#  How saccadic vision might help with the interpretability of deep networks
††thanks: The study was supported by the Ministry of Education and Science of
Russia (Project No. 14.Y26.31.0022).

1st Iana Sereda  Research and Education Center
Mathematics of Future Technologies
Nizhny Novgorod, Russia
<EMAIL_ADDRESS>2nd Grigory Osipov  Research and Education Center
Mathematics of Future Technologies
Nizhny Novgorod, Russia
<EMAIL_ADDRESS>
###### Abstract

We describe how some problems (interpretability, lack of object-orientedness)
of modern deep networks potentially could be solved by adapting a biologically
plausible saccadic mechanism of perception. A sketch of such a saccadic vision
model is proposed. Proof of concept experimental results are provided to
support the proposed approach.

###### Index Terms:

deep learning, interpretability by design, saccades, ECG

##  I Introduction

The topic of interpretability of data representation arising in an intelligent
agent is interesting for many reasons. For example, medical applications
require decision-making to be transparent for a human because the cost of a
mistake is high. Again, the higher the interpretability of the representation
for a human, the more it is similar to how a human thinks, and, therefore, the
closer we are to solving the problem of modeling intelligence in general.

The converse is also true: data on the mechanisms of the biological brain can
be useful in the sense of introducing their principles into artificial
intelligence models. Thus, some features of the structure of the visual cortex
inspired the development of convolutional networks[1].

Figure 1: A schematic example of the presence of non-interpretable zones in
the code space of a trained convolutional neural network. Blue dots are
projections of real electrocardiogram (ECG) fragments into the code space of
the model. The blue area around them is the area of interpretable codes, the
white area is of not interpretable codes. ECG pictures are generated by a 2d
convolutional generative model from points in code space located on a dashed
line segment. The input data for the convolutional autoencoder were images of
the cardiac cycle, considered as a two-dimensional low-resolution black and
white picture. It can be seen that pictures 2 and 3 show impossible heart
cycles.

Deep convolutional networks are often used today in applied problems as one of
the basic components of learning systems. On some tasks, for example, the task
of modeling faces, it is possible to achieve representations with good
interpretability[2]. However, this success does not translate to all types of
data equally. Problems begin in those tasks where generalization is required
for a small amount of data; where datasets are poorly balanced; where active
learning is required and catastrophic forgetting must be avoided[3][4]; A
separate type of difficulty is the selection of architecture
hyperparameters[5]. Finally, it is known that the representation of deep
convolutional networks is not object-oriented(details below), while in human
representation proto-objects appear already at the early stages of processing
of visual information[6]. This paper will examine how the use of the
principles of saccadic vision might be helpful in fixing some of the above
problems.

##  II Related Work

One of the most human-readable representations of a visual scene is the
semantic scene graph: if it is present, the task of generating the text
describing the scene is trivial[7]. The nodes of such a graph are usually
nouns that name objects on the stage. The node can be assigned its coordinates
on the image. There are approaches to generate a scene graph using deep
learning techniques. The first step of the algorithm is usually to select
areas of the image in which the presence of objects is likely [8] [9][10].
Such regions are searched using a separate convolutional network, for example,
Faster R-CNN[11], which returns the coordinates of the region of interest and
the corresponding feature tensor. However, with this approach, graph
representations arise not at every level of abstraction, but only at the
highest, i.e. at the level of whole objects, but not at the level of their
parts. Before the appearance of this graph, intermediate scene representations
are activation tensors that arise in a neural network (or a group of neural
networks), often convolutional.

Convolutional networks associate an image area with a feature tensor in which
the concept of coordinates does not appear explicitly. When an attempt is made
to obtain the coordinates of an object from the convolutional network in an
explicit form, overfitting inevitably occurs, which is shown in [12].

Meanwhile, the concept of objects is inseparable from the concept of their
coordinates. When modeling the world by physics, distances appear in the model
of the world in an explicit form. The fact that distances do not explicitly
occur in standard convolutional networks representations means that it is not
guaranteed for representation to be object-oriented.

Distances are more explicitly generated when deep learning models are equipped
with attention mechanisms. For example, in the model of recurrent attention a
sequence of views is formed, whose coordinates can be obtained explicitly if
desired[13]. However, training models with attention, both old and newer[14],
requires a lot of data, since work is still being done with large networks.
Thus, the task of creating a deep agent, whose data representation is object-
oriented at all levels of abstraction and at the same time allows learning
from several samples, has not yet been solved.

###  II-A Eye movements

The retina of the eye has an uneven distribution of receptors, and therefore
at any given time a person sees clearly only a relatively small part of the
picture. Thus, in order to see the whole scene, a person makes many quick eye
movements, saccades. At least in part, these movements are involuntary. Human
saccades occur several times per second[15]. The generated sequence of
saccades depends on the image being viewed and on the problem solved by the
subject[16]. The length of the focus movement during saccades is different,
sometimes microsaccades are distinguished[17].

From the above it follows that recognition of a scene by a human is a sequence
of controls (of refocusing of the eyes), each of which leads to new sensory
data. The brain that recognizes the scene does not see the new scene
immediately, but learns it through the controls and their results. This is
different from how the input image is represented when working with deep
learning networks: the image is often thought of as a point in
multidimensional vector space.

