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| bshanks@0 | 1 Specific aims
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| bshanks@53 | 2 Massivenew datasets obtained with techniques such as in situ hybridization (ISH), immunohistochemistry, in situ transgenic
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| bshanks@53 | 3 reporter, microarray voxelation, and others, allow the expression levels of many genes at many locations to be compared.
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| bshanks@53 | 4 Our goal is to develop automated methods to relate spatial variation in gene expression to anatomy. We want to find marker
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| bshanks@53 | 5 genes for specific anatomical regions, and also to draw new anatomical maps based on gene expression patterns. We have
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| bshanks@53 | 6 three specific aims:
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| bshanks@30 | 7 (1) develop an algorithm to screen spatial gene expression data for combinations of marker genes which selectively target
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| bshanks@30 | 8 anatomical regions
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| bshanks@84 | 9 (2) develop an algorithm to suggest new ways of carving up a structure into anatomically distinct regions, based on
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| bshanks@84 | 10 spatial patterns in gene expression
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| bshanks@33 | 11 (3) create a 2-D “flat map” dataset of the mouse cerebral cortex that contains a flattened version of the Allen Mouse
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| bshanks@35 | 12 Brain Atlas ISH data, as well as the boundaries of cortical anatomical areas. This will involve extending the functionality of
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| bshanks@35 | 13 Caret, an existing open-source scientific imaging program. Use this dataset to validate the methods developed in (1) and (2).
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| bshanks@84 | 14 Although our particular application involves the 3D spatial distribution of gene expression, we anticipate that the methods
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| bshanks@84 | 15 developed in aims (1) and (2) will generalize to any sort of high-dimensional data over points located in a low-dimensional
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| bshanks@84 | 16 space.
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| bshanks@84 | 17 In terms of the application of the methods to cerebral cortex, aim (1) is to go from cortical areas to marker genes,
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| bshanks@84 | 18 and aim (2) is to let the gene profile define the cortical areas. In addition to validating the usefulness of the algorithms,
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| bshanks@84 | 19 the application of these methods to cortex will produce immediate benefits, because there are currently no known genetic
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| bshanks@84 | 20 markers for most cortical areas. The results of the project will support the development of new ways to selectively target
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| bshanks@84 | 21 cortical areas, and it will support the development of a method for identifying the cortical areal boundaries present in small
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| bshanks@84 | 22 tissue samples.
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| bshanks@53 | 23 All algorithms that we develop will be implemented in a GPL open-source software toolkit. The toolkit, as well as the
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| bshanks@30 | 24 machine-readable datasets developed in aim (3), will be published and freely available for others to use.
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| bshanks@30 | 25 Background and significance
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| bshanks@84 | 26 Aim 1: Given a map of regions, find genes that mark the regions
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| bshanks@85 | 27 Machine learning terminology The task of looking for marker genes for known anatomical regions means that one is
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| bshanks@85 | 28 looking for a set of genes such that, if the expression level of those genes is known, then the locations of the regions can be
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| bshanks@85 | 29 inferred.
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| bshanks@85 | 30 If we define the regions so that they cover the entire anatomical structure to be divided, we may say that we are using
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| bshanks@85 | 31 gene expression to determine to which region each voxel within the structure belongs. We call this a classification task,
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| bshanks@85 | 32 because each voxel is being assigned to a class (namely, its region). An understanding of the relationship between the
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| bshanks@85 | 33 combination of their expression levels and the locations of the regions may be expressed as a function. The input to this
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| bshanks@85 | 34 function is a voxel, along with the gene expression levels within that voxel; the output is the regional identity of the target
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| bshanks@85 | 35 voxel, that is, the region to which the target voxel belongs. We call this function a classifier. In general, the input to a
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| bshanks@85 | 36 classifier is called an instance, and the output is called a label (or a class label).
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| bshanks@30 | 37 The object of aim 1 is not to produce a single classifier, but rather to develop an automated method for determining a
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| bshanks@30 | 38 classifier for any known anatomical structure. Therefore, we seek a procedure by which a gene expression dataset may be
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| bshanks@85 | 39 analyzed in concert with an anatomical atlas in order to produce a classifier. The initial gene expression dataset used in
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| bshanks@85 | 40 the construction of the classifier is called training data. In the machine learning literature, this sort of procedure may be
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| bshanks@85 | 41 thought of as a supervised learning task, defined as a task in which the goal is to learn a mapping from instances to labels,
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| bshanks@85 | 42 and the training data consists of a set of instances (voxels) for which the labels (regions) are known.
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| bshanks@30 | 43 Each gene expression level is called a feature, and the selection of which genes1 to include is called feature selection.
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| bshanks@33 | 44 Feature selection is one component of the task of learning a classifier. Some methods for learning classifiers start out with
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| bshanks@33 | 45 a separate feature selection phase, whereas other methods combine feature selection with other aspects of training.
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| bshanks@30 | 46 One class of feature selection methods assigns some sort of score to each candidate gene. The top-ranked genes are then
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| bshanks@30 | 47 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
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| bshanks@30 | 48 procedure may be used in which features are added and subtracted from the selected set depending on how much they raise
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| bshanks@30 | 49 the score. Such procedures are called “stepwise” or “greedy”.
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| bshanks@30 | 50 Although the classifier itself may only look at the gene expression data within each voxel before classifying that voxel, the
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| bshanks@85 | 51 algorithm which constructs the classifier may look over the entire dataset. We can categorize score-based feature selection
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| bshanks@85 | 52 methods depending on how the score of calculated. Often the score calculation consists of assigning a sub-score to each voxel,
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| bshanks@85 | 53 and then aggregating these sub-scores into a final score (the aggregation is often a sum or a sum of squares or average). If
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| bshanks@85 | 54 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
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| bshanks@85 | 55 information from the voxel itself is used to calculate a voxel’s sub-score, then we say it is a pointwise scoring method.
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| bshanks@85 | 56 Our strategy for Aim 1
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| bshanks@85 | 57 Key questions when choosing a learning method are: What are the instances? What are the features? How are the features
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| bshanks@85 | 58 chosen? Here are four principles that outline our answers to these questions.
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| bshanks@84 | 59 Principle 1: Combinatorial gene expression
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| bshanks@84 | 60 It is too much to hope that every anatomical region of interest will be identified by a single gene. For example, in the
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| bshanks@84 | 61 cortex, there are some areas which are not clearly delineated by any gene included in the Allen Brain Atlas (ABA) dataset.
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| bshanks@84 | 62 However, at least some of these areas can be delineated by looking at combinations of genes (an example of an area for
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| bshanks@84 | 63 which multiple genes are necessary and sufficient is provided in Preliminary Studies, Figure 4). Therefore, each instance
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| bshanks@84 | 64 should contain multiple features (genes).
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| bshanks@84 | 65 Principle 2: Only look at combinations of small numbers of genes
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| bshanks@84 | 66 When the classifier classifies a voxel, it is only allowed to look at the expression of the genes which have been selected
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| bshanks@84 | 67 as features. The more data that are available to a classifier, the better that it can do. For example, perhaps there are weak
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| bshanks@84 | 68 correlations over many genes that add up to a strong signal. So, why not include every gene as a feature? The reason is that
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| bshanks@84 | 69 we wish to employ the classifier in situations in which it is not feasible to gather data about every gene. For example, if we
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| bshanks@84 | 70 want to use the expression of marker genes as a trigger for some regionally-targeted intervention, then our intervention must
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| bshanks@84 | 71 contain a molecular mechanism to check the expression level of each marker gene before it triggers. It is currently infeasible
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| bshanks@84 | 72 to design a molecular trigger that checks the level of more than a handful of genes. Similarly, if the goal is to develop a
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| bshanks@84 | 73 procedure to do ISH on tissue samples in order to label their anatomy, then it is infeasible to label more than a few genes.
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| bshanks@84 | 74 Therefore, we must select only a few genes as features.
