:
1 2022 NeurIPS 2022 Gaze Meets ML Workshop

# Electrode Clustering and Bandpass Analysis of EEG Data for Gaze Estimation

Ard Kastrati1<EMAIL_ADDRESS>
Martyna Beata Płomecka2<EMAIL_ADDRESS>
Joël Küchler1<EMAIL_ADDRESS>
Nicolas Langer2<EMAIL_ADDRESS>
Roger Wattenhofer1<EMAIL_ADDRESS>
1ETH Zurich  Switzerland
2University of Zurich  Switzerland

###### Abstract

In this study, we validate the findings of previously published papers,
showing the feasibility of an Electroencephalography (EEG) based gaze
estimation. Moreover, we extend previous research by demonstrating that with
only a slight drop in model performance, we can significantly reduce the
number of electrodes, indicating that a high-density, expensive EEG cap is not
necessary for the purposes of EEG-based eye tracking. Using data-driven
approaches, we establish which electrode clusters impact gaze estimation and
how the different types of EEG data preprocessing affect the models’
performance. Finally, we also inspect which recorded frequencies are most
important for the defined tasks.

###### keywords:

EEG, Clustering, Deep Learning, Gaze Estimation, Bandpassing

##  1 Introduction

The ability to track eye movement patterns offers insights into the cognitive
processes underlying a wide variety of human behaviour. Eye tracking allows
researchers to recognize and quantify visual attention, fatigue and
performance in various scientific studies ([Eriksson and
Papanikotopoulos(1997), Holmqvist et al.(2011)Holmqvist, Nyström, Andersson,
Dewhurst, Jarodzka, and Van de Weijer, Liu and Heynderickx(2011)]). Nowadays,
infrared video-based eye trackers are the most common approach in research
labs ([Cornelissen et al.(2002)Cornelissen, Peters, and Palmer]). This eye
tracking technique uses infrared light to create a dark pupil and a corneal
reflection to provide contrast in locating the center of the pupil ([Holmqvist
et al.(2011)Holmqvist, Nyström, Andersson, Dewhurst, Jarodzka, and Van de
Weijer]). Although accurate, there are various limitations ([Holmqvist et
al.(2012)Holmqvist, Nyström, and Mulvey]). Examples include individual
differences in the contrast of the pupil and iris, time-consuming setup and
calibration for each scanning session ([Carter and Luke(2020)]). Moreover,
installing such a system involves setting up an optical path to align infrared
light to the cornea without interference from the visual paradigm display.

Another line of research demonstrated the feasibility of
Electroencephalography (EEG) and Electrooculography (EOG) signal decoding for
gaze estimation purposes. EOG is often concurrently measured with EEG by using
electrode pairs placed horizontally and/or vertically around the eye to record
changes in electric potentials that originate from movements of the eye
muscles ([Martínez-Cerveró et al.(2020)Martínez-Cerveró, Ardali, Jaramillo-
Gonzalez, Wu, Tonin, Birbaumer, and Chaudhary]). However, gaze estimation
based on Electroencephalography (EEG) data was mainly overlooked in practice
([Manabe et al.(2015)Manabe, Fukumoto, and Yagi, Hládek et al.(2018)Hládek,
Porr, and Brimijoin]), even though the solution has been known for years. One
potential reason is the necessity to obtain a rich enough set of data with
concurrently recorded EEG data and infrared video-based eye tracking data
serving as ground truth. In addition, this solution requires equipment and
expertise for both EEG acquisition and eye tracking ([Dimigen et
al.(2011)Dimigen, Sommer, Hohlfeld, Jacobs, and Kliegl]). Moreover, the
noisiness of EEG data poses an additional challenge leaving the research and
development in the EEG-based gaze estimation area behind ([Grech et
al.(2008)Grech, Cassar, Muscat, Camilleri, Fabri, Zervakis, Xanthopoulos,
Sakkalis, and Vanrumste]). Nonetheless, with the rapid progress in Machine
Learning and the increase in the collection of datasets, the EEG modality
became more approachable and easier to process ([Kastrati et
al.(2021)Kastrati, Płomecka, Pascual, Wolf, Gillioz, Wattenhofer, and
Langer]).

The primary goal of this study is to investigate which spatio-spectral brain
signal components are most relevant for decoding eye movement and
understanding what is the minimal and best placement of the electrodes. More
importantly, we show that good-performing models with high accuracy can be
achieved even when the number of electrodes is significantly reduced compared
to a high-density, 128-electrodes EEG cap. Additionally, we demonstrate that
when using a standard pipeline for EEG preprocessing that includes independent
component analysis (ICA) ([Pion-Tonachini et al.(2019)Pion-Tonachini, Kreutz-
Delgado, and Makeig]), one can infer gaze direction also from electrodes
placed in the occipital part of the head, albeit with much lower accuracy.
Finally, we include an experimental analysis of the importance of different
frequency intervals of the EEG signals for gaze direction.

##  2 Related Work

To date, only a few EEG studies attempted to estimate the actual eye gaze
position on a computer screen, and those resulted in high inaccuracies
(estimation error 15°) ([Borji and Itti(2012)]), and complicated analyses
([Manabe et al.(2015)Manabe, Fukumoto, and Yagi]). Moreover, there are a
number of limitations associated with EEG and specifically EOG electrodes,
including a variety of metabolic activities over time and potential drifts
([Martinsen and Grimnes(2011)]). There has been some investigation into the
traditional methods of supervised machine learning; for instance, ([Bulling et
al.(2010)Bulling, Ward, Gellersen, and Tröster]) categorized the various
directions and durations of saccades with a mean accuracy of 76.1%. Another
study ([Vidal et al.(2011)Vidal, Bulling, and Gellersen]) developed an EOG
feature-based approach that discriminated between saccades, smooth pursuits,
and vestibulo-ocular reflex movement achieving quite good results, all of them
being above 80%. Latest approaches in machine learning (ML) have shown promise
for the development of more precise EEG/EOG-based eye tracking systems. For
example, the EEGEyeNet ([Kastrati et al.(2021)Kastrati, Płomecka, Pascual,
Wolf, Gillioz, Wattenhofer, and Langer]) benchmark, along with the rich
dataset of simultaneous Electroencephalography and eye tracking recordings,
has demonstrated promising results for further development of EEG-EOG-based
gaze estimation using deep learning frameworks. Recently, ([Wolf et
al.(2022)Wolf, Kastrati, Płomecka, Li, Klebe, Veicht, Wattenhofer, and
Langer]) proposed a novel framework for time-series segmentation, creating
ocular event detectors that rely solely on EEG data. This solution achieved
state-of-the-art performance in ocular event detection across diverse eye
tracking experiment paradigms, showing the high potential of EEG-based eye
tracking solution, used not only as a complimentary modality but in some
instances, it can also be beneficial when classic eye tracker hardware is
unavailable.

##  3 Methods

###  3.1 Dataset, Benchmarking Tasks and Models

We use EEGEyeNet dataset and benchmark ([Kastrati et al.(2021)Kastrati,
Płomecka, Pascual, Wolf, Gillioz, Wattenhofer, and Langer]) to run our
studies. EEGEyeNet consists of synchronized EEG and eye tracking data and
consists of collected from three different experimental paradigms. It also
establishes a benchmark consisting of three tasks with an increasing level of
difficulty:

  * •

Left-Right task (LR): This is a binary classification task, where the goal is
determining the direction of the subject’s gaze along the horizontal axis.

  * •

Direction task: The task is to regress the two target values, i.e., angle and
amplitude of the relative change of the gaze position during the saccade.

  * •

Absolute position task: The goal of the task is determining the absolute
position of the subject’s gaze in the screen, described in terms of XY-
coordinates.

In each task, a window of 1 second (with sampling rate of 500Hz) of EEG data
is given as an input sample, and the goal is to classify or regress (depending
on the task) the target value. The data acquisition, models used,
preprocessing methods, and the whole pipeline is explained in much more detail
in the original work ([Kastrati et al.(2021)Kastrati, Płomecka, Pascual, Wolf,
Gillioz, Wattenhofer, and Langer]).

###  3.2 EEG data preprocessing

In this manuscript, we use two types of EEG -data preprocessing. The first
one, from now on referred to as ”Minimal Preprocessing”, includes algorithms
implemented in the EEGlab plugin: to identify bad channels
(clean_rawdata111http://sccn.ucsd.edu/wiki/Plugin_list_process), to reduce the
noise (Zapline) and bandpass filtering (0.5-40Hz). Detected bad channels were
automatically removed and later interpolated using a spherical spline
interpolation. The details of the preprocessing pipeline can be found in the
EEGEyeNet paper ([Kastrati et al.(2021)Kastrati, Płomecka, Pascual, Wolf,
Gillioz, Wattenhofer, and Langer]). The second preprocessing type, i.e.
”Maximal preprocessing”, is the state-of-the-art preprocessing used for
neuroscientific applications. In addition to the ”Minimal Preprocessing”
pipeline, it includes a Non-Brain artifactual source components removal based
on the automatic classification result as provided by Independent Component
Label (ICLabel) ([Pion-Tonachini et al.(2019)Pion-Tonachini, Kreutz-Delgado,
and Makeig]) algorithm.

###  3.3 Gradient-Based Feature Importance

In order to indicate whether a feature (in our case an electrode) contributes
significantly to the prediction, the gradient concerning the input can be
computed ([Simonyan et al.(2013)Simonyan, Vedaldi, and Zisserman]). This shows
how much the model relies on the feature. In our case, we are interested in
aggregating this score over an electrode to get its importance. First, the
resulting gradients are normalized for each input separately. Next, all
absolute gradient values of an electrode are summed up. If an electrode is
left with a higher score comparatively, it indicates that it has a more
significant contribution to the input.

##  4 Electrode Clustering on Minimally Preprocessed Data

A dense, 128-electrode EEG cap is often infeasible to be used in practice. In
this section, we show that most of the important information for eye movement
is highly concentrated in the frontal electrodes.

###  4.1 Finding Important Electrodes

We use gradient-based methods to rank the most important electrodes. From the
spatial distribution (topoplots) in Figure 1 we can see a clear symmetrical
structure and that the most information used for decoding eye movement is in
the frontal electrode. Interestingly, the topoplots show that the central
electrode (Cz), which was the recording reference electrode also carries some
relevant information. This can be explained, by the fact that the data were
offline re-referenced to average reference and thus Cz was interpolated by all
other electrodes. To produce the topoplots we used the average performance
across all models and for each task presented in the benchmark ([Kastrati et
al.(2021)Kastrati, Płomecka, Pascual, Wolf, Gillioz, Wattenhofer, and
Langer]).