###  II-B Active exploration of a situation

In some works, for example[18], it is proposed to consider saccadic vision as
a process of sequential generation of hypotheses by the brain and their
immediate experimental verification. In this formulation, the saccade is the
most informative experiment aimed at reducing the uncertainty about the
structure of the situation.

An active exploration of the situation by the biological brain occurs not only
during the movement of the gaze, but also during the movement of the body, for
example, when a rat is exploring a new environment. The representation of the
environment in this case is built on the basis of place cells and grid
cells[19]. The module, consisting of grid cells, renews its activity as the
animal moves around the environment and ”marks on the map” where the animal is
now. There are grounds[20] for the hypothesis that the place cells and grid
cells are a particular version of a more general mechanism for modeling of the
world by the mammalian brain[18]. If this is the case, then not only the
embeddings of sensory data, but also the controls that led to this data,
should be included in the representation of data at all levels of abstraction
in the deep agent.

##  III Electrocardiograms

Our numerical modeling was carried out on human electrocardiograms (ECG) as
training data. Such data was chosen because in has a lower dimensionality than
images of real scenes, but at the same time ECG still represents a scene with
objects of different types placed on it. The objects here are not people,
animals, trees, as it happens in photographs, but components of the human
cardiac cycle: various waves and wave complexes. The appearance of these
”objects” varies under the influence of many factors[21]: the patient’s
diseases, sliding of the electrodes on the skin, breathing, muscle contraction
under the electrodes, age, gender, and even drinking cold water before the
examination.

Deep networks have shown their applicability to many tasks that arise when
working with ECG: the task of delineation[22], diagnosing certain pathological
conditions[23], compression [24]. However, studies of the representation of
ECG in convolutional networks show problems when interpolating between points
in the code space[25].

##  IV The proposed approach

An interesting point about grid cells is the lack of influence of external
sensory data (for example, visual data) on the activity of cells of that type.
Grid cells work only on information about the animal’s own movements[26]. Thus
the mechanism of grid cells can be considered as the assignment of equivalence
classes on the set of available controls. I.e. being at A, the animal can
reach location B, performing different sequences of movements. The fact that
they all lead to the same outcome, unifies them into an outcome equivalence
class.

This data prompts us to consider the assumption that the representation of a
data in the brain is a simulation of sensory results of controls. Both types
of data - data on controls and on their sensory results - must be present in
the representation of the intectual agent in order to increase its
interpretability. In other words problem of dimensionality reduction of
sensory data probably should not be considered in isolation from the controls
at which such sensory data are observed. In the case of vision, controls are
saccades, in the case of navigation in space of body in space controls are
complicated series of body movements.

###  IV-A Displacement of an object within the receptive field of the network

Many types of data people perceive as a scene of objects that interact,
forming a hierarchical structure. Objects inside the scene can change their
relative positions from situation to situation. The response of the deep
convolutional networks to the shift (of the object inside the receptive field)
has negative characteristics. The conceptual problem is depicted in Fig. 2.

Figure 2: An illustration of the appearance of uninterpretability in the deep
layer. In the first case (left), the specific pattern is in the center of the
receptive field. In the second case, it is shifted (right). Despite the fact
that the object remains the same, in the general case, the codes in the deep
layers may be different. This is not necessarily always the case, but in
standard deep convolutional network there is no guarantee against this
behavior. Thus, morphology information becomes inherently entangled with
displacement information.

In [27] it is shown that its reasons, at least in part, lie in the method of
dimensionality reduction. I.e. in the pooling operations that are a standard
part of convolutional architectures.

As a conceptual way to solve the above problem, we propose the separation of
information about ”what is observed” and ”under what control” into two types
of variables inside the representation of an intelligent agent. Fig. 3. shows
how, using the example of the situation from Fig. 2, it is possible to correct
problematic response to a shift.

Figure 3: Representation in which information about the attributes of an
object and about its location in scene coordinates is divided into two
different variables.

In the above figure, there is an instance of a certain class of objects - a
splash of a characteristic shape. There is also a ”blue device” - a specialist
in this type of objects (it can be a small neural network or another tool for
dimensionality reduction). The device gives an informative description of the
object seen, ”123”. The description it gives is informative if and only if the
object being described is in the center of its receptive field. Therefore, one
needs to apply the correct control to move the blue device so that it is in
the right place. The pair ”control” — ”sensory result” here is analogous to
the visual concepts: saccade — fixation.

###  IV-B Uncertainty of controls

Objects in the real world are not always located at fixed distances from each
other. Consider photographs of people. Sometimes there may be a tree next to
the person in the photo. If our gaze is looking at the object named ”the left
eye of a person”, it is impossible to predict where to move the gaze so that a
tree falls into its center. After all, the tree may not be present in this
photo. Even if it is present, it can be anywhere.

But let’s ask another question. Where do we need to shift our gaze so that to
see the nose of this man? This question can be answered in contrast to the
question about a tree. The answer will not be deterministic, it will be
approximate.

Thus, the ”saccadic” face model does not have to generate an accurate
prediction of where to move the gaze in order for it to capture the next
landmark. Usually accurate coordinate prediction is not possible even in the
case of such a rigid structure as a face. However, the model could suggest not
the exact coordinate, but the region the next one is in. If this region is
small enough, each point can be checked for the presence of the center of the
next predicted next landmark using the brute-force search. According to our
hypothesis this could correspond to mammalian microsaccades.

