# Deep Dynamic Effective Connectivity Estimation from Multivariate Time Series

Usman Mahmood 1,2 1, Zening Fu1,2, Vince Calhoun1,2,3,4 and Sergey Plis1,2 1
Tri-institutional Center for Translational Research in Neuroimaging and Data
Science (TReNDS),
2 Georgia State University, 3 Georgia Institute of Technology, 4Emory
University
Correspondence<EMAIL_ADDRESS>
###### Abstract

Recently, methods that represent data as a graph, such as graph neural
networks (GNNs) have been successfully used to learn data representations and
structures to solve classification and link prediction problems. The
applications of such methods are vast and diverse, but most of the current
work relies on the assumption of a static graph. This assumption does not hold
for many highly dynamic systems, where the underlying connectivity structure
is non-stationary and is mostly unobserved. Using a static model in these
situations may result in sub-optimal performance. In contrast, modeling
changes in graph structure with time can provide information about the system
whose applications go beyond classification. Most work of this type does not
learn effective connectivity and focuses on cross-correlation between nodes to
generate undirected graphs. An undirected graph is unable to capture direction
of an interaction which is vital in many fields, including neuroscience. To
bridge this gap, we developed dynamic effective connectivity estimation via
neural network training (DECENNT), a novel model to learn an interpretable
directed and dynamic graph induced by the downstream classification/prediction
task. DECENNT outperforms state-of-the-art (SOTA) methods on five different
tasks and infers interpretable task-specific dynamic graphs. The dynamic
graphs inferred from functional neuroimaging data align well with the existing
literature and provide additional information. Additionally, the temporal
attention module of DECENNT identifies time-intervals crucial for predictive
downstream task from multivariate time series data.

###### Index Terms:

dynamic ENC, fMRI, FNC, time series

##  I Introduction

Many classification/prediction problems can be solved by learning the
underlying structure/pattern of the data and how different components are co-
related with each other. Datasets from different fields are often represented
as a graph. Graph networks [1, 2] are proposed to work on such datasets.
Recently, methods such as, graph neural networks (GNNs) have been extensively
used to learn representations on graph-structured data [3, 4, 5, 6]. GNNs take
nodes from data and update representations of nodes with the help of different
aggregating functions. The aggregate functions work using a message-passing
system, where a node receives messages from its neighbors, which are defined
by edges. The representations can then be used for node classification, graph
classification, or predicting edges between nodes by using an existing true
graph structure or learning the graph [7, 8, 9, 5, 10, 11, 12, 13, 14]. For
any of the mentioned tasks, most of the existing work (classification, link
prediction) has been done on static graphs, e.g., Parisot et al. [6], creates
a static graph based on representation and phenotype information of subjects,
Mahmood et al. [15] learns a static graph between brain regions, Kipf et al.
[13] learns a static graph in an interacting system. In reality, many fields
(social networks, brain connectivity, traffic data, speech) are dynamically
changing and cannot be completely represented using a static graph. We propose
that learning a dynamic graph for such systems can increase our understanding
of these highly dynamic systems and may also yield higher classification
performance based on the task. For example, learning the dynamic connectivity
of the brain’s networks can help researchers to understand brain dynamics and
the causes of brain disorders by learning how connectivity changes while
performing tasks, or with age. Dynamic graphs for social network data can help
to understand users’ patterns, peak traffic times, retention time, and many
other vital aspects of the network.

Even though graph networks have excelled in many areas, we see a couple of
shortcomings in the current work regarding graph-structured data. 1) Most of
the work done, whether for graph classification (node or graph) or graph
learning (link prediction), work on un-directed graphs and assume that the
graph structure of the data is available or easily created and thus work
directly on the graph-structured data [16, 5]. This assumption is improbable
in many different datasets across many fields. 2) The semi or unsupervised
methods developed to learn the graph structure (link-prediction) create
embeddings/representations based on the learned graph structure, and use these
embeddings to either predict future embeddings or perform classification tasks
where loss is the error in prediction or classification [13, 17, 15]. The
problem with this approach is that the embeddings are used for
prediction/classification. The learned structure is not tested directly,
especially where the true graph structure is never available (e.g., brain
functional network connectivity (FNC)) and thus is unreliable. The
unreliability increases in systems with relatively easy tasks and noiseless
data (real or simulated). Kipf et al. [13] shows that using a full graph leads
to almost the same or better performance (in terms of loss) as the learned
static graph, thus questioning the correctness/importance of the learned graph
structure. 3) The graph structure is assumed to be static, which is highly
unlikely in many datasets such as a) social network, where a node can
join/leave at any time or create/drop an edge. b) Brain functional
connectivity, where the connectivity between brain regions is always dynamic.
Xu et al. [18] shows that using a static graph learning method for a dynamic
system/graph can lead to lower classification performance, Kipf et al. [13]
shows improved results by just dynamically re-evaluating the static learned
graph during testing. The improved performance for the relevant task is
understandable as the dynamic connectivity provides essential information
about the system.

