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| bshanks@0 | 1 Specific aims
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| bshanks@33 | 2 Massive new datasets obtained with techniques such as in situ hybridization (ISH) and BAC-transgenics allow the expres-
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| bshanks@33 | 3 sion levels of many genes at many locations to be compared. Our goal is to develop automated methods to relate spatial
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| bshanks@33 | 4 variation in gene expression to anatomy. We want to find marker genes for specific anatomical regions, and also to draw
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| bshanks@33 | 5 new anatomical maps based on gene expression patterns. We have three specific aims:
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| bshanks@30 | 6 (1) develop an algorithm to screen spatial gene expression data for combinations of marker genes which selectively target
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| bshanks@30 | 7 anatomical regions
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| bshanks@30 | 8 (2) develop an algorithm to suggest new ways of carving up a structure into anatomical subregions, based on spatial
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| bshanks@30 | 9 patterns in gene expression
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| bshanks@33 | 10 (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 | 11 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 | 12 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@30 | 13 In addition to validating the usefulness of the algorithms, the application of these methods to cerebral cortex will produce
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| bshanks@30 | 14 immediate benefits, because there are currently no known genetic markers for many cortical areas. The results of the project
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| bshanks@33 | 15 will support the development of new ways to selectively target cortical areas, and it will support the development of a
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| bshanks@33 | 16 method for identifying the cortical areal boundaries present in small tissue samples.
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| bshanks@33 | 17 All algorithms that we develop will be implemented in an open-source software toolkit. The toolkit, as well as the
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| bshanks@30 | 18 machine-readable datasets developed in aim (3), will be published and freely available for others to use.
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| bshanks@30 | 19 Background and significance
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| bshanks@30 | 20 Aim 1
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| bshanks@30 | 21 Machine learning terminology: supervised learning
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| bshanks@30 | 22 The task of looking for marker genes for anatomical subregions means that one is looking for a set of genes such that, if
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| bshanks@30 | 23 the expression level of those genes is known, then the locations of the subregions can be inferred.
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| bshanks@30 | 24 If we define the subregions so that they cover the entire anatomical structure to be divided, then instead of saying that we
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| bshanks@30 | 25 are using gene expression to find the locations of the subregions, we may say that we are using gene expression to determine
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| bshanks@30 | 26 to which subregion each voxel within the structure belongs. We call this a classification task, because each voxel is being
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| bshanks@30 | 27 assigned to a class (namely, its subregion).
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| bshanks@30 | 28 Therefore, an understanding of the relationship between the combination of their expression levels and the locations of
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| bshanks@33 | 29 the subregions may be expressed as a function. The input to this function is a voxel, along with the gene expression levels
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| bshanks@30 | 30 within that voxel; the output is the subregional identity of the target voxel, that is, the subregion to which the target voxel
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| bshanks@30 | 31 belongs. We call this function a classifier. In general, the input to a classifier is called an instance, and the output is called
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| bshanks@30 | 32 a label (or a class label).
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| bshanks@30 | 33 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 | 34 classifier for any known anatomical structure. Therefore, we seek a procedure by which a gene expression dataset may be
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| bshanks@30 | 35 analyzed in concert with an anatomical atlas in order to produce a classifier. Such a procedure is a type of a machine learning
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| bshanks@33 | 36 procedure. The construction of the classifier is called training (also learning), and the initial gene expression dataset used
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| bshanks@33 | 37 in the construction of the classifier is called training data.
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| bshanks@30 | 38 In the machine learning literature, this sort of procedure may be thought of as a supervised learning task, defined as a
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| bshanks@30 | 39 task in which the goal is to learn a mapping from instances to labels, and the training data consists of a set of instances
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| bshanks@30 | 40 (voxels) for which the labels (subregions) are known.
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| bshanks@30 | 41 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 | 42 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 | 43 a separate feature selection phase, whereas other methods combine feature selection with other aspects of training.
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| bshanks@30 | 44 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 | 45 chosen. Some scoring measures can assign a score to a set of selected genes, not just to a single gene; in this case, a dynamic
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| bshanks@30 | 46 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 | 47 the score. Such procedures are called “stepwise” or “greedy”.
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| bshanks@30 | 48 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@30 | 49 learning algorithm which constructs the classifier may look over the entire dataset. We can categorize score-based feature
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| bshanks@30 | 50 selection methods depending on how the score of calculated. Often the score calculation consists of assigning a sub-score to
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| bshanks@30 | 51 each voxel, and then aggregating these sub-scores into a final score (the aggregation is often a sum or a sum of squares). If
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| bshanks@30 | 52 only information from nearby voxels is used to calculate a voxel’s sub-score, then we say it is a local scoring method. If only
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| bshanks@30 | 53 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@30 | 54 Key questions when choosing a learning method are: What are the instances? What are the features? How are the
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| bshanks@30 | 55 features chosen? Here are four principles that outline our answers to these questions.
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| bshanks@30 | 56 Principle 1: Combinatorial gene expression It is too much to hope that every anatomical region of interest will be
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| bshanks@30 | 57 identified by a single gene. For example, in the cortex, there are some areas which are not clearly delineated by any gene
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| bshanks@30 | 58 included in the Allen Brain Atlas (ABA) dataset. However, at least some of these areas can be delineated by looking at
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| bshanks@30 | 59 combinations of genes (an example of an area for which multiple genes are necessary and sufficient is provided in Preliminary
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| bshanks@30 | 60 Results). Therefore, each instance should contain multiple features (genes).