###  IV-C Characteristic controls

In the above example one could roughly predict where the nose is. But it is
almost impossible to predict what would be seen outside of the facial area.
There may be a tree, an ocean, a wall, etc.

That means that for each landmark there are, roughly, two types of controls:
with a predictable result and with an unpredictable one. The controls with
predictable result we will call characteristic controls for that landmark.

###  IV-D Uncertainty of predicted result of a control

Control results can be predicted with varying degrees of accuracy even in the
case of characteristic controls. For example, seeing a nose at a given moment,
we can predict where the eye will be on the face in question, but we cannot
predict its color, its iris pattern, its shape, the length of the eyelashes.
That means that the accuracy of the prediction cannot be measured, say, as the
pixel-by-pixel difference between the predicted data and actually discovered
during the saccade.

It is reasonable to measure the predictability of control results using the
concept of entropy. For a given landmark $A$ and control $u(A)$, one can ask
the question: how much is the entropy of its results lower than the entropy of
the results of an arbitrary control $v(A)$?

###  IV-E Recurrent definition

If $H(u(A))<<H(v(A))$, than $u(A)$ is a characteristic for $A$. Here A is an
indicator function that has the meaning of a detector of a given type of
landmarks. The set $u(A)$ is a set of $n$-dimensional vectors, each of which
is obtained as a sensory result of control $u$, performed ”from A”. I.e.
relative to the coordinate system specified by the triggering of the indicator
function A.

If a vector random variable (of dimensionality $n$) has a low entropy of the
empirically observed sample, then it is reasonable to make a hypothesis that
the process of its sampling can be approximated as the generation of points
from the vicinity of a low-dimensional structure embedded in $R^{n}$. In the
simplest case, this low-dimensional structure is a point from n-dimensional
space. In a more complex way, this may be a piecewise linear approximation of
a manifold by a small ReLU network.

Hence, in order to find out whether the control $u$ is characteristic for
indicator $A$, we can solve the dimensionality reduction problem, trying to
minimize the accompanying loss of information. With luck, we get an
approximation of this low-dimensional structure around which the results for
$u(A)$ are centered. For simplicity, but without loss of generality, we assume
that the results $u(A)$ are concentrated around the point $b\in R^{n}$.

Then the execution of the saccade $u$ from the indicator $A$ is equivalent to
testing the hypothesis of the following form ”if from object $A$ we move our
gaze to point $u(A)$, then what we see there will be a point located close to
$b$. Here, roughly, there are two options: the observed point $b^{\prime}$
will be close to $b$ or not close. The presence of two options gives a direct
reason to introduce the indicator function $B$:

$B(b^{\prime})=\begin{cases}0,dist(b,b^{\prime})>threshold\\\
1,dist(b,b^{\prime})\leq threshold\end{cases}$

We have described how to obtain new indicator from existing indicator through
finding of characteristic control. This gives the rule of recursive formation
of the proposed structure of ”control - results of control” memory of an
intelligent agent. For a number of ECGs, the statement $AuB$ will be true
after some correction for $u$ (i.e., after a series of microsaccades in the
vicinity of $u$). For a part of the ECGs, the statement will turn out to be
incorrect. For example, instead of it, $Au!B$ will work. Binary coding
success/failure is a property of indicators, and thus we got a composite
indicator $AuB$, which takes on the values true or false. Now we can ”ask the
world” for all situations in which this indicator is triggered, and then look
for characteristic controls for it …

###  IV-F Initial indicator

To define a recurrent process, in addition to the stepwise rule of obtaining
one indicator from another, an initial indicator is required. One can think of
different heuristic ways to get it. A particular option is to get it from the
teacher as an outer pointer of attention. For example, in textbooks on ECG
interpretation, arrows often show where to look at a given picture of ECG in
order to learn a new morphological feature.

##  V ECG experiments

ECG is a type of data to which deep neural networks are successfully applied
in the following sense: with a sufficiently large training dataset for some
types of pathologies of the cardiovascular system, it is possible to obtain
good quality indicators for some accepted formal quality metrics. It is shown
that a convolutional network can effectively reduce the dimensionality of the
ECG signal, but at the same time it has problems with representation in hidden
layers[25]. The representation of data is not sufficiently interpretable, and
the functionalist doctors are not replaced yet. The successes of convolutional
networks on formal metrics and the failure of interpretability on this data
type prompts the study of new ways of constructing deep representations that
would take this into account. So this data type was chosen by us for
conducting minimal proof-of-concept experiments.

The experiments were carried out on the LUDB dataset[28]. It contains
approximately 200 ECGs with delineation of the cardiac cycle structure. In
particular, the highest points of the R-peaks are marked. A small subset of
healthy patients was selected.

###  V-A Indicator A

First, we experimentally demonstrated that the precise centering of an object
in the center of the receptive field when decreasing the dimensionality is
very important. The R-peaks were the test object in this experiment.

R-peaks of healthy people are similar, but not exactly. Even in the same ECG
the R-peaks may slightly change from cycle to cycle. However, due to their
similarity the dimensionality reduction algorithms should work effectively if
small ECG fragments containing the R-peak are fed to them as a data set.