Classification gains by using a dynamic graph depend on the downstream task;
e.g., a social network’s dynamic graph may not be too helpful to predict a
user’s gender but can provide additional information, which itself is
extremely important for understanding the dynamic system and its working. For
example, studies like [19, 20, 21] show that dynamic functional connectivity
(FC) show re-occurring patterns which cannot be captured in static FC. Thus,
we want to learn dynamic and directed graph structure representing the time
series data and use that for interpretation, understanding and prediction
tasks.

We present a novel method called - dynamic effective connectivity estimation
via neural network training - (DECENNT). DECENNT is a semi-supervised method
(unknown graph structure, known graph labels) which we use to 1) learn dynamic
directed graph structure using embeddings for better understanding of the
underlying system and 2) perform graph classification based on the learned
graph structure/connectivity alone. We propose that high classification
results based on the learned graph structure alone and not the representations
greatly reduce the uncertainty regarding the usefulness of the learned
structure and produce more useful and reliable graphs, which is the key
objective of our study. These graphs can then be used for understanding the
underlying system and interpreting the cause(s) of classification/prediction.

Recently, graph structure has been used to represent the brain. Brain
connectivity is highly dynamic and changes with functionality being performed.
Understanding the dynamic functionality would help to understand the
functionality and connectivity of the brain. Thus, we apply our model to learn
the brain’s dynamic effective connectivity (EC) using the functional magnetic
resonance imaging (fMRI) data. fMRI is an imaging method used to capture
blood-oxygenation level dependent (BOLD) signals in the brain, which measures
neural activity between brain regions.

In this study, we learn dynamic EC of the brain and use that to 1) predict the
presence of a disease or predict the gender of the subject and 2) learn
dynamics of brain networks’ connectivity related to the downstream task. Many
recent studies have been proposed to learn or compute FC of the brain [22, 23,
24, 25, 26] and use it to predict the gender or disease/disorder [27, 22, 28,
23, 26, 25, 29] using GNNs or other such methods but have limitations as
discussed above. Methods that incorporate dynamic FC (either learned or
computed) are mostly window-based [30, 31, 32, 33], which partition the data
into multiple windows, each consisting of data from multiple time-points. As
the structure/connectivity can change at any time, we create an instantaneous
dynamic structure. Existing studies most commonly use symmetric estimates of
relationship, e.g., Pearson correlation coefficients (PCC) matrix, to
represent FC. The symmetric correlation matrix does not capture effective
connectivity, and does not reveal the direction of flow of information. To
incorporate direction, some studies [34, 35] use methods like transfer entropy
[36] or Granger causality [37]. The former is notoriously difficult to
estimate reliably, while the latter has problems with latent confounders.
Moreover, these approaches do not directly fit the requirements for an
interpretable differential layer.

We also incorporate temporal attention in our model to provide better
interpretable results and further understand the working of brain
functionality. We apply our model to both resting state-fMRI (rs-fMRI), where
the essential time-points (putative biomarkers) are not known and to speech
data. In the latter, we predict the presence of a specific target word in the
speech and use the attention module to mark the time-points of the occurrence.

Contributions: Our study has the following contributions.

  1. 1.

Without the availability brain’s true E/FC structure, by using fMRI data, we
learn a directed connectivity structure of the brain that provides additional
details than existing literature. Thus, it removes the need to use a separate
method to compute connectivity before applying classification.

  2. 2.

Based on the learned dynamic EC, our model outperforms other SOTA methods in
classification tasks (disorders, gender, and keyword detection) across
multiple datasets and pre-processing.

  3. 3.

Our temporal attention module finds the essential time-points for the
downstream task with very high accuracy and is stable/consistent across
multiple trials. It improves classification performance and finds important
bio-markers related to the downstream task. This in turn can lead to reducing
the temporal dimensions and discarding time-points that are unrelated to the
downstream task.

##  II DECENNT

We use the proposed DECENNT model to learn a dynamic directed graph
structure/pattern for any multivariate time series data. The dynamic directed
graph is essential to learning and understanding the system and can be used in
different ways to perform classification on the downstream task. We learn a
distinct dynamic graph $G$ for the complete time series where $G$ is a set of
$T$ graphs with $T$ being the total time-points of the time series. We define
G as:

$G=\\{g_{1},....g_{t},....g_{T}|T=$ time-points} and $g_{t}=(V_{t},E_{t})$,
where, $V_{t}$ and $E_{t}$ represent the vertices and edges present at time-
point $t$. After computing the set $G$ we use our temporal attention module to
focus on the important time-points and generate a single final graph $G_{f}$
representing the complete time series. $G_{f}$ is used for downstream
classification. To create the embedding $h_{i,t}$ for the $i_{th}$ component
at time $t$ we use a bidirectional long short-term memory (biLSTM) [38] which
takes the time series for the component $i$ and produces $h_{t}$ for each
component. To create the connectivity matrix (adjacency matrix/graph) between
the components (nodes) at each time-point $t$ we use a self-attention module
[39]. We explain both parts separately in the following sections. Fig. 1 shows
the complete architecture of the model.

