# Using Shallow Neural Networks with Functional Connectivity from EEG signals
for Early Diagnosis of Alzheimer’s and Frontotemporal Dementia

Zaineb Ajra1, Binbin Xu2, Gérard Dray2, Jacky Montmain2, Stéphane Perrey1

(1EuroMov Digital Health in Motion, Univ. Montpellier, IMT Mines Ales,
Montpellier, France.
2EuroMov Digital Health in Motion, Univ. Montpellier, IMT Mines Ales, Ales,
France.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Introduction:  Dementia is a neurological disorder associated with aging that
can cause a loss of cognitive functions, impacting daily life. Alzheimer’s
disease (AD) is the most common cause of dementia, accounting for 50–70% of
cases, while frontotemporal dementia (FTD) affects social skills and
personality. Electroencephalography (EEG) provides an effective tool to study
the effects of AD on the brain. Methods:  In this study, we propose to use
shallow neural networks applied to two sets of features: spectral-temporal and
functional connectivity using four methods. We compare three supervised
machine learning techniques to the CNN models to classify EEG signals of AD /
FTD and control cases. We also evaluate different measures of functional
connectivity from common EEG frequency bands considering multiple thresholds.
Results and Discussion:  Results showed that the shallow CNN-based models
achieved the highest accuracy of 94.54% with AEC in test dataset when
considering all connections, outperforming conventional methods and providing
potentially an additional early dementia diagnosis tool.
https://doi.org/10.3389%2Ffneur.2023.1270405

##  1 Introduction

Dementia is a neurodegenerative disease that results in the destruction of
nerve cells in the brain, leading to various symptoms that affect cognition,
emotion and movement [1]. It becomes more common as people age, and can have
many different effects, some of which may be reversible. Early-onset dementia,
which occurs before the age of 65 years, is most frequently caused by
Alzheimer’s disease (AD) or frontotemporal dementia (FTD). In contrast, late-
onset dementia occurs after the age of 65 [2]. AD is characterized by amnesia,
fluent aphasia, and visuospatial difficulties. On the other hand, FTD is
characterized by changes in personality and behavior [3]. AD affects neurons
and disrupts neurotransmitters responsible for storing memories and
transmitting messages, while FTD causes degeneration in the frontal and
anterior temporal lobes. However, neuropsychological tests aiming to
differentiate between FTD and AD often yield uncertain or contradictory
results [4]. It is crucial to distinguish between the two, as they impact
different cortical regions and exhibit distinct clinical findings.

Neuroimaging techniques have made a significant contribution to the
identification of AD and FTD [5]. Over the past two decades,
electroencephalography (EEG) has gained significant interest in clinical
practice and research as a non-invasive tool for diagnosing dementia and
determining its severity. EEG signals record the electrical activity of the
brain, and have the potential to serve as a biomarker for AD and other
neurodegenerative diseases [6]. EEG analysis involves extracting useful
information from EEG signals using various features such as time, frequency,
and time-frequency. The time domain analysis involves statistical features
such as mean, median, and standard deviation. The frequency domain features
involve decomposing the signal into different frequency sub-bands such as
delta, theta, alpha, beta, and Low-gamma. These features are commonly used in
machine learning algorithms for AD classification. Time-frequency features
involve using a spectrogram image as an alternative method for representing
the characteristics of raw EEG data. The spectrogram displays the variation in
energy values and frequency responses over time, using different magnitudes.
In comparison to other manual or supervised feature extraction techniques,
spectrograms are more effective in classifying signal/time-series data since
they include more unknown and valuable features [7].

Functional connectivity (FC) [8] is among the most commonly used techniques
for studying brain function. It aims to characterize the observational
similarity between different brain regions and how such similarity changes due
to patients’ pathology or even under different mental tasks. This kind of
analysis has been applied in various experimental contexts, ranging from high-
resolution fMRI data to more recently, EEG data, to provide more detailed
temporal information. There is mounting evidence suggesting that Alzheimer’s
disease and various psychiatric disorders are associated with disruptions or
enhancements in FC [9].

The aim of this work is to improve diagnostic accuracy of dementia by
exploring EEG signals using shallow Convolutional Neural Network (CNN) to
classify subjects with AD, FTD, and healthy control (HC) group. Our approach
involves feature extraction in both the time-frequency domain and functional
connectivity analysis. We also compared our results with conventional
classification methods, such as Linear discriminant Analysis (LDA), Support
Vector Machine (SVM), and K-nearest neighbors (kNN), which rely on temporal
and frequency feature extraction.

##  2  Data and models

###  2.1 Dataset

In this study, we explored the publicly available dataset collected by
Miltiadous et al. in [10]. The original dataset consisted of 19 EEG channels,
with a sampling rate of 500 Hz, recorded from the scalp of 88 participants,
including 36 AD patients, 23 FTD patients, and 29 healthy control subjects.
The mean age and standard deviation (SD) for the AD group were 66.4 (SD =
7.9), for the FTD group were 63.6 (SD = 8.2), and for the HC group were 67.9
(SD = 5.4). The cognitive decline and functional performance of patients with
AD were evaluated using the Mini Mental State Examination (MMSE) score. The
EEG signals were obtained from participants who were seated, relaxed, and had
their eyes closed, following a clinical protocol. The released data were
subjected to initial pre-processing steps, including band-pass filtering
within the frequency range of 0.5 to 45 Hz and the signals were re-referenced
to A1-A2. Then, Artifact Subspace Reconstruction (ASR) was applied to remove
bad data periods exceeding the maximum acceptable standard deviation of 17 for
the 0.5 second window. To further investigate signal enhancement, Independent
Component Analysis (ICA) was applied to the EEG signals, resulting in 19
distinct ICA components. As part of this process, elements classified as ’eye
artifacts’ or ’jaw artifacts’ were automatically identified and removed. The
average recordings duration for AD and FTD groups is approximately 13 minutes,
ranging from 11 to 17 minutes. The recordings for the HC subjects lasted for
an average of 13.8 minutes, ranging from 12.5 to 16.5 minutes.

The main question of this work is how to improve the accuracy of dementia
diagnosis by investigating EEG signals, so that we could achieve better
classification rate on three groups: AD, FTD, and HC. We focus on feature
extraction through functional connectivity methods using various thresholding
techniques. Additionally, the study compares the outcomes with CNN based time-
frequency analysis and traditional classification approaches as benchmarking
against state-of-the-art methods. The overall workflow of this study is
illustrated in Figure 1. In the following section, we will detail the feature
extraction steps and the classification methods.

Figure 1: Main workflow of this study.

###  2.2 Feature Extraction Methods

####  2.2.1 Spectral-temporal feature extraction

The time-frequency analysis allows to obtain richer information which are more
appropriate to neural network [11]. In the work of [10], the EEG data was at
first divided into epochs of 5 seconds with a 2.5 second overlap, features in
the time and frequency domains were extracted for the classification. In this
study, we use different spectral-temporal feature extraction method: the
preprocessed data was epoched of duration 10 seconds, then spectral-temporal
features are extracted using Fourier Transform and Hanning window tapering
approach with EEGLab (see examples in Figure 2).

Figure 2: Examples of the three classes (AD, FTD, HC), and three electrode
channels (Fz, Cz, Pz)

In total, $130\,150$ spectrograms (subjects $\times$ channels $\times$ epochs)
were extracted from the cleaned dataset. The choice of the spectral-temporal
features is supported by their ability to capture the temporal and spectral
information of EEG signals. This approach is consistent with the existing
literature, where similar methods have demonstrated effectiveness in
classifying various forms of dementia. In [12], the authors used a time-
frequency features by applying both Fourier and Wavelet transforms to classify
patients with Alzheimer’s disease versus healthy controls. This specific study
achieved $83\%$ accuracy using decision trees, highlighting the potential of
time-frequency features in dementia classification.

####  2.2.2 Connectivity feature extraction

##### Functional Connectivity Measures

Functional connectivity refers to the interactions between brain regions,
which can be quantified using measures of dependency between their temporal
dynamics. By thresholding the connectivity values between all pairs of brain
regions, one can identify functional connectivity networks that show which
brain regions interact with each other. In EEG-based functional connectivity
networks, nodes are represented by EEG channels, and links are connections
between channel pairs [9]. The relationship between channels can be quantified
using various methods, such as phase synchronization index (PSI), imaginary
part of coherency (ImCoh), Pearson correlation (Corr) and Amplitude Envelope
Correlation (AEC) [13] [14].

Functional connectivity features offer an effective way to extract relevant
information from EEG signals, as demonstrated by previous studies [15] and
[16] which used functional connectivity methods to discriminate between
different groups, including AD/HC, AD/FTD, and FTD/HC, using SVM models. These
studies provided valuable insights, with accuracies ranging from $72.2\%$ to
$87.67\%$. The use of functional connectivity features is motivated by their
ability to unveil complex interactions within EEG data, enabling the
identification of distinct patterns that can differentiate between various
neurological conditions. This study employed four functional connectivity
estimation methods, namely PSI, ImCoh, Corr and AEC to compare the
performances of time- and frequency-domain measures in classifying AD, FTD and
HC. For each participant, we generated a connectivity matrix between all pairs
of electrodes using the four functional connectivity measures separately. We
applied three thresholding strategies on each connectivity matrix:

  1. 1.

an absolute threshold with one commonly fixed value of 0.7,

  2. 2.

a proportional threshold varied from $10\%$ to $90\%$ with steps of $10\%$,
and

  3. 3.

no thresholding (raw connectivity matrix without any thresholding). These
resulting matrices were then compared to evaluate the effects of thresholding
on the classification performance.

We evaluated the effects of thresholding on classification performance by
comparing the resulting matrices. A total of 6850 connectivity matrices were
then generated for each functional connectivity method, and these matrices
were used for classification using a shallow Convolutional Neural Network.

