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changeset 93:9f36acf8d9a8
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author | bshanks@bshanks.dyndns.org |
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date | Tue Apr 21 14:50:10 2009 -0700 (16 years ago) |
parents | b4b79f107b2a |
children | e460569c21d4 |
files | grant.doc grant.html grant.odt grant.pdf grant.txt |
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2.4 from gene expression data.
2.5 Our project is guided by a concrete application with a well-specified criterion of success (how well we can find marker
2.6 genes for / reproduce the layout of cortical areas), which will provide a solid basis for comparing different methods.
2.7 -_________________________________________
2.8 - 11Outside of isocortex, the number of layers varies.
2.9 +Significance
2.10 +________________________
2.11 +11Outside of isocortex, the number of layers varies.
2.12 12The sagittal data do not cover the entire cortex, and also have greater registration error[13]. Genes were selected by the Allen Institute for
2.13 coronal sectioning based on, “classes of known neuroscientific interest... or through post hoc identification of a marked non-ubiquitous expression
2.14 pattern”[13].
2.15 @@ -298,51 +299,6 @@
2.16 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
2.17 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
2.18 chooses the ROI for which genes will be found, and it creates that ROI by (pairwise voxel correlation) clustering around the seed.
2.19 -Significance
2.20 -The method developed in aim (1) will be applied to each cortical area to find a set of marker genes such that the combinatorial
2.21 -expression pattern of those genes uniquely picks out the target area. Finding marker genes will be useful for drug discovery
2.22 -as well as for experimentation because marker genes can be used to design interventions which selectively target individual
2.23 -cortical areas.
2.24 -The application of the marker gene finding algorithm to the cortex will also support the development of new neuroanatom-
2.25 -ical methods. In addition to finding markers for each individual cortical areas, we will find a small panel of genes that can
2.26 -find many of the areal boundaries at once. This panel of marker genes will allow the development of an ISH protocol that
2.27 -will allow experimenters to more easily identify which anatomical areas are present in small samples of cortex.
2.28 -The method developed in aim (2) will provide a genoarchitectonic viewpoint that will contribute to the creation of a
2.29 -better map. The development of present-day cortical maps was driven by the application of histological stains. If a different
2.30 -set of stains had been available which identified a different set of features, then today’s cortical maps may have come out
2.31 -differently. It is likely that there are many repeated, salient spatial patterns in the gene expression which have not yet been
2.32 -captured by any stain. Therefore, cortical anatomy needs to incorporate what we can learn from looking at the patterns of
2.33 -gene expression.
2.34 -While we do not here propose to analyze human gene expression data, it is conceivable that the methods we propose
2.35 -to develop could be used to suggest modifications to the human cortical map as well. In fact, the methods we will develop
2.36 -will be applicable to other datasets beyond the brain. We will provide an open-source toolbox to allow other researchers
2.37 -to easily use our methods. With these methods, researchers with gene expression for any area of the body will be able to
2.38 -efficiently find marker genes for anatomical regions, or to use gene expression to discover new anatomical patterning. As
2.39 -described above, marker genes have a variety of uses in the development of drugs and experimental manipulations, and in
2.40 -the anatomical characterization of tissue samples. The discovery of new ways to carve up anatomical structures into regions
2.41 -may lead to the discovery of new anatomical subregions in various structures, which will widely impact all areas of biology.
2.42 -Although our particular application involves the 3D spatial distribution of gene expression, we anticipate that the
2.43 -methods developed in aims (1) and (2) will not be limited to gene expression data, but rather will generalize to any sort of
2.44 -high-dimensional data over points located in a low-dimensional space.
2.45 -The approach: Preliminary Studies
2.46 -Format conversion between SEV, MATLAB, NIFTI
2.47 -We have created software to (politely) download all of the SEV files16 from the Allen Institute website. We have also created
2.48 -software to convert between the SEV, MATLAB, and NIFTI file formats, as well as some of Caret’s file formats.
2.49 -Flatmap of cortex
2.50 -We downloaded the ABA data and applied a mask to select only those voxels which belong to cerebral cortex. We divided
2.51 -the cortex into hemispheres.
2.52 -Using Caret[7], we created a mesh representation of the surface of the selected voxels. For each gene, for each node of
2.53 -the mesh, we calculated an average of the gene expression of the voxels “underneath” that mesh node. We then flattened
2.54 -the cortex, creating a two-dimensional mesh.
2.55 -We sampled the nodes of the irregular, flat mesh in order to create a regular grid of pixel values. We converted this grid
2.56 -into a MATLAB matrix.
2.57 -We manually traced the boundaries of each of 49 cortical areas from the ABA coronal reference atlas slides. We then
2.58 -converted these manual traces into Caret-format regional boundary data on the mesh surface. We projected the regions
2.59 -onto the 2-d mesh, and then onto the grid, and then we converted the region data into MATLAB format.
2.60 -At this point, the data are in the form of a number of 2-D matrices, all in registration, with the matrix entries representing
2.61 -a grid of points (pixels) over the cortical surface:
2.62 -_
2.63 - 16SEV is a sparse format for spatial data. It is the format in which the ABA data is made available.
2.64
2.65
2.66
2.67 @@ -359,36 +315,67 @@
2.68 The red outline is the boundary of region
2.69 SS. Pixels are colored according to correla-
2.70 tion, with red meaning high correlation and
2.71 -blue meaning low.
2.72 -∙A 2-D matrix whose entries represent the regional label associated with each surface pixel
2.73 -∙For each gene, a 2-D matrix whose entries represent the average expression level underneath each surface pixel
2.74 +blue meaning low. The method developed in aim (1) will be applied to each cortical area to find
2.75 + a set of marker genes such that the combinatorial expression pattern of those
2.76 + genes uniquely picks out the target area. Finding marker genes will be useful
2.77 + for drug discovery as well as for experimentation because marker genes can be
2.78 + used to design interventions which selectively target individual cortical areas.
2.79 + The application of the marker gene finding algorithm to the cortex will
2.80 + also support the development of new neuroanatomical methods. In addition
2.81 + to finding markers for each individual cortical areas, we will find a small panel
2.82 + of genes that can find many of the areal boundaries at once. This panel of
2.83 + marker genes will allow the development of an ISH protocol that will allow
2.84 + experimenters to more easily identify which anatomical areas are present in
2.85 + small samples of cortex.
2.86 + The method developed in aim (2) will provide a genoarchitectonic viewpoint
2.87 + that will contribute to the creation of a better map. The development of
2.88 + present-day cortical maps was driven by the application of histological stains.
