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From YouTube: Jeff Hawkins Reviews "Grid Cell Firing Fields in a Volumetric Space" Paper - December 9, 2020
Description
Jeff Hawkins reviews the paper “Grid Cell Firing Fields in A Volumetric Space” by Roddy Grieves, et al.. He first goes through the premise of the paper where the authors recorded grid cells in rats as they go through a 2D arena and 3D maze. The team then explores different ways grid cell modules can encode high dimensional information. Lastly, Marcus discusses a talk by Benjamin Dunn showing simultaneous recordings from over 100 neurons in a grid cell module.
Paper reviewed: https://www.biorxiv.org/content/10.1101/2020.12.06.413542v1
Marcus’s paper: https://www.biorxiv.org/content/10.1101/578641v2
Talk by Benjamin Dunn: https://www.youtube.com/watch?v=Hlzqvde3h0M
A
A
So
this
paper
added
a
lot.
I
thought
to
that.
The
authors
you
can
see
them
here.
I
guess
out
of
kate
jeffrey's
lab
I'm
gonna.
I
just
highlighted
some
things
here
and
I'll.
Just
I
will
jump
around
and
talk
about
the
highlight
of
things.
First
of
all,
this
is
the
first
figure
this
you
can
see
highlight
in
the
lower
highlight
here.
It's
basically
experimental
setup.
They
have
a
rat.
A
That's
got
you
know:
they're
recording
from
grid
cells
in
the
rats
and
reading
onto
rhino
cortex
and
the
rats
spend
some
time
in
a
typical
2d
maze
and
they
characterize
the
place
fields
for
those
grid
cells.
Then
the
rat
is
transferred
to
this
three-dimensional
maze.
Just
this
looks
like
a
lot
of
fun
to
play
in
big
play
structure,
and
so
then
they
map
the
receptive
fields
to
those
same
cells
in
this
three-dimensional
maze
and
then
the
rat
goes
back
again
spend
some
time
in
the
two-dimensional
again.
A
So
that's
the
experimental
paradigm
and
they're
recording
the
cells
in
all
three
of
those.
The
question
is:
is
how
do
grid
fields
mean
that
if
you
were
measuring
a
grid
cell,
how
do
its
fields
respond?
There's
a
lot
of
data.
We
know
how
good
the
good
cell
fields
are
in
two-dimensional,
many
of
the
boxes
and
mazes
and
there's
conflicting
data
about
three-dimensional
spaces.
A
So
this
I'm
not
sure
if
this
is
the
first
experiment
that
did
a
true
real
grid
like
a
three-dimensional
box,
because
there's
other
experiments
where
the
rats
are
crawling
up,
walls
and
ramps,
and
things
like
that,
then
they,
this
upper
figure
here,
is
basically
suggesting
well
you're
saying
you
know
this.
This
is
like
the
place
fields.
We
see
for
a
grid
cell.
This
on
the
left
here
number
one
in
a
2d
maze
and
the
question.
A
Maze,
you
know
what
are
the:
how
are
these
cells
responding?
What
are
the
fields?
What
are
the
good
stuff
fields
like
and
if
you
look
at
number
three
here
these
are
two
sort
of
these.
Are
the
two
well
known,
I
guess
ways
you
can
optimally
stack
spheres
ones.
One's
called.
I
pick
up
what
these
acronyms
are
for,
but
they're
pretty
much.
A
A
So
you
know
where
do
cells
respond
would
be
random,
so
these
are
the
options
that
they
talked
about,
but
then
they
record
the
animal
and-
and
these
on
the
on
here
on
this
under
g
here,
these
these
ones
on
the
the
first
three
columns
here-
are
the
classic
grid
cell
receptive.
You
know,
field
response
properties
that
you
see,
and
I
said
why
am
I
just
I
should
just
go
full
screen
here.
I
haven't
done
that.
A
I'm
sorry,
I
know
you're
seeing
my
buttons
you're,
seeing
my
stuff
on
the
bottom
of
the
screen
anyway
anyway.
So
these
are
the
classic
receptive
fields
that
you
see
in
in
almost
all
good
cell
experiments
yeah
as
the
map
moves
through
the
the
two-dimensional
maze,
and
you
can
see
different
scales
of
good
cell
responses.
These
are
the
actual
traces.
A
These
are
the
the
different
ways
they
statistically
show
them,
and
then
they
can
do
the
same
thing
for
the
three-dimensional
maze,
so
they
can
show
where
the
rat
moves
around
in
the
three-dimensional
maze
where
and
where
did
the
cells
respond,
and
then
they
do.
The
same
sort
of
you
know
a
clustering
analysis
on
these,
and
then
these
over
here
they're
projecting
the
three
dimensional
volume
onto
this
onto
the
edges
of
the
of
the
cube.
A
So
if
you
want
to
project
like
you,
have
this
big
cloud
of
deceptive
fields
here
you
can
project
them
onto
one
side
or
another
side
or
on
the
bottom,
that
kind
of
thing
and
already
you're
going
to
see.
You
can
start
seeing
some
regularities
in
this
which
we'll
talk
about
in
a
moment.
So
if
you
have
questions
along
the
way-
just
let
me
know
so-
I
just
highlighted
I
read
through
the
whole
thing.
I'm
just
gonna
highlight
some
things
that
jumped
out
at
me.
A
This
first
highly
surprising,
the
number
of
fields
exhibited
in
the
arena
and
the
lattice
are
not
different
meaning
when
they
total
the
number
of
receptive
these
grid
fields
like
where
do
they
find
these
cells
responding.
The
number
of
fields
in
the
two-dimensional
space
was
was
the
same
as
the
number
of
fields
in
the
three-dimensional
space.
So
obviously
the
three-dimensional
space
is
gonna
have
different
coverage,
but
it
wasn't
like
there
were
more
fields
in
three-dimensional
space,
it's
just
like
no
they're
just
different,
so
that
was
very
interesting.
C
A
Yeah,
so
they
do
this
over
a
population
of
cells
right,
so
you
do
this
over
population
cells,
and
you
can
say:
okay,
given
that
population
of
cells,
how
many
place
fields
that
were
you
know
how
many
grid
fields?