###  II-A biLSTM

biLSTMs have been used very successfully for time series data. LSTMs take one
input (e.g., word) from a sequence (e.g., sentence) and provide embeddings for
data at each point in the sequence. The effectiveness of LSTMs comes from the
memory and forget gates, which help the model to learn relationships between
input at different time-points. In time series data, e.g., a sentence, each
input is not independent of previous or future values. Thus it makes very
crucial to find these effective relationships between the data. As the effect
of input at a time, $t$ onto other inputs, is unknown and can vary across
different time series and components of the same time series, it is crucial to
learn these relationships based on the downstream task. The working of the
LSTMs can be explained by the following equations. $\sigma$ represents sigmoid
activation, $b$ are the biases, and $\odot$ is the Hadamard product [40].

$\begin{array}[]{ll}\mathbf{i_{t}}=\sigma(\mathbf{W}_{ii}\mathbf{x}_{t}+b_{ii}+\mathbf{W}_{hi}\mathbf{h}_{t-1}+b_{hi})\\\
\mathbf{f_{t}}=\sigma(\mathbf{W}_{if}\mathbf{x}_{t}+b_{if}+\mathbf{W}_{hf}\mathbf{h}_{t-1}+b_{hf})\\\
\mathbf{g_{t}}=\tanh(\mathbf{W}_{ig}\mathbf{x}_{t}+b_{ig}+\mathbf{W}_{hg}\mathbf{h}_{t-1}+b_{hg})\\\
\mathbf{o_{t}}=\sigma(\mathbf{W}_{io}\mathbf{x}_{t}+b_{io}+\mathbf{W}_{ho}\mathbf{h}_{t-1}+b_{ho})\\\
\mathbf{c_{t}}=\mathbf{f_{t}}\odot\mathbf{c_{t-1}}+\mathbf{i_{t}}\odot\mathbf{g_{t}}\\\
\mathbf{h}_{t}=\mathbf{o_{t}}\odot\tanh(\mathbf{c_{t}})\\\ \end{array}$ (1)

Here $\mathbf{h}_{t}$ represent the embedding for the input at $t$. We use a
biLSTM to create embeddings $\mathbf{h}_{t}$ for each component $i$. Thus
$\mathbf{h}_{t}^{f}=LSTM(x_{t},\mathbf{h}_{t-1})$,
$\mathbf{h}_{t}^{b}=LSTM(x_{t},\mathbf{h}_{t+1})$ and
$\mathbf{h}_{t}=concatenate(\mathbf{h}_{t}^{f},\mathbf{h}_{t}^{b})$. Here
$\mathbf{h}_{t}^{f}$ and $\mathbf{h}_{t}^{b}$ are representation for forward
and backward pass. We use LSTM for each component individually, sharing
weights of LSTM among the components. We give $x_{t}^{i}$ as input to the LSTM
along with hidden vector and receive $\mathbf{h}_{t}^{i}$ for the component
$i$. This allows us to later compute connectivity matrix (links/edges) between
the components/nodes.

##  III Self-Attention

Self-attention creates new embeddings for each $\mathbf{x^{i}}$ depending on
$n$ other embeddings in the sequence. Self-attention tries to find the
relationship of each input with all other inputs denoted by weights and
updates $\mathbf{x^{i}}$ accordingly. Self-attention can be considered a
special case of a typical GNN with $1$ layer/hop where a node receives input
from the neighbors that are one hop away. Because of the ability of the self-
attention module to create weights by learning the relationship between
different embeddings, we create a connectivity matrix between components at
each time-point $t$ by giving
{$\mathbf{h}{}_{t}^{i},.....\mathbf{h}{}_{t}^{n}$ }, $n$ = total components,
as input to the self-attention module and creating new embeddings
{$\tilde{\mathbf{h}{}}_{t}^{i},.....\tilde{\mathbf{h}{}}_{t}^{n}$} and the
weight matrix $\mathbf{W}{}_{t}$, where each
$\mathbf{W}{}_{t}\in\mathbb{R}{}^{n*n}$. The self-attention module creates
three embeddings, namely, key ($\mathbf{k}{}$), value ($\mathbf{v}{}$), and
query ($\mathbf{q}{}$) and creates new embeddings for each input using these
embeddings. The set of equations in (2) can sum up the whole process. For
simplicity, we omit the $t$ from these equations. ⊺ represents transpose and
$\oplus$ represents concatenation.

