cg
view grant.html @ 29:5e2e4732b647
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author | bshanks@bshanks.dyndns.org |
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date | Mon Apr 13 03:43:51 2009 -0700 (16 years ago) |
parents | 01c118d1074b |
children | 6ec3230fe1dc |
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1 Specific aims
2 Massive new datasets obtained with techniques such as in situ hybridization
3 (ISH) and BAC-transgenics allow the expression levels of many genes at many
4 locations to be compared. Our goal is to develop automated methods to relate
5 spatial variation in gene expression to anatomy. We want to find marker genes
6 for specific anatomical regions, and also to draw new anatomical maps based on
7 gene expression patterns. We have three specific aims:
8 (1) develop an algorithm to screen spatial gene expression data for combi-
9 nations of marker genes which selectively target anatomical regions
10 (2) develop an algorithm to suggest new ways of carving up a structure into
11 anatomical subregions, based on spatial patterns in gene expression
12 (3) create a 2-D “flat map” dataset of the mouse cerebral cortex that con-
13 tains a flattened version of the Allen Mouse Brain Atlas ISH data, as well as
14 the boundaries of cortical anatomical areas. Use this dataset to validate the
15 methods developed in (1) and (2).
16 In addition to validating the usefulness of the algorithms, the application of
17 these methods to cerebral cortex will produce immediate benefits, because there
18 are currently no known genetic markers for many cortical areas. The results
19 of the project will support the development of new ways to selectively target
20 cortical areas, and it will support the development of a method for identifying
21 the cortical areal boundaries present in small tissue samples.
22 All algorithms that we develop will be implemented in an open-source soft-
23 ware toolkit. The toolkit, as well as the machine-readable datasets developed
24 in aim (3), will be published and freely available for others to use.
25 1
27 Background and significance
28 Aim 1
29 Machine learning terminology: supervised learning
30 The task of looking for marker genes for anatomical subregions means that
31 one is looking for a set of genes such that, if the expression level of those genes
32 is known, then the locations of the subregions can be inferred.
33 If we define the subregions so that they cover the entire anatomical structure
34 to be divided, then instead of saying that we are using gene expression to find
35 the locations of the subregions, we may say that we are using gene expression to
36 determine to which subregion each voxel within the structure belongs. We call
37 this a classification task, because each voxel is being assigned to a class (namely,
38 its subregion).
39 Therefore, an understanding of the relationship between the combination of
40 their expression levels and the locations of the subregions may be expressed as
41 a function. The input to this function is a voxel, along with the gene expression
42 levels within that voxel; the output is the subregional identity of the target
43 voxel, that is, the subregion to which the target voxel belongs. We call this
44 function a classifier. In general, the input to a classifier is called an instance,
45 and the output is called a label (or a class label).
46 The object of aim 1 is not to produce a single classifier, but rather to develop
47 an automated method for determining a classifier for any known anatomical
48 structure. Therefore, we seek a procedure by which a gene expression dataset
49 may be analyzed in concert with an anatomical atlas in order to produce a
50 classifier. Such a procedure is a type of a machine learning procedure. The
51 construction of the classifier is called training (also learning), and the initial
52 gene expression dataset used in the construction of the classifier is called training
53 data.
54 In the machine learning literature, this sort of procedure may be thought
55 of as a supervised learning task, defined as a task in which the goal is to learn
56 a mapping from instances to labels, and the training data consists of a set of
57 instances (voxels) for which the labels (subregions) are known.
58 Each gene expression level is called a feature, and the selection of which
59 genes1 to include is called feature selection. Feature selection is one component
60 of the task of learning a classifier. Some methods for learning classifiers start
61 out with a separate feature selection phase, whereas other methods combine
62 feature selection with other aspects of training.
63 One class of feature selection methods assigns some sort of score to each
64 candidate gene. The top-ranked genes are then chosen. Some scoring measures
65 can assign a score to a set of selected genes, not just to a single gene; in this
66 case, a dynamic procedure may be used in which features are added and sub-
67 tracted from the selected set depending on how much they raise the score. Such
68 procedures are called “stepwise” or “greedy”.
69 __________________________
70 1Strictly speaking, the features are gene expression levels, but we’ll call them genes.