You
know
the
blobs
that
were
you
know,
the
a
grid
cell
responded
occurred
in
the
two-dimensional
maze
and
then
in
the
three-dimensional
machine,
so
three-dimensional
maze
is
the
same
area,
I
believe
as
the
two-dimensional
rays.
A
So
it's
it's
just
taller,
and
so
you
might
think
that
if
you
have
a
now
a
three-dimensional
volume
you're
going
to
have
more
place
fields
in
there,
like
more
there's
going
to
be
more
places
that
you
know
you
can
say,
oh
there's
going
to
be
that's
going
to
be
tile
up.
If
all,
if
all
place
fields
were
just
little
cubes
those
little
spheres
and
you
had
them
arranged
on
the
surface
of
the
two-dimensional
space,
then
you
go
into
three-dimensional
space.
You
might
think
there's
you
know.
You
know
right
many,
many
more,
but.
C
D
A
The
arena
is
the
two-dimensional
space,
so
they're
saying:
okay,
well,
there's
the
same
number
of
fields,
but
these
fields
are
bigger
now
right
and
they
also
here
the
next
one
is:
they
also
exhibit
a
significantly
larger,
spacing
so
they're
much
more
spread
out
between
them,
which
makes
sense
too,
but
but
one
could
you
didn't
have
to
have
both
of
those
right,
but
so
there
were
bigger
fields,
response,
bigger
fields
and
they're
more
widely.
Spaced
out
is
that
is
that
clear,
yeah,
okay,
okay,
there's
a
lot
of
figures
here.
A
But
the
point
of
this
figure
is
so
the
grid
cells
map,
the
ladders
with
large
and
wider
space,
but
stable
fields.
So
that's
we
just
already
talked
about
that
they're
large
they're,
widely
spaced
but
they're,
stable,
meaning
that
the
same
cell
responds
in
the
same
spot
every
time.
So
it's
that's
what
you
would
expect
from
a
grid
cell,
but
they
just
want
to
make
sure
that's
the
case
right.
You
know
the
whole
point
is
the
grid
cells
always
supposed
to
be
strong,
we're
never
in
some
location.
Well,
that's
happening
here
too
this
this.
A
I
highlighted
this
figure
j
here
because
I
thought
it
was.
They
just
threw
this
in,
and
I
thought
it
was
a
little
bit
odd
that
they
didn't
talk
about
a
little
bit
more
and
it
was
a
little
confusing
to
me
and-
and
so
this
is
the
hot.
This
is
the
technique
says
the
grid
cell
spacing,
and
the
arena
in
the
lattice
was
uncorrelated.
A
Okay
and
the
arena
grid
modules-
the
bottom
histogram
here
were
just
disrupted
disrupted
in
the
latter.
So
what
they're,
showing
here,
I
believe,
is
they're
showing
the
spacing
between
grid
cell,
a
population
of
grid
cells
and
what
their
ideal
spacing
is
so
like.
Oh,
this
may
be
35
centimeters,
something
like
that
here
follow
and
this
one
may
be
like
70
centimeters
or
something
like
that,
and
they
show
this.
They
call
this
disrupted.
I'm
not
sure
why
to
me.
I
thought
these
are
like
two
separate
grid
cell
modules.
A
It's
like
that's
what
I
expect
to
see.
I
would
expect
to
see
some
cells
that
are
responding
at
one
spacing
and
a
bunch
of
other
cells
responding
on
another
spacing
and
that's
kind
of
in
the
realm
of
what
we
might
see
for
two
bits
homologous,
but
they
didn't
make
that
connection.
They
they
just
they
just
basically
labeled
in
a
weird
way.
A
A
A
It
almost
looks
like
it's.
It's
like
two
separate
things
that
are
bumping
into
each
other
right.
It's
almost
like
two
peaks
that
are
very
close.
You
got
this
little
dip
in
the
middle
here
and
anyway.
I
thought
this
is.
This
is
one
of
those
figures
where
you
look
at
it
and
going.
I
wonder:
maybe
it's
something
here:
okay,
I
I
just
didn't
want
to
forget
this.
It's
like
this
is
interesting
and
I
don't
I
thought
their
interpretation
was
too
quick.
A
C
C
A
A
Saw
it
briefly,
but
I
didn't
I
didn't
read
it
before.
I
haven't
read
it
carefully.
I
just
figured
I'd,
read
the
paper
first
and
I
haven't
gone
back
and
read
it
anyway.
There
is
a
supplemental
figure.
Six
first,
you
know.
Maybe
I
I
didn't
get
there.
I
didn't
read
any
of
the
supplemental
material
I
apologize
for
that,
but
I
just
thought:
there's
a
so:
maybe
they
addressed
it
here,
but
this
just
jumped
right
out
of
me
like.
Well,
that's
an
interesting
result.
Why
didn't
they
talk
about
that?
More
so.
D
It
just
just
to
comment
if,
if
I
took
those
two
sequences,
those
two
histograms
and
I
correlated
against
each
other,
there
is
a
very
poor
correlation
between
them.
Where
you
have
you
have
you
have
they're
almost
they're,
not
even
anti-correlated,
where
you
have
dips
and
you
have
peaks
on
from
one
to
the
other.
D
A
A
So
so
I'm
not
expecting
that
I
would
not.
In
fact,
this
whole
paper
is
that
these
vertical
spacings
are
really
different
and
weird,
so
I'm
not
expecting
there
to
be
a
bump
at
50.,
but
whatever
the
bump
is
it's
interesting.
I
mean
this
almost
looks
like
there's
two
bumps
one
around
70
and
one
around
in
80,
or
something
like
that
and
I'm
not
expecting
it
to
be
these
to
be
the
same.
I'm
not
you
know
you
might
have.
A
You
might
have
come
in
here
saying:
oh
we're
going
to
have
these
vertical
spacing
as
grid
cells.
It's
going
to
be
a
nice
cubic.
You
know
that
was
that
was
the
idea
of
this
ideally
packed
spherical
array.
Then
you
would
expect
to
see
the
the
this
spacing
to
be
equivalent.