2.89 + If a different set of stains had been available which identified a different set of
2.90 + features, then today’s cortical maps may have come out differently. It is likely
2.91 + that there are many repeated, salient spatial patterns in the gene expression
2.92 + which have not yet been captured by any stain. Therefore, cortical anatomy
2.93 + needs to incorporate what we can learn from looking at the patterns of gene
2.94 + expression.
2.95 + While we do not here propose to analyze human gene expression data, it is
2.96 + conceivable that the methods we propose to develop could be used to suggest
2.97 + modifications to the human cortical map as well. In fact, the methods we
2.98 + will develop will be applicable to other datasets beyond the brain. We will
2.99 + provide an open-source toolbox to allow other researchers to easily use our
2.100 + methods. With these methods, researchers with gene expression for any area
2.101 + of the body will be able to efficiently find marker genes for anatomical regions,
2.102 +or to use gene expression to discover new anatomical patterning. As described above, marker genes have a variety of uses in
2.103 +the development of drugs and experimental manipulations, and in the anatomical characterization of tissue samples. The
2.104 +discovery of new ways to carve up anatomical structures into regions may lead to the discovery of new anatomical subregions
2.105 +in various structures, which will widely impact all areas of biology.
2.106
2.107 Figure 2: Gene Pitx2
2.108 is selectively underex-
2.109 -pressed in area SS. We created a normalized version of the gene expression data by subtracting each gene’s mean
2.110 - expression level (over all surface pixels) and dividing the expression level of each gene by its
2.111 - standard deviation.
2.112 - The features and the target area are both functions on the surface pixels. They can be referred
2.113 - to as scalar fields over the space of surface pixels; alternately, they can be thought of as images
2.114 - which can be displayed on the flatmapped surface.
2.115 - To move beyond a single average expression level for each surface pixel, we plan to create a
2.116 - separate matrix for each cortical layer to represent the average expression level within that layer.
2.117 - Cortical layers are found at different depths in different parts of the cortex. In preparation for
2.118 - extracting the layer-specific datasets, we have extended Caret with routines that allow the depth
2.119 - of the ROI for volume-to-surface projection to vary.
2.120 - In the Research Plan, we describe how we will automatically locate the layer depths. For
2.121 -validation, we have manually demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.
2.122 -Feature selection and scoring methods
2.123 -Underexpression of a gene can serve as a marker Underexpression of a gene can sometimes serve as a marker. See,
2.124 -for example, Figure 2.
2.125 -Correlation Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance
2.126 -as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the
2.127 -surface pixels.
2.128 -One class of feature selection scoring methods contains methods which calculate some sort of “match” between each gene
2.129 -image and the target image. Those genes which match the best are good candidates for features.
2.130 -One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between
2.131 -each gene and each cortical area. The top row of Figure 1 shows the three genes most correlated with area SS.
2.132 -
2.133 +pressed in area SS. Although our particular application involves the 3D spatial distribution of gene expression, we
2.134 + anticipate that the methods developed in aims (1) and (2) will not be limited to gene expression
2.135 + data, but rather will generalize to any sort of high-dimensional data over points located in a
2.136 + low-dimensional space.
2.137 + The approach: Preliminary Studies
2.138 + Format conversion between SEV, MATLAB, NIFTI
2.139 + We have created software to (politely) download all of the SEV files16 from the Allen Institute
2.140 + website. We have also created software to convert between the SEV, MATLAB, and NIFTI file
2.141 + formats, as well as some of Caret’s file formats.
2.142 + Flatmap of cortex
2.143 + We downloaded the ABA data and applied a mask to select only those voxels which belong to
2.144 + cerebral cortex. We divided the cortex into hemispheres.
2.145 +Using Caret[7], we created a mesh representation of the surface of the selected voxels. For each gene, for each node of
2.146 +the mesh, we calculated an average of the gene expression of the voxels “underneath” that mesh node. We then flattened
2.147 +the cortex, creating a two-dimensional mesh.
2.148 +We sampled the nodes of the irregular, flat mesh in order to create a regular grid of pixel values. We converted this grid
2.149 +into a MATLAB matrix.
2.150 +We manually traced the boundaries of each of 49 cortical areas from the ABA coronal reference atlas slides. We then
2.151 +converted these manual traces into Caret-format regional boundary data on the mesh surface. We projected the regions
2.152 +onto the 2-d mesh, and then onto the grid, and then we converted the region data into MATLAB format.
2.153 +_________________________________________
2.154 + 16SEV is a sparse format for spatial data. It is the format in which the ABA data is made available.
2.155 +At this point, the data are in the form of a number of 2-D matrices, all in registration, with the matrix entries representing
2.156 +a grid of points (pixels) over the cortical surface:
2.157 +∙A 2-D matrix whose entries represent the regional label associated with each surface pixel
2.158 +∙For each gene, a 2-D matrix whose entries represent the average expression level underneath each surface pixel
2.159
2.160
2.161 Figure 3: The top row shows the two genes
2.162 @@ -398,7 +385,39 @@
2.163 vidually) best match area AUD, according
2.164 to gradient similarity. From left to right and
2.165 top to bottom, the genes are Ssr1, Efcbp1,
2.166 -Ptk7, and Aph1a. Conditional entropy An information-theoretic scoring method is to find
2.167 +Ptk7, and Aph1a. We created a normalized version of the gene expression data by subtracting
2.168 + each gene’s mean expression level (over all surface pixels) and dividing the
2.169 + expression level of each gene by its standard deviation.
2.170 + The features and the target area are both functions on the surface pix-
2.171 + els. They can be referred to as scalar fields over the space of surface pixels;
2.172 + alternately, they can be thought of as images which can be displayed on the
2.173 + flatmapped surface.
2.174 + To move beyond a single average expression level for each surface pixel, we
2.175 + plan to create a separate matrix for each cortical layer to represent the average
2.176 + expression level within that layer. Cortical layers are found at different depths
2.177 + in different parts of the cortex. In preparation for extracting the layer-specific
2.178 + datasets, we have extended Caret with routines that allow the depth of the
2.179 + ROI for volume-to-surface projection to vary.
2.180 + In the Research Plan, we describe how we will automatically locate the
2.181 + layer depths. For validation, we have manually demarcated the depth of the
2.182 + outer boundary of cortical layer 5 throughout the cortex.