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| bshanks@30 | 61 Principle 2: Only look at combinations of small numbers of genes When the classifier classifies a voxel, it is
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| bshanks@30 | 62 only allowed to look at the expression of the genes which have been selected as features. The more data that is available to
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| bshanks@30 | 63 a classifier, the better that it can do. For example, perhaps there are weak correlations over many genes that add up to a
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| bshanks@30 | 64 strong signal. So, why not include every gene as a feature? The reason is that we wish to employ the classifier in situations
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| bshanks@30 | 65 in which it is not feasible to gather data about every gene. For example, if we want to use the expression of marker genes as
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| bshanks@30 | 66 a trigger for some regionally-targeted intervention, then our intervention must contain a molecular mechanism to check the
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| bshanks@30 | 67 expression level of each marker gene before it triggers. It is currently infeasible to design a molecular trigger that checks the
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| bshanks@33 | 68 level of more than a handful of genes. Similarly, if the goal is to develop a procedure to do ISH on tissue samples in order
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| bshanks@30 | 69 to label their anatomy, then it is infeasible to label more than a few genes. Therefore, we must select only a few genes as
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| bshanks@30 | 70 features.
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| bshanks@30 | 71 Principle 3: Use geometry in feature selection
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| bshanks@33 | 72 _________________________________________
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| bshanks@33 | 73 1Strictly speaking, the features are gene expression levels, but we’ll call them genes.
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| bshanks@30 | 74 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 | 75 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 | 76 about the geometric relations between each voxel and its neighbors; this requires non-pointwise, local scoring methods. See
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| bshanks@30 | 77 Preliminary Results for evidence of the complementary nature of pointwise and local scoring methods.
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| bshanks@30 | 78 Principle 4: Work in 2-D whenever possible
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| bshanks@30 | 79 There are many anatomical structures which are commonly characterized in terms of a two-dimensional manifold. When
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| bshanks@30 | 80 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 | 81 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@33 | 82 data.
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| bshanks@30 | 83 Therefore, when possible, the instances should represent pixels, not voxels.
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| bshanks@30 | 84 Aim 2
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| bshanks@30 | 85 Machine learning terminology: clustering
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| bshanks@30 | 86 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 | 87 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@30 | 88 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@30 | 89 clustering or cluster analysis.
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| bshanks@30 | 90 The task of deciding how to carve up a structure into anatomical subregions can be put into these terms. The instances
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| bshanks@33 | 91 are once again voxels (or pixels) along with their associated gene expression profiles. We make the assumption that voxels
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| bshanks@33 | 92 from the same subregion have similar gene expression profiles, at least compared to the other subregions. This means that
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| bshanks@33 | 93 clustering voxels is the same as finding potential subregions; we seek a partitioning of the voxels into subregions, that is,
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| bshanks@33 | 94 into clusters of voxels with similar gene expression.
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| bshanks@30 | 95 It is desirable to determine not just one set of subregions, but also how these subregions relate to each other, if at all;
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| bshanks@33 | 96 perhaps some of the subregions are more similar to each other than to the rest, suggesting that, although at a fine spatial
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| bshanks@33 | 97 scale they could be considered separate, on a coarser spatial scale they could be grouped together into one large subregion.
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| bshanks@33 | 98 This suggests the outcome of clustering may be a hierarchial tree of clusters, rather than a single set of clusters which
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| bshanks@33 | 99 partition the voxels. This is called hierarchial clustering.
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| bshanks@30 | 100 Similarity scores
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| bshanks@30 | 101 A crucial choice when designing a clustering method is how to measure similarity, across either pairs of instances, or
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| bshanks@33 | 102 clusters, or both. There is much overlap between scoring methods for feature selection (discussed above under Aim 1) and
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| bshanks@30 | 103 scoring methods for similarity.
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| bshanks@30 | 104 Spatially contiguous clusters; image segmentation
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| bshanks@33 | 105 We have shown that aim 2 is a type of clustering task. In fact, it is a special type of clustering task because we have
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| bshanks@33 | 106 an additional constraint on clusters; voxels grouped together into a cluster must be spatially contiguous. In Preliminary
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| bshanks@33 | 107 Results, we show that one can get reasonable results without enforcing this constraint, however, we plan to compare these
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| bshanks@33 | 108 results against other methods which guarantee contiguous clusters.
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| bshanks@30 | 109 Perhaps the biggest source of continguous clustering algorithms is the field of computer vision, which has produced a
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| bshanks@33 | 110 variety of image segmentation algorithms. Image segmentation is the task of partitioning the pixels in a digital image into
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| bshanks@30 | 111 clusters, usually contiguous clusters. Aim 2 is similar to an image segmentation task. There are two main differences; in
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| bshanks@30 | 112 our task, there are thousands of color channels (one for each gene), rather than just three. There are imaging tasks which
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| bshanks@33 | 113 use more than three colors, however, for example multispectral imaging and hyperspectral imaging, which are often used
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| bshanks@33 | 114 to process satellite imagery. A more crucial difference is that there are various cues which are appropriate for detecting
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| bshanks@33 | 115 sharp object boundaries in a visual scene but which are not appropriate for segmenting abstract spatial data such as gene
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| bshanks@33 | 116 expression. Although many image segmentation algorithms can be expected to work well for segmenting other sorts of
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| bshanks@33 | 117 spatially arranged data, some of these algorithms are specialized for visual images.
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| bshanks@30 | 118 Dimensionality reduction
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| bshanks@33 | 119 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@30 | 120 instances. However, some clustering algorithms perform better on small numbers of features. There are techniques which
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| bshanks@30 | 121 “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 | 122 extraction or dimensionality reduction. The small set of features that such a technique yields is called the reduced feature
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| bshanks@30 | 123 set. After the reduced feature set is created, the instances may be replaced by reduced instances, which have as their features
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| bshanks@30 | 124 the reduced feature set rather than the original feature set of all gene expression levels. Note that the features in the reduced
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| bshanks@30 | 125 feature set do not necessarily correspond to genes; each feature in the reduced set may be any function of the set of gene
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| bshanks@30 | 126 expression levels.
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| bshanks@30 | 127 Another use for dimensionality reduction is to visualize the relationships between subregions. For example, one might
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| bshanks@33 | 128 want tomake a 2-D plot upon which each subregion is represented by a single point, and with the property that subregions
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| bshanks@30 | 129 with similar gene expression profiles should be nearby on the plot (that is, the property that distance between pairs of points
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| bshanks@33 | 130 in the plot should be proportional to some measure of dissimilarity in gene expression). It is likely that no arrangement of
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| bshanks@30 | 131 the points on a 2-D plan will exactly satisfy this property – however, dimensionality reduction techniques allow one to find
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| bshanks@30 | 132 arrangements of points that approximately satisfy that property. Note that in this application, dimensionality reduction
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| bshanks@30 | 133 is being applied after clustering; whereas in the previous paragraph, we were talking about using dimensionality reduction
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| bshanks@30 | 134 before clustering.