Figure 1: Topoplots for (a) LR Task, (b) Position Task, (c) Amplitude Task and
(d) Angular Task. Colors of higher intensity represent a higher importance.
The colors are shown on a logarithmic scale. The score is averaged over all
five deep learning architectures presented in ([Kastrati et al.(2021)Kastrati,
Płomecka, Pascual, Wolf, Gillioz, Wattenhofer, and Langer]).

###  4.2 Choice of Electrodes

Due to the symmetrical results on the topoplots, the most important electrodes
are chosen as follows: first, all electrodes are ranked according to gradient-
based analysis (see Figure 2 for the importance of each electrode). The best
electrode, which is not yet in the cluster, is added together with its
symmetrical counterpart. This method responds to the brain’s natural symmetry
and focuses on localized head areas that we expect to be more resilient and
insightful than single isolated electrodes. This procedure is repeated until
the loss stops increasing. With this method, we converged to 23 electrodes
which are mainly on the frontal part (as can be observed in the topoplots and
Figure 3). In order to compare models with each other and for convenience, it
is ensured that the same configurations are used across all tasks and models.

Figure 2: Gradient Based Feature Importance for (a) LR Task, (b) Position
Task, (c) Amplitude Task and (d) Angular Task. The normalized gradient results
for each electrode and each task from the experiments on the minimally
preprocessed data.

###  4.3 Choice of Clusters

We observed that the best electrodes are of similar significance in all
architectures for all the tasks and the accuracy. In addition, the choice of
23 electrodes achieves the same accuracy that one achieves with all 128
electrodes. This high accuracy encouraged us to train models on an even more
reduced number of electrodes and this way we created several different and
smaller clusters. We choose clusters of sizes 2, 3, 8, and 23. For all tasks
and the models in EEGEyeNet benchmark, we converged to the same clusters shown
below.

Figure 3: Electrode Clustering Visualisation. This figure shows the electrode placement. Colour-coded electrodes belong to a configuration. Pink nodes form the Top2 configuration. For Top3, the blue electrode is added. By combining Top3 with the teal electrodes, we get the Top8 configuration. The final 23 electrode composition consists of all coloured nodes. Name | Electrodes | Quantity | Colour Combination
---|---|---|---
Top2 | {125,128} | 2 | Pink
Top3 | Top2 $\cup$ {17} | 3 | Pink + Blue
Top8 | Top3 $\cup$ {1,32,38,121,129} | 8 | Pink + Blue + Teal
SideFronts | Top8 $\cup$ {4,5,6,8,12,14,19,21,25} | 23 | Pink + Blue + Teal + Yellow
|  $\cup$ {33,43,120,122,126,127} |  |
Table 1: Electrode Clustering. Each electrode is given a colour as shown in
Figure 3. The colours refer to the hues of the used electrodes.

###  4.4 Evaluation

Cluster | Model | LR. (%) $\uparrow$ | Amp. (mm) $\downarrow$ | Ang. (rad) $\downarrow$ | Pos. (mm) $\downarrow$
---|---|---|---|---|---
All (128) | InceptionTime | $97.07\pm 0.2$ | $27.25\pm 0.94$ | $0.36\pm 0.06$ | $64.6\pm 4.88$
EEGNet | $97.86\pm 0.27$ | $29.14\pm 1.92$ | $0.38\pm 0.12$ | $70.36\pm 1.11$
CNN | $97.35\pm 0.48$ | $27.81\pm 3.39$ | $0.33\pm 0.03$ | $62.85\pm 0.79$
PyramidalCNN | $\mathbf{98.12}\pm 0.18$ | $\mathbf{25.79}\pm 1.25$ | $\mathbf{0.27}\pm 0.08$ | $\mathbf{61.34}\pm 1.28$
Xception | $97.05\pm 0.4$ | $28.19\pm 2.14$ | $0.4\pm 0.05$ | $63.49\pm 1.19$
Top23 | InceptionTime | $97.54\pm 0.91$ | $27.82\pm 5.48$ | $0.32\pm 0.03$ | $62.92\pm 1.66$
EEGNet | $97.96\pm 0.24$ | $29.82\pm 1.31$ | $\mathbf{0.26}\pm 0.01$ | $69.22\pm 1.21$
CNN | $98.16\pm 0.31$ | $25.97\pm 3.94$ | $0.3\pm 0.06$ | $62.88\pm 1.35$
PyramidalCNN | $\mathbf{98.38}\pm 0.2$ | $\mathbf{24.08}\pm 1.08$ | $0.32\pm 0.18$ | $\mathbf{61.32}\pm 0.67$
Xception | $97.96\pm 0.63$ | $28.35\pm 2.08$ | $0.37\pm 0.17$ | $63.2\pm 1.51$
Top8 | InceptionTime | $97.92\pm 0.68$ | $26.08\pm 2.34$ | $0.35\pm 0.12$ | $65.57\pm 0.84$
EEGNet | $98.1\pm 0.13$ | $29.62\pm 1.61$ | $0.27\pm 0.02$ | $71.76\pm 1.22$
CNN | $97.92\pm 0.47$ | $27.87\pm 3.17$ | $0.32\pm 0.03$ | $65.37\pm 1.29$
PyramidalCNN | $98.12\pm 0.32$ | $26.21\pm 1.75$ | $\mathbf{0.26}\pm 0.03$ | $\mathbf{65.33}\pm 1.76$
Xception | $97.98\pm 0.24$ | $\mathbf{25.75}\pm 1.03$ | $0.34\pm 0.07$ | $66.2\pm 1.66$
Top3 | InceptionTime | $98.0\pm 0.35$ | $31.94\pm 3.27$ | $0.36\pm 0.08$ | $72.28\pm 1.87$
EEGNet | $\mathbf{98.46}\pm 0.15$ | $33.64\pm 1.68$ | $0.38\pm 0.1$ | $76.12\pm 0.92$
CNN | $98.34\pm 0.24$ | $\mathbf{28.12}\pm 3.92$ | $0.32\pm 0.03$ | $71.77\pm 0.75$
PyramidalCNN | $98.1\pm 0.67$ | $29.11\pm 1.98$ | $\mathbf{0.27}\pm 0.04$ | $\mathbf{70.98}\pm 0.59$
Xception | $97.72\pm 0.3$ | $30.46\pm 3.15$ | $0.35\pm 0.07$ | $74.46\pm 1.71$
Top2 | InceptionTime | $98.18\pm 0.49$ | $\mathbf{32.15}\pm 3.31$ | $0.36\pm 0.05$ | $77.07\pm 3.68$
EEGNet | $\mathbf{98.57}\pm 0.11$ | $38.23\pm 1.55$ | $0.34\pm 0.01$ | $77.55\pm 0.89$
CNN | $97.39\pm 1.88$ | $33.08\pm 4.28$ | $0.34\pm 0.05$ | $75.29\pm 2.52$
PyramidalCNN | $98.48\pm 0.21$ | $33.16\pm 2.01$ | $\mathbf{0.28}\pm 0.03$ | $\mathbf{75.47}\pm 0.74$
Xception | $97.84\pm 0.44$ | $34.94\pm 2.43$ | $0.47\pm 0.24$ | $79.21\pm 1.61$
Table 2: Benchmark. The performance of all models in EEGEyeNet benchmark for
each task and each chosen cluster. The error of left-right task (LR) is
measured in accuracy, amplitude and position task is measured in pixels, and
angle in radians.

We evaluate all proposed electrode configurations on all deep learning models.
The benchmark is run for all the tasks separately on NVIDIA GeForce GTX 1080
Ti. We see that on average equal or better results (in comparison with
networks that have access to the full information) can be achieved by just
using a fraction of electrodes.

We can observe that in Table 2, running the benchmark with only 23 electrodes
the models perform equally and sometimes even better than training the models
with all 128 electrodes. This can be since many of the other electrodes do not
add any useful information but just noise. This can be seen for example in the
Left-Right task where PyramidalCNN achieves a score of 98.46 which is better
than any model trained with 128 electrodes. Similar behavior can be seen also
in the amplitude task. We can also observe that decreasing the number of
electrodes to 8, decreases the performance of the models in the position task,
however, the performance of all the other tasks with 8 electrodes still
remains equally good compared to the performance with 128 electrodes.
Decreasing the number of electrodes to 3 and 2, leads to a decrease in
performance on all the tasks except the left-right task which can be decoded
with high accuracy with only 2 electrodes. Nonetheless, with only 2-3
electrodes the performance on the amplitude task and the position task
decreases significantly.

##  5 Electrode Clustering on Maximally Preprocessed Data

In the previous section, we saw that the most important electrodes for
decoding eye movement are the frontal electrodes. More specifically, if we
choose 23 frontal electrodes then we achieve the same performance as with the
full cap which consists of 128 electrodes.

In this section, we investigate how state-of-the-art preprocessing methods
used for neuroscientific applications (maximally preprocessing) affect these
results. Maximally preprocessing steps exclude ocular artifacts and include
ideally only neurophysiological information. This makes the estimation of gaze
position harder, however, it also reveals other insights how brain activity is
related to eye movement.

Interestingly, the analysis of the maximmally preprocessed data shows that the
electrodes in the occipital part of the brain are also important for inferring
gaze direction if one considers only the neurophysiological information.
However, even after maximally preprocessing the data, the frontal electrodes
can still be used for inferring gaze direction, indicating that the
preprocessing might be suboptimal in removing ocular artifacts from the
recorded EEG signal. Alternatively one could also speculate that the actual
neuronal activity in the frontal electrodes (e.g. frontal eye fields) actually
entails information about the eye movement.

###  5.1 Finding Important Electrodes

We use again gradient-based methods to find important electrodes.

Figure 4: Topoplots for (a) LR Task, (b) Position Task, (c) Amplitude Task and
(d) Angular Task. Colors of higher intensity represent a higher importance.
The colors are shown on a logarithmic scale. The score is averaged over all
five deep learning architectures presented in ([Kastrati et al.(2021)Kastrati,
Płomecka, Pascual, Wolf, Gillioz, Wattenhofer, and Langer]).

We observe in Figure 4 again symmetry but, compared to minimally preprocessed
data, we can see more sparsity in the distribution of the important brain
regions. In particular, in contrast to minimally preprocessed data, we can
identify in left-right and direction task an additional important region. Most
of the importance is still located in the frontal electrodes, but now the
second region of interest around the occipital region can also be identified.

###  5.2 Choice of Electrodes

Due to the symmetry in the topoplots we use the same technique as in the
minimally preprocessed data. That is, we rank the electrodes according to the
gradient-based analysis (see Figure 5 for the importance of each electrode).
With maximally preprocessed data we identified 40 most important electrodes,
which achieve almost the same performance compared to the dense 128-electrode
cap.

