Polar histogram

Purpose

The polar histogram describes the overall direction of tree or vessel growth.

  • The growth is displayed in the form of a round directional histogram that uses pie shaped wedges to describe values (unlike the traditional histogram that uses rectangles to describe values).

  • Length is plotted as a function of direction.

  • The Polar Histogram is not affected by translation. If several processes or vessels are included in the analysis, their relative position is unimportant.

The meaning of length in polar histograms is important: Projection collapses Z, but length in Z is used in the length calculation for the polar histogram. The total length in the polar histogram is the same as the total length of all processes or vessels used in the polar histogram analysis.

General information

Use to avoid the disconnection problem associated with the traditional histogram (i.e., the right side of the histogram is connected to the left side) and to make the information easier to interpret visually.

Perform for dendrites, axons, apical dendrites, and vessels.

Apply to the results of the Fan In Diagram transform by checking the Apply Fan-in Transform checkbox.

While it is generated, a Polar Histogram breaks each segment of a tree or vessel into a series of straight line-segments.

  • These segments are collected in bins according to directional orientation.
  • The total length in a given bin is summed as a wedge with the same orientation as the line segments, and represented in the circular polar histogram graph.

The 3D tracing is reduced to a 2D Polar Histogram, that is, the spatial geometry is transformed into length and direction. The Polar Histogram considers direction to be a local property of the process or vessel (while the Wedge analysis describes direction relative to a fixed point for all of the processes or vessels).

The right side of a circle has a tangent that runs vertically.

  • If the circle is drawn clockwise, then the tangent points down; the angle is 270 degrees.
  • Move along the circle to the bottom of the circle and the tangent now points to the left; the angle is 180 degrees.
  • Continue further around the circle and the tangent now points up; the angle is 90 degrees.
  • Continue on to the top of the circle and the tangent now points to the right; the angle is 0 degrees.

Throughout this circumnavigation of the circle, all directions are equally represented. A Polar Histogram of a circle is a solid circle around the center of the graph.

Although projection collapses Z, length in Z is used in the length calculation for the Polar Histogram. The total length in the Polar Histogram is the same as the total length of all processes or vessels used in the Polar Histogram analysis.

Analysis can be performed for dendrites, axons, apical dendrites, and vessels. The Polar Histogram can also be applied to the results of the Fan-In transform by checking the Apply Fan-in Transform box.

Procedure

  1. Select Polar Histogram from the Orientation dropdown menu on the Analyze ribbon to open the Polar Histogram Analysis window.

  2. Branched Structures: Check boxes for the structures you wish to analyze, or click Select All to select all structures. Click Select None to clear these check boxes.

  3. Select Apply Fan-in Transform to have this method applied to the data.

  4. Click Display to display a Polar Histogram window for each type of structure .

Polar Histogram menu options

The instructions below refer to menu in the Polar Histogram display window (not the general Neurolucida Explorer menu).

File:

  • Print: Select to print the Polar Histogram.

  • Save to Excel: Select to save the Polar Histogram to a new Excel file.

Edit:

  • Copy Graph to Clipboard: Copy the Polar Histogram display to the clipboard.

  • Copy Histogram Data to Clipboard: Copy a table containing histogram data to the clipboard. The data includes: radians start, radians end, radians length, degrees start, degrees end, and degrees length.

  • From the Clipboard, paste the graph or histogram data into another program such as Microsoft Word or Microsoft Excel.

Display:

  • Radial Length: Adds or removes radial length markers from the display.

  • Degree Ticks: Adds or removes degree ticks from the display.

  • Outline Pie Segments: Adds or removes a black outline around the pie segments.

  • Settings: Click to open the Polar Histogram Settings window and choose from the following:

    • Number of intervals in histogram: Type a number between 2 and 120. The number of intervals determines the number of bins.

    • Use cubic spline smoothing: Select to minimize the effects of artifacts from the data collection process (especially if fine continuous tracing was used).

      The discrete coordinates of the computer are similar to a checkerboard.

      • A square on a checkerboard has 8 neighbors and therefore only 8 available directions.
      • A polar histogram reveals this artifact by having prominent wedges at the 8 principal directions of 0, 45, 90, 135, 180, 225, 270, and 315 degrees.
      • The tracing can be smoothed to remove the 8-neighbor artifact. Cubic splines are used for the smoothing process because they are the simplest form of curve that is capable of representing 3-dimensional curves of the type present in the tracing.

      Example:

      The example illustrates the difference that cubic spline smoothing makes in the analysis of the data.

      This tracing was analyzed without (left) and with (right) cubic spline smoothing.

      • The histogram on the left (without) has prominent wedges at 0, 45, and 90 degrees. These 3 directions are the neighbors used to draw a line that trends up, right, and diagonally up and right.
      • The histogram on the right (with) shows a more accurate representation of the directional growth of the line.

    • Change Color: Click to open the Select Color window and choose a color.

 

Reference

McMullen, N.T., Glaser, E.M., and Tagamets, M (1984). Morphometry of spine-free nonpyramidal neurons in rabbit auditory cortex. Journal of Comparative Neurology 222:383-395.