this scenario as the model involving a double integral to convert between a "current-density" scale of the model's parameters and the "voltages-through-space" scale of the observed data.

 

[Mike Lawrence]

On Wed, Aug 12, 2015 at 12:05 PM, Mike Lawrence <mike....@gmail.com> wrote:

Ignoring for the moment the issue of the topology being on the surface of a sphere (which is being discussed elsewhere, [2]), is there a recommended way to model this scenario in stan?

Not sure. Do you have a way to calculate the 2nd derivative?

I don't know; I actually think that maybe it might be better to think of this scenario as the model involving a double integral to convert between a "current-density" scale of the model's parameters and the "voltages-through-space" scale of the observed data.

 

Oops, I think my paucity of calculus betrays me; I'm not sure if "double integral" is the proper term here. I meant that if current density can be computed as the 2nd derivative of a given voltage topography, whatever transform is needed to go backward and derive the expected voltage topography from a given current density.