$\begin{array}[]{ll}{\mathbf{k}{}^{i}}={{\mathbf{h}{}^{i}}^{\intercal}\mathbf{W}{}^{(k)}},\hskip
7.11317pt{\mathbf{v}{}^{i}}={{\mathbf{h}{}^{i}}^{\intercal}\mathbf{W}{}^{(v)}},\hskip
7.11317pt{\mathbf{q}{}^{i}}={{\mathbf{h}{}^{i}}^{\intercal}\mathbf{W}{}^{(q)}}\\\
{\mathbf{K}{}}=\oplus_{i=1}^{n}{\mathbf{k}{}^{i}}^{\intercal},\hskip
7.11317pt{\mathbf{w}{}^{i}}=\operatorname{softmax}({\mathbf{q}{}^{i}}{\mathbf{K}{}})\\\
{\tilde{\mathbf{h}{}}^{i}}=\sum_{j}^{n}{({w^{i}_{j}}{\mathbf{v}{}^{j}})},\mathbf{W}=\oplus_{i=1}^{n}\mathbf{w}{}^{i}\end{array}$
(2)

Here $\mathbf{W}{}\in\mathbb{R}{}^{n*n}$ is the connectivity matrix between
$n$ components/nodes in the graph. We use $\mathbf{W}{}$ for downstream
classification and not the embeddings. As the true graph is never available in
many applications to directly compare with, we propose that a connectivity
matrix leading to state-of-the-art classification performance makes it more
reliable than using the embeddings ($\tilde{\mathbf{h}{}}$) for
classification.

Figure 1: DECENNT architecture using biLSTM, self-attention and temporal
attention.

##  IV Temporal Attention

Since we get a set of the matrices $\mathbf{W}{}$, one for each time-point, an
easy and standard way is to average the $T$ matrices, however, not all time-
points can be equally crucial for the downstream task. Therefore, we introduce
a new temporal attention mechanism to focus on important time-points. The
temporal attention module is essential for the downstream classification and
for finding important timepoints/biomarkers in the data, thus making it
crucial to our model. We name our attention model - global temporal attention
(GTA) - that attends to crucial time-points and is stable and consistent
across randomly seeded trials.

###  IV-A GTA

To give the attention module a global view of the graph, we present GTA. The
global view allows the model to learn how each connectivity matrix contributes
to the global graph or structure of the data in the downstream task. We create
a sum of all the $T$ connectivity matrices and call it $\mathbf{W}{}^{global}$
representing the global view. We then compare the similarity of each local
$\mathbf{W}_{t}$ with the global view and use them to create the temporal
attention vector $\bm{\alpha}$.

$\begin{array}[]{ll}\mathbf{W}{}^{global}=\sum_{t=1}^{T}(\mathbf{W}{}_{t}),\widetilde{\mathbf{W}{}_{t}}=\mathbf{W}{}_{t}\odot\mathbf{W}{}^{global}\\\
\bm{\alpha}=(\oplus_{t=1}^{T}(((\operatorname{flat}(\widetilde{\mathbf{W}{}_{t}}))\mathbf{W}{}^{MLP_{l1}})\mathbf{W}{}^{MLP_{l2}})\\\
\mathbf{W}{}^{f}=\sum_{t=1}^{T}(\mathbf{W}{}_{t}{\alpha_{t}})\end{array}$ (3)

Here $\odot$ is the Hadamard product [40] between matrices, $\oplus$
represents concatenation, and $\mathbf{W}^{f}$ is the final weight matrix.
Equation (3) shows the equations for GTA.

TABLE I: Details of the neuroimaging datasets used. We tried different CV folds in our experiments but that did not have a significant effect on results. We report the results with CV folds that match comparing studies. Name | Category | Preprocessing | Parcellation | Subjects | 0 Class | 1 Class | CV Folds | TP
---|---|---|---|---|---|---|---|---
FBIRN | Schizophrenia | SPM12 | ICA | 311 | 151 | 160 | 4, 6, 18 | 157
OASIS | Dementia | SPM12 | ICA | 912 | 651 | 261 | 4, 10 | 157
ABIDE | Autism | SPM12 | ICA | 569 (TR=2) | 255 | 314 | 5, 10 | 140
ABIDE | Autism | SPM12 | ICA | 869 | 398 | 471 | 5, 10 | 140
HCP | Gender | SPM12 | ICA | 833 | 390 | 443 | 5, 15 | 980
FBIRN | Schizophrenia | SPM12 | Shaefer 200 | 311 | 151 | 160 | 18 | 157
HCP | Gender | Glassier | Shaeffer 200 | 942 | 411 | 531 | 10 | 1200
ABIDE | Autism | C-PAC | Shaeffer 200 | 871 | 403 | 468 | 10 | 83-316

##  V Experiments

This section reports the training process of our model, details about hyper-
parameters, and datasets used.

