# Extensions of Granger Causality Calculations on Brain Networks for Efficient and Accurate Seizure Focus Identification via iEEGs

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## Abstract

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## 1. Introduction

^{−20}) [1]. GC analyses assign a metric for every pair of electrodes to measure the degree to which activity at one electrode dictates activity at another. This creates a graph of the patient’s brain, also known as a GC map. From the GC map, summation of all the GC values originating at each electrode results in computation of a total GC outdegree metric for that electrode. Nodes are then ranked by their total GC outdegree and the ranks of nodes belonging to the actual SOZ and RZ are summed to create a rank order sum metric. Via comparison of this sum to a random distribution of rank order sums of the same size subset from the same starting number of ranks, it is shown that total GC outdegree results in a statistically significant rank order sum of actual SOZ and RZ nodes with an aggregate probability across all 25 patients much lower than predicted by chance [1]. However, because the total GC outdegree metric was previously applied without evaluation of alternatives, we start with the GC map and aim to create a more precise algorithm that informs selection of a RZ/SOZ. For ease of comparison, we also use the rank order sum method to evaluate algorithms’ predictive capability. It is important to point out that the Granger statistical approach demonstrates that a particular signal in one channel statistically follows a signal in a different channel. This does not in fact prove that one event is causal, but the term ‘Granger Causality’ is widely used as the name of the statistic. We will keep the name for clarity with this caveat, understanding that the current study begins with the connectivity data derived and reported earlier [1].

## 2. Materials and Methods

#### 2.1. Patients

#### 2.2. Granger Causality Map Computation

#### 2.3. Graph Algorithms

#### 2.3.1. Monte Carlo Sampling Approach

#### 2.3.2. PageRank Approach

#### 2.3.3. Centrality Approach

#### 2.4. Statistical Analysis

^{5}times, thus creating the normal distribution [17]. In order to determine significance of the experimental rank order sum from the algorithm being tested, if the experimental sum is significantly less than the sum as expected by chance from the normal distribution of sums, then it can be concluded that the algorithm reveals information about the causality of each node and the makeup of the RZ and SOZ [1].

## 3. Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Results of one run of any of the graph algorithms tested can be visually depicted as shown. In this example, displayed are the Indegree Centrality results for one patient for one run of the algorithm. Axes here are arbitrary units from the GC map. Each number labels a node, with a square representing those in the SOZ and a circle representing all others. The top eight nodes by Indegree Centrality metric are yellow. Directed edges are shown. Figure produced using matplotlib.

**Figure 2.**Results for the Monte Carlo sampling algorithm (1000 random restarts, 1000 samples per random restart) run on one patient are shown. Shown are the neurologist-determined SOZ (panel

**A**) and surgical RZ (panel

**B**) determined through means currently utilized for epilepsy surgery. The top 3 (panel

**C**) and top 10 (panel

**D**) ranked nodes according to the sampling algorithm are displayed. Note the similarities between top sampling nodes and neurologist-determined SOZ and RZ nodes. (Panel

**E**) shows the most influential edges of the GC map for reference. It is along such edges that tokens travel in the sampling algorithm. (Panel

**F**) shows the top 3 and 10 nodes as determined by the total GC outdegree method, also for reference [1].

**Figure 3.**Similar to Figure 2, results for the Monte Carlo sampling algorithm (1000 random restarts, 1000 samples per random restart) run on one patient are shown.

**Figure 5.**The RZ rank order sums of all algorithms and their variants are displayed here, with total GC outdegree included for comparison. SE bars are shown from sample sizes of five runs for each algorithm. Note the results of in/outdegree centrality algorithms, which exceed that of the total GC outdegree method from the literature while avoiding stochasticity of results found in other tested graph algorithms.

**Figure 6.**In this chart, sampling algorithms are described by the number of random restarts and the number of samples taken per random restart. Algorithms utilizing GC maps with forward edges include downhill sampling and indegree centrality. Algorithms utilizing GC maps with reverse edges include uphill sampling and outdegree centrality. Across nearly all algorithms tested, graph algorithms run on a reversed GC map had higher SOZ predictive capability, suggesting that interictal brain regions that cause neural activity in other regions better corelate with the SOZ rather than brain regions with the resulting neural activity itself.

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**MDPI and ACS Style**

Yang, V.B.; Madsen, J.R.
Extensions of Granger Causality Calculations on Brain Networks for Efficient and Accurate Seizure Focus Identification via iEEGs. *Brain Sci.* **2021**, *11*, 1167.
https://doi.org/10.3390/brainsci11091167

**AMA Style**

Yang VB, Madsen JR.
Extensions of Granger Causality Calculations on Brain Networks for Efficient and Accurate Seizure Focus Identification via iEEGs. *Brain Sciences*. 2021; 11(9):1167.
https://doi.org/10.3390/brainsci11091167

**Chicago/Turabian Style**

Yang, Victor B., and Joseph R. Madsen.
2021. "Extensions of Granger Causality Calculations on Brain Networks for Efficient and Accurate Seizure Focus Identification via iEEGs" *Brain Sciences* 11, no. 9: 1167.
https://doi.org/10.3390/brainsci11091167