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

26

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 genes 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 Although the classifier itself may only look at the gene expression data within

70 2

71

72 each voxel before classifying that voxel, the learning algorithm which constructs

73 the classifier may look over the entire dataset. We can categorize score-based

74 feature selection methods depending on how the score of calculated. Often

75 the score calculation consists of assigning a sub-score to each voxel, and then

76 aggregating these sub-scores into a final score (the aggregation is often a sum or

77 a sum of squares). If only information from nearby voxels is used to calculate a

78 voxel’s sub-score, then we say it is a local scoring method. If only information

79 from the voxel itself is used to calculate a voxel’s sub-score, then we say it is a

80 pointwise scoring method.

81 Key questions when choosing a learning method are: What are the instances?

82 What are the features? How are the features chosen? Here are four principles

83 that outline our answers to these questions.

84 Principle 1: Combinatorial gene expression

85 Above, we defined an “instance” as the combination of a voxel with the

86 “associated gene expression data”. In our case this refers to the expression level

87 of genes within the voxel, but should we include the expression levels of all

88 genes, or only a few of them?

89 It is too much to hope that every anatomical region of interest will be iden-

90 tified by a single gene. For example, in the cortex, there are some areas which

91 are not clearly delineated by any gene included in the Allen Brain Atlas (ABA)

92 dataset. However, at least some of these areas can be delineated by looking

93 at combinations of genes (an example of an area for which multiple genes are

94 necessary and sufficient is provided in Preliminary Results).

95 Principle 2: Only look at combinations of small numbers of genes

96 When the classifier classifies a voxel, it is only allowed to look at the expres-

97 sion of the genes which have been selected as features. The more data that is

98 available to a classifier, the better that it can do. For example, perhaps there

99 are weak correlations over many genes that add up to a strong signal. So, why

100 not include every gene as a feature? The reason is that we wish to employ

101 the classifier in situations in which it is not feasible to gather data about every

102 gene. For example, if we want to use the expression of marker genes as a trigger

103 for some regionally-targeted intervention, then our intervention must contain a

104 molecular mechanism to check the expression level of each marker gene before

105 it triggers. It is currently infeasible to design a molecular trigger that checks

106 the level of more than a handful of genes. Similarly, if the goal is to develop a

107 procedure to do ISH on tissue samples in order to label their anatomy, then it

108 is infeasible to label more than a few genes. Therefore, we must select only a

109 few genes as features.

110 Principle 3: Use geometry in feature selection

111 When doing feature selection with score-based methods, the simplest thing

112 to do would be to score the performance of each voxel by itself and then com-

113 bine these scores (pointwise scoring). A more powerful approach is to also use

114 information about the geometric relations between each voxel and its neighbors;

115 3

116

117 this requires non-pointwise, local scoring methods. See Preliminary Results for

118 evidence of the complementary nature of pointwise and local scoring methods.

119 Principle 4: Work in 2-D whenever possible

120 There are many anatomical structures which are commonly characterized in

121 terms of a two-dimensional manifold. When it is known that the structure that

122 one is looking for is two-dimensional, the results may be improved by allowing

123 the analysis algorithm to take advantage of this prior knowledge. In addition,

124 it is easier for humans to visualize and work with 2-D data.

125 Therefore, when possible, the instances should represent pixels, not voxels.

126 Aim 2

127 Machine learning terminology: clustering

128 If one is given a dataset consisting merely of instances, with no class labels,

129 then analysis of the dataset is referred to as unsupervised learning in the jargon

130 of machine learning. One thing that you can do with such a dataset is to group

131 instances together. A set of similar instances is called a cluster, and the activity

132 of finding grouping the data into clusters is called clustering or cluster analysis.

133 The task of deciding how to carve up a structure into anatomical subregions

134 can be put into these terms. The instances are once again voxels (or pixels)

135 along with their associated gene expression profiles. We make the assumption

136 that voxels from the same subregion have similar gene expression profiles, at

137 least compared to the other subregions. This means that clustering voxels is

138 the same as finding potential subregions; we seek a partitioning of the voxels

139 into subregions, that is, into clusters of voxels with similar gene expression.

140 It is desirable to determine not just one set of subregions, but also how

141 these subregions relate to each other, if at all; perhaps some of the subregions

142 are more similar to each other than to the rest, suggesting that, although at a

143 fine spatial scale they could be considered separate, on a coarser spatial scale

144 they could be grouped together into one large subregion. This suggests the

145 outcome of clustering may be a hierarchial tree of clusters, rather than a single

146 set of clusters which partition the voxels. This is called hierarchial clustering.