2.183 + Feature selection and scoring methods
2.184 + Underexpression of a gene can serve as a marker Underexpression of a
2.185 + gene can sometimes serve as a marker. See, for example, Figure 2.
2.186 + Correlation Recall that the instances are surface pixels, and consider the
2.187 + problem of attempting to classify each instance as either a member of a partic-
2.188 + ular anatomical area, or not. The target area can be represented as a boolean
2.189 + mask over the surface pixels.
2.190 +One class of feature selection scoring methods contains methods which calculate some sort of “match” between each gene
2.191 +image and the target image. Those genes which match the best are good candidates for features.
2.192 +One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between
2.193 +each gene and each cortical area. The top row of Figure 1 shows the three genes most correlated with area SS.
2.194 +
2.195 +
2.196 +Figure 4: Upper left: wwc1. Upper right:
2.197 +mtif2. Lower left: wwc1 + mtif2 (each
2.198 +pixel’s value on the lower left is the sum of
2.199 +the corresponding pixels in the upper row). Conditional entropy An information-theoretic scoring method is to find
2.200 features such that, if the features (gene expression levels) are known, uncer-
2.201 tainty about the target (the regional identity) is reduced. Entropy measures
2.202 uncertainty, so what we want is to find features such that the conditional dis-
2.203 @@ -417,56 +436,9 @@
2.204 This finds pairs of genes which are most informative (at least at these dis-
2.205 cretization thresholds) relative to the question, “Is this surface pixel a member
2.206 of the target area?”. Its advantage over linear methods such as logistic regres-
2.207 - sion is that it takes account of arbitrarily nonlinear relationships; for example,
2.208 - if the XOR of two variables predicts the target, conditional entropy would
2.209 - notice, whereas linear methods would not.
2.210 -
2.211 -
2.212 -Figure 4: Upper left: wwc1. Upper right:
2.213 -mtif2. Lower left: wwc1 + mtif2 (each
2.214 -pixel’s value on the lower left is the sum of
2.215 -the corresponding pixels in the upper row). Gradient similarity We noticed that the previous two scoring methods,
2.216 - which are pointwise, often found genes whose pattern of expression did not
2.217 - look similar in shape to the target region. For this reason we designed a
2.218 - non-pointwise local scoring method to detect when a gene had a pattern of
2.219 - expression which looked like it had a boundary whose shape is similar to the
2.220 - shape of the target region. We call this scoring method “gradient similarity”.
2.221 - One might say that gradient similarity attempts to measure how much the
2.222 - border of the area of gene expression and the border of the target region over-
2.223 - lap. However, since gene expression falls off continuously rather than jumping
2.224 - from its maximum value to zero, the spatial pattern of a gene’s expression often
2.225 - does not have a discrete border. Therefore, instead of looking for a discrete
2.226 - border, we look for large gradients. Gradient similarity is a symmetric function
2.227 - over two images (i.e. two scalar fields). It is is high to the extent that matching
2.228 - pixels which have large values and large gradients also have gradients which
2.229 - are oriented in a similar direction. The formula is:
2.230 - ∑
2.231 - pixel<img src="cmsy7-32.png" alt="∈" />pixels cos(abs(∠∇1 -∠∇2)) ⋅|∇1| + |∇2|
2.232 - 2 ⋅ pixel_value1 + pixel_value2
2.233 - 2
2.234 - where ∇1 and ∇2 are the gradient vectors of the two images at the current
2.235 -pixel; ∠∇i is the angle of the gradient of image i at the current pixel; |∇i| is the magnitude of the gradient of image i at
2.236 -the current pixel; and pixel_valuei is the value of the current pixel in image i.
2.237 -The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar,
2.238 -then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a
2.239 -similar direction (because the borders are similar).
2.240 -Most of the genes in Figure 5 were identified via gradient similarity.
2.241 -Gradient similarity provides information complementary to correlation
2.242 -To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider
2.243 -Fig. 3. The top row of Fig. 3 displays the 3 genes which most match area AUD, according to a pointwise method17. The
2.244 -bottom row displays the 3 genes which most match AUD according to a method which considers local geometry18 The
2.245 -pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is
2.246 -that this includes many areas which don’t have a salient border matching the areal border. The geometric method identifies
2.247 -_________________________________________
2.248 - 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
2.249 -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
2.250 -they predict area AUD.
2.251 - 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,
2.252 -was calculated, and this was used to rank the genes.
2.253 -genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes
2.254 -genes which don’t express over the entire area. Genes which have high rankings using both pointwise and border criteria,
2.255 -such as Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker
2.256 -for AUD; we deliberately chose a “difficult” area in order to better contrast pointwise with geometric methods.
2.257 +sion is that it takes account of arbitrarily nonlinear relationships; for example, if the XOR of two variables predicts the
2.258 +target, conditional entropy would notice, whereas linear methods would not.
2.259 +
2.260
2.261
2.262
2.263 @@ -490,59 +462,81 @@
2.264 bors, but not from the entire rest of the
2.265 cortex). The genes are Pitx2, Aldh1a2,
2.266 Ppfibp1, Slco1a5, Tshz2, Trhr, Col12a1,
2.267 -Ets1. Areas which can be identified by single genes Using gradient simi-
2.268 - larity, we have already found single genes which roughly identify some areas
2.269 - and groupings of areas. For each of these areas, an example of a gene which
2.270 - roughly identifies it is shown in Figure 5. We have not yet cross-verified these
2.271 - genes in other atlases.
2.272 - In addition, there are a number of areas which are almost identified by single
2.273 - genes: COAa+NLOT (anterior part of cortical amygdalar area, nucleus of the
2.274 - lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior cingulate),
2.275 - VIS (visual), AUD (auditory).
2.276 - These results validate our expectation that the ABA dataset can be ex-
2.277 - ploited to find marker genes for many cortical areas, while also validating the
2.278 - relevancy of our new scoring method, gradient similarity.
2.279 - Combinations of multiple genes are useful and necessary for some
2.280 - areas
2.281 - In Figure 4, we give an example of a cortical area which is not marked by
2.282 - any single gene, but which can be identified combinatorially. Acccording to
2.283 - logistic regression, gene wwc1 is the best fit single gene for predicting whether
2.284 - or not a pixel on the cortical surface belongs to the motor area (area MO).
