►
Description
Marcus Lewis presents an unsupervised learning technique that represents inputs using magnitudes and phases in relation to grid cells. He proposes an alternate view of grid cells that enables the creation of maps of novel environments and objects in a predictive basis.
Marcus first gives an overview and his assumptions on the core pieces of the algorithm and provides examples to support his viewpoint on grid cells in mini columns. He then presents a simulation and shows how the technique can be implemented in artificial neural networks and biological tissues.
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A
Okay,
we're
started
so
hey,
I'm
here.
Here's
the
the
topic
I'm
presenting
today,
so
I've
been
kind
of
building
this
up
in
the
back
of
my
mind
for
a
while,
a
couple
different
papers
in
the
past
have
influenced
me
on
this,
and
and
I'm
broadly,
what
I'm
presenting
today
is
just
an
unsupervised
learning
technique
that
uses
some
good
cell
tricks
and
kind
of
it
uses
the
fundamentals
of
grid
cells
in
some
kind
of
unexpected
ways.
But
then
I
loop
back
and
say
like
wait.
A
So
what
I'm
presenting
today
I'll
just
read
off
the
first
bullet:
it's
an
unsupervised
learning
technique
that
represents
inputs
using
magnitudes,
for
example,
firing
rates
and
phases.
For
example,
the
point
on
a
ring
or
the
point
on
a
torus
or
a
rhombus
like
good
cells,
and
these
phases
can
be
spatial
phases
like
grid
cells
or
they
can
be
temporal
sort
of
like
a
velocity,
controlled
oscillator.
There's
there's
a
lot
of
flexibility
on
exactly
what
these
systems
are
and
I'll
talk
a
little
bit
more
about
that.
A
But
it's
an
unsupervised
learning
system
that
fundamentally
has
magnitudes
and
faces,
and
the
pitch
as
like
a
machine
learning
or
an
ai
system,
is
that
it
represents
input
in
a
way
that
incorporates
how
the
input
input
may
change
over
time
and
the
neuroscience
motivation
is
that
it's
an
unsupervised
algorithm
that
may
underlie
grid
cells
and
potentially
other
cells.
I
draw
a
connection
to
visual
cortex
here
as
well,
and
also
it
shows
a
use
case
for
unexplained
properties
of
grid
cells.
A
A
So
so
this
this,
what
I've
labeled
the
conventional
view
on
the
left
is,
is
how
most
people,
how
a
lot
of
people
think
of
good
cells
and
certainly
a
way
we've
framed.
It
is
that
grid
cells
are
sort
of
representing
a
pure
location
that
doesn't
really
depend
directly
on
the
sensory
input
and
and
and
I've
made
this
figure
to
look
a
lot
like
our
models.
This
this
intentionally
resembles
the
spatial
pooler,
but
but
other
people
who
have
modeled
this
would
would
describe
it
in
similar
ways.
A
I
am
adopting
an
alternate
view
that
the
the
grid
cells
themselves
are
are
creating
a
basis
they're
they're
they're,
representing
input
in
a
basis
where
the
like,
with
magnitudes
and
phases,
collectively
representing,
what's
out
there
in
a
way
that
general
generalizes
to
novel
environments.
So
on
the
bottom
I've
written,
I
have
to
move
around
zoom
ui.
As
I
talk
about
this,
this
is
strange
so
on
the
bottom,
I've
written
the
the
focus
of
the
model
on
the
left.
A
We
sort
of
frame
the
task
at
solving
as
learning
and
recognizing
environments
and
predicting
and
learned
environments,
and
with
the
view
on
the
right,
I'm
more
taking
on
the
view
of
representing
novel
environments
and
being
able
to
predict
in
novel
environments
and
and
by
the
way
you
can.
Of
course,
it
goes
without
staying
with
this
crowd,
but
you
can
substitute
environments
with
objects,
etc.
A
So
I've
I've
drawn
this
below
on
the
left
two
options.
You
have
rings,
you
have
tauruses
or
rhombuses,
and-
and
you
you'll
see
that,
like
I've,
subtly
color-coded
these
so
as
if
they're,
some
of
them
are
responding,
they're
responding
more
vigorously
than
others
and
this.
The
second
piece
of
this
algorithm
is
the
objective
that
a
time
series
input
coming
into
the
network
should
tend
to
cause
smooth
linear
updates
to
the
phases,
and
I've
depicted
this
as
a
set
of
dots
that
are
equally
spaced
apart
within
a
module,
but
they
can
be
different
across
modules.
A
B
B
A
B
You
know
this
is
you
know,
it's
very
interesting
because
you
know
a
whole
bunch
of
research
talking
about
how
the
hippocampal
complex
processes
and
remembers
time-based
sequences.
You
know
your
your
memories
and
yet,
at
the
same
time
it
can
recognize
objects,
and
it's
on
my
reading
list
to
try
to
figure
out
how
do
they?
How
does
it
do
both?
And
I
it
seems
like
you're,
trying
to
do
something
like
that
here
too
right,
yeah,
yeah,
yeah,
okay,.
A
So
I
I
list
these
as
the
two
pieces,
but
now
I'm
going
to
list
a
third
kind
of
optional
piece,
a
third,
an
obvious
extension
to
this
is
rather
than
using
time
use
movement
as
the
transition.
Sorry,
if
you
hear
a
siren
behind
me,
so
so
I'll
read
the
text
here,
an
additional
alternative
objective,
using
motor
input
or
optic
flow
input
or
something
equivalent
it.
A
B
B
A
C
B
A
C
A
So
moving
on
my
wordiest
slide,
where
I
state
evidence
from
grid
cells-
and
that
has
influenced
me
toward
this-
and
I
also
point
out
some
nice
theoretical
insights
from
other
people.
So
so,
let's
see
this
is
divided
into
two
sections
and
this
this
top
one,
the
computational
substrate.
So
the
fact
that
I'm
using
these
rings
with
magnitudes
the
first
bullet
point,
argues
why
I'm
using
rings
or
toruses.
So
the
the
thing
that
I've
found
a
really
important
fact
that
that
I
would
say,
is
not
conclusive.
A
Yet
I
haven't
seen
a
paper
that
utterly
convinces
me
of
this,
but
people
all
seem
to
be
suggesting
it,
and
that's
that
even
when
grid
cell
rate
maps
are
distorted,
the
relative
phases
of
the
cells
are
preserved.