147 Similarity scores

148 A crucial choice when designing a clustering method is how to measure

149 similarity, across either pairs of instances, or clusters, or both. There is much

150 overlap between scoring methods for feature selection (discussed above under

151 Aim 1) and scoring methods for similarity.

152 Spatially contiguous clusters; image segmentation

153 We have shown that aim 2 is a type of clustering task. In fact, it is a

154 special type of clustering task because we have an additional constraint on

155 clusters; voxels grouped together into a cluster must be spatially contiguous.

156 In Preliminary Results, we show that one can get reasonable results without

157 4

158

159 enforcing this constraint, however, we plan to compare these results against

160 other methods which guarantee contiguous clusters.

161 Perhaps the biggest source of continguous clustering algorithms is the field

162 of computer vision, which has produced a variety of image segmentation algo-

163 rithms. Image segmentation is the task of partitioning the pixels in a digital

164 image into clusters, usually contiguous clusters. Aim 2 is similar to an image

165 segmentation task. There are two main differences; in our task, there are thou-

166 sands of color channels (one for each gene), rather than just three. There are

167 imaging tasks which use more than three colors, however, for example multispec-

168 tral imaging and hyperspectral imaging, which are often used to process satellite

169 imagery. A more crucial difference is that there are various cues which are ap-

170 propriate for detecting sharp object boundaries in a visual scene but which are

171 not appropriate for segmenting abstract spatial data such as gene expression.

172 Although many image segmentation algorithms can be expected to work well

173 for segmenting other sorts of spatially arranged data, some of these algorithms

174 are specialized for visual images.

175 Dimensionality reduction

176 Unlike aim 1, there is no externally-imposed need to select only a handful

177 of informative genes for inclusion in the instances. However, some clustering

178 algorithms perform better on small numbers of features. There are techniques

179 which “summarize” a larger number of features using a smaller number of fea-

180 tures; these techniques go by the name of feature extraction or dimensionality

181 reduction. The small set of features that such a technique yields is called the

182 reduced feature set. After the reduced feature set is created, the instances may

183 be replaced by reduced instances, which have as their features the reduced fea-

184 ture set rather than the original feature set of all gene expression levels. Note

185 that the features in the reduced feature set do not necessarily correspond to

186 genes; each feature in the reduced set may be any function of the set of gene

187 expression levels.

188 Another use for dimensionality reduction is to visualize the relationships

189 between subregions. For example, one might want to make a 2-D plot upon

190 which each subregion is represented by a single point, and with the property

191 that subregions with similar gene expression profiles should be nearby on the

192 plot (that is, the property that distance between pairs of points in the plot

193 should be proportional to some measure of dissimilarity in gene expression). It

194 is likely that no arrangement of the points on a 2-D plan will exactly satisfy

195 this property – however, dimensionality reduction techniques allow one to find

196 arrangements of points that approximately satisfy that property. Note that

197 in this application, dimensionality reduction is being applied after clustering;

198 whereas in the previous paragraph, we were talking about using dimensionality

199 reduction before clustering.

200 Clustering genes rather than voxels

201 Although the ultimate goal is to cluster the instances (voxels or pixels), one

202 strategy to achieve this goal is to first cluster the features (genes). There are

203 5

204

205 two ways that clusters of genes could be used.

206 Gene clusters could be used as part of dimensionality reduction: rather than

207 have one feature for each gene, we could have one reduced feature for each gene

208 cluster.

209 Gene clusters could also be used to directly yield a clustering on instances.

210 This is because many genes have an expression pattern which seems to pick

211 out a single, spatially continguous subregion. Therefore, it seems likely that an

212 anatomically interesting subregion will have multiple genes which each individ-

213 ually pick it out1. This suggests the following procedure: cluster together genes

214 which pick out similar subregions, and then to use the more popular common

215 subregions as the final clusters. In the Preliminary Data we show that a num-

216 ber of anatomically recognized cortical regions, as well as some “superregions”

217 formed by lumping together a few regions, are associated with gene clusters in

218 this fashion.

219 Aim 3

220 Background

221 The cortex is divided into areas and layers. To a first approximation, the

222 parcellation of the cortex into areas can be drawn as a 2-D map on the surface of

223 the cortex. In the third dimension, the boundaries between the areas continue

224 downwards into the cortical depth, perpendicular to the surface. The layer

225 boundaries run parallel to the surface. One can picture an area of the cortex as

226 a slice of many-layered cake.