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| bshanks@63 | 75 The requirement to find combinations of only a small number of genes limits us from straightforwardly applying many
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| bshanks@63 | 76 of the most simple techniques from the field of supervised machine learning. In the parlance of machine learning, our task
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| bshanks@63 | 77 combines feature selection with supervised learning.
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| bshanks@85 | 78 _________________________________________
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| bshanks@85 | 79 1Strictly speaking, the features are gene expression levels, but we’ll call them genes.
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| bshanks@30 | 80 Principle 3: Use geometry in feature selection
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| bshanks@30 | 81 When doing feature selection with score-based methods, the simplest thing to do would be to score the performance of
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| bshanks@30 | 82 each voxel by itself and then combine these scores (pointwise scoring). A more powerful approach is to also use information
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| bshanks@30 | 83 about the geometric relations between each voxel and its neighbors; this requires non-pointwise, local scoring methods. See
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| bshanks@84 | 84 Preliminary Studies, figure 3 for evidence of the complementary nature of pointwise and local scoring methods.
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| bshanks@30 | 85 Principle 4: Work in 2-D whenever possible
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| bshanks@30 | 86 There are many anatomical structures which are commonly characterized in terms of a two-dimensional manifold. When
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| bshanks@30 | 87 it is known that the structure that one is looking for is two-dimensional, the results may be improved by allowing the analysis
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| bshanks@33 | 88 algorithm to take advantage of this prior knowledge. In addition, it is easier for humans to visualize and work with 2-D
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| bshanks@85 | 89 data. Therefore, when possible, the instances should represent pixels, not voxels.
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| bshanks@43 | 90 Related work
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| bshanks@44 | 91 There is a substantial body of work on the analysis of gene expression data, most of this concerns gene expression data
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| bshanks@84 | 92 which are not fundamentally spatial2.
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| bshanks@43 | 93 As noted above, there has been much work on both supervised learning and there are many available algorithms for
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| bshanks@43 | 94 each. However, the algorithms require the scientist to provide a framework for representing the problem domain, and the
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| bshanks@43 | 95 way that this framework is set up has a large impact on performance. Creating a good framework can require creatively
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| bshanks@43 | 96 reconceptualizing the problem domain, and is not merely a mechanical “fine-tuning” of numerical parameters. For example,
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| bshanks@84 | 97 we believe that domain-specific scoring measures (such as gradient similarity, which is discussed in Preliminary Studies) may
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| bshanks@43 | 98 be necessary in order to achieve the best results in this application.
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| bshanks@53 | 99 We are aware of six existing efforts to find marker genes using spatial gene expression data using automated methods.
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| bshanks@85 | 100 [11 ] mentions the possibility of constructing a spatial region for each gene, and then, for each anatomical structure of
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| bshanks@53 | 101 interest, computing what proportion of this structure is covered by the gene’s spatial region.
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| bshanks@85 | 102 GeneAtlas[5] and EMAGE [23] allow the user to construct a search query by demarcating regions and then specifing
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| bshanks@53 | 103 either the strength of expression or the name of another gene or dataset whose expression pattern is to be matched. For the
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| bshanks@53 | 104 similiarity score (match score) between two images (in this case, the query and the gene expression images), GeneAtlas uses
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| bshanks@53 | 105 the sum of a weighted L1-norm distance between vectors whose components represent the number of cells within a pixel3
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| bshanks@85 | 106 whose expression is within four discretization levels. EMAGE uses Jaccard similarity4. Neither GeneAtlas nor EMAGE
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| bshanks@53 | 107 allow one to search for combinations of genes that define a region in concert but not separately.
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| bshanks@85 | 108 [13 ] describes AGEA, ”Anatomic Gene Expression Atlas”. AGEA has three components. Gene Finder: The user
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| bshanks@85 | 109 selects a seed voxel and the system (1) chooses a cluster which includes the seed voxel, (2) yields a list of genes which are
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| bshanks@85 | 110 overexpressed in that cluster. (note: the ABA website also contains pre-prepared lists of overexpressed genes for selected
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| bshanks@85 | 111 structures). Correlation: The user selects a seed voxel and the system then shows the user how much correlation there is
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| bshanks@85 | 112 between the gene expression profile of the seed voxel and every other voxel. Clusters: will be described later
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| bshanks@43 | 113 Gene Finder is different from our Aim 1 in at least three ways. First, Gene Finder finds only single genes, whereas we
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| bshanks@43 | 114 will also look for combinations of genes. Second, gene finder can only use overexpression as a marker, whereas we will also
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| bshanks@85 | 115 search for underexpression. Third, Gene Finder uses a simple pointwise score5, whereas we will also use geometric scores
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| bshanks@84 | 116 such as gradient similarity (described in Preliminary Studies). Figures 4, 2, and 3 in the Preliminary Studies section contains
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| bshanks@84 | 117 evidence that each of our three choices is the right one.
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| bshanks@85 | 118 [6 ] looks at the mean expression level of genes within anatomical regions, and applies a Student’s t-test with Bonferroni
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| bshanks@51 | 119 correction to determine whether the mean expression level of a gene is significantly higher in the target region. Like AGEA,
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| bshanks@51 | 120 this is a pointwise measure (only the mean expression level per pixel is being analyzed), it is not being used to look for
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| bshanks@51 | 121 underexpression, and does not look for combinations of genes.
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| bshanks@85 | 122 [9 ] describes a technique to find combinations of marker genes to pick out an anatomical region. They use an evolutionary
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| bshanks@46 | 123 algorithm to evolve logical operators which combine boolean (thresholded) images in order to match a target image. Their
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| bshanks@51 | 124 match score is Jaccard similarity.
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| bshanks@84 | 125 In summary, there has been fruitful work on finding marker genes, but only one of the previous projects explores
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| bshanks@51 | 126 combinations of marker genes, and none of these publications compare the results obtained by using different algorithms or
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| bshanks@51 | 127 scoring methods.
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| bshanks@84 | 128 Aim 2: From gene expression data, discover a map of regions
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| bshanks@30 | 129 Machine learning terminology: clustering
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| bshanks@85 | 130 _
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| bshanks@85 | 131 2By “fundamentally spatial” we mean that there is information from a large number of spatial locations indexed by spatial coordinates; not
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| bshanks@85 | 132 just data which have only a few different locations or which is indexed by anatomical label.
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| bshanks@85 | 133 3Actually, many of these projects use quadrilaterals instead of square pixels; but we will refer to them as pixels for simplicity.
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| bshanks@85 | 134 4the number of true pixels in the intersection of the two images, divided by the number of pixels in their union.
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| bshanks@85 | 135 5“Expression energy ratio”, which captures overexpression.
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| bshanks@30 | 136 If one is given a dataset consisting merely of instances, with no class labels, then analysis of the dataset is referred to as
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| bshanks@30 | 137 unsupervised learning in the jargon of machine learning. One thing that you can do with such a dataset is to group instances
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| bshanks@46 | 138 together. A set of similar instances is called a cluster, and the activity of finding grouping the data into clusters is called
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| bshanks@46 | 139 clustering or cluster analysis.
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| bshanks@84 | 140 The task of deciding how to carve up a structure into anatomical regions can be put into these terms. The instances
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| bshanks@84 | 141 are once again voxels (or pixels) along with their associated gene expression profiles. We make the assumption that voxels
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| bshanks@84 | 142 from the same anatomical region have similar gene expression profiles, at least compared to the other regions. This means
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| bshanks@84 | 143 that clustering voxels is the same as finding potential regions; we seek a partitioning of the voxels into regions, that is, into
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| bshanks@84 | 144 clusters of voxels with similar gene expression.
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| bshanks@85 | 145 It is desirable to determine not just one set of regions, but also how these regions relate to each other. The outcome of
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| bshanks@85 | 146 clustering may be a hierarchial tree of clusters, rather than a single set of clusters which partition the voxels. This is called
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| bshanks@85 | 147 hierarchial clustering.