71 2
73 Although the classifier itself may only look at the gene expression data within
74 each voxel before classifying that voxel, the learning algorithm which constructs
75 the classifier may look over the entire dataset. We can categorize score-based
76 feature selection methods depending on how the score of calculated. Often
77 the score calculation consists of assigning a sub-score to each voxel, and then
78 aggregating these sub-scores into a final score (the aggregation is often a sum or
79 a sum of squares). If only information from nearby voxels is used to calculate a
80 voxel’s sub-score, then we say it is a local scoring method. If only information
81 from the voxel itself is used to calculate a voxel’s sub-score, then we say it is a
82 pointwise scoring method.
83 Key questions when choosing a learning method are: What are the instances?
84 What are the features? How are the features chosen? Here are four principles
85 that outline our answers to these questions.
86 Principle 1: Combinatorial gene expression It is too much to hope
87 that every anatomical region of interest will be identified by a single gene. For
88 example, in the cortex, there are some areas which are not clearly delineated
89 by any gene included in the Allen Brain Atlas (ABA) dataset. However, at
90 least some of these areas can be delineated by looking at combinations of genes
91 (an example of an area for which multiple genes are necessary and sufficient
92 is provided in Preliminary Results). Therefore, each instance should contain
93 multiple features (genes).
94 Principle 2: Only look at combinations of small numbers of genes
95 When the classifier classifies a voxel, it is only allowed to look at the expression of
96 the genes which have been selected as features. The more data that is available
97 to a classifier, the better that it can do. For example, perhaps there are weak
98 correlations over many genes that add up to a strong signal. So, why not include
99 every gene as a feature? The reason is that we wish to employ the classifier in
100 situations in which it is not feasible to gather data about every gene. For
101 example, if we want to use the expression of marker genes as a trigger for some
102 regionally-targeted intervention, then our intervention must contain a molecular
103 mechanism to check the expression level of each marker gene before it triggers.
104 It is currently infeasible to design a molecular trigger that checks the level of
105 more than a handful of genes. Similarly, if the goal is to develop a procedure to
106 do ISH on tissue samples in order to label their anatomy, then it is infeasible
107 to label more than a few genes. Therefore, we must select only a few genes as
108 features.
109 Principle 3: Use geometry in feature selection
110 When doing feature selection with score-based methods, the simplest thing
111 to do would be to score the performance of each voxel by itself and then com-
112 bine these scores (pointwise scoring). A more powerful approach is to also use
113 information about the geometric relations between each voxel and its neighbors;
114 this requires non-pointwise, local scoring methods. See Preliminary Results for
115 evidence of the complementary nature of pointwise and local scoring methods.
116 3
118 Principle 4: Work in 2-D whenever possible
119 There are many anatomical structures which are commonly characterized in
120 terms of a two-dimensional manifold. When it is known that the structure that
121 one is looking for is two-dimensional, the results may be improved by allowing
122 the analysis algorithm to take advantage of this prior knowledge. In addition,
123 it is easier for humans to visualize and work with 2-D data.
124 Therefore, when possible, the instances should represent pixels, not voxels.
125 Aim 2
126 Machine learning terminology: clustering
127 If one is given a dataset consisting merely of instances, with no class labels,
128 then analysis of the dataset is referred to as unsupervised learning in the jargon
129 of machine learning. One thing that you can do with such a dataset is to group
130 instances together. A set of similar instances is called a cluster, and the activity
131 of finding grouping the data into clusters is called clustering or cluster analysis.
132 The task of deciding how to carve up a structure into anatomical subregions
133 can be put into these terms. The instances are once again voxels (or pixels)
134 along with their associated gene expression profiles. We make the assumption
135 that voxels from the same subregion have similar gene expression profiles, at
136 least compared to the other subregions. This means that clustering voxels is
137 the same as finding potential subregions; we seek a partitioning of the voxels
138 into subregions, that is, into clusters of voxels with similar gene expression.
139 It is desirable to determine not just one set of subregions, but also how
140 these subregions relate to each other, if at all; perhaps some of the subregions
141 are more similar to each other than to the rest, suggesting that, although at a
142 fine spatial scale they could be considered separate, on a coarser spatial scale
143 they could be grouped together into one large subregion. This suggests the
144 outcome of clustering may be a hierarchial tree of clusters, rather than a single
145 set of clusters which partition the voxels. This is called hierarchial clustering.
146 Similarity scores
147 A crucial choice when designing a clustering method is how to measure
148 similarity, across either pairs of instances, or clusters, or both. There is much
149 overlap between scoring methods for feature selection (discussed above under
150 Aim 1) and scoring methods for similarity.
151 Spatially contiguous clusters; image segmentation
152 We have shown that aim 2 is a type of clustering task. In fact, it is a
153 special type of clustering task because we have an additional constraint on
154 clusters; voxels grouped together into a cluster must be spatially contiguous.