227 Although it is known that different cortical areas have distinct roles in both

228 normal functioning and in disease processes, there are no known marker genes

229 for many cortical areas. When it is necessary to divide a tissue sample into

230 cortical areas, this is a manual process that requires a skilled human to combine

231 multiple visual cues and interpret them in the context of their approximate

232 location upon the cortical surface.

233 Even the questions of how many areas should be recognized in cortex, and

234 what their arrangement is, are still not completely settled. A proposed division

235 of the cortex into areas is called a cortical map. In the rodent, the lack of a

236 single agreed-upon map can be seen by contrasting the recent maps given by

237 Swanson?? on the one hand, and Paxinos and Franklin?? on the other. While

238 the maps are certainly very similar in their general arrangement, significant

239 differences remain in the details.

240 Significance

241 The method developed in aim (1) will be applied to each cortical area to find

242 a set of marker genes such that the combinatorial expression pattern of those

243 __________________________

244 1This would seem to contradict our finding in aim 1 that some cortical areas are combina-

245 torially coded by multiple genes. However, it is possible that the currently accepted cortical

246 maps divide the cortex into subregions which are unnatural from the point of view of gene

247 expression; perhaps there is some other way to map the cortex for which each subregion can

248 be identified by single genes.

249 6

250

251 genes uniquely picks out the target area. Finding marker genes will be useful

252 for drug discovery as well as for experimentation because marker genes can be

253 used to design interventions which selectively target individual cortical areas.

254 The application of the marker gene finding algorithm to the cortex will

255 also support the development of new neuroanatomical methods. In addition to

256 finding markers for each individual cortical areas, we will find a small panel

257 of genes that can find many of the areal boundaries at once. This panel of

258 marker genes will allow the development of an ISH protocol that will allow

259 experimenters to more easily identify which anatomical areas are present in

260 small samples of cortex.

261 The method developed in aim (3) will provide a genoarchitectonic viewpoint

262 that will contribute to the creation of a better map. The development of present-

263 day cortical maps was driven by the application of histological stains. It is

264 conceivable that if a different set of stains had been available which identified

265 a different set of features, then the today’s cortical maps would have come out

266 differently. Since the number of classes of stains is small compared to the number

267 of genes, it is likely that there are many repeated, salient spatial patterns in

268 the gene expression which have not yet been captured by any stain. Therefore,

269 current ideas about cortical anatomy need to incorporate what we can learn

270 from looking at the patterns of gene expression.

271 While we do not here propose to analyze human gene expression data, it is

272 conceivable that the methods we propose to develop could be used to suggest

273 modifications to the human cortical map as well.

274 Related work

275 There does not appear to be much work on the automated analysis of spatial

276 gene expression data.

277 There is a substantial body of work on the analysis of gene expression data,

278 however, most of this concerns gene expression data which is not fundamentally

279 spatial.

280 As noted above, there has been much work on both supervised learning and

281 clustering, and there are many available algorithms for each. However, the

282 completion of Aims 1 and 2 involves more than just choosing between a set of

283 existing algorithms, and will constitute a substantial contribution to biology.

284 The algorithms require the scientist to provide a framework for representing the

285 problem domain, and the way that this framework is set up has a large impact

286 on performance. Creating a good framework can require creatively reconcep-

287 tualizing the problem domain, and is not merely a mechanical “fine-tuning”

288 of numerical parameters. For example, we believe that domain-specific scoring

289 measures (such as gradient similarity, which is discussed in Preliminary Work)

290 may be necessary in order to achieve the best results in this application.

291 We are aware of two existing efforts to relate spatial gene expression data to

292 anatomy through computational methods.

293 [?] describes an analysis of the anatomy of the hippocampus using the ABA

294 dataset. In addition to manual analysis, two clustering methods were employed,

295 7

296

297 a modified Non-negative Matrix Factorization (NNMF), and a hierarchial bifur-

298 cation clustering scheme based on correlation as the similarity score. The paper

299 yielded impressive results, proving the usefulness of such research. We have run

300 NNMF on the cortical dataset and while the results are promising (see Prelim-

301 inary Data), we think that it will be possible to find a better method2 (we also

302 think that more automation of the parts that this paper’s authors did manually

303 will be possible).

304 and [?] describes AGEA. todo

305 __________________________

306 2We ran “vanilla” NNMF, whereas the paper under discussion used a modified method.

307 Their main modification consisted of adding a soft spatial contiguity constraint. However,

308 on our dataset, NNMF naturally produced spatially contiguous clusters, so no additional

309 constraint was needed. The paper under discussion mentions that they also tried a hierarchial

310 variant of NNMF, but since they didn’t report its results, we assume that those result were

311 not any more impressive than the results of the non-hierarchial variant.

