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closeOn interpretation and statistics.
Posted by landsgevaer on 17 Oct 2013 at 12:39 GMT
The authors calculated gradients and divergences of fMRI activation maps (beta-maps), which in itself I find interesting. However:
- The authors claim that the direction of gradients correspond with "energy flow", where "energy" refers to "free energy". The latter is a highly abstract concept. In particular, there presently is no means of measuring the free energy related to a brain state, and certainly fMRI activation is not a measure of free energy (how the authors make the link in their section on "Biophysical interpretation" is not clear to me; I cannot follow the argument). Therefore, the authors interpretation in terms of "propagation of neural activity" is not meaningful. Similarly, divergence (i.e. the laplacian of the activation map) relates to the second derivative, i.e. whether activity exceeds the average activity in neighbouring sites. Activation peaks have positive divergence, so it is closely related to "peakiness". Activation troughs (or extrema of *de*activation!) show negative divergence. How that signifies a "source/sink of energy" is again unclear to me. The authors also allude to the "stable" nature of the emerging "flow". However, they do not provide any information about how these pattern develop over time, so these claims seem unfounded (perhaps they mean to refer to reproducibility or consistency across subjects, but that is an entirely different interpretation).
- The authors seem to have tested the significance of gradients by submitting their x-, y-, and z-components to separate univariate tests. This defies the tensorial nature of gradients and is inappropriate. Gradient-vectors are 3-dimensional entities that require a multivariate statistical assessment (e.g. Manova, or Hotelling T^2-test). The employed univariate tests depend upon the chosen (arbitrary) coordinate frame as much as on the gradient field in which one might be interested. Furthermore, despite their claim that no validation is required, the random field that is obtained from a gradient (whether the complete vectorial outcome, or a single component) does not have the same spatial correlation structure as the underlying activation field. This further renders the parametric statistics invalid.
In summary, I believe that a multivariate analysis of gradient maps may have its uses, but the present paper provides neither a proper statistical framework to assess it, nor a solid interpretation of its outcomes.
(Similar issues I encountered in http://www.ncbi.nlm.nih.g... which I read in an effort to better understand the presented reasoning.)
RE: On interpretation and statistics.
strelkuz replied to landsgevaer on 18 Oct 2013 at 13:19 GMT
Thank you for the comment, which raises a number of interesting issues for the discussion. We will try to reply to each of the raised points.
- We agree that the conception of free energy is very abstract, however it does not preclude the attempts to apply it to the real data. Besides, it is already applied to the fMRI studies of brain activations as discussed in the recent reviews of K. Friston. Still, the existence of the stable gradients in brain activity seems to contradict the free energy principle, this is the main idea of our biophysical discussion. In our personal correspondence with K. Friston, he said that minimising variational free energy is not the same as destroying thermodynamic free energy gradients; these would only occur when the brain was deprived of sensory input for a very long time and this is an interesting perspective on sleep where one might expect the activity gradients to be attenuated.
We also agree that the interpretation of the found stable gradients in terms of activity propagation (activity/energy flows) is hypothetical. However, the hypothesis of the existence of such gradients is confirmed. The main idea is that at the place of the functional inputs to a specialized neuroglial population, there is a a more abrupt change of activity as elsewhere in this population. Thus, the gradient vector reflects the direction of activity propagation in the places of functional inputs to specialized brain centres. It is not a flow of activity per se but it may reflect the direction of the flow.
- Concerning the notion of stability of the gradient vectors, it is based on the double reasoning. First of all, these gradients were found in the contrast images, which are the weighted means during a rather long stimulation time. Thus, these gradients are maintained throughout the long stimulation period. This is one aspect of their stability. Another aspect is that they are stable between subjects as shown by the statistical analysis.
-We doubt that MANOVA could be a good approach to estimate vector projections. Vector projections are intercorrelated, e.g., if we turn a vector in the xy plane, the increase of the projections on one axis is accompanied b the decrease of the projections on the other axis. This purely mathematical correlation of vector projections can bias the results of the MANOVA analysis irrespectively of brain physiology (see Cole et al., Psychological Bulletin, Vol 115(3), 1994, 465-474). We did not construct a GLM model with different projections in one model because the variables are interdependent. As for the analysis of divergence, which provides one value per voxel, the resulting data can be easily analysed with conventional methods.
We agree that the non-parametric analysis would be more cautious than the parametric one, this is a problem for most of the fMRI studies.
- The analysis of the projections on the conventional in neuroimaging axes provides an idea of the main directions along these axes. It is very difficult to visualize a vector per voxel in the brain, projections are quite helpful. We do not see any problem in the choice of this particular set of the axes for the projections given that they are well defined in the brain and are widely used.