155 In Preliminary Results, we show that one can get reasonable results without
156 enforcing this constraint, however, we plan to compare these results against
157 other methods which guarantee contiguous clusters.
158 Perhaps the biggest source of continguous clustering algorithms is the field
159 of computer vision, which has produced a variety of image segmentation algo-
160 4
162 rithms. Image segmentation is the task of partitioning the pixels in a digital
163 image into clusters, usually contiguous clusters. Aim 2 is similar to an image
164 segmentation task. There are two main differences; in our task, there are thou-
165 sands of color channels (one for each gene), rather than just three. There are
166 imaging tasks which use more than three colors, however, for example multispec-
167 tral imaging and hyperspectral imaging, which are often used to process satellite
168 imagery. A more crucial difference is that there are various cues which are ap-
169 propriate for detecting sharp object boundaries in a visual scene but which are
170 not appropriate for segmenting abstract spatial data such as gene expression.
171 Although many image segmentation algorithms can be expected to work well
172 for segmenting other sorts of spatially arranged data, some of these algorithms
173 are specialized for visual images.
174 Dimensionality reduction
175 Unlike aim 1, there is no externally-imposed need to select only a handful
176 of informative genes for inclusion in the instances. However, some clustering
177 algorithms perform better on small numbers of features. There are techniques
178 which “summarize” a larger number of features using a smaller number of fea-
179 tures; these techniques go by the name of feature extraction or dimensionality
180 reduction. The small set of features that such a technique yields is called the
181 reduced feature set. After the reduced feature set is created, the instances may
182 be replaced by reduced instances, which have as their features the reduced fea-
183 ture set rather than the original feature set of all gene expression levels. Note
184 that the features in the reduced feature set do not necessarily correspond to
185 genes; each feature in the reduced set may be any function of the set of gene
186 expression levels.
187 Another use for dimensionality reduction is to visualize the relationships
188 between subregions. For example, one might want to make a 2-D plot upon
189 which each subregion is represented by a single point, and with the property
190 that subregions with similar gene expression profiles should be nearby on the
191 plot (that is, the property that distance between pairs of points in the plot
192 should be proportional to some measure of dissimilarity in gene expression). It
193 is likely that no arrangement of the points on a 2-D plan will exactly satisfy
194 this property – however, dimensionality reduction techniques allow one to find
195 arrangements of points that approximately satisfy that property. Note that
196 in this application, dimensionality reduction is being applied after clustering;
197 whereas in the previous paragraph, we were talking about using dimensionality
198 reduction before clustering.
199 Clustering genes rather than voxels
200 Although the ultimate goal is to cluster the instances (voxels or pixels), one
201 strategy to achieve this goal is to first cluster the features (genes). There are
202 two ways that clusters of genes could be used.
203 Gene clusters could be used as part of dimensionality reduction: rather than
204 have one feature for each gene, we could have one reduced feature for each gene
205 cluster.
206 5
208 Gene clusters could also be used to directly yield a clustering on instances.
209 This is because many genes have an expression pattern which seems to pick
210 out a single, spatially continguous subregion. Therefore, it seems likely that an
211 anatomically interesting subregion will have multiple genes which each individ-
212 ually pick it out2. This suggests the following procedure: cluster together genes
213 which pick out similar subregions, and then to use the more popular common
214 subregions as the final clusters. In the Preliminary Data we show that a num-
215 ber of anatomically recognized cortical regions, as well as some “superregions”
216 formed by lumping together a few regions, are associated with gene clusters in
217 this fashion.
218 Aim 3
219 Background
220 The cortex is divided into areas and layers. To a first approximation, the
221 parcellation of the cortex into areas can be drawn as a 2-D map on the surface of
222 the cortex. In the third dimension, the boundaries between the areas continue
223 downwards into the cortical depth, perpendicular to the surface. The layer
224 boundaries run parallel to the surface. One can picture an area of the cortex as
225 a slice of many-layered cake.
226 Although it is known that different cortical areas have distinct roles in both
227 normal functioning and in disease processes, there are no known marker genes
228 for many cortical areas. When it is necessary to divide a tissue sample into
229 cortical areas, this is a manual process that requires a skilled human to combine
230 multiple visual cues and interpret them in the context of their approximate
231 location upon the cortical surface.
