1 Specific aims

2 Massivenew datasets obtained with techniques such as in situ hybridization (ISH), immunohistochemistry, or in situ trans-

3 genic reporter allow the expression levels of many genes at many locations to be compared. Our goal is to develop automated

4 methods to relate spatial variation in gene expression to anatomy. We want to find marker genes for specific anatomical

5 regions, and also to draw new anatomical maps based on gene expression patterns. We have three specific aims:

6 (1) develop an algorithm to screen spatial gene expression data for combinations of marker genes which selectively target

7 anatomical regions

8 (2) develop an algorithm to suggest new ways of carving up a structure into anatomical regions, based on spatial patterns

9 in gene expression

10 (3) create a 2-D “flat map” dataset of the mouse cerebral cortex that contains a flattened version of the Allen Mouse

11 Brain Atlas ISH data, as well as the boundaries of cortical anatomical areas. This will involve extending the functionality of

12 Caret, an existing open-source scientific imaging program. Use this dataset to validate the methods developed in (1) and (2).

13 In addition to validating the usefulness of the algorithms, the application of these methods to cerebral cortex will produce

14 immediate benefits, because there are currently no known genetic markers for many cortical areas. The results of the project

15 will support the development of new ways to selectively target cortical areas, and it will support the development of a

16 method for identifying the cortical areal boundaries present in small tissue samples.

17 All algorithms that we develop will be implemented in an open-source software toolkit. The toolkit, as well as the

18 machine-readable datasets developed in aim (3), will be published and freely available for others to use.

19 Background and significance

20 Aim 1

21 Machine learning terminology: supervised learning

22 The task of looking for marker genes for anatomical regions means that one is looking for a set of genes such that, if the

23 expression level of those genes is known, then the locations of the regions can be inferred.

24 If we define the regions so that they cover the entire anatomical structure to be divided, then instead of saying that we

25 are using gene expression to find the locations of the regions, we may say that we are using gene expression to determine to

26 which region each voxel within the structure belongs. We call this a classification task, because each voxel is being assigned

27 to a class (namely, its region).

28 Therefore, an understanding of the relationship between the combination of their expression levels and the locations of

29 the regions may be expressed as a function. The input to this function is a voxel, along with the gene expression levels

30 within that voxel; the output is the regional identity of the target voxel, that is, the region to which the target voxel belongs.

31 We call this function a classifier. In general, the input to a classifier is called an instance, and the output is called a label

32 (or a class label).

33 The object of aim 1 is not to produce a single classifier, but rather to develop an automated method for determining a

34 classifier for any known anatomical structure. Therefore, we seek a procedure by which a gene expression dataset may be

35 analyzed in concert with an anatomical atlas in order to produce a classifier. Such a procedure is a type of a machine learning

36 procedure. The construction of the classifier is called training (also learning), and the initial gene expression dataset used

37 in the construction of the classifier is called training data.

38 In the machine learning literature, this sort of procedure may be thought of as a supervised learning task, defined as a

39 task in which the goal is to learn a mapping from instances to labels, and the training data consists of a set of instances

40 (voxels) for which the labels (regions) are known.

41 Each gene expression level is called a feature, and the selection of which genes1 to include is called feature selection.

42 Feature selection is one component of the task of learning a classifier. Some methods for learning classifiers start out with

43 a separate feature selection phase, whereas other methods combine feature selection with other aspects of training.

44 One class of feature selection methods assigns some sort of score to each candidate gene. The top-ranked genes are then

45 chosen. Some scoring measures can assign a score to a set of selected genes, not just to a single gene; in this case, a dynamic

46 procedure may be used in which features are added and subtracted from the selected set depending on how much they raise

47 the score. Such procedures are called “stepwise” or “greedy”.

48 Although the classifier itself may only look at the gene expression data within each voxel before classifying that voxel, the

49 learning algorithm which constructs the classifier may look over the entire dataset. We can categorize score-based feature

50 selection methods depending on how the score of calculated. Often the score calculation consists of assigning a sub-score to

51 each voxel, and then aggregating these sub-scores into a final score (the aggregation is often a sum or a sum of squares). If

52 only information from nearby voxels is used to calculate a voxel’s sub-score, then we say it is a local scoring method. If only

53 information from the voxel itself is used to calculate a voxel’s sub-score, then we say it is a pointwise scoring method.

54 Key questions when choosing a learning method are: What are the instances? What are the features? How are the

55 features chosen? Here are four principles that outline our answers to these questions.