###### Phase Synchronization Index

Phase Synchronization Index measures how two or more signals are synchronized
in terms of their phase relationship. It ignores the effect of amplitude and
detects the correlation between different signal pairs only takes only into
account the instantaneous phase relationship between the signals. Supposing
the instantaneous phases of two signals $x(t)$ and $y(t)$ being $\phi_{x}(t)$
and $\phi_{y}(t)$, then the phase synchronization index (PSI) is defined as:

$PSI=\frac{1}{T}\left|\sum_{t=1}^{T}\exp\\{j(\phi_{x}(t)-\phi_{y}(t))\\}\right|$
(1)

The PSI is sensitive to phase change and its value ranges from 0 to 1. The PSI
value of 1 indicates strict phase locking between the signals. On the other
hand, a value of 0 indicates that the phases are uniformly distributed and
there is no synchronization between the signals.

Examples of connectivity matrices with these four methods can be found in the
following Figure 3.

Figure 3: Examples of connectivity matrices assessed by Corr, ImCoh, PSI and
AEC for AD, FTD, and HC subjects, thresholded at 0.7 and _z-score_ normalized.

###### Imaginary Part of Coherency

Coherency is a standard method to determine the spectral similarity between
two signals. The coherency can be split into real and imaginary parts and
different quantities can be analyzed further. Volume conduction in EEG
recordings significantly affects coherence estimators. Electrical activity of
the cortex disparately spreads across scalp electrodes at some distance from
its generators allowing the same cortical activity to be measured by multiple
neighboring electrodes at the same time i.e., with zero phases [17]. This
method is calculated from the coherency measure and is defined by :

$ImCoh=\frac{\left|I_{\text{mag}}(P_{xy})\right|}{\sqrt{\left|P_{xx}\right|\left|P_{yy}\right|}}$
(2)

where $I_{\text{mag}}$ is the imaginary part of the power spectral density
between the x and y signals, $P_{xy}$ and $P_{yy}$ denote the power spectral
densities of x and y with themselves. This measure is between 0 and 1. If the
value is close to 1, it means that there is a real link between the two
signals. If the value is close to 0, it indicates that the signals are
independent. Indeed, this link is realized by the volume conduction and
therefore a false positive.

###### Pearson correlation coefficient

Correlation (Corr) is used to estimate the level of linear dependence between
two signals $x(t)$ and $y(t)$ in the time domain. The correlation is given by
the following:

$Corr=\frac{Cov(x,y)}{\sigma_{x}\sigma_{y}}$ (3)

where $Cov(x,y)$ is the covariance between electrodes $x$ and $y$;
$\sigma_{x}$ and $\sigma_{y}$ are the standard deviations of $x$ and $y$,
respectively. Corr value varies between 1 and $-1$, where 1 is a complete
positive linear correlation, 0 is no linear correlation, and $-1$ is a total
negative linear correlation. The greater the absolute value of Corr becomes,
the stronger the correlation is.

###### Amplitude Envelope Correlation

The Amplitude Envelope Correlation (AEC) [18] is a commonly used method to
measure the synchrony of cortical oscillations. This method involves
calculating the Amplitude Envelope (AE) of a given cortical oscillation, which
represents the energy fluctuations of the oscillation over time and is defined
as the absolute value of the Hilbert transform. The AEC is then computed by
correlating the AE of two oscillatory brain signals. High AEC values indicate
synchronous AE fluctuations between oscillations or networks. Therefore, the
AEC can provide important insights into the functional connectivity of
different brain regions.

##### Graph Analysis

The core idea of functional connectivity can actually be considered as a
network. One type of the most common tools to evaluate network is based on
graph theory. Graph theory utilizes a mathematical framework to represent the
connections between objects, where the objects are referred to as vertices,
and the links that connect them are known as edges. The application of Graph
theory to brain imaging data has demonstrated its potential as an
understandable and adaptable method for representing brain networks [9].
Regarding brain networks in sensor space, the vertices in a graph can
symbolize electrodes, and the edges can represent a certain measure of
connectivity between these electrodes. The most common metrics to evaluate a
graph are: mean degree (measures how interconnected the neighbors of a node
are), clustering coefficient (a measure of local connectivity), efficiency
(measures how efficiency information is transmitted across the graph) and
betweeness centrality (quantifies the importance of a node in facilitating
communication between other nodes). Usually, before computing graph metrics,
to eliminate the background noise or other perturbations, random threshold or
other justified thresholding is selected, and edge weights falling below this
threshold are then adjusted to zero (and removed from the final graph) [19].

####  2.2.3 Conventional feature extraction

Although the EEG signal’s complexity makes it challenging to achieve
clinically acceptable classification performance using feature engineering
alone, it is still worthwhile to benchmark the performance of conventional
machine learning models, as their computation costs are lower than those of
deep learning models. In this study, we extracted the conventional time and
frequency domain features from each epoch to setup the classification dataset.
Five basic EEG frequency bands are considered: delta, theta, alpha, beta, and
Low-gamma. For the frequency domain features, we calculated the energy of each
frequency band, while for the time-domain feature, we computed the minimum,
maximum, mean, median, variance, standard deviation, kurtosis, and skewness.
We considered these features because they have demonstrated their relevance in
the EEG-based dementia classification task and their potential to capture
distinct patterns. Many studies have used temporal analysis and energy
assessments of EEG rhythms. For instance,[20] used temporal features from EEG
signals (including maximum, minimum, mean, skewness, and kurtosis) as well as
changes in signal energy. The authors achieved $83.1\%$ accuracy in binary
classification (distinguishing Alzheimer’s patients from healthy controls)
using the random forest classifier. Furthermore, our selection of frequency-
domain features is supported by previous studies demonstrating their potential
as informative markers for Alzheimer’s disease classification. In particular,
[10], [21] and [22] have used dominant EEG frequency bands in their
classification frameworks. These approaches yielded remarkable accuracies in
distinguishing Alzheimer’s patients from healthy subjects or frontotemporal
dementia patients reaching $93\%$ for a binary classification.

###  2.3 Classification

####  2.3.1 Convolutional Neural Network based models

Convolutional neural network (CNN) models are powerful tools for automatic
feature extraction and classification, outperforming conventional machine
learning methods [11]. The effectiveness of CNN model design depends on
several factors, including the number of layers, configurations, and training
requirements. 2D CNN models excel at extracting two-dimensional features from
images, potentially leading to superior performance. In this study, we aim to
classify two types of input:

  1. 1.

two-dimensional spectral-temporal features of dimension $224\times 224$ from
EEG signals, which can be treated as images. This size is the common size of
state of the art CNN based models.

  2. 2.

functional connectivity matrices of dimension $19\times 19$ extracted from
each FC (19 EEG electrodes).

To achieve this, we propose a shallow CNN model consisting of two blocks of 2D
convolutional layers (Conv2D) that extract the most relevant features from the
inputs. Each Conv2D layer is followed by a batch normalization layer, which
speeds up training and improves model convergence. Both Conv2D layers use a
Rectified Linear Unit (ReLU) activation and are equipped with 50 filters with
a kernel size of $5\times 5$ to disrupt the network’s linear structure and
make it sparse. The two blocks of Conv2D layers are connected by a max pooling
layer with a size of $2\times 2$, which reduces the number of parameters in
the network and improves computational efficiency.

####  2.3.2 Multi-frequency bands functional connectivity classification

In our study, we utilized a weighted approach to construct functional
connectivity matrices. This involved applying a thresholding technique to the
generated connectivity matrices. Thresholding is a widely employed method in
research literature to eliminate weaker connections, as they are more
susceptible to experimental noise [23]. Two common approaches for thresholding
in functional connectivity analysis are the absolute threshold and the
proportional threshold. The absolute threshold approach involves choosing
edges with connectivity values exceeding a specified threshold value. On the
other hand, the proportional threshold approach involves selecting the
strongest percentage of connections within each network.

To classify the three groups (AD, FTD, and HC), we initially calculated the
functional connectivity within the three groups using four different methods.
We employ a thresholding technique on the resulting connectivity matrices,
retaining only the highest connectivity values. In the case of a proportional
threshold, we explore nine thresholding values ranging from $10\%$ to $90\%$
in increment of $10\%$. Subsequently, based on the CNN model classification
results, we determine the optimal threshold value from the nine options. Using
this optimal threshold value, we generate four functional connectivity
matrices in the four frequency bands (Theta (4–8 Hz), alpha (8–12 Hz), beta
(12–30 Hz) and Low-gamma (30-45 Hz)) for the functional connectivity methods
that showed the best classification performance (AEC, Corr and PSI). We then
compare the classification performance of the optimal proportional value with
that of the absolute threshold and with no thresholding applied.

####  2.3.3 Reference methods with Conventional machine learning algorithms

In order to further evaluate the performances of the studies shallow CNN
models, we also utilized conventional machine learning techniques to conduct
epoch-based classification of three groups: AD, FTD and healthy control group.
We utilized both temporal and frequency features in our approach. To address
three distinct classification problems, we tested three commonly used
classification algorithms in EEG studies: Linear Discriminant Analysis (LDA),
Support Vector Machines (SVM), and K-Nearest Neighbors (kNN). We used a
10-fold cross-validation testing method to evaluate the performance of each
algorithm. These three methods server as the baseline methods.