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| bshanks@85 | 148 Similarity scores A crucial choice when designing a clustering method is how to measure similarity, across either pairs
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| bshanks@85 | 149 of instances, or clusters, or both. There is much overlap between scoring methods for feature selection (discussed above
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| bshanks@85 | 150 under Aim 1) and scoring methods for similarity.
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| bshanks@85 | 151 Spatially contiguous clusters; image segmentation We have shown that aim 2 is a type of clustering task. In fact,
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| bshanks@85 | 152 it is a special type of clustering task because we have an additional constraint on clusters; voxels grouped together into a
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| bshanks@85 | 153 cluster must be spatially contiguous. In Preliminary Studies, we show that one can get reasonable results without enforcing
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| bshanks@85 | 154 this constraint; however, we plan to compare these results against other methods which guarantee contiguous clusters.
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| bshanks@85 | 155 Image segmentation is the task of partitioning the pixels in a digital image into clusters, usually contiguous clusters. Aim
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| bshanks@85 | 156 2 is similar to an image segmentation task. There are two main differences; in our task, there are thousands of color channels
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| bshanks@85 | 157 (one for each gene), rather than just three6. A more crucial difference is that there are various cues which are appropriate
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| bshanks@85 | 158 for detecting sharp object boundaries in a visual scene but which are not appropriate for segmenting abstract spatial data
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| bshanks@85 | 159 such as gene expression. Although many image segmentation algorithms can be expected to work well for segmenting other
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| bshanks@85 | 160 sorts of spatially arranged data, some of these algorithms are specialized for visual images.
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| bshanks@51 | 161 Dimensionality reduction In this section, we discuss reducing the length of the per-pixel gene expression feature
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| bshanks@51 | 162 vector. By “dimension”, we mean the dimension of this vector, not the spatial dimension of the underlying data.
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| bshanks@33 | 163 Unlike aim 1, there is no externally-imposed need to select only a handful of informative genes for inclusion in the
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| bshanks@85 | 164 instances. However, some clustering algorithms perform better on small numbers of features7. There are techniques which
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| bshanks@30 | 165 “summarize” a larger number of features using a smaller number of features; these techniques go by the name of feature
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| bshanks@30 | 166 extraction or dimensionality reduction. The small set of features that such a technique yields is called the reduced feature
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| bshanks@85 | 167 set. Note that the features in the reduced feature set do not necessarily correspond to genes; each feature in the reduced set
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| bshanks@85 | 168 may be any function of the set of gene expression levels.
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| bshanks@85 | 169 Clustering genes rather than voxels Although the ultimate goal is to cluster the instances (voxels or pixels), one
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| bshanks@85 | 170 strategy to achieve this goal is to first cluster the features (genes). There are two ways that clusters of genes could be used.
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| bshanks@30 | 171 Gene clusters could be used as part of dimensionality reduction: rather than have one feature for each gene, we could
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| bshanks@30 | 172 have one reduced feature for each gene cluster.
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| bshanks@30 | 173 Gene clusters could also be used to directly yield a clustering on instances. This is because many genes have an expression
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| bshanks@53 | 174 pattern which seems to pick out a single, spatially continguous region. Therefore, it seems likely that an anatomically
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| bshanks@85 | 175 interesting region will have multiple genes which each individually pick it out8. This suggests the following procedure:
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| bshanks@42 | 176 cluster together genes which pick out similar regions, and then to use the more popular common regions as the final clusters.
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| bshanks@84 | 177 In Preliminary Studies, Figure 7, we show that a number of anatomically recognized cortical regions, as well as some
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| bshanks@84 | 178 “superregions” formed by lumping together a few regions, are associated with gene clusters in this fashion.
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| bshanks@51 | 179 The task of clustering both the instances and the features is called co-clustering, and there are a number of co-clustering
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| bshanks@51 | 180 algorithms.
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| bshanks@85 | 181 ________________________________
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| bshanks@85 | 182 6There are imaging tasks which use more than three colors, for example multispectral imaging and hyperspectral imaging, which are often
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| bshanks@85 | 183 used to process satellite imagery.
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| bshanks@85 | 184 7First, because the number of features in the reduced dataset is less than in the original dataset, the running time of clustering algorithms
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| bshanks@85 | 185 may be much less. Second, it is thought that some clustering algorithms may give better results on reduced data.
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| bshanks@85 | 186 8This would seem to contradict our finding in aim 1 that some cortical areas are combinatorially coded by multiple genes. However, it is
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| bshanks@85 | 187 possible that the currently accepted cortical maps divide the cortex into regions which are unnatural from the point of view of gene expression;
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| bshanks@85 | 188 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
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| bshanks@85 | 189 the cluster prototype fits an anatomical region, the individual genes are each somewhat different from the prototype.
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| bshanks@43 | 190 Related work
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| bshanks@85 | 191 Some researchers have attempted to parcellate cortex on the basis of non-gene expression data. For example, [15], [2], [16],
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| bshanks@85 | 192 and [1 ] associate spots on the cortex with the radial profile9 of response to some stain ([10] uses MRI), extract features from
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| bshanks@85 | 193 this profile, and then use similarity between surface pixels to cluster. Features used include statistical moments, wavelets,
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| bshanks@85 | 194 and the excess mass functional. Some of these features are motivated by the presence of tangential lines of stain intensity
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| bshanks@85 | 195 which correspond to laminar structure. Some methods use standard clustering procedures, whereas others make use of the
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| bshanks@85 | 196 spatial nature of the data to look for sudden transitions, which are identified as areal borders.
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| bshanks@85 | 197 [20 ] describes an analysis of the anatomy of the hippocampus using the ABA dataset. In addition to manual analysis,
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| bshanks@43 | 198 two clustering methods were employed, a modified Non-negative Matrix Factorization (NNMF), and a hierarchial recursive
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| bshanks@44 | 199 bifurcation clustering scheme based on correlation as the similarity score. The paper yielded impressive results, proving
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| bshanks@85 | 200 the usefulness of computational genomic anatomy. We have run NNMF on the cortical dataset10 and while the results are
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| bshanks@84 | 201 promising, they also demonstrate that NNMF is not necessarily the best dimensionality reduction method for this application
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| bshanks@84 | 202 (see Preliminary Studies, Figure 6).
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| bshanks@85 | 203 AGEA[13] includes a preset hierarchial clustering of voxels based on a recursive bifurcation algorithm with correlation
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| bshanks@85 | 204 as the similarity metric. EMAGE[23] allows the user to select a dataset from among a large number of alternatives, or by
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| bshanks@53 | 205 running a search query, and then to cluster the genes within that dataset. EMAGE clusters via hierarchial complete linkage
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| bshanks@53 | 206 clustering with un-centred correlation as the similarity score.
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| bshanks@85 | 207 [6 ] clustered genes, starting out by selecting 135 genes out of 20,000 which had high variance over voxels and which were
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| bshanks@53 | 208 highly correlated with many other genes. They computed the matrix of (rank) correlations between pairs of these genes, and
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| bshanks@53 | 209 ordered the rows of this matrix as follows: “the first row of the matrix was chosen to show the strongest contrast between
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| bshanks@53 | 210 the highest and lowest correlation coefficient for that row. The remaining rows were then arranged in order of decreasing
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| bshanks@53 | 211 similarity using a least squares metric”. The resulting matrix showed four clusters. For each cluster, prototypical spatial
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| bshanks@53 | 212 expression patterns were created by averaging the genes in the cluster. The prototypes were analyzed manually, without
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| bshanks@85 | 213 clustering voxels.