So
it
really
is
like
a
population
of
cells
that
is
being
compressed
and
shaped
into
different
environments.
A
Yeah,
as
as
opposed
to
other
viewpoints
on
grid
cells,
don't
think
of
them
as
being
a
a
module
of
cells
with
different
phases.
The
more
think
of
it,
as
this
is
a
bunch
of
cells
that
have
independently
learned
principal
components
and
they
create
an
over
complete
basis
set,
but
it's,
but
it's
not
doing.
There's
no
rhombus
or
there's
no
there's.
C
A
A
I
I
think
that
I
think
the
crowd
is
leaning
toward
this
side
of
things,
but
I
don't
think
it's
conclusive
yet
so
the
the
next
three
bullet
points
are
why
I'm
bringing
in
magnitudes.
So
one
of
them
is
the
the
study.
We've
talked
about
this
a
little
bit
before
this
rat
from
this
this
group.
It's
called
grid
cells
and
code,
local
positional
information
and
that's
the
and
in
it
they
showed
that,
like
rats
rats,
moving
around
an
environment
in
different
firing
fields,
a
given
grid
cell
will
fire
differently.
A
You
could
you
could
say
that
they
fire
different
firing
rates
in
the
fields
or
maybe
maybe
that's
well,
they
just
don't
fire
at
all.
Yeah
yeah,
that's
true
too,
in
general.
Extra
information
can
be
decoded
from
it.
So
I
don't
want
it.
I
don't
want
to
prescribe
that.
It
is
not
the
firing
rates,
it
might
be
something
else
about
the
spikes
but
but
but
they
they
do
seem
to
encode
extra
information
and.
A
Yeah,
I
think
that
those
are
two
two
examples
of
the
same
phenomenon:
the
tank
paper,
this
one
I
particularly
like,
because
it's
a
rat
navigating
in
2d,
where
the
tank
paper
is
mice,
moving
on
1d
tracks
virtual
and
they
reconstruct
the
2d
map
from
that
so
which
is
probably
true
it's
just.
This
has
less
statistics
going
into
it.
It's
just
a
basic
observable
effect.
A
So
now
now
two
more
cases
for
for
magnitudes.
One
is
just
straight
from
things:
I've
presented
before
that.
If
you,
if
you
draw
an
analog
from
grid
cells
to
the
fourier
transform
you
can
you
get
an
idea,
it
just
kind
of
gives
you
some
intuition
about
how
good
cells
could
do
more
things
if
they
had
both
a
magnitude
and
a
phase.
I
I'm
not
here
to
say
that
grid
cells
are
doing
exactly
a
fourier
transform,
but
but
it's
a
useful
analogy
to
help.
A
You
just
see
things
and
it
kind
of
inspired
the
simulation
that
I
end
up
running
here.
Oh
and
now
the
the
last
point
I
I'll
read
it
magnitudes
enable
a
connection
to
orientation,
columns
and
v1
this
one.
This
motivation
comes
straight
from
I
mean
jeff
has
proposed
literally
like
40
times
that,
like
somehow
these
are
the
these
mini
columns
with
orientation
selectivity.
A
We
want
to
find
a
way
for
them
to
be
good
cell
modules.
They
seem
like
they
could
be,
and-
and
this
viewpoint
I
think,
makes
let's
see
it
now.
I
see
a
way
that
that
could
be
true
and
and
I'll
show
some
like
nice.
Some
nice
initial
results
that
point
toward
that
now
now
the
the
objective.
These
final
two
bullet
points,
the
so
back
to
the
topic
of
distortions.
A
This
is
more
of
a
theory
paper,
but
it
really
influenced
my
thinking
on
this
and
and
they
they
had
a
certain
model
that
leads
to
distorted
grid
cells,
I'm
not
using
their
model,
as
is
because
they're
they're
an
example
of
the
people
who
kind
of
moved
away
from
the
rings
or
the
torus
and
and
I've
kind
of
moved
back
in
that
direction
of
bringing
the
ring
and
the
taurus
back.
But
but
this
paper
really
did
influence
my
thinking
and
it
kind
of
prepared
me
for
this.
The
second
paper
can.
B
A
I
think
you
can
make
cases
for
both.
I
would
say
it's
I'd
say
it's
mostly
whether
it's
possible
to
take
it's
weird,
because
it
depends
like
if
this
animal
has
has
done
random,
walking
in
an
environment
like
this
before
it
might
do
one
thing,
but
if
it
hasn't
it
might
do
another
thing,
it's
complicated.
You
know
I
can
make
cases
refer,
but
for
both
I
can't
give
you
a
stroke.
B
Okay,
well
good,
you
said
you,
you
opened
the
idea,
it's
possible
because
that
that
it's
a
sort
of
a
relationship,
an
idea
I've
been
thinking
about
which
is
like
you
know,
the
space
is
defined
by
movements.
It's
not
the
spine
by
some
defined
by
some.
You
know,
euclidean
metric,
it's
defined
by
movements
and
therefore
the
movements
can
what
movements
are
available
at
any
point
in
time
can
vary
depending
on
where
you
are,
and
so
that
that
that
is
a
way
of
explaining
distortions
in
grid
fields.
But
but
I.
B
D
B
D
A
D
A
B
So
I
I
still
think
it's
it's
a
map
of
the
environment.
It's
getting
back
to
what
I'm
saying
before
you,
the
map
of
something
includes
the
sensor
or
the
motor
affordances
of
it
right
it.
It's
it
doesn't
make
it
less
of
a
map.
It
just
makes
it
like
an
it
makes
it
non.
You
move
away
from
a
you
know,
strict
metric
space
of
euclidean
geometry,
or
something
like
that
right
it,
but
it's
still
a
map.
It's
just.
You
can't
separate
the
map
or
the
the
model
from
the
behaviors.
B
That's
how
that's
how
I've
been
thinking
about?
I'm
not
sure.
If
that's
where
you're
going
here,
it
sounds
like
that
you're
going
the
same
direction,
so
I
don't
think
this
throws
away
the
idea
that
we
had
before
I
just
before
we
had
sort
of
a
naive
these
in
my
mind,
I
had
a
naive
idea
that
the
you
know
these
metric
spaces
of
department.
B
A
A
So
this
brings
me
to
the
the
final
point.