56 Principle 1: Combinatorial gene expression It is too much to hope that every anatomical region of interest will be

57 identified by a single gene. For example, in the cortex, there are some areas which are not clearly delineated by any gene

58 included in the Allen Brain Atlas (ABA) dataset. However, at least some of these areas can be delineated by looking at

59 combinations of genes (an example of an area for which multiple genes are necessary and sufficient is provided in Preliminary

60 Results). Therefore, each instance should contain multiple features (genes).

61 Principle 2: Only look at combinations of small numbers of genes When the classifier classifies a voxel, it is

62 only allowed to look at the expression of the genes which have been selected as features. The more data that is available to

63 a classifier, the better that it can do. For example, perhaps there are weak correlations over many genes that add up to a

64 strong signal. So, why not include every gene as a feature? The reason is that we wish to employ the classifier in situations

65 in which it is not feasible to gather data about every gene. For example, if we want to use the expression of marker genes as

66 a trigger for some regionally-targeted intervention, then our intervention must contain a molecular mechanism to check the

67 expression level of each marker gene before it triggers. It is currently infeasible to design a molecular trigger that checks the

68 level of more than a handful of genes. Similarly, if the goal is to develop a procedure to do ISH on tissue samples in order

69 to label their anatomy, then it is infeasible to label more than a few genes. Therefore, we must select only a few genes as

70 features.

71 Principle 3: Use geometry in feature selection

72 _________________________________________

73 1Strictly speaking, the features are gene expression levels, but we’ll call them genes.

74 When doing feature selection with score-based methods, the simplest thing to do would be to score the performance of

75 each voxel by itself and then combine these scores (pointwise scoring). A more powerful approach is to also use information

76 about the geometric relations between each voxel and its neighbors; this requires non-pointwise, local scoring methods. See

77 Preliminary Results for evidence of the complementary nature of pointwise and local scoring methods.

78 Principle 4: Work in 2-D whenever possible

79 There are many anatomical structures which are commonly characterized in terms of a two-dimensional manifold. When

80 it is known that the structure that one is looking for is two-dimensional, the results may be improved by allowing the analysis

81 algorithm to take advantage of this prior knowledge. In addition, it is easier for humans to visualize and work with 2-D

82 data.

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

84 Related work

85 There is a substantial body of work on the analysis of gene expression data, most of this concerns gene expression data

86 which is not fundamentally spatial2.

87 As noted above, there has been much work on both supervised learning and there are many available algorithms for

88 each. However, the algorithms require the scientist to provide a framework for representing the problem domain, and the

89 way that this framework is set up has a large impact on performance. Creating a good framework can require creatively

90 reconceptualizing the problem domain, and is not merely a mechanical “fine-tuning” of numerical parameters. For example,

91 we believe that domain-specific scoring measures (such as gradient similarity, which is discussed in Preliminary Work) may

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

93 We are aware of five existing efforts to find marker genes using spatial gene expression data using automated methods.

94 GeneAtlas[1] and EMAGE [11] allow the user to construct a search query by demarcating regions and then specifing

95 either the strength of expression or the name of another gene or dataset whose expression pattern is to be matched. For

96 the similiarity score (match score), GeneAtlas appears to use strength of expression, and EMAGE uses Jaccard similarity,

97 which is equal to the number of true pixels in the intersection of the two images, divided by the number of pixels in their

98 union. Neither GeneAtlas nor EMAGE allow one to search for combinations of genes that together match a region.

99 [6 ] describes AGEA, ”Anatomic Gene Expression Atlas”. AGEA has three components:

100 * Gene Finder: The user selects a seed voxel and the system (1) chooses a cluster which includes the seed voxel, (2)

101 yields a list of genes which are overexpressed in that cluster. (note: the ABA website also contains pre-prepared lists of

102 overexpressed genes for selected structures)

103 * Correlation: The user selects a seed voxel and the shows the user how much correlation there is between the gene

104 expression profile of the seed voxel and every other voxel.

105 * Clusters: AGEA includes a precomputed hierarchial clustering of voxels based on a recursive bifurcation algorithm

106 with correlation as the similarity metric.

107 Gene Finder is different from our Aim 1 in at least three ways. First, Gene Finder finds only single genes, whereas we

108 will also look for combinations of genes. Second, gene finder can only use overexpression as a marker, whereas we will also

109 search for underexpression. Third, Gene Finder uses a simple pointwise score3, whereas we will also use geometric scores

110 such as gradient similarity. The Preliminary Data section contains evidence that each of our three choices is the right one.

111 [? ] looks at the mean expression level of genes within anatomical regions, and applies a Student’s t-test with Bonferroni

112 correction to determine whether the mean expression level of a gene is significantly higher in the target region. Like AGEA,

113 this is a pointwise measure (only the mean expression level per pixel is being analyzed), it is not being used to look for

114 underexpression, and does not look for combinations of genes.