Figure 5: Gradient Based Feature Importance for (a) LR Task, (b) Position
Task, (c) Amplitude Task and (d) Angular Task. The normalized gradient results
for each electrode and each task from the experiments on the maximally
preprocessed data.

###  5.3 Choice of clusters

If we decrease the number of electrodes below 40, then the performance starts
decreasing. In this section, we investigate how the performance decreases for
several different smaller clusters. Motivated by the fact that there are two
different regions in maximally preprocessed data, we distinguish between the
following clusters: front, back, front extended, and back extended. The
extended version includes more electrodes in each region which results in more
stable training. In Figure 6, we show all the main clusters for the best 40
electrodes. In Figure 6 the most important front electrodes are marked with
pink color. The front extended cluster is marked with a purple cluster,
composed of less important front electrodes. The back cluster is marked in
yellow, which is extended with the green cluster (back extended cluster).

Figure 6: Electrode Clustering Visualisation. The best 40 electrodes according
to our gradient feature importance method are marked. The pink-colored nodes
correspond to the front part. For the extended version, the purple electrodes
are added. The same is done for back electrodes. Its base consists of the
yellow nodes, the extended version consists of the green ones.

###  5.4 Evaluation

We evaluate all proposed electrode configurations on all deep learning models
for the maximally preprocessed data as well. In Table 3, we can see that with
only 40 electrodes the models perform equally well and except for LR task, the
models perform even better. For example, in the angle task, PyramidalCNN
achieves a score of 0.68 if trained with 40 electrodes compared to the score
of 0.76 radians if trained with 128 electrodes. Similar behavior can be seen
also for the amplitude task. We can also observe that the models achieve
competitive performance compared to the trained models with the full cap, also
if they are trained with only the front (extended) cluster. Here we can also
see for the angle task, PyramidalCNN trained only on the front extended task
achieving a score of 0.67 radian, which is better than all models trained on
the full 128-electrode cap.

Cluster | Model | LR. (%) $\uparrow$ | Amp. (mm) $\downarrow$ | Ang. (rad) $\downarrow$ | Pos. (mm) $\downarrow$
---|---|---|---|---|---
All | InceptionTime | $\mathbf{93.61}\pm 0.67$ | $61.42\pm 3.55$ | $0.78\pm 0.08$ | $125.07\pm 3.43$
EEGNet | $86.31\pm 1.47$ | $60.42\pm 0.7$ | $1.18\pm 0.05$ | $\mathbf{114.06}\pm 0.38$
CNN | $90.48\pm 1.87$ | $64.31\pm 2.77$ | $0.89\pm 0.14$ | $122.59\pm 3.81$
(128) | PyramidalCNN | $93.08\pm 0.49$ | $\mathbf{57.73}\pm 1.44$ | $\mathbf{0.76}\pm 0.09$ | $133.59\pm 1.62$
| Xception | $89.36\pm 3.01$ | $64.2\pm 1.41$ | $0.9\pm 0.19$ | $125.34\pm 2.79$
Top40 | InceptionTime | $\mathbf{93.59}\pm 1.05$ | $60.42\pm 2.31$ | $0.75\pm 0.06$ | $122.48\pm 1.98$
EEGNet | $87.95\pm 0.09$ | $59.66\pm 0.58$ | $1.09\pm 0.03$ | $\mathbf{112.9}\pm 0.19$
CNN | $90.64\pm 1.09$ | $62.38\pm 1.4$ | $0.83\pm 0.09$ | $119.28\pm 0.62$
PyramidalCNN | $92.01\pm 1.66$ | $57.04\pm 0.82$ | $\mathbf{0.68}\pm 0.02$ | $134.6\pm 2.45$
Xception | $92.12\pm 0.8$ | $\mathbf{63.0}\pm 1.5$ | $0.87\pm 0.13$ | $125.71\pm 1.96$
| InceptionTime | $\mathbf{92.6}\pm 0.17$ | $59.34\pm 3.23$ | $0.86\pm 0.19$ | $121.48\pm 2.17$
Front | EEGNet | $87.26\pm 0.55$ | $60.52\pm 0.68$ | $1.12\pm 0.02$ | $\mathbf{112.91}\pm 0.38$
& | CNN | $90.99\pm 0.8$ | $61.86\pm 2.52$ | $0.82\pm 0.03$ | $118.94\pm 1.69$
Back | PyramidalCNN | $91.36\pm 1.0$ | $\mathbf{58.63}\pm 0.47$ | $\mathbf{0.75}\pm 0.03$ | $134.0\pm 2.99$
| Xception | $90.07\pm 0.15$ | $63.15\pm 3.3$ | $0.96\pm 0.11$ | $125.17\pm 1.59$
Front | InceptionTime | $\mathbf{92.47}\pm 0.81$ | $\mathbf{60.72}\pm 1.78$ | $0.8\pm 0.15$ | $124.68\pm 2.12$
EEGNet | $83.94\pm 1.41$ | $63.35\pm 0.92$ | $1.08\pm 0.01$ | $\mathbf{113.83}\pm 0.36$
CNN | $91.42\pm 0.48$ | $69.73\pm 5.59$ | $0.8\pm 0.03$ | $119.23\pm 0.92$
Ext. | PyramidalCNN | $91.28\pm 1.69$ | $60.94\pm 1.98$ | $\mathbf{0.67}\pm 0.03$ | $135.0\pm 2.27$
| Xception | $91.76\pm 0.25$ | $66.74\pm 3.45$ | $0.81\pm 0.09$ | $124.06\pm 1.94$
Front | InceptionTime | $\mathbf{91.68}\pm 0.19$ | $\mathbf{61.31}\pm 2.5$ | $0.91\pm 0.15$ | $123.05\pm 0.9$
EEGNet | $83.33\pm 0.63$ | $63.25\pm 0.27$ | $1.12\pm 0.02$ | $\mathbf{114.19}\pm 0.39$
CNN | $90.57\pm 0.82$ | $61.84\pm 2.28$ | $0.85\pm 0.04$ | $120.67\pm 3.23$
PyramidalCNN | $89.54\pm 1.8$ | $61.36\pm 1.2$ | $\mathbf{0.78}\pm 0.02$ | $135.15\pm 3.52$
Xceptio | $91.04\pm 0.69$ | $62.6\pm 2.98$ | $0.85\pm 0.07$ | $125.44\pm 2.74$
Back | InceptionTime | $71.42\pm 2.48$ | $78.46\pm 3.53$ | $\mathbf{1.52}\pm 0.08$ | $130.57\pm 1.05$
EEGNet | $\mathbf{74.32}\pm 0.43$ | $\mathbf{67.93}\pm 0.19$ | $1.81\pm 0.01$ | $\mathbf{119.88}\pm 0.1$
CNN | $71.39\pm 1.02$ | $75.04\pm 1.81$ | $1.54\pm 0.07$ | $127.8\pm 1.55$
Ext. | PyramidalCNN | $70.91\pm 1.32$ | $70.85\pm 0.58$ | $1.56\pm 0.03$ | $145.75\pm 2.04$
| Xception | $70.67\pm 1.29$ | $81.64\pm 1.15$ | $1.66\pm 0.06$ | $132.16\pm 0.66$
Back | InceptionTime | $69.21\pm 1.85$ | $75.45\pm 2.48$ | $1.67\pm 0.08$ | $127.42\pm 1.63$
EEGNet | $\mathbf{72.98}\pm 0.49$ | $\mathbf{68.72}\pm 0.37$ | $1.78\pm 0.07$ | $\mathbf{119.87}\pm 0.18$
CNN | $70.02\pm 0.63$ | $79.54\pm 8.81$ | $\mathbf{1.54}\pm 0.04$ | $126.45\pm 1.93$
PyramidalCNN | $67.88\pm 3.94$ | $73.03\pm 2.25$ | $1.61\pm 0.02$ | $145.6\pm 1.69$
Xception | $68.19\pm 2.03$ | $81.93\pm 2.57$ | $1.7\pm 0.05$ | $132.81\pm 2.23$
Table 3: Benchmark. The performance of all models in EEGEyeNet benchmark for
each task and each chosen cluster for maximally preprocessed data. The error
of left-right task (LR) is measured in accuracy, amplitude and position task
is measured in pixels, and angle is measured in radians.

The position task for maximally preprocessed data is difficult for all the
models even with the full cap, and as stated in ([Kastrati et
al.(2021)Kastrati, Płomecka, Pascual, Wolf, Gillioz, Wattenhofer, and Langer])
it is not clear how good this task can be solved with only neurophysiological
data. Finally, compared to the minimally preprocessed data, we can also see
eye movement information can be decoded with only the electrodes on the
occipital regions of the scalp, however, the performance decreases
significantly. For instance, the best model trained with frontal electrodes
achieves an accuracy of 92.58%, whereas the best performing model trained with
the electrodes on the occipital part achieves an accuracy of 74.53%. If only
the occipital electrodes are used the performance on the other tasks decreases
significantly. This can be seen for the angle task where the performance of
the models changes from 0.67 to 1.55 radians. This error is close to the naive
baselines reported in [Kastrati et al.(2021)Kastrati, Płomecka, Pascual, Wolf,
Gillioz, Wattenhofer, and Langer], showing the difficulty in learning more
complex eye movements other than left-right classification solely from
occipital electrodes.

##  6 Bandpassing

The final analysis of the impact of preprocessing consists of bandpassing our
data before training.

###  6.1 Choice of frequency bands

We decided to focus our attention on a limited number of intervals, roughly
based on historically pre-defined frequency bands ([Newson and
Thiagarajan(2019)]). We define four frequency intervals: Delta : 1-4Hz, Theta
: 4-8Hz, Alpha : 8-13Hz and Beta : 13-32Hz. The raw EEG measurements are
bandpass for each subject before preparation of the input data, to allow
precise frequency selection before reducing the signals to 1-second intervals
to feed our model. This bandpassing is performed on both the maximally and
minimally preprocessed data, before training respectively on each band.

###  6.2 Results

We present the results below, including two frequency intervals obtained by
merging frequency intervals. We report the average results of the EEGEyeNet
benchmark for the left-right task. In Figure 7, we observe an important drop
in accuracy for each of our bandpass datasets, with expectedly higher results
when combined.

Figure 7: Bandpass analysis. The relation between the frequency intervals and
the performance of the models in the LR task for the (a) maximally
preprocessed data and (b) minimally preprocessed data.

Interestingly, the maximally preprocessed dataset seems to contain most of its
helpful information in the higher frequencies, namely the 13-32Hz dataset,
with an accuracy above the combination of the three other intervals. It seems
to be the contrary for our minimally preprocessed dataset, with a high
accuracy in 4-8Hz frequency band and 1-13Hz frequency band prediction. We see
no significant improvement from the inclusion of the 13-32Hz interval. We
hypothesize that the significant front electrodes’ importance observed during
clustering on the minimally preprocessed data contains mainly sub-13Hz signals
used for prediction. A potential reason for this is that saccades happen only
every 200-300ms. Additionally, the fact that 13-32Hz frequency band is
important for maximally preprocessed data indicates that actual neuronal
activity is used for decoding.