All
the
you
know
these
different
dimensions,
but
we
know
that's
not
the
case.
A
So
the
question
now
is
like
what
is
it
and
I-
and
I
think
this
is
just
showing
I
think,
you're
right
there's
this
is
saying
hey
this
says
it's
not
just
a
nice
vertical.
You
know
it's
not
just
a
stack
of
stack
spheres
that
are
ideally
stacked.
That's
what
you're
saying
kevin.
I
agree
that
it
says
it's
not
that
way
yeah,
but
there
is
some
structure
here.
I
think
it's
more
than
that
was
worthy
of
investigating.
D
A
Yeah
yeah,
so
I
thought
I
thought
this
figure
was
suggestive
enough,
especially
this
almost
looks
like
two
peaks
here.
I
just
thought
it
was
like.
I
just
thought,
as
I'm
trying
to
understand
this,
I
every
once
in
a
while.
I
might
come
back
and
think
about
that.
That's
it's
really
a
sort
of
a
tangent.
Then
this
this
figure
here,
it's
titled
grid
cells.
The
grid
fields
were
randomly
distributed
in
the
lattice,
so
they
looked
at
different
statistical
ways.
A
You
could
try
to
figure
out
how
the
grid
fields
might
be
distributed,
and
so
I
highlight
up
here
the
the
first
two
of
these
are
well
they
they
say
these
are
actually.
The
first
thing
is
actual
good
cells.
The
second,
the
next
one
here
this
hcp
and
the
fcc
are
the
two
ideal
hexagonal
stacking
methods,
then
there's
a
column
method
which
is
like.
A
Oh
what,
if
they
just
these
fields,
are
all
stacked
on
top
of
one
another,
like
a
cell
responds
in
the
same
spot
but
vertically
in
the
column,
and
then
the
fourth
one
is
randomized.
You
know
there's
no,
there's
no
structure
to
where
these
fields
occur,
and
so
the
point
of
this
figure
and
a
bunch
of
data
here
is
that
the
data
looks
like
the
fields
are
random.
A
C
A
B
A
B
A
B
I
think
it's
likely
they
would
have
visually
caught
that
they
would
have
by
looking
at
the
firing
fields.
They
would
have
noticed
this
distinctive
diagonal
column.
I
don't
know
if
their
statistical
technique
right
in
this
particular
figure
would
have
caught
that,
but
they're
aware
of
the
model
I
mean
they,
they
cite
it
and
they.
A
A
So
I'll
come
back
to
it
later
because
when
they
wrote
about,
I
didn't
understand
what
they
said
about
it,
so
you
can
help
me
on
that
one.
Well,
that's
it's
an
easy
question.
I
mean
now
that
now
that
I
think
about
it.
Reading
through
this
paper,
I
would
think.
Oh
maybe
it
would
be
obvious
that
there's
these,
like
you,
know,
columns
of
different
diagrams.
I
guess,
but.
B
A
C
A
A
C
A
Are
diagonal:
this
is
the
projection
on
the
on
the
bottom
surface
of
the
ladder.
So
that
looks
like
a
traditional
down
here
this.
This
would
be
like
the
the
classic
2d
grid
cell
in
a
maze-
and
this
is
the
volumetric
extension
where
these
look
like
sort
of
column
like
well,
that's
a
really
interesting
question.
It
seems.
A
C
A
A
A
But
if
you
project
them
all
down
onto
the
onto
the
bottom
surface,
it's
going
to
look
more
like
a
traditional
2d.
It's
like
what
we
think
of
as
a
grid
cell
module
is
really
a
projection
of
a
higher
dimensional
space,
which
is
something
I've
been
toying
around
and
arguing
for
a
while
too.
So
the
question
is:
is
it
really
random
up
here
or
is
it
it's
just?
They
didn't
see
the
structure
because
they
weren't
looking
at
it
the
right
way,
and
so
I
think
the
point
of
your
marcus.
C
C
Right,
I
guess
here's
the
here's,
the
other
key
question
for
me
is
you
know
we
and
marcus.
I
guess
we'll
talk
about
this
other
paper
too,
but
you
know
the
theory
is
that
any
grid
cell
one
grid
cell
module
doesn't
give
you
unique
location,
but
multiple
modules
together,
give
you
a
unique
location,
right,
yeah,
and
so
can
you
extrapolate
that
property
to
three
dimensions?
C
A
B
A
C
A
A
I
guess:
okay,
let's
go!
Let's
come
back
now,
because
I'm
really
I'm
still
a
little
bit
confused
by
your
comments,
because
you
know.
On
the
one
hand
I
thought
like,
if
you
think
about
the
paper
you
wrote
with
with
marco,
it's
like
okay,
there's
a
bunch
of
these
2d
grid
cell
modules,
they're
just
like
every
other
2d
grid
cell
module,
but
they're
at
different
orientations
in
space
and
therefore
together
they
work
right.
A
D
Well,
one
of
the
things
that
bothers
me
about
their
the
diagrams,
where
they're
trying
to
get
chi
coefficients
out
there
I
mean
they're,
trying
to
fit
a
couple
of
3d
classic
3d
close
packing
structures
to
it,
and
but
they,
but
they
they
have
no
provision
for
that.
The
projection
could
be
at
some
angle
or
others.
It's
basically
does
it
fit
orthogonally.
D
A
B
A
D
A
Okay,
so
I
guess
the
question
is
the
methods
they
use,
which
are
all
statistical
methods
that
most
of
them,
I'm
not
familiar
with
to
determine
these
this
issue.
Would
they
have
caught
that
sort
of
regularity
that
we're
talking
about,
or
is
it
truly
random
as
they
claim?
I
don't
know.
I
I
mean
I
just
don't
know.
I
don't
know
enough
about
the
mathematics.
They
used
to
do
this
to
to
make
that.
D
A
Well,
I
know
again
they're
using
they're,
not
just
projecting
and
saying
what
does
it
look
like
they're
doing
various
techniques
to
talk
to
that
measure,
grittiness
and
and
they're
all
statistical
and
I'm
just
there.