###  2.4 Training and evaluation

In summary, three feature sets were tested to discriminate between AD, FTD
patients, and HC subjects: (1) statistical features, (2) functional
connectivity features, and (3) time-frequency representation. Each set of
features was used independently as input for the CNN model, which was trained
using the same configuration for all sets of features. We utilized the
Stochastic Gradient Descent with Momentum (SGDM) optimizer, with a learning
rate fixed at $0.001$. The maximum number of training epochs was set to 50,
with a batch size of 128. The same early stopping rule was applied, with a
validation patience of 20 and a validation frequency of every 8 iterations.
Keeping the original subject / trial ratios from the raw data, samples from
all features sets were randomly split into training ($70\%$), validation
($15\%$) and test ($15\%$). To prevent over-fitting, 10 random triple-sets
were generated and used for the CNN models. The median values of four metrics
(Accuracy, Specificity, Sensitivity and F1-score) on test sets are reported
from the 50 epochs-long training.

The four specific quantitative indexes of the model’s classification
performance are as follows:

Accuracy $\displaystyle=\frac{TP+TN}{TP+TN+FP+FN}$ (4) Specificity
$\displaystyle=\frac{TN}{TN+FP}$ Sensitivity $\displaystyle=\frac{TP}{FN+TP}$
F1-score
$\displaystyle=2\times\frac{\text{Precision}\times\text{Sensitivity}}{\text{Precision}+\text{Sensitivity}}$

Where Precision is defined as:

$\text{Precision}=\frac{TP}{TP+FP}$ (5)

In these equations, TP denotes the correct classifications of positive cases,
TN denotes the correct classifications of negative cases, FP denotes the
incorrect classifications of negative cases into class positive, and FN
denotes the incorrect classifications of positive cases into class negative.

##  3  Results

###  3.1 Classification performances: absolute threshold

Table 1 presents the classification performance of different methods, where an
absolute threshold of 0.7 for FC was used, and median accuracy is reported.
The three conventional machine learning models, SVM, LDA, and kNN, were tested
under the same training conditions, and the results showed poor performance,
with a global median accuracy of only at $58.68\%$, $57.89\%$, and $59.77\%$,
respectively. These values were significantly lower than the results obtained
with the CNN model. The CNN-based models were able to achieve an accuracy of
$83.14\%$, $87.82\%$, $73.29\%$, $85.87\%$ and $81.97\%$ for Time-Frequency
analysis, PSI, ImCoh , Corr and AEC methods respectively. However, it is
important to note that different approaches may be necessary to determine the
brain regions involved since there’s no universal method for determining
functional connectivity [24]. In fact, identifying the optimal method to infer
these connections between brain regions remains a challenge in the field of
network neuroscience [25]. Regarding functional connectivity, the PSI method
provided the best performance in the CNN model.

Table 1: The classification performance of different methods with an absolute threshold of 0.7 for FC (median $\%$). TF denotes Time-Frequency. The bold values represent the best results obtained by the proposed CNN. Methods | ACC | Sensitivity | Specificity | F1 score
---|---|---|---|---
LDA | 57.89 | 35.93 | 68.03 | 35.91
SVM | 58.68 | 37.03 | 67.98 | 36.94
KNN | 59.77 | 35.41 | 67.77 | 33.93
CNN-TF | 83.14 | 82.41 | 91.28 | 82.66
CNN-PSI | 87.82 | 86.89 | 93.43 | 87.69
CNN-ImCoh | 73.29 | 71.07 | 86.87 | 72.60
CNN-Corr | 85.87 | 87.56 | 92.24 | 86.44
CNN-AEC | 81.97 | 84.83 | 90.15 | 83.24

Further result details regarding Table 1 can be found in Table 2. As observed,
the choice of the FC used to create the matrices of connections has a
significant impact on the overall predictive performance. For the AD group,
Corr, PSI, and AEC yielded the best performance in the classification task
with a sensitivity of $86.62\%$, $86.15\%$, and $85.21\%$, respectively.
Conversely, in the case of HC subjects, Corr outperformed the other methods.
Additionally, among all functional connectivity methods, the FTD group
exhibited the lowest classification performance, with sensitivities ranging
between $76.64\%$ and $66.39\%$. Given that there’s no universal approach to
deduce connections and achieve precise results, it is crucial to explore
various methods to establish a dependable framework for automatically
diagnosing dementia. Our proposed method, which involves using a matrix of
connections as input to a CNN, provides more accurate results than
conventional classification methods that require a prior feature engineering
step. Since EEG signals are highly complex in nature, conventional techniques
cannot always guarantee satisfactory classification performance. The reasons
behind the high performance of our CNN-based approach can be attributed to the
synergy between CNN and functional connectivity features. Functional
connectivity features, which incorporate the complex interactions between
distinct brain regions, offer a comprehensive representation of brain
dynamics. Conventional methods often struggle to take advantage of this lack
of information, relying primarily on manual feature engineering that can
ignore crucial patterns. In contrast, CNN’s hierarchical architecture excels
at detecting patterns that consistently align with the complexities inherent
in functional connectivity features. In addition, the integration of time-
frequency features into the CNN learning process enriches its ability to
understand the temporal and spectral aspects of EEG data. Overall, the
strength of the CNN model lies in its capacity to capture features from a
variety of data representations, eliminating the need for explicit feature
engineering, which is particularly beneficial for complex signals such as EEG.
The CNN’s flexibility and ability to adapt to different types of data
underline its superiority over traditional methods in this complex context.

Table 2: The performances by groups with different methods, threshold of 0.7 for FC (median $\%$) Models | Sensitivity | Specificity | F1 score
---|---|---|---
AD | HC | FTD | AD | HC | FTD | AD | HC | FTD
LDA | 29.06 | 41.68 | 36.06 | 74.66 | 63.23 | 66.16 | 27.90 | 42.95 | 36.89
SVM | 47.25 | 28.83 | 35.02 | 58.68 | 80.33 | 64.95 | 45.58 | 29.36 | 35.89
KNN | 38.67 | 57.21 | 10.34 | 63.38 | 46.28 | 93.66 | 37.69 | 48.63 | 15.48
CNN-TF | 85.81 | 83.30 | 77.67 | 88.02 | 90.68 | 94.69 | 84.84 | 83.15 | 79.95
CNN-PSI | 86.15 | 86.52 | 76.64 | 87.33 | 92.39 | 95.27 | 84.35 | 86.15 | 79.45
CNN-ImCoh | 77.58 | 71.91 | 66.39 | 81.50 | 84.93 | 92.20 | 75.99 | 71.91 | 69.68
CNN-Corr | 86.62 | 87.92 | 74.59 | 88.00 | 92.09 | 95.52 | 84.52 | 86.46 | 78.51
CNN-AEC | 85.21 | 84.27 | 73.36 | 85.67 | 91.12 | 95.14 | 82.63 | 83.73 | 77.14

###  3.2 Classification performances: proportional threshold

In functional connectivity analysis, absolute thresholding involves selecting
edges with connectivity values above a fixed threshold and setting other edges
to 0. This can lead to different network densities across subjects [19].
However, since the absolute threshold is fixed across all subjects, it may not
be suitable for datasets with varying connectivity strengths or individual
differences. To address this issue and ensure more comparable networks across
subjects, we adopted in this work a proportional threshold. Therefore, by
using a proportional threshold, we can achieve more consistent results and
better account for individual differences in connectivity strengths, leading
to a more reliable and interpretable comparison of functional connectivity
networks across subjects.

Table 3 presents the classification performance of different methods using a
proportional threshold of the top $20\%$ of connection values. For this
method, we applied thresholding to the obtained real-valued connectivity
matrices by selecting only the highest $20\%$ of connectivity values. This
method was used to ensure that we provide equal amounts of information to all
functional connectivity matrices. Indeed, absolute thresholding used
previously is a method of selecting a fixed threshold value to determine which
connections in a network should be considered significant. However, the
threshold selection is generally far from being an automated procedure,
especially when using different functional connectivity metrics. Using a fixed
threshold value across all analyses may not be appropriate for all methods and
could potentially bias the results. Proportional thresholding, on the other
hand, is a flexible thresholding method that adjusts the threshold value based
on the distribution of the data. In this method, a percentage of the highest
values of connections is selected as the threshold. The highest classification
performance is observed with the Corr method, achieving an accuracy of
$92.59\%$, as shown in this table. On the other hand, the ImCoh shows the
lowest classification performance with an accuracy of only $77.19\%$.

Table 3: The performances of different FC using a proportional threshold of keeping the highest $20\%$ of values (median $\%$). The bold values represent the best results obtained by the proposed CNN. Methods | ACC | Sensitivity | Specificity | F1 score
---|---|---|---|---
CNN-PSI | 87.23 | 89.04 | 93.28 | 88.30
CNN-ImCoh | 77.19 | 74.59 | 86.27 | 76.47
CNN-Corr | 92.59 | 94.38 | 95.22 | 92.82
CNN-AEC | 89.86 | 89.91 | 95.07 | 90.12

###  3.3 Classification performances: varying thresholds

In this section, we aimed to determine the most effective threshold for each
connectivity method to enhance the classification accuracy of the CNN model,
ultimately improving the accuracy and reliability of our functional
connectivity-based classification approach. To achieve this, we applied a
proportional thresholding on the obtained connectivity matrices from $10\%$ to
$90\%$ with steps of $10\%$. The performance of the CNN classifier in
discriminating AD, FTD, and HC with the four functional connectivity methods
is presented in Table 4. Notably, the Corr and AEC methods demonstrated the
highest classification performance, with accuracy values ranging from
$82.55\%$ to $92.59\%$ across the nine proportional thresholding values.
Particularly, we observe that retaining only $20\%$ of the strongest
connections yields the best classification performance, achieving an accuracy
of $92.59\%$ with Corr method. The comparison of the performance of functional
connectivity methods across various proportional thresholding values is
illustrated in Figure 4.