###  V-A Training

We ran our experiments using RTX 2080 using PyTorch. The hidden dimensions for
the LSTM, self-attention including key, query, and value modules, were all set
to $64$. Both LSTM and self-attention modules had only one layer. We tried to
incorporate multiple layers, but it did not help in terms of classification
performance nor interpretation. The dimensions of MLP layer for calculating
temporal attention vector were $\gamma*len(flat(\mathbf{W}_{t}))$ and $1$ with
$\gamma=0.25$. We used batch normalization after the first MLP layer. ReLU
activation was used in our model between the MLP layers. A final two-layer MLP
was used to get logits for binary classification problem with $\mathbf{W}_{f}$
as input with dimensions $64$ and $2$. We used cross-entropy loss with Adam
optimizer. Let $\theta$ represent all the parameters of the architecture,
$\mathbf{\hat{y}}$ being the prediction and $\mathbf{y}$ is the true labels,
the loss is calculated as:

$\displaystyle
loss=\operatorname{CrossEntropy}(\mathbf{\hat{y}},\mathbf{y})+\lambda\|\bm{\theta}\|_{1}$
(4)
$\displaystyle\bm{\theta^{*}}=\operatorname*{arg\,min}_{\bm{\theta}}(loss)$
(5)

$\lambda$ (regularization weight) was set to $1e^{-6}$ and learning rate
($\eta$) was $1e^{-4}$. We reduced the learning rate by a factor of $0.5$ when
validation loss reached plateau. Early stopping was used to stop training the
model based on validation loss and patience of $15$. For each dataset, to have
a fair result, we perform n-fold cross validation, depending on the size of
the data with 10 randomly seeded trials for each fold. We report the mean area
under curve - receiver operating characteristic (AUC-ROC) and many other
metrics to show classification performance. For region based experiments,
$\gamma$ was reduced to $0.005$, $\eta$ was set to $5e-3$ for HCP dataset and
$3e-4$ for others. Batch size was set to $32$.

###  V-B Datasets

To test our model for a) classification b) learned connectivity matrix and c)
learned temporal attention we use five different datasets across two fields;
neuroimaging, and natural language processing (NLP). Refer to Tab. I for
details of the neuroimaging datasets. Validation and test size was kept same.

####  V-B1 NeuroImaging

The neuroimaging datasets can be further divided into two sub-tasks; brain
disorder and gender prediction.

##### Disorder Prediction

Three datasets used in this study include FBIRN (Function Biomedical
Informatics Research Network111We are using fBIRN phase III.) [41] project,
release 1.0 of ABIDE (Autism Brain Imaging Data
Exchange222http://fcon_1000.projects.nitrc.org/indi/abide/) [42] and release
3.0 of OASIS (Open Access Series of Imaging Studies333https://www.oasis-
brains.org/) [43] to predict schizophrenia, autism and dementia respectively.

##### Gender Prediction

Healthy controls from the HCP [44] are used for gender prediction.

##### Preprocessing

We use different brain parcellation techniques which can be divided into two
sub-categories; ICA and region based. The preprocessing method applied depends
on the parcellation technique used and the methods used in SOTA studies for
the particular dataset.

ICA parcellation: All experiments used a fully automated independent component
analysis (ICA) as a brain parcellation technique. We first preprocess the fMRI
data using statistical parametric mapping (SPM12,
http://www.fil.ion.ucl.ac.uk/spm/) within MATLAB 2020. Subjects were included
in the analysis if the subjects have head motion $\leq 3^{\circ}$ and $\leq 3$
mm, and with functional data providing near full brain successful
normalization [45]. For each subject, $100$ ICA components are estimated using
the Neuromark template and used in experiments following the same procedure
described in [45]. We use ICA timecourses as input to the model. For ABIDE1,
we conduct two ICA based experiments using all subjects and subjects with
$TR=2$.

Region parcellation: SOTA methods use different preprocessing pipelines for
HCP and ABIDE dataset. For comparison with these SOTA methods on HCP and ABIDE
dataset, we preprocess these datasets following existing studies. HCP [44] was
first minimally pre-processed following [46], and then FIX-ICA based denoising
was applied to reduce noise in the data [47, 48]. After denoising, $152$
subjects were discarded based on head motion following [23] which results into
$942$ subjects. ABIDE1 [42] was pre-processed using cpac [49], out of $1112$
subjects $871$ were selected following [50, 6, 51]. To divide the data into
regions, we use Shaefer [52] and Harvard Oxford (HO) [53] atlas depending on
the experiment. Refer to Tab. I for details about the datasets.

####  V-B2 NLP

To show the broad implications of our method, we apply our method for keyword
detection in audio files. We choose this problem because it has many practical
applications (e.g., virtual assistants in smartphones and robots). We use
Speech Commands Dataset [54] for predicting the occurrence of a keyword in
speech. The audio files are combined with a background noise of a coffee shop
[55] to make prediction harder. We use this dataset to test the temporal
attention weights we get from our model because important time-points
(location of word cat in the noise) is known. We match this experiment with
classifying brain disorder. The keyword ”cat” can be thought as the presence
of a disease and the background noise can be considered as the noise and other
data present in the fMRI time-courses.