I
don't
know
what
those
are
so,
okay,
I
I
don't
want
to
go.
I
don't
want
to
conclude
that
they
did
or
didn't
do
something.
A
I
already
pointed
out
these
comments
earlier,
I
said
surprising.
The
number
of
fields
exhibited
in
the
arena
us
did
not
differ,
so
you
had
fewer
grid
cell
modules
in
the
lattice
per
square
cubic
millimeter
than
you
did
in
the
arenas
and
the
radius
was
significantly
larger
and
and
they
had
significantly
larger
space
in
the
lattice
okay.
We
already
did
this
one.
Oh
yeah,
we're
already
down
here.
We
did
all
this
here,
I'm
getting
lost.
C
A
Oh,
I
think
yeah
I
mean
you
know,
I
have
a
basic
understanding
of
it,
but
my
understanding
that
we
just
we're
just
taking
grid
cell
modules
and
all
the
property
grids
modules,
but
if
we
assume
they're
not
all
coplanar
and
they're
at
different
angles
or
another,
then
you'd
be
able
to
map
three-dimensional
spaces
and
but
all
the
properties,
the
path
integration
would
work.
Just
fine
right.
I
did
never.
I
never
asked
myself
what
you
know
if
I
were
measuring
an
individual
cell.
A
You
know
you
know
I
you
know,
I
guess
you
know
what
would
it
do
in
other?
You
know
in
it
within
the
plane
of
a
projection
that
cell,
that
cell
would
be
just
like
a
grid
cell
right
in
that
plane
of
whatever
that
plane
is
and
would
that
cell
ever
fire
in
other
planes,
another
module.
I
guess
not,
I
mean
marcus.
You
tell
me,
I
guess
every
the
connection
between
what
they're
showing
here
and
what
that
what
your
paper
is.
I
hadn't
thought
about
it.
A
B
Well,
no
nr's,
they
could
all
be
coplanar,
the
the
the
the
it
could
be
hexagonal
on
the
plane.
All
of
the
modules
can
be
hexagons
on
the
same
plane.
The
big
question
is,
as
you
go
up
in
the
third
dimension,
what
happens
and
and
they
could
shoot
if
one
module
shoots
off
in
one
angle,
with
columns
in
one
angle,
one
direction:
another
module
shoots
off
in
another
direction:
great
it
works,
but.
A
Why
would
they,
but
if
I
just
think
of
of
space,
is
without
any
boundaries
on
it?
Well,
why
would
it?
Why
would
I
expect
it
to
be?
Like
a
you
know,
all
coplanar
at
one
point
and
then
not
complain
around
another
point
I
mean:
wouldn't
it
wouldn't
a
module,
be
at
some
angle
to
the
other
modules
throughout
all
the
space?
A
A
As
long
as
that's,
if
I
don't
assume,
there's
a
preferential
bottom
plane,
wouldn't
they
just
wouldn't-
I
just
think
of
these
gridsole
modules
of
different
just
forever
being
at
some
angle
to
each
other
and
therefore
but
then,
but
then
they
wouldn't
project
in
a
hexagonal
array,
both
of
them
on
this
on
the
same
hexagon
all
right,
I
guess
I'm
missing
some.
No.
C
A
Have
a
bunch
of
why
would
they
be
on
the
same
plane?
Wouldn't
they
be?
If
I
took
if
I
had
a,
I
had
a
if
I
had
a
plane
going
up
to
some
45
degree
angles,
and
it
was
like
it's
hexagonal
on
that
plane.
But
if
I
project
it
down
onto
a
base
plane
which
is
horizontal,
then
I
wouldn't
have
a
regular
hexagonal
projection
right.
I'd
have
a
squish
projection,
I'm
missing
something,
I'm
not
understanding
them.
B
A
B
The
thing
I'm
questioning
is
whether
it's
the
thing
we
should
be
focusing
on
right
now,
like
it's
diving
into
the
details
of
the
different
types
of
projections,
you
can
do.
Yes,
we
you
can
limit
yourself
to
the
types
of
projections
you're
talking
about,
but
that's
not
all
of
them
and
you
there
are
ones
you
can
do
where
your
what's,
the
oblique
oblique
projections
allow
you
to
have
allow
you
to
have
everything,
share
the
same
plane
but
be
projected
in
different
angles.
A
But
let's
get
to
the
end
of
the
paper
because
the
end
of
the
paper,
I've
got
some
highlighted
things
which
I
think
get
back
to
the
question.
We'll
talk
about
here
so
and
your
papers
represent
the
end
of
the
paper.
This
this
figure
four
is.
These
figures
are
pretty
well
described.
The
descriptive
good
cell,
the
good
fields
in
the
lattice
were
vertically
elongated
and
some
form
diagonal
columns.
So
so
I
this
little
some
of
these
figures
I
didn't
get,
but
you
know
a
just
says
you
know.
C
A
The
the
you
know,
the
the
rat
is
running
around
on
two
dimensions
and
it's
just
cross-sectioning
this
big
sort
of
oval
thing
in
half
so
but
anyway,
this
was
another
observation
that
these
fields
are
vertically
elongated
and
the
hexagonal
columns
think
I
I
need
to
go
back
and
read
that
again,
it
was
confusing
to
me,
but
this
I've
hi.
This
is
my
high
here
on
on
d,
where
they
were
showing
these
these.
A
These
blue
things
are
the
different,
you
know,
rate
maps
of
the
cells
firing
and
then
they
projected
it
down
onto
the
xy
plane.
So
they
were
saying
hey.
These
are
like
random
up
here,
but
when
I
project
them
down
to
the
xy
plane,
it's
like
hexagonal
columns,
you
know
so
it's
like
they
have
an
array.
So
again,
this
gets
back
to
the
idea
that
you
have
a
that.
What
we
see
this,
this
gets
back.
A
The
idea
that
you
know
there's
some,
that
that
there's
a
sort
of
a
projection
of
in
volumetric
in
3d
space
down
onto
a
2d
plane
and
and
and
it's
not
the
the
the
the
space
is
not
isometric
in
all
dimensions.
You
know
it's
got
some
preferential
planes
to
it.