Table 4: Performance Evaluation of CNN Model with Proportional Thresholding (PT): Exploring 9 PT Values from 10% to 90% across Different Functional Connectivity Methods. The bold values represent the best results obtained by the proposed CNN. Threshold | Methods | ACC | Sensitivity | Specificity | F1 score
---|---|---|---|---|---
10% | CNN-PSI | 80.51 | 79.49 | 92.58 | 82.46
CNN-ImCoh | 75.15 | 72.75 | 87.46 | 74.11
CNN-Corr | 89.18 | 90.17 | 94.76 | 90.53
CNN-AEC | 85.19 | 86.52 | 93.73 | 86.91
20% | CNN-PSI | 87.23 | 89.04 | 93.28 | 88.3
CNN-ImCoh | 77.19 | 74.59 | 86.27 | 76.47
CNN-Corr | 92.59 | 94.38 | 95.22 | 92.82
CNN-AEC | 89.86 | 89.91 | 95.07 | 90.12
30% | CNN-PSI | 86.74 | 86.62 | 93.43 | 86.82
CNN-ImCoh | 76.12 | 76.97 | 85.67 | 74.36
CNN-Corr | 89.77 | 88.48 | 95.37 | 89.74
CNN-AEC | 89.96 | 88.20 | 96.27 | 90.36
40% | CNN-PSI | 87.62 | 87.08 | 94.18 | 87.94
CNN-ImCoh | 75.54 | 73.77 | 87.01 | 74.84
CNN-Corr | 91.03 | 92.42 | 95.82 | 91.73
CNN-AEC | 89.67 | 88.97 | 84.17 | 90.24
50% | CNN-PSI | 86.65 | 88.2 | 92.84 | 87.3
CNN-ImCoh | 77.68 | 75 | 86.57 | 75.63
CNN-Corr | 90.35 | 88.20 | 96.93 | 90.89
CNN-AEC | 89.38 | 89.89 | 94.63 | 89.89
60% | CNN-PSI | 84.8 | 85.92 | 91.83 | 85.25
CNN-ImCoh | 76.61 | 76.97 | 86 | 74.36
CNN-Corr | 88.79 | 88.76 | 93.88 | 88.64
CNN-AEC | 88.5 | 87.64 | 95.37 | 89.27
70% | CNN-PSI | 82.26 | 81.92 | 91.79 | 82.51
CNN-ImCoh | 77.88 | 73.88 | 89.85 | 76.56
CNN-Corr | 87.52 | 87.56 | 93.73 | 88.18
CNN-AEC | 86.26 | 86.85 | 94.33 | 86.45
80% | CNN-PSI | 82.46 | 80.9 | 92.97 | 83.43
CNN-ImCoh | 77 | 72.75 | 88.36 | 74.75
CNN-Corr | 85.96 | 87.09 | 92.69 | 86.89
CNN-AEC | 86.45 | 87.08 | 93.88 | 87.02
90% | CNN-PSI | 79.92 | 82.02 | 89.4 | 81.22
CNN-ImCoh | 78.17 | 76.12 | 87.16 | 77.16
CNN-Corr | 81.77 | 80.34 | 93.99 | 83.72
CNN-AEC | 82.55 | 80.05 | 90.50 | 82.77
Figure 4: Performance comparison of functional connectivity methods at
different proportional thresholding values.

###  3.4 Classification performances without any thresholding on connectivity
matrix

Table 5 reports the classification performance of different weighted FC
without any thresholding. Interestingly, using raw (non-thresholded)
connectivity matrices led to the highest classification performance, compared
to using thresholded matrices. This means that setting a threshold may cause
important information about connections between brain regions to be lost.
Discarding weaker connections by setting a threshold can reduce the
classifier’s accuracy in distinguishing between groups. Additionally, the
optimal threshold value can vary depending on the used FC method, making it
difficult to choose an appropriate threshold. Therefore, using non-thresholded
connectivity matrices may better represent the connectivity patterns and
capture subtle differences between groups. We compared four functional
connectivity methods and found that the AEC method was the most effective,
while the ImCoh method was the least effective, in distinguishing between AD,
FTD, and HC subjects.

Table 5: The performances of different weighted FC without any thresholding (median $\%$). The bold values represent the best results obtained by the proposed CNN. Methods | ACC | Sensitivity | Specificity | F1 score
---|---|---|---|---
CNN-PSI | 93.37 | 93.66 | 96.59 | 93.22
CNN-ImCoh | 77.10 | 78.09 | 85.50 | 75.85
CNN-Corr | 93.47 | 92.02 | 96.17 | 93.22
CNN-AEC | 94.54 | 95.22 | 96.72 | 94.56

Figure 5: Mean FC matrices across all epochs (upper) and topologically
significant local connectivity patterns among three groups: AD, FTD, and HC
(lower)

When we compare FC matrices using AEC among AD, FTD, and HC subjects, we
observed that FTD patients had stronger connections in frontal and temporal
regions than HC subjects (Figure 5). These results align with a study that
used fMRI to investigate functional connectivity in mild cognitive impairment
(MCI) and AD patients compared to healthy controls by [26]. They found that AD
patients had increased functional connectivity in the prefrontal cortex
compared to healthy controls. Additionally, FTD patients showed disruptions in
functional connectivity that were more widespread, particularly in regions
affected by the disease, such as the frontal and temporal lobes [27].

###  3.5 Comparison of FC methods across various frequency bands

The division of EEG signals into frequency bands is a widely adopted approach
in Alzheimer’s disease research due to the distinct neural activity patterns
observed at different frequency ranges. Certain frequency bands have been
associated with specific cognitive functions and brain regions affected by the
disease. For instance, abnormalities in the low-gamma frequency range have
been linked to memory impairment and cognitive decline, while changes in delta
and theta bands may indicate disruptions in attention and executive function.
By examining EEG signals in frequency bands, several studies [28, 29], aim to
capture these specific neural signatures related to Alzheimer’s disease
pathology, thus enhancing the ability of classification models to detect
disease-related patterns.

For the delta frequency band, the corresponding connectivity matrix has only
extremely high values which make the classification fail. Therefore, we focus
solely on the four other frequency bands (theta, alpha, beta, low-gamma) for
analysis.

Tables 6–8 present the classification performance of the CNN model across
different functional connectivity methods, considering various frequency bands
and thresholding techniques. Among the three threshold techniques, the CNN
classification model utilizing functional connectivity measures without any
thresholding demonstrated superior performance, surpassing the accuracy of the
model employing functional connectivity methods with both absolute and
proportional thresholds, achieving an accuracy of $60.33\%$.

Table 6: Performance comparison of FC methods across various frequency bands _without any thresholding_ (median $\%$). The bold values represent the best results obtained by the proposed CNN. FC method | Frequency band | ACC | Sensitivity | Specificity | F1 score
---|---|---|---|---|---
AEC | Theta | 52.92 | 58.15 | 68.66 | 53.56
Alpha | 51.66 | 52.11 | 69.33 | 52.37
Beta | 54.09 | 49.53 | 62.61 | 55.31
Low-gamma | 60.33 | 56.10 | 80.33 | 61.05
Corr | Theta | 47.86 | 32.58 | 83.73 | 39.93
Alpha | 50.97 | 29.49 | 88.96 | 39.25
Beta | 51.36 | 51.4 | 72 | 49.39
Low-gamma | 53.51 | 53.29 | 71.50 | 55.10
PSI | Theta | 52.24 | 47.47 | 76.72 | 49.63
Alpha | 54.58 | 47.89 | 78.33 | 53.68
Beta | 57.12 | 52.53 | 80.6 | 55.57
Low-gamma | 58.09 | 63.48 | 70.75 | 58.10
Table 7: Performance comparison of FC methods across various frequency bands _with an absolute threshold of 0.7_ (median $\%$). The bold values represent the best results obtained by the proposed CNN. FC method | Frequency band | ACC | Sensitivity | Specificity | F1 score
---|---|---|---|---|---
AEC | Theta | 48.64 | 48.03 | 68.96 | 46.53
Alpha | 52.05 | 52.11 | 71 | 53.74
Beta | 50.1 | 47.19 | 71.34 | 46.93
Low-gamma | 58.48 | 49.72 | 82.84 | 54.63
Corr | Theta | 51.46 | 50.56 | 70.6 | 49.11
Alpha | 46.98 | 50.41 | 74.3 | 43.31
Beta | 46.1 | 37.36 | 75.52 | 40.74
Low-gamma | 49.81 | 41.85 | 77.46 | 45.43
PSI | Theta | 48.83 | 42.02 | 75.17 | 47.48
Alpha | 50.39 | 42.96 | 75.67 | 48.48
Beta | 50.39 | 39.75 | 88.49 | 45.01
Low-gamma | 52.92 | 42.98 | 80.60 | 47.89
Table 8: Performance comparison of FC methods across various frequency bands _using a proportional threshold of keeping the highest $20\%$ of values_ (median $\%$). The bold values represent the best results obtained by the proposed CNN. FC method | Frequency band | ACC | Sensitivity | Specificity | F1 score
---|---|---|---|---|---
AEC | Theta | 44.74 | 30.28 | 82.33 | 39.09
Alpha | 42.11 | 38.2 | 65.67 | 37.67
Beta | 50.1 | 54.92 | 78.01 | 48.73
Low-gamma | 56.82 | 41.80 | 86.42 | 50.62
Corr | Theta | 46 | 27.93 | 85.67 | 37.72
Alpha | 45.32 | 22.47 | 84.33 | 29.57
Beta | 48.25 | 53.05 | 67 | 48.37
Low-gamma | 50.88 | 44.37 | 77.33 | 50.33
PSI | Theta | 45.61 | 29.81 | 84 | 39.14
Alpha | 36.55 | 36.52 | 71.94 | 38.58
Beta | 47.66 | 21.07 | 89.7 | 30
Low-gamma | 48.44 | 50.84 | 67.76 | 48.07

In terms of classification performance of the three groups, the EEG Low-gamma
band information exhibited the highest performance across all functional
connectivity measures. Notably, when using the AEC method without any
thresholding, a significant accuracy improvement was observed, resulting in an
accuracy of $60.33\%$. However, when comparing the classification of EEG
frequency bands and the whole EEG spectrum, we can conclude that the
classification of all EEG spectrum frequencies ranging from 0.5 to 45 Hz
outperformed the classification based on individual EEG frequency bands. This
suggests that considering the entire EEG spectrum provides valuable
information for accurate discrimination of the three groups - AD, FTD, and HC
- using the CNN model with functional connectivity measures.