115 [4 ] describes a technique to find combinations of marker genes to pick out an anatomical region. They use an evolutionary

116 algorithm to evolve logical operators which combine boolean (thresholded) images in order to match a target image. Their

117 match score is Jaccard similarity.

118 In summary, there has been fruitful work on finding marker genes, however, only one of the previous projects explores

119 combinations of marker genes, and none of these publications compare the results obtained by using different algorithms or

120 scoring methods.

121 Aim 2

122 Machine learning terminology: clustering

123 If one is given a dataset consisting merely of instances, with no class labels, then analysis of the dataset is referred to as

124 unsupervised learning in the jargon of machine learning. One thing that you can do with such a dataset is to group instances

125 _________________________________________

126 2By “fundamentally spatial” we mean that there is information from a large number of spatial locations indexed by spatial coordinates; not

127 just data which has only a few different locations or which is indexed by anatomical label.

128 3“Expression energy ratio”, which captures overexpression.

129 together. A set of similar instances is called a cluster, and the activity of finding grouping the data into clusters is called

130 clustering or cluster analysis.

131 The task of deciding how to carve up a structure into anatomical regions can be put into these terms. The instances are

132 once again voxels (or pixels) along with their associated gene expression profiles. We make the assumption that voxels from

133 the same region have similar gene expression profiles, at least compared to the other regions. This means that clustering

134 voxels is the same as finding potential regions; we seek a partitioning of the voxels into regions, that is, into clusters of voxels

135 with similar gene expression.

136 It is desirable to determine not just one set of regions, but also how these regions relate to each other, if at all; perhaps

137 some of the regions are more similar to each other than to the rest, suggesting that, although at a fine spatial scale they

138 could be considered separate, on a coarser spatial scale they could be grouped together into one large region. This suggests

139 the outcome of clustering may be a hierarchial tree of clusters, rather than a single set of clusters which partition the voxels.

140 This is called hierarchial clustering.

141 Similarity scores

142 A crucial choice when designing a clustering method is how to measure similarity, across either pairs of instances, or

143 clusters, or both. There is much overlap between scoring methods for feature selection (discussed above under Aim 1) and

144 scoring methods for similarity.

145 Spatially contiguous clusters; image segmentation

146 We have shown that aim 2 is a type of clustering task. In fact, it is a special type of clustering task because we have

147 an additional constraint on clusters; voxels grouped together into a cluster must be spatially contiguous. In Preliminary

148 Results, we show that one can get reasonable results without enforcing this constraint, however, we plan to compare these

149 results against other methods which guarantee contiguous clusters.

150 Perhaps the biggest source of continguous clustering algorithms is the field of computer vision, which has produced a

151 variety of image segmentation algorithms. Image segmentation is the task of partitioning the pixels in a digital image into

152 clusters, usually contiguous clusters. Aim 2 is similar to an image segmentation task. There are two main differences; in

153 our task, there are thousands of color channels (one for each gene), rather than just three. There are imaging tasks which

154 use more than three colors, however, for example multispectral imaging and hyperspectral imaging, which are often used

155 to process satellite imagery. A more crucial difference is that there are various cues which are appropriate for detecting

156 sharp object boundaries in a visual scene but which are not appropriate for segmenting abstract spatial data such as gene

157 expression. Although many image segmentation algorithms can be expected to work well for segmenting other sorts of

158 spatially arranged data, some of these algorithms are specialized for visual images.

159 Dimensionality reduction In this section, we discuss reducing the length of the per-pixel gene expression feature

160 vector. By “dimension”, we mean the dimension of this vector, not the spatial dimension of the underlying data.

161 Unlike aim 1, there is no externally-imposed need to select only a handful of informative genes for inclusion in the

162 instances. However, some clustering algorithms perform better on small numbers of features. There are techniques which

163 “summarize” a larger number of features using a smaller number of features; these techniques go by the name of feature

164 extraction or dimensionality reduction. The small set of features that such a technique yields is called the reduced feature

165 set. After the reduced feature set is created, the instances may be replaced by reduced instances, which have as their features

166 the reduced feature set rather than the original feature set of all gene expression levels. Note that the features in the reduced

167 feature set do not necessarily correspond to genes; each feature in the reduced set may be any function of the set of gene

168 expression levels.

169 Dimensionality reduction before clustering is useful on large datasets. First, because the number of features in the

170 reduced data set is less than in the original data set, the running time of clustering algorithms may be much less. Second,

171 it is thought that some clustering algorithms may give better results on reduced data.