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| bshanks@85 | 214 [9 ] applies their technique for finding combinations of marker genes for the purpose of clustering genes around a “seed
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| bshanks@85 | 215 gene”. They do this by using the pattern of expression of the seed gene as the target image, and then searching for other
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| bshanks@85 | 216 genes which can be combined to reproduce this pattern. Other genes which are found are considered to be related to the
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| bshanks@85 | 217 seed. The same team also describes a method[22] for finding “association rules” such as, “if this voxel is expressed in by
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| bshanks@85 | 218 any gene, then that voxel is probably also expressed in by the same gene”. This could be useful as part of a procedure for
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| bshanks@85 | 219 clustering voxels.
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| bshanks@46 | 220 In summary, although these projects obtained clusterings, there has not been much comparison between different algo-
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| bshanks@85 | 221 rithms or scoring methods, so it is likely that the best clustering method for this application has not yet been found. The
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| bshanks@85 | 222 projects using gene expression on cortex did not attempt to make use of the radial profile of gene expression. Also, none of
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| bshanks@85 | 223 these projects did a separate dimensionality reduction step before clustering pixels, none tried to cluster genes first in order
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| bshanks@85 | 224 to guide automated clustering of pixels into spatial regions, and none used co-clustering algorithms.
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| bshanks@85 | 225 Aim 3: apply the methods developed to the cerebral cortex
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| bshanks@30 | 226 Background
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| bshanks@84 | 227 The cortex is divided into areas and layers. Because of the cortical columnar organization, the parcellation of the cortex
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| bshanks@84 | 228 into areas can be drawn as a 2-D map on the surface of the cortex. In the third dimension, the boundaries between the
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| bshanks@84 | 229 areas continue downwards into the cortical depth, perpendicular to the surface. The layer boundaries run parallel to the
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| bshanks@85 | 230 surface. One can picture an area of the cortex as a slice of a six-layered cake11.
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| bshanks@85 | 231 It is known that different cortical areas have distinct roles in both normal functioning and in disease processes, yet there
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| bshanks@85 | 232 are no known marker genes for most cortical areas. When it is necessary to divide a tissue sample into cortical areas, this is
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| bshanks@85 | 233 a manual process that requires a skilled human to combine multiple visual cues and interpret them in the context of their
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| bshanks@85 | 234 approximate location upon the cortical surface.
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| bshanks@33 | 235 Even the questions of how many areas should be recognized in cortex, and what their arrangement is, are still not
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| bshanks@53 | 236 completely settled. A proposed division of the cortex into areas is called a cortical map. In the rodent, the lack of a single
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| bshanks@85 | 237 agreed-upon map can be seen by contrasting the recent maps given by Swanson[19] on the one hand, and Paxinos and
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| bshanks@85 | 238 Franklin[14] on the other. While the maps are certainly very similar in their general arrangement, significant differences
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| bshanks@85 | 239 remain.
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| bshanks@36 | 240 The Allen Mouse Brain Atlas dataset
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| bshanks@85 | 241 __
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| bshanks@85 | 242 9A radial profile is a profile along a line perpendicular to the cortical surface.
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| bshanks@85 | 243 10We ran “vanilla” NNMF, whereas the paper under discussion used a modified method. Their main modification consisted of adding a soft
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| bshanks@85 | 244 spatial contiguity constraint. However, on our dataset, NNMF naturally produced spatially contiguous clusters, so no additional constraint was
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| bshanks@85 | 245 needed. The paper under discussion also mentions that they tried a hierarchial variant of NNMF, which we have not yet tried.
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| bshanks@85 | 246 11Outside of isocortex, the number of layers varies.
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| bshanks@84 | 247 The Allen Mouse Brain Atlas (ABA) data were produced by doing in-situ hybridization on slices of male, 56-day-old
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| bshanks@36 | 248 C57BL/6J mouse brains. Pictures were taken of the processed slice, and these pictures were semi-automatically analyzed
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| bshanks@85 | 249 to create a digital measurement of gene expression levels at each location in each slice. Per slice, cellular spatial resolution
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| bshanks@85 | 250 is achieved. Using this method, a single physical slice can only be used to measure one single gene; many different mouse
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| bshanks@85 | 251 brains were needed in order to measure the expression of many genes.
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| bshanks@85 | 252 An automated nonlinear alignment procedure located the 2D data from the various slices in a single 3D coordinate
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| bshanks@36 | 253 system. In the final 3D coordinate system, voxels are cubes with 200 microns on a side. There are 67x41x58 = 159,326
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| bshanks@85 | 254 voxels in the 3D coordinate system, of which 51,533 are in the brain[13].
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| bshanks@85 | 255 Mus musculus is thought to contain about 22,000 protein-coding genes[25]. The ABA contains data on about 20,000
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| bshanks@85 | 256 genes in sagittal sections, out of which over 4,000 genes are also measured in coronal sections. Our dataset is derived from
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| bshanks@85 | 257 only the coronal subset of the ABA12.
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| bshanks@85 | 258 The ABA is not the only large public spatial gene expression dataset13. With the exception of the ABA, GenePaint, and
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| bshanks@85 | 259 EMAGE, most of the other resources have not (yet) extracted the expression intensity from the ISH images and registered
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| bshanks@85 | 260 the results into a single 3-D space, and to our knowledge only ABA and EMAGE make this form of data available for public
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| bshanks@85 | 261 download from the website14. Many of these resources focus on developmental gene expression.
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| bshanks@46 | 262 Significance
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| bshanks@43 | 263 The method developed in aim (1) will be applied to each cortical area to find a set of marker genes such that the
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| bshanks@42 | 264 combinatorial expression pattern of those genes uniquely picks out the target area. Finding marker genes will be useful for
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| bshanks@30 | 265 drug discovery as well as for experimentation because marker genes can be used to design interventions which selectively
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| bshanks@30 | 266 target individual cortical areas.
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| bshanks@30 | 267 The application of the marker gene finding algorithm to the cortex will also support the development of new neuroanatom-
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| bshanks@33 | 268 ical methods. In addition to finding markers for each individual cortical areas, we will find a small panel of genes that can
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| bshanks@33 | 269 find many of the areal boundaries at once. This panel of marker genes will allow the development of an ISH protocol that
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| bshanks@30 | 270 will allow experimenters to more easily identify which anatomical areas are present in small samples of cortex.
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| bshanks@85 | 271 The method developed in aim (2) will provide a genoarchitectonic viewpoint that will contribute to the creation of a
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| bshanks@85 | 272 better map. The development of present-day cortical maps was driven by the application of histological stains. If a different
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| bshanks@85 | 273 set of stains had been available which identified a different set of features, then today’s cortical maps may have come out
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| bshanks@85 | 274 differently. It is likely that there are many repeated, salient spatial patterns in the gene expression which have not yet been
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| bshanks@85 | 275 captured by any stain. Therefore, cortical anatomy needs to incorporate what we can learn from looking at the patterns of
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| bshanks@85 | 276 gene expression.
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| bshanks@63 | 277 While we do not here propose to analyze human gene expression data, it is conceivable that the methods we propose to
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| bshanks@63 | 278 develop could be used to suggest modifications to the human cortical map as well.
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| bshanks@63 | 279 Related work
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| bshanks@85 | 280 [13 ] describes the application of AGEA to the cortex. The paper describes interesting results on the structure of correlations
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| bshanks@63 | 281 between voxel gene expression profiles within a handful of cortical areas. However, this sort of analysis is not related to either
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| bshanks@46 | 282 of our aims, as it neither finds marker genes, nor does it suggest a cortical map based on gene expression data. Neither of
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| bshanks@46 | 283 the other components of AGEA can be applied to cortical areas; AGEA’s Gene Finder cannot be used to find marker genes
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| bshanks@85 | 284 for the cortical areas; and AGEA’s hierarchial clustering does not produce clusters corresponding to the cortical areas15.
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| bshanks@46 | 285 In summary, for all three aims, (a) only one of the previous projects explores combinations of marker genes, (b) there has
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| bshanks@43 | 286 been almost no comparison of different algorithms or scoring methods, and (c) there has been no work on computationally
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| bshanks@43 | 287 finding marker genes for cortical areas, or on finding a hierarchial clustering that will yield a map of cortical areas de novo
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| bshanks@43 | 288 from gene expression data.