##  7 Conclusion

The study’s major finding shows that minimally preprocessed data used for
prediction contains most of its information for the task in a limited number
of electrodes in the frontal part. Therefore, by reducing their number to one-
quarter of its original number, we decreased the input data size significantly
and stabilized the prediction results with special cases where the performance
even increases. Moreover, the short fit of our data and the high stability of
the results might suggest the possibility of reducing the model’s complexity
without accuracy loss.

It is interesting to note that also, for the maximally preprocessed dataset,
the frontal electrodes are most important for gaze prediction. Since this
dataset was treated with a non-brain artifactual source components removal
based on the automatic classification result as provided by Independent
Component Analysis, we can speculate the signal stems from the frontal eye
field area, which plays an essential role in controlling visual attention and
eye movements ([Armstrong et al.(2009)Armstrong, Chang, and Moore]). The
oculomotor artefact removal in the maximally preprocessed dataset makes the
gaze estimation position task more challenging. Nevertheless, this dataset
contains mostly neurophysiological information and reveals other insights into
how brain activity relates to eye movement. In particular, it is interesting
to note how the second region of interest, located at the occipital part of
the head, was important for inferring gaze direction. Since, most likely, the
electrodes located in this part of the head are not influenced by any residual
oculomotor noise, we can conclude that this signal contains information
measured in the region of the visual cortex, revealing how neurophysiological
brain activity is related to eye movement.

Finally, we analyzed the impact of bandpassing our data before training. As
expected, the low frequencies were most significant for the minimally
preprocessed dataset, as they are related to ocular artefacts. However, the
maximally preprocessed dataset revealed a different pattern, showing that the
13-32Hz frequencies contained the most meaningful information for our tasks. A
current limitation in this work is that the bandpass analysis is performed
only on the left-right task and for limited windows of frequency bands.
Extensions of this work call for a fine-grained spectral analysis and how the
performance changes across different eye movement patterns (other than left-
right).

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# A Mathematical Reconstruction of Endothelial Cell Networks

Okezue Bell  Stanford University, Department of Undergraduate Mathematics,
Stanford, CA, 94305<EMAIL_ADDRESS>Anthony Bell  University of
Pennsylvania School of Engineering and Applied Science, Philadelphia, PA,
19104

###### Abstract

Endothelial cells form the linchpin of vascular and lymphatic systems,
creating intricate networks that are pivotal for angiogenesis, controlling
vessel permeability, and maintaining tissue homeostasis. Despite their
critical roles, there is no rigorous mathematical framework to represent the
connectivity structure of endothelial networks. Here, we develop a pioneering
mathematical formalism called $\pi$-graphs to model the multi-type junction
connectivity of endothelial networks. We define $\pi$-graphs as abstract
objects consisting of endothelial cells and their junction sets, and introduce
the key notion of $\pi$-isomorphism that captures when two $\pi$-graphs have
the same connectivity structure. We prove several propositions relating the
$\pi$-graph representation to traditional graph-theoretic representations,
showing that $\pi$-isomorphism implies isomorphism of the corresponding
unnested endothelial graphs, but not vice versa. We also introduce a temporal
dimension to the $\pi$-graph formalism and explore the evolution of
topological invariants in spatial embeddings of $\pi$-graphs. Finally, we
outline a topological framework to represent the spatial embedding of
$\pi$-graphs into geometric spaces. The $\pi$-graph formalism provides a novel
tool for quantitative analysis of endothelial network connectivity and its
relation to function, with the potential to yield new insights into vascular
physiology and pathophysiology.

###### keywords:

Endothelial, Isomorphism, $\pi$-graph, junction sets

##  1 Introduction

The endothelium, a monolayer of endothelial cells lining the blood and
lymphatic vessels, plays a central role in vascular homeostasis [1]. Far from
a passive conduit for blood flow, the endothelium actively regulates vascular
permeability, blood coagulation, angiogenesis, and inflammatory responses.
These functions arise from the ability of endothelial cells to form complex
networks with specialized cell-cell junctions. Adherens junctions and tight
junctions are ubiquitous structures that mechanically and chemically couple
endothelial cells, while gap junctions mediate the passage of ions and small
signaling molecules between cells. More recently discovered structures, such
as nectin junctions, also contribute to endothelial connectivity.
Collectively, endothelial cell-cell junctions enable the coordination of
endothelial responses across vascular networks.

Endothelial network integrity is critical for vascular health, as disruption
of interendothelial junctions and resulting increases in vascular permeability
are key events in atherosclerosis, ischemia-reperfusion injury, sepsis, and
other disease states [2, 3, 4]. Aberrant endothelial connectivity also enables
the pathological angiogenesis seen in cancer and eye diseases [5].
Quantitative analysis of endothelial network structure therefore has the
potential to yield new biomarkers and therapeutic strategies for vascular
dysfunction [6, 7, 8, 9, 10].

Network representations have become an invaluable tool for understanding the
structure and function of complex biological systems. Prominent examples
include neuron maps [11], metabolic system representations [12], cell-cell
communication models [13, 14], and protein-protein interaction networks [15].
In these contexts, network models have revealed fundamental organizing
principles such as small-world topology, modularity, and centrality. A
unifying mathematical language for biological networks has emerged from graph
theory, which captures connectivity patterns between large numbers of
interacting elements at a high fidelity.

However, standard graphs are not sufficient to capture the rich junction
architecture connecting endothelial cells into vascular networks. A single
edge between two cells cannot represent the multiple junction types that may
exist between them[16]. Instead, endothelial network connectivity is
inherently a multi-relation structure. Furthermore, the concept of the line
graph, which can represent multi-edges, does not naturally capture the
distinct types of endothelial junctions and their set-like organization around
cells. As a result, a more sophisticated mathematical representation is
needed.

Here we develop a mathematical formalism called $\pi$-graphs to meet this
need. $\pi$-graphs are abstract structures that faithfully represent the
multi-junction connectivity of endothelial networks using an intuitive set-
based language. We define $\pi$-graphs and their morphisms, and prove several
results linking $\pi$-graphs to their underlying unnested endothelial graphs.
We also describe a framework based on topological realization to represent the
spatial embedding of endothelial networks. Our work provides an expressive,
tractable, and generalizable framework to quantitatively interrogate
endothelial networks.

##  2 Models and Methods

###  2.1 Basic Definitions

We begin by formally defining the notion of an endothelial $\pi$-graph. Let
$\mathcal{E}$ be a finite set, whose elements we call endothelial cells (ECs).
We consider four types of intercellular junctions that can connect ECs:

  1. 1.

Adherens junctions (AJs): Cell-cell adhesion structures that mechanically link
adjacent ECs via homophilic interactions between transmembrane VE-cadherin
proteins[17].

  2. 2.

Tight junctions (TJs): Multiprotein complexes composed of claudins, occludins,
and adaptor proteins that form a seal between adjacent ECs to regulate
paracellular permeability[18].

  3. 3.

Gap junctions (GJs): Intercellular channels formed by connexin proteins that
allow direct exchange of ions and small molecules (<1 kDa) between the
cytoplasm of adjacent ECs[19].

  4. 4.

Nectin junctions (NJs): Adhesive structures formed by heterophilic
interactions between nectin proteins that are distinct from but colocalize
with AJs and help initiate cell-cell contacts[20].

We represent each of these junction types as a distinct symmetric binary
relation on the set $\mathcal{E}$:

###### Definition 1

Let $\sim_{AJ},\sim_{TJ},\sim_{GJ},\sim_{NJ}$ be symmetric binary relations on
$\mathcal{E}$ representing adherens junctions, tight junctions, gap junctions,
and nectin junctions respectively. The set of all junction relations is
denoted
$\mathcal{J}_{\mathcal{E}}:={\sim_{AJ},\sim_{TJ},\sim_{GJ},\sim_{NJ}}$.

The relations in $\mathcal{J}_{\mathcal{E}}$ jointly determine the
connectivity of the endothelial network. To capture the set of junctions
incident at each EC, we introduce the notion of $\pi$-incidence:

###### Definition 2

A $\pi$-incidence on $\mathcal{E}$ is a function
$\pi:\mathcal{E}\to\mathcal{P}(\mathcal{J}_{\mathcal{E}})$ satisfying:

  1. 1.

(Junction Consistency) For all $x\in\mathcal{E}$ and
$\sim\in\mathcal{J}_{\mathcal{E}}$,

$\sim\in\pi(x)\iff\exists y\in\mathcal{E}\textrm{ s.t. }x\sim y$

  2. 2.

(Nondegeneracy) For all $x\in\mathcal{E}$, $\pi(x)\neq\emptyset$

The Junction Consistency condition states that a cell $x$ is $\pi$-incident to
a junction relation $\sim$ if and only if $x$ is connected by $\sim$ to some
other cell. The Nondegeneracy condition requires that each cell participates
in at least one junction. Note that distinct elements $x,y\in\mathcal{E}$ may
share multiple junction relations, i.e. $|\pi(x)\cap\pi(y)|>1$. This multi-
relational structure is a key feature of endothelial networks that motivates
the $\pi$-graph formalism.

With these ingredients, we can now define the central object of study:

###### Definition 3

An endothelial $\pi$-graph is a tuple $G=(\mathcal{E},\pi)$ where
$\mathcal{E}$ is a finite set of ECs and $\pi$ is a $\pi$-incidence on
$\mathcal{E}$. The set of all $\pi$-graphs is denoted $\mathbf{\varpi}$.

Thus, a $\pi$-graph $G\in\mathbf{\varpi}$ encodes the network connectivity of
an endothelial monolayer as a set $\mathcal{E}$ of ECs and a $\pi$-incidence
reflecting the junction architecture connecting the ECs. Figure 1 illustrates
a simple example.

Figure 1: Example of an endothelial $\pi$-graph $G=(\mathcal{E},\pi)$. (A)
Schematic of an EC monolayer with ECs (labeled $x_{1},\dots,x_{5}$) connected
by adherens junctions (red), tight junctions (blue), and gap junctions
(green). (B) The $\pi$-graph representation of the monolayer, with
$\pi$-incidence displayed as junction sets associated to each cell. For
instance, $\pi(x_{1})={\sim_{AJ},\sim_{TJ}}$.

###  2.2 Elementary Properties

$\pi$-graphs satisfy a number of basic properties that reflect natural
features of endothelial networks. We highlight a few key properties here.