Let’s collect the dataset as follows: we select short signal fragments
containing R-peaks from the ECGs of healthy people (from the first lead).
Let’s choose the fragments short enough so that little gets there, except for
the R-peak and its small vicinity (for the LUDB dataset that signal duration
is about 40 numbers). Thus, we got a small dataset of 40-dimensional points,
each of which contains an R-peak, and the center of the peak is located in the
20-th component of the vector, that is, in the center of the clipped fragment.
If the saccade (shift of the gaze) of the agent each time brings his gaze
exactly to the center of the R-peak, then the dataset for the training of an
indicator specialized in R-peak morphologies would be just that.

We compared the fragments selected in this way with the fragments of the same
duration (40), but obtained not by shifts of the gaze to the R-peaks, but by
shifts of the gaze to random places of the ECGs.

The entropy of this second set should be higher, because there is more
uniformity among R-peaks than among random fragments. This can be seen when
decreasing the dimensionality of the geminate dataset, as shown in Fig. 4,
left: R-peaks form a pronounced cluster when the dimensionality decreases from
40 to 2.

Since this cluster is well expressed, it is easy to create an indicator $A$ on
its basis in the terminology of the discussed framework: the indicator $A$
will be considered triggered if the signal fragment received at its input
falls into a circle approximating the blue cluster.

However, the situation with dimensionality reduction quickly changes if, when
cutting fragments with R-peaks, the center of the R-peak is not located
exactly in the same place of the fragment, but approximately in the same
place.

To demonstrate that we added gaussian noise of varying strength to the R-peaks
coordinates given by the human doctor and studied how this affects the quality
of dimensionality reduction. Figure 4 shows that with increasing noise in
control $u$, the possibility of dimensionality reduction effectively degrades.
Even with modern methods of dimensionality reduction, linear separability
(blue from red in fig.4) is becoming increasingly difficult to achieve.

Figure 4: The result of clustering of 40-point ECG segments, lead i. All dots
are projections of these segments after dimensionality reduction by tSNE. Red
dots corresponds to the ”general population”, i.e. to ECG segments cutted out
without any system. The blue dots correspond to the purposefully cutted ECG
segments: always in the vicinity of the R-wave in 7 healthy patients. The
clustering result varies depending on how the ”blue” segments were cutted out.
In the far left picture the cutting occurs so that the center of the selected
segment always coincides with actual center of the R-peak, which is indicated
by the human expert. In the rest of the pictures, the cutting takes place in
such a way that the center of the cutted segment is drawn from the normal
distribution with the expectation at the actual center of the R-peak, but with
non-zero variance. An increase in dispersion rapidly leads to degradation of
the pattern of separable clusters.  Figure 5: Results of clustering ECG
segments depending on control. Control here means the offset length to the
left of the R-peak. The blue dots in each figure correspond to ECG segments,
the center of which is calculated as the center of the R-peak + offset $u$.
The offset length at which the next set of ”blue” segments was selected is
shown above the arrows. Red color, as before, marks the ECG segments, selected
completely at random from random ECGs, lead i. The length of the section (for
both blue and red) is 40. This figure illustrates the fact that with most
controls the cluster structure is not clearly expressed. However, under some
controls it is clearly expressed and even is close to the linear separability
of the red and blue clusters. In particular, clusters are well distinguishable
at small $u$ (see $u=20$) and in the region of 120 (see $u=120$). Also
clustering at large shifts is different from clustering at the small shifts.
At large shifts (see $u=170,400$) red and blue dots are distributed in a more
similar way than in clustering at small shifts (see $u=20,40,120$), where blue
dots follow clearly different distribution compared to the red ones.

###  V-B Characteristic control for indicator A

Next, we conducted an experiment to find characteristic controls $u$ for this
initial indicator $A$. We investigated the effectiveness of dimensionality
reduction for the results of different controls for the above mentioned R-peak
indicator. The qualitative result is shown in Fig.5. In this case, controls
are gaze shifts on different distances from the center of the coordinate
system (which is specified by the triggering of the $A$-indicator). The
experiment investigated the question of which controls will show themselves as
characteristic.

The results are as follows:

  * •

controls representing very small displacements are characteristic (this is not
shown in Fig.5).

  * •

controls that represent very large displacements are not characteristic (shown
in Fig.5, for example $u=170$)

This reflects the locality of relationships in the structure of the ECG
signal: what is close is more predictable than what is far away. Although even
what is close is not completely predictable.

The main interest in this experiment, however, is the question, what is the
maximum characteristic control? That is, which gaze shift is the largest among
those that still lead to predictable results. As can be seen from the Fig.5,
such a displacement among the displacements to the right is $u=120$.

It turns out that this shift relative to the R-peak corresponds to the
location of the T-wave of this cardiac cycle (T-wave is another stable
component of the cardiac cycle along with the R-peak). The search for
characteristic control, thus, naturally leads to the modeling of the structure
of the cardiac cycle.

###  V-C One-shot learning of indicator B

Here we demonstrate that learning by one example within the the proposed
framework is natural and does not require any modifications.

A control was chosen in which the sample of its sensory results showed a
tendency to differ from the general sample of ECG fragments (characteristic
control form previous step, $u=120$ ). In this cloud of 40-dimensional points
one point was randomly chosen. This point, $b$, was used to create a new
indicator as described in section IV-E, i.e. $B(b^{\prime})$.

One shot learning was organized as follows: in the vicinity of the initial
indicator $A$, we searched with a sliding window for such ECG fragments that
would trigger that indicator $B(b^{\prime})$. In this way, we collected a
point cloud. After its dimensionality reduction this cloud became dense, shown
in Fig.6 on right. The center of this cloud was the ideal T-wave of a healthy
person, also shown in Fig.6.