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| bshanks@53 | 289 Our project is guided by a concrete application with a well-specified criterion of success (how well we can find marker
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| bshanks@53 | 290 genes for / reproduce the layout of cortical areas), which will provide a solid basis for comparing different methods.
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| bshanks@53 | 291 _________________________________________
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| bshanks@85 | 292 12The sagittal data do not cover the entire cortex, and also have greater registration error[13]. Genes were selected by the Allen Institute for
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| bshanks@85 | 293 coronal sectioning based on, “classes of known neuroscientific interest... or through post hoc identification of a marked non-ubiquitous expression
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| bshanks@85 | 294 pattern”[13].
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| bshanks@85 | 295 13Other such resources include GENSAT[8], GenePaint[24], its sister project GeneAtlas[5], BGEM[12], EMAGE[23], EurExpress (http://www.
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| bshanks@85 | 296 eurexpress.org/ee/; EurExpress data are also entered into EMAGE), EADHB (http://www.ncl.ac.uk/ihg/EADHB/database/EADHB_database.
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| bshanks@85 | 297 html), MAMEP (http://mamep.molgen.mpg.de/index.php), Xenbase (http://xenbase.org/), ZFIN[18], Aniseed (http://aniseed-ibdm.
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| bshanks@85 | 298 univ-mrs.fr/), VisiGene (http://genome.ucsc.edu/cgi-bin/hgVisiGene ; includes data from some of the other listed data sources), GEISHA[4],
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| bshanks@85 | 299 Fruitfly.org[21], COMPARE (http://compare.ibdml.univ-mrs.fr/), GXD[17], GEO[3] (GXD and GEO contain spatial data but also non-spatial
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| bshanks@85 | 300 data. All GXD spatial data are also in EMAGE.)
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| bshanks@85 | 301 14without prior offline registration
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| bshanks@85 | 302 15In both cases, the cause is that pairwise correlations between the gene expression of voxels in different areas but the same layer are often stronger
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| bshanks@85 | 303 than pairwise correlations between the gene expression of voxels in different layers but the same area. Therefore, a pairwise voxel correlation
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| bshanks@85 | 304 clustering algorithm will tend to create clusters representing cortical layers, not areas (there may be clusters which presumably correspond to the
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| bshanks@85 | 305 intersection of a layer and an area, but since one area will have many layer-area intersection clusters, further work is needed to make sense of
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| bshanks@85 | 306 these). The reason that Gene Finder cannot the find marker genes for cortical areas is that, although the user chooses a seed voxel, Gene Finder
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| bshanks@85 | 307 chooses the ROI for which genes will be found, and it creates that ROI by (pairwise voxel correlation) clustering around the seed.
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| bshanks@84 | 308 Preliminary Studies
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| bshanks@85 | 309
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| bshanks@85 | 310
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| bshanks@85 | 311 Figure 1: Top row: Genes Nfic and
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| bshanks@85 | 312 A930001M12Rik are the most correlated
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| bshanks@85 | 313 with area SS (somatosensory cortex). Bot-
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| bshanks@85 | 314 tom row: Genes C130038G02Rik and
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| bshanks@85 | 315 Cacna1i are those with the best fit using
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| bshanks@85 | 316 logistic regression. Within each picture, the
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| bshanks@85 | 317 vertical axis roughly corresponds to anterior
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| bshanks@85 | 318 at the top and posterior at the bottom, and
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| bshanks@85 | 319 the horizontal axis roughly corresponds to
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| bshanks@85 | 320 medial at the left and lateral at the right.
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| bshanks@85 | 321 The red outline is the boundary of region
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| bshanks@85 | 322 SS. Pixels are colored according to correla-
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| bshanks@85 | 323 tion, with red meaning high correlation and
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| bshanks@85 | 324 blue meaning low. Format conversion between SEV, MATLAB, NIFTI
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| bshanks@85 | 325 We have created software to (politely) download all of the SEV files16 from
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| bshanks@85 | 326 the Allen Institute website. We have also created software to convert between
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| bshanks@85 | 327 the SEV, MATLAB, and NIFTI file formats, as well as some of Caret’s file
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| bshanks@85 | 328 formats.
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| bshanks@85 | 329 Flatmap of cortex
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| bshanks@85 | 330 We downloaded the ABA data and applied a mask to select only those voxels
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| bshanks@85 | 331 which belong to cerebral cortex. We divided the cortex into hemispheres.
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| bshanks@85 | 332 Using Caret[7], we created a mesh representation of the surface of the se-
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| bshanks@85 | 333 lected voxels. For each gene, for each node of the mesh, we calculated an
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| bshanks@85 | 334 average of the gene expression of the voxels “underneath” that mesh node. We
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| bshanks@85 | 335 then flattened the cortex, creating a two-dimensional mesh.
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| bshanks@85 | 336 We sampled the nodes of the irregular, flat mesh in order to create a regular
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| bshanks@85 | 337 grid of pixel values. We converted this grid into a MATLAB matrix.
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| bshanks@85 | 338 We manually traced the boundaries of each of 49 cortical areas from the
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| bshanks@85 | 339 ABA coronal reference atlas slides. We then converted these manual traces
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| bshanks@85 | 340 into Caret-format regional boundary data on the mesh surface. We projected
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| bshanks@85 | 341 the regions onto the 2-d mesh, and then onto the grid, and then we converted
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| bshanks@85 | 342 the region data into MATLAB format.
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| bshanks@85 | 343 At this point, the data are in the form of a number of 2-D matrices, all in
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| bshanks@85 | 344 registration, with the matrix entries representing a grid of points (pixels) over
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| bshanks@85 | 345 the cortical surface:
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| bshanks@85 | 346 ∙ A 2-D matrix whose entries represent the regional label associated with each
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| bshanks@85 | 347 surface pixel
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| bshanks@85 | 348 ∙ For each gene, a 2-D matrix whose entries represent the average expression
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| bshanks@85 | 349 level underneath each surface pixel
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| bshanks@75 | 350
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| bshanks@78 | 351 Figure 2: Gene Pitx2
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| bshanks@75 | 352 is selectively underex-
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| bshanks@77 | 353 pressed in area SS. We created a normalized version of the gene expression data by subtracting each gene’s mean
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| bshanks@84 | 354 expression level (over all surface pixels) and dividing the expression level of each gene by its
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| bshanks@84 | 355 standard deviation.
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| bshanks@75 | 356 The features and the target area are both functions on the surface pixels. They can be referred
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| bshanks@75 | 357 to as scalar fields over the space of surface pixels; alternately, they can be thought of as images
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| bshanks@75 | 358 which can be displayed on the flatmapped surface.
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| bshanks@75 | 359 To move beyond a single average expression level for each surface pixel, we plan to create a
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| bshanks@75 | 360 separate matrix for each cortical layer to represent the average expression level within that layer.
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| bshanks@75 | 361 Cortical layers are found at different depths in different parts of the cortex. In preparation for
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| bshanks@75 | 362 extracting the layer-specific datasets, we have extended Caret with routines that allow the depth
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| bshanks@75 | 363 of the ROI for volume-to-surface projection to vary.
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| bshanks@75 | 364 In the Research Plan, we describe how we will automatically locate the layer depths. For
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| bshanks@85 | 365 validation, we have manually demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.
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| bshanks@77 | 366 Feature selection and scoring methods
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| bshanks@75 | 367 Underexpression of a gene can serve as a marker Underexpression of a gene can sometimes serve as a marker. See,
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| bshanks@75 | 368 for example, Figure 2.
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| bshanks@75 | 369 Correlation Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance
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| bshanks@75 | 370 as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the
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| bshanks@75 | 371 surface pixels.