You
know
what
I'm
saying
like
okay,
it's
like
it's
like,
maybe
maybe
he
said,
oh
major,
started
by
saying
hey.
A
I
have
this
two
dimensional
grid
cell
array,
but
I
need
to
represent
three
dimensional
spaces,
so
we're
going
to
do
that
somehow
by
extending
things
up
in
these
columns
and
these
elongated
receptive
fields,
it's
hard
to
interpret
what
this
all
means-
and
I
just
think
that
that's
their
observation.
So
I
love
those
questions
about
that.
D
A
C
D
C
A
Maybe
I
highlighted
later
that
they
point
out
that
the
all
the
other
things
you
expect
to
see
with
grid
cells
like
the
theta
procession,
that
happens
here
as
well
face
procession
in
these
cells,
and
I
don't
I'm
not
sure
I
think
I
don't
think
I'm
not
sure
your
interpretation
kevin,
I
think,
you're
just
pointing
out.
This
is
a
classic
way
of
representing
where.
A
D
D
A
B
D
B
B
Hertz's
about
frequencies,
like
when
you
deal
when
you
look
at
these
rate
maps,
it's
like
an
average
over
a
five
minute
trial
or
longer
and
hurts
like
they
call
it
a
hurt,
but
you
can't
interpret
it
as
a
firing
rate.
You
really
need
to
interpret
it
more
as
a
probability
that
the
cell
fires
at
that
location,
yeah
and
try,
because
sometimes
it
might
that
be
there
firing
a
lot
other
times
it
might
not
fire
at
all.
A
A
If,
if
they're,
probably,
if
there's
a
wide
spread
to
the
probability,
cell's
gonna
fire,
then
I
have
to
have
a
whole
bunch
of
them
to
to
get
a
reliable
readout
or
you
could.
Maybe
someone
could
argue?
Oh
no.
It's
actually
something
very
precise
that
cell's
not
firing
this
time,
because
remember
the
the
that
picture.
I
always
like
from
the
tank
paper
where
that's
where
I
first
saw
it,
but
it
was
published
elsewhere.
First,
that
grid
cells
will
sometimes
reliably
not
fire
in
certain
locations.
A
You
know
reliably
every
time
this
cell
on
this
location,
where
you
would
expect
it
the
fire,
does
not
fire
and
it
fires
every
player
themselves,
so
that
that
couldn't
that
was
that
wasn't
just
a
probability
I
was
like.
No,
this
thing
is
telling
you
something:
this
is
reliably
not
firing
in
this
location
and
reliably
finding
another
location.
So
that
would
argue
that
perhaps
this
you
know
this,
it's
not
really
a
probability.
It
might
be
encoding,
something
you
know.
Does
that
make
sense
so.
D
A
Okay,
so
let
me
just
I'm
gonna,
let's
go
through
these
different
text.
Things
here,
I
thought
were
useful,
so
just
remind
grid
fields
are
significantly
more
elongated
in
the
lattice
than
the
arena,
so
that
was
just
an
observation
that
these
these
these
cells,
these
fields
stretch
out
and
they're
no
longer
like
spherical
they're,
like
stretched
out.
They
also
said
they
found
evidence
for
square
firing.
Patterns
was
observed
in
some
grid
cells.
A
Now
this
is
interesting
because
we've
talked
about
different
grid
cell
mechanisms
and
what
what
you
know
how
grid
cells
are
derived
and
there's
some
situations
where
you
might
get
a
square
pattern,
not
a
hexagonal
pattern,
so
they're
observing
that
in
some
cells,
so
that
was
a
nice
little
clue
there
and
then
here's
the
summary
of
the
paper.
I
just
I
just
sort
of
tacked
together
the
words
that
I
think
somebody
read
of
just
the
highlighter
which,
in
summary
grid
fields,
formed
random
configurations.
A
They
call
them
random
of
slightly
vertically
now
they
say
slightly
vertical
elongated
fields,
theta,
head
direction,
speed
coding,
spike
dynamics
and
spatial
information
properties
were
largely
preserved
in
the
lattice
maze.
I
don't
know
what
space
I
mean.
I
don't
know
what
that
all
means,
but
but
the
thing
I
picked
up
is
this
head
direction,
theta
coding
and
so
on.
These
are
all
preserved,
so
they
didn't.
A
A
They're
basically
correlating
speed
of
movement
with
field
size,
and
then
they
said,
the
grid
cell
modules
broke
down.
Findings
reminiscent
of
those
on
verticals,
seen
on
vertical
surface,
a
failure
to
integrate
speed.
The
distance
traveled
may
explain
these
results.
It
may
be
why
place
cells
were
not
disrupted
in
the
same
maze,
so
they're,
it's
this
something
I
didn't.
I
didn't
follow
this
reference
here.
I
probably
should
number
ten
but
they're
pointing
out
that
this.
A
The
slow
path
integration
component
here
is
is
tied
to
speed
of
movement
and
at
some
speeds
the
whole
thing
fails,
and
we,
if
we
want
to
do
a
little
bit
of
introspection,
it
might
be
something
like
I
don't
want
to
get
too
much
on
this,
but
you
know
imagine
you
were
climbing
through
a
maze
of
some
sort
and
you
started
going
really
fast.
Well,
you
might
get
lost.
A
A
And
I
was
like:
oh
that's,
weird,
alternative
explanations
for
column
of
firing
fields
could
so
again
now
they're
going
back
to
this
column
of
filing
fields
that
that
that
that
you,
a
cell,
if
you
take
its
different
positions
where
it
fires
it
fires
in
vertical
space,
they
kind
of
line
up,
even
though
they're
not
equally
spaced
up
there.
Alternative
experiences
for
common
foreign
fields
include
behavioral
biases
for
horizontal
movements
or
that
overlapping
columns
represent
an
efficient
way
to
map
higher
dimensional
space
and
19.
Is
your
college,
your
paper?
A
So
that's.
This
is
the
reference
to
the
clucas
and
lewis
paper.
So
I
don't
know
if
you
want
to
comment
on
that.
I
didn't.
I
did
not
understand
this
marcus.