###  3.6 Evaluation of metrics from Graph analysis

In addition to evaluating the performances of functional connectivity
classification using the proposed CNN model, we have explored an alternative
approach using the graph analysis, where graph metrics are employed to assess
functional connectivity. This approach allows us to gain insights into the
network-level properties and dynamics, providing a comprehensive understanding
of the brain’s functional organization in our study. Here, we chose to present
only the results of the Corr, PSI, and AEC methods for assessing graph
properties in each EEG frequency band across the three groups. The decision to
exclude the ImCoh method from the presentation of results is based on its
lower classification performance when compared to the other three methods for
all thresholding techniques and across all three groups. In order to analyze
the properties and characteristics of the brain network, Table 9 displays the
graph metrics for functional connectivity networks in different frequency
bands with an absolute threshold of 0.7 using AEC method for three distinct
groups: AD patients, HC and FTD subjects. The graph metrics provide valuable
insights into the topology and efficiency of brain networks in each group.
Figure 6 illustrates the network connectivity between 19 electrodes of AD, FTD
and HC groups on the Theta, Alpha, Beta and Low-gamma bands using AEC method
and with an absolute threshold of 0.7.

Figure 6: Connectivity network of the AD (left), FTD (middle) and HC (right) groups on the a) theta, b) alpha, c) beta and d) Low-gamma bands with an absolute threshold of 0.7 Table 9: Comparison of Graph Metrics using AEC Method with an absolute threshold in AD, HC, and FTD Groups. The bold values represent the best results achieved for different graph metrics.  Frequency bands | Mean Degree | Clustering Coefficient | Efficiency | Betweenness Centrality
---|---|---|---|---
| AD | HC | FTD | AD | HC | FTD | AD | HC | FTD | AD | HC | FTD
Theta | 11.89 | 10.95 | 17.05 | 0.93 | 0.89 | 0.91 | 0.65 | 0.62 | 0.83 | 13.05 | 13.15 | 2.94
Alpha | 13.68 | 15.89 | 12.84 | 0.92 | 0.99 | 0.98 | 0.68 | 0.80 | 0.68 | 5.36 | 5.26 | 10.73
Beta | 13.26 | 10.84 | 14.32 | 0.94 | 0.83 | 0.99 | 0.69 | 0.62 | 0.73 | 9.57 | 12.10 | 8.10
Low-gamma | 13.05 | 12.42 | 13.47 | 0.91 | 0.88 | 1.00 | 0.64 | 0.61 | 0.64 | 3.89 | 5.15 | 2.94

Regarding the clustering coefficient metric, in the Alpha frequency band, the
HC group shows the highest clustering coefficient, indicating a stronger
tendency for nodes to form clusters in their brain networks. However, in the
Beta and Low-gamma frequency bands, the FTD group have high clustering
coefficients, indicating significant clustering in their brain networks.
Concerning efficiency metric, the FTD group exhibits the highest efficiency in
the Theta and Beta frequency bands, implying that their brain networks
facilitate information transmission more effectively compared to the AD and HC
groups. With reference to betweenness centrality metric, in Theta, Beta and
Low-gamma frequency bands, the HC group consistently shows higher betweenness
centrality than the AD and FTD groups, indicating a relatively higher
influence of specific nodes in their brain networks.

Overall, the results suggest that the FTD group tends to have more
interconnected brain networks (higher mean degree) with efficient information
transmission (higher efficiency) compared to the AD and HC groups. On the
other hand, the HC group shows stronger clustering tendencies (higher
clustering coefficient) and more prominent nodes facilitating communication
between other nodes (higher betweenness centrality) in their brain networks.
These findings highlight the distinct characteristics of brain networks in
each group, potentially providing valuable insights into the underlying
neurodegenerative processes and the healthy brain’s functional organization.

###  3.7 Exploring the Clinical Implications of Functional Connectivity
Analysis

In this paper, our analysis of functional connectivity holds significant
promise for clinical applications in the early diagnosis and management of
dementia. By examining brain network interactions via EEG signals, we can
potentially identify subtle changes preceding cognitive decline, thus serving
as sensitive biomarkers for early detection. The high accuracy achieved by our
shallow neural network model in classifying AD/FTD/HC cases underlines the
diagnostic potential of functional connectivity analysis.

Figure 7: Relationship between MMSE Scores and Prediction Probabilities for
AD, HC, and FTD Groups, at trial level. All healthy subjects / patients
included.

In Figure 7, we illustrate the relationship between the prediction
probabilities of the AD, HC, and FTD groups and the MMSE scores identified in
the testing set results. In the AD group (left), a higher MMSE score was
linked to a lower prediction probability, while among HC subjects (middle), a
higher MMSE score was associated with an increased prediction probability. For
the FTD group (right), a higher prediction probability was observed when the
MMSE score ranged between 18 and 27. These findings align with clinical
observations, where a lower MMSE score typically indicates a more pronounced
cognitive decline. Notably, for the present dataset, the average MMSE score
was 17.75 (SD = 4.5) for the AD group, 22.17 (SD = 8.22) for the FTD group,
and 30 for the HC group.

Consequently, the classification probabilities assigned to each group could
serve as a measurable indicator of dementia progression, providing clinicians
with an objective index for monitoring disease progression and evaluating
treatment. This information could not only enhance clinical decision-making,
but also facilitate personalized treatment strategies and improve patient care
in the field of dementia. Beyond diagnosis, these results hold promise for
monitoring disease progression and evaluating treatment efficacy, offering
clinicians with a versatile set of tools to improve early detection and
patient-centered care in dementia disorders.

###  3.8 Compared with Previous Studies

Table 10 illustrates the comparison between the classification result in our
study and that of previous EEG studies. In recent years, several studies have
explored the classification of dementia using EEG, employing various
approaches. However, a particular study [30], stands out as it tested the
performance of SVM and LDA models on a group of 890 subjects during their
resting state. The study aimed to classify individuals into three levels:
healthy controls (HC), mild cognitive impairment (MCI), and Alzheimer’s
disease (AD). The SVM model, utilizing features extracted from EEG such as
absolute and relative power, Hajorth metrics, and time-frequency properties,
achieved the highest accuracy of $70.2\%$. It’s worth noting that most other
studies focused on binary classification between different groups, such as
AD/HC, FTD/HC, and AD/FTD.

Table 10: Comparison of the classification accuracy of AD patients with other previous studies Study | Feature set | Classifier | Classes | Best Acc
---|---|---|---|---
2023 [30] |  Absolute power, relative power, Hjorth metrics (activity, mobility and complexity) and time-frequency property (STFT) |  LDA, SVM |  AD/MCI/HC |  Acc = 70%, (LDA) Acc = 70.2% (SVM)
2023 [20] |  Temporal features from EEG (maximum, minimum, mean, skewness and kurtosis), signal energy changes and TEP peaks |  Random Forest |  AD/HC |  Acc = 83.1%
2023 [31] |  Functional connectivity features in source space from EEG |  eXtreme Gradient Boosting |  AD/HC AD/FTD |  Acc = 87.1% Acc = 86.7%
2022 [32] |  High order functional connectivity features from EEG source space |  Random Forest |  FTD/HC AD/HC |  Acc = 93.15% Acc = 89%
2022 [10] |  Energy of EEG rythms, mean, variance and IQR features |  Decision Trees, Random Forest, ANN, SVM, Naive Bayes and KNN |  AD/HC FTD/HC AD/FTD |  Acc = 99.1% Acc = 98% Acc = 91%
2021 [22] |  Hjorth parameters EEG frequency bands, using DWT |  SVM, RLDA and KNN |  AD/HC |  Acc = 97.64% (KNN)
2020 [15] |  Functional connectivity features |  SVM |  AD/HC |  Acc = 87.67%
2018 [12] |  Time-frequency features by applying both the Fourier and Wavelet transforms |  Decision Trees |  AD/HC |  Acc = 83%
2017 [16] |  Functional connectivity features |  SVM |  FTD/HC |  Acc = 72.7%
|  |  |  AD/FTD |  Acc = 72.2%
|  |  |  AD/HC |  Acc = 44.9%
2011 [33] |  EEG rythms energy from source space |  KNN |  FTD/HC AD/FHC FTD/AD |  Acc = 85.8% Acc = 92.8% Acc = 89.9%
2003 [21] |  EEG frequency bands features (from delta and theta bands) |  Logistic Regression |  AD/FTD |  Acc = 93.3%

##  4  Conclusion

In this work, we propose an automatic diagnosis method for AD, FTD, and HC
subjects using EEG time series and deep learning. We use four different
approaches to infer the matrix of connections between brain areas: Phase
Synchronization Index, Pearson’s correlation, Imaginary part of Coherency, and
Amplitude Envelope Correlation. These matrices are trained with CNN-based
models. In addition, we compared the performance of conventional methods,
including SVM, LDA, and KNN, with the shallow CNN model for different feature
extraction paradigms – time-frequency features and functional connectivity
features. Our comparisons revealed that the CNN model outperformed
conventional methods in dementia classification for both time-frequency and
functional connectivity features. The comparison of the four different
approaches shows that our CNN-based method is more accurate, demonstrating the
importance of network topology in describing brain data. Our findings revealed
that the CNN-AEC without any thresholding method is the most effective among
the methods we studied, reaching $94.54\%$ cross-validation accuracy. The
results suggest that EEG-based measures of functional connectivity, when
combined with convolutional neural network, provide an accurate, reliable and
rapid method of dementia classification and can significantly improve the
efficiency of AD diagnosis. The high performance of the basic CNN model
suggests that a simple neural network architecture may be adequate for
classifying dementia diseases. The pipeline is general and could be used for
other mental disorder in which EEG time series can be recorded. In future
work, it would be interesting to convert the EEG recordings captured at the
scalp level into EEG time series data in the source space using source
reconstruction. The CNN classifier will be assessed by comparing its
performance in both the sensor space and the source space.