312 8

313

314 Preliminary work

315 Format conversion between SEV, MATLAB, NIFTI

316 todo

317 Flatmap of cortex

318 todo

319 Using combinations of multiple genes is necessary and sufficient to

320 delineate some cortical areas

321 Here we give an example of a cortical area which is not marked by any

322 single gene, but which can be identified combinatorially. according to logistic

323 regression, gene wwc13 is the best fit single gene for predicting whether or not a

324 pixel on the cortical surface belongs to the motor area (area MO). The upper-left

325 picture in Figure shows wwc1’s spatial expression pattern over the cortex. The

326 lower-right boundary of MO is represented reasonably well by this gene, however

327 the gene overshoots the upper-left boundary. This flattened 2-D representation

328 does not show it, but the area corresponding to the overshoot is the medial

329 surface of the cortex. MO is only found on the lateral surface (todo).

330 Gnee mtif24 is shown in figure the upper-right of Fig. . Mtif2 captures MO’s

331 upper-left boundary, but not its lower-right boundary. Mtif2 does not express

332 very much on the medial surface. By adding together the values at each pixel

333 in these two figures, we get the lower-left of Figure . This combination captures

334 area MO much better than any single gene.

335 Correlation todo

336 Conditional entropy todo

337 Gradient similarity todo

338 Geometric and pointwise scoring methods provide complementary

339 information

340 To show that local geometry can provide useful information that cannot be

341 detected via pointwise analyses, consider Fig. . The top row of Fig. displays the

342 3 genes which most match area AUD, according to a pointwise method5. The

343 bottom row displays the 3 genes which most match AUD according to a method

344 which considers local geometry6 The pointwise method in the top row identifies

345 __________________________

346 3“WW, C2 and coiled-coil domain containing 1”; EntrezGene ID 211652

347 4“mitochondrial translational initiation factor 2”; EntrezGene ID 76784

348 5For each gene, a logistic regression in which the response variable was whether or not a

349 surface pixel was within area AUD, and the predictor variable was the value of the expression

350 of the gene underneath that pixel. The resulting scores were used to rank the genes in terms

351 of how well they predict area AUD.

352 6For each gene the gradient similarity (see section ??) between (a) a map of the expression

353 of each gene on the cortical surface and (b) the shape of area AUD, was calculated, and this

354 was used to rank the genes.

355 9

356

357

358

359 Figure 1: Upper left: wwc1. Upper right: mtif2. Lower left: wwc1 + mtif2

360 (each pixel’s value on the lower left is the sum of the corresponding pixels in

361 the upper row). Within each picture, the vertical axis roughly corresponds to

362 anterior at the top and posterior at the bottom, and the horizontal axis roughly

363 corresponds to medial at the left and lateral at the right. The red outline is

364 the boundary of region MO. Pixels are colored approximately according to the

365 density of expressing cells underneath each pixel, with red meaning a lot of

366 expression and blue meaning little.

367 10

368

369

370

371 Figure 2: The top row shows the three genes which (individually) best predict

372 area AUD, according to logistic regression. The bottom row shows the three

373 genes which (individually) best match area AUD, according to gradient similar-

374 ity. From left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a,

375 Ptk7, Aph1a again, and Lepr

376 genes which express more strongly in AUD than outside of it; its weakness is that

377 this includes many areas which don’t have a salient border matching the areal

378 border. The geometric method identifies genes whose salient expression border

379 seems to partially line up with the border of AUD; its weakness is that this

380 includes genes which don’t express over the entire area. Genes which have high

381 rankings using both pointwise and border criteria, such as Aph1a in the example,

382 may be particularly good markers. None of these genes are, individually, a

383 perfect marker for AUD; we deliberately chose a “difficult” area in order to

384 better contrast pointwise with geometric methods.

385 Areas which can be identified by single genes

386 todo

387 Specific to Aim 1 (and Aim 3)

388 Forward stepwise logistic regression todo

389 SVM on all genes at once

390 In order to see how well one can do when looking at all genes at once, we

391 ran a support vector machine to classify cortical surface pixels based on their

392 gene expression profiles. We achieved classification accuracy of about 81%7.

393 As noted above, however, a classifier that looks at all the genes at once isn’t

394 practically useful.

395 ____________

396 75-fold cross-validation.

397 11

398

399 The requirement to find combinations of only a small number of genes limits

400 us from straightforwardly applying many of the most simple techniques from

401 the field of supervised machine learning. In the parlance of machine learning,

402 our task combines feature selection with supervised learning.