##### Preprocessing

For prediction, we collect samples of audios for the keyword ”cat” from speech
command dataset [54] which has 1515 audio files for the keyword. To create
”cat” class examples, we superimpose each of the keyword audios with the
length of one second onto a two seconds long background noise at a random
location, resulting in a two seconds long audio consisting of background noise
and keyword cat. To make it a difficult problem, we do the following things;
a) the audio of cat is mixed with the noise when creating the final audio,
which means the timepoints where the word cat is added has noise as well and
b) before mixing the two audio files, we match the amplitudes of the two
audios by normalizing both audios (background and cat) to same scale and
finally c) we normalize the 1-second long sum of both files so that the sum
does not have higher values than the rest of the audio file which only has
noise. As a result of the points (a-c) metioned above, the model cannot
perform classification based on amplitude. Furthermore, the model receives the
mel-spectogram of the audio files as input rather than actual audio files.
Fig. 2 shows the mel-spectogram of the three audio files. To create ”no-cat”
class examples, we use another 1515 two seconds long audio files containing
only the background noise. Thus, we use 3030 audio files for the downstream
task (”cat”/”no-cat” classification). For all of the 3030 audio files, we
compute mel-spectrogram to convert each of them into a matrix of size
_components $\times$ time courses_. To test our model on multiple keywords, we
create another dataset with keyword ’nine’ following the same method.

Figure 2: Mel-spectogram of ’background noise’, ’cat audio’ and ’superimposed
audio’. The model receives background and superimposed files as input.

##  VI Results

We show three different results, one for each of the paper contributions. We
compare our results with SOTA DL methods [56, 57, 15, 32, 23, 58, 59, 22, 51,
6, 26] depending on the task, and ML methods such as support vector machine
(SVM) and logistic regression (LR). To be fair to the other papers, we report
directly from the results mentioned in the papers. Not all methods were
applicable to each of the dataset/task, or the code/results of other methods
were not available. All figures are generated using multiple test subjects
across at least $10$ randomly seeded trials. Our experiments show that our
model beats SOTA methods on classification/prediction tasks but more
importantly our learned EC structures are a) similar to existing studies, b)
provides knowledge not present in existing methods; FC, c) captures direction
of connectivity and d) finds important temporal bio-markers relevant to the
downstream task.

Figure 3: AUC comparision of DECENNT model with four different methods (MILC [57], STDIM [56], LR, SVM), over five different datasets on ICA time courses (Ref to section V-B1). Our method significantly outperforms SOTA methods. We performed Autism experiments with 869 subjects (DECENNT ALL - all TRs) as well. As we do not have a pre-training step we compare with not-pre-trained (NPT) version of MILC and STDIM. Input to ML models was same ICA time courses. TABLE II: Classification performance comparison of DECENNT with other DL methods on region based data of HCP and FBIRN datasets (Ref to section V-B1). Our DECENNT model outperforms all other methods in almost every metric. The best two scores are shown as bold and italic respectively. Note: As we use all the regions in the atlas we report the mean accuracy for SVM-RBF [59]. The results for GCN [22] on HCP data are reported in GIN paper [23]. HCP | FBIRN
---|---
| DECENNT | GIN | SVM-RBF | GCN | ST-GCN | PLS | DECENNT | BrainGNN
AUC | 93.6 | NA | NA | NA | NA | 88.125 | 82.5 | 78.8
ACC | 86.0 | 84.6 | 68.7 | 83.98 | 83.7 | 79.9 | NA | NA
Precision | 87.2 | 86.19 | NA | 84.59 | NA | NA | NA | NA
Recall | 88.6 | 86.81 | NA | 87.78 | NA | NA | NA | NA
Parcellation |  | Shaefer
---
200
| Shaefer
---
400
| Shaefer 400
---
\+ Fan 39
| Shaefer
---
400
| Multi-moda
---
22
| Dosenbach
---
160
| Shaefer
---
200
| AAL
---
116
Validation | 10 | 10 | 10 | 10 | 5 | 10 | 18 | 18
Subjects | 942 | 942 | 434 | 942 | 1091 | 820 | 311 | 311
Study | Our | [23] | [59] | [22] | [32] | [58] | Our | [15]
TABLE III: Comparison of AUC score on ABIDE1 region based dataset (Ref to section V-B1). Existing methods use Harvard Oxford (HO) parcellation with $111$ brain regions. Unlike GCN [6] and DeepGCN [51] we use only fMRI data.  Method | Parcellation | Input | AUC
---|---|---|---
DECENNT | Shaefer | fMRI data | 0.70
DECENNT | HO | fMRI data | 0.69
GCN [6] | HO |  | fMRI +
---
phenotypic data
0.75
DeepGCN [51] | HO |  | fMRI +
---
phenotypic data
0.75
Metric Learning [26] | HO | fMRI data | 0.58

###  VI-A Classification

Our method turns out to be the best performing model against SOTA methods,
giving the highest AUC score for all the datasets used for classification
(disorder, gender, speech) with ICA data. Even with region-based data our
model performs better than existing methods on HCP and FBIRN dataset. As our
model does not use phenotypic information about subjects, our model lacks
behind GCN [6] and DeepGCN [51] on ABIDE.