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| bshanks@30 | 135 Clustering genes rather than voxels
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| bshanks@30 | 136 Although the ultimate goal is to cluster the instances (voxels or pixels), one strategy to achieve this goal is to first cluster
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| bshanks@30 | 137 the features (genes). There are two ways that clusters of genes could be used.
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| bshanks@30 | 138 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 | 139 have one reduced feature for each gene cluster.
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| bshanks@30 | 140 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@30 | 141 pattern which seems to pick out a single, spatially continguous subregion. Therefore, it seems likely that an anatomically
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| bshanks@30 | 142 interesting subregion will have multiple genes which each individually pick it out2. This suggests the following procedure:
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| bshanks@33 | 143 cluster together genes which pick out similar subregions, and then to use the more popular common subregions as the
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| bshanks@33 | 144 final clusters. In the Preliminary Data we show that a number of anatomically recognized cortical regions, as well as some
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| bshanks@30 | 145 “superregions” formed by lumping together a few regions, are associated with gene clusters in this fashion.
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| bshanks@30 | 146 Aim 3
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| bshanks@30 | 147 Background
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| bshanks@33 | 148 The cortex is divided into areas and layers. To a first approximation, the parcellation of the cortex into areas can
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| bshanks@33 | 149 be drawn as a 2-D map on the surface of the cortex. In the third dimension, the boundaries between the areas continue
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| bshanks@33 | 150 downwards into the cortical depth, perpendicular to the surface. The layer boundaries run parallel to the surface. One can
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| bshanks@33 | 151 picture an area of the cortex as a slice of many-layered cake.
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| bshanks@30 | 152 Although it is known that different cortical areas have distinct roles in both normal functioning and in disease processes,
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| bshanks@30 | 153 there are no known marker genes for many cortical areas. When it is necessary to divide a tissue sample into cortical areas,
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| bshanks@30 | 154 this is a manual process that requires a skilled human to combine multiple visual cues and interpret them in the context of
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| bshanks@30 | 155 their approximate location upon the cortical surface.
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| bshanks@33 | 156 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@33 | 157 completely settled. A proposed division of the cortex into areas is called a cortical map. In the rodent, the lack of a
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| bshanks@36 | 158 single agreed-upon map can be seen by contrasting the recent maps given by Swanson[4] on the one hand, and Paxinos
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| bshanks@36 | 159 and Franklin[3] on the other. While the maps are certainly very similar in their general arrangement, significant differences
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| bshanks@30 | 160 remain in the details.
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| bshanks@36 | 161 The Allen Mouse Brain Atlas dataset
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| bshanks@36 | 162 The Allen Mouse Brain Atlas (ABA) data was produced by doing in-situ hybridization on slices of male, 56-day-old
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| bshanks@36 | 163 C57BL/6J mouse brains. Pictures were taken of the processed slice, and these pictures were semi-automatically analyzed
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| bshanks@36 | 164 in order to create a digital measurement of gene expression levels at each location in each slice. Per slice, cellular spatial
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| bshanks@36 | 165 resolution is achieved. Using this method, a single physical slice can only be used to measure one single gene; many different
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| bshanks@36 | 166 mouse brains were needed in order to measure the expression of many genes.
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| bshanks@36 | 167 Next, an automated nonlinear alignment procedure located the 2D data from the various slices in a single 3D coordinate
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| bshanks@36 | 168 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@36 | 169 voxels in the 3D coordinate system, of which 51,533 are in the brain[2].
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| bshanks@36 | 170 Mus musculus, the common house mouse, is thought to contain about 22,000 protein-coding genes[6]. The ABA contains
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| bshanks@36 | 171 data on about 20,000 genes in sagittal sections, out of which over 4,000 genes are also measured in coronal sections. Our
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| bshanks@36 | 172 dataset is derived from only the coronal subset of the ABA, because the sagittal data does not cover the entire cortex,
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| bshanks@36 | 173 and has greater registration error[2]. Genes were selected by the Allen Institute for coronal sectioning based on, “classes of
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| bshanks@36 | 174 known neuroscientific interest... or through post hoc identification of a marked non-ubiquitous expression pattern”[2].
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| bshanks@30 | 175 Significance
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| bshanks@30 | 176 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@33 | 177 combinatorial expression pattern of those genes uniquely picks out the target area. Finding marker genes will be useful for
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| bshanks@36 | 178 _________________________________________
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| bshanks@36 | 179 2This 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@36 | 180 possible that the currently accepted cortical maps divide the cortex into subregions which are unnatural from the point of view of gene expression;
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| bshanks@36 | 181 perhaps there is some other way to map the cortex for which each subregion can be identified by single genes.
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| bshanks@30 | 182 drug discovery as well as for experimentation because marker genes can be used to design interventions which selectively
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| bshanks@30 | 183 target individual cortical areas.
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| bshanks@30 | 184 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 | 185 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 | 186 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 | 187 will allow experimenters to more easily identify which anatomical areas are present in small samples of cortex.
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| bshanks@33 | 188 The method developed in aim (3) will provide a genoarchitectonic viewpoint that will contribute to the creation of
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| bshanks@33 | 189 a better map. The development of present-day cortical maps was driven by the application of histological stains. It is
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| bshanks@33 | 190 conceivable that if a different set of stains had been available which identified a different set of features, then the today’s
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| bshanks@33 | 191 cortical maps would have come out differently. Since the number of classes of stains is small compared to the number of
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| bshanks@33 | 192 genes, 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@33 | 193 captured by any stain. Therefore, current ideas about cortical anatomy need to incorporate what we can learn from looking
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| bshanks@33 | 194 at the patterns of gene expression.