172 Another use for dimensionality reduction is to visualize the relationships between regions after clustering. For example,

173 one might want to make a 2-D plot upon which each region is represented by a single point, and with the property that regions

174 with similar gene expression profiles should be nearby on the plot (that is, the property that distance between pairs of points

175 in the plot should be proportional to some measure of dissimilarity in gene expression). It is likely that no arrangement of

176 the points on a 2-D plan will exactly satisfy this property – however, dimensionality reduction techniques allow one to find

177 arrangements of points that approximately satisfy that property. Note that in this application, dimensionality reduction

178 is being applied after clustering; whereas in the previous paragraph, we were talking about using dimensionality reduction

179 before clustering.

180 Clustering genes rather than voxels

181 Although the ultimate goal is to cluster the instances (voxels or pixels), one strategy to achieve this goal is to first cluster

182 the features (genes). There are two ways that clusters of genes could be used.

183 Gene clusters could be used as part of dimensionality reduction: rather than have one feature for each gene, we could

184 have one reduced feature for each gene cluster.

185 Gene clusters could also be used to directly yield a clustering on instances. This is because many genes have an expression

186 patternwhich seems to pick out a single, spatially continguous region. Therefore, it seems likely that an anatomically

187 interesting region will have multiple genes which each individually pick it out4. This suggests the following procedure:

188 cluster together genes which pick out similar regions, and then to use the more popular common regions as the final clusters.

189 In the Preliminary Data we show that a number of anatomically recognized cortical regions, as well as some “superregions”

190 formed by lumping together a few regions, are associated with gene clusters in this fashion.

191 The task of clustering both the instances and the features is called co-clustering, and there are a number of co-clustering

192 algorithms.

193 Related work

194 We are aware of five existing efforts to cluster spatial gene expression data.

195 [9 ] describes an analysis of the anatomy of the hippocampus using the ABA dataset. In addition to manual analysis,

196 two clustering methods were employed, a modified Non-negative Matrix Factorization (NNMF), and a hierarchial recursive

197 bifurcation clustering scheme based on correlation as the similarity score. The paper yielded impressive results, proving

198 the usefulness of computational genomic anatomy. We have run NNMF on the cortical dataset5 and while the results are

199 promising (see Preliminary Data), we think that it will be possible to find an even better method.

200 AGEA’s[6] hierarchial clustering was described above. EMAGE[11] allows the user to select a dataset from among a

201 large number of alternatives, or by running a search query, and then to cluster the genes within that dataset. Clustering is

202 hierarchial complete linkage clustering with un-centred correlation as the similarity score.

203 todo [?]

204 In an interesting twist, [4] applies their technique for finding combinations of marker genes for the purpose of clustering

205 genes around a “seed gene”. The way they do this is by using the pattern of expression of the seed gene as the target image,

206 and then searching for other genes which can be combined to reproduce this pattern. Those other genes which are found

207 are considered to be related to the seed. The same team also describes a method[10] for finding “association rules” such as,

208 “if this voxel is expressed in by any gene, then that voxel is probably also expressed in by the same gene”. This could be

209 useful as part of a procedure for clustering voxels.

210 In summary, although these projects obtained clusterings, there has not been much comparison between different algo-

211 rithms or scoring methods, so it is likely that the best clustering method for this application has not yet been found. Also,

212 none of these projects did a separate dimensionality reduction step before clustering pixels, or tried to cluster genes first in

213 order to guide the clustering of pixels into spatial regions, or used co-clustering algorithms.

214 Aim 3

215 Background

216 The cortex is divided into areas and layers. To a first approximation, the parcellation of the cortex into areas can

217 be drawn as a 2-D map on the surface of the cortex. In the third dimension, the boundaries between the areas continue

218 downwards into the cortical depth, perpendicular to the surface. The layer boundaries run parallel to the surface. One can

219 picture an area of the cortex as a slice of many-layered cake.

220 Although it is known that different cortical areas have distinct roles in both normal functioning and in disease processes,

221 there are no known marker genes for many cortical areas. When it is necessary to divide a tissue sample into cortical areas,

222 this is a manual process that requires a skilled human to combine multiple visual cues and interpret them in the context of

223 their approximate location upon the cortical surface.

224 Even the questions of how many areas should be recognized in cortex, and what their arrangement is, are still not

225 completely settled. A proposed division of the cortex into areas is called a cortical map. In the rodent, the lack of a

226 single agreed-upon map can be seen by contrasting the recent maps given by Swanson[8] on the one hand, and Paxinos

227 and Franklin[7] on the other. While the maps are certainly very similar in their general arrangement, significant differences

228 remain in the details.