- Concerning the relation of the divergences with the peakedness suggested in the comment, it is not so straightforward. We also had this idea, that is why we compared the classical activation peaks with the analysis of divergences (Tables 1, 2), the found areas were completely different. This is an interesting subject for further investigation to see how positive and negative divergences are related to the classical activation results.
In general, it would be more exact to say that this analysis just studies the differences between the neighbouring voxels. However, while writing this article we thought that it would be better to put it in a larger theoretical frame with neurophysiological hypotheses behind. Perhaps, the chosen style was not the optimal one and the article should have been closer to the concrete results of the analysis than to the abstract line of reasoning.
RE: On interpretation and statistics.
landsgevaer replied to strelkuz on 23 Oct 2013 at 13:16 GMT
- I realise that the concept of free energy has been used in relation to fMRI data. However, to my knowledge it has not been equated with activation levels (i.e. beta maps). One obvious reason why this would be problematic is that "baseline" is not a "minimum". Baseline already involves much ongoing activity, and the occurrence of deactivation proves that. In other words, if the brain were minimizing free energy, and fMRI measured free energy, then it would not end up at the fMRI baseline. Therefore, if free energy is useful, then activation levels cannot form its measurable correlate.
- A similar point can be made about the alleged confirmation of the hypothesis of energy flow. Even if the authors show that gradients are "stable" (I read: reproducible), then this is not substantiating evidence that this corresponds with a direction of "activity propagation". This seems pretty obvious when knowledge about neural connectivity (e.g. including long tracts) is taken into consideration. (IMHO, the fact that activity "flows" outside of the brain, acknowledged by the authors, would further substantiate this: if the interpretation cannot be trusted in one direction, why could it be trusted in another?)
- The authors conclude that if an activation pattern survives after weighting over a long period, that it must be stable during that period. That is flawed reasoning. It merely states that is has non-zero mean; how the dynamics in time evolve is impossible to say. (For instance, if in a block design a region is activated 50% of the time but not the other 50% of the time, the average activation will be non-zero and significant, but the activity is very "unstable".)
- The authors argue that MANOVA may be ill suited to test vector projections. My intended point was that the authors should ideally not look at projections at all, because these depend upon the projection axis, but consider the 3-D vectors themselves. Although projections are not per se "wrong", they are arbitrary; that holds for standard x,y,z, directions as well (these are convention, but do not have any special meaning as far as the brain is concerned). In a separate reply the authors suggest to consider the lenght of the gradients only. I would advise against this because, first, differences in direction are equally interesting and, second, for non-isotropic fields this leads to unclear statistics (however, note that Hoteling's T-square contains a measure of vector length, taking into account the covariance structure, akin a Mahalonobis distance). Other examples of applications where 3-D vector fields form the outcome of interest are for example diffusion tensor imaging (principal axis of diffusion) or deformation-based morphology (translation fields), so the authors may be inspired by such developments.
- I agree with the authors' final assessment regarding their "larger theoretical frame with neurophysiological hypotheses". I would actually like to encourage the authors to continue their methodological work, and make it more rigorous and general, but refrain from diluting it with unsubstantiable and poorly defined terminology like "activation flow". I wish them well.
RE: RE: On interpretation and statistics.
strelkuz replied to landsgevaer on 23 Oct 2013 at 15:54 GMT
In fact, we did not mean that gradient vectors can point ouside the brain: this is impossible because signal values can not be greater outside the brain than inside (if they reflect brain activity and are not some artefacts). What we meant is that non-zero values of the signal can exist outside the brain. These are to be considered as artefacts and replaced by the NaN values in the analysis of the gradient vectors.
Concerning the baseline activity, the point of K. Friston is that given a long time of the absence of stimulation, activity of the brain will be minimized. This is the way it corresponds to the free energy minimisation principle. Given frequent stimulation as in usual life, brain activity even at "rest" needs to adapt to the input, to create coding (internal representations), thus its energy is not minimal in this case. Discussing this problem in the recent review (Plos comp biology), the authors show that free energy can be represented as the difference between the complexity of coding and the accuracy of coding. By coding they mean predictive coding: internal representations, concepts. It follows from this definition that to minimize free energy, the brain should either increase the accuracy of coding or decrease its complexity. It is evident that at the same level of accuracy the brain should favour less complex coding to save energy. Thus, the relation of the free energy minimisation principle to baseline activity can be considered from this perspective.
RE: On interpretation and statistics.
strelkuz replied to landsgevaer on 21 Oct 2013 at 14:05 GMT
In fact, if we are not interested in the directions but just in a general effect per voxel, it would be better to calculate the length of the vector per voxel from the projections. This is a more solid way to combine the projections that entering them into MANOVA. In this case, the information about the directions will be lost, only the general effect per voxel.