Accuracy of Estimating the Area of Cortical Muscle Representations from TMS Mapping Data Using Voronoi Diagrams

Abstract

Motor evoked potentials (MEPs) caused by transcranial magnetic stimulation (TMS) provide a possibility of noninvasively mapping cortical muscle representations for clinical and research purposes. The interpretation of such results is complicated by the high variability in MEPs and the lack of a standard optimal mapping protocol. Comparing protocols requires the determination of the accuracy of estimated representation parameters (such as the area), which is problematic without ground truth data. We addressed this problem and obtained two main results: (1) the development of a bootstrapping-based approach for estimating the within-session variability and bias of representation parameters and (2) estimations of the area and amplitude-weighted area accuracies for motor representations using this approach. The method consists in the simulation of TMS mapping results by subsampling MEPs from a single map with a large number of stimuli. We studied the extensor digitorum communis (EDC) and flexor digitorum superficialis (FDS) muscle maps of 15 healthy subjects processed using Voronoi diagrams. We calculated the (decreasing) dependency of the errors in the area and weighted area on the number of stimuli. This result can be used to choose a number of stimuli sufficient for studying the effects of a given size (e.g., the protocol with 150 stimuli leads to relative errors of 7% for the area and 11% for the weighted area in 90% of the maps). The approach is applicable to other parameters (e.g., the center of gravity) and other map processing methods, such as spline interpolation.

Appendices

Appendix A: Using the Voronoi Tessellation Method to Estimate the Area of Cortical Muscle Representations

Before the formal definition, it is useful to discuss the concept of the Voronoi diagram method by considering a common demonstrative example. Consider an area on the Earth with airports. Suppose an airplane must find the nearest airport. To do so, the pilots must determine the distances to all the airports and choose the shortest distance. However, this process can be optimized. Specifically, we can prepare a map in which all the locations with the same nearest airport are identically colored, and zones corresponding to different airports have different colors. Now, we can find a location on the map, look at its color, and instantly find the nearest airport. Such splitting of the plane is called a Voronoi diagram (or a Voronoi tessellation).

The method can be formally defined in the following way.

Consider a finite set $Y$ of points $y_{i}$ in a plane. The Voronoi diagram splits the plane as follows: each region $V_{i}$ contains points for which the point $y_{i}$ is the nearest among $Y,$ or formally $x \in V_{i} \Rightarrow ||x - y_{i} || = min_{j} ||x - y_{j} ||.$

Spherical Voronoi Tessellation

In the present study, the Voronoi tessellation method was applied to stimulation points located on a cortical surface. The approximation of this surface by a plane limits the corresponding accuracy and can lead to the considerable distortion of the Voronoi cell areas at the periphery of the muscle representation. Thus, we used a more accurate approximation by a sphere and employed a spherical modification of the Voronoi tessellation (Na et al. 2002).

Processing of the Boundary Voronoi Cells

The planar Voronoi diagram splits the entire plane into a finite number of cells, so the outer cells have an infinite area. In the case of the spherical approach and TMS mapping, we defined the corresponding cells as those that intersect the half of the sphere opposite the muscle representation position. In most cases, such cells correspond to stimuli with MEPs below the threshold. In our computational algorithm, we do not count such cells, including those with positive responses (this is a rare situation). In addition to “infinite” cells, enormously large cells with positive MEPs can appear at the periphery if the border is poorly outlined (it is common to obtain 2–3 such cells in a reduced map generated by bootstrapping). Including the full areas of large border cells leads to a positive area bias, and removing these cells from computations leads to a negative bias. To avoid this problem, we identified cells with areas larger than $Q3 + 2(Q3 - Q1),$ where $Q1$ is the first and $Q3$ is the third quartile of all the areas of the “finite” Voronoi cells with positive MEPs and took $Q3$ as the areas of these cells in further computations of the parameters.

Appendix B: Introduction to Statistical Bootstrapping

A simple example can be given to illustrate to the bootstrapping method. Consider a coin (potentially an unfair one) flipped ten times, with a result of three heads (one point for each) and seven tails (zero points for each). Now, we have a statistical sample of ten results, and we can calculate some simple statistical characteristics, such as the average value (0.3) or standard deviation (0.483). However, if we must determine more complex characteristics, such as the probability of the average value of 93 tosses being higher than 0.54, a different approach is required. Statistical bootstrapping is a simple computational method for such tasks. For the sample of three heads and seven tails, let us randomly choose one of the results of the sample 93 times. By iterating this random choice of 93 results and comparing each average value with 0.54, we obtain a sample of positive/negative answers. Using this data, we can compute the desired probability. Therefore, the core concept of bootstrapping is the expansion of a statistical sample by Monte Carlo random sampling.

Although the presented example can be easily solved analytically, this may not be the case for more complex problems in which the use of bootstrapping is appropriate. In the present study, a method based on this approach was applied to evaluate the accuracy of the estimates of muscle representation parameters computed from TMS mapping data.

Appendix C: Implementation of the Bootstrapping-Based Analysis of the Accuracy of Cortical Representation Parameters Obtained Using TMS Mappings

Figure 10 contains the flow-chart diagram for the bootstrapping-based analysis of the data from a single TMS map. Figure 11 contains the flow-chart diagram of the subroutine for the generation of a single reduced map.

Fig. 10

The flow-chart diagram of the bootstrapping-based analysis of the data from a single TMS map

Fig. 11

The flow-chart diagram of the subroutine for the generation of a single reduced map

The Matlab implementation of the developed approach, the mapping data, and an example of the analysis of the accuracy for a sample cortical representation parameter (mean MEP amplitude multiplied by the maximal distance to the center of gravity) are available at the GitHub repository https://github.com/a-chernyavskiy/TMS-BS/. The computation of the parameters based on the Voronoi tessellation and other functions related to TMS mapping are planned to be added to the repository.