It
said
that
overlapping
columns
represent
an
efficient
way
to
map
higher
dimensional
space
overlapping
columns.
I
didn't
understand
what
that
meant.
Do
you
know
what
that
means.
A
A
Oh
well,
it's
interesting,
so
maybe
is
it
possible
that
they
were
just
assuming
that
they're
going
to
be
vertical.
Somehow,
let's
just
go
this
way
for
a
second
and
and
then
they
say
well,
they're
not
really
vertical,
but
they're
overlapping,
they're
overlapping,
but
but
that's
what
you'd
see
if
they,
if
they
were
slightly
off
in
vertical.
I.
A
Some
super
oblique
angle,
then
they
wouldn't
be
viewed
as
overlapping
columns.
They
would
be
viewed
as
just
like
you
know
lines
going
across
the
surface
it
would
you
know
what
I'm
saying
if
it's,
if
it's
a
perfectly
vertical
column,
it'll,
be
a
point
and
then,
if
it's
like,
then,
if
they
were
all
slightly
tilting
in
different
directions,
then
those
columns
would
be
overlapping,
but
if
they
were
arguing
really
all
these
oblique
angles
you
wouldn't
see.
That
is
that,
is
that
a
correct
interpretation.
B
Strictly
speaking,
they
are
kind
of
getting
at
the
essence
of
of
the
of
it
when
they
say.
If
you
have
two
good
cell
modules,
they
each
have
diagonal
columns
pointing
off
in
different
directions
that
gives
you
overlapping
columns
the
two
grid
cell
modules,
their
columns
now
overlap.
You
now
have
a
3d
location
code.
They
manage
to
compress
something
down
into
like
20
words,
and,
and
that
might
be
why
it's
confusing,
because
they
just
compressed
it,
did
that.
A
A
Imagine
I
have
if
I,
if
I'm
doing
this
correct,
imagine
I
have
a
couple
of
good
cell
modules
with
the
cells
which
are
like.
I
got
to
read
your
paper
again.
I'm
sorry,
but
you
were
talking
about
the
the
cells
respond
in
a
column
of
fashion
off
orthogonal
to
the
plane
of
the
of
the
grid.
Is
that
right.
A
A
What
you
would
see
is
you'd
see
elongated
fields
vertically,
because
if
I
just
have
two
planes
that
are
those
two,
those
two
columns
are
intersecting,
the
intersection
would
be
elongated
space
and
and
and
then
so.
This
would
be
consistent
with
your
model,
where
the
with
the
multiple
good
cell
modules
are
off
by
just
a
slight
angle,
not
off
by
a
big
angle
that
they
they
could
all
be
just
tilted
slightly.
C
C
A
Let's
go
down
to
this
next,
this
last
big
block
of
texture,
so
their
their
paper
suggests
that
place
cells
and
spatial
mapping
can
function
when
grid
cells
are
irregular,
meaning
they're,
saying
vigils
are
irregular
they're,
not
they're,
saying
these
are
not
regular
patterns
that
they're
firing,
meaning.
I
interpret
this
meaning
they're
saying
path.
Integration
is
not
working
throughout
grid
zone
modules,
place.
A
A
How
the
how
the
grid
cell
is
working
as
long
as
I
get
good
visual
clues,
I
can
know
where
I
am,
which
is
we
know-
that's
true
grid
cell
inputs
may
not
need
to
be
regular
to
support
place
cells.
I
guess
again
regular
meaning
path,
integration
wouldn't
work
everywhere.
That's
how
I
interpret
that
the
grid.
A
A
A
This
is
the
this
is
the
last
thing
I've
got
right
here.
I
think
this
is
the
end
of
the
video.
So
you
know,
grid
cell
inputs
may
not
need
to
be
regular
to
support
play
cells.
The
grid
symmetry
seen
in
the
laboratory
again.
What
undefined
the
grid
symmetry
seen
in
the
laboratory
may
be
a
consequence
of
the
symmetric
geometry
and
homogeneous
behavior
of
laboratory
settings,
meaning
in
they're,
saying
in
in
more
realistic
settings
like
this
3d
setting.
A
Okay,
well,
I
guess
now
that
we
could
get
down
into
the
details.
We
have
to
look
more
carefully
at
how
they
did
their
statistical
analysis.
I
find
it
very
intriguing
which
is
really
really
interesting
today,
that
I
get
a
better
understanding
of
how
the
marcus
here
and
merco's
paper
could
solve.
This
could
be
consistent
with
this.
If
it's
it's
really
interesting,
you
know
what
what
what
we're
saying
here
is.
Let
me
just
go
back
to
this:
either.
A
B
A
B
No,
that's
you're
totally
right.
I
I
mean
I've
a
whole
like
bulleted
list
in
my
head
of
way
of
thoughts
on
this
and
directions.
You
could
go
and
conclusions
you
could
draw
and
ways
you
could
rescue
the
paper,
mine
and
mercos
that
make
it
fit
this
in
ways
that
maybe
that
maybe
maybe
it
doesn't
deserve
to
be
rescued.
Maybe
this
is
right
and
showing
we've
gone
in
the
wrong
direction.
A
A
There
are
projection
of
fields
that
are
somewhat
random
in
3d
space.
That's
that's
what
they're
saying
here.
I
I've
never
thought.
I
just
thought
that
I've
always
felt
that
it's
possible
that
what
we
see
is
a
2d
grid.
Why
can't
I
come
back
up
here?
What
we
see
is
a
2d
grid
cell
module,
it's
really
not
the
whole
story,
and-
and
we
know
that
because
of
the
the
cells
that
don't
fire
reliably
at
different
locations,
we
know
that
because
we
have
to
represent
3d
space.
A
So
one
general
theme
here
is
that
what
we
think
about
the
2d
grid
cell
module
is
just
a
projection
of
a
bunch
of
something
else
elsewhere
and
along
those
lines
that
you
know
if,
if
they
were
really
random
in
3d
space,
which
is
inconsistent
with
your
paper,
but
if
they're
really
random
3d
spaces,
these
people
projected
then
perhaps
path
integration
only
occurs
in
2d
production.