232 Even the questions of how many areas should be recognized in cortex, and
233 what their arrangement is, are still not completely settled. A proposed division
234 of the cortex into areas is called a cortical map. In the rodent, the lack of a
235 single agreed-upon map can be seen by contrasting the recent maps given by
236 Swanson?? on the one hand, and Paxinos and Franklin?? on the other. While
237 the maps are certainly very similar in their general arrangement, significant
238 differences remain in the details.
239 Significance
240 The method developed in aim (1) will be applied to each cortical area to find
241 a set of marker genes such that the combinatorial expression pattern of those
242 genes uniquely picks out the target area. Finding marker genes will be useful
243 for drug discovery as well as for experimentation because marker genes can be
244 used to design interventions which selectively target individual cortical areas.
245 __________________________
246 2This would seem to contradict our finding in aim 1 that some cortical areas are combina-
247 torially coded by multiple genes. However, it is possible that the currently accepted cortical
248 maps divide the cortex into subregions which are unnatural from the point of view of gene
249 expression; perhaps there is some other way to map the cortex for which each subregion can
250 be identified by single genes.
251 6
253 The application of the marker gene finding algorithm to the cortex will
254 also support the development of new neuroanatomical methods. In addition to
255 finding markers for each individual cortical areas, we will find a small panel
256 of genes that can find many of the areal boundaries at once. This panel of
257 marker genes will allow the development of an ISH protocol that will allow
258 experimenters to more easily identify which anatomical areas are present in
259 small samples of cortex.
260 The method developed in aim (3) will provide a genoarchitectonic viewpoint
261 that will contribute to the creation of a better map. The development of present-
262 day cortical maps was driven by the application of histological stains. It is
263 conceivable that if a different set of stains had been available which identified
264 a different set of features, then the today’s cortical maps would have come out
265 differently. Since the number of classes of stains is small compared to the number
266 of genes, it is likely that there are many repeated, salient spatial patterns in
267 the gene expression which have not yet been captured by any stain. Therefore,
268 current ideas about cortical anatomy need to incorporate what we can learn
269 from looking at the patterns of gene expression.
270 While we do not here propose to analyze human gene expression data, it is
271 conceivable that the methods we propose to develop could be used to suggest
272 modifications to the human cortical map as well.
273 Related work
274 There does not appear to be much work on the automated analysis of spatial
275 gene expression data.
276 There is a substantial body of work on the analysis of gene expression data,
277 however, most of this concerns gene expression data which is not fundamentally
278 spatial.
279 As noted above, there has been much work on both supervised learning and
280 clustering, and there are many available algorithms for each. However, the
281 completion of Aims 1 and 2 involves more than just choosing between a set of
282 existing algorithms, and will constitute a substantial contribution to biology.
283 The algorithms require the scientist to provide a framework for representing the
284 problem domain, and the way that this framework is set up has a large impact
285 on performance. Creating a good framework can require creatively reconcep-
286 tualizing the problem domain, and is not merely a mechanical “fine-tuning”
287 of numerical parameters. For example, we believe that domain-specific scoring
288 measures (such as gradient similarity, which is discussed in Preliminary Work)
289 may be necessary in order to achieve the best results in this application.
290 We are aware of two existing efforts to relate spatial gene expression data to
291 anatomy through computational methods.
292 [?] describes an analysis of the anatomy of the hippocampus using the ABA
293 dataset. In addition to manual analysis, two clustering methods were employed,
294 a modified Non-negative Matrix Factorization (NNMF), and a hierarchial bifur-
295 cation clustering scheme based on correlation as the similarity score. The paper
296 yielded impressive results, proving the usefulness of such research. We have run
297 7
299 NNMF on the cortical dataset and while the results are promising (see Prelim-
300 inary Data), we think that it will be possible to find a better method3 (we also
301 think that more automation of the parts that this paper’s authors did manually
302 will be possible).
303 and [?] describes AGEA. todo
304 __________________________
305 3We ran “vanilla” NNMF, whereas the paper under discussion used a modified method.
306 Their main modification consisted of adding a soft spatial contiguity constraint. However,
307 on our dataset, NNMF naturally produced spatially contiguous clusters, so no additional
308 constraint was needed. The paper under discussion mentions that they also tried a hierarchial
309 variant of NNMF, but since they didn’t report its results, we assume that those result were
310 not any more impressive than the results of the non-hierarchial variant.