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| bshanks@84 | 372 One class of feature selection scoring methods contains methods which calculate some sort of “match” between each gene
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| bshanks@84 | 373 image and the target image. Those genes which match the best are good candidates for features.
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| bshanks@85 | 374 _________________________________________
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| bshanks@85 | 375 16SEV is a sparse format for spatial data. It is the format in which the ABA data is made available.
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| bshanks@75 | 376 One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between
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| bshanks@75 | 377 each gene and each cortical area. The top row of Figure 1 shows the three genes most correlated with area SS.
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| bshanks@85 | 378
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| bshanks@85 | 379
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| bshanks@85 | 380 Figure 3: The top row shows the two genes
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| bshanks@85 | 381 which (individually) best predict area AUD,
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| bshanks@85 | 382 according to logistic regression. The bot-
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| bshanks@85 | 383 tom row shows the two genes which (indi-
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| bshanks@85 | 384 vidually) best match area AUD, according
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| bshanks@85 | 385 to gradient similarity. From left to right and
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| bshanks@85 | 386 top to bottom, the genes are Ssr1, Efcbp1,
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| bshanks@85 | 387 Ptk7, and Aph1a. Conditional entropy An information-theoretic scoring method is to find
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| bshanks@85 | 388 features such that, if the features (gene expression levels) are known, uncer-
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| bshanks@85 | 389 tainty about the target (the regional identity) is reduced. Entropy measures
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| bshanks@85 | 390 uncertainty, so what we want is to find features such that the conditional dis-
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| bshanks@85 | 391 tribution of the target has minimal entropy. The distribution to which we are
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| bshanks@85 | 392 referring is the probability distribution over the population of surface pixels.
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| bshanks@85 | 393 The simplest way to use information theory is on discrete data, so we
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| bshanks@85 | 394 discretized our gene expression data by creating, for each gene, five thresholded
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| bshanks@85 | 395 boolean masks of the gene data. For each gene, we created a boolean mask of
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| bshanks@85 | 396 its expression levels using each of these thresholds: the mean of that gene, the
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| bshanks@85 | 397 mean minus one standard deviation, the mean minus two standard deviations,
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| bshanks@85 | 398 the mean plus one standard deviation, the mean plus two standard deviations.
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| bshanks@85 | 399 Now, for each region, we created and ran a forward stepwise procedure
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| bshanks@85 | 400 which attempted to find pairs of gene expression boolean masks such that the
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| bshanks@85 | 401 conditional entropy of the target area’s boolean mask, conditioned upon the
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| bshanks@85 | 402 pair of gene expression boolean masks, is minimized.
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| bshanks@85 | 403 This finds pairs of genes which are most informative (at least at these dis-
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| bshanks@85 | 404 cretization thresholds) relative to the question, “Is this surface pixel a member
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| bshanks@85 | 405 of the target area?”. Its advantage over linear methods such as logistic regres-
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| bshanks@85 | 406 sion is that it takes account of arbitrarily nonlinear relationships; for example,
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| bshanks@85 | 407 if the XOR of two variables predicts the target, conditional entropy would
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| bshanks@85 | 408 notice, whereas linear methods would not.
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| bshanks@85 | 409 Gradient similarity We noticed that the previous two scoring methods,
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| bshanks@85 | 410 which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. For
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| bshanks@85 | 411 this reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which
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| bshanks@85 | 412 looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method “gradient
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| bshanks@85 | 413 similarity”.
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| bshanks@85 | 414
|
| bshanks@85 | 415
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| bshanks@85 | 416 Figure 4: Upper left: wwc1. Upper right:
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| bshanks@85 | 417 mtif2. Lower left: wwc1 + mtif2 (each
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| bshanks@85 | 418 pixel’s value on the lower left is the sum of
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| bshanks@85 | 419 the corresponding pixels in the upper row). One might say that gradient similarity attempts to measure how much the
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| bshanks@85 | 420 border of the area of gene expression and the border of the target region over-
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| bshanks@85 | 421 lap. However, since gene expression falls off continuously rather than jumping
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| bshanks@85 | 422 from its maximum value to zero, the spatial pattern of a gene’s expression often
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| bshanks@85 | 423 does not have a discrete border. Therefore, instead of looking for a discrete
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| bshanks@85 | 424 border, we look for large gradients. Gradient similarity is a symmetric function
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| bshanks@85 | 425 over two images (i.e. two scalar fields). It is is high to the extent that matching
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| bshanks@85 | 426 pixels which have large values and large gradients also have gradients which
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| bshanks@85 | 427 are oriented in a similar direction. The formula is:
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| bshanks@85 | 428 ∑
|
| bshanks@85 | 429 pixel<img src="cmsy7-32.png" alt="∈" />pixels cos(abs(∠∇1 -∠∇2)) ⋅|∇1| + |∇2|
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| bshanks@41 | 430 2 ⋅ pixel_value1 + pixel_value2
|
| bshanks@41 | 431 2
|
| bshanks@85 | 432 where ∇1 and ∇2 are the gradient vectors of the two images at the current
|
| bshanks@85 | 433 pixel; ∠∇i is the angle of the gradient of image i at the current pixel; |∇i| is
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| bshanks@85 | 434 the magnitude of the gradient of image i at the current pixel; and pixel_valuei
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| bshanks@85 | 435 is the value of the current pixel in image i.
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| bshanks@85 | 436 The intuition is that we want to see if the borders of the pattern in the
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| bshanks@85 | 437 two images are similar; if the borders are similar, then both images will have
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| bshanks@85 | 438 corresponding pixels with large gradients (because this is a border) which are
|
| bshanks@85 | 439 oriented in a similar direction (because the borders are similar).
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| bshanks@69 | 440 Most of the genes in Figure 5 were identified via gradient similarity.
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| bshanks@43 | 441 Gradient similarity provides information complementary to correlation
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| bshanks@41 | 442 To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider
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| bshanks@85 | 443 Fig. 3. The top row of Fig. 3 displays the 3 genes which most match area AUD, according to a pointwise method17. The
|
| bshanks@85 | 444 _________________________________________
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| bshanks@85 | 445 17For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor
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| bshanks@85 | 446 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
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| bshanks@85 | 447 they predict area AUD.
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| bshanks@85 | 448 bottom row displays the 3 genes which most match AUD according to a method which considers local geometry18 The
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| bshanks@46 | 449 pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is
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| bshanks@46 | 450 that this includes many areas which don’t have a salient border matching the areal border. The geometric method identifies
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| bshanks@46 | 451 genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes
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| bshanks@46 | 452 genes which don’t express over the entire area. Genes which have high rankings using both pointwise and border criteria,
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| bshanks@46 | 453 such as Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker
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| bshanks@46 | 454 for AUD; we deliberately chose a “difficult” area in order to better contrast pointwise with geometric methods.
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| bshanks@85 | 455
|
| bshanks@85 | 456
|
| bshanks@85 | 457
|
| bshanks@85 | 458
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| bshanks@85 | 459 Figure 5: From left to right and top
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| bshanks@85 | 460 to bottom, single genes which roughly
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| bshanks@85 | 461 identify areas SS (somatosensory primary
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| bshanks@85 | 462 + supplemental), SSs (supplemental so-
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| bshanks@85 | 463 matosensory), PIR (piriform), FRP (frontal
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| bshanks@85 | 464 pole), RSP (retrosplenial), COApm (Corti-
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| bshanks@85 | 465 cal amygdalar, posterior part, medial zone).