I
feel
like
this
first
paper
kind
of
prepared
my
mind
for
this,
and
this
final
bullet
point
gave
me
a
good
objective
that
I
could
that
I
could
use
this
objective
of
of
of
time-based
sequences
should
tend
to
cause
straight
paths.
Is,
is
something
that
I
got
from
this
yube
and
dylan
payton
and
bruno's
group
at
berkeley.
They
did
the
sparse
manifold,
transform
and-
and
my
objective
is
very
similar
to
theirs.
They
didn't
use.
A
A
They
were
more
focused
on
successive
successive
operations
and
visual
cortex
and
how
it
they
call
it
flattening
the
manifold
and
how
that
enables
you
to
perform
classification
and
regression
and
which
it
was
a
really
interesting
paper,
but
it's
a
slightly
different
motivation,
but
but
but
their
objective
was
thought-provoking
to
me
and
I'm
using
it
basically
so
this
this
computational
substrate
this
ring
with
a
dot
that
has
a
certain
magnitude.
There
are
many
ways
to
implement
it,
and,
and
so
so
you
can
almost
think
of
it
as
like.
A
A
You
could
do
it
with
mini
bursts,
which
is
a
little
bit
like
firing
rates,
but
I
still
think
of
it
as
separate.
It's
like
a
qualitatively
super
spike.
It's
like
it's
like
a
different
kind
of
spike
or
you
could
do
it
with
a
population
code.
If
you
have
multiple
cells
at
the
given
phase
and
more
more
of
them
fire
giving
given
a
higher
magnitude,
that
could
do
it
too,
and
that
really
would
work
with
the
vco
approach.
A
A
This
is
clearly
the
real
and
imaginary
components
of
this
like
in
some
ways.
These
are
two
ways
of
saying
the
same
thing.
There
are
always
two
ways
to
express
complex
numbers,
and
these
are
those
two
ways,
but
but
you
don't
you
don't
have
to
bring
in
the
notion
of
complex
numbers
to
get
this
to
work.
So
so
in
my
experiment,
I
actually
just
do
this.
I
take
pairs
of
units
and
call
those
collectively
a
ring
and
I'll
show
that
in
a
figure
soon.
E
A
C
A
So,
in
some
ways
like
this,
this
slide
is
like
I.
I
basically
said
this
at
the
beginning
of
this
slide
that
that,
in
some
ways
you
this
this
is
like
the
the
the
actual
substrate
that
provides
this
at
this
abstraction,
and
now
I'm
going
to
build
things
with
this
abstraction.
Now,
I'm
going
to
accomplish
things
with
this
abstraction,
so
just
to
kind
of
prime
your
mind
for
for
an
experiment
with
rats
moving
around
in
a
more
realistic
environment
with
with
like
foliage
with
with
with
leaves
and
plants
all
around
it.
A
I
I
just
thought
I'd,
prime
you
with
this.
Okay,
that's
that's!
That's
all
I'm
going
for
this
slide,
so
so
the
the
simulation
that
I
ran
here
is
an
agent
moving
through
randomly
generated
open
environments
with
landmarks
in
it
and
I'm
actually
calling
the
landmarks
foliage
and
the
inputs
to
the
system.
A
I
I
I
intentionally
chose
these
object,
vector
cells,
because
because
they
have
this
fun
property
for
this
demo,
that
that
even
that,
okay,
first
of
all,
for
those
of
you
who
don't
know
or
don't
remember,
an
object
vector
cell
fires
when
the
rat
is
at
a
certain
location
relative
to
an
object.
And
it's
it's
not
a
it's.
Not
it's
not
a
specific
object.
It's
more
just
objects
in
general
and
and
and
the
rat
doesn't
have
to
be
facing
it.
It
can
be
facing
any
direction.
A
So
it's
like
it's
allocentric
location
relative
to
the
object,
and
so
I
I
basically
I
end
up
using
these
as
my
input
to
the
network
and
they
basically
say
where
the
foliage
is
relative
to
the
animal,
but
in
an
allocentric
direction,
and
the
objective
as
this
animal
runs
around
the
runs
around
the
environment
is
the
first
sentence
is
a
little
hand
wavy
I'll,
come
back
to
it.
Do
a
good
job
of
representing
the
input.
A
A
A
I
could
have
framed
this
an
image
being
translated
task
or
a
camera
moving
across
an
image
they're
all
the
same
experiment,
because,
because
object
vector
cells
are
in
an
allocentric
reference
frame,
you
can
actually
just
treat
them
all
as
the
same
experiment,
and
so
so
you
so
I've
described
it
as
an
agent,
but
you
could
you
could
reinterpret
it
to
think
of
this
as
a
visual
cortex
task
as
well,
and
so
so
the
way
the
network
works.
So
once
again,
super
simple:
this
is
a
one
layer
network.
A
B
Okay,
I
mean
that
was
a
very
intriguing
thing.
You
said
earlier
about
how
you're
going
to
take
advantage
of
these
magnitudes,
and
so
I
didn't-
I
didn't
know
what
I
meant
now
in
this
situation.
Are
you
suggesting
that
the
magnitudes
themselves
are
the
object,
encoding
or
the
environment?
Encoding
means
that
that
that
so
the
rings
could
be
shared
among
the
the
rings
themselves.
The
phases
can
be
shared
among
different
environments,
but
the
but
there'd
be
unique
coatings
by
magnitude
per
environment.
Is
that
what
you
just
said.
A
That's
what
that's
what
happens
in
this
experiment?
Yes,
that's
what
happens
in
this
experiment.
I
don't
want
to
push
that
on
you
too
hard,
because
you'll
see.
One
thing
I
made
easy
on
myself
in
this
experiment
is
that
the
image
the
the
foliage
never
goes
outside
the
bounds
of
this
viewport,
that
the
animal
can
always
see
the
whole
thing.
B
A
A
A
A
But
but
that's
maybe
that
will
prove
useful,
but
that
I
didn't
want
to
push
that
on
you
too
hard,
because,
as
I
mean
I
pointed
to
like
you
brought
up,
I
pointed
to
that
experiment
with
the
wrap
moving
and
how
the
different
fields
have
different
firing
rates,
which
is
not
what
I'm
showing
here.
So
so
that's.
B
Well,
it's
consistent.
Those
different
firing
rates
are
consistent
at
any
location
in
the
in
the
rats.
Maze,
yeah
right,
so,
okay
grid
cell
number
42
is
not
flying
right
now.