Parisot et al. [6] reports a decrease of $\sim 2.5$ AUC by using a different
phenotypic information which clearly shows the dependence on phenotypic data.
[26] reports much lower AUC score by using only fMRI data. Fig. 3 shows the
classification results on ICA data. The machine learning methods fail due to
high data dimensions ($m$), and relatively smaller number of subjects($n$),
$m>>n$. Tab. II and Tab. III show region based classification results. We
would like to point that GIN [23] and ST-GCN [32] use test data for hyper-
parameter tuning and early stopping, whereas we use validation data for both
and test data is used only to test the model. Kim et al. [60] reports lower
results for GIN ($81.34$ ACC and $89.55$ AUC) and ST-GCN when not using test
data as validation data.

###  VI-B Connectivity Matrix

We first present the learned connectivity for the relatively easier task of
NLP. We show the difference in the learned EC for the two keywords (’cat’ and
’nine’) in Fig. 4. Fig. 4a show high connectivity between higher channels,
whereas Fig. 4b show high connectivity for relatively lower channels which
follows the high frequency sounds in ’cat’ and relatively lower frequency
sounds in ’nine’.

(a) Cat EC

(b) Nine EC

Figure 4: EC learned by our model for keywords ’cat’ and ’nine’ superimposed
with noise. We used a test fold of $16$ subjects and computed mean EC with 10
trials per subject. Our model accurately gives high attention values to
medium-to-high channels for ’cat’ and low-to-medium channels for ’nine’
samples. Average values: Inside green box: $0.89$ for Fig. 4a and $0.78$ for
Fig. 4b, outisde box: $0.69$ for Fig. 4a and $0.63$ for Fig. 4b. X and y axis
denote the frequency channels in hertz (HZ).

Next we compare the connectivity matrix learned by our model on neuroimaging
dataset with FNC computed PCC, which is probably the most popular method for
computing connectivity matrix. Fig. 5 shows that the two matrices are
comparable, but our effective network connectivity (ENC) Fig. 5a is directed
and provides additional details. We also see that our ENC Fig. 5a has more
inter-network connectivity which is missing in Fig. 5b. The effect of visual
(VI) network onto other networks is seen only in Fig. 5a. We group the ICA
components according to [61] into seven domains based on anatomical and
functional properties. 53 components out of 100 fall into the seven domains
and the rest are marked as noise. The connectivity matrix clearly shows that
the components have high intra-domain connectivity, which matches the existing
literature [61].

(a) DECENNT ENC

(b) PCC FNC

Figure 5: Fig. 5a is the connectivity matrix generated by our model for FBIRN
dataset. We used a test fold of $16$ subjects ($10$ trials each) and computed
mean ENC for all subjects. Fig. 5b is the mean FNC of the same subjects
generated by PCC. Both figures are strikingly similar, which verifies the
correctness of the connectivity matrix learned by our model. To match the
positive weights of our model, we normalize the FNC from 0 to 1 instead of -1
to 1.

Furthermore, as our model learns ENC, we use Fig. 6 to show the importance of
direction. Fig. 6 (left) shows edges from $a$ to $b$, where $a>b$ For example,
the edge (8,21) means the edge is from $21$ to $8$. It is observable that the
components in visual (VI) heavily affect components in sensorimotor (SM). The
direction is reversed in Fig. 6 (right) and SM does not affect VI. Similar
direction can be seen between cognitive control (CC) and SM. The presence of
direction is of paramount importance and is missing from FNC. It can
potentially help to make and answer interventions in data.

Figure 6: Top 10$\%$ directed edges of FBIRN ENC 5a. The numbers represent the
$53$ crucial components. The figure shows the direction of connectivity.
Visual (VI) affects other domains, cognitive control affects sensorimotor.
Edges: VI $\rightarrow$ other: 79, other $\rightarrow$ VI: 25. CC
$\rightarrow$ SM: 9, SM $\rightarrow$ CC: 3.

###  VI-C Temporal Attention

As in rs-fMRI the subjects are not performing any specific task at any time-
point, there is no available true knowledge of important time-points. Because
of this reason, we show keyword detection experiments where the precise
location of the keyword is available. Fig. 7 shows attention weights for 8
test subjects. The attended time-points match with the time-points of the
keyword. This is extremely significant and proves the model can accurately
find important time-points, as the location of the keyword was never given to
the model. We compute the statistical values such as (precision, recall) of
the temporal attention, mentioned in the caption of Fig. 7. We assign label
’1’ to time-points where ’cat’ audio is superimposed and label ’0’ to all
other time-points which gives us the true labels. For predicted labels, we
assign label ’1’ to time-points with attention value greater than 0 and label
’1’ to all other time-points. The stats shows that the model a) assigns high
attention values to ’cat’ time-points, b) does not attend to ’non-cat’ time-
points and c) does not attend to all ’cat’ time-points. Although, we would
have liked the model to attend to all cat time-points, we think the model does
not do that because of two reasons; 1) The 1-sec long ’cat’ audio files on
average have the ’cat’ sound for only 0.5 seconds or less whereas, when
creating Fig. 7 and the stats, we used the complete 1-sec time-points. 2) The
model maybe looking for a part of the keyword ’cat’ which is distinct from the
noise.

