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 four existing efforts to find marker genes using spatial gene expression data using automated methods.

94 [1 ] describes GeneAtlas. GeneAtlas allows the user to construct a search query by freely demarcating one or two 2-D

95 regions on sagittal slices, and then to specify either the strength of expression or the name of another gene whose expression

96 pattern is to be matched. GeneAtlas differs from our Aim 1 in at least two ways. First, GeneAtlas finds only single genes,

97 whereas we will also look for combinations of genes3. Second, at least for the custom spatial search, Gene Atlas appears to

98 use a simple pointwise scoring method (strength of expression), whereas we will also use geometric metrics such as gradient

99 similarity.

100 [6 ] describes AGEA, ”Anatomic Gene Expression Atlas”. AGEA has three components:

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

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

103 overexpressed genes for selected structures)

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

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

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

107 with correlation as the similarity metric.

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

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

110 search for underexpression. Third, Gene Finder uses a simple pointwise score4, whereas we will also use geometric scores

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

112 [10 ] todo

113 [4 ] todo

114 In summary, none of the previous projects explores combinations of marker genes, and none of their publications compare

115 the results obtained by using different algorithms or scoring methods.

116 Aim 2

117 Machine learning terminology: clustering

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

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

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

121 clustering or cluster analysis.

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

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

124 _________________________________________

125 2By “fundamentally spatial” we mean that there is information from a large number of spatial locations; not just data which has only a few

126 different locations.

127 3See Preliminary Data for an example of an area which cannot be marked by any single gene in the dataset, but which can be marked by a

128 combination.

129 4“Expression energy ratio”, which captures overexpression.

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

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

132 with similar gene expression.

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

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

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

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

137 This is called hierarchial clustering.

138 Similarity scores

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

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

141 scoring methods for similarity.

142 Spatially contiguous clusters; image segmentation

143 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

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

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

146 results against other methods which guarantee contiguous clusters.

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

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

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

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

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

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

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

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

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

156 Dimensionality reduction

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

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

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

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

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

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

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

164 expression levels.

165 Another use for dimensionality reduction is to visualize the relationships between regions. For example, one might want

166 to make a 2-D plot upon which each region is represented by a single point, and with the property that regions with similar

167 gene expression profiles should be nearby on the plot (that is, the property that distance between pairs of points in the plot

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

169 a 2-D plan will exactly satisfy this property – however, dimensionality reduction techniques allow one to find arrangements

170 of points that approximately satisfy that property. Note that in this application, dimensionality reduction is being applied

171 after clustering; whereas in the previous paragraph, we were talking about using dimensionality reduction before clustering.

172 Clustering genes rather than voxels

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

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

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

176 have one reduced feature for each gene cluster.

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

178 pattern which seems to pick out a single, spatially continguous region. Therefore, it seems likely that an anatomically

179 interesting region will have multiple genes which each individually pick it out5. This suggests the following procedure:

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

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

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

183 _________________________________________

184 5This would seem to contradict our finding in aim 1 that some cortical areas are combinatorially coded by multiple genes. However, it is

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

186 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

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

188 Related work

189 We are aware of three existing efforts to cluster spatial gene expression data.

190 [9 ] describes an analysis of the anatomy of the hippocampus using the ABA dataset. In addition to manual analysis,

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

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

193 the usefulness of computational genomic anatomy. We have run NNMF on the cortical dataset6 and while the results are

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

195 AGEA’s[6] hierarchial clustering differs from our Aim 2 in at least two ways. First, AGEA uses perhaps the simplest

196 possible similarity score (correlation), and does no dimensionality reduction before calculating similarity. While it is possible

197 that a more complex system will not do any better than this, we believe further exploration of alternative methods of scoring

198 and dimensionality reduction is warranted. Second, AGEA did not look at clusters of genes; in Preliminary Data we have

199 shown that clusters of genes may identify interesting spatial regions such as cortical areas.

200 [10 ] todo

201 In summary, although these projects obtained hierarchial clusterings, there has not been much comparison between

202 different algorithms or scoring methods, so it is likely that the best clustering method for this application has not yet been

203 found.

204 Aim 3

205 Background

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

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

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

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

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

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

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

213 their approximate location upon the cortical surface.

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

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

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

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

218 remain in the details.

219 The Allen Mouse Brain Atlas dataset

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

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

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

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

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

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

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

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

228 Mus musculus, the common house mouse, is thought to contain about 22,000 protein-coding genes[12]. The ABA contains

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

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

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

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

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

234 GenePaint[11], its sister project GeneAtlas[1], BGEM[5], EMAGE[?], EurExpress (http://www.eurexpress.org/ee/; Eu-

235 rExpress data is also entered into EMAGE), todo. With the exception of the ABA, GenePaint, and EMAGE, most of these

236 resources, have not (yet) extracted the expression intensity from the ISH images and registered the results into a single 3-D

237 space, and only ABA and EMAGE make this form of data available for public download from the website. Many of these

238 resources focus on developmental gene expression.