First, the local junction architecture around an EC is encoded by its
$\pi$-incidence:

###### Proposition 1

Let $G=(\mathcal{E},\pi)$ be a $\pi$-graph. For any $x\in\mathcal{E}$, the set
$\pi(x)$ uniquely determines the set of all ECs sharing a junction with $x$,
i.e.

${y\in\mathcal{E}:\exists\sim\in\pi(x)\textrm{ s.t. }x\sim
y}=\bigcup_{\sim\in\pi(x)}[x]{\sim}\setminus{x}$

where $[x]{\sim}$ denotes the equivalence class of $x$ under the relation
$\sim$.

###### Proof 1

Let $y\in\mathcal{E}$ with $y\neq x$. If $y$ shares a junction with $x$, then
there exists $\sim\in\pi(x)$ such that $x\sim y$. By definition of equivalence
class, this implies $y\in[x]\sim\setminus{x}$.

Conversely, suppose $y\in[x]{\sim}\setminus{x}$ for some $\sim\in\pi(x)$. By
definition of equivalence class, we have $x\sim y$, so $x$ and $y$ share a
junction. Therefore

${y\in\mathcal{E}:\exists\sim\in\pi(x)\textrm{ s.t. }x\sim
y}=\bigcup_{\sim\in\pi(x)}[x]\sim\setminus{x}$

as desired.

This property allows us to recover the local "junction neighborhood" of a cell
from its $\pi$-incidence alone. In biological terms, the junction neighborhood
reflects the set of all cells that can directly communicate with or
mechanically influence a given cell.

Next, we consider the global connectivity of $\pi$-graphs. Define a path of
length $n$ from $x$ to $y$ in $G$ as a sequence of cells
$x=x_{0},x_{1},\dots,x_{n}=y$ such that for each $0\leq i<n$, there exists
$\sim\in\pi(x_{i})\cap\pi(x_{i+1})$ with $x_{i}\sim x_{i+1}$. We say $G$ is
connected if there exists a path between any two cells:

###### Proposition 2

A $\pi$-graph $G=(\mathcal{E},\pi)$ is connected if and only if for all
$x,y\in\mathcal{E}$, there exists a path from $x$ to $y$.

Connectivity of the $\pi$-graph captures the ability of ECs to communicate
across the endothelial monolayer via a combination of junction types. This
communication may take the form of mechanical forces transmitted via AJs,
chemical and electrical signals propagated by GJs, or regulation of
paracellular transport through the coordination of TJs and NJs. In this sense,
$\pi$-graph connectivity abstractly reflects integrated endothelial function.

Finally, we define an operation called $\pi$-union that models the merging of
two $\pi$-graphs over a common subset of ECs:

###### Definition 4

Let $G_{1}=(\mathcal{E}_{1},\pi_{1})$ and $G_{2}=(\mathcal{E}_{2},\pi_{2})$ be
$\pi$-graphs with $\mathcal{E}_{1}\cap\mathcal{E}_{2}\neq\emptyset$. The
$\pi$-union of $G_{1}$ and $G_{2}$ is the $\pi$-graph
$G_{1}\sqcup_{\pi}G_{2}=(\mathcal{E}_{1}\cup\mathcal{E}_{2},\pi_{1\cup 2})$
where for $x\in\mathcal{E}_{1}\cup\mathcal{E}_{2}$,

$\pi_{1\cup 2}(x)=\begin{cases}\pi_{1}(x)&\text{if
}x\in\mathcal{E}_{1}\setminus\mathcal{E}_{2}\\\ \pi_{2}(x)&\text{if
}x\in\mathcal{E}_{2}\setminus\mathcal{E}_{1}\\\
\pi_{1}(x)\cup\pi_{2}(x)&\text{if
}x\in\mathcal{E}_{1}\cap\mathcal{E}_{2}\end{cases}$

Intuitively, the $\pi$-union glues together two $\pi$-graphs along their
common ECs and unions the $\pi$-incidences of the common ECs (Figure 2). This
models the formation of a larger endothelial tissue by merging two smaller
tissues with some overlapping cells. The notation $\sqcup_{\pi}$ was chosen to
suggest a modified disjoint union operation that preserves the common elements
and their $\pi$-incidences.

Figure 2: Schematic illustration of $\pi$-union. (A) Two $\pi$-graphs $G_{1}$
and $G_{2}$ with overlapping ECs $x_{3},x_{4}$ (purple). (B) The $\pi$-union
$G_{1}\sqcup_{\pi}G_{2}$ merges the graphs along the common cells. The
$\pi$-incidence of each common cell unions those from $G_{1}$ and $G_{2}$
(purple sets).

The $\pi$-union provides a natural way to build up complex $\pi$-graphs from
simpler building blocks, and to decompose $\pi$-graphs into components. One
can show that $\pi$-union is associative and commutative up to
$\pi$-isomorphism (defined in the next section), giving the set of
$\pi$-graphs the algebraic structure of a commutative monoid.

Having established the basic definition and properties of $\pi$-graphs, we
turn next to the notion of $\pi$-graph isomorphism to characterize when two
$\pi$-graphs have the same connectivity structure.

##  3 Results

###  3.1 $\pi$-graph Isomorphisms

A fundamental question in the construction of such $\pi$-graphs is when two
$\pi$-graphs have the same connectivity structure. To formalize this notion,
we introduce the concept of $\pi$-graph isomorphism:

###### Definition 5

Let $G_{1}=(\mathcal{E}_{1},\pi_{1})$ and $G_{2}=(\mathcal{E}_{2},\pi_{2})$ be
$\pi$-graphs. A $\pi$-graph isomorphism from $G_{1}$ to $G_{2}$ is a bijective
function $\varphi:\mathcal{E}_{1}\to\mathcal{E}_{2}$ such that for all
$x\in\mathcal{E}_{1}$ and $\sim\in\mathcal{J}_{\mathcal{E}_{1}}$,

$\sim\in\pi_{1}(x)\iff\varphi(\sim)\in\pi_{2}(\varphi(x))$

where $\varphi(\sim):={(\varphi(a),\varphi(b)):a\sim b}$.

If such an isomorphism exists, we say $G_{1}$ and $G_{2}$ are $\pi$-isomorphic
and write $G_{1}\cong_{\pi}G_{2}$. The set of all $\pi$-isomorphisms from
$G_{1}$ to $G_{2}$ is denoted $\text{Iso}_{\pi}(G_{1},G_{2})$.

Intuitively, a $\pi$-isomorphism is a bijection between the EC sets of two
$\pi$-graphs that preserves the $\pi$-incidence structure. The condition
$\sim\in\pi_{1}(x)\iff\varphi(\sim)\in\pi_{2}(\varphi(x))$ ensures that if two
ECs are connected by a junction in $G_{1}$, then their images under $\varphi$
are connected by the same type of junction in $G_{2}$, and vice versa.

Figure 3 illustrates an example of $\pi$-isomorphic graphs, with the dashed
arrows indicating the bijection $\varphi$ that preserves the color-coded
$\pi$-incidence structure.

Figure 3: Example of $\pi$-isomorphic graphs $G_{1}$ and $G_{2}$. The
bijection $\varphi:\mathcal{E}_{1}\to\mathcal{E}_{2}$ (dashed arrows)
preserves the $\pi$-incidence structure, i.e. $x_{i}\sim x_{j}$ in $G_{1}$ if
and only if $\varphi(x_{i})\sim\varphi(x_{j})$ in $G_{2}$ for each junction
type $\sim$. The coloring of the graph edges and corresponding junction type
is consistent with Figure 1.

The following proposition establishes basic properties of $\pi$-isomorphisms:

###### Proposition 3

Let $\mathbf{\varpi}$ be the class of $\pi$-graphs.

  1. 1.

For any $G\in\mathbf{\varpi}$, the identity function
$\text{id}_{\mathcal{E}}:\mathcal{E}\to\mathcal{E}$ is a $\pi$-isomorphism
from $G$ to itself.

  2. 2.

For any $\pi$-isomorphism $\varphi:G_{1}\to G_{2}$, the inverse function
$\varphi^{-1}:\mathcal{E}_{2}\to\mathcal{E}_{1}$ is a $\pi$-isomorphism from
$G_{2}$ to $G_{1}$.

  3. 3.

For any $\pi$-isomorphisms $\varphi:G_{1}\to G_{2}$ and $\psi:G_{2}\to G_{3}$,
the composition $\psi\circ\varphi:\mathcal{E}_{1}\to\mathcal{E}_{3}$ is a
$\pi$-isomorphism from $G_{1}$ to $G_{3}$.

Therefore, $\pi$-isomorphism defines an equivalence relation on
$\mathbf{\varpi}$.

###### Proof 2

(1) For any $x\in\mathcal{E}$ and $\sim\in\mathcal{J}_{\mathcal{E}}$, we have
$\sim\in\pi(x)\iff\text{id}_{\mathcal{E}}(\sim)\in\pi(\text{id}_{\mathcal{E}}(x))$
since $\text{id}_{\mathcal{E}}(\sim)=\sim$ and $\text{id}_{\mathcal{E}}(x)=x$.
Thus $\text{id}_{\mathcal{E}}$ is a $\pi$-isomorphism.

(2) Let $\varphi:(\mathcal{E}_{1},\pi_{1})\to(\mathcal{E}_{2},\pi_{2})$ be a
$\pi$-isomorphism. For any $y\in\mathcal{E}_{2}$ and
$\sim\in\mathcal{J}_{\mathcal{E}_{2}}$, we have

$\displaystyle\sim\in\pi_{2}(y)$
$\displaystyle\iff\varphi^{-1}(\sim)\in\pi_{1}(\varphi^{-1}(y))$
$\displaystyle\iff{(\varphi^{-1}(a),\varphi^{-1}(b)):a\sim
b}\in\pi_{1}(\varphi^{-1}(y))$

since $\varphi$ is a bijection. Thus $\varphi^{-1}$ is a $\pi$-isomorphism.

(3) Let $\varphi:(\mathcal{E}_{1},\pi_{1})\to(\mathcal{E}_{2},\pi_{2})$ and
$\psi:(\mathcal{E}_{2},\pi_{2})\to(\mathcal{E}_{3},\pi_{3})$ be
$\pi$-isomorphisms. For any $x\in\mathcal{E}_{1}$ and
$\sim\in\mathcal{J}_{\mathcal{E}_{1}}$, we have

$\displaystyle\sim\in\pi_{1}(x)$
$\displaystyle\iff\varphi(\sim)\in\pi_{2}(\varphi(x))$
$\displaystyle\iff\psi(\varphi(\sim))\in\pi_{3}(\psi(\varphi(x)))$
$\displaystyle\iff(\psi\circ\varphi)(\sim)\in\pi_{3}((\psi\circ\varphi)(x))$

Thus $\psi\circ\varphi$ is a $\pi$-isomorphism.