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| bshanks@30 | 195 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@30 | 196 develop could be used to suggest modifications to the human cortical map as well.
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| bshanks@30 | 197 Related work
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| bshanks@30 | 198 There does not appear to be much work on the automated analysis of spatial gene expression data.
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| bshanks@30 | 199 There is a substantial body of work on the analysis of gene expression data, however, most of this concerns gene expression
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| bshanks@30 | 200 data which is not fundamentally spatial.
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| bshanks@30 | 201 As noted above, there has been much work on both supervised learning and clustering, and there are many available
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| bshanks@30 | 202 algorithms for each. However, the completion of Aims 1 and 2 involves more than just choosing between a set of existing
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| bshanks@33 | 203 algorithms, and will constitute a substantial contribution to biology. The algorithms require the scientist to provide a
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| bshanks@30 | 204 framework for representing the problem domain, and the way that this framework is set up has a large impact on performance.
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| bshanks@30 | 205 Creating a good framework can require creatively reconceptualizing the problem domain, and is not merely a mechanical
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| bshanks@30 | 206 “fine-tuning” of numerical parameters. For example, we believe that domain-specific scoring measures (such as gradient
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| bshanks@30 | 207 similarity, which is discussed in Preliminary Work) may be necessary in order to achieve the best results in this application.
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| bshanks@30 | 208 We are aware of two existing efforts to relate spatial gene expression data to anatomy through computational methods.
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| bshanks@36 | 209 [5 ] describes an analysis of the anatomy of the hippocampus using the ABA dataset. In addition to manual analysis,
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| bshanks@32 | 210 two clustering methods were employed, a modified Non-negative Matrix Factorization (NNMF), and a hierarchial recursive
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| bshanks@32 | 211 bifurcation clustering scheme based on correlation as the similarity score. The paper yielded impressive results, proving the
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| bshanks@34 | 212 usefulness of such research. We have run NNMF on the cortical dataset3 and while the results are promising (see Preliminary
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| bshanks@34 | 213 Data), we think that it will be possible to find a better method (we also think that more automation of the parts that this
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| bshanks@32 | 214 paper’s authors did manually will be possible).
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| bshanks@33 | 215 [2 ] describes AGEA, ”Anatomic Gene Expression Atlas”. AGEA is an analysis tool for the ABA dataset. AGEA has
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| bshanks@32 | 216 three components:
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| bshanks@33 | 217 * Gene Finder: The user selects a seed voxel and the system (1) chooses a cluster which includes the seed voxel, (2)
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| bshanks@33 | 218 yields a list of genes which are overexpressed in that cluster.
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| bshanks@32 | 219 * Correlation: The user selects a seed voxel and the shows the user how much correlation there is between the gene
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| bshanks@32 | 220 expression profile of the seed voxel and every other voxel.
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| bshanks@33 | 221 * Clusters: AGEA includes a precomputed hierarchial clustering of voxels based on a recursive bifurcation algorithm
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| bshanks@33 | 222 with correlation as the similarity metric.
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| bshanks@34 | 223 Gene Finder is different from our Aim 1 in at least four ways. First, although the user chooses a seed voxel, Gene
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| bshanks@34 | 224 Finder, not the user, chooses the cluster for which genes will be found, and in our experience it never chooses cortical areas,
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| bshanks@34 | 225 instead preferring cortical layers4. Therefore, Gene Finder cannot be used to find marker genes for cortical areas. Second,
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| bshanks@34 | 226 Gene Finder finds only single genes, whereas we will also look for combinations of genes5. Third, gene finder can only use
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| bshanks@34 | 227 overexpression as a marker, whereas in the Preliminary Data we show that underexpression can also be used. Fourth, Gene
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| bshanks@34 | 228 Finder uses a simple pointwise score6, whereas we will also use geometric metrics such as gradient similarity.
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| bshanks@36 | 229 _________________________________________
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| bshanks@30 | 230 3We 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@30 | 231 spatial contiguity constraint. However, on our dataset, NNMF naturally produced spatially contiguous clusters, so no additional constraint was
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| bshanks@33 | 232 needed. The paper under discussion mentions that they also tried a hierarchial variant of NNMF, but since they didn’t report its results, we
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| bshanks@33 | 233 assume that those result were not any more impressive than the results of the non-hierarchial variant.
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| bshanks@34 | 234 4Because of the way in which Gene Finder chooses a cluster, layers will always be preferred to areas if pairwise correlations between the gene
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| bshanks@34 | 235 expression of voxels in different areas but the same layer are stronger than pairwise correlatios between the gene expression of voxels in different
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| bshanks@34 | 236 layers but the same area. This appears to be the case.
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| bshanks@34 | 237 5See Preliminary Data for an example of an area which cannot be marked by any single gene in the dataset, but which can be marked by a
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| bshanks@34 | 238 combination.
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| bshanks@34 | 239 6“Expression energy ratio”, which captures overexpression.
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| bshanks@36 | 240 The hierarchial clustering is different from our Aim 2 in at least three ways. First, the clustering finds clusters cor-
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| bshanks@36 | 241 responding to layers, but no clusters corresponding to areas7 8 Our Aim 2 will not be accomplished until a clustering is
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| bshanks@36 | 242 produced which yields areas. Second, AGEA uses perhaps the simplest possible similarity score (correlation), and does no
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| bshanks@36 | 243 dimensionality reduction before calculating similarity. While it is possible that a more complex system will not do any better
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| bshanks@36 | 244 than this, we believe further exploration of alternative methods of scoring and dimensionality reduction is warranted. Third,
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| bshanks@36 | 245 AGEA did not look at clusters of genes; in Preliminary Data we have shown that clusters of genes may identify intersting
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| bshanks@36 | 246 spatial subregions such as cortical areas.
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| bshanks@36 | 247 _______
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| bshanks@36 | 248 7This is for the same reason as in footnote 4.
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| bshanks@34 | 249 8There are clusters which presumably correspond to the intersection of a layer and an area, but since one area will have many layer-area
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| bshanks@34 | 250 intersection clusters, further work is needed to make sense of these.