Like
you,
you
perform
path,
integration
in
2d,
but
the
actual
spacing
of
the
the.
A
A
I
think
in
this
case
I
was
saying
something
else
I
was
saying
this
is
it
occurs
in
2d
in
the
in
one
spot
in
the
projection.
Like
remember,
we
have
neural
tissue
that
is
two-dimensional
right
and
so
there's
a
lot
of
constraints
that
puts
on
the
system
that
there
are
certain
things
that
can
only
be
computed
in
the
two-dimensional
system,
and
so
this
is
not
saying
that
there's
a
bunch
of
2d
modules,
so
they're
at
different
angles.
This
is
my
comment.
Here's
a
different
comment.
A
C
C
If
you
were
to
project
the
positions
you
know,
ren
in
each
module
has
some
random
projection
to
3d,
even
though
you're
doing
path
integration
in
2d,
it
still
works
in
3d
through
because
of
the
math.
It
still
works,
even
though
it
actually
is
only
doing
path.
Integration
in
the
2d
physical
space
in
the
layout
right.
So
the
path
integration
doesn't
change.
The
modules.
C
Physically
planar,
but
at
maybe
different
orientations
at
different
scales,
let's
say
well,
okay,
right,
the
different
modules
would
be
the
two
modules.
Would
they
wouldn't
be
exactly
the
same
scale
in
orientation?
There
would
be
slightly
different
scales
and
orientation.
That's
what
we've
assumed,
although.
A
That's
that's
what
we've
assumed
all
along,
but
how
do
I
get
the
3d
aspect?
Out
of
it
to
me
to
get
the
3d
aspect,
I
thought
we
had
to
have
to
say
that
those
those
grid
cell
modules
are
are
representing
their
their
their
path,
integrating
differently
because
they
have
to
be
at
an
angle.
No,
no!
No.
How
do
they?
How
do
you.
B
A
Yes,
okay,
that's
it
that's
the
bottom
line.
That's
it
right!
The
bottom
line
is
given
mode
of
command.
Some
of
these
guys
are
going
to
move
their
bumps
more
and
some
are
going
to
move
them
less
right
depending
on
the
orientation
and
the
movement
right.
So
I'm
going
to
move
one
dimension,
I'm
going
to
be
one
unit
in
some
direction.
Well,
some
modules
are
going
to
change.
Some
won't
change.
Some
will
change
a
bit
here
and
all
they
all
matter.
I'm
trying
to.
I
still
I'm
struggling
how
to.
A
C
Do
so
the
the
module?
Doesn't
it's
all
in
2d,
it's
a
physical
2d
sheet
and
it
path
integrates
just
like
what
we
the
way.
We
think
it
doesn't
really
know
anything
about
3d
space
or
4d
space
or
anything.
Instead,
what
happens
is
when
you
make
a
3d
movement
that
3d
movement
vector
is
projected
to
some
2d
movement
vector
so
that
you
can
move
around
in
this
2d
space
right
and
so
module
1
will
have
some
random
projection
of
the
3d
movement
into
2d,
and
it's
still
going
to
a
path
integrated
in
2d.
B
A
Okay,
so
this
now
all
right
this
I
got
it.
I
got
it.
This
is
very
helpful
so,
prior
to
sending
this
paper
out,
I
was
writing
up
notes
for
a
problem.
I've
been
working
on
and
I'm
struggling
with,
and
I
wanted
help
and
it
gets
exactly
to
this
issue
the
problem
I
was
doing,
as
I
say.
Okay,
I
have
a
good
sense
for
how
columns,
specifically
meaning
columns,
can
learn
to
represent
a
movement
vector.
Remember
the
whole
minikin
hypothesis.
I
have
the
each
of
these.
A
These
complex
cells
are
simply
representing
movement
in
some
in
some
direction,
and
the
question
then
comes
down
to
how
is
orientation
integrated
into
this.
So
I
would
I
was
trying
to
solve
a
different
problem.
I
was
trying
to
solve
the
problem
of
how
do
I
do
a
reference
stream,
a
reference
frame
transformation.
A
Given
I
have
a
set
of
movements
in
one.
I
have
a
set
of
movements
of
ego
space.
Like
my
I'm
flexing
my
finger,
and
I
want
to
turn
it
into
a
movement
vector
in
an
object
space
like
the
coffee
cup,
and
so
I
need
to
have
some
representation
orientation
and
that
representation
orientation
tells
me
how
movement
in
one
reference
frame
is
going
to
change
to
movement
in
another
reference
frame
and
I'm
trying
to
figure
out
a
neural
mechanism
doing
this
and
I'm
struggling
with
it.
This
is
this
is
related.
What
you
just
told
me
here.
A
I
have
now
a
set
of
a
set
of
grid
cells
that
are
all
laying
in
the
same
plane
of
the
tissue
and
they
actually
represent.
They
represent
different
modules
in
the
sense
that
how
they
respond
to
a
movement
vectoring
virus
right.
If
I
have
a
movement
vector,
some
of
these
cells
are
going
to
pass,
integrate.
One.
B
A
And
some
are
going
to
path
integrate
another
way,
but
that
way
is
going
to
be
dependent
on
the
orientation
variable.
It's
not.
There
is
because
that
that's
important,
you
have
to
know
the
orientation
yeah
so
so
now
the
question
comes.
If,
if
you
follow
me
with
the
question,
comes
and
says,
okay,
I've
got
a
set
of
movement,
vectors
that
are
in
ego
space.
That's
consistent!
That's
what
it
is,
and
now
I
have
to
do
unique
path
integrations
on
different
modules,
good
cell
modules
that
are
coplanar
based
on
an
orientation.
C
C
A
A
They're
all
going
to
get
the
same
movement
vector
but
they're
going
to
respond
differently.
Are
you
imagining
these
different
grid
cell
modules
are
interleaved
into
like
every
other
cell
might
be
a
different
orientation,
or
do
you
imagine
they're
like
they're,
physically
separated
like
oh,
like
like
they
are
with
spacing,
like
you
know
the
size
of
this,
like?