Well,
that's
weird,
but
it
never
fires
at
that
location,
but
it
you
know
so
it
is.
It
is
it's
very
specific
to
location.
So
you
know
that's
a
a
big
clue
and
yeah
okay.
A
Okay,
so
in
my
next
slide
I
have
an
actual
video
of
this
running.
What
should
I
tell
you
yeah,
but
before
I
run
the
video
this
is
I've
labeled
it,
but
just
to
make
sure
you
know
here
I'll,
even
show
it
running
and
then
I'll
repeat
one
as
I
explain
it.
So
here's
the
actual
input
to
the
network,
a
set
of
pixels
being
input
to
the
network.
Here's
a
basic
reconstruction.
A
It's
it's,
I
wouldn't
say
the
reconstruction's,
overly
impressive,
because
it's
just
a
linear
network
with
just
a
small
number
of
cells,
but
I
wasn't
really
pushing
it
hard
to
do
a
good
job
on
reconstruction,
but
but
anyway
that
so
this
could
do
better
if
I
pushed
it
harder.
But
what
I
really
wanted
to
focus
on
this
last
figure.
It
might
be
intuitive
to
you
immediately
or
you
might
need
some
just
some
explanation.
A
B
So
so
what
I'm
missing
here
is
I,
I
guess
I
you
know
you
had
the
picture.
Okay,
we
got
the
a
and
b
neurons
representing
a
ring
and
you're
plotting
that
here
I
get
that.
I
don't
understand
how
the
system
takes
its
input
and
learns
to
do
this.
I
mean
what
how
does
that
visual
input
lead
to
this
these
these
movements?
A
B
B
And
when
you,
when
you
showed
the
when
you
showed
the
the
very
first
picture
in
your
presentation,
you
had
that.
But
then,
when
you
showed
the
idea
of
two
cells
representing
a
ring,
you
didn't
show
that
input
to
those
cells.
So
I
forgot
that
so
all
right,
so
these
cells
already
are
being
driven
by
sensory
input,
and
this
is
what
their
filters
look
like.
Yes,.
A
Yes,
so,
and
the
another
point
that
I
would
be,
I
would
expand
the
slide
into
multiple
slides
with
multiple
figures.
So
you'll
have
to
imagine
this
figure
that
here
I'm
plotting.
Yes,
these
two
artificial
units
and
they're
they're
they're
the
same
filter
translated.
This
is
the
same
filter
translated
now,
if
you,
if
you
switch
your
mind
over
to
the
biological
ring
version
of
this,
where
it
doesn't
just
have
two
units,
it
has
a
bunch
of
units
at
different
points
of
the
ring.
A
A
D
A
I'm
doing
it
because
I'm
not
forcing
anything,
but
what
I'm
doing
is
my
so
so
those
rings
I
showed
here.
I
can
go
back
to
the
previous
slide.
Just
these
rings
I'll
just
be
like
very
specific
right
now.
My
my
loss
function
is
based
on
the
values
of
these
rings
is
based
on.
I
actually
compute.
These
phases
figure
out
how
the
phase
is
changing
and
my
loss
function,
optimizes
that.
A
D
B
A
B
A
Experiment:
it's
it's
more!
It's
it's!
It's
it's
allocentric,
because
if
you
think
of
these
okay,
like
I'll,
give
you
two
answers.
If
you
have
your
mind
right
now
in
the
space
of
v1-
and
this
is
a
camera
receiving
input
totally,
it's
egocentric,
if
you
have
your
mind
in
enter
rhinocortex
with
object,
vector
cells.
Being
these
pixels
because
object,
vector
cells
are
basically
pixels
as
seen
overhead.
B
C
D
A
A
B
A
Yeah
bringing
motor
input
into
it
well
like
bringing
motor
into
input
to.
It
is
certainly
a
to
do
item
here,
but
I
also
think
it
has
some
sort
of
obvious
ways
we
can
do
it.
We
updating
rings
using
motor
input
is
not
a
is
not
a
hard
thing
to
think
about,
like
this
kind
of
provides
a
natural
extension
point
for
that.
It's
not
like
it's
a
big
shortcoming.
It's
just
not
doing
it
yet.
B
A
B
A
B
A
a
complex
cell
in
v1.
Well,
this
far
is
faster.
The
faster
the
thing
is
moving.
The
line
is
moving
right,
so
I
was
just
I
was
just
saying:
oh
so
it
just
seemed
like
you
had
all
the
ingredients
to
do.
Speed
like
you
could
speed
it
up
and
slow
it
down,
and
these
filters
would
get
higher
and
lower
magnitude,
and
I
didn't
walk
you
through,
and
I
said
oh
does
this:
does
this
all
work,
does
it
does
it
make
the
correct
prediction?
B
B
Oh,
we
move
faster
through
the
ring
yeah,
but
why
I
mean
I
imagine
those
those
two
cells
representing
the
two
dimensions
of
your
ring
right,
you're,
whether
you
know
the
in
the
in
the
non
complex
number
example
yeah.
I
would
just
assume
that
the
ring
would
get
larger.
I
don't
know
I'm
confused.
I
have
to
think
about
it
more
all
right
anyway.
What
really
motive
me,
I'm
saying?
Oh,
how
would
I
incorporate
changes
in
orientation?
We're
talking
about
movements?
Well
the
movement's
really
just
two
things
you.
B
If
I'm
a
sensor,
I
can
change
my
orientation.
I
go
faster
and
slower,
you
know
and
I
can
change
my
orientation
faster
and
slower,
and
I
can
go
you
know,
and
so
I
was
trying
to
imagine.
Well,
there's
no
orientation
changes
in
here.
Yet
you
don't
have
anything
like
that.
We
talked
about
that
and
then
I
was
wondering
well
does
it
handle
the
speed
like
speed,
cells
and
hippocampus
and
and
I'm
just
thinking
out
loud,
so
maybe
we
should
just
go
on
yeah.
E
E
E
Okay,
so
when
you
say
the
six
modules
filters,
I'm
sorry
I'm
just
having
a
hard
time
mapping
those
filters
to
the
representation
of
the
state,
so
the
six
modules
filters
are
basically
a
a
mapping
over
this
this
over
the
field
of
these
grid
cells.
Is
that
what
I'm
hearing
or
what
is
it
their
response
over
this
field?.
A
A
E
So
so
that's
okay,
so
you
had
a
64
over
64.
input.
