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Description
Recorded talk (15 minutes) for NetSci 2022 Conference. Slides are available here: https://figshare.com/articles/presentation/Hypergraphs_Demonstrate_Anastomoses_During_Divergent_Integration/20362773
A
A
So
the
big
idea
of
this
presentation
is
to
capture
embryo
dynamics,
and
you
can
see
two
examples
of
embryos
and
animations
below,
and
the
idea
is
to
take
this
biological
data,
these
dynamical
data
and
create
a
time
series
of
static
embryo
networks
with
spatially
localized
differentiation
and
we're
going
to
start
from
where
we've
been
classically
and
then
move
to
some
more
recent
work.
So,
first
of
all,
we'll
look
at
the
classic
embryo
network,
which
our
discrete
temporal
representations
ordered
axially
by
spatial
position,
and
this
has
been
presented
at
different
network
conferences
in
the
past.
A
So
this
is
a
three-dimensional
point
cloud
of
an
embryo.
This
was
published
in
biosystems
in
2018,
and
this
work
shows
that
there's
a
point
you
can
create
a
point
cloud
of
the
c
elegans
embryo.
You
can
use
cell
tracking
to
define
the
centroid
of
each
cell
and
you
can
create
plots
and
the
position
in
cell
lineage
data
can
be
derived
from
real
data.
That
collected
from
a
c
elegans
through
microscopy
and
these
create
spatiotemporal
models
of
embryogenesis.
A
The
connectivity
in
these
networks
are
based
on
distance
or
single
cell
interactions,
so
things
like
an
interactome
or
proximity
to
one
another
or
other
signaling
mechanisms,
and
so
you
can
see
that
in
this
case
we
have
a
zebrafish
embryo
and
we
have
a
number
of
different
criteria
we
can
use
for
connectivity.
So
these
are
all
the
cells
that
are
tracked
in
the
239
cell
embryo.
A
A
So
one
thing
that
came
out
of
the
earlier
work
was
this
model
called
the
density
bifurcation
model,
and
this
involves
four
steps.
So
the
process
of
increasing
connectivity
and
development
as
follows:
cells
might
divide
and
migrate
and,
as
such
connectivity
increases
and
the
graphs
that
I
showed
you
before
cell
migration,
which
is
the
movement
of
cells,
enriches
local
communities
and
cliques.
So
these
networks
are
enriched
in
different
ways
as
cells
move
around
over
time.
A
The
function
of
cells
diverge,
which
is
differentiation
in
a
biological
sense,
and
because
of
that
two
interconnected
networks
emerge,
then
interconnected
networks
provide
weak
ties,
so
the
connections
between
these
two
new
networks
provide
weak
ties
between
them
and
these
are
functional
interdependencies
and
these
can
be
thought
of,
as
maybe
functional
interdependencies
between
emerging
tissues
or
emerging
systems
or
subsystems,
and
so
this
is
where
we've
left
it.
Now,
I'm
going
to
present
work
that
kind
of
pull
pushes
this
forward
in
new
directions.
A
So
this
is
an
example
of
our
disembodied
networks.
We
have
you
can
see
that
the
connectivity
increases
over
time.
We
switch
the
threshold
at
some
point
so
that
we
don't
have
too
dense
a
network,
but
you
can
see
that
there
are
these
patterns
of
connectivity
that
emerge
over
developmental
time,
and
so
this
brings
us
to
this
idea
of
anastomoses.
A
So
an
anastomosis
is
something
that
connects
two
divergent
structures.
So
in
the
heart,
for
example,
it's
the
connection
between
heart
valves
or
in
the
gastrointestinal
system.
There
are
the
sphincters
at
the
entrance
and
the
exit
to
the
stomach,
and
so
these
different
systems,
the
esophagus,
this
stomach,
the
intestine
they're
all
connected
by
these
anastomoses,
so
they're
cross
connections
between
subgraphs.
A
So
this
is
the
developmental
hyper
graph
method.
This
is
present
in
a
preprint.
That's
just
been
recently
released
on
researchgate,
so
we're
trying
to
model
cell
division
and
differentiation
in
a
c
elegans
like
organism,
it's
not
c
elegans
per
se,
but
we're
building
something
that's
close
to
an
idealized
c
elegans,
and
the
reason
for
that
is
because
it's
easy
to
track
the
cells
mathematically
and
then
build
these
sub
networks
and
demonstrate
how
this
method
works.