Properties (1)-(3) are the reflexive, symmetric, and transitive properties of
an equivalence relation[21], respectively. Therefore, $\pi$-isomorphism is an
equivalence relation on the class of $\pi$-graphs.

This proposition justifies the notation $G_{1}\cong_{\pi}G_{2}$ for
$\pi$-isomorphic graphs, and allows us to partition the class of $\pi$-graphs
into equivalence classes $[G]{\pi}={H\in\mathbf{\varpi}:H\cong{\pi}G}$
represented by a distinguished element $G\in\mathbf{\varpi}$.

###  3.2 Relationship to Standard Graph Isomorphism

We explore the relationship between $\pi$-isomorphism and the standard notion
of graph isomorphism. To make this relationship precise, we introduce an
"unnested" representation of an endothelial $\pi$-graph.

Given a $\pi$-graph $G=(\mathcal{E},\pi)$, define the unnested endothelial
junction graph (or simply unnested graph) of $G$ as
$U_{G}=(\mathcal{E},\bigcup_{x\in\mathcal{E}}\pi(x))$. In other words, $U_{G}$
is the graph whose vertices are the ECs of $G$ and whose edges represent the
individual junctions, without the $\pi$-incidence structure. Figure 4
illustrates this construction.

Figure 4: Unnested graph construction. (A) $\pi$-graph $G$ with ECs
$x_{1},\dots,x_{5}$ connected by multiple junction types (colored edges). (B)
The unnested graph $U_{G}$ retains the EC vertices but replaces the
$\pi$-incidences with individual junction edges. Distinct junction types
between two ECs yield multiple edges (e.g. the AJ and GJ between $x_{1}$ and
$x_{2}$).

We can now relate $\pi$-isomorphisms of $\pi$-graphs to standard graph
isomorphisms of their unnested graphs:

###### Proposition 4

Let $G_{1}=(\mathcal{E}_{1},\pi_{1})$ and $G_{2}=(\mathcal{E}_{2},\pi_{2})$ be
$\pi$-graphs with unnested graphs $U_{G_{1}}=(\mathcal{E}_{1},R_{1})$ and
$U_{G_{2}}=(\mathcal{E}_{2},R_{2})$, respectively. If
$\varphi:\mathcal{E}_{1}\to\mathcal{E}_{2}$ is a $\pi$-isomorphism from
$G_{1}$ to $G_{2}$, then $\varphi$ is a graph isomorphism from $U_{G_{1}}$ to
$U_{G_{2}}$.

###### Proof 3

Let $x,y\in\mathcal{E}_{1}$. We must show that $(x,y)\in R_{1}$ if and only if
$(\varphi(x),\varphi(y))\in R_{2}$. Suppose $(x,y)\in R_{1}$. By definition of
unnested graph, there exists $\sim\in\pi_{1}(x)$ such that $x\sim y$. Since
$\varphi$ is a $\pi$-isomorphism, this implies
$\varphi(\sim)\in\pi_{2}(\varphi(x))$ and $\varphi(x)\sim\varphi(y)$.
Therefore $(\varphi(x),\varphi(y))\in R_{2}$. The converse follows by a
symmetric argument.

This proposition shows that $\pi$-isomorphism is a stronger condition than
unnested graph isomorphism, i.e. $\pi$-isomorphic $\pi$-graphs necessarily
have isomorphic unnested graphs. However, the converse is not true in general:

###### Proposition 5

There exist $\pi$-graphs $G_{1}$ and $G_{2}$ such that $U_{G_{1}}$ and
$U_{G_{2}}$ are isomorphic as graphs but $G_{1}\not\cong_{\pi}G_{2}$.

###### Proof 4

We construct a counterexample. Let $\mathcal{E}={x_{1},x_{2},x_{3}}$ and
consider the following $\pi$-graphs:

$\displaystyle G_{1}$ $\displaystyle=(\mathcal{E},\pi_{1})\text{ where
}\pi_{1}(x_{1})={\sim_{AJ}},\pi_{1}(x_{2})={\sim_{AJ},\sim_{GJ}},\pi_{1}(x_{3})={\sim_{GJ}}$
$\displaystyle G_{2}$ $\displaystyle=(\mathcal{E},\pi_{2})\text{ where
}\pi_{2}(x_{1})={\sim_{AJ},\sim_{GJ}},\pi_{2}(x_{2})={\sim_{GJ}},\pi_{2}(x_{3})={\sim_{AJ}}$

and junction relations given by: $x_{1}\sim_{AJ}x_{2}$, $x_{2}\sim_{AJ}x_{1}$,
$x_{2}\sim_{GJ}x_{3}$, $x_{3}\sim_{GJ}x_{2}$, $x_{1}\sim_{AJ}x_{3}$. The
unnested graphs are:

$\displaystyle U_{G_{1}}$
$\displaystyle=(\mathcal{E},{(x_{1},x_{2}),(x_{2},x_{1}),(x_{2},x_{3}),(x_{3},x_{2})})$
$\displaystyle U_{G_{2}}$
$\displaystyle=(\mathcal{E},{(x_{1},x_{2}),(x_{2},x_{1}),(x_{1},x_{3}),(x_{3},x_{1})})$

which are isomorphic via the bijection $\varphi:\mathcal{E}\to\mathcal{E}$
defined by $\varphi(x_{1})=x_{1},\varphi(x_{2})=x_{3},\varphi(x_{3})=x_{2}$.
However, $G_{1}\not\cong_{\pi}G_{2}$ since there is no bijection
$\psi:\mathcal{E}\to\mathcal{E}$ that preserves the $\pi$-incidence structure,
i.e. $\sim\in\pi_{1}(x)\iff\psi(\sim)\in\pi_{2}(\psi(x))$ for all
$x\in\mathcal{E}$ and $\sim\in\mathcal{J}_{\mathcal{E}}$. Intuitively, this is
because the distribution of AJ and GJ among the ECs differs between $G_{1}$
and $G_{2}$, even though they have the same total number of junctions.

This proposition highlights a key feature of the $\pi$-graph formalism: it
captures connectivity information beyond what is represented by the unnested
graph. The additional information is precisely the $\pi$-incidence structure
that encodes which subsets of ECs are connected by each junction type.

Despite this negative result, there are some conditions under which
$\pi$-isomorphism and unnested graph isomorphism coincide:

###### Proposition 6

Let $G_{1}=(\mathcal{E}_{1},\pi_{1})$ and $G_{2}=(\mathcal{E}_{2},\pi_{2})$ be
$\pi$-graphs satisfying the following conditions:

  1. 1.

$\forall x,y\in\mathcal{E}_{i}$, $|\pi_{i}(x)\cap\pi_{i}(y)|\leq 1$ ($i=1,2$).

  2. 2.

$\forall x\in\mathcal{E}_{i}$, $\forall\sim,\approx\in\pi_{i}(x)$ with
$\sim\neq\approx$, $[x]{\sim}\cap[x]{\approx}={x}$ ($i=1,2$).

If $\varphi:\mathcal{E}_{1}\to\mathcal{E}_{2}$ is an unnested graph
isomorphism (i.e. a graph isomorphism from $U_{G_{1}}$ to $U_{G_{2}}$), then
$\varphi$ is also a $\pi$-isomorphism from $G_{1}$ to $G_{2}$.

###### Proof 5

Assume the hypotheses and let $\varphi:\mathcal{E}_{1}\to\mathcal{E}_{2}$ be
an unnested graph isomorphism. We must show that $\varphi$ preserves the
$\pi$-incidence structure, i.e. for all $x\in\mathcal{E}_{1}$ and
$\sim\in\mathcal{J}_{\mathcal{E}_{1}}$,

$\sim\in\pi_{1}(x)\iff\varphi(\sim)\in\pi_{2}(\varphi(x))$

($\implies$) Suppose $\sim\in\pi_{1}(x)$. By definition of $\pi_{1}$, there
exists $y\in\mathcal{E}_{1}$ with $y\neq x$ such that $x\sim y$. Since
$\varphi$ is an unnested graph isomorphism, this implies
$(\varphi(x),\varphi(y))\in R_{2}$, i.e. there exists
$\approx\in\pi_{2}(\varphi(x))$ such that $\varphi(x)\approx\varphi(y)$. We
claim that $\approx=\varphi(\sim)$.

Suppose for contradiction that $\approx\neq\varphi(\sim)$. Then by condition
(2), $[\varphi(x)]{\approx}\cap[\varphi(x)]{\varphi(\sim)}={\varphi(x)}$. But
$\varphi(y)\in[\varphi(x)]{\approx}$ by definition of $\approx$, and
$\varphi(y)\in[\varphi(x)]{\varphi(\sim)}$ since $y\in[x]{\sim}$ and $\varphi$
is a graph isomorphism. This contradicts
$[\varphi(x)]{\approx}\cap[\varphi(x)]_{\varphi(\sim)}={\varphi(x)}$.
Therefore $\approx=\varphi(\sim)$, so $\varphi(\sim)\in\pi_{2}(\varphi(x))$.

($\impliedby$) Conversely, suppose $\varphi(\sim)\in\pi_{2}(\varphi(x))$. By
definition of $\pi_{2}$, there exists $z\in\mathcal{E}_{2}$ with
$z\neq\varphi(x)$ such that $\varphi(x)\sim z$. Since $\varphi$ is a
bijection, $z=\varphi(y)$ for some $y\in\mathcal{E}_{1}$ with $y\neq x$.
Moreover, since $\varphi$ is an unnested graph isomorphism,
$(\varphi(x),\varphi(y))\in R_{2}$ implies $(x,y)\in R_{1}$, i.e. there exists
$\approx\in\pi_{1}(x)$ such that $x\approx y$.

We claim that $\approx=\sim$. Suppose for contradiction that
$\approx\neq\sim$. By condition (1), $|\pi_{1}(x)\cap\pi_{1}(y)|\leq 1$, so
$\approx$ is the unique junction relation between $x$ and $y$ in $G_{1}$. But
by the forward direction of the proof, $\approx\in\pi_{1}(x)$ implies
$\varphi(\approx)\in\pi_{2}(\varphi(x))$, and
$\varphi(\approx)\neq\varphi(\sim)$ since $\approx\neq\sim$ and $\varphi$ is
injective. This contradicts condition (1) for $G_{2}$, since
$\varphi(\sim),\varphi(\approx)\in\pi_{2}(\varphi(x))\cap\pi_{2}(\varphi(y))$.
Therefore $\approx=\sim$, so $\sim\in\pi_{1}(x)$.