This
is
this:
is
the
weightings
of
okay,
the
response
or
whatever?
So
those
are.
Those
are
static.
Placement
of
those
filters
so
that
they're
they're
constant,
no
matter
where
what
position
of
the
the
environment's
in
is
that
is
that
what
I'm
getting.
B
B
A
Yeah,
I
mean
the
only
the
only
part
where
I
actually
demonstrate
the
prediction
is
what
I
just
talked
about
with
how
you
can
just
extrapolate
oops
you
can
just
extrapolate.
You
just
saw
a
spoiler
of
what's
coming,
extrapolating
where
the
bumps
are
all
going.
That's
the
that's
the
extent
of
the
prediction
I
see,
I
see.
Okay,
thank
you!
A
So
now
I'm
going
to
jump
into
the
the
tying
this
to
v1
thing.
So
the
what
what
happens
here
when
you
train
like
this
and
you
use
the
biological
view
of
the
ring.
What
you're
getting
is
these
clusters
of
cells
with
shared
orientations
that
are
grid
like
or
that
are
that
are
grid
cells,
that
are
one
d,
good
cells
in
a
sense?
And
so
it's
it's
fun
to
draw
the
connection
of.
Could
this?
Could
this
explain
something.
B
A
B
B
Yeah,
and
so
I
kind
of
thought
you
need
two
of
it
and
so
to
have
a
to
have
a
module,
a
repetitive
sort
of
grid
cell
response.
He
had
to
combine
the
inputs
from
two
cells,
so
I
might
be
getting
confused
here.
Mark
I'm
sorry,
but
I
I
heart,
I
have
trouble
seeing
those
individual
cells
as
grid
cells.
I
mean.
B
A
You
finish
that
slide.
Okay,
I'm
going
to
go
back
to
this.
So
what
yeah?
What
the,
if,
if
you're
a
mini
column,
then
so
so
we're
now
in
v1,
land,
yeah
and
you're,
seeing
well
like
what
one
example
is
you
literally
see
a
spatial
frequency
that
is
oriented
directly
with
the
orientation
of
that
column?
A
B
B
Okay,
all
right
so,
but
if
I
think
about
a
mini
column
as
a
a
an
orientation
column,
well,
the
orientation's
not
going
to
cycle
on
and
off
right
you're
saying
I
right
it.
You
know
it
takes
a
set
of
cells
to
make
a
module,
so
it
would
have
to
be
multiple
cells
in
the
mini
column.
That
are
taking
turns.
Is
that.
B
C
A
A
Yeah,
so
I
I'll
go
ahead,
so
I
this
is
I'll,
read
this
bullet.
I
basically
said
it
already,
but
if
mini
columns
are
ring
attractors
and
if,
if
we
take
this
experiment
literally
because
I
do
think
this
is
a
bit
of
a
naive
basic
experiment,
this
would
suggest
that
every
cell,
in
a
mini
column,
represents
the
same
orientation
of
input.
I
didn't
phrase
that
very
well:
it
represent,
has
the
same
orientation,
the
same
the
same
type
of
stimulus,
but
but
it's
the
optimal
stimulus.
It's
like
different
translated
versions
of
it.
A
But,
but
I
do
think
the
fact
that
I
just
I
mean
I
literally
trained
on
a
translation
task
here
and
I'm
only
doing
a
basic
linear
mapping.
I
think
that,
like
that,
this
experiment
only
provides
just
like
a
broad
sense.
It
doesn't
I
I
would
want
to
do
a
more
more
advanced
simulation
before
really
predicting
what
the
different
cells
in
a
module
should
respond
to
and
are.
B
B
A
A
E
A
A
A
Sure,
that's,
I
would
say
that's
what
grid
cells
seem
to
be
a
2d
grid
cells
seem
to
each
have
a
2d
phase.
That's
what
okay!
That's?
Why
I'm
showing
a
toroid?
That's
why
I'm
showing
these
rhombuses
is
because
grid
cells
seem
to
be
doing
that.
So
the
answer
is
they
do
it
by
doing
it
just
spatially
where
each
each
good
cell
has
a
specific
2d
phase
and
they
kind
of
cover
the
set
of
possible
phases.
E
So
what
I'm,
what
I'm
trying
to
extrapolate
to
is
that
if
there
there
are
things
in
the
environment
where
you
know
you
want,
you
know
there
to
be
in
a
edge
that
you
fall
off
so
that
then
you
have.
You
know
the
magnitude
and
then
the
phase
where
the
boundary
you
know
has
a
certain.
You
know
falling
off
point
as
opposed
to
a
wrapping
around
point,
but
I'm
just
wondering
taking
your
phase
idea.
E
You
don't
have
to
restrict
it
to
2d.
You
could
the
phase.
If,
if
phase
is,
is
basically
the
the
fundamental
thing
that's
being
represented
here?
You
you
don't
have
to
stop
at
two
dimension.
I
mean
it's
kind
of
like
jeff's
idea
of
going.
You
know
one
d,
but
then
many
of
them
you
could
do
the
same
thing
with
phases.
If
you
need
that
wrap
around
capability,
it
just
becomes.
You
know
a
weird.
E
You
know
it's
hard
to
visualize
because
it's
going
beyond
3d,
but
you
know
if,
if
these
mechanisms
all
work
this
way,
then
you
don't
need
to
necessarily
stop.
If
there's
a
motivation,
if
you
actually
have
a
these
carriers
in
there
that
allow
you
to
have
orthogonal
phases
that
can
combine
in
these
interesting
ways,
I'm
just
throwing
that
out
as
a
you
know
whether
that's
biologically
there
or
not.
I
don't
know.
A
At
the
moment,
the
first
thing
I'm
reaching
for
when
I
when
I
want
to
be
able
to
represent
multiple
dimensions,
arbitrary
numbers
of
dimensions,
I'm
still
leaning
toward
doing
it
with
multiple
rings
and
and
giving
the
system
give
it
a
hundred
rings,
and
let
it
decide
how
many
dimensions
it
wants
to
represent
with
those
and
that's
kind
of
my
starting
point.
But
there
is
just
going
back
to
what
you
first
said
or
back
to
what
we
first
said.
There's
this
open
question
of.
A
B
Mechanism-
and
so
you
project
somehow
these
high
dimensional
1d
vectors
onto
a
2d
strum
straight.