229 The Allen Mouse Brain Atlas dataset

230 The Allen Mouse Brain Atlas (ABA) data was produced by doing in-situ hybridization on slices of male, 56-day-old

231 C57BL/6J mouse brains. Pictures were taken of the processed slice, and these pictures were semi-automatically analyzed

232 in order to create a digital measurement of gene expression levels at each location in each slice. Per slice, cellular spatial

233 _________________________________________

234 4This would seem to contradict our finding in aim 1 that some cortical areas are combinatorially coded by multiple genes. However, it is

235 possible that the currently accepted cortical maps divide the cortex into regions which are unnatural from the point of view of gene expression;

236 perhaps there is some other way to map the cortex for which each region can be identified by single genes. Another possibility is that, although

237 the cluster prototype fits an anatomical region, the individual genes are each somewhat different from the prototype.

238 5We ran “vanilla” NNMF, whereas the paper under discussion used a modified method. Their main modification consisted of adding a soft

239 spatial contiguity constraint. However, on our dataset, NNMF naturally produced spatially contiguous clusters, so no additional constraint was

240 needed. The paper under discussion also mentions that they tried a hierarchial variant of NNMF, which we have not yet tried.

241 resolution is achieved. Using this method, a single physical slice can only be used to measure one single gene; many different

242 mouse brains were needed in order to measure the expression of many genes.

243 Next, an automated nonlinear alignment procedure located the 2D data from the various slices in a single 3D coordinate

244 system. In the final 3D coordinate system, voxels are cubes with 200 microns on a side. There are 67x41x58 = 159,326

245 voxels in the 3D coordinate system, of which 51,533 are in the brain[6].

246 Mus musculus, the common house mouse, is thought to contain about 22,000 protein-coding genes[13]. The ABA contains

247 data on about 20,000 genes in sagittal sections, out of which over 4,000 genes are also measured in coronal sections. Our

248 dataset is derived from only the coronal subset of the ABA, because the sagittal data does not cover the entire cortex, and

249 also has greater registration error[6]. Genes were selected by the Allen Institute for coronal sectioning based on, “classes of

250 known neuroscientific interest... or through post hoc identification of a marked non-ubiquitous expression pattern”[6].

251 The ABA is not the only large public spatial gene expression dataset. Other such resources include GENSAT[3],

252 GenePaint[12], its sister project GeneAtlas[1], BGEM[5], EMAGE[11], EurExpress6, EADHB7, MAMEP8, Xenbase9, ZFIN[?],

253 Aniseed10, VisiGene11, GEISHA[?], Fruitfly.org[?], COMPARE12 todo. With the exception of the ABA, GenePaint, and

254 EMAGE, most of these resources have not (yet) extracted the expression intensity from the ISH images and registered the

255 results into a single 3-D space, and only ABA and EMAGE make this form of data available for public download from the

256 website13. Many of these resources focus on developmental gene expression.

257 Significance

258 The method developed in aim (1) will be applied to each cortical area to find a set of marker genes such that the

259 combinatorial expression pattern of those genes uniquely picks out the target area. Finding marker genes will be useful for

260 drug discovery as well as for experimentation because marker genes can be used to design interventions which selectively

261 target individual cortical areas.

262 The application of the marker gene finding algorithm to the cortex will also support the development of new neuroanatom-

263 ical methods. In addition to finding markers for each individual cortical areas, we will find a small panel of genes that can

264 find many of the areal boundaries at once. This panel of marker genes will allow the development of an ISH protocol that

265 will allow experimenters to more easily identify which anatomical areas are present in small samples of cortex.

266 The method developed in aim (3) will provide a genoarchitectonic viewpoint that will contribute to the creation of

267 a better map. The development of present-day cortical maps was driven by the application of histological stains. It is

268 conceivable that if a different set of stains had been available which identified a different set of features, then the today’s

269 cortical maps would have come out differently. Since the number of classes of stains is small compared to the number of

270 genes, it is likely that there are many repeated, salient spatial patterns in the gene expression which have not yet been

271 captured by any stain. Therefore, current ideas about cortical anatomy need to incorporate what we can learn from looking

272 at the patterns of gene expression.

273 While we do not here propose to analyze human gene expression data, it is conceivable that the methods we propose to

274 develop could be used to suggest modifications to the human cortical map as well.

275 Related work

276 [6 ] describes the application of AGEA to the cortex. The paper describes interesting results on the structure of correlations

277 between voxel gene expression profiles within a handful of cortical areas. However, this sort of analysis is not related to either

278 of our aims, as it neither finds marker genes, nor does it suggest a cortical map based on gene expression data. Neither of

279 the other components of AGEA can be applied to cortical areas; AGEA’s Gene Finder cannot be used to find marker genes

280 for the cortical areas; and AGEA’s hierarchial clustering does not produce clusters corresponding to the cortical areas14.