A
So
this
is
why
we're
doing
it
this
way,
and
so
we
have
this
mosaic
development,
which
is
the
form
of
development.
That's
deterministic,
where
cells
divide-
and
we
know
what
the
fate
of
the
cell
and
the
fate
of
the
tissues
are.
We
don't
worry
about
signaling
or
anything
else
and
in
this
case,
embryogenesis
begins
as
a
single
hypernode
of
developmental
stem
cells.
So
each
one
of
these
nodes
with
the
numbers
in
them
those
nodes
are
hypernodes
and
inside
the
hypernodes
are
a
certain
number
of
cells
and
that's
what
that
number
represents.
A
A
But
then
you
get
differentiation
events
and
you
can
see
that
with
this
gray
lineage
at
the
top
right
on
top
right
hand,
side
of
this
slide,
and
you
can
see
that
you
have
this
gray
lineage
that
splits
off
from
this
yellow,
lineage
and
down
here.
You
have
this
gray
lineage
that
splits
into
a
orange
lineage
in
a
white
lineage,
and
so
you
can
see
that
there's
a
division
there
and
we
refer
to
that
as
a
differentiation
event,
and
so
we
can
see
that
there's
a
bifurcation
into
subgraphs
as
differentiation
occurs.
A
But
the
other
thing
too,
about
this
hypergraph
method
is
that
cells
can
transition
between
subgraphs.
So
they
can
form
these
anastomoses.
They
can
cells
can
start
in
one
sort
of
sublineage
and
move
to
another
sublineage.
This
is
analogous
to
differentiation,
redifferentiation
or
some
other
type
of
specification
and
development.
A
A
So
we
have
these
anastomoses
between
neural
precursors
and
embryo
networks,
as
opposed
to
the
germline
which
remains
separate
and
doesn't
have
any
anastomoses
with
anything
else.
Then
we
can
actually
create
a
spatiotemporal
analog
to
this
too.
So
we
don't
just
need
to
worry
about
time
in
this
version
of
the
graph.
We
have
time
and
we
have
these
anastomoses
and
we
have
these
subgraphs.
In
this
case
we
have
a
spatiotemporal
hypergraph.
So
what
does
that
look
like?
A
A
You
have
the
256
cell
layer
and
you
have
the
lineage
markers
going
downwards
from
one
set
of
categories
or
set
of
hypernodes
to
the
next,
and
so
you
can
see
that
the
128
cells
are
divided
up
quite
differently
in
the
in
the
120
in
the
in
what
we
call
layer.
Seven,
because
it's
seven
steps
away
from
the
two
cell
stage.
A
It's
very
distributed
very
different
differently
and
has
very
a
lot
fewer
categories
or
or
hypernodes
than
layer
eight.
And
so
you
can
see
the
left
right
axis
going
from
top
to
bottom.
It's
a
little
bit
strange
because
we
don't
have
that
third
dimension
and
then
the
anterior
posterior
axis,
which
is
front
to
back,
and
so
you
can
see
that
you
have
many
more
of
these
orange
hypernodes
in
the
256
cell
stage.
You
get
definition
of
different
different
cell
types
and
subtypes,
and
this
is
flowing
over
developmental
time.
A
But
you
can
see
the
logic
here
now
when
I
say
I
started
to
say
category,
and
so
when
I
started
to
say
category
I
mean
one
of
these
hypernodes,
and
why
do
I
say
category
well,
as
it
turns
out
cell
cell
type,
is
a
very
hard
problem
in
biology
developmental
biology,
in
particular,
because
it's
hard
to
really
understand
what
a
cell
type
is.
You
can
use
genotypic
markers,
you
can
use
different
types
of
phenotypic
markers,
but
ultimately
we
have
many
cell
types
that
we
don't
really
know
what
they
are.
A
So
as
a
nice
heuristic,
we
use
categories
and
we
map
those
categories
to
these
subgraph.
These
these
hypernodes
and
they'll
form
sub
groups
of
them
will
form
some
graphs.
So
what
we
need
to
do
is
introduce
some
sort
of
categorical
representation,
and
so,
in
this
case,
we're
looking
at
categories
x
and
y
that
change
their
membership
by
what
we
call
functors.
A
So
we're
actually
applying
category
theory
to
this,
and
so
this
is
a
map
here,
a
categorical
model
of
a
single
lineage
tree
layer,
and
so
we
have
sources
coming
up
from
the
bottom
they're
coming
up
to
this
layer
and
then
they're
going
to
the
next
layer.