That's
because
that's
what
you
got
in
the
brain,
you
have
2d
surfaces,
so
it's
like
it's
like
you
can
somehow
you
could
try
to
make
make
it
work.
Do
you
have
a
whole
bunch
of
1d
modules,
but
the
2d
when
we
can
see
these
cells
that
look
like
to
these
they're?
Really
not
it's.
It's
sort
of
an
artifact
of
how
the
it's
not
how
the
system
works.
B
It's
an
artifact
of
how
the
the
system
has
to
maintain
stability,
it's
a
very
hand,
waving
argument,
but
but,
and
that
that
also
kind
of
explains
why
these
grid
cells,
these
2d
grid
cells,
don't
always
fire
reliably,
you
know
they
or
they
sometimes
don't
fire
when
you
expect
them
to
anyway.
It's
just
I'm
just
pointing
out.
There
is
a
you:
can
I'm
the
best
been
trying
to
figure
out
how
to
make
it?
How
do
you
make
a
2d
module
show
up,
even
when
you've
got
a
lot
of
1d
modules?
A
D
So
so
the
input
to
the
agent
is
is
a
perspective
from
the
top
right
and
the
the
rats
just
moving
on
straight
line.
So
if
you
you
take
kind
of
a
like
a
mixed
perspective
and
if
you
consider
the
agent's
not
going
through
a
straight
line,
he's
taking
a
turn
yeah
and
the
environment's
also
going
to
rotate
as
the
agent
moves.
A
So
that's
that's
the
trick
here,
though,
object
vector
cells,
it's
not
going
to
object,
vector
cells
are
in
an
allocentric
reference
frame,
so
this
picture,
let
me
go
back
to
it.
The
rat
can
be
faced
if
the
rat
is
standing
in
this
point
relative
to
the
object,
this
little
square
right
here,
the
rat
can
be
facing
any
direction
and
the
cell
will
continue
firing.
A
So
that
that's
how
I
kind
of
basically
dodged
that
that
problem,
I
I
avoided
having
to
do
the
egocentric
to
al
centric
transition,
because
I
avoided
all
hard
problems
except
just
so.
I
could
get
the
core
insight
across
your
right
to
bring
that
up
that
the
actual
visual
problem
of
an
agent
moving
around
an
environment
is
gonna
have
to
deal
with
that,
and
I've
not
done
that
here.
I
have
points
of
view
on
how
to
do
that,
but
it
was
something
I
intentionally
put
out
of
scope
for
this
experiment.
D
But
it
is
a
simple
step
as
well.
I
mean,
if
you
consider
the
ego
center
perspective
and
instead
of
just
translating
you're,
also
rotating
the
input,
as
as
the
red
you
know
just
consider
like
he
has
a
fixed
head,
head's,
not
moving,
but
the
agent
is
turning
some
degrees
during
his
trajectory
and
then
the
environment
is
rotating.
From
that
perspective
as
well.
Is
that.
D
D
A
B
I
don't,
I
think,
it's
a
hard
problem,
maybe
I'm
thinking
about
it
wrong.
It's
when
you
go
to
multiple
dimensions.
If
you
assume
you
have
a
bunch
of
one-dimensional
modules,
then
your
orientation,
what
what
is
your
orientation?
It's
a
very
complex
function
right
and
it's
got
also
multiple
dimensions
and
so
how
to
figure
out
how
to
update
it.
It
becomes
a
difficult
problem.
I.
B
You're
dealing
with
you
know
lots
of
1d
good
cell
modules
that
it
feels
like
it's
a
difficult
problem.
I've
been
trying
to
figure
out
how
to
solve
that
problem
because
it's
not
it's
not
a
simple
one.
You
know
we
think
about
head
direction.
Oh
it's
it's
a
simple
representation,
but
to
really
do
this
correct
in
3d
or
4d
or
whatever
it's.
B
A
D
B
A
So
here
this
is
just
my
slide,
where
I
can,
where
I
just
say:
there's
a
lot,
you
could
go.
Do
a
lot
more.
You
could
do
past
this.
This
was
just
a
starting
point.
First
of
all,
this
experiment,
I
made
it
easy.
You
can
make
it
harder
by
having
true
random
walks
that
aren't
always
straight
lines.
You
could
have
foliage
entering
or
exiting
the
viewport
coming
out
into
site
and
auto
site
and
dealing
with
that
and
and
there's
just
a
lot
more.
I
could
talk
about
here
using
time
during
inference
here.
A
A
So
what
I
actually
mean
by
that
is
time
is
not
being
used
and
deciding
what
to
activate
it's
being
used
to
decide
how
to
update
the
weights,
but
but
the
activations
are
just
a
simple
forward
network
of
the
current
instantaneous
input
and
so
like
that,
making
this
into
something
that
actually
uses
time
uses
input
over
time
would
make
it
a
more
powerful
model,
and
it's
just
not
that
I'll
just
go
through
this
fast,
because
you
can
see
the
last
bullet
point.
I
could
go
on.
A
This
presentation
is
probably
right
now
so,
but
I
do
want
to
bring
like
this.
The
second
to
last
bullet
point,
I
think,
is
going
to
be
a
is
a
fun
thing,
because
this
gets
to
the
question
of
taking
these.
You
know
100
mini
columns
or
something
and
mapping
them
to
multiple
dimensional
input
and
and
so
so,
using
inputs
for
more
than
two
degrees
of
freedom.
A
A
You
don't
it's
not
like
humans
occupy
every
single
point
in
their
joint
configuration
space
and
then,
but
going
past
skeletons
talk
about
a
t-shirt
and
how
it
has
a
sort
of
configuration
space
and
how
this
gets
mapped
to
different
points
and
the
rings
different
points
in
the
the
various
rings.
It's
it's
really
fun
to
consider
how
you
would
take
this
and
let
an
unsupervised
learning
system
just
decide
how
to
do
that,
so
that
that
is
one
thing
that
that
that
is
very
thought-provoking
to
me
here
now.
A
My
next
slide
is
truly
a
summary
side.
It
says
summary
and
so
yeah.
To
summarize,
this
is
a
technique
for
unsupervised
learning
of
a
basis
and
the
basis
represents
input
in
a
way
that
incorporates
how
the
input
is
likely
to
change
over
time,
and
this
this
offers
a
sort
of
alternative
view
of
grid
cells
that
enables
creating
maps
of
novel
environments.