2.285 - The upper-left picture in Figure 4 shows wwc1’s spatial expression pattern over
2.286 - the cortex. The lower-right boundary of MO is represented reasonably well by
2.287 - this gene, but the gene overshoots the upper-left boundary. This flattened 2-D
2.288 - representation does not show it, but the area corresponding to the overshoot is
2.289 - the medial surface of the cortex. MO is only found on the dorsal surface. Gene
2.290 - mtif2 is shown in the upper-right. Mtif2 captures MO’s upper-left boundary,
2.291 - but not its lower-right boundary. Mtif2 does not express very much on the
2.292 - medial surface. By adding together the values at each pixel in these two figures,
2.293 - we get the lower-left image. This combination captures area MO much better
2.294 - than any single gene.
2.295 - This shows that our proposal to develop a method to find combinations of
2.296 - marker genes is both possible and necessary.
2.297 - Feature selection integrated with prediction As noted earlier, in gen-
2.298 - eral, any predictive method can be used for feature selection by running it
2.299 - inside a stepwise wrapper. Also, some predictive methods integrate soft con-
2.300 - straints on number of features used. Examples of both of these will be seen in
2.301 - the section “Multivariate Predictive methods”.
2.302 - Multivariate Predictive methods
2.303 - Forward stepwise logistic regression Logistic regression is a popular
2.304 - method for predictive modeling of categorial data. As a pilot run, for five
2.305 - cortical areas (SS, AUD, RSP, VIS, and MO), we performed forward stepwise
2.306 - logistic regression to find single genes, pairs of genes, and triplets of genes
2.307 - which predict areal identify. This is an example of feature selection integrated
2.308 - with prediction using a stepwise wrapper. Some of the single genes found
2.309 - were shown in various figures throughout this document, and Figure 4 shows
2.310 - a combination of genes which was found.
2.311 - We felt that, for single genes, gradient similarity did a better job than
2.312 - logistic regression at capturing our subjective impression of a “good gene”.
2.313 -SVM on all genes at once
2.314 -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
2.315 -surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%19. This shows that
2.316 +Ets1. Gradient similarity We noticed that the previous two scoring methods,
2.317 + which are pointwise, often found genes whose pattern of expression did not
2.318 + look similar in shape to the target region. For this reason we designed a
2.319 + non-pointwise local scoring method to detect when a gene had a pattern of
2.320 + expression which looked like it had a boundary whose shape is similar to the
2.321 + shape of the target region. We call this scoring method “gradient similarity”.
2.322 + One might say that gradient similarity attempts to measure how much the
2.323 + border of the area of gene expression and the border of the target region over-
2.324 + lap. However, since gene expression falls off continuously rather than jumping
2.325 + from its maximum value to zero, the spatial pattern of a gene’s expression often
2.326 + does not have a discrete border. Therefore, instead of looking for a discrete
2.327 + border, we look for large gradients. Gradient similarity is a symmetric function
2.328 + over two images (i.e. two scalar fields). It is is high to the extent that matching
2.329 + pixels which have large values and large gradients also have gradients which
2.330 + are oriented in a similar direction. The formula is:
2.331 + ∑
2.332 + pixel<img src="cmsy7-32.png" alt="∈" />pixels cos(abs(∠∇1 -∠∇2)) ⋅|∇1| + |∇2|
2.333 + 2 ⋅ pixel_value1 + pixel_value2
2.334 + 2
2.335 + where ∇1 and ∇2 are the gradient vectors of the two images at the current
2.336 + pixel; ∠∇i is the angle of the gradient of image i at the current pixel; |∇i| is
2.337 + the magnitude of the gradient of image i at the current pixel; and pixel_valuei
2.338 + is the value of the current pixel in image i.
2.339 + The intuition is that we want to see if the borders of the pattern in the
2.340 + two images are similar; if the borders are similar, then both images will have
2.341 + corresponding pixels with large gradients (because this is a border) which are
2.342 + oriented in a similar direction (because the borders are similar).
2.343 + Most of the genes in Figure 5 were identified via gradient similarity.
2.344 + Gradient similarity provides information complementary to cor-
2.345 + relation
2.346 + To show that gradient similarity can provide useful information that cannot
2.347 + be detected via pointwise analyses, consider Fig. 3. The top row of Fig. 3
2.348 + displays the 3 genes which most match area AUD, according to a pointwise
2.349 + method17. The bottom row displays the 3 genes which most match AUD ac-
2.350 + cording to a method which considers local geometry18 The pointwise method
2.351 + in the top row identifies genes which express more strongly in AUD than out-
2.352 + side of it; its weakness is that this includes many areas which don’t have a
2.353 + salient border matching the areal border. The geometric method identifies
2.354 + genes whose salient expression border seems to partially line up with the bor-
2.355 + der of AUD; its weakness is that this includes genes which don’t express over
2.356 + the entire area. Genes which have high rankings using both pointwise and bor-
2.357 + der criteria, such as Aph1a in the example, may be particularly good markers.
2.358 + None of these genes are, individually, a perfect marker for AUD; we deliberately
2.359 + chose a “difficult” area in order to better contrast pointwise with geometric
2.360 + methods.
2.361 + Areas which can be identified by single genes Using gradient simi-
2.362 +larity, we have already found single genes which roughly identify some areas and groupings of areas. For each of these areas,
2.363 +an example of a gene which roughly identifies it is shown in Figure 5. We have not yet cross-verified these genes in other
2.364 +atlases.
2.365 +In addition, there are a number of areas which are almost identified by single genes: COAa+NLOT (anterior part of
2.366 +cortical amygdalar area, nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior cingulate), VIS
2.367 +(visual), AUD (auditory).
2.368 +These results validate our expectation that the ABA dataset can be exploited to find marker genes for many cortical
2.369 +areas, while also validating the relevancy of our new scoring method, gradient similarity.
2.370 _________________________________________
2.371 - 195-fold cross-validation.
2.372 -the genes included in the ABA dataset are sufficient to define much of cortical anatomy. However, as noted above, a classifier
2.373 -that looks at all the genes at once isn’t as practically useful as a classifier that uses only a few genes.
2.374 + 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
2.375 +variable was the value of the expression of the gene underneath that pixel. The resulting scores were used to rank the genes in terms of how well
2.376 +they predict area AUD.
2.377 + 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,
2.378 +was calculated, and this was used to rank the genes.
2.379 +Combinations of multiple genes are useful and necessary for some areas
2.380 +In Figure 4, we give an example of a cortical area which is not marked by any single gene, but which can be identified
2.381 +combinatorially. Acccording to logistic regression, gene wwc1 is the best fit single gene for predicting whether or not a
2.382 +pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure 4 shows wwc1’s spatial
2.383 +expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, but the
2.384 +gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding
2.385 +to the overshoot is the medial surface of the cortex. MO is only found on the dorsal surface. Gene mtif2 is shown in the
2.386 +upper-right. Mtif2 captures MO’s upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much
2.387 +on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left image. This
2.388 +combination captures area MO much better than any single gene.