239 Significance

240 ___________________________

241 6We ran “vanilla” NNMF, whereas the paper under discussion used a modified method. Their main modification consisted of adding a soft

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

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

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

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

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

247 target individual cortical areas.

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

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

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

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

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

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

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

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

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

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

258 at the patterns of gene expression.

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

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

261 Related work

262 [6 ] describes the application of AGEA to the cortex. The paper describes interesting results on the structure of correlations

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

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

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

266 genes for most cortical areas; and AGEA’s hierarchial clustering does not produce clusters corresponding to most cortical

267 areas7 .

268 In summary, for all three aims, (a) none of the previous projects explores combinations of marker genes, (b) there has

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

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

271 from gene expression data.

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

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

274 _________________________________________

275 7In both cases, the root cause is that pairwise correlations between the gene expression of voxels in different areas but the same layer are

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

277 correlation clustering algorithm will often create clusters representing cortical layers, not areas. This is why the hierarchial clustering does not

278 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

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

280 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,

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

282 Preliminary work

283 Format conversion between SEV, MATLAB, NIFTI

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

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

286 Flatmap of cortex

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

288 the cortex into hemispheres.

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

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

291 the cortex, creating a two-dimensional mesh.

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

293 into a MATLAB matrix.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

311 Feature selection and scoring methods

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

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

314 surface pixels.

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

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

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

318 each gene and each cortical area.

319 todo: fig

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

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

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

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

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

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

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

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

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

329 binary masks such that the conditional entropy of the target area’s binary mask, conditioned upon the pair of gene expression

330 binary masks, is minimized.

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

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

333

334

335

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

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

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

339 todo: fig

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

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

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

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

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

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

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

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

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

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

350 ∑

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

352 2 &#x22C5; pixel_value1 + pixel_value2

353 2

354 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

355 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

356 value of the current pixel in image i.

357 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,

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

359 similar direction (because the borders are similar).

360 Gradient similarity provides information complementary to correlation

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

362 Fig. . The top row of Fig. displays the 3 genes which most match area AUD, according to a pointwise method8. The bottom

363 row displays the 3 genes which most match AUD according to a method which considers local geometry9 The pointwise

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

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

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

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

368 Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker for AUD;

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

370 Combinations of multiple genes are useful

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

372 natorially. according to logistic regression, gene wwc110 is the best fit single gene for predicting whether or not a pixel on

373 _________________________________________

374 8For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor

375 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

376 they predict area AUD.

377 9For each gene the gradient similarity (see section ??) between (a) a map of the expression of each gene on the cortical surface and (b) the

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

379 10&#8220;WW, C2 and coiled-coil domain containing 1&#8221;; EntrezGene ID 211652

380

381

382

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

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

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

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

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

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

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

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

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

392 Gene mtif211 is shown in figure the upper-right of Fig. . Mtif2 captures MO&#8217;s upper-left boundary, but not its lower-right

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

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

395 Areas which can be identified by single genes

396 todo

397 Underexpression of a gene can serve as a marker

398 todo

399 Specific to Aim 1 (and Aim 3)

400 Forward stepwise logistic regression todo

401 SVM on all genes at once

402 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

403 surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%12. As noted above,

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

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

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

407 combines feature selection with supervised learning.

408 Decision trees

409 todo

410 Specific to Aim 2 (and Aim 3)

411 Raw dimensionality reduction results

412 todo

413 (might want to incld nnMF since mentioned above)

414 _________________________________________

415 11&#8220;mitochondrial translational initiation factor 2&#8221;; EntrezGene ID 76784

416 125-fold cross-validation.

417 Dimensionality reduction plus K-means or spectral clustering

418 Many areas are captured by clusters of genes

419 todo

420 todo

421 Research plan

422 Further work on flatmapping

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

424 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

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

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

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

428 mappings which preserve angle (conformal maps).

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

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

431 structure seems to be wrong.

432 todo amongst other things:

433 Develop algorithms that find genetic markers for anatomical regions

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

435 geometric, and information-theoretic measures.

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

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

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

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

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

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

442 machines.

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

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

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

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

447 a larger area which can be fit.

448 Apply these algorithms to the cortex

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

450 between SEV, NIFTI and MATLAB formats.

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

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

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

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

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

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

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

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

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

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

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

462 Bibliography &amp; References Cited

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527

528 _______________________________________________________________________________________________________

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

530 Principle 4: Work in 2-D whenever possible

531 &#8212;

532 note:

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

534 two hemis

535

536