A
The
grid
cells,
in
this
view,
are
learning
a
basis
for
environments.
It
walks
into
it.
You
walk
into
an
environment,
see
the
surroundings
activate
a
certain
set
of
grid
cells,
and
from
that
point
you
can
predict
what
grid
cells
are
going
to
be
active
in
it
at
all
the
all
the
surrounding
locations
you
can.
You
can
anticipate
what
the
input
is
going
to
do
to
the
grid
cells
elsewhere,
which
is
just
a
fun
idea.
It
kind
of
gets
both.
B
B
I
mean
I
mean
I
have
this
view
of
grid
cells
and
how
they
work
and-
and
I
think
you
know
what
that
view
is
largely
I'm
trying
to
understand
how
this
is
different
from
that
and
one
way
you
can
say
well,
this
is
a
way
of
training
grid
cells.
Okay,
you
keep
focusing
on
that.
We're
training
the
system,
but
it's
also,
I
think
it's
a
way
of
a
different
way
of
way.
Grid
cells
represent
space.
I
think
you
said
that
too
right.
B
I
I'm
not
sure
if
that's
true
but
it
but
you
like
you
just
moment
ago,
you
said:
oh,
I
don't
have
to
represent
all
the
different
configurations
of
the
skeleton
or
something
like
that,
but
I'm
really
I'm
getting
lost
on
that.
I'm,
maybe
maybe
you
could
try.
I'm
trying
to
give
you
some
clues
that
you
might
be
able
to
give
me
some
more
words
to
help
me
understand
what
you've
presented
and.
B
A
B
As
we
thought
about
grid
cells,
we
would
say:
oh
when
I
walk
into
environment,
I
can
pick
a
random
anchoring
of
all
the
grid
cell
modules,
and
then
I
can
associate
that
with
the
sensory
input
yep.
What
you
just
said
is
not
that
way.
What
you
just
said
is
I
take
a
sensory
input
and
it's
going
to
anchor
the
grid
cells
based
on
the
sensory
input.
B
So
what
happens
then
so,
like
imagine,
the
the
animal
goes
into
a
box
or
the
foliage
and
says:
oh
I've
seen
this
arrangement
before
I'm
going
to
re-anchor
my
grid
cells
like
I've
done
it
before,
but
then
it
quickly
says.
Oh
no.
This
is
a
new
environment
that
was
only
part
of
it.
Looked
like
something
I've
seen
before.
B
What
happens
then?
Do
the
grid
cells
get
re-anchored?
Would
I,
if
I
go
back
to
that
same
point
again,
would
they
would
they
have
two
different
representations?
You
know
the
old
way
of
thinking
about
it.
I
could
be
in
two
rooms
and
if
the
two
rooms
had
a
similar
section
to
them,
that,
if
I'm
in
that
section
of
the
room
they
sing
the
same.
If
all
I
saw
I
would
re-anchor
the
same.
But
if
I
knew
that
if
I
somehow
knew
that
there
were
different
rooms,
I
would
re-anchor
it
differently.
A
B
A
C
B
Trying
to
understand
what
it
is
you
I
mean
it's
very
intriguing,
but
I
don't
understand
what
to
propose
it's,
not
I'm
not
claiming
you
haven't
proposed
something
really
interesting.
I
just
I
don't
understand
it
yet.
So
I'm
asking
you
to
help
me
figure
out
trying
to
come
up
with
clues,
so
you
can
help
me
understand
where
I'm
wrong
or
where
I'm
confused.
C
C
If
we
went
into
a
new
environment
like
jeff
said:
we'd
pick
a
random
set
of
cells
to
anchor
on
and
from
that
point
on,
we
could
always
predict
new
locations,
even
though
we
had
never
been
to
them
in
the
new
environment
right,
and
so
we
could
still
predict
them
through
movement,
but
we
couldn't
predict
the
sensory
input
because
we
hadn't
learned
them
before.
Are
you
saying
here
that
you
have
a
representation
that
you
come
into
a
new
environment?
C
B
That's
that's
assuming
then
it's
assuming
that
the
parts
of
the
world
I
haven't
experienced
yet
are
going
to
be
similar
to
the
parts
I
saw
before
it's
right,
you're.
In
some
sense,
you
formed
a
representation
for
the
new
room.
That's
has
knowledge
about
the
previous
room,
so
you
can
predict
things.
Otherwise,
how
could
you
predict
anything.
A
C
C
E
E
A
A
So
yeah,
and
so
the
things
that
super
type
pointed
out
are
useful.
Yes,
that's
totally
what
I'm
going
for
is,
as
you
walk
into
a
room,
activate
a
set
of
grid
cells,
and
now
you
can
predict
sensory
input
elsewhere
and
and
that
may
sound
impossible,
but
it's
based
on
just
simple
things
like
saying
like:
I
can
see
a
a
wall
over
there,
and
so
I
know
I'm
still
gonna
see
a
wall
over
there.
When
I,
when
I
move
it's,
it's
encoding,
just
simple
knowledge
like
that.
C
C
B
Mean
the
alternate
way
of
doing
this,
you
say
I
walk
into
a
novel
room.
I
see
a
wall
over
there
and
you
say:
okay,
I'm
gonna
start
learning
the
model
of
this
room
that
and
that
model
the
room
has
a
wall
at
some
location
in
the
room,
and
now,
when
I
turn
and
turn
back
or
whatever
I
turn
I
will
based
on
my
model,
I
would
say:
oh
well,
I've
learned,
what's
over
there,
there's
a
wall
over
there.
I
mean
I
mean
that
seems.
B
Yeah
yeah
yeah,
so
so
in
the
back
of
my
mind,
I'm
thinking
along
the
line
here
that
you
know
everything
we
have
to
do
is
is
based
on
displacement
cells
and
this
composition,
and
and
so
that's
part
of
the
complexity
of
what
we
have
to
solve,
and
I'm
I'm
just
intrigued
by
the
idea
that
you
may
have
come
up
with
a
sort
of
an
alternate
way
of
achieving
the
same
results.
But
I
don't
understand
it.
It's
like
it's
like.
B
But
the
basic
idea
is
yes:
you'd
build
this
compositional
model,
this
allocentric
model
of
the
world
that
you
you
you
you
just
walked
into
and
that's
how
you
would
predict
that
the
wall
was
still
there
now
that
that's
a
simple
engineering,
it's
simple,
but
it's
a
engineering
way
of
thinking
about
it
and
and
maybe
you've
come
up
with
an
alternate
way
which
is
intriguing
but
but
I'm
confused
as
to
how
it's
different.