2.389 +This shows that our proposal to develop a method to find combinations of marker genes is both possible and necessary.
2.390 +Feature selection integrated with prediction As noted earlier, in general, any predictive method can be used for
2.391 +feature selection by running it inside a stepwise wrapper. Also, some predictive methods integrate soft constraints on number
2.392 +of features used. Examples of both of these will be seen in the section “Multivariate Predictive methods”.
2.393 +Multivariate Predictive methods
2.394
2.395
2.396
2.397 @@ -555,39 +549,52 @@
2.398 from left: NNMF. Right: Landmark Isomap. Additional details: In the
2.399 third and fourth rows, 7 dimensions were found, but only 6 displayed. In
2.400 the last row: for PCA, 50 dimensions were used; for NNMF, 6 dimensions
2.401 -were used; for landmark Isomap, 7 dimensions were used. Data-driven redrawing of the cor-
2.402 +were used; for landmark Isomap, 7 dimensions were used. Forward stepwise logistic regression Lo-
2.403 + gistic regression is a popular method for pre-
2.404 + dictive modeling of categorial data. As a pi-
2.405 + lot run, for five cortical areas (SS, AUD, RSP,
2.406 + VIS, and MO), we performed forward stepwise
2.407 + logistic regression to find single genes, pairs of
2.408 + genes, and triplets of genes which predict areal
2.409 + identify. This is an example of feature selec-
2.410 + tion integrated with prediction using a stepwise
2.411 + wrapper. Some of the single genes found were
2.412 + shown in various figures throughout this doc-
2.413 + ument, and Figure 4 shows a combination of
2.414 + genes which was found.
2.415 + We felt that, for single genes, gradient simi-
2.416 + larity did a better job than logistic regression at
2.417 + capturing our subjective impression of a “good
2.418 + gene”.
2.419 + SVM on all genes at once
2.420 + In order to see how well one can do when
2.421 + looking at all genes at once, we ran a support
2.422 + vector machine to classify cortical surface pix-
2.423 + els based on their gene expression profiles. We
2.424 + achieved classification accuracy of about 81%19.
2.425 + This shows that the genes included in the ABA
2.426 + dataset are sufficient to define much of cortical
2.427 + anatomy. However, as noted above, a classifier
2.428 + that looks at all the genes at once isn’t as prac-
2.429 + tically useful as a classifier that uses only a few
2.430 + genes.
2.431 + Data-driven redrawing of the cor-
2.432 tical map
2.433 - We have applied the following dimensionality
2.434 - reduction algorithms to reduce the dimension-
2.435 - ality of the gene expression profile associated
2.436 - with each voxel: Principal Components Anal-
2.437 - ysis (PCA), Simple PCA (SPCA), Multi-Dimensional
2.438 - Scaling (MDS), Isomap, Landmark Isomap, Lapla-
2.439 - cian eigenmaps, Local Tangent Space Alignment
2.440 - (LTSA), Hessian locally linear embedding, Dif-
2.441 - fusion maps, Stochastic Neighbor Embedding
2.442 - (SNE), Stochastic Proximity Embedding (SPE),
2.443 - Fast Maximum Variance Unfolding (FastMVU),
2.444 - Non-negative Matrix Factorization (NNMF). Space
2.445 - constraints prevent us from showing many of
2.446 - the results, but as a sample, PCA, NNMF, and
2.447 - landmark Isomap are shown in the first, second,
2.448 - and third rows of Figure 6.
2.449 - After applying the dimensionality reduction,
2.450 - we ran clustering algorithms on the reduced data.
2.451 - To date we have tried k-means and spectral
2.452 - clustering. The results of k-means after PCA,
2.453 - NNMF, and landmark Isomap are shown in the
2.454 - last row of Figure 6. To compare, the left-
2.455 - most picture on the bottom row of Figure 6
2.456 - shows some of the major subdivisions of cor-
2.457 - tex. These results clearly show that different di-
2.458 - mensionality reduction techniques capture dif-
2.459 - ferent aspects of the data and lead to differ-
2.460 - ent clusterings, indicating the utility of our pro-
2.461 - posal to produce a detailed comparion of these
2.462 - techniques as applied to the domain of genomic
2.463 - anatomy.
2.464 +We have applied the following dimensionality reduction algorithms to reduce the dimensionality of the gene expression
2.465 +profile associated with each voxel: Principal Components Analysis (PCA), Simple PCA (SPCA), Multi-Dimensional Scaling
2.466 +(MDS), Isomap, Landmark Isomap, Laplacian eigenmaps, Local Tangent Space Alignment (LTSA), Hessian locally linear
2.467 +embedding, Diffusion maps, Stochastic Neighbor Embedding (SNE), Stochastic Proximity Embedding (SPE), Fast Maximum
2.468 +Variance Unfolding (FastMVU), Non-negative Matrix Factorization (NNMF). Space constraints prevent us from showing
2.469 +_________________________________________
2.470 + 195-fold cross-validation.
2.471 +many of the results, but as a sample, PCA, NNMF, and landmark Isomap are shown in the first, second, and third rows of
2.472 +Figure 6.
2.473 +After applying the dimensionality reduction, we ran clustering algorithms on the reduced data. To date we have tried
2.474 +k-means and spectral clustering. The results of k-means after PCA, NNMF, and landmark Isomap are shown in the last
2.475 +row of Figure 6. To compare, the leftmost picture on the bottom row of Figure 6 shows some of the major subdivisions of
2.476 +cortex. These results clearly show that different dimensionality reduction techniques capture different aspects of the data
2.477 +and lead to different clusterings, indicating the utility of our proposal to produce a detailed comparion of these techniques
2.478 +as applied to the domain of genomic anatomy.
2.479
2.480 Figure 7: Prototypes corresponding to sample gene clusters,
2.481 clustered by gradient similarity. Region boundaries for the
2.482 @@ -657,10 +664,6 @@
2.483 4.Explore clustering algorithms applied to genes: including gene shaving, TODO
2.484 5.Develop an algorithm to use dimensionality reduction and/or hierarchial clustering to create anatomical maps
2.485 6.Run this algorithm on the cortex: present a hierarchial, genoarchitectonic map of the cortex
2.486 -_________________________________________
2.487 - 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
2.488 -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
2.489 -trees that use fewer genes
2.490 # Linear discriminant analysis
2.491 # jbt, coclustering
2.492 # self-organizing map
2.493 @@ -683,6 +686,10 @@
2.494 test out various dimensionality reduction schemes in combination with supervised learning. create or extend supervised
2.495 learning frameworks which use multivariate versions of the best scoring methods.