Figure 6: An illustration of one shot learning in the proposed framework. In
the left it is shown the result of dimensionality reduction conducted as in
fig.4. The red dots, as usual, represent a sample of random ECG fragments of
length 40. The blue ones are chosen according to the control $u=120$ relative
to the R-peak in healthy patients. An arbitrary point from the blue points is
selected (shown by a black arrow). It corresponds to the ECG section shown in
the figure slightly to the right, and it plays a role of an indicator B. The
second ”clustering” is obtained in a slightly more complicated way, described
in text. Red dots again correspond to random ECG fragments, and blue ones are
obtained by cutting out a certain segment of ECG from the vicinity of point B.
The rightmost graph shows the center of the new blue cluster. It looks like a
part of a perfect T-wave.

The experiment demonstrates that one shot active learning in the proposed
framework is also natural.

##  VI Results and Discussion

A sketch of adapting the principle of saccadic vision to correct some of the
problems inherent in modern deep learning was considered. It has been shown
that the phenomenon of microsaccades occurs naturally in this model, which is
consistent with neuroiological data.

The theoretical part describes a learning framework in which the structure of
a deep agent grows as new patterns are discovered in the data. The search for
new patterns is active - through testing hypotheses about characteristic
controls. The growth of the model occurs in the course of an iterative
procedure for the formation of new indicators upon detection of characteristic
controls for indicators already existing in long-term memory. The long-term
memory of the described agent contains information about indicators and
controls, thus having a graph-like structure: indicators are nodes, controls
are edges, two indicators connected by an edge themselves become a new
indicator, that is, a node.

The experimental part shows how the $i+1$-th indicator ”grows” from the $i$-th
indicator using the example of the ECG. It is shown that if the first
indicator is on one of the characteristic waves, which are identified by
doctors, then the second also turns out to be on one of the characteristic
waves, which has its own name in the medical literature.

If we interpret the proposed recognition process from a Bayesian point of
view, then the agent always chooses the hypothesis with the highest
likelihood, so that if it is rejected, the Bayesian surprise will be great. An
agent built in the framework of the described approach (when controls are
explicitly stored in the agent’s memory) knows ”when it does not know”. This
is true because the recognition process is a series of hypothesis tests. If
each of the hypotheses is true, then the agent’s representation can be
trusted. If one of the hypotheses was not fulfilled in the experiment, then
this fragment of the signal (for which the saccade was intended) contains a
surprise, which is an instance of an unknown phenomenon to be studied (i.e.,
to create a new indicator).

Another possible advantage of the proposed learning ideology is that not the
entire model is trained at any given time. There is no need to set the
structure of an agent in advance, unlike many modern networks. The absence of
a pre-built network structure opens up the potential for overcoming
catastrophic forgetting. This is because it is possible to add new indicators
on top of the others into the any place of adaptively growing network
structure, since the layered structure no longer exists.

From a philosophical point of view, the advantage of this sketch is that at
each step of the agent’s recognition of the scene, the result of this
recognition represents a natural narrative, just as a path in a semantic graph
is a sentence[7] of a natural language.

The practical results presented in this work are the initial in a series of
experiments with ECG and other types of data in the direction of creating
learning algorithms in which the agent’s data representation contains
predictions of sensory control results, and the controls are explicitly stored
in the agent’s memory.

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# From cell intercalation to flow, the importance of T1 transitions

Harish P. Jain  Njord Centre, Physics Department, University of Oslo  Axel
Voigt  Institute of Scientific Computing, TU Dresden, 01069 Dresden, Germany
Center for Systems Biology Dresden, Pfotenhauerstr. 108, 01307 Dresden,
Germany  Cluster of Excellence, Physics of Life, TU Dresden, Arnoldstr. 18,
01307 Dresden, Germany  Luiza Angheluta  Njord Centre, Physics Department,
University of Oslo

###### Abstract

Within the context of epithelial monolayers, T1 transitions, also known as
cell-intercalations, are topological rearrangements of cells that contribute
to fluidity of the epithelial monolayers. We use a multi-phase field model to
show that the ensemble-averaged flow profile of a T1 transition exhibits a
saddle point structure, where large velocities are localised near cells
undergoing T1 transitions, contributing to vortical flow. This tissue
fluidisation corresponds to the dispersion of cells relative to each other.
While the temporal evolution of the mean pair-separation distance between
initially neighbouring cells depends on specific model details, the mean pair-
separation distance increases linearly with the number of T1 transitions, in a
way that is robust to model parameters.

††preprint: APS/123-QED

Introduction.-  Spontaneous flows within epithelial monolayers play a pivotal
role in many biological processes including tissue development, wound healing,
angiogenesis and invasion of cancer cells [1, 2, 3]. In confluency, cells may
migrate collectively while maintaining contact with their neighboring cells,
thus preserving cell-cell junctions [1]. However, cells may also migrate
relative to each other through structural rearrangements mediated by local
events of remodelling of cell-cell junctions. Three types of such events are
identified [4, 5, 6]. A T1 transition is an event whereby two neighbouring
cells move apart while two of their neighbours move towards each other and
make contact. A T2 transition occurs during extrusion/apoptosis events where
cells are eliminated from the monolayer, and a T3 transition is an event
associated with cell division. Notably, T1 events preserve the total cell
count within the monolayer, distinguishing them from T2 and T3 transitions.