281 In summary, for all three aims, (a) only one of the previous projects explores combinations of marker genes, (b) there has

282 been almost no comparison of different algorithms or scoring methods, and (c) there has been no work on computationally

283 finding marker genes for cortical areas, or on finding a hierarchial clustering that will yield a map of cortical areas de novo

284 from gene expression data.

285 ___________________

286 6http://www.eurexpress.org/ee/; EurExpress data is also entered into EMAGE

287 7http://www.ncl.ac.uk/ihg/EADHB/database/EADHB_database.html

288 8http://mamep.molgen.mpg.de/index.php

289 9http://xenbase.org/

290 10http://aniseed-ibdm.univ-mrs.fr/

291 11http://genome.ucsc.edu/cgi-bin/hgVisiGene ; includes data from some the other listed data sources

292 12http://compare.ibdml.univ-mrs.fr/

293 13without prior offline registration

294 14In both cases, the root cause is that pairwise correlations between the gene expression of voxels in different areas but the same layer are

295 often stronger than pairwise correlations between the gene expression of voxels in different layers but the same area. Therefore, a pairwise voxel

296 correlation clustering algorithm will tend to create clusters representing cortical layers, not areas. This is why the hierarchial clustering does not

297 find most cortical areas (there are clusters which presumably correspond to the intersection of a layer and an area, but since one area will have

298 many layer-area intersection clusters, further work is needed to make sense of these). The reason that Gene Finder cannot find marker genes for

299 most cortical areas is that in Gene Finder, although the user chooses a seed voxel, Gene Finder chooses the ROI for which genes will be found,

300 and it creates that ROI by (pairwise voxel correlation) clustering around the seed.

301 Our project is guided by a concrete application with a well-specified criterion of success (how well we can find marker

302 genes for / reproduce the layout of cortical areas), which will provide a solid basis for comparing different methods.

303 Preliminary work

304 Format conversion between SEV, MATLAB, NIFTI

305 We have created software to (politely) download all of the SEV files from the Allen Institute website. We have also created

306 software to convert between the SEV, MATLAB, and NIFTI file formats, as well as some of Caret’s file formats.

307 Flatmap of cortex

308 We downloaded the ABA data and applied a mask to select only those voxels which belong to cerebral cortex. We divided

309 the cortex into hemispheres.

310 Using Caret[2], we created a mesh representation of the surface of the selected voxels. For each gene, for each node of

311 the mesh, we calculated an average of the gene expression of the voxels “underneath” that mesh node. We then flattened

312 the cortex, creating a two-dimensional mesh.

313 We sampled the nodes of the irregular, flat mesh in order to create a regular grid of pixel values. We converted this grid

314 into a MATLAB matrix.

315 We manually traced the boundaries of each cortical area from the ABA coronal reference atlas slides. We then converted

316 these manual traces into Caret-format regional boundary data on the mesh surface. We projected the regions onto the 2-d

317 mesh, and then onto the grid, and then we converted the region data into MATLAB format.

318 At this point, the data is in the form of a number of 2-D matrices, all in registration, with the matrix entries representing

319 a grid of points (pixels) over the cortical surface:

320 ∙A 2-D matrix whose entries represent the regional label associated with each surface pixel

321 ∙For each gene, a 2-D matrix whose entries represent the average expression level underneath each surface pixel

322 We created a normalized version of the gene expression data by subtracting each gene’s mean expression level (over all

323 surface pixels) and dividing each gene by its standard deviation.

324 The features and the target area are both functions on the surface pixels. They can be referred to as scalar fields over

325 the space of surface pixels; alternately, they can be thought of as images which can be displayed on the flatmapped surface.

326 To move beyond a single average expression level for each surface pixel, we plan to create a separate matrix for each

327 cortical layer to represent the average expression level within that layer. Cortical layers are found at different depths in

328 different parts of the cortex. In preparation for extracting the layer-specific datasets, we have extended Caret with routines

329 that allow the depth of the ROI for volume-to-surface projection to vary.

330 In the Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually

331 demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.

332 Feature selection and scoring methods

333 Correlation Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance

334 as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the

335 surface pixels.

336 One class of feature selection scoring method are those which calculate some sort of “match” between each gene image

337 and the target image. Those genes which match the best are good candidates for features.

338 One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between

339 each gene and each cortical area.

340 todo: fig

341 Conditional entropy An information-theoretic scoring method is to find features such that, if the features (gene

342 expression levels) are known, uncertainty about the target (the regional identity) is reduced. Entropy measures uncertainty,

343 so what we want is to find features such that the conditional distribution of the target has minimal entropy. The distribution

344 to which we are referring is the probability distribution over the population of surface pixels.