The conditions in this proposition have a natural biological interpretation.
Condition (1) states that any two ECs share at most one type of junction.
Condition (2) states that distinct junction relations on a common EC connect
that EC to disjoint sets of ECs. In other words, the conditions preclude
having multiple redundant junctions between ECs and ensure that each junction
relation represents a unique connectivity pattern.

Under these biologically plausible assumptions[16], the $\pi$-graph and
unnested graph representations are equivalent up to isomorphism. This suggests
that for endothelial networks satisfying the assumptions, one can work with
the simpler unnested graph representation without losing connectivity
information. However, the $\pi$-graph representation is still valuable for
encoding the multiplicity of junction types and explicitly representing the
junction architecture around each EC.

###  3.3 Spatial Representation

We now introduce a framework to represent the spatial embedding and geometry
of $\pi$-graphs, extending the abstract connectivity structure to incorporate
biologically relevant spatial information. Our approach is based on the theory
of topological graphs and their embeddings into geometric spaces.

###  3.4 Topological $\pi$-graphs

We begin by defining a topological $\pi$-graph, which endows an abstract
$\pi$-graph with a topology inherited from its vertex and edge sets.

###### Definition 6

A topological $\pi$-graph is a tuple
$\mathcal{G}=(G,\mathcal{T}_{\mathcal{E}},\mathcal{T}_{\mathcal{J}})$ where:

  1. 1.

$G=(\mathcal{E},\pi)$ is an abstract $\pi$-graph.

  2. 2.

$\mathcal{T}_{\mathcal{E}}$ is a topology on the vertex set $\mathcal{E}$.

  3. 3.

$\mathcal{T}_{\mathcal{J}}$ is a topology on the set of junctions
$\mathcal{J}_{G}:=\bigcup{x\in\mathcal{E}}\pi(x)$.

  4. 4.

(Continuity) The $\pi$-incidence
$\pi:\mathcal{E}\to\mathcal{P}(\mathcal{J}_{G})$ is continuous with respect to
$\mathcal{T}_{\mathcal{E}}$ and the subspace topology on
$\mathcal{P}(\mathcal{J}_{G})$ induced by $\mathcal{T}_{\mathcal{J}}$.

Intuitively, a topological $\pi$-graph equips the vertex and edge sets of an
abstract $\pi$-graph with topologies that are compatible with the
$\pi$-incidence structure, in the sense that the $\pi$-incidence map is
continuous. This allows us to treat $\pi$-graphs as geometric objects and
study their topological properties.

A natural choice for the vertex topology $\mathcal{T}_{\mathcal{E}}$ is the
discrete topology, which captures the intuition that ECs are distinct,
separated objects. For the junction topology $\mathcal{T}_{\mathcal{J}}$,
there are several biologically motivated options, such as the discrete
topology (treating junctions as distinct objects), the Euclidean topology
(embedding junctions into Euclidean space), or a more general geometric
topology (e.g. representing junctions as simplicial complexes). The choice of
junction topology depends on the specific geometric properties one wishes to
model.

Figure 5 illustrates an example of a topological $\pi$-graph with discrete
vertex topology and Euclidean junction topology.

Figure 5: Example of a topological $\pi$-graph. (A) Abstract $\pi$-graph with
vertices $x_{1},\dots,x_{5}$ and junctions represented by colored edges. (B)
Topological $\pi$-graph embedding into $\mathbb{R}^{2}$. Vertices are embedded
as discrete points (black dots) and junctions are embedded as line segments
connecting vertices (colored lines). The $\pi$-incidence is represented by the
adjacency of vertex points and junction segments.

###  3.5 Spatial Embeddings of $\pi$-graphs

We now turn to the problem of embedding topological $\pi$-graphs into
geometric spaces in a way that respects the $\pi$-incidence structure and
junction geometry. We focus on embeddings into Euclidean space
$\mathbb{R}^{d}$, but the framework can be generalized to other spaces (e.g.
manifolds) as needed.

###### Definition 7

Let $\mathcal{G}=(G,\mathcal{T}_{\mathcal{E}},\mathcal{T}_{\mathcal{J}})$ be a
topological $\pi$-graph and $d\geq 1$. A spatial embedding of $\mathcal{G}$
into $\mathbb{R}^{d}$ is a pair of continuous maps
$(\varphi_{\mathcal{E}},\varphi_{\mathcal{J}})$ where:

  1. 1.

$\varphi_{\mathcal{E}}:(\mathcal{E},\mathcal{T}_{\mathcal{E}})\to\mathbb{R}^{d}$
embeds the vertices.

  2. 2.

$\varphi_{\mathcal{J}}:(\mathcal{J}_{G},\mathcal{T}_{\mathcal{J}})\to\mathcal{K}(\mathbb{R}^{d})$
embeds the junctions, where $\mathcal{K}(\mathbb{R}^{d})$ is the hyperspace of
nonempty compact subsets of $\mathbb{R}^{d}$ with the Vietoris topology.

  3. 3.

(Incidence Compatibility) For all $x\in\mathcal{E}$ and $j\in\mathcal{J}_{G}$,

$j\in\pi(x)\iff\varphi_{\mathcal{E}}(x)\in\varphi_{\mathcal{J}}(j)$

Most crucially, vertices are embedded as points in $\mathbb{R}^{d}$ via the
continuous map $\varphi_{\mathcal{E}}$, junctions are embedded as nonempty
compact subsets of $\mathbb{R}^{d}$ via the continuous map
$\varphi_{\mathcal{J}}$. The compactness requirement ensures that junctions
are bounded and closed, which are natural geometric constraints. The
hyperspace $\mathcal{K}(\mathbb{R}^{d})$ is equipped with the Vietoris
topology, which provides a notion of convergence for sequences of compact
sets, and the Incidence Compatibility condition ensures that the spatial
embedding respects the $\pi$-incidence structure, i.e. a vertex $x$ is
incident to a junction $j$ in the $\pi$-graph if and only if the embedded
vertex point $\varphi_{\mathcal{E}}(x)$ lies in the embedded junction region
$\varphi_{\mathcal{J}}(j)$.

Figure 6 illustrates spatial embedding for a simple $\pi$-graph.

Figure 6: Example of spatial embedding. (A) Topological $\pi$-graph with
discrete vertex topology and Euclidean junction topology. (B) Spatial
embedding into $\mathbb{R}^{2}$. Vertices are embedded as points and junctions
are embedded as line segments (AJ and GJ) or circular disks (TJ and NJ).
Incidence compatibility is depicted by vertex points lying in their incident
junction regions.

The definition of spatial embedding provides a general framework to represent
the geometry of $\pi$-graphs. By choosing appropriate vertex and junction
topologies and embedding maps, one can model a variety of biologically
relevant scenarios, such as:

  * •

ECs as point particles and junctions as line segments or polygonal chains
representing the physical connections between cells.

  * •

ECs as extended spatial regions (e.g. polygons or ellipsoids) and junctions as
overlap regions between cells.

  * •

ECs as point particles and junctions as probability distributions over
$\mathbb{R}^{d}$ representing the likelihood of a junction occurring at each
point in space.

In each case, the embedding maps $\varphi_{\mathcal{E}}$ and
$\varphi_{\mathcal{J}}$ can be tailored to capture the desired geometric
properties, such as continuity, smoothness, or distance constraints.

###  3.6 Topological Invariants

An important aspect of spatial embeddings is their behavior under continuous
deformations [22], which capture the idea of elastic transformations that
preserve intrinsic structure. This leads to the notion of topological
invariants, which are properties of spatial embeddings that are preserved by
continuous deformations.

###### Definition 8

Let $\mathcal{G}$ be a topological $\pi$-graph and let
$(\varphi_{\mathcal{E}},\varphi_{\mathcal{J}})$ and
$(\psi_{\mathcal{E}},\psi_{\mathcal{J}})$ be spatial embeddings of
$\mathcal{G}$ into $\mathbb{R}^{d}$. A continuous deformation from
$(\varphi_{\mathcal{E}},\varphi_{\mathcal{J}})$ to
$(\psi_{\mathcal{E}},\psi_{\mathcal{J}})$ is a pair of continuous maps
$(H_{\mathcal{E}},H_{\mathcal{J}})$ where:

  1. 1.

$H_{\mathcal{E}}:\mathcal{E}\times[0,1]\to\mathbb{R}^{d}$ is a homotopy from
$\varphi_{\mathcal{E}}$ to $\psi_{\mathcal{E}}$, i.e.
$H_{\mathcal{E}}(x,0)=\varphi_{\mathcal{E}}(x)$ and
$H_{\mathcal{E}}(x,1)=\psi_{\mathcal{E}}(x)$ for all $x\in\mathcal{E}$.

  2. 2.

$H_{\mathcal{J}}:\mathcal{J}_{G}\times[0,1]\to\mathcal{K}(\mathbb{R}^{d})$ is
a homotopy from $\varphi_{\mathcal{J}}$ to $\psi_{\mathcal{J}}$, i.e.
$H_{\mathcal{J}}(j,0)=\varphi_{\mathcal{J}}(j)$ and
$H_{\mathcal{J}}(j,1)=\psi_{\mathcal{J}}(j)$ for all $j\in\mathcal{J}_{G}$.

  3. 3.

(Incidence Compatibility) For all $x\in\mathcal{E}$, $j\in\mathcal{J}_{G}$,
and $t\in[0,1]$,

$j\in\pi(x)\iff H_{\mathcal{E}}(x,t)\in H_{\mathcal{J}}(j,t)$

If such a deformation exists, we say
$(\varphi_{\mathcal{E}},\varphi_{\mathcal{J}})$ and
$(\psi_{\mathcal{E}},\psi_{\mathcal{J}})$ are topologically equivalent
embeddings.

Intuitively, a continuous deformation is a continuous path in the space of
spatial embeddings that preserves the incidence compatibility condition at
each point along the path. Topological equivalence of embeddings is then the
equivalence relation generated by the existence of a continuous deformation.

A topological invariant of a spatial embedding is a property that is preserved
by topological equivalence. Figure 7 illustrates some examples of topological
invariants for simple $\pi$-graph embeddings.

Figure 7: Examples of topological invariants. (A) Two topologically equivalent
embeddings with the same connected component, cycle, and intersection pattern.
(B) Two embeddings that are not topologically equivalent, differing by the
presence of a cycle (left) versus a void (right).

The study of topological invariants provides a powerful tool to classify and
compare the intrinsic structure of spatial embeddings, independent of the
specific geometry. By computing invariants for different endothelial networks
and comparing them across conditions, one can gain insight into the
topological organization of the endothelium and how it relates to function.

###  3.7 Temporal Dynamics of $\pi$-graphs

To capture the dynamic nature of endothelial networks, we extend the
$\pi$-graph formalism by introducing a temporal dimension. Let
$T\subseteq\mathbb{R}$ be a time interval of interest. We define a temporal
$\pi$-graph as a tuple $\mathcal{G}=(G,\tau)$ where:

  1. 1.