T1 transitions have been observed in both epithelial [7] and mesenchymal [8]
tissues, and play an active role in development of an embryo from early
gastrulation to late-stage organogenesis [9], as well as during cancer
metastasis [10]. Empirical evidence shows that Myosin II contributes to the
build-up of tension at cell-cell junctions, thereby controlling T1 transitions
in epithelial tissues [11, 12]. However, the interplay between mechanical and
biochemical signaling in these events remains a topic of debate [13]. There
have been several studies that are aimed at characterising the mechanical
influence of T1 transitions in tissues [14, 6, 15, 16, 5, 17, 18] and in
simulations [15, 19, 20, 21].

Tissue flow has been explored mostly using the coarse-grained approaches from
active nematics or active polar matter or, more generally, active p-atic
liquid crystals, whereby spontaneous flows are influenced by their topological
defects [22, 23, 24]. Several experimental studies report different discrete
orientational order (nematic, polar or in general p-atic) of cell tissues,
corresponding to different lowest-energy topological defects [25, 26, 24].
This system dependency may be attributed to different types of cell lines, but
may also signal that the system is far from a hydrodynamic limit with a well-
defined discrete symmetry, thus making the identification of relevant group
symmetries and corresponding topological defects a matter of debate. On the
other hand, it is becoming more evident that T1 transitions are important
sources of tissue flow [27, 19]. Qualitative differences between flows
occurring at hydrodynamic scales and those induced by structural re-
arrangements of cells have been identified by comparing hydrodynamic
(continuum) simulations with cell-resolved (discrete) simulations [23, 28].
How the flows originating at the discrete cell level influence the tissue flow
at hydrodynamic scales is far less explored and understood.

In this paper, we aim to bridge this gap to better understand the generic flow
patterns due to cell neighbor rearrangements. Within a two-dimensional multi-
phase field model, resolving the dynamics of each individual cell, we quantify
the average flow profile of a T1 transition. We demonstrate that T1
transitions are short-lived and localised events of cells with high speeds
generating a $4$-fold vorticity with alternating chirality. Considering the
number of cells and their size to remain constant, by using a Lagrangian
approach, we quantify how the cell pair dispersion is mediated solely by T1
transitions. We predict a robust scaling of the mean pair separation with the
number of T1 transitions which hints at a generic mechanism of relative
dispersion by structural rearrangements. The details of the model are in the
Supplementary Material (SM) which includes the Refs. [29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39].

Figure 1: Configuration of $4$ cells (a) before, (b) during, and (c) after a
T1 transition. The 3-way vertices involved in the T1 are marked by red dots.
Angle $\zeta$ is the orientation of the newly-formed junction after a T1
transition measured counterclockwise from the positive $x$ semi-axis.
Streamlines of the average flow profile at the end of a T1 transition (d) and
(e). The colormap in (d) represents the magnitude of the polarisation, $p$.
$p=1$ at a location if for all T1 transitions in the ensemble, the velocity
points in the same direction at that location, and $p=0$ if velocity is
randomly oriented. The colormap in (e) represents the average speed. The flow
is visualized using LIC (line integral convolution), and the arrows indicate
the direction. The evolution of these fields before and after the T1
transition are shown in the Supplementary Movies 1-4, see SM [40]. Figure 2:
Temporal snapshots of the evolution of a cell triplet. T1 transitions break
the contact between cells in the triplet, and increase the area spanned by the
triplet.

## T1 transitions.-

A T1 transition re-configures the junctions between neighboring cells as
illustrated in Fig. (1a, 1b, 1c). The two 3-way vertex represented by red dots
move towards each other shrinking the corresponding junction (Fig. 1a). Within
multi-phase field models, a T1 transition is a spatial-temporal event with
finite duration [19], that starts when the junction vanishes, i.e. two 3-way
vertices merge creating a transient extracellular gap (approximated as a four-
way vertex), as shown in Fig. 1b. The T1 transition concludes when a new pair
of 3-way vertices nucleate forming a new junction. We define the orientation,
$\zeta$, of the T1 transition by the angle of the newly formed junction with
respect to the x-axis (Fig. 1a). We use this angle $\zeta$ to reorient the
different T1 transitions and the corresponding velocity profile in order to
compute ensemble-averaged properties as detailed in the SM [40]. The
corresponding flow field is reoriented such that the junction created after
the T1 transition aligns with the horizontal axis, see Fig. (1d, 1e). In these
figures, we illustrate the average flow profile at the end of a T1 transition.
This shows the saddle point structure with two orthogonal directions of
attraction (along the vertical axis) and repulsion (along the horizontal
axis), respectively. In turn, this is a source of quadruple vortices with
alternating circulations. The colormap in Fig. (1d) corresponds to the
magnitude of polarization, $p(\mathbf{x},t)$, which varies spatially between
$0$ and $1$. The polarization is $1$ when on average cell velocities point in
the same direction at a given position $\mathbf{x}$. Conversely,
$p(\mathbf{x},t)=0$, if there is no preferred direction at $\mathbf{x}$.
Notice that the non-zero polarization is localised around the T1 transition in
such a way that we can discern the ensemble-averaged shape of the $4$ cells
involved in the T1 transition. The polarisation field informs on the
likelihood of the observed streamlines, where a higher polarisation indicates
that the streamlines shown are more likely to occur. The colormap in Fig. (1e)
corresponds to the ensemble-average of the magnitude of the velocity field
around a T1 transition. Higher magnitudes are localised around the center,
where the cells involved in the T1 transition are present, and decreases in
the rest of the domain [41, 19]. The movies $1-4$ from the SM [40] show the
evolution of flow profiles before and after a T1 transition. The temporal
behavior is consistent with the cell speed profile before and after a T1
transition [19], whereby the typical cell speed increases during a T1
transition, and decays back to the average speed of all cells after the event.
This property along with the flow profile suggests that T1 transitions are
localised burst of high speeds, leading to vortical flow [19]. Cells that are
caged among their neighbours, undergo small shape fluctuations and experience
resistance to their motion due to intercellular interaction. However, these
small fluctuations may build up and can bring two 3-way vertices closer
together. This leads to accumulation of energy near those vertices.
Eventually, this can induce a T1 transition releasing energy, and thereby
helping uncage the cells. This is akin to the buildup prior to a slip event in
stick-slip dynamics [42]. Provided, there is no preferred orientation of the
T1 transition, the cells would all move away from their initial neighbours as
elucidated in the next section.