## Acknowledgments

ZA received a doctoral fellowship from AXIAUM Université Montpellier-ISDM
(ANR-20-THIA-0005-01) and ED I2S in France.

## References

  * [1] A. M. d. Oliveira, M. Radanovic, P. C. H. d. Mello, P. C. Buchain, A. D. B. Vizzotto, D. L. Celestino, F. Stella, C. V. Piersol, and O. V. Forlenza, “Nonpharmacological interventions to reduce behavioral and psychological symptoms of dementia: a systematic review,” BioMed research international, vol. 2015, 2015.
  * [2] S. Lattante, S. Ciura, G. A. Rouleau, and E. Kabashi, “Defining the genetic connection linking amyotrophic lateral sclerosis (als) with frontotemporal dementia (ftd),” Trends in Genetics, vol. 31, no. 5, pp. 263–273, 2015\.
  * [3] M. F. Mendez, K. M. Perryman, B. L. Miller, and J. L. Cummings, “Behavioral differences between frontotemporal dementia and alzheimer’s disease: a comparison on the behave-ad rating scale,” International Psychogeriatrics, vol. 10, no. 2, pp. 155–162, 1998.
  * [4] R. J. Perry and J. R. Hodges, “Differentiating frontal and temporal variant frontotemporal dementia from alzheimer’s disease,” Neurology, vol. 54, no. 12, pp. 2277–2284, 2000.
  * [5] V. Mantzavinos and A. Alexiou, “Biomarkers for alzheimer’s disease diagnosis,” Current Alzheimer Research, vol. 14, no. 11, pp. 1149–1154, 2017.
  * [6] G. Biagetti, P. Crippa, L. Falaschetti, S. Luzzi, and C. Turchetti, “Classification of alzheimer’s disease from eeg signal using robust-pca feature extraction,” Procedia Computer Science, vol. 192, pp. 3114–3122, 2021.
  * [7] R. Ramos-Aguilar, J. A. Olvera-López, I. Olmos-Pineda, and S. Sánchez-Urrieta, “Feature extraction from eeg spectrograms for epileptic seizure detection,” Pattern Recognition Letters, vol. 133, pp. 202–209, 2020.
  * [8] K. J. Friston, “Functional and effective connectivity: a review,” Brain connectivity, vol. 1, no. 1, pp. 13–36, 2011.
  * [9] E. Bullmore and O. Sporns, “Complex brain networks: graph theoretical analysis of structural and functional systems,” Nature reviews neuroscience, vol. 10, no. 3, pp. 186–198, 2009.
  * [10] A. Miltiadous, K. D. Tzimourta, N. Giannakeas, M. G. Tsipouras, T. Afrantou, P. Ioannidis, and A. T. Tzallas, “Alzheimer’s disease and frontotemporal dementia: a robust classification method of eeg signals and a comparison of validation methods,” Diagnostics, vol. 11, no. 8, p. 1437, 2021.
  * [11] Z. Ajra, B. Xu, G. Dray, J. Montmain, and S. Perrey, “Mental arithmetic task classification with convolutional neural network based on spectral-temporal features from EEG,” in 2022 44th Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC) (Glasgow), pp. 52–55, July 2022.
  * [12] G. Fiscon, E. Weitschek, A. Cialini, G. Felici, P. Bertolazzi, S. De Salvo, A. Bramanti, P. Bramanti, and M. C. De Cola, “Combining eeg signal processing with supervised methods for alzheimer’s patients classification,” BMC medical informatics and decision making, vol. 18, no. 1, pp. 1–10, 2018\.
  * [13] A. Bruns, R. Eckhorn, H. Jokeit, and A. Ebner, “Amplitude envelope correlation detects coupling among incoherent brain signals,” Neuroreport, vol. 11, no. 7, pp. 1509–1514, 2000.
  * [14] S. Sun, X. Li, J. Zhu, Y. Wang, R. La, X. Zhang, L. Wei, and B. Hu, “Graph theory analysis of functional connectivity in major depression disorder with high-density resting state eeg data,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 27, no. 3, pp. 429–439, 2019.
  * [15] A. T. Adebisi, V. Gonuguntla, H.-W. Lee, and K. C. Veluvolu, “Classification of dementia associated disorders using eeg based frequent subgraph technique,” in 2020 International Conference on Data Mining Workshops (ICDMW) (Sorrento), pp. 613–620, IEEE, 2020.
  * [16] M. Dottori, L. Sedeño, M. Martorell Caro, F. Alifano, E. Hesse, E. Mikulan, A. M. García, A. Ruiz-Tagle, P. Lillo, A. Slachevsky, et al., “Towards affordable biomarkers of frontotemporal dementia: a classification study via network’s information sharing,” Scientific reports, vol. 7, no. 1, p. 3822, 2017.
  * [17] D. Sakellariou, A. M. Koupparis, V. Kokkinos, M. Koutroumanidis, and G. K. Kostopoulos, “Connectivity measures in eeg microstructural sleep elements,” Frontiers in Neuroinformatics, vol. 10, p. 5, 2016.
  * [18] J. F. Hipp, D. J. Hawellek, M. Corbetta, M. Siegel, and A. K. Engel, “Large-scale cortical correlation structure of spontaneous oscillatory activity,” Nature neuroscience, vol. 15, no. 6, pp. 884–890, 2012.
  * [19] M. Abazid, N. Houmani, J. Boudy, B. Dorizzi, J. Mariani, and K. Kinugawa, “A comparative study of functional connectivity measures for brain network analysis in the context of ad detection with eeg,” Entropy, vol. 23, no. 11, p. 1553, 2021.
  * [20] A.-M. Tăutan, E. P. Casula, M. C. Pellicciari, I. Borghi, M. Maiella, S. Bonni, M. Minei, M. Assogna, A. Palmisano, C. Smeralda, et al., “Tms-eeg perturbation biomarkers for alzheimer’s disease patients classification,” Scientific Reports, vol. 13, no. 1, p. 7667, 2023.
  * [21] M. Lindau, V. Jelic, S.-E. Johansson, C. Andersen, L.-O. Wahlund, and O. Almkvist, “Quantitative eeg abnormalities and cognitive dysfunctions in frontotemporal dementia and alzheimer’s disease,” Dementia and geriatric cognitive disorders, vol. 15, no. 2, pp. 106–114, 2003.
  * [22] M. S. Safi and S. M. M. Safi, “Early detection of alzheimer’s disease from eeg signals using hjorth parameters,” Biomedical Signal Processing and Control, vol. 65, p. 102338, 2021.
  * [23] C. v. Stam and E. Van Straaten, “The organization of physiological brain networks,” Clinical neurophysiology, vol. 123, no. 6, pp. 1067–1087, 2012\.
  * [24] A. M. Bastos and J.-M. Schoffelen, “A tutorial review of functional connectivity analysis methods and their interpretational pitfalls,” Frontiers in systems neuroscience, vol. 9, p. 175, 2016.
  * [25] S. G. Shandilya and M. Timme, “Inferring network topology from complex dynamics,” New Journal of Physics, vol. 13, no. 1, p. 013004, 2011.
  * [26] L. Penalba-Sánchez, P. Oliveira-Silva, A. L. Sumich, and I. Cifre, “Increased functional connectivity patterns in mild alzheimer’s disease: A rsfmri study,” Frontiers in Aging Neuroscience, vol. 14, 2022.
  * [27] A. Hafkemeijer, C. Möller, E. G. Dopper, L. C. Jiskoot, A. A. Van Den Berg-Huysmans, J. C. Van Swieten, W. M. Van Der Flier, H. Vrenken, Y. A. Pijnenburg, F. Barkhof, et al., “A longitudinal study on resting state functional connectivity in behavioral variant frontotemporal dementia and alzheimer’s disease,” Journal of Alzheimer’s Disease, vol. 55, no. 2, pp. 521–537, 2017.
  * [28] M. Abazid, N. Houmani, B. Dorizzi, J. Boudy, J. Mariani, and K. Kinugawa, “Weighted brain network analysis on different stages of clinical cognitive decline,” Bioengineering, vol. 9, no. 2, p. 62, 2022.
  * [29] C. Babiloni, K. Blinowska, L. Bonanni, A. Cichocki, W. De Haan, C. Del Percio, B. Dubois, J. Escudero, A. Fernández, G. Frisoni, et al., “What electrophysiology tells us about alzheimer’s disease: a window into the synchronization and connectivity of brain neurons,” Neurobiology of aging, vol. 85, pp. 58–73, 2020.
  * [30] B. Jiao, R. Li, H. Zhou, K. Qing, H. Liu, H. Pan, Y. Lei, W. Fu, X. Wang, X. Xiao, et al., “Neural biomarker diagnosis and prediction to mild cognitive impairment and alzheimer’s disease using eeg technology,” Alzheimer’s research & therapy, vol. 15, no. 1, pp. 1–14, 2023.
  * [31] P. Prado, S. Moguilner, J. A. Mejía, A. Sainz-Ballesteros, M. Otero, A. Birba, H. Santamaria-Garcia, A. Legaz, S. Fittipaldi, J. Cruzat, et al., “Source space connectomics of neurodegeneration: one-metric approach does not fit all,” Neurobiology of Disease, vol. 179, p. 106047, 2023.
  * [32] R. Herzog, F. E. Rosas, R. Whelan, S. Fittipaldi, H. Santamaria-Garcia, J. Cruzat, A. Birba, S. Moguilner, E. Tagliazucchi, P. Prado, et al., “Genuine high-order interactions in brain networks and neurodegeneration,” Neurobiology of Disease, vol. 175, p. 105918, 2022.
  * [33] K. Nishida, M. Yoshimura, T. Isotani, T. Yoshida, Y. Kitaura, A. Saito, H. Mii, M. Kato, Y. Takekita, A. Suwa, et al., “Differences in quantitative eeg between frontotemporal dementia and alzheimer’s disease as revealed by loreta,” Clinical Neurophysiology, vol. 122, no. 9, pp. 1718–1725, 2011\.


implies that $\ker\textrm{coker}f=\ker(F\to F_{Q})$ is isomorphic to
$\textrm{coker}\ker f=\textrm{coker}(0\to F_{1})=F_{1}$ if and only if
$T_{Q}=\ker(Q\to F_{Q})\simeq 0$, i.e. $Q\simeq F_{Q}$ is an object in
${\mathcal{F}}$.

