Density Correlation Analysis of the Brain Cortex Architecture

We develop a new technique for quantitative analysis of the brain cortex architecture. This new technique is analogous to the calculation of the density correlation function g(x,y) used in condensed matter physics to describe the structure of crystals and anisotropic liquids. Usually, the function g(x,y) is obtained experimentally as a Fourier transform of the angular intensity of the scattered neutron or x-ray beam. In our technique we directly use coordinates of neurons obtained by optical microscope. The analogy to neutron or x-ray scattering stems from the fact that we compute the contribution to the image from the neighborhoods of every neuron in the cortex similarly to the way that neutrons or x-rays are scattered from every atom of the liquid.

The density correlation function g(x,y) describes the neighborhood of a typical neuron in the cortex as a two-dimensional landscape. The neurons in the cortex are not located at random. If we know that a neuron is placed at the origin (0,0), we can predict the probability density g(x,y) of finding another neuron with coordinates (x,y). Obviously there are no neurons that are too close to a given one. That is why g(x,y) has almost zero in the circle [Ö(x2+y2)] < d, where d is the neuron diameter. The peak of g(x,y) at a certain point (x,y) indicates that there is a higher probability of finding a neuron if we move from a given neuron by x in a horizontal direction, and by y in a vertical direction.

For example, a peak at (0,20mm) indicates that the neurons prefer to form vertical columns in which the vertical spacing is about 20mm. Such observations as this are difficult to make with the naked eye, since in the vicinity of each particular neuron there are only a few other neurons that at first glance are situated at random. If we average the vicinities of each neuron using our analysis, we can quantitatively characterize the neighborhood of each neuron, which is of great importance in understanding the interactions among neurons.

The Java applet Density Correlation Applet allows us to analyze any data file that contains x and y coordinates which may represent neurons or any other objects such as molecules in a solid or liquid, cars on a highway, or stars in the sky. Since the Java applet cannot read the files from your directory, we provide a menu of sample files (located on our server) that you can analyze. If you want to analyze your own file, you can download the Java application Density Correlation and Sample Files onto your own computer. Your file should be a text file that contain N lines, each with two numbers x and y. The last line should end with a carriage return.

In using the Java applet, select Choose File from the File menu and press the Open File button. Filenames that begin with C indicate that the file contains positions of neurons in the superior temporal sulcus (STS) cortex of control individuals. The pial surface is always parallel to th horizontal axis and goes above the sample. The gray-white boundary is below the sample. Letters D and L indicate analogous files for individuals with Alzheimer's disease (AD) and individuals with Lewy Body disease. The file marked by A shows the auditory cortex from the first temporal gyros. We also provide examples of a two-dimensional ideal gas (file random) in which the points have random coordinates, a real gas with an excluded volume(real) , a liquid, a square crystal (square), a quasicrystal (quasi), a gas at the condensation point (condens), and an anisotropic liquid that consists of polymeric chains oriented in one direction (chains). The applet reads the coordinates of N objects from the file, labels them with index i, and draws them in the upper left corner of the window. The unit of measurement for anatomical files is the micrometer. For other files it is the Angstrom unit Å or the manometer.

The applet determines the number of objects in the file N, and the width and height of the sample, defined as the difference between the maximal and minimal x and y coordinates of the object. If you want to change the position of the data, select Flip x/y, Reflect x, or Reflect y. If you want to compute the correlation function g(x,y) for a given file, you must specify, using the appropriate scroll-bars, the grid cell size D, the horizontal cell number nx, and the vertical cell number ny. Use the default values of these parameters to get started. Press the Start button to perform the analysis.

To produce calculations, we carry out the following procedure. We place a square grid of (2nx+1)×(2ny+1) cells over the sample in such a way that object i is at the center of the central cell of the lattice with exactly nx columns to the right, nx columns to the left, and exactly ny rows above and below. Each grid cell is the square of the specified size D and the coordinates of its corners are

 x- D 2 , x+ D 2 ,       y- D 2 , y+ D 2 ,
where x and y are multiples of D, i.e.,

 x = -Dnx,D(-nx+1)¼-D,0,D,¼D(-nx+1),Dnx;

 y = -Dny,D(-ny+1)¼-D,0,D,¼D(-ny+1),Dny.
Thus Dnx and Dny are the maximal distances for which the function g(x,y) is calculated.

Next we count the number of objects Ni(x,y) in each grid cell and we average these numbers over all N possible positions of the grid (each position corresponds to an object i at the origin). Finally, we define the average object density

 g(x,y) = 1 ND2 N å i = 1 Ni(x,y),
and plot it below the graph as a two-dimensional topographical map in which shades of gray indicate the value of g(x,y). The darkest areas are the cells with maximal density and the lightest are those with minimal density. The program also computes the average density, which is simply the number of objects divided by the area of the sample analyzed. You can change the color scheme by changing the lower and upper thresholds, which by default coincide with the minimal and maximal densities-shown also as percentages of the average value. The upper threshold shows the value of the density above which the cell shading is completely black. The lower threshold shows the value of the density below which the cell shading is completely white. Between these upper and lower thresholds, the brightness changes linearly with density.

The switch from the gray scale to an artificial color scheme, check the Colors box. When Colors is checked, the density is indicated by color hue-as in a conventional topographical map. Red cells correspond to the highest ``mountains'' and blue cells correspond to the deepest parts of the ocean.

When there is a lack of statistics in the sample, the density g(x,y) often fluctuates rapidly. The estimated error of g(x,y) is s2 = A/D2N2, where A = W×h is the total area of the sample. The default value of the grid cell size is selected in such a way that s = 10 percent of the value g(x,y).

In order to improve the statistics, we can analyze several analogous file, adding them from the Choose File menu to the list of already analyzed files File List using the Add File button. The average density landscape is shown in the lower right corner of the window.

If only one file is analyzed, the average coincides with the data for this file. However, we can manipulate the colors of the average independently from the colors of the sample. Thus we can compare the different color schemes and choose the one that is better. To discard the average and initiate an analysis of the new values of D, nx, and ny, select the Open File button instead of the Add File button.

The microcolumns are most visible for the default value of the parameters as the vertical strips of high density in the center of the density landscapes for control individuals (files C). They are significantly disrupted in Alzheimer's and Lew Body diseases (files D, and L). To see the lamina structure, select D = 100 and nx = 30,ny = 30. You then see several horizontal strips corresponding to the dense layers of the neurons. The width of the strips shows the average width of the lamina and the distance between strips corresponds to the average distance between the adjacent lamina.

The wide columnar structure (in addition to microcolumns) is obvious in the auditory cortex file A384.

The size of the objects on the density landscape can be found by taking into account the fact that each cell has a size D and that the corners of the landscape map correspond to the value of x = ±nxD and y = ±nyD. Thus we can see that the lamina width is about 500mm and that the inter-lamina distance is 1000mm. Similarly, the width of the microcolumns is about 50mm and the inter-columnar distance is about 80mm.

Density Correlation Applet

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