345 The simplest way to use information theory is on discrete data, so we discretized our gene expression data by creating,

346 for each gene, five thresholded boolean masks of the gene data. For each gene, we created a boolean mask of its expression

347 levels using each of these thresholds: the mean of that gene, the mean minus one standard deviation, the mean minus two

348 standard deviations, the mean plus one standard deviation, the mean plus two standard deviations.

349 Now, for each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression

350 boolean masks such that the conditional entropy of the target area’s boolean mask, conditioned upon the pair of gene

351 expression boolean masks, is minimized.

352 This finds pairs of genes which are most informative (at least at these discretization thresholds) relative to the question,

353 “Is this surface pixel a member of the target area?”.

354

355

356

357 Figure 1: The top row shows the three genes which (individually) best predict area AUD, according to logistic regression.

358 The bottom row shows the three genes which (individually) best match area AUD, according to gradient similarity. From

359 left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a, Ptk7, Aph1a again, and Lepr

360 todo: fig

361 Gradient similarity We noticed that the previous two scoring methods, which are pointwise, often found genes whose

362 pattern of expression did not look similar in shape to the target region. Fort his reason we designed a non-pointwise local

363 scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar

364 to the shape of the target region. We call this scoring method “gradient similarity”.

365 One might say that gradient similarity attempts to measure how much the border of the area of gene expression and

366 the border of the target region overlap. However, since gene expression falls off continuously rather than jumping from its

367 maximum value to zero, the spatial pattern of a gene’s expression often does not have a discrete border. Therefore, instead

368 of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images

369 (i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have

370 gradients which are oriented in a similar direction. The formula is:

371 ∑

372 pixel<img src="cmsy7-32.png" alt="&#x2208;" />pixels cos(abs(&#x2220;&#x2207;1 -&#x2220;&#x2207;2)) &#x22C5;|&#x2207;1| + |&#x2207;2|

373 2 &#x22C5; pixel_value1 + pixel_value2

374 2

375 where &#x2207;1 and &#x2207;2 are the gradient vectors of the two images at the current pixel; &#x2220;&#x2207;i is the angle of the gradient of

376 image i at the current pixel; |&#x2207;i| is the magnitude of the gradient of image i at the current pixel; and pixel_valuei is the

377 value of the current pixel in image i.

378 The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar,

379 then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a

380 similar direction (because the borders are similar).

381 Gradient similarity provides information complementary to correlation

382 To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider

383 Fig. . The top row of Fig. displays the 3 genes which most match area AUD, according to a pointwise method15. The

384 bottom row displays the 3 genes which most match AUD according to a method which considers local geometry16 The

385 pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is

386 that this includes many areas which don&#8217;t have a salient border matching the areal border. The geometric method identifies

387 genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes

388 genes which don&#8217;t express over the entire area. Genes which have high rankings using both pointwise and border criteria,

389 such as Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker

390 for AUD; we deliberately chose a &#8220;difficult&#8221; area in order to better contrast pointwise with geometric methods.

391 Combinations of multiple genes are useful

392 Here we give an example of a cortical area which is not marked by any single gene, but which can be identified combi-

393 natorially. according to logistic regression, gene wwc117 is the best fit single gene for predicting whether or not a pixel on

394 _________________________________________

395 15For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor

396 variable was the value of the expression of the gene underneath that pixel. The resulting scores were used to rank the genes in terms of how well

397 they predict area AUD.

398 16For each gene the gradient similarity (see section ??) between (a) a map of the expression of each gene on the cortical surface and (b) the

399 shape of area AUD, was calculated, and this was used to rank the genes.

400 17&#8220;WW, C2 and coiled-coil domain containing 1&#8221;; EntrezGene ID 211652

401

402

403

404 Figure 2: Upper left: wwc1. Upper right: mtif2. Lower left: wwc1 + mtif2 (each pixel&#8217;s value on the lower left is the sum

405 of the corresponding pixels in the upper row). Within each picture, the vertical axis roughly corresponds to anterior at the

406 top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right.

407 The red outline is the boundary of region MO. Pixels are colored approximately according to the density of expressing cells

408 underneath each pixel, with red meaning a lot of expression and blue meaning little.

409 the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure shows wwc1&#8217;s spatial expression

410 pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene

411 overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the

412 overshoot is the medial surface of the cortex. MO is only found on the lateral surface (todo).

413 Gene mtif218 is shown in figure the upper-right of Fig. . Mtif2 captures MO&#8217;s upper-left boundary, but not its lower-right

414 boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these

415 two figures, we get the lower-left of Figure . This combination captures area MO much better than any single gene.