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| bshanks@30 | 251 Preliminary work
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| bshanks@30 | 252 Format conversion between SEV, MATLAB, NIFTI
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| bshanks@35 | 253 We have created software to (politely) download all of the SEV files from the Allen Institute website. We have also created
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| bshanks@38 | 254 software to convert between the SEV, MATLAB, and NIFTI file formats, as well as some of Caret’s file formats.
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| bshanks@30 | 255 Flatmap of cortex
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| bshanks@36 | 256 We downloaded the ABA data and applied a mask to select only those voxels which belong to cerebral cortex. We divided
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| bshanks@36 | 257 the cortex into hemispheres.
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| bshanks@36 | 258 Using Caret[1], we created a mesh representation of the surface of the selected region. For each gene, for each node of
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| bshanks@37 | 259 the mesh, we used Caret to calculate an average of the gene expression of the voxels “underneath” that mesh node. We
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| bshanks@37 | 260 then used Caret to flatten the cortex, creating a two-dimensional mesh.
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| bshanks@36 | 261 We sampled the nodes of the irregular, flat mesh in order to create a regular grid of pixel values. We converted this grid
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| bshanks@36 | 262 into a MATLAB matrix.
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| bshanks@36 | 263 We manually traced the boundaries of each cortical area from the ABA coronal reference atlas slides. We then converted
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| bshanks@36 | 264 these manual traces into Caret-format regional boundary data on the mesh surface. Using Caret, we projected the regions
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| bshanks@36 | 265 onto the 2-d mesh, and then onto the grid, and then we converted the region data into MATLAB format.
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| bshanks@37 | 266 At this point, the data is in the form of a number of 2-D matrices, all in registration, with the matrix entries representing
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| bshanks@37 | 267 a grid of points (pixels) over the cortical surface:
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| bshanks@36 | 268 ∙A 2-D matrix whose entries represent the regional label associated with each surface pixel
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| bshanks@36 | 269 ∙For each gene, a 2-D matrix whose entries represent the average expression level underneath each surface pixel
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| bshanks@38 | 270 We created a normalized version of the gene expression data by subtracting each gene’s mean expression level (over all
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| bshanks@38 | 271 surface pixels) and dividing each gene by its standard deviation.
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| bshanks@37 | 272 To move beyond a single average expression level for each surface pixel, we plan to create a separate matrix for each
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| bshanks@37 | 273 cortical layer to represent the average expression level within that layer. Cortical layers are found at different depths in
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| bshanks@37 | 274 different parts of the cortex. In preparation for extracting the layer-specific datasets, we have extended Caret with routines
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| bshanks@37 | 275 that allow the depth of the ROI for volume-to-surface projection to vary.
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| bshanks@36 | 276 In the Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually
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| bshanks@36 | 277 demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.
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| bshanks@38 | 278 Feature selection and scoring methods
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| bshanks@38 | 279 Correlation Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance
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| bshanks@38 | 280 as either a member of a particular anatomical area, or not. The target area can be represented as a binary mask over the
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| bshanks@38 | 281 surface pixels.
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| bshanks@38 | 282 The features and the target area are both functions on the surface pixels; alternately, they can be thought of as images
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| bshanks@38 | 283 which can be displayed on the flatmapped surface. One class of feature selection scoring method are those which calculate
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| bshanks@38 | 284 some sort of “match” between each gene image and the target image. Those genes which match the best are good candidates
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| bshanks@38 | 285 for features.
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| bshanks@38 | 286 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@38 | 287 each gene and each cortical area.
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| bshanks@39 | 288 todo: fig
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| bshanks@38 | 289 Conditional entropy An information-theoretic scoring method is to find features such that, if the features (gene
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| bshanks@38 | 290 expression levels) are known, uncertainty about the target (the regional identity) is reduced. Entropy measures uncertainty,
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| bshanks@38 | 291 so what we want is to find features such that the conditional distribution of the target has minimal entropy. The distribution
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| bshanks@38 | 292 to which we are referring is the probability distribution over the population of surface pixels.
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| bshanks@38 | 293 The simplest way to use information theory is on discrete data, so we discretized our gene expression data by creating,
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| bshanks@38 | 294 for each gene, five thresholded binary masks of the gene data. For each gene, we created a binary mask of its expression
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| bshanks@38 | 295 levels over pixels using each of these thresholds: the mean of that gene, the mean minus one standard deviation, the mean
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| bshanks@38 | 296 minus two standard deviations, the mean plus one standard deviation, the mean plus two standard deviations.
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| bshanks@39 | 297 Now, for each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression
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| bshanks@39 | 298 binary masks such that the conditional entropy of the target area’s binary mask, conditioned upon the pair of gene expression
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| bshanks@39 | 299 binary masks, is minimized.
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| bshanks@39 | 300 This finds pairs of genes which are most informative (at least at these discretization thresholds) relative to the question,
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| bshanks@39 | 301 “Is this surface pixel a member of the target area?”.
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| bshanks@38 | 302
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| bshanks@38 | 303
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| bshanks@38 | 304
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| bshanks@38 | 305 Figure 1: Upper left: wwc1. Upper right: mtif2. Lower left: wwc1 + mtif2 (each pixel’s value on the lower left is the sum
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| bshanks@38 | 306 of the corresponding pixels in the upper row). Within each picture, the vertical axis roughly corresponds to anterior at the
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| bshanks@38 | 307 top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right.
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| bshanks@38 | 308 The red outline is the boundary of region MO. Pixels are colored approximately according to the density of expressing cells
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| bshanks@38 | 309 underneath each pixel, with red meaning a lot of expression and blue meaning little.
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| bshanks@39 | 310 todo: fig
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| bshanks@39 | 311 Gradient similarity We noticed that the previous two scoring methods, which are pointwise, often found genes whose
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| bshanks@39 | 312 pattern of expression did not look similar in shape to the target region. Fort his reason we designed a non-pointwise local
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| bshanks@39 | 313 scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar
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| bshanks@39 | 314 to the shape of the target region.