Relative Dispersion.-  The vortical flow induced by T1 transitions is
reflected in the dispersion of cells relative to each other, and thus in the
mixing of cells. We can quantify the flow properties indirectly through the
relative dispersion of cells. We use the center of mass denoted by
$\mathbf{r}_{i}$ for the $i$-th cell to track cell migration, see SM [40] for
details. To quantify single particle dispersion, we use the mean-squared
displacement of the cells given as

$\sigma^{2}(t)=\frac{1}{N}\sum\limits_{i=1}^{N}|\mathbf{r}_{i}(t+t_{0})-\mathbf{r}_{i}(t_{0})|^{2},$
(1)

which measures how far a cell has migrated from its initial location at
$t_{0}$ in a time-lag $t$. $N$ is the total number of cells. Likewise, to
quantify relative dispersion, we define the mean pair-separation distance in a
time-lag $t$ as

$\Pi(t)=\frac{1}{N_{P}(t_{0})}\sum\limits_{j=1}^{N_{P}(t_{0})}|\mathbf{r}_{j,1}(t)-\mathbf{r}_{j,2}(t)|,$
(2)

where $N_{P}(t_{0})$ is the number of pairs of cells that were neighbours at
an initial time $t_{0}$. $\mathbf{r}_{j,1}$ and $\mathbf{r}_{j,2}$ are the
positions of the two cells corresponding to the $j^{th}$ neighbour pair at
$t_{0}$. To better quantify the dynamics of the 3-way vertices, we also
consider the dispersion of triplets. A triplet consists of $3$ cells where
each cell is a neighbour to the other two cells. Let $\mathbf{r}_{k,1}$,
$\mathbf{r}_{k,2}$, and $\mathbf{r}_{k,3}$ be the positions of the $3$ cells
corresponding to the $k^{th}$ triplet. The corresponding lengths of the edges
in the triplet triangle are $\bar{a}=|\mathbf{r}_{k,1}-\mathbf{r}_{k,2}|$,
$\bar{b}=|\mathbf{r}_{k,2}-\mathbf{r}_{k,3}|$, and
$\bar{c}=|\mathbf{r}_{k,3}-\mathbf{r}_{k,1}|$. Given the perimeter of the
triangle, $\bar{s}=\bar{a}+\bar{b}+\bar{c}$, the triangle area of the $k^{th}$
triplet is $A_{k}=\bar{s}(\bar{s}-\bar{a})(\bar{s}-\bar{b})(\bar{s}-\bar{c})$.
Thus, we can also define the mean triplet-separation area as

$\Lambda(t)=\frac{1}{N_{T}(t_{0})}\sum\limits_{k=1}^{N_{T}(t_{0})}A_{k}(t+t_{0}),$
(3)

where $N_{T}(t_{0})$ is the number of triplets of cells at time $t_{0}$.
$\Lambda(t)$ is the average area of all triangles made by triplet of cells
that were mutually adjacent at time $t_{0}$. Snapshots from the evolution of
one such triplet triangle are shown in Fig 2. Geometrically, the dispersion of
cells is associated with deformations, translations and rotations of the
triplet triangle.

The initial area of the triplet triangles is non-zero and is related to the
cell size. Notice that the cells remain mutually adjacent for a transient time
until the 3-way vertex shared between them is destroyed in a T1 transition by
merging with another 3-way vertex. From hereon, the cells are likely to
further separate from each other due to subsequent T1 transitions. The triplet
area could also collapse to $0$ when the $3$ cells become collinear (see Fig.
2c-d).

Figure 3: Evolution of $\sigma^{2}$ (a), $\Pi$ (b) and $\Lambda$ (c) plotted
against time. $\sigma^{2}$ (d) and $\Pi$ (e) plotted against the cumulative
number of T1 transitions within the tissue. $\sqrt{\Lambda}$ plotted against
$\Pi$ (f). Different colours correspond to different cell activities defined
by the self-propulsion speed $v_{0}$ (see SM [40]), as per the legend at the
top.