2.496 ∙January 2010 (milestone): Submit a publication on single marker genes for cortical areas
2.497 +_________________________________________
2.498 + 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
2.499 +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
2.500 +trees that use fewer genes
2.501 ∙February-July 2010: Continue to develop scoring methods and supervised learning frameworks. Explore the best way
2.502 to integrate radial profiles with supervised learning. Explore the best way to make supervised learning techniques
2.503 robust against incorrect labels (i.e. when the areas drawn on the input cortical map are slightly off). Quantitatively
3.1 Binary file grant.odt has changed
4.1 Binary file grant.pdf has changed
5.1 --- a/grant.txt Tue Apr 21 14:28:12 2009 -0700
5.2 +++ b/grant.txt Tue Apr 21 14:50:10 2009 -0700
5.3 @@ -222,6 +222,21 @@
5.4
5.5 == Significance ==
5.6
5.7 +\begin{wrapfigure}{L}{0.35\textwidth}\centering
5.8 +%%\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_3_654_jet.eps}
5.9 +%%\\
5.10 +%%\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_3_724_jet.eps}
5.11 +%%\caption{Top row: Genes Nfic, A930001M12Rik, C130038G02Rik are the most correlated with area SS (somatosensory cortex). Bottom row: Genes C130038G02Rik, Cacna1i, Car10 are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.}
5.12 +
5.13 +\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps}
5.14 +\\
5.15 +\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps}
5.16 +
5.17 +\caption{Top row: Genes $Nfic$ and $A930001M12Rik$ are the most correlated with area SS (somatosensory cortex). Bottom row: Genes $C130038G02Rik$ and $Cacna1i$ are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.}
5.18 +\label{SScorrLr}\end{wrapfigure}
5.19 +
5.20 +
5.21 +
5.22 The method developed in aim (1) will be applied to each cortical area to find a set of marker genes such that the combinatorial expression pattern of those genes uniquely picks out the target area. Finding marker genes will be useful for drug discovery as well as for experimentation because marker genes can be used to design interventions which selectively target individual cortical areas.
5.23
5.24 The application of the marker gene finding algorithm to the cortex will also support the development of new neuroanatomical methods. In addition to finding markers for each individual cortical areas, we will find a small panel of genes that can find many of the areal boundaries at once. This panel of marker genes will allow the development of an ISH protocol that will allow experimenters to more easily identify which anatomical areas are present in small samples of cortex.
5.25 @@ -247,6 +262,12 @@
5.26 new anatomical subregions in various structures, which will widely
5.27 impact all areas of biology.
5.28
5.29 +\begin{wrapfigure}{L}{0.2\textwidth}\centering
5.30 +\includegraphics[scale=.27]{holeExample_2682_SS_jet.eps}
5.31 +\caption{Gene $Pitx2$ is selectively underexpressed in area SS.}
5.32 +\label{hole}\end{wrapfigure}
5.33 +
5.34 +
5.35 Although our particular application involves the 3D spatial
5.36 distribution of gene expression, we anticipate that the methods
5.37 developed in aims (1) and (2) will not be limited to gene expression
5.38 @@ -257,19 +278,6 @@
5.39
5.40
5.41 == The approach: Preliminary Studies ==
5.42 -\begin{wrapfigure}{L}{0.35\textwidth}\centering
5.43 -%%\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_3_654_jet.eps}
5.44 -%%\\
5.45 -%%\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_3_724_jet.eps}
5.46 -%%\caption{Top row: Genes Nfic, A930001M12Rik, C130038G02Rik are the most correlated with area SS (somatosensory cortex). Bottom row: Genes C130038G02Rik, Cacna1i, Car10 are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.}
5.47 -
5.48 -\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps}
5.49 -\\
5.50 -\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps}
5.51 -
5.52 -\caption{Top row: Genes $Nfic$ and $A930001M12Rik$ are the most correlated with area SS (somatosensory cortex). Bottom row: Genes $C130038G02Rik$ and $Cacna1i$ are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.}
5.53 -\label{SScorrLr}\end{wrapfigure}
5.54 -
5.55
5.56
5.57
5.58 @@ -290,10 +298,6 @@
5.59
5.60 At this point, the data are in the form of a number of 2-D matrices, all in registration, with the matrix entries representing a grid of points (pixels) over the cortical surface:
5.61
5.62 -\begin{wrapfigure}{L}{0.2\textwidth}\centering
5.63 -\includegraphics[scale=.27]{holeExample_2682_SS_jet.eps}
5.64 -\caption{Gene $Pitx2$ is selectively underexpressed in area SS.}
5.65 -\label{hole}\end{wrapfigure}
5.66
5.67
5.68 * A 2-D matrix whose entries represent the regional label associated with each surface pixel
5.69 @@ -301,38 +305,6 @@
5.70
5.71
5.72
5.73 -
5.74 -We created a normalized version of the gene expression data by subtracting each gene's mean expression level (over all surface pixels) and dividing the expression level of each gene by its standard deviation.
5.75 -
5.76 -The features and the target area are both functions on the surface pixels. They can be referred to as scalar fields over the space of surface pixels; alternately, they can be thought of as images which can be displayed on the flatmapped surface.
5.77 -
5.78 -To move beyond a single average expression level for each surface pixel, we plan to create a separate matrix for each cortical layer to represent the average expression level within that layer. Cortical layers are found at different depths in different parts of the cortex. In preparation for extracting the layer-specific datasets, we have extended Caret with routines that allow the depth of the ROI for volume-to-surface projection to vary.
5.79 -
5.80 -In the Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.
5.81 -
5.82 -
5.83 -
5.84 -
5.85 -
5.86 -
5.87 -
5.88 -=== Feature selection and scoring methods ===
5.89 -
5.90 -
5.91 -
5.92 -\vspace{0.3cm}**Underexpression of a gene can serve as a marker**
5.93 -Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}.
5.94 -
5.95 -
5.96 -
5.97 -\vspace{0.3cm}**Correlation**
5.98 -Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the surface pixels.