311 8
313 Preliminary work
314 Format conversion between SEV, MATLAB, NIFTI
315 todo
316 Flatmap of cortex
317 todo
318 Using combinations of multiple genes is necessary and sufficient to
319 delineate some cortical areas
320 Here we give an example of a cortical area which is not marked by any
321 single gene, but which can be identified combinatorially. according to logistic
322 regression, gene wwc14 is the best fit single gene for predicting whether or not a
323 pixel on the cortical surface belongs to the motor area (area MO). The upper-left
324 picture in Figure shows wwc1’s spatial expression pattern over the cortex. The
325 lower-right boundary of MO is represented reasonably well by this gene, however
326 the gene overshoots the upper-left boundary. This flattened 2-D representation
327 does not show it, but the area corresponding to the overshoot is the medial
328 surface of the cortex. MO is only found on the lateral surface (todo).
329 Gnee mtif25 is shown in figure the upper-right of Fig. . Mtif2 captures MO’s
330 upper-left boundary, but not its lower-right boundary. Mtif2 does not express
331 very much on the medial surface. By adding together the values at each pixel
332 in these two figures, we get the lower-left of Figure . This combination captures
333 area MO much better than any single gene.
334 Correlation todo
335 Conditional entropy todo
336 Gradient similarity todo
337 Geometric and pointwise scoring methods provide complementary
338 information
339 To show that local geometry can provide useful information that cannot be
340 detected via pointwise analyses, consider Fig. . The top row of Fig. displays the
341 3 genes which most match area AUD, according to a pointwise method6. The
342 bottom row displays the 3 genes which most match AUD according to a method
343 which considers local geometry7 The pointwise method in the top row identifies
344 __________________________
345 4“WW, C2 and coiled-coil domain containing 1”; EntrezGene ID 211652
346 5“mitochondrial translational initiation factor 2”; EntrezGene ID 76784
347 6For each gene, a logistic regression in which the response variable was whether or not a
348 surface pixel was within area AUD, and the predictor variable was the value of the expression
349 of the gene underneath that pixel. The resulting scores were used to rank the genes in terms
350 of how well they predict area AUD.
351 7For each gene the gradient similarity (see section ??) between (a) a map of the expression
352 of each gene on the cortical surface and (b) the shape of area AUD, was calculated, and this
353 was used to rank the genes.
354 9
358 Figure 1: Upper left: wwc1. Upper right: mtif2. Lower left: wwc1 + mtif2
359 (each pixel’s value on the lower left is the sum of the corresponding pixels in
360 the upper row). Within each picture, the vertical axis roughly corresponds to
361 anterior at the top and posterior at the bottom, and the horizontal axis roughly
362 corresponds to medial at the left and lateral at the right. The red outline is
363 the boundary of region MO. Pixels are colored approximately according to the
364 density of expressing cells underneath each pixel, with red meaning a lot of
365 expression and blue meaning little.
366 10
370 Figure 2: The top row shows the three genes which (individually) best predict
371 area AUD, according to logistic regression. The bottom row shows the three
372 genes which (individually) best match area AUD, according to gradient similar-
373 ity. From left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a,
374 Ptk7, Aph1a again, and Lepr
375 genes which express more strongly in AUD than outside of it; its weakness is that
376 this includes many areas which don’t have a salient border matching the areal
377 border. The geometric method identifies genes whose salient expression border
378 seems to partially line up with the border of AUD; its weakness is that this
379 includes genes which don’t express over the entire area. Genes which have high
380 rankings using both pointwise and border criteria, such as Aph1a in the example,
381 may be particularly good markers. None of these genes are, individually, a
382 perfect marker for AUD; we deliberately chose a “difficult” area in order to
383 better contrast pointwise with geometric methods.
384 Areas which can be identified by single genes
385 todo
386 Specific to Aim 1 (and Aim 3)
387 Forward stepwise logistic regression todo
388 SVM on all genes at once
389 In order to see how well one can do when looking at all genes at once, we
390 ran a support vector machine to classify cortical surface pixels based on their
391 gene expression profiles. We achieved classification accuracy of about 81%8.
392 As noted above, however, a classifier that looks at all the genes at once isn’t
393 practically useful.
394 ____________
395 85-fold cross-validation.
396 11
398 The requirement to find combinations of only a small number of genes limits
399 us from straightforwardly applying many of the most simple techniques from
400 the field of supervised machine learning. In the parlance of machine learning,
401 our task combines feature selection with supervised learning.