The mean dispersion of single cells ($\sigma^{2}$), cell pairs ($\Pi$) and
triplets ($\Lambda$) as functions of time-lag $t$ and for different activities
$v_{0}$, see SM [40], are shown in Fig. 3a, 3b, and 3c. Within the considered
computational time, we see that cells are super-diffusive with a scaling
exponent that lies between $1$ (normal diffusion limit of Brownian motion) and
$2$ (ballistic limit of self-propulsion). For all three quantities, the net
dispersion increases with cell activity which is consistent with the
observation that the fluctuations in the tissue flow increase with activity
[22]. At low activities, we notice that there is almost no dispersion of cell
pairs and triplets. While, the pair-separation $\Pi$ increases with time-lag,
the triplet area $\Lambda(t)$ remains constant for a initial transient period.
A single T1 transition would destroy the 3-way vertex shared by the cells in
the triplet, but $2$ pairs out of $3$ initial pairs in the triplet remain. The
triplet area starts to diverge when at least one of the cells breaks contact
with the other $2$ cells in the triplet, which requires a minimum of $2$ T1
transitions. Thus, in this initial period, the area of the triplet triangle
$A_{k}(t)$ fluctuates around $A_{k}(t_{0})$. As the cells undergo further T1
transitions and move apart, the area spanned by the triplet $\Lambda$
increases monotonically with the time-lag.

Fig. 3d and 3e show the net dispersion of single cells ($\sigma^{2}(t)$) and
cell pairs ($\Pi(t)$) plotted against the cumulative number of T1 transitions
within the tissue in time-lag $t$. Interestingly, the curves for $\Pi$ for
different $v_{0}$ collapse into a linear master curve. A similar collapse is
also present for the triplet area, which shows that these two quantities are
directly related to the number of T1 transitions. However, this is not the
case for $\sigma^{2}$, because cells can also migrate in local flocks, which
affects $\sigma^{2}$, but does not change $\Pi$ and $\Lambda$ since the cell-
cell connectivity is preserved [1]. In the absence of T1 transitions, cells
are topologically caged within their neighbours, and this makes it difficult
to generate flow fluctuations (when T2 and T3 transitions are also absent). It
takes approximately $250$ T1 transitions across the tissue of $100$ cells, or
equivalently $10$ T1 transitions per cell, for the mean-pair separation to
double its value. Two neighbouring cells have to undergo around $10$ T1
transitions each, such that their cell centers could be two cell apart. When
$\sqrt{\Lambda}$ is plotted against $\Pi$, initially $\sqrt{\Lambda}$ is
constant, and later it varies linearly with $\Pi$ (see Fig. 3f).

To further explore the robustness of this data collapse, we also vary other
model parameters i.e. cell deformability $Ca$, rotational noise $D$, shape
alignment $\alpha$, adhesion parameter $a_{a}$, and activity ratio
$r_{active}$. For high values of $Ca$, the cells are more deformable. The
rotational diffusion coefficient $D$, and the rate $\alpha$ at which cells
tend to align their preferred direction of motion with the direction of their
elongation control the dynamics of the self-propulsion mechanism. For
$a_{a}=0$, the interactions between cells are purely repulsive, while
$a_{a}>0$ corresponds to additional attraction interaction between cells.
$r_{active}$ is the fraction of cells which have nonzero activity. See SM for
details [40]. The mean-pair separation distance as a function of cumulative
number of T1 transitions for these scenarios is plotted in Fig. (4). In all
these scenarios, we see similar scaling behaviour suggesting that T1
transitions affect relative dispersion in similar ways irrespective of model
parameters.

Figure 4: Data collapse of the relative dispersion curves as function of the
cumulative T1 transitions for various model parameters and setups: (a) cell
deformability ($Ca$) is varied for $v_{0}=1$. (b) rotational noise $D$ and
shape alignment $\alpha$ are varied for $v_{0}=1$. (c) repulsion vs repulsion
and adhesion cell-cell interaction. (d) mixtures of active and passive cells
with a fraction $r_{active}$ of active cells with $v_{0}=1$.

Discussion.-  In summary, we have studied how T1 transitions contribute to the
fluidization of an epithelial monolayer. We have shown that T1 transitions, as
topological events of neighbour exchanges, are sources of transient saddle-
point flows generating 4-fold vortices. The average polarization is high
around the principle axes of the saddle point, which corresponds to underlying
cells losing and gaining neighbours. Due to these flows, T1 transitions
promote cell mixing. While the temporal behavior of the cell dispersion
properties (single, pair or triplets) depends on the model parameters, we
found an underlying robust scaling law of the _relative_ dispersion (pairs and
triplets) as function of the number of T1 transitions. The data collapse of
the different dispersion curves onto a linear master curve suggests that the
rate of relative dispersion is directly proportional of the occurrence rate of
T1 transitions regardless of the underlying nucleation mechanism. This unveils
a deeper connection between topological rearrangements and mixing in densely
packed systems, in general. As $\Pi(t)$ and the cumulative number of T1
transitions are measurable within epithelial tissues and other computational
models, our results are accessible for experimental validation.

###### Acknowledgements.

L.A. and H.P.J. acknowledge funding by the European Union’s Horizon $2020$
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 945371. A.V. acknowledges funding by the German Research
Foundation (DFG) within grant FOR3013. We are grateful for discussions with
Amin Doostmohammadi, Richard D.J.G. Ho, Per Arne Rikvold and Dag K. Dysthe.

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