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| bshanks@85 | 466 Grouping some areas together, we have
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| bshanks@85 | 467 also found genes to identify the groups
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| bshanks@85 | 468 ACA+PL+ILA+DP+ORB+MO (anterior
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| bshanks@85 | 469 cingulate, prelimbic, infralimbic, dorsal pe-
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| bshanks@85 | 470 duncular, orbital, motor), posterior and lat-
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| bshanks@85 | 471 eral visual (VISpm, VISpl, VISI, VISp; pos-
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| bshanks@85 | 472 teromedial, posterolateral, lateral, and pri-
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| bshanks@85 | 473 mary visual; the posterior and lateral vi-
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| bshanks@85 | 474 sual area is distinguished from its neigh-
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| bshanks@85 | 475 bors, but not from the entire rest of the
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| bshanks@85 | 476 cortex). The genes are Pitx2, Aldh1a2,
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| bshanks@85 | 477 Ppfibp1, Slco1a5, Tshz2, Trhr, Col12a1,
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| bshanks@85 | 478 Ets1. Areas which can be identified by single genes Using gradient simi-
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| bshanks@85 | 479 larity, we have already found single genes which roughly identify some areas
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| bshanks@85 | 480 and groupings of areas. For each of these areas, an example of a gene which
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| bshanks@85 | 481 roughly identifies it is shown in Figure 5. We have not yet cross-verified these
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| bshanks@85 | 482 genes in other atlases.
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| bshanks@85 | 483 In addition, there are a number of areas which are almost identified by single
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| bshanks@85 | 484 genes: COAa+NLOT (anterior part of cortical amygdalar area, nucleus of the
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| bshanks@85 | 485 lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior cingulate),
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| bshanks@85 | 486 VIS (visual), AUD (auditory).
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| bshanks@85 | 487 These results validate our expectation that the ABA dataset can be ex-
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| bshanks@85 | 488 ploited to find marker genes for many cortical areas, while also validating the
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| bshanks@85 | 489 relevancy of our new scoring method, gradient similarity.
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| bshanks@85 | 490 Combinations of multiple genes are useful and necessary for some
|
| bshanks@85 | 491 areas
|
| bshanks@85 | 492 In Figure 4, we give an example of a cortical area which is not marked by
|
| bshanks@85 | 493 any single gene, but which can be identified combinatorially. Acccording to
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| bshanks@85 | 494 logistic regression, gene wwc1 is the best fit single gene for predicting whether
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| bshanks@85 | 495 or not a pixel on the cortical surface belongs to the motor area (area MO).
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| bshanks@85 | 496 The upper-left picture in Figure 4 shows wwc1’s spatial expression pattern over
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| bshanks@85 | 497 the cortex. The lower-right boundary of MO is represented reasonably well by
|
| bshanks@85 | 498 this gene, but the gene overshoots the upper-left boundary. This flattened 2-D
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| bshanks@85 | 499 representation does not show it, but the area corresponding to the overshoot is
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| bshanks@85 | 500 the medial surface of the cortex. MO is only found on the dorsal surface. Gene
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| bshanks@85 | 501 mtif2 is shown in the upper-right. Mtif2 captures MO’s upper-left boundary,
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| bshanks@85 | 502 but not its lower-right boundary. Mtif2 does not express very much on the
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| bshanks@85 | 503 medial surface. By adding together the values at each pixel in these two figures,
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| bshanks@85 | 504 we get the lower-left image. This combination captures area MO much better
|
| bshanks@85 | 505 than any single gene.
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| bshanks@85 | 506 This shows that our proposal to develop a method to find combinations of
|
| bshanks@85 | 507 marker genes is both possible and necessary.
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| bshanks@85 | 508 Feature selection integrated with prediction As noted earlier, in gen-
|
| bshanks@85 | 509 eral, any predictive method can be used for feature selection by running it
|
| bshanks@85 | 510 inside a stepwise wrapper. Also, some predictive methods integrate soft con-
|
| bshanks@85 | 511 straints on number of features used. Examples of both of these will be seen in
|
| bshanks@85 | 512 the section “Multivariate Predictive methods”.
|
| bshanks@85 | 513 Multivariate Predictive methods
|
| bshanks@85 | 514 Forward stepwise logistic regression Logistic regression is a popular
|
| bshanks@85 | 515 method for predictive modeling of categorial data. As a pilot run, for five
|
| bshanks@85 | 516 cortical areas (SS, AUD, RSP, VIS, and MO), we performed forward stepwise
|
| bshanks@85 | 517 logistic regression to find single genes, pairs of genes, and triplets of genes
|
| bshanks@85 | 518 which predict areal identify. This is an example of feature selection integrated
|
| bshanks@85 | 519 with prediction using a stepwise wrapper. Some of the single genes found
|
| bshanks@85 | 520 were shown in various figures throughout this document, and Figure 4 shows
|
| bshanks@85 | 521 a combination of genes which was found.
|
| bshanks@85 | 522 We felt that, for single genes, gradient similarity did a better job than
|
| bshanks@85 | 523 logistic regression at capturing our subjective impression of a “good gene”.
|
| bshanks@85 | 524 _________________________________________
|
| bshanks@85 | 525 18For each gene the gradient similarity between (a) a map of the expression of each gene on the cortical surface and (b) the shape of area AUD,
|
| bshanks@84 | 526 was calculated, and this was used to rank the genes.
|
| bshanks@84 | 527
|
| bshanks@60 | 528
|
| bshanks@69 | 529
|
| bshanks@69 | 530
|
| bshanks@69 | 531
|
| bshanks@69 | 532 Figure 6: First row: the first 6 reduced dimensions, using PCA. Second
|
| bshanks@69 | 533 row: the first 6 reduced dimensions, using NNMF. Third row: the first
|
| bshanks@69 | 534 six reduced dimensions, using landmark Isomap. Bottom row: examples
|
| bshanks@69 | 535 of kmeans clustering applied to reduced datasets to find 7 clusters. Left:
|
| bshanks@69 | 536 19 of the major subdivisions of the cortex. Second from left: PCA. Third
|
| bshanks@69 | 537 from left: NNMF. Right: Landmark Isomap. Additional details: In the
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| bshanks@69 | 538 third and fourth rows, 7 dimensions were found, but only 6 displayed. In
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| bshanks@69 | 539 the last row: for PCA, 50 dimensions were used; for NNMF, 6 dimensions
|
| bshanks@85 | 540 were used; for landmark Isomap, 7 dimensions were used. SVM on all genes at once
|
| bshanks@85 | 541 In order to see how well one can do when
|
| bshanks@85 | 542 looking at all genes at once, we ran a support
|
| bshanks@85 | 543 vector machine to classify cortical surface pix-
|
| bshanks@85 | 544 els based on their gene expression profiles. We
|
| bshanks@85 | 545 achieved classification accuracy of about 81%19.
|
| bshanks@85 | 546 This shows that the genes included in the ABA
|
| bshanks@85 | 547 dataset are sufficient to define much of cortical
|
| bshanks@85 | 548 anatomy. However, as noted above, a classifier
|
| bshanks@85 | 549 that looks at all the genes at once isn’t as prac-
|
| bshanks@85 | 550 tically useful as a classifier that uses only a few
|
| bshanks@85 | 551 genes.
|
| bshanks@85 | 552 Data-driven redrawing of the cor-
|
| bshanks@85 | 553 tical map
|
| bshanks@85 | 554 We have applied the following dimensional-
|
| bshanks@85 | 555 ity reduction algorithms to reduce the dimen-
|
| bshanks@85 | 556 sionality of the gene expression profile associ-
|
| bshanks@85 | 557 ated with each voxel: Principal Components
|
| bshanks@85 | 558 Analysis (PCA), Simple PCA (SPCA), Multi-
|
| bshanks@85 | 559 Dimensional Scaling (MDS), Isomap, Land-
|
| bshanks@85 | 560 mark Isomap, Laplacian eigenmaps, Local Tan-
|
| bshanks@85 | 561 gent Space Alignment (LTSA), Hessian locally
|
| bshanks@85 | 562 linear embedding, Diffusion maps, Stochastic
|
| bshanks@85 | 563 Neighbor Embedding (SNE), Stochastic Prox-
|
| bshanks@85 | 564 imity Embedding (SPE), Fast Maximum Vari-
|
| bshanks@85 | 565 ance Unfolding (FastMVU), Non-negative Ma-
|
| bshanks@85 | 566 trix Factorization (NNMF). Space constraints
|
| bshanks@85 | 567 prevent us from showing many of the results,
|
| bshanks@85 | 568 but as a sample, PCA, NNMF, and landmark
|
| bshanks@85 | 569 Isomap are shown in the first, second, and third
|
| bshanks@85 | 570 rows of Figure 6.