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| bshanks@39 | 315 had shape of the pattern of expression did not seem to match the shape of the target area.
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| bshanks@39 | 316 todo
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| bshanks@30 | 317 Using combinations of multiple genes is necessary and sufficient to delineate some cortical areas
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| bshanks@30 | 318 Here we give an example of a cortical area which is not marked by any single gene, but which can be identified combi-
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| bshanks@34 | 319 natorially. according to logistic regression, gene wwc19 is the best fit single gene for predicting whether or not a pixel on
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| bshanks@30 | 320 the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure shows wwc1’s spatial expression
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| bshanks@30 | 321 pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene
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| bshanks@33 | 322 overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the
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| bshanks@30 | 323 overshoot is the medial surface of the cortex. MO is only found on the lateral surface (todo).
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| bshanks@38 | 324 Gene mtif210 is shown in figure the upper-right of Fig. . Mtif2 captures MO’s upper-left boundary, but not its lower-right
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| bshanks@33 | 325 boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these
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| bshanks@33 | 326 two figures, we get the lower-left of Figure . This combination captures area MO much better than any single gene.
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| bshanks@30 | 327 Geometric and pointwise scoring methods provide complementary information
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| bshanks@36 | 328 To show that local geometry can provide useful information that cannot be detected via pointwise analyses, consider Fig.
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| bshanks@36 | 329 . The top row of Fig. displays the 3 genes which most match area AUD, according to a pointwise method11. The bottom
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| bshanks@36 | 330 row displays the 3 genes which most match AUD according to a method which considers local geometry12 The pointwise
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| bshanks@36 | 331 method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is that this
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| bshanks@36 | 332 includes many areas which don’t have a salient border matching the areal border. The geometric method identifies genes
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| bshanks@38 | 333 whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes genes
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| bshanks@38 | 334 which don’t express over the entire area. Genes which have high rankings using both pointwise and border criteria, such as
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| bshanks@38 | 335 Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker for AUD;
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| bshanks@38 | 336 we deliberately chose a “difficult” area in order to better contrast pointwise with geometric methods.
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| bshanks@38 | 337 Areas which can be identified by single genes
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| bshanks@39 | 338 _________________________________________
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| bshanks@34 | 339 9“WW, C2 and coiled-coil domain containing 1”; EntrezGene ID 211652
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| bshanks@34 | 340 10“mitochondrial translational initiation factor 2”; EntrezGene ID 76784
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| bshanks@36 | 341 11For 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@36 | 342 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@36 | 343 they predict area AUD.
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| bshanks@36 | 344 12For each gene the gradient similarity (see section ??) between (a) a map of the expression of each gene on the cortical surface and (b) the
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| bshanks@36 | 345 shape of area AUD, was calculated, and this was used to rank the genes.
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| bshanks@33 | 346
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| bshanks@30 | 347
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| bshanks@30 | 348
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| bshanks@33 | 349 Figure 2: The top row shows the three genes which (individually) best predict area AUD, according to logistic regression.
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| bshanks@30 | 350 The bottom row shows the three genes which (individually) best match area AUD, according to gradient similarity. From
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| bshanks@30 | 351 left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a, Ptk7, Aph1a again, and Lepr
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| bshanks@39 | 352 todo
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| bshanks@39 | 353 Areas can sometimes be marked by underexpression
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| bshanks@39 | 354 todo
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| bshanks@39 | 355 Specific to Aim 1 (and Aim 3)
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| bshanks@39 | 356 Forward stepwise logistic regression todo
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| bshanks@30 | 357 SVM on all genes at once
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| bshanks@30 | 358 In order to see how well one can do when looking at all genes at once, we ran a support vector machine to classify cortical
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| bshanks@34 | 359 surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%13. As noted above,
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| bshanks@30 | 360 however, a classifier that looks at all the genes at once isn’t practically useful.
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| bshanks@30 | 361 The requirement to find combinations of only a small number of genes limits us from straightforwardly applying many
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| bshanks@33 | 362 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@30 | 363 combines feature selection with supervised learning.
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| bshanks@30 | 364 Decision trees
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| bshanks@30 | 365 todo
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| bshanks@30 | 366 Specific to Aim 2 (and Aim 3)
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| bshanks@30 | 367 Raw dimensionality reduction results
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| bshanks@30 | 368 todo
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| bshanks@30 | 369 (might want to incld nnMF since mentioned above)
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| bshanks@30 | 370 Dimensionality reduction plus K-means or spectral clustering
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| bshanks@30 | 371 Many areas are captured by clusters of genes
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| bshanks@30 | 372 todo
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| bshanks@30 | 373 todo
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| bshanks@36 | 374 _________________________________________
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| bshanks@36 | 375 135-fold cross-validation.
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| bshanks@30 | 376 Research plan
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| bshanks@30 | 377 todo amongst other things:
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| bshanks@30 | 378 Develop algorithms that find genetic markers for anatomical regions
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| bshanks@30 | 379 1.Develop scoring measures for evaluating how good individual genes are at marking areas: we will compare pointwise,
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| bshanks@30 | 380 geometric, and information-theoretic measures.
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| bshanks@30 | 381 2.Develop a procedure to find single marker genes for anatomical regions: for each cortical area, by using or combining
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| bshanks@30 | 382 the scoring measures developed, we will rank the genes by their ability to delineate each area.
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| bshanks@30 | 383 3.Extend the procedure to handle difficult areas by using combinatorial coding: for areas that cannot be identified by any
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| bshanks@30 | 384 single gene, identify them with a handful of genes. We will consider both (a) algorithms that incrementally/greedily
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| bshanks@30 | 385 combine single gene markers into sets, such as forward stepwise regression and decision trees, and also (b) supervised
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| bshanks@33 | 386 learning techniques which use soft constraints to minimize the number of features, such as sparse support vector
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| bshanks@30 | 387 machines.