What
are
the
essences
of
its
difference?
B
B
Okay,
let's,
let's
just
take
that,
I
don't
I
know,
I
know
the
words,
I
don't
understand
what
you're
saying
about
it.
So,
okay,
I'm
trying
to
learn
a
basis,
so
I
think
of
I
think
of
1d
grid
cell
modules
as
a
set
of
them
as
a
basis
for
an
environment
and
but
I
don't
think
they're
learned,
I
think
everyone's
like
oh
there's,
a
structural
neural
tissue.
That
does
that
and
we
just
have
to
build
upon
it.
C
Am
I
learning
I
think,
instead
of
if
I
understand
it's
sort
of
how
sensory
inputs
tend
to
change
as
you
move
around
in
in
like
a
in
the
single
representation?
So
if
you,
if
you're
at
a
location,
you
know
you
know
the
location
and
you
see,
there's
a
wall
in
front
of
you.
That's
five
meters
away,
five
meters
in
front
of
you
now
I
move
one
meter
towards
it.
C
The
representation
changes
in
some
way,
but
you
you,
given
that
that
you've
seen
things
move
around
in
the
world
or
that
you've
moved
around
the
world
before
you
should
be
able
to
predict
how
anything
any
sensory
input
will
change
as
you
move
closer
to
it
or
further
away.
That's
kind
of
an
encoding
of
that
knowledge,
which
we
never
really
had
before
in
our
models.
Displacement
cells
didn't
really
capture
that.
B
Well,
you
could
you
could
get
to
that
answer
in
a
very
roundabout
way
by
building
well
and
then-
and
I
argued
I
argued-
that
in
a
novel
environment
a
set.
If
you
attend
a
set
of
objects,
you
create
a
set
of
displacement
vectors
and
it
just
by
attending
that
set
of
objects.
You,
you,
then,
could
see
the
similarity
between
those
set
of
displacement,
vectors
and
other
places
you've
been
in
the
world
that
had
a
similar
set
of
attended
displacement.
B
B
D
C
B
E
B
A
So
one
one
thing
just
in
case
just
in
case
this
helps
so
I
we're
using
the
word
basis
in
ways
that
I
can't
tell
if
they're
the
same
so
so,
because
when
I
think
of
our
models
and
where
a
basis
is
being
learned,
I
think
of
the
spatial
cooler
the
space,
the
the
spatial
cooler
is
learning
a
way
to
represent
arbitrary
inputs.
That's.
B
Okay,
but
in
the
way
I
currently
think
I
also
think.
Okay,
the
spatial
pool
is
like
a
basis
of
the
sensory,
but
but
then
I
would
think
of
like
a
bunch
of
grid
cell
modules
or
bases
for
representing
space,
especially
if
they're
like
one-dimensional
vectors.
You
can
think
of
oh
that's
a
basis
for
representing
space.
B
So
now
we
had
to
we
had
to
match
up
between
these
two
bases,
and
that
was
the
connections
between
the
sensory
and
the
grid
cells,
so
so,
okay,
so
you're
going
to
do
that
all
in
one
step,
but
I'm
confused
exactly
how
it's
happening.
I'm
just
confused
by
that.
I'm
learning
am
I
learning
these.
These
v1
like
filters
and
and
if
I
do,
how
do
I
sign
them
to
a
movement
vector
I'm
just
I'm
sorry,
I'm
just
so
confused.
C
So
maybe
that
could
be
a
good
exercise
for
another
meeting
is
okay
marcus?
I
don't
know
if
you
want
it
like,
should
be
able
to
deconstruct
that
and
show
step
by
step
how
that's
happening
through
these
visualizations,
and
I
don't
know
if
there's
a
good
way
to
do
that
or
not,
and
you
haven't
really
done
the
predictive
part
of
it
here
yet.
C
B
Yeah,
I
remember
this
reminds
me
marcus
when,
when
we
first
started
learning
about
grid
cells,
he
went
off
and
did
some
research
on
him
and
you
came
back
and
you
talked
and
I
was
very
confused
by
him
for
a
while.
I
just
could
just
totally
confused.
I
just
didn't
understand
it
and
then
eventually
I
got
it
and
I
think
I
understand
pretty
well.
I
feel
like
it's
not
here.
I
think
you've
got
some
idea
here
that
you
understand
completely
and
I'm
just
missing
it,
and
so
just
I'm
just
trying
to
figure.
B
A
B
A
A
B
All
right,
but
you
didn't
make
the
case,
so
that's
the
thing
that
it's
really
motivating
here.
One
of
the
things
that's
really
motivating
and
I
think
it's
a
great
motivation
somehow
I
have
to
make.
Maybe
you
can
make
that
connection
like
okay,
because
because
that's
bothersome
to
me,
too,
right
and
and
the
dropping
out
of
the
grid
cells
or
the
different
firing
rates
is
also
bothered
to
me.
But
I
didn't
I'm
not
sure
you
made
that
connection
to
this.
What
you
presented
here.
A
So
that's,
but
that's
just
one
of
the
motivations.
The
second
is
a
second
second
and
third
I'd
say
like
novelty
in
general
and
the
and
the
connection
to
to
to
v1
the
connection
to
wanting
to
find
a
way
for
a
cell
to
both
have
a
stimulus
tuning
like
like
v1
units
and
be
somehow
a
location
that
that
was
that's
another
puzzling
thing.
B
Yeah,
I
guess
maybe
I
didn't
follow
that
connection
completely
enough.
I've
been
using
I've
been
working
on
a
similar
trying
to
solve
the
same
problem,
but
I
couldn't
I
couldn't
translate
the
way
I
didn't
think
about
it.
The
way
you're
thinking
about
it
so
well,
maybe
I
think
super
tight
suggested
I'm
running
out
of
time
here,
and
so
we
just
have
to
continue
this.
C
C
That
I
think
things
to
understand
it
better
would
be
good
for
me
also.
I
was
also
missing
some
of
the
like
the
details,
like
exactly
what
the
network
looks
like
what
the
loss
function
is
and
how
it
was
trained
and
all
that-
and
just
you
know-
maybe
this
is
not
the
best
venue
for
it,
but
I
was
missing
some
of
that
too.