(a) All trials

(b) Averaged

Figure 7: Fig. 7a is the normalized temporal attention weights for keyword
detection task for $8$ subjects, with $10$ trials for each. To keep the lines
separate, we added the trial number to the weights. Fig. 7b is the mean
weights for the $10$ trials of the same subjects. The top red line marks the
actual time-points for the keyword. Statistics: True Positive=56, False
Positive=18, True Negative=331, False Negative=291, Precision=0.76,
Sensitivity=0.16, Specificity=0.95.

To further check the correctness of the time-points selected by our model and
the affects on classification performance, we perform an experiment where
after training the model, we compute $\mathbf{W}^{f}$ using only top $5\%$
attended time-points for training data to train an LR model and then use the
top $5\%$ time-points for the test data to test the model. Similarly we
perform experiments for bottom $5\%$ values as well. Tab. IV shows the
comparison using three brain datasets. The results show that the LR model
provides high AUC score by just using top $5\%$ of the time-points attended by
the model. Thus, it proves that a) not all time-points are important for
classification of the downstream task and b) our model accurately finds the
important time-points. We use an LR model for this experiment to show that the
learned top/bottom $5\%$ time-points are not limited to our model but is
generalized such that an independent LR module gives high classification
results using the top $5\%$ attended time-points and does not learn on the low
$5\%$ data. In our experiments, we also note upto $5\%$ drop in AUC when not
using temporal-attention.

TABLE IV: AUC score comparison on brain datasets with ICA components by using all, top 5 % and bottom 5% time-points only. | Method | FBIRN | OASIS | ABIDE
---|---|---|---|---
100 % | DECENNT | 0.844 | 0.72 | 0.65
Top 5 % | LR | 0.835 | 0.713 | 0.642
Bottom 5 % | LR | 0.566 | 0.548 | 0.532

##  VII Conclusion

Our model demonstrates the importance of learning dynamic temporal graphs for
any multivariate time series, which is currently missing from the existing
literature. Using dynamic graphs, our model outperforms SOTA methods across
five different tasks, proving that the model is applicable across different
fields and tasks. By learning the correct graph structure/connectivity matrix
for the data, our model eliminates the need for existing external methods such
as PCC, K-means. Our model learns a directed graph structure that provides
more detail than a symmetric correlation matrix which does not capture
effective connectivity. As seen in results, our learned EC matrices give the
direction of connectivity between brain regions. The temporal attention module
proves to be highly effective in terms of classification. As shown in the
paper, it provides stable attention weights and accurately finds the critical
time-points depending on the downstream task. Both self and temporal attention
modules result into stable, consistent attention values and increase the
classification performance across tasks. These attributes address the
questions regarding explainability of attention mentioned in [62, 63]. Many
tasks across many fields are ever dynamic and have missing graph structure,
e.g. (Brain functional networks, social networks, self-driving cars etc.)
which increases the need of methods like DECENNT. Temporal attention used in
brain connectivity can help us find important bio-markers relative to the
disorder/disease which in turn help us understand the disorder and its causes.
For future work, we plan to extensively interpret the learned connectivity
structures, and see the differences in them across controls and patients and
across multiple brain disorders. We plan to compare our model with other
methods of capturing brain’s network connectivity such as transfer entropy and
Granger causality. We also want to incorporate a form of spatial attention,
which like temporal attention, could help identify essential nodes/components
that are sometimes unavailable in many fields. We also for each class of
subject want to examine how ENC changes overtime and if/how the direction of
flow of information changes through time.

## Acknowledgements

This work was funded in part by NIH RF1MH121885, R01MH123610, R01EB006841 and
NSF 2112455 grants. Data for healthy subjects was provided [in part] by the
Human Connectome Project, WU-Minn Consortium (Principal Investigators: David
Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and
Centers that support the NIH Blueprint for Neuroscience Research; and by the
McDonnell Center for Systems Neuroscience at Washington University. Data for
Schizophrenia used in this study were downloaded from the Function BIRN Data
Repository (http://bdr.birncommunity.org:8080/BDR/), supported by grants to
the Function BIRN (U24-RR021992) Testbed funded by the National Center for
Research Resources at the National Institutes of Health, U.S.A. Data for
Alzheimer’s was provided by OASIS-3: Principal Investigators: T. Benzinger, D.
Marcus, J. Morris; NIH P50AG00561, P30NS09857781, P01AG026276, P01AG003991,
R01AG043434, UL1TR000448, R01EB009352. AV-45 doses were provided by Avid
Radiopharmaceuticals, a wholly owned subsidiary of Eli Lilly. Autism data was
provided by ABIDE. We acknowledge primary support for the work by Adriana Di
Martino provided by the (NIMH K23MH087770) and the Leon Levy Foundation and
primary support for the work by Michael P. Milham and the INDI team was
provided by gifts from Joseph P. Healy and the Stavros Niarchos Foundation to
the Child Mind Institute, as well as by an NIMH award to MPM (NIMH
R03MH096321).

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