5.99 -
5.100 -One class of feature selection scoring methods contains methods which calculate some sort of "match" between each gene image and the target image. Those genes which match the best are good candidates for features.
5.101 -
5.102 -One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between each gene and each cortical area. The top row of Figure \ref{SScorrLr} shows the three genes most correlated with area SS.
5.103 -
5.104 -
5.105 \begin{wrapfigure}{L}{0.35\textwidth}\centering
5.106 %%\includegraphics[scale=.27]{singlegene_AUD_lr_top_1_3386_jet.eps}\includegraphics[scale=.27]{singlegene_AUD_lr_top_2_1258_jet.eps}\includegraphics[scale=.27]{singlegene_AUD_lr_top_3_420_jet.eps}
5.107 %%
5.108 @@ -344,14 +316,36 @@
5.109 \caption{The top row shows the two genes which (individually) best predict area AUD, according to logistic regression. The bottom row shows the two genes which (individually) best match area AUD, according to gradient similarity. From left to right and top to bottom, the genes are $Ssr1$, $Efcbp1$, $Ptk7$, and $Aph1a$.}
5.110 \label{AUDgeometry}\end{wrapfigure}
5.111
5.112 -\vspace{0.3cm}**Conditional entropy**
5.113 -An information-theoretic scoring method is to find features such that, if the features (gene expression levels) are known, uncertainty about the target (the regional identity) is reduced. Entropy measures uncertainty, so what we want is to find features such that the conditional distribution of the target has minimal entropy. The distribution to which we are referring is the probability distribution over the population of surface pixels.
5.114 -
5.115 -The simplest way to use information theory is on discrete data, so we discretized our gene expression data by creating, for each gene, five thresholded boolean masks of the gene data. For each gene, we created a boolean mask of its expression levels using each of these thresholds: the mean of that gene, the mean minus one standard deviation, the mean minus two standard deviations, the mean plus one standard deviation, the mean plus two standard deviations.
5.116 -
5.117 -Now, for each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression boolean masks such that the conditional entropy of the target area's boolean mask, conditioned upon the pair of gene expression boolean masks, is minimized.
5.118 -
5.119 -This finds pairs of genes which are most informative (at least at these discretization thresholds) relative to the question, "Is this surface pixel a member of the target area?". Its advantage over linear methods such as logistic regression is that it takes account of arbitrarily nonlinear relationships; for example, if the XOR of two variables predicts the target, conditional entropy would notice, whereas linear methods would not.
5.120 +We created a normalized version of the gene expression data by subtracting each gene's mean expression level (over all surface pixels) and dividing the expression level of each gene by its standard deviation.
5.121 +
5.122 +The features and the target area are both functions on the surface pixels. They can be referred to as scalar fields over the space of surface pixels; alternately, they can be thought of as images which can be displayed on the flatmapped surface.
5.123 +
5.124 +To move beyond a single average expression level for each surface pixel, we plan to create a separate matrix for each cortical layer to represent the average expression level within that layer. Cortical layers are found at different depths in different parts of the cortex. In preparation for extracting the layer-specific datasets, we have extended Caret with routines that allow the depth of the ROI for volume-to-surface projection to vary.
5.125 +
5.126 +In the Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.
5.127 +
5.128 +
5.129 +
5.130 +
5.131 +
5.132 +
5.133 +
5.134 +=== Feature selection and scoring methods ===
5.135 +
5.136 +
5.137 +
5.138 +\vspace{0.3cm}**Underexpression of a gene can serve as a marker**
5.139 +Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}.
5.140 +
5.141 +
5.142 +
5.143 +\vspace{0.3cm}**Correlation**
5.144 +Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the surface pixels.
5.145 +
5.146 +One class of feature selection scoring methods contains methods which calculate some sort of "match" between each gene image and the target image. Those genes which match the best are good candidates for features.
5.147 +
5.148 +One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between each gene and each cortical area. The top row of Figure \ref{SScorrLr} shows the three genes most correlated with area SS.
5.149 +
5.150
5.151
5.152 \begin{wrapfigure}{L}{0.35\textwidth}\centering
5.153 @@ -361,23 +355,15 @@
5.154 \caption{Upper left: $wwc1$. Upper right: $mtif2$. Lower left: wwc1 + mtif2 (each pixel's value on the lower left is the sum of the corresponding pixels in the upper row).}
5.155 \label{MOcombo}\end{wrapfigure}
5.156
5.157 -
5.158 -\vspace{0.3cm}**Gradient similarity**
5.159 -We noticed that the previous two scoring methods, which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. For this reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method "gradient similarity".
5.160 -
5.161 -One might say that gradient similarity attempts to measure how much the border of the area of gene expression and the border of the target region overlap. However, since gene expression falls off continuously rather than jumping from its maximum value to zero, the spatial pattern of a gene's expression often does not have a discrete border. Therefore, instead of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images (i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have gradients which are oriented in a similar direction. The formula is:
5.162 -
5.163 -\begin{align*}
5.164 -\sum_{pixel \in pixels} cos(abs(\angle \nabla_1 - \angle \nabla_2)) \cdot \frac{\vert \nabla_1 \vert + \vert \nabla_2 \vert}{2} \cdot \frac{pixel\_value_1 + pixel\_value_2}{2}
5.165 -\end{align*}
5.166 -
5.167 -where $\nabla_1$ and $\nabla_2$ are the gradient vectors of the two images at the current pixel; $\angle \nabla_i$ is the angle of the gradient of image $i$ at the current pixel; $\vert \nabla_i \vert$ is the magnitude of the gradient of image $i$ at the current pixel; and $pixel\_value_i$ is the value of the current pixel in image $i$.
5.168 -
5.169 -The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar, then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a similar direction (because the borders are similar).
5.170 -
5.171 -Most of the genes in Figure \ref{singleSoFar} were identified via gradient similarity.
5.172 -
5.173 -\vspace{0.3cm}**Gradient similarity provides information complementary to correlation**
5.174 +\vspace{0.3cm}**Conditional entropy**
5.175 +An information-theoretic scoring method is to find features such that, if the features (gene expression levels) are known, uncertainty about the target (the regional identity) is reduced. Entropy measures uncertainty, so what we want is to find features such that the conditional distribution of the target has minimal entropy. The distribution to which we are referring is the probability distribution over the population of surface pixels.