416 Areas which can be identified by single genes

417 todo

418 Underexpression of a gene can serve as a marker

419 todo

420 Specific to Aim 1 (and Aim 3)

421 Forward stepwise logistic regression todo

422 SVM on all genes at once

423 In order to see how well one can do when looking at all genes at once, we ran a support vector machine to classify cortical

424 surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%19. As noted above,

425 however, a classifier that looks at all the genes at once isn&#8217;t practically useful.

426 The requirement to find combinations of only a small number of genes limits us from straightforwardly applying many

427 of the most simple techniques from the field of supervised machine learning. In the parlance of machine learning, our task

428 combines feature selection with supervised learning.

429 Decision trees

430 todo

431 Specific to Aim 2 (and Aim 3)

432 Raw dimensionality reduction results

433 todo

434 (might want to incld nnMF since mentioned above)

435 _________________________________________

436 18&#8220;mitochondrial translational initiation factor 2&#8221;; EntrezGene ID 76784

437 195-fold cross-validation.

438 Dimensionality reduction plus K-means or spectral clustering

439 Many areas are captured by clusters of genes

440 todo

441 todo

442 Research plan

443 Further work on flatmapping

444 In anatomy, the manifold of interest is usually either defined by a combination of two relevant anatomical axes (todo),

445 or by the surface of the structure (as is the case with the cortex). In the former case, the manifold of interest is a plane, but

446 in the latter case it is curved. If the manifold is curved, there are various methods for mapping the manifold into a plane.

447 In the case of the cerebral cortex, it remains to be seen which method of mapping the manifold into a plane is optimal

448 for this application. We will compare mappings which attempt to preserve size (such as the one used by Caret[2]) with

449 mappings which preserve angle (conformal maps).

450 Although there is much 2-D organization in anatomy, there are also structures whose shape is fundamentally 3-dimensional.

451 If possible, we would like the method we develop to include a statistical test that warns the user if the assumption of 2-D

452 structure seems to be wrong.

453 todo amongst other things:

454 Develop algorithms that find genetic markers for anatomical regions

455 1.Develop scoring measures for evaluating how good individual genes are at marking areas: we will compare pointwise,

456 geometric, and information-theoretic measures.

457 2.Develop a procedure to find single marker genes for anatomical regions: for each cortical area, by using or combining

458 the scoring measures developed, we will rank the genes by their ability to delineate each area.

459 3.Extend the procedure to handle difficult areas by using combinatorial coding: for areas that cannot be identified by any

460 single gene, identify them with a handful of genes. We will consider both (a) algorithms that incrementally/greedily

461 combine single gene markers into sets, such as forward stepwise regression and decision trees, and also (b) supervised

462 learning techniques which use soft constraints to minimize the number of features, such as sparse support vector

463 machines.

464 4.Extend the procedure to handle difficult areas by combining or redrawing the boundaries: An area may be difficult

465 to identify because the boundaries are misdrawn, or because it does not &#8220;really&#8221; exist as a single area, at least on the

466 genetic level. We will develop extensions to our procedure which (a) detect when a difficult area could be fit if its

467 boundary were redrawn slightly, and (b) detect when a difficult area could be combined with adjacent areas to create

468 a larger area which can be fit.

469 # Linear discriminant analysis

470 Apply these algorithms to the cortex

471 1.Create open source format conversion tools: we will create tools to bulk download the ABA dataset and to convert

472 between SEV, NIFTI and MATLAB formats.

473 2.Flatmap the ABA cortex data: map the ABA data onto a plane and draw the cortical area boundaries onto it.

474 3.Find layer boundaries: cluster similar voxels together in order to automatically find the cortical layer boundaries.

475 4.Run the procedures that we developed on the cortex: we will present, for each area, a short list of markers to identify

476 that area; and we will also present lists of &#8220;panels&#8221; of genes that can be used to delineate many areas at once.

477 Develop algorithms to suggest a division of a structure into anatomical parts

478 1.Explore dimensionality reduction algorithms applied to pixels: including TODO

479 2.Explore dimensionality reduction algorithms applied to genes: including TODO

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

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

482 5.Develop an algorithm to use dimensionality reduction and/or hierarchial clustering to create anatomical maps

483 6.Run this algorithm on the cortex: present a hierarchial, genoarchitectonic map of the cortex

484 # Linear discriminant analysis

485 # jbt, coclustering

486 # self-organizing map

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554

555 _______________________________________________________________________________________________________

556 stuff i dunno where to put yet (there is more scattered through grant-oldtext):

557 Principle 4: Work in 2-D whenever possible

558 &#8212;

559 note:

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

561 two hemis

562

563