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| bshanks@33 | 388 4.Extend the procedure to handle difficult areas by combining or redrawing the boundaries: An area may be difficult
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| bshanks@33 | 389 to identify because the boundaries are misdrawn, or because it does not “really” exist as a single area, at least on the
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| bshanks@30 | 390 genetic level. We will develop extensions to our procedure which (a) detect when a difficult area could be fit if its
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| bshanks@30 | 391 boundary were redrawn slightly, and (b) detect when a difficult area could be combined with adjacent areas to create
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| bshanks@30 | 392 a larger area which can be fit.
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| bshanks@30 | 393 Apply these algorithms to the cortex
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| bshanks@30 | 394 1.Create open source format conversion tools: we will create tools to bulk download the ABA dataset and to convert
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| bshanks@30 | 395 between SEV, NIFTI and MATLAB formats.
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| bshanks@30 | 396 2.Flatmap the ABA cortex data: map the ABA data onto a plane and draw the cortical area boundaries onto it.
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| bshanks@30 | 397 3.Find layer boundaries: cluster similar voxels together in order to automatically find the cortical layer boundaries.
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| bshanks@30 | 398 4.Run the procedures that we developed on the cortex: we will present, for each area, a short list of markers to identify
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| bshanks@30 | 399 that area; and we will also present lists of “panels” of genes that can be used to delineate many areas at once.
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| bshanks@30 | 400 Develop algorithms to suggest a division of a structure into anatomical parts
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| bshanks@30 | 401 1.Explore dimensionality reduction algorithms applied to pixels: including TODO
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| bshanks@30 | 402 2.Explore dimensionality reduction algorithms applied to genes: including TODO
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| bshanks@30 | 403 3.Explore clustering algorithms applied to pixels: including TODO
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| bshanks@30 | 404 4.Explore clustering algorithms applied to genes: including gene shaving, TODO
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| bshanks@30 | 405 5.Develop an algorithm to use dimensionality reduction and/or hierarchial clustering to create anatomical maps
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| bshanks@30 | 406 6.Run this algorithm on the cortex: present a hierarchial, genoarchitectonic map of the cortex
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| bshanks@33 | 407 Bibliography & References Cited
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| bshanks@33 | 408 [1]D C Van Essen, H A Drury, J Dickson, J Harwell, D Hanlon, and C H Anderson. An integrated software suite for surface-
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| bshanks@33 | 409 based analyses of cerebral cortex. Journal of the American Medical Informatics Association: JAMIA, 8(5):443–59, 2001.
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| bshanks@33 | 410 PMID: 11522765.
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| bshanks@33 | 411 [2]Lydia Ng, Amy Bernard, Chris Lau, Caroline C Overly, Hong-Wei Dong, Chihchau Kuan, Sayan Pathak, Susan M Sunkin,
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| bshanks@33 | 412 Chinh Dang, Jason W Bohland, Hemant Bokil, Partha P Mitra, Luis Puelles, John Hohmann, David J Anderson, Ed S
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| bshanks@33 | 413 Lein, Allan R Jones, and Michael Hawrylycz. An anatomic gene expression atlas of the adult mouse brain. Nat Neurosci,
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| bshanks@33 | 414 12(3):356–362, March 2009.
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| bshanks@36 | 415 [3]George Paxinos and Keith B.J. Franklin. The Mouse Brain in Stereotaxic Coordinates. Academic Press, 2 edition, July
|
| bshanks@36 | 416 2001.
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| bshanks@36 | 417 [4]Larry Swanson. Brain Maps: Structure of the Rat Brain. Academic Press, 3 edition, November 2003.
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| bshanks@36 | 418 [5]Carol L. Thompson, Sayan D. Pathak, Andreas Jeromin, Lydia L. Ng, Cameron R. MacPherson, Marty T. Mortrud,
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| bshanks@33 | 419 Allison Cusick, Zackery L. Riley, Susan M. Sunkin, Amy Bernard, Ralph B. Puchalski, Fred H. Gage, Allan R. Jones,
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| bshanks@33 | 420 Vladimir B. Bajic, Michael J. Hawrylycz, and Ed S. Lein. Genomic anatomy of the hippocampus. Neuron, 60(6):1010–
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| bshanks@33 | 421 1021, December 2008.
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| bshanks@36 | 422 [6]Robert H Waterston, Kerstin Lindblad-Toh, Ewan Birney, Jane Rogers, Josep F Abril, Pankaj Agarwal, Richa Agarwala,
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| bshanks@36 | 423 Rachel Ainscough, Marina Alexandersson, Peter An, Stylianos E Antonarakis, John Attwood, Robert Baertsch, Jonathon
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| bshanks@36 | 424 Bailey, Karen Barlow, Stephan Beck, Eric Berry, Bruce Birren, Toby Bloom, Peer Bork, Marc Botcherby, Nicolas Bray,
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| bshanks@36 | 425 Michael R Brent, Daniel G Brown, Stephen D Brown, Carol Bult, John Burton, Jonathan Butler, Robert D Campbell,
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| bshanks@36 | 426 Piero Carninci, Simon Cawley, Francesca Chiaromonte, Asif T Chinwalla, Deanna M Church, Michele Clamp, Christopher
|
| bshanks@36 | 427 Clee, Francis S Collins, Lisa L Cook, Richard R Copley, Alan Coulson, Olivier Couronne, James Cuff, Val Curwen, Tim
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| bshanks@36 | 428 Cutts, Mark Daly, Robert David, Joy Davies, Kimberly D Delehaunty, Justin Deri, Emmanouil T Dermitzakis, Colin
|
| bshanks@36 | 429 Dewey, Nicholas J Dickens, Mark Diekhans, Sheila Dodge, Inna Dubchak, Diane M Dunn, Sean R Eddy, Laura Elnitski,
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| bshanks@36 | 430 Richard D Emes, Pallavi Eswara, Eduardo Eyras, Adam Felsenfeld, Ginger A Fewell, Paul Flicek, Karen Foley, Wayne N