So
you
have
events
hereditary
events
where
they're
coming
from
the
most
recent
ancestral,
hypernode
they're,
passing
on
to
the
daughter,
hypernode
and
then
they're
passing
on
to
its
daughter's
hypernode.
A
But
additionally,
you
can
have
these
fx
and
fy
functors,
which
are
exchanges
between
x
and
y.
So
cells
can
change
their
change
their
state.
They
can
change
your
identity
even
within
the
same
division
event,
and
so
for
each
layer
specific
category
x
y.
They
can
exchange
cells
with
self-similar
categories
in
the
next
layer
of
the
tree,
using
your
heredity
functor,
which
these
arrows
outward
from
this
diagram.
A
They
can
also
exchange
cells
with
each
other,
using
a
transformation
functor
which
is
within
this
layer,
and
so
now
the
future
directions
for
this
work.
There
isn't
really
mapped
to
a
lot
of
data,
but
it
does
have
a
lot
of
use
and
it
has
particular
use
in
terms
of
embodying
these
networks
in
an
input
output
setting.
A
So
we
can
look
at
growth,
form,
growth
in
form
of
of
neuronal
networks
or
connectomes,
and
so
network
neuroscience
2022.
I
presented
on
some
work
on
embodied
networks,
and
so
these
are
originally
supposed
to
be
embedded
in
some
sort
of
agent
like
a
breitenberg
vehicle
or
some
sort
of
agent
that
has
a
sensory
organs
and
effector
organs,
so
you
can
think
of
them
as,
like.
A
You
know
limbs
as
the
effector
organs
and
eyes
or
ears
as
the
sensory
organs.
So
the
way
to
read
through
these
graphs
is
to
say
I,
which
is
the
input,
is
a
hypernode
for
a
single
organ.
So
you
have
two
eyes
basically,
and
each
eye
has
a
number
of
cells
within
it,
and
then
o
is
the
output
and
there
are
a
number
of
cells
within
that
hyper
hypernode
and
that's
representative
of
like
a
foot
or
a
hand
or
something
like
that
or
a
wheel
in
the
case
of
a
breitenberg
vehicle.
A
Each
of
these
three
graphs
increase
in
complexity,
so
we
start
with
a
simple
mapping
between
input
and
output.
We
end
up,
then,
with
a
bunch
of
interneurons
loosely
defined
as
these
hypernodes
that
sit
in
between
the
input
and
output,
and
then
we
have
a
network
of
intermediate
hypernodes
or
interneurons
in
between
the
input
and
output.
A
I
showed
you
so
this
is
another
example
where
we
can
use
these
networks
to
characterize
the
growth
of
cells,
the
growth
of
number
of
cells
and
different
parts
of
the
phenotype,
how
they're
connected
together
and
then
these
anastomoses
between
different
parts
of
the
network,
so
that
sub-graphs
and
these
sub-graphs
represent
like
different
parts
of
a
brain,
different
nuclei
or
centers
in
the
brain,
and
they
all
are
kind
of
semi-independent
but
they're
all
linked.
And
so
you
can
actually
do
a
lot
of
things
with
these
graphs.
A
The
number
of
cells
in
each
of
these
nodes
grow
at
a
differential
rate
and
then
in
this
case
we
have
a
mother,
hypernode
and
daughter
hypernodes.
The
daughter
hypernodes
are
divisions
of
the
mother
hypernode
and
they
contain
they
inherit
the
cells
from
those,
but
they
also
divide
as
well.
So
you
have
division
of
number
of
division
events
that
occur
from
the
mother
to
the
daughter,
but
they
inherit
those
cells
and
their
identity,
and
so
you
can
trace
out
these
hereditary
lineages
across
this.
A
So
it's
a
very
busy
diagram,
but
that's
what's
going
on,
and
so
we
can
actually
characterize
a
lot
of
different
phenomena.
We
can
characterize
growth
heterocrony,
which
is
where
the
growth
rate
changes
over
developmental
time,
sequence,
heterochronic,
where
the
sequence
of
events
changes
over
time
and
then
critical
periods
where
things
are
required,
where
things
have
to
be
in
place
at
a
certain
point
in
time
in
the
nervous
system,
develops
sort
of
this
enhanced
plasticity.
So
we
can
capture
all
of
these.