403 Decision trees

404 todo

405 Specific to Aim 2 (and Aim 3)

406 Raw dimensionality reduction results

407 todo

408 (might want to incld nnMF since mentioned above)

409 Dimensionality reduction plus K-means or spectral clustering

410 Many areas are captured by clusters of genes

411 todo

412 todo

413 12

414

415 Research plan

416 todo amongst other things:

417 Develop algorithms that find genetic markers for anatomical re-

418 gions

419 1. Develop scoring measures for evaluating how good individual genes are at

420 marking areas: we will compare pointwise, geometric, and information-

421 theoretic measures.

422 2. Develop a procedure to find single marker genes for anatomical regions: for

423 each cortical area, by using or combining the scoring measures developed,

424 we will rank the genes by their ability to delineate each area.

425 3. Extend the procedure to handle difficult areas by using combinatorial cod-

426 ing: for areas that cannot be identified by any single gene, identify them

427 with a handful of genes. We will consider both (a) algorithms that incre-

428 mentally/greedily combine single gene markers into sets, such as forward

429 stepwise regression and decision trees, and also (b) supervised learning

430 techniques which use soft constraints to minimize the number of features,

431 such as sparse support vector machines.

432 4. Extend the procedure to handle difficult areas by combining or redrawing

433 the boundaries: An area may be difficult to identify because the bound-

434 aries are misdrawn, or because it does not “really” exist as a single area,

435 at least on the genetic level. We will develop extensions to our procedure

436 which (a) detect when a difficult area could be fit if its boundary were

437 redrawn slightly, and (b) detect when a difficult area could be combined

438 with adjacent areas to create a larger area which can be fit.

439 Apply these algorithms to the cortex

440 1. Create open source format conversion tools: we will create tools to bulk

441 download the ABA dataset and to convert between SEV, NIFTI and MAT-

442 LAB formats.

443 2. Flatmap the ABA cortex data: map the ABA data onto a plane and draw

444 the cortical area boundaries onto it.

445 3. Find layer boundaries: cluster similar voxels together in order to auto-

446 matically find the cortical layer boundaries.

447 4. Run the procedures that we developed on the cortex: we will present, for

448 each area, a short list of markers to identify that area; and we will also

449 present lists of “panels” of genes that can be used to delineate many areas

450 at once.

451 13

452

453 Develop algorithms to suggest a division of a structure into anatom-

454 ical parts

455 1. Explore dimensionality reduction algorithms applied to pixels: including

456 TODO

457 2. Explore dimensionality reduction algorithms applied to genes: including

458 TODO

459 3. Explore clustering algorithms applied to pixels: including TODO

460 4. Explore clustering algorithms applied to genes: including gene shaving,

461 TODO

462 5. Develop an algorithm to use dimensionality reduction and/or hierarchial

463 clustering to create anatomical maps

464 6. Run this algorithm on the cortex: present a hierarchial, genoarchitectonic

465 map of the cortex

466 ______________________________________________

467 stuff i dunno where to put yet (there is more scattered through grant-

468 oldtext):

469 Principle 4: Work in 2-D whenever possible

470 In anatomy, the manifold of interest is usually either defined by a combina-

471 tion of two relevant anatomical axes (todo), or by the surface of the structure

472 (as is the case with the cortex). In the former case, the manifold of interest is

473 a plane, but in the latter case it is curved. If the manifold is curved, there are

474 various methods for mapping the manifold into a plane.

475 The method that we will develop will begin by mapping the data into a

476 2-D plane. Although the manifold that characterized cortical areas is known

477 to be the cortical surface, it remains to be seen which method of mapping the

478 manifold into a plane is optimal for this application. We will compare mappings

479 which attempt to preserve size (such as the one used by Caret??) with mappings

480 which preserve angle (conformal maps).

481 Although there is much 2-D organization in anatomy, there are also struc-

482 tures whose shape is fundamentally 3-dimensional. If possible, we would like

483 the method we develop to include a statistical test that warns the user if the

484 assumption of 2-D structure seems to be wrong.

485 if we need citations for aim 3 significance, http://www.sciencedirect.

486 com/science?_ob=ArticleURL&_udi=B6WSS-4V70FHY-9&_user=4429&_coverDate=

487 12%2F26%2F2008&_rdoc=1&_fmt=full&_orig=na&_cdi=7054&_docanchor=&_acct=

488 C000059602&_version=1&_urlVersion=0&_userid=4429&md5=551eccc743a2bfe6e992eee0c3194203#

489 app2 has examples of genetic targeting to specific anatomical regions

490 —

491 note:

492 do we need to cite: no known markers? impressive results?

493 14

494

495