402 Decision trees
403 todo
404 Specific to Aim 2 (and Aim 3)
405 Raw dimensionality reduction results
406 todo
407 (might want to incld nnMF since mentioned above)
408 Dimensionality reduction plus K-means or spectral clustering
409 Many areas are captured by clusters of genes
410 todo
411 todo
412 12
414 Research plan
415 todo amongst other things:
416 Develop algorithms that find genetic markers for anatomical re-
417 gions
418 1. Develop scoring measures for evaluating how good individual genes are at
419 marking areas: we will compare pointwise, geometric, and information-
420 theoretic measures.
421 2. Develop a procedure to find single marker genes for anatomical regions: for
422 each cortical area, by using or combining the scoring measures developed,
423 we will rank the genes by their ability to delineate each area.
424 3. Extend the procedure to handle difficult areas by using combinatorial cod-
425 ing: for areas that cannot be identified by any single gene, identify them
426 with a handful of genes. We will consider both (a) algorithms that incre-
427 mentally/greedily combine single gene markers into sets, such as forward
428 stepwise regression and decision trees, and also (b) supervised learning
429 techniques which use soft constraints to minimize the number of features,
430 such as sparse support vector machines.
431 4. Extend the procedure to handle difficult areas by combining or redrawing
432 the boundaries: An area may be difficult to identify because the bound-
433 aries are misdrawn, or because it does not “really” exist as a single area,
434 at least on the genetic level. We will develop extensions to our procedure
435 which (a) detect when a difficult area could be fit if its boundary were
436 redrawn slightly, and (b) detect when a difficult area could be combined
437 with adjacent areas to create a larger area which can be fit.
438 Apply these algorithms to the cortex
439 1. Create open source format conversion tools: we will create tools to bulk
440 download the ABA dataset and to convert between SEV, NIFTI and MAT-
441 LAB formats.
442 2. Flatmap the ABA cortex data: map the ABA data onto a plane and draw
443 the cortical area boundaries onto it.
444 3. Find layer boundaries: cluster similar voxels together in order to auto-
445 matically find the cortical layer boundaries.
446 4. Run the procedures that we developed on the cortex: we will present, for
447 each area, a short list of markers to identify that area; and we will also
448 present lists of “panels” of genes that can be used to delineate many areas
449 at once.
450 13
452 Develop algorithms to suggest a division of a structure into anatom-
453 ical parts
454 1. Explore dimensionality reduction algorithms applied to pixels: including
455 TODO
456 2. Explore dimensionality reduction algorithms applied to genes: including
457 TODO
458 3. Explore clustering algorithms applied to pixels: including TODO
459 4. Explore clustering algorithms applied to genes: including gene shaving,
460 TODO
461 5. Develop an algorithm to use dimensionality reduction and/or hierarchial
462 clustering to create anatomical maps
463 6. Run this algorithm on the cortex: present a hierarchial, genoarchitectonic
464 map of the cortex
465 ______________________________________________
466 stuff i dunno where to put yet (there is more scattered through grant-
467 oldtext):
468 Principle 4: Work in 2-D whenever possible
469 In anatomy, the manifold of interest is usually either defined by a combina-
470 tion of two relevant anatomical axes (todo), or by the surface of the structure
471 (as is the case with the cortex). In the former case, the manifold of interest is
472 a plane, but in the latter case it is curved. If the manifold is curved, there are
473 various methods for mapping the manifold into a plane.
474 The method that we will develop will begin by mapping the data into a
475 2-D plane. Although the manifold that characterized cortical areas is known
476 to be the cortical surface, it remains to be seen which method of mapping the
477 manifold into a plane is optimal for this application. We will compare mappings
478 which attempt to preserve size (such as the one used by Caret??) with mappings
479 which preserve angle (conformal maps).
480 Although there is much 2-D organization in anatomy, there are also struc-
481 tures whose shape is fundamentally 3-dimensional. If possible, we would like
482 the method we develop to include a statistical test that warns the user if the
483 assumption of 2-D structure seems to be wrong.
484 if we need citations for aim 3 significance, http://www.sciencedirect.
485 com/science?_ob=ArticleURL&_udi=B6WSS-4V70FHY-9&_user=4429&_coverDate=
486 12%2F26%2F2008&_rdoc=1&_fmt=full&_orig=na&_cdi=7054&_docanchor=&_acct=
487 C000059602&_version=1&_urlVersion=0&_userid=4429&md5=551eccc743a2bfe6e992eee0c3194203#
488 app2 has examples of genetic targeting to specific anatomical regions
489 —
490 note:
491 do we need to cite: no known markers, impressive results?
492 14