$G=(\mathcal{E},\pi)$ is an abstract $\pi$-graph.

  2. 2.

$\tau:T\to\varpi$ is a continuous function that assigns a $\pi$-graph
$\tau(t)=(\mathcal{E}_{t},\pi_{t})$ to each time point $t\in T$.

Intuitively, a temporal $\pi$-graph represents the evolution of an endothelial
network over time, with the topology of the network encoded by the time-
varying $\pi$-incidence function $\pi_{t}$. We require that the vertex set
$\mathcal{E}_{t}$ and junction set $\mathcal{J}_{\mathcal{E}_{t}}$ vary
continuously with time to ensure well-defined dynamics.

The temporal $\pi$-graph formalism enables us to study the dynamics of
endothelial network connectivity, such as the formation and dissolution of
junctions, the migration of ECs, and the remodeling of network topology. By
considering the time-dependent properties of $\pi$-graphs, we can develop more
realistic models of angiogenesis, vascular permeability, and other dynamic
processes in the endothelium (Figure 8).

Figure 8: Schematic illustration of a temporal $\pi$-graph evolving over time.
The network topology, encoded by the $\pi$-incidence, changes continuously as
junctions form, dissolve, and remodel.

###  3.8 Evolution of Topological Invariants

The spatial embedding framework introduced in Section 3.6 naturally extends to
temporal $\pi$-graphs. We define a spatiotemporal embedding of a temporal
$\pi$-graph $\mathcal{G}=(G,\tau)$ into $\mathbb{R}^{d}$ as a pair of
continuous maps $(\Phi_{\mathcal{E}},\Phi_{\mathcal{J}})$ where:

  1. 1.

$\Phi_{\mathcal{E}}:\mathcal{E}\times T\to\mathbb{R}^{d}$ embeds the vertices.

  2. 2.

$\Phi_{\mathcal{J}}:\mathcal{J}_{\mathcal{G}}\times
T\to\mathcal{K}(\mathbb{R}^{d})$ embeds the junctions, where
$\mathcal{J}_{\mathcal{G}}:=\bigcup_{t\in T}\mathcal{J}_{\mathcal{E}_{t}}$ is
the set of all junctions across time.

The spatiotemporal embedding maps vertices and junctions into $\mathbb{R}^{d}$
continuously over time, respecting the $\pi$-incidence structure at each time
point.

With this framework in place, we can study the evolution of topological
invariants of endothelial networks. For instance, we can track the number of
connected components, cycles, or higher-dimensional holes as the network
evolves under physiological or pathological conditions (Figure 9). The
persistence of these invariants over time provides insight into the robustness
and adaptability of the network topology.

Figure 9: Evolution of topological invariants of an endothelial network
embedded in $\mathbb{R}^{3}$. The Betti numbers $\beta_{0}$, $\beta_{1}$, and
$\beta_{2}$, which count the number of connected components, cycles, and voids
respectively, change over time as the network remodels.

To quantify the evolution of topological invariants, we can employ techniques
from persistent homology, which provide a multiscale description of the
topology of a spatial embedding and its change over time. Persistent homology
computes the birth and death times of topological features as a filtration
parameter (such as a distance threshold or time) varies, yielding a barcode or
persistence diagram that summarizes the topological structure of the embedding
across scales.

By applying persistent homology to spatiotemporal embeddings of $\pi$-graphs,
we can track the emergence, persistence, and disappearance of topological
features in evolving endothelial networks. This provides a powerful tool to
compare the topological dynamics of different networks, detect critical
transitions or bifurcations, and identify topological biomarkers of vascular
dysfunction.

##  4 Discussion

In this paper, we have introduced a novel mathematical framework called
$\pi$-graphs to represent the complex multi-scale connectivity structure of
endothelial networks. $\pi$-graphs capture the essence of endothelial
connectivity by abstracting the detailed physiology of intercellular junctions
into a concise, mathematically tractable formalism based on set theory,
topology, and graph theory.

The key ingredients of the $\pi$-graph framework are:

  * •

A finite set $\mathcal{E}$ of endothelial cells (ECs).

  * •

A collection
$\mathcal{J}_{\mathcal{E}}={\sim_{AJ},\sim_{TJ},\sim_{GJ},\sim_{NJ}}$ of
symmetric binary relations on $\mathcal{E}$ representing different types of
intercellular junctions.

  * •

A $\pi$-incidence map
$\pi:\mathcal{E}\to\mathcal{P}(\mathcal{J}_{\mathcal{E}})$ assigning to each
EC the set of junctions it participates in, subject to basic consistency and
nondegeneracy conditions.

The $\pi$-incidence map is the heart of the formalism, encoding the local
junction architecture around each EC in a manner that is both biologically
expressive and mathematically elegant. In contrast to standard graph
representations, $\pi$-graphs natively capture the multiplicity of junction
types and their combinatorial arrangement around ECs.

We have developed the basic theory of $\pi$-graphs, including notions of
$\pi$-graph isomorphism, $\pi$-graph union, and $\pi$-graph connectivity, that
highlight the rich mathematical structure of the formalism. In particular, we
have shown that $\pi$-graph isomorphism is a strictly stronger notion than
isomorphism of the underlying "unnested" EC graphs obtained by forgetting the
$\pi$-incidence structure (Proposition 2). This underscores the importance of
the $\pi$-incidence in capturing the full complexity of endothelial
connectivity.

At the same time, we have established conditions under which $\pi$-graph
isomorphism and EC graph isomorphism coincide (Proposition 3), namely when any
two ECs share at most one type of junction and different junction types link
ECs to disjoint sets of neighbors. These conditions have a natural biological
interpretation and suggest that in certain regimes, one can work with the
simpler EC graph representation without losing essential connectivity
information.

To further ground the $\pi$-graph formalism in biological reality, we have
introduced a spatial representation framework that allows embedding
$\pi$-graphs into Euclidean space in a geometry-preserving manner. The key
idea is to represent ECs as points or regions in space and junctions as lines,
surfaces, or more general compact subsets, constrained by an incidence
compatibility condition that ensures the geometric embedding respects the
$\pi$-incidence structure.

The spatial embedding framework is highly flexible and can accommodate a
variety of biologically relevant geometric features, such as the size and
shape of ECs, the thickness and tortuosity of junctions, and the density of
junctions per unit volume. Mathematically, the framework draws on concepts
from topological graph theory and the theory of abstract cell complexes,
providing a rigorous foundation for studying the interplay between the
combinatorial connectivity and geometric arrangement of endothelial networks.

A central theme that emerges from the spatial embedding framework is the
importance of topological invariants - properties of embedded $\pi$-graphs
that are preserved by continuous deformations of the embedding. Examples of
key topological invariants include the number of connected components, the
number and type of cycles, and the homology groups that capture higher-
dimensional "holes" in the embedding.

The study of topological invariants provides a powerful tool to classify and
compare endothelial networks across different biological contexts, independent
of the precise geometric details. For instance, one could use topological
invariants to quantify the degree of network remodeling during angiogenesis,
or to characterize the topological defects that arise in pathological
conditions such as vascular leakage or tumor angiogenesis.

The introduction of a temporal dimension to the $\pi$-graph formalism and the
study of evolving topological invariants open up new avenues to investigate
the dynamics of endothelial networks. By representing the time-varying
connectivity and spatial embedding of endothelial networks, temporal
$\pi$-graphs provide a natural framework to model angiogenesis, vascular
remodeling, and other dynamic processes in the endothelium.

The application of persistent homology to track the evolution of topological
invariants offers a principled way to quantify the multiscale topology of
endothelial networks and its change over time. Persistent homology can detect
the birth, persistence, and death of topological features such as connected
components, cycles, and higher-dimensional holes, which may reflect the
formation, stability, and regression of vascular structures. By comparing the
persistence diagrams or barcodes of different endothelial networks, we can
gain insight into the topological basis of vascular function and dysfunction.

From a broader perspective, the temporal $\pi$-graph formalism and evolving
topological invariants contribute to the growing field of dynamical systems
and network evolution. Endothelial networks provide a rich biological context
to study the coupling between network topology, geometry, and dynamics, and to
explore the emergence of complex behavior from simple local rules. The
$\pi$-graph framework also has potential applications to other biological
systems that exhibit time-varying connectivity, such as neural networks,
cellular signaling pathways, and ecological communities.

To fully realize the potential of the temporal $\pi$-graph formalism, several
challenges and opportunities for future work remain. Some key directions
include:

  * •

Developing efficient algorithms to compute and update $\pi$-graphs and their
topological invariants from time-series data, such as live-cell imaging of
endothelial junctions.

  * •

Integrating temporal $\pi$-graphs with models of endothelial cell mechanics,
migration, and signaling to predict the dynamics of network remodeling and its
feedback on cell behavior.

  * •

Investigating the role of junction plasticity, EC heterogeneity, and
microenvironmental cues in shaping the temporal topology of endothelial
networks.

  * •

Applying the temporal $\pi$-graph framework to study the topological basis of
vascular patterning, anastomosis, and pruning during development and disease.

  * •

Comparing the topological dynamics of endothelial networks across different
tissues, species, and experimental conditions to identify conserved and
divergent features.

  * •

Constructing temporal $\pi$-graphs from experimental data, such as imaging of
junction protein localization or freeze-fracture electron microscopy, to
bridge the gap between molecular and tissue-scale dynamics.

The temporal $\pi$-graph formalism and evolving topological invariants in
particular provide a powerful lens to study the dynamics of endothelial
networks across scales. By integrating tools from algebraic topology,
dynamical systems, and network science, this framework can drive new
discoveries and applications in vascular biology and beyond. As the field of
mathematical biology continues to grow, we anticipate that temporal
$\pi$-graphs will find broader utility in modeling and analyzing the
structure, function, and dynamics of complex biological systems.

As the explosion of experimental data on endothelial network structure
continues, the $\pi$-graph formalism can provide a unifying mathematical
language to organize, analyze, and interpret this data in a principled way. At
the same time, the study of $\pi$-graphs as mathematical objects in their own
right may inspire new theoretical questions and constructions that enrich the
broader field of applied topology and network science.

In conclusion, the $\pi$-graph framework introduced here represents an
exciting new direction in the systems biology of the endothelium that can
bridge the gap between molecular-level mechanisms and tissue-level function.
By providing a rigorous mathematical foundation for studying the connectivity,
geometry, and topology of endothelial networks, $\pi$-graphs have the
potential to catalyze new discoveries and advances in vascular biology and
beyond. As such, we believe the $\pi$-graph formalism will be a valuable
addition to the rapidly growing toolkit of mathematical methods in biology and
medicine.

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