5.176 +
5.177 +The simplest way to use information theory is on discrete data, so we discretized our gene expression data by creating, for each gene, five thresholded boolean masks of the gene data. For each gene, we created a boolean mask of its expression levels using each of these thresholds: the mean of that gene, the mean minus one standard deviation, the mean minus two standard deviations, the mean plus one standard deviation, the mean plus two standard deviations.
5.178 +
5.179 +Now, for each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression boolean masks such that the conditional entropy of the target area's boolean mask, conditioned upon the pair of gene expression boolean masks, is minimized.
5.180 +
5.181 +This finds pairs of genes which are most informative (at least at these discretization thresholds) relative to the question, "Is this surface pixel a member of the target area?". Its advantage over linear methods such as logistic regression is that it takes account of arbitrarily nonlinear relationships; for example, if the XOR of two variables predicts the target, conditional entropy would notice, whereas linear methods would not.
5.182 +
5.183
5.184
5.185 \begin{wrapfigure}{L}{0.35\textwidth}\centering
5.186 @@ -389,6 +375,25 @@
5.187 \caption{From left to right and top to bottom, single genes which roughly identify areas SS (somatosensory primary \begin{latex}+\end{latex} supplemental), SSs (supplemental somatosensory), PIR (piriform), FRP (frontal pole), RSP (retrosplenial), COApm (Cortical amygdalar, posterior part, medial zone). Grouping some areas together, we have also found genes to identify the groups ACA+PL+ILA+DP+ORB+MO (anterior cingulate, prelimbic, infralimbic, dorsal peduncular, orbital, motor), posterior and lateral visual (VISpm, VISpl, VISI, VISp; posteromedial, posterolateral, lateral, and primary visual; the posterior and lateral visual area is distinguished from its neighbors, but not from the entire rest of the cortex). The genes are $Pitx2$, $Aldh1a2$, $Ppfibp1$, $Slco1a5$, $Tshz2$, $Trhr$, $Col12a1$, $Ets1$.}
5.188 \label{singleSoFar}\end{wrapfigure}
5.189
5.190 +\vspace{0.3cm}**Gradient similarity**
5.191 +We noticed that the previous two scoring methods, which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. For this reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method "gradient similarity".
5.192 +
5.193 +One might say that gradient similarity attempts to measure how much the border of the area of gene expression and the border of the target region overlap. However, since gene expression falls off continuously rather than jumping from its maximum value to zero, the spatial pattern of a gene's expression often does not have a discrete border. Therefore, instead of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images (i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have gradients which are oriented in a similar direction. The formula is:
5.194 +
5.195 +\begin{align*}
5.196 +\sum_{pixel \in pixels} cos(abs(\angle \nabla_1 - \angle \nabla_2)) \cdot \frac{\vert \nabla_1 \vert + \vert \nabla_2 \vert}{2} \cdot \frac{pixel\_value_1 + pixel\_value_2}{2}
5.197 +\end{align*}
5.198 +
5.199 +where $\nabla_1$ and $\nabla_2$ are the gradient vectors of the two images at the current pixel; $\angle \nabla_i$ is the angle of the gradient of image $i$ at the current pixel; $\vert \nabla_i \vert$ is the magnitude of the gradient of image $i$ at the current pixel; and $pixel\_value_i$ is the value of the current pixel in image $i$.
5.200 +
5.201 +The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar, then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a similar direction (because the borders are similar).
5.202 +
5.203 +Most of the genes in Figure \ref{singleSoFar} were identified via gradient similarity.
5.204 +
5.205 +\vspace{0.3cm}**Gradient similarity provides information complementary to correlation**
5.206 +
5.207 +
5.208 +
5.209 To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider Fig. \ref{AUDgeometry}. The top row of Fig. \ref{AUDgeometry} displays the 3 genes which most match area AUD, according to a pointwise method\footnote{For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor 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 they predict area AUD.}. The bottom row displays the 3 genes which most match AUD according to a method which considers local geometry\footnote{For 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, was calculated, and this was used to rank the genes.} The pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is that this includes many areas which don't have a salient border matching the areal border. The geometric method identifies genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes genes which don't express over the entire area. Genes which have high rankings using both pointwise and border criteria, such as $Aph1a$ in the example, may be particularly good markers. None of these genes are, individually, a perfect marker for AUD; we deliberately chose a "difficult" area in order to better contrast pointwise with geometric methods.
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5.211
5.212 @@ -459,6 +464,7 @@
5.213 After applying the dimensionality reduction, we ran clustering algorithms on the reduced data. To date we have tried k-means and spectral clustering. The results of k-means after PCA, NNMF, and landmark Isomap are shown in the last row of Figure \ref{dimReduc}. To compare, the leftmost picture on the bottom row of Figure \ref{dimReduc} shows some of the major subdivisions of cortex. These results clearly show that different dimensionality reduction techniques capture different aspects of the data and lead to different clusterings, indicating the utility of our proposal to produce a detailed comparion of these techniques as applied to the domain of genomic anatomy.
5.214
5.215
5.216 +
5.217 \begin{wrapfigure}{L}{0.5\textwidth}\centering
5.218 \includegraphics[scale=.2]{cosine_similarity1_rearrange_colorize.eps}
5.219 \caption{Prototypes corresponding to sample gene clusters, clustered by gradient similarity. Region boundaries for the region that most matches each prototype are overlayed.}
5.220 @@ -500,6 +506,8 @@
5.221
5.222 We will develop scoring methods for evaluating how good individual genes are at marking areas. We will compare pointwise, geometric, and information-theoretic measures. We already developed one entirely new scoring method (gradient similarity), but we may develop more. Scoring measures that we will explore will include the L1 norm, correlation, expression energy ratio, conditional entropy, gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such as Student's t-test, and the Mann-Whitney U test (a non-parametric test). In addition, any predictive procedure induces a scoring measure on genes by taking the prediction error when using that gene to predict the target.
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5.224 +
5.225 +
5.226 Using some combination of these measures, we will develop a procedure to find single marker genes for anatomical regions: for each cortical area, we will rank the genes by their ability to delineate each area.
5.227
5.228 Some cortical areas have no single marker genes but can be identified by combinatorial coding. This requires multivariate scoring measures and feature selection procedures. Many of the measures, such as expression energy, gradient similarity, Jaccard, Dice, Hough, Student's t, and Mann-Whitney U are univariate. We will extend these scoring measures for use in multivariate feature selection, that is, for scoring how well combinations of genes, rather than individual genes, can distinguish a target area. There are existing multivariate forms of some of the univariate scoring measures, for example, Hotelling's T-square is a multivariate analog of Student's t.