An epimorphism $f\colon F\to F_{2}$ in ${\mathcal{F}}$ considered as a
morphism in ${\mathcal{A}}$ has cokernel $Q$ which is an object of
${\mathcal{T}}$. Let $K=\ker_{{\mathcal{A}}}f$. The short exact sequence
$0\to\textrm{Im}_{{\mathcal{A}}}f\to F_{2}\to Q\to 0$ in ${\mathcal{A}}$ shows
that $\ker\textrm{coker}f=\ker(F_{2}\to 0)=F_{2}$ is isomorphic to
$\textrm{coker}\ker f=\textrm{coker}(K\to F)=\textrm{Im}_{{\mathcal{A}}}f$ if
and only if $Q=0$, i.e. $f$ is an epimorphism in ${\mathcal{A}}$.

The statement for strict mono- and epimorphisms in ${\mathcal{T}}$ follows by
considering the opposite categories. ∎

###  A.3. Quasi-abelian categories and cotilting torsion pairs

Recall that an additive category ${\mathcal{E}}$ with kernels and cokernels is
_quasi-abelian_ if the family of kernel-cokernel sequences defines an exact
structure on ${\mathcal{E}}$. We say that it is the _canonical exact
structure_ and denote it by $({\mathcal{E}},{\mathcal{S}}_{\textrm{can}})$.

The following theorem gives (the only) examples of quasi-abelian categories.

######  Proposition A.4.

Let $({\mathcal{T}},{\mathcal{F}})$ be a torsion pair in an abelian category
${\mathcal{A}}$. Then

  1. (1)

category ${\mathcal{F}}$ is quasi-abelian and the canonical exact structure is
the one induced from ${\mathcal{A}}$,

  2. (2)

category ${\mathcal{T}}$ is quasi-abelian and the canonical exact structure is
the one induced from ${\mathcal{A}}$.

###### Proof.

In a kernel-cokernel sequence both morphisms are strict. Hence, by Proposition
A.3, kernel-cokernel sequences $F_{1}\xrightarrow{i}F\xrightarrow{d}F_{2}$, in
${\mathcal{T}}$ or in ${\mathcal{F}}$, correspond to short exact sequences
$0\to F_{1}\xrightarrow{i}F\xrightarrow{d}F_{2}\to 0$ in ${\mathcal{A}}$. It
follows that the family of kernel-cokernel sequences defines an exact
structure on ${\mathcal{T}}$ or on ${\mathcal{F}}$ and this exact structure is
induced from ${\mathcal{A}}$. ∎

By [Sch99, Proposition 1.2.32] any quasi-abelian category
$({\mathcal{E}},{\mathcal{S}}_{\textrm{can}})$ with the canonical exact
structure admits the right abelian envelope. The category
${\mathcal{A}}_{r}({\mathcal{E}},{\mathcal{S}}_{\textrm{can}})$ is defined as
the heart of a $t$-structure in the derived category of ${\mathcal{E}}$. More
precisely, ${\mathcal{A}}_{r}({\mathcal{E}},{\mathcal{S}}_{\textrm{can}})$ is
the full subcategory of the homotopy category of ${\mathcal{E}}$ with objects
complexes $0\to E_{1}\xrightarrow{d_{E}}E_{0}\to 0$ given by monomorphisms
$d_{E}$, localised by the multiplicative system formed by homotopy classes of
morphisms of complexes for which the square

$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{E}}$$\textstyle{E_{0}}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{F}}$$\textstyle{F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

is both cartesian and co-cartesian, [Sch99, Corollary 1.2.20].

Let $\mathscr{Q}uasi\textrm{-}ab$ denote the groupoid of quasi-abelian
categories with equivalences up to functorial isomorphisms as morphisms and
$\mathscr{C}otilt$ the groupoid of cotilting torsion pairs with equivalences
up to functorial isomorphisms as morphisms. Define

(25) $\varphi\colon\mathscr{C}otilt\to\mathscr{Q}uasi\textrm{-}ab$

which maps a cotilting torsion pair $({\mathcal{T}},{\mathcal{F}})$ to the
category ${\mathcal{F}}$. Proposition A.4 ensures that $\varphi$ is well-
defined.

######  Proposition A.5.

_cf._ [BVdB03, Proposition B.3] Functor $\varphi$ (25) is an equivalence. Its
quasi-inverse maps a quasi-abelian category ${\mathcal{E}}$ to the torsion
pair $({\mathcal{T}},{\mathcal{E}})$ in the right abelian envelope
${\mathcal{A}}_{r}({\mathcal{E}})$.

##  Appendix B Reconstructing a scheme $X$ from
$\mathop{\operatorname{Coh}}\nolimits(X)$ and
$\mathop{\textrm{QCoh}}\nolimits(X)$

For completeness we briefly recall how to reconstruct a Noetherian scheme $X$
from the category of coherent sheaves following [Gab62], _cf._ [Rou10].

First, we argue that one can equivalently reconstruct $X$ from
$\mathop{\textrm{QCoh}}\nolimits(X)$, as
$\mathop{\operatorname{Coh}}\nolimits(X)$ and
$\mathop{\textrm{QCoh}}\nolimits(X)$ determine each other.

Recall [KS06] that an _ind-object_ over a category ${\mathcal{C}}$ is an
object of
$\textrm{Fun}({\mathcal{C}}^{\mathop{\textrm{op}}\nolimits},\textrm{Sets})$
which is isomorphic to an inductive limit of representable functors. By [KS06,
Corollary 8.6.3] if ${\mathcal{C}}$ is a small abelian category then the
category of ind-objects is equivalent to the category
$\mathop{\textrm{Lex}}\nolimits({\mathcal{C}}^{\mathop{\textrm{op}}\nolimits},\mathop{\mathcal{A}\textrm{b}}\nolimits)$
of contravariant left exact functors from ${\mathcal{C}}$ to the category of
abelian groups.

Recall that an abelian category is _locally Noetherian_ if is has exact
inductive limits and a family of Noetherian generators, _cf._ [Gab62, Section
II.4]. By [Gab62, Theorem VI.2.1] the category of ${\mathcal{O}}_{X}$–modules
on a Noetherian scheme is locally Noetherian. Hence, so is its full
subcategory $\mathop{\textrm{QCoh}}\nolimits(X)$ of quasi-coherent
${\mathcal{O}}_{X}$–modules.

######  Proposition B.1.

Let $X$ be a Noetherian scheme. Then $\mathop{\textrm{QCoh}}\nolimits(X)$ is
equivalent to the category of ind-objects over
$\mathop{\operatorname{Coh}}\nolimits(X)$ while
$\mathop{\operatorname{Coh}}\nolimits(X)\subset\mathop{\textrm{QCoh}}\nolimits(X)$
is equivalent to the subcategory of Noetherian objects.

###### Proof.

Let us first show that
$\mathop{\operatorname{Coh}}\nolimits(X)\subset\mathop{\textrm{QCoh}}\nolimits(X)$
is the subcategory of Noetherian objects. Let $F$ be a coherent sheaf and
$X=\bigcup_{i=1}^{n}U_{i}$ an affine open cover of $X$. An ascending chain
$\\{F_{i}\\}_{i}\subset F$ stabilizes when restricted to any $U_{i}$, as
coherent modules over Noetherian rings are Noetherian (see [AM69, Proposition
6.5]). Hence, $\\{F_{i}\\}_{i}$ stabilizes.

Let now $F\in\mathop{\textrm{QCoh}}\nolimits(X)$ be a Noetherian object. We
check that for any open inclusion $j\colon U\to X$, the sheaf $j^{*}F$ is also
Noetherian. Given $\\{G_{i}\\}\subset j^{*}F$, $\\{j_{*}G_{i}\\}_{i}\subset
j_{*}j^{*}F$ and $\\{F_{i}:=F\times_{j_{*}j^{*}F}j_{*}G_{i}\\}\subset F$ are
ascending chains, where the fiber product is taken over the adjunction unit
$F\to j_{*}j^{*}F$. As $F$ is Noetherian, the chain $\\{F_{i}\\}_{i}$
stabilizes. Hence, so does $\\{G_{i}\simeq j^{*}F_{i}\\}$. In particular, if
$U\subset X$ is affine, $j^{*}F$ is Noetherian hence coherent. It follows that
$F\in\mathop{\operatorname{Coh}}\nolimits(X)$.

