Spatial Point Pattern Analysis of Neurons Using Ripley's K-Function in 3D

Abstract

The aim of this paper is to apply a non-parametric statistical tool, Ripley's K-function, to analyze the 3-dimensional distribution of pyramidal neurons. Ripley's K-function is a widely used tool in spatial point pattern analysis. There are several approaches in 2D domains in which this function is executed and analyzed. Drawing consistent inferences on the underlying 3D point pattern distributions in various applications is of great importance as the acquisition of 3D biological data now poses lesser of a challenge due to technological progress. As of now, most of the applications of Ripley's K-function in 3D domains do not focus on the phenomenon of edge correction, which is discussed thoroughly in this paper. The main goal is to extend the theoretical and practical utilization of Ripley's K-function and corresponding tests based on bootstrap resampling from 2D to 3D domains.

Keywords: Ripley's K-function; bootstrap resampling; edge correction in 3D.

Figure 1

Probability density function for 500 simulated K-functions for an arbitrary t and an arbitrary observed value from one of the simulations using the edge correction method in Eq. 11.

Figure 2

(A) Scatterplot of a raw glt sample. (B) Same sample after executing the station function. (C) After executing the divide function.

Figure 3

A demonstration of how caps and wedges may occur where a and b are the Euclidean distances from the event to the sample domain boundaries, t is the radius of the sphere as in K(t), and h and r are the height and radius of the outside-of-domain cap respectively.

Figure 4

500 estimations of the K-function based on simulations under CSR for samples identical in geometry to etv-pyramids (A,C) and glt-pyramids (B,D). The simulations demonstrate the expected outcome when estimating the K-function without (A,B) and with (C,D) the edge correction term under CSR (having $\widehat{K}(t)-4\pi t^3/3$ on the y-axis).

Figure 5

Groupwise average (weighted) $\widehat{K}(\text{t})-4\pi {\text{t}}^3/3$ for samples in etv-pyramids and glt-pyramids.

Figure 6

etv (blue) and glt (red) p-values for different values of t. The drop for values larger than 10 μm translates well to cell diameters of 15–20 μm.

Figure 7

Estimated variance of $\widehat{K}(\text{t})$ for etv and glt.

Figure 8

Groupwise average (weighted) $\widehat{K}(t)-4\pi t^3/3$ for etv- (left) and glt-pyramids (right) and confidence intervals based on the bootstrap procedure (dashed).

Figure 9

Probability density function of BTSS for s = 1000 resamples and the observed BTSS (filled bar) when w(t) = t⁻² and t= 20,…,60 μm.