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| bshanks@36 | 431 Frankel, Lucinda A Fulton, Robert S Fulton, Terrence S Furey, Diane Gage, Richard A Gibbs, Gustavo Glusman, Sante
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| bshanks@36 | 432 Gnerre, Nick Goldman, Leo Goodstadt, Darren Grafham, Tina A Graves, Eric D Green, Simon Gregory, Roderic Guig,
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| bshanks@36 | 433 Mark Guyer, Ross C Hardison, David Haussler, Yoshihide Hayashizaki, LaDeana W Hillier, Angela Hinrichs, Wratko
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| bshanks@36 | 434 Hlavina, Timothy Holzer, Fan Hsu, Axin Hua, Tim Hubbard, Adrienne Hunt, Ian Jackson, David B Jaffe, L Steven
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| bshanks@36 | 435 Johnson, Matthew Jones, Thomas A Jones, Ann Joy, Michael Kamal, Elinor K Karlsson, Donna Karolchik, Arkadiusz
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| bshanks@36 | 436 Kasprzyk, Jun Kawai, Evan Keibler, Cristyn Kells, W James Kent, Andrew Kirby, Diana L Kolbe, Ian Korf, Raju S
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| bshanks@36 | 437 Kucherlapati, Edward J Kulbokas, David Kulp, Tom Landers, J P Leger, Steven Leonard, Ivica Letunic, Rosie Levine, Jia
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| bshanks@36 | 438 Li, Ming Li, Christine Lloyd, Susan Lucas, Bin Ma, Donna R Maglott, Elaine R Mardis, Lucy Matthews, Evan Mauceli,
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| bshanks@36 | 439 John H Mayer, Megan McCarthy, W Richard McCombie, Stuart McLaren, Kirsten McLay, John D McPherson, Jim
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| bshanks@36 | 440 Meldrim, Beverley Meredith, Jill P Mesirov, Webb Miller, Tracie L Miner, Emmanuel Mongin, Kate T Montgomery,
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| bshanks@36 | 441 Michael Morgan, Richard Mott, James C Mullikin, Donna M Muzny, William E Nash, Joanne O Nelson, Michael N
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| bshanks@36 | 442 Nhan, Robert Nicol, Zemin Ning, Chad Nusbaum, Michael J O’Connor, Yasushi Okazaki, Karen Oliver, Emma Overton-
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| bshanks@36 | 443 Larty, Lior Pachter, Gens Parra, Kymberlie H Pepin, Jane Peterson, Pavel Pevzner, Robert Plumb, Craig S Pohl, Alex
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| bshanks@36 | 444 Poliakov, Tracy C Ponce, Chris P Ponting, Simon Potter, Michael Quail, Alexandre Reymond, Bruce A Roe, Krishna M
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| bshanks@36 | 445 Roskin, Edward M Rubin, Alistair G Rust, Ralph Santos, Victor Sapojnikov, Brian Schultz, Jrg Schultz, Matthias S
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| bshanks@36 | 446 Schwartz, Scott Schwartz, Carol Scott, Steven Seaman, Steve Searle, Ted Sharpe, Andrew Sheridan, Ratna Shownkeen,
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| bshanks@36 | 447 Sarah Sims, Jonathan B Singer, Guy Slater, Arian Smit, Douglas R Smith, Brian Spencer, Arne Stabenau, Nicole Stange-
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| bshanks@36 | 448 Thomann, Charles Sugnet, Mikita Suyama, Glenn Tesler, Johanna Thompson, David Torrents, Evanne Trevaskis, John
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| bshanks@36 | 449 Tromp, Catherine Ucla, Abel Ureta-Vidal, Jade P Vinson, Andrew C Von Niederhausern, Claire M Wade, Melanie Wall,
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| bshanks@36 | 450 Ryan J Weber, Robert B Weiss, Michael C Wendl, Anthony P West, Kris Wetterstrand, Raymond Wheeler, Simon
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| bshanks@36 | 451 Whelan, Jamey Wierzbowski, David Willey, Sophie Williams, Richard K Wilson, Eitan Winter, Kim C Worley, Dudley
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| bshanks@36 | 452 Wyman, Shan Yang, Shiaw-Pyng Yang, Evgeny M Zdobnov, Michael C Zody, and Eric S Lander. Initial sequencing and
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| bshanks@36 | 453 comparative analysis of the mouse genome. Nature, 420(6915):520–62, December 2002. PMID: 12466850.
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| bshanks@33 | 454
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| bshanks@33 | 455 _______________________________________________________________________________________________________
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| bshanks@30 | 456 stuff i dunno where to put yet (there is more scattered through grant-oldtext):
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| bshanks@16 | 457 Principle 4: Work in 2-D whenever possible
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| bshanks@33 | 458 In anatomy, the manifold of interest is usually either defined by a combination of two relevant anatomical axes (todo),
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| bshanks@33 | 459 or by the surface of the structure (as is the case with the cortex). In the former case, the manifold of interest is a plane, but
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| bshanks@33 | 460 in the latter case it is curved. If the manifold is curved, there are various methods for mapping the manifold into a plane.
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| bshanks@33 | 461 The method that we will develop will begin by mapping the data into a 2-D plane. Although the manifold that
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| bshanks@33 | 462 characterized cortical areas is known to be the cortical surface, it remains to be seen which method of mapping the manifold
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| bshanks@33 | 463 into a plane is optimal for this application. We will compare mappings which attempt to preserve size (such as the one used
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| bshanks@33 | 464 by Caret[1]) with mappings which preserve angle (conformal maps).
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| bshanks@33 | 465 Although there is much 2-D organization in anatomy, there are also structures whose shape is fundamentally 3-dimensional.
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| bshanks@30 | 466 If possible, we would like the method we develop to include a statistical test that warns the user if the assumption of 2-D
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| bshanks@30 | 467 structure seems to be wrong.
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| bshanks@33 | 468 —
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| bshanks@33 | 469 note:
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| bshanks@33 | 470 do we need to cite: no known markers, impressive results?
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| bshanks@36 | 471 two hemis
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| bshanks@33 | 472
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| bshanks@33 | 473
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