By [Gab62, Theorem II.4.1], given a Noetherian category ${\mathcal{C}}$, the
category
$\mathop{\textrm{Lex}}\nolimits({\mathcal{C}}^{\mathop{\textrm{op}}\nolimits},\mathop{\mathcal{A}\textrm{b}}\nolimits)$
is the unique up to an equivalence locally Noetherian category ${\mathcal{D}}$
such that ${\mathcal{C}}$ is equivalent to the category of Noetherian objects
in ${\mathcal{D}}$. As the category $\mathop{\textrm{QCoh}}\nolimits(X)$ is
locally Noetherian, the above argument shows that
$\mathop{\textrm{QCoh}}\nolimits(X)\simeq\mathop{\textrm{Lex}}\nolimits(\mathop{\operatorname{Coh}}\nolimits(X)^{\mathop{\textrm{op}}\nolimits},\mathop{\mathcal{A}\textrm{b}}\nolimits)$
is equivalent to the category of ind-objects over
$\mathop{\operatorname{Coh}}\nolimits(X)$. ∎

The reconstruction of a scheme from the category of (quasi-)coherent sheaves
relies on the notion of a spectrum of an abelian category ${\mathcal{A}}$. The
_spectrum_ $\mathop{\textrm{Spec}}\nolimits({\mathcal{A}})$ is the set of
isomorphism classes of indecomposable injective objects in ${\mathcal{A}}$. By
[Gab62, Theorem VI.1.1], for a Noetherian scheme $X$, indecomposable injective
objects in $\mathop{\textrm{QCoh}}\nolimits(X)$ are in bijection with the set
of irreducible closed subsets of $X$ (to an irreducible closed $Z\subset X$
the bijection assigns the injective envelope of ${\mathcal{O}}_{Z}$).

A localising subcategory ${\mathcal{B}}$ of an abelian category
${\mathcal{A}}$ allows one to represent
$\mathop{\textrm{Spec}}\nolimits({\mathcal{A}})$ as a disjoint union of
$\mathop{\textrm{Spec}}\nolimits({\mathcal{B}})$ and
$\mathop{\textrm{Spec}}\nolimits({\mathcal{A}}/{\mathcal{B}})$, see [Gab62,
Section IV.1]. We say that a localising subcategory
${\mathcal{B}}\subset{\mathcal{A}}$ is _of finite type_ if there exists an
object $M\in{\mathcal{A}}$ such that ${\mathcal{B}}$ is the minimal localising
subcategory containing $M$.

To a Grothendieck category ${\mathcal{A}}$ one assigns the ringed space
$R({\mathcal{A}})$ with $\mathop{\textrm{Spec}}\nolimits({\mathcal{A}})$ as
the underlying set. Open subsets in $R({\mathcal{A}})$ are of the form
$\mathop{\textrm{Spec}}\nolimits({\mathcal{A}}/{\mathcal{B}})$, for localising
subcategories of finite type ${\mathcal{B}}\subset{\mathcal{A}}$ while the
sheaf ${\mathcal{O}}$ is defined via centers,
${\mathcal{O}}(\mathop{\textrm{Spec}}\nolimits({\mathcal{A}}/{\mathcal{B}})):=Z({\mathcal{A}}/{\mathcal{B}})$.

######  Theorem B.2.

[Gab62] A Noetherian scheme $X$ is isomorphic to
$R(\mathop{\textrm{QCoh}}\nolimits(X))$.

If the scheme $X$ is separated of finite type over $k$, one can reconstruct
$X$ from $\mathop{\operatorname{Coh}}\nolimits(X)$ directly, _cf._ [Rou10].

Define a Serre subcategory ${\mathcal{B}}\subset{\mathcal{A}}$ to be _of
finite type_ if it is the smallest Serre subcategory in ${\mathcal{A}}$
containing $M$, for some object $M\in{\mathcal{A}}$. It is _irreducible_ if it
is not equal to zero and it is not generated by two proper Serre
subcategories.

By [Rou10, Theorem 4.1], the map
$Z\mapsto\mathop{\operatorname{Coh}}\nolimits_{Z}(X)$ is a bijection between
closed subsets $Z\subset X$ and Serre subcategories of finite type in
$\mathop{\operatorname{Coh}}\nolimits(X)$. Here
$\mathop{\operatorname{Coh}}\nolimits_{Z}(X)\subset\mathop{\operatorname{Coh}}\nolimits(X)$
is the full subcategory of sheaves supported on $Z$.

To an abelian category ${\mathcal{A}}$ one assigns the ringed space
$r({\mathcal{A}})$. The underlying set is the set of irreducible Serre
subcategories of finite type in ${\mathcal{A}}$. Open subsets in
$r({\mathcal{A}})$ are the sets $D({\mathcal{B}})$ of points of
$r({\mathcal{A}})$ not contained in a given Serre subcategory
${\mathcal{B}}\subset{\mathcal{A}}$ of finite type while the sheaf
${\mathcal{O}}$ is defined as
${\mathcal{O}}(D({\mathcal{B}})):=Z({\mathcal{A}}/{\mathcal{B}})$.

######  Theorem B.3.

[Rou10] A separated Noetherian scheme $X$ of finite type over $k$ is
isomorphic to $r(\mathop{\operatorname{Coh}}\nolimits(X))$.

## References

  * [AM69] M. F. Atiyah and I. G. Macdonald.  Introduction to commutative algebra.  Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969\.
  * [Art70] M. Artin.  Algebraization of formal moduli. II. Existence of modifications.  Ann. of Math. (2), 91:88–135, 1970.
  * [Art71] M. Artin.  Algebraic spaces, volume 3.  Yale University Press, New Haven, Conn.-London, 1971.  A James K. Whittemore Lecture in Mathematics given at Yale University, 1969.
  * [Aus86] M. Auslander.  Rational singularities and almost split sequences.  Trans. Amer. Math. Soc., 293(2):511–531, 1986.
  * [AZ13] I. Arzhantsev and M. Zaidenberg.  Acyclic curves and group actions on affine toric surfaces.  In Affine algebraic geometry, pages 1–41. World Sci. Publ., Hackensack, NJ, 2013.
  * [BB20] A. Bodzenta and A. Bondal.  Abelian envelopes of exact categories and highest weight categories.  arXiv: 2012.15707[math.RT], 2020.
  * [BBD82] A. A. Beĭlinson, J. Bernstein, and P. Deligne.  Faisceaux pervers.  In Analysis and topology on singular spaces, I (Luminy, 1981), volume 100 of Astérisque, pages 5–171. Soc. Math. France, Paris, 1982.
  * [Bre02] H. Brenner.  On superheight conditions for the affineness of open subsets.  J. Algebra, 247(1):37–56, 2002.
  * [BVdB03] A. Bondal and M. Van den Bergh.  Generators and representability of functors in commutative and noncommutative geometry.  Mosc. Math. J., 3(1):1–36, 258, 2003.
  * [CP21] J. Calabrese and R. Pirisi.  Gabriel’s theorem and birational geometry.  Proc. Amer. Math. Soc., 149(3):907–922, 2021.
  * [Gab62] P. Gabriel.  Des catégories abéliennes.  Bull. Soc. Math. France, 90:323–448, 1962.
  * [GD71] A. Grothendieck and J. A. Dieudonné.  Eléments de géométrie algébrique. I, volume 166 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 1971.
  * [Gro61] A. Grothendieck.  Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes.  Inst. Hautes Études Sci. Publ. Math., (8):222, 1961.
  * [GW10] U. Görtz and T. Wedhorn.  Algebraic geometry I.  Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010.  Schemes with examples and exercises.
  * [Har80] R. Hartshorne.  Stable reflexive sheaves.  Math. Ann., 254(2):121–176, 1980.
  * [Huy06] D. Huybrechts.  Fourier-Mukai transforms in algebraic geometry.  Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford, 2006.
  * [Kel90] B. Keller.  Chain complexes and stable categories.  Manuscripta Math., 67(4):379–417, 1990.
  * [KS06] M. Kashiwara and P. Schapira.  Categories and sheaves, volume 332 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 2006.
  * [MP14] S. Meinhardt and H. Partsch.  Quotient categories, stability conditions, and birational geometry.  Geom. Dedicata, 173:365–392, 2014.
  * [Nag56] M. Nagata.  A treatise on the 14-th problem of Hilbert.  Mem. Coll. Sci. Univ. Kyoto Ser. A. Math., 30:57–70, 1956.
  * [Pol07] A. Polishchuk.  Constant families of $t$-structures on derived categories of coherent sheaves.  Mosc. Math. J., 7(1):109–134, 167, 2007.
  * [Qui73] Daniel Quillen.  Higher algebraic $K$-theory. I.  pages 85–147. Lecture Notes in Math., Vol. 341, 1973.
  * [Rou10] R. Rouquier.  Derived categories and algebraic geometry.  In Triangulated categories, volume 375 of London Math. Soc. Lecture Note Ser., pages 351–370. Cambridge Univ. Press, Cambridge, 2010\.
  * [Sch99] J.-P. Schneiders.  Quasi-abelian categories and sheaves.  Mém. Soc. Math. Fr. (N.S.), (76):vi+134, 1999.
  * [Sch00] S. Schröer.  On contractible curves on normal surfaces.  J. Reine Angew. Math., 524:1–15, 2000.
  * [Sta22] The Stacks Project Authors.  Stacks project, 2022.
  * [SV04] S. Schröer and G. Vezzosi.  Existence of vector bundles and global resolutions for singular surfaces.  Compos. Math., 140(3):717–728, 2004.
  * [Yos90] Yu. Yoshino.  Cohen-Macaulay modules over Cohen-Macaulay rings, volume 146 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 1990.