|
| bshanks@71 | 571
|
| bshanks@85 | 572 Figure 7: Prototypes corresponding to sample gene clusters,
|
| bshanks@85 | 573 clustered by gradient similarity. Region boundaries for the
|
| bshanks@85 | 574 region that most matches each prototype are overlayed. After applying the dimensionality reduction, we ran clus-
|
| bshanks@85 | 575 tering algorithms on the reduced data. To date we have tried
|
| bshanks@85 | 576 k-means and spectral clustering. The results of k-means af-
|
| bshanks@85 | 577 ter PCA, NNMF, and landmark Isomap are shown in the
|
| bshanks@85 | 578 last row of Figure 6. To compare, the leftmost picture on
|
| bshanks@85 | 579 the bottom row of Figure 6 shows some of the major sub-
|
| bshanks@85 | 580 divisions of cortex. These results clearly show that differ-
|
| bshanks@85 | 581 ent dimensionality reduction techniques capture different as-
|
| bshanks@85 | 582 pects of the data and lead to different clusterings, indicating
|
| bshanks@85 | 583 the utility of our proposal to produce a detailed comparion
|
| bshanks@85 | 584 of these techniques as applied to the domain of genomic
|
| bshanks@85 | 585 anatomy.
|
| bshanks@85 | 586 Many areas are captured by clusters of genes We
|
| bshanks@85 | 587 also clustered the genes using gradient similarity to see if
|
| bshanks@85 | 588 the spatial regions defined by any clusters matched known
|
| bshanks@85 | 589 anatomical regions. Figure 7 shows, for ten sample gene clusters, each cluster’s average expression pattern, compared to
|
| bshanks@85 | 590 a known anatomical boundary. This suggests that it is worth attempting to cluster genes, and then to use the results to
|
| bshanks@85 | 591 cluster voxels.
|
| bshanks@85 | 592 _____________________________
|
| bshanks@85 | 593 195-fold cross-validation.
|
| bshanks@84 | 594 Research Design and Methods
|
| bshanks@42 | 595 Further work on flatmapping
|
| bshanks@85 | 596 Often the surface of a structure serves as a natural 2-D basis for anatomical organization. Even when the shape of the
|
| bshanks@85 | 597 surface is known, there are multiple ways to map it into a plane. We will compare mappings which attempt to preserve
|
| bshanks@85 | 598 size (such as the one used by Caret[7]) with mappings which preserve angle (conformal maps). Although there is much 2-D
|
| bshanks@85 | 599 organization in anatomy, there are also structures whose anatomy is fundamentally 3-dimensional. We plan to include a
|
| bshanks@85 | 600 statistical test that warns the user if the assumption of 2-D structure seems to be wrong.
|
| bshanks@85 | 601 Automatic segmentation of cortical layers
|
| bshanks@85 | 602 Extension to probabalistic maps Presently, we do not have a probabalistic atlas which is registered to the ABA
|
| bshanks@85 | 603 space. However, in anticipation of the availability of such maps, we would like to explore extensions to our Aim 1 techniques
|
| bshanks@85 | 604 which can handle probabalistic maps.
|
| bshanks@30 | 605 Develop algorithms that find genetic markers for anatomical regions
|
| bshanks@30 | 606 1.Develop scoring measures for evaluating how good individual genes are at marking areas: we will compare pointwise,
|
| bshanks@30 | 607 geometric, and information-theoretic measures.
|
| bshanks@30 | 608 2.Develop a procedure to find single marker genes for anatomical regions: for each cortical area, by using or combining
|
| bshanks@30 | 609 the scoring measures developed, we will rank the genes by their ability to delineate each area.
|
| bshanks@30 | 610 3.Extend the procedure to handle difficult areas by using combinatorial coding: for areas that cannot be identified by any
|
| bshanks@30 | 611 single gene, identify them with a handful of genes. We will consider both (a) algorithms that incrementally/greedily
|
| bshanks@30 | 612 combine single gene markers into sets, such as forward stepwise regression and decision trees, and also (b) supervised
|
| bshanks@33 | 613 learning techniques which use soft constraints to minimize the number of features, such as sparse support vector
|
| bshanks@30 | 614 machines.
|
| bshanks@33 | 615 4.Extend the procedure to handle difficult areas by combining or redrawing the boundaries: An area may be difficult
|
| bshanks@33 | 616 to identify because the boundaries are misdrawn, or because it does not “really” exist as a single area, at least on the
|
| bshanks@30 | 617 genetic level. We will develop extensions to our procedure which (a) detect when a difficult area could be fit if its
|
| bshanks@30 | 618 boundary were redrawn slightly, and (b) detect when a difficult area could be combined with adjacent areas to create
|
| bshanks@30 | 619 a larger area which can be fit.
|
| bshanks@51 | 620 # Linear discriminant analysis
|
| bshanks@64 | 621 Decision trees todo
|
| bshanks@85 | 622 20.
|
| bshanks@30 | 623 Apply these algorithms to the cortex
|
| bshanks@30 | 624 1.Create open source format conversion tools: we will create tools to bulk download the ABA dataset and to convert
|
| bshanks@30 | 625 between SEV, NIFTI and MATLAB formats.
|
| bshanks@30 | 626 2.Flatmap the ABA cortex data: map the ABA data onto a plane and draw the cortical area boundaries onto it.
|
| bshanks@30 | 627 3.Find layer boundaries: cluster similar voxels together in order to automatically find the cortical layer boundaries.
|
| bshanks@30 | 628 4.Run the procedures that we developed on the cortex: we will present, for each area, a short list of markers to identify
|
| bshanks@30 | 629 that area; and we will also present lists of “panels” of genes that can be used to delineate many areas at once.
|
| bshanks@30 | 630 Develop algorithms to suggest a division of a structure into anatomical parts
|
| bshanks@60 | 631 # mixture models, etc
|
| bshanks@30 | 632 1.Explore dimensionality reduction algorithms applied to pixels: including TODO
|
| bshanks@30 | 633 2.Explore dimensionality reduction algorithms applied to genes: including TODO
|
| bshanks@30 | 634 3.Explore clustering algorithms applied to pixels: including TODO
|
| bshanks@85 | 635 _________________________________________
|
| bshanks@85 | 636 20Already, for each cortical area, we have used the C4.5 algorithm to find a decision tree for that area. We achieved good classification accuracy
|
| bshanks@85 | 637 on our training set, but the number of genes that appeared in each tree was too large. We plan to implement a pruning procedure to generate
|
| bshanks@85 | 638 trees that use fewer genes
|
| bshanks@30 | 639 4.Explore clustering algorithms applied to genes: including gene shaving, TODO
|
| bshanks@30 | 640 5.Develop an algorithm to use dimensionality reduction and/or hierarchial clustering to create anatomical maps
|
| bshanks@30 | 641 6.Run this algorithm on the cortex: present a hierarchial, genoarchitectonic map of the cortex
|
| bshanks@51 | 642 # Linear discriminant analysis
|
| bshanks@51 | 643 # jbt, coclustering
|
| bshanks@51 | 644 # self-organizing map
|
| bshanks@85 | 645 # confirm with EMAGE, GeneAtlas, GENSAT, etc, to fight overfitting, two hemis
|
| bshanks@53 | 646 # compare using clustering scores
|
| bshanks@64 | 647 # multivariate gradient similarity
|
| bshanks@66 | 648 # deep belief nets
|
| bshanks@66 | 649 # note: slice artifact
|
| bshanks@33 | 650 Bibliography & References Cited
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