Oh
there's
a
module
over
here
on
to
the
left
that
responds
to
one
orientation
and
the
module
next
to
it,
responds
to
a
different
orientation,
and
the
modulation
responds
with
different
orientations.
C
Could
be
interesting,
they
could
be
an
early,
but
it's
simpler
if
they're,
not
because
the
path
integration,
the
connections,
the
physical
connections
that
allow
the
pad
integration
should
be
all
within
a
module.
So
I
guess
they
could
be
interleaved,
but
then
this
connection
would
be
like
crossing
over
one
another.
Physically.
C
A
If
what
if
we
said,
we
had
1d
modules
that
were
corresponding
to
two
meaning
columns,
yeah
and,
and
so
now
it's
much
easier
for
you
to
imagine.
I
could
have
okay
remember.
We
talked
about
slabs
so
like
in
v1.
You
have
a
set
of
many
columns
that
all
have
the
same
basic
quote:
receptive
field,
but
they're
aligned
in
a
slab
and
then
the
next
set
of
mini
columns.
B
A
Slab
you're,
basically
you
you,
you
have
a
changing
of
orientation.
This
is
not
the
orientation
like
like
a
like
a
receptive
field
orientation.
This
is
the
orientation
that
is
dynamically
determined
like
head
direction
cells.
It's
such
a
different
orientation,
it's
it's!
The
it
is
the
it
is.
It
is
the
the
the
it
is,
the
the
factor
we
have
to
use
to
say
how
should
a
movement,
a
movement
in
ecospace
affect
this
column
this
this
1d
grid
cell
module
based
on
an
orientation
that
could
work.
A
So
I
don't
like
the
idea
that
you
have
these
multiple
modules
spatially
aligned
like
like
2d
modules
like
in
the
internal
cortex,
because
there's
just
not
enough
of
them.
You
know,
there's
just
never
enough.
2D
grid
cell
modules
in
the
anterior
cortex,
you
know
we
need
lots
of
them,
and-
and
just
you
know
a
couple
of
two
or
three
or
four
is
just
never
enough.
A
So
so
the
idea
that
again
now
we're
now
we're
going
back
to
I'm
starting
to
think
about
that
here,
but
we're
going
back
to
I'm
going
back
to
the
idea
that
okay,
let's
say
we
have
a
set
of
minicoms
that
represent
1d
modules.
Those
modules
are
all
those
each
one
is
is:
is
basically
they're
all
responding
to
the
same
in
motor
input
but
they're
varying
each
one's
getting
a
different
orient.
Each
one
has
a
different
orientation
projection
to
it
in
some
sense,
and
then
they
update
each
other
appropriately.
A
B
And
so
that
that's
one
way
to
to
go
about
like
to
to
anyway,
that
that
is,
that
is
one
of
the
ideas
that
is
not,
let's
see
I'll,
say
two
things
about
it.
This
paper
is
compatible
with
that
idea.
On
the
other
hand,
that
idea
is
a
little
bit
hand
wavy,
and
it's
being
almost
anti-scientific,
it's
just
coming
up
with
another
explanation
like
eventually
you
want
to
find
proof
for
this
mechanism,
it's
kind
of
anyway,
it's
being
unfalsifiable,
it's
running
away,
because
you
want
this
theory.
B
A
I
I
think
yeah
just
from
the
end,
I
agree
with
you,
I'm
not
sure
I
which
exactly
you
mean
marina,
but
it
it.
I
agree
with
you
in
some
sense,
I
always
thought
that
grid
cells
were
not
the
right
answer.
They
were.
They
were
like
this
artifact
along
the
way
and
because
I
felt
that
good
cells
that
modules
have
to
be
one
dimensional-
and
I
felt
you
know,
there's
all
these
other
things
that
they
have
to
do.
We
just
and
there's
not
enough
good.
So
much
so
I
think
we're
in
agreement
about
that.
A
B
B
A
But
wait
I
remember
when,
when
you
were
reviewing
your
paper,
I
kept
coming
back.
I
believe
saying:
hey,
isn't
this
consistent
with
1d2?
Can
this
all
work
in
one
day,
and
I
think
you
said
yes,
so
I'm
not
sure
why
you
feel
you
need
to
rescue,
because
I
don't
think
the
core
of
your
paper
was
2d.
The
core
of
your
paper
was
an
idea
that
could
work
in
any
dimensional.
B
True,
I
would
say
I'd
say
I'd
say
the
paper
had
kind
of
two
bets
and
one
of
them
is
that
it
works
in
any
dimensions.
Two
is
that,
like
we
also
did
emphasize
2d
so
so
the
paper
would
be
more
right
if
it
were
if
the
2d
thing
were
correct,
but
so
so
so
the
paper
you
just
presented-
and
I
should
disclaim
that,
like
the
the
results
you
just
showed
in
that
paper,
they'd
been
showing
them
in
talks
up
to
that
point
and
we
knew
about
it
as
we
wrote
this
paper
so.
A
B
So
so,
like
we
ahead
of
time,
we're
trying
to
kind
of
decide,
how
is
this
theory
compatible
with
this?
With
this
data
and
and
one
thing
we
pointed
out-
you've
kind
of
alluded
to
it.
One
one
thing
that
we
pointed
out
is
that
if
you
have
neurons
that
read
out
from
two
modules
you'll,
if
you
have
neurons
that
read
out
from
two
modules,
you.
A
B
A
B
It
again
yeah
this
this
was
this
was
a
nice
insight
from
marco
when
he
showed
this,
because
because
this,
this
figure
really
shows
two
different
things.
It
illustrates
how
you
can
how
you
can
decode
locations
with
more
and
more
modules
better,
but
it
also
illustrates
what
you
would
what
cells
firing
fields
would
look
like
if
they're
conjunctive
representations
of
two
modules
and
in
some
sense
these
this
m
equals
two
almost
kinda
sort
of
resembles
the
fields
you've
showed.
A
B
If,
if,
if,
if
I'm
module,
if
you
have
one
module
another
module,
another
module,
the
middle
one
is
receiving
some
input
from
the
ones
next
to
it
that
modulate
its
firing
rate
you'll
get
something
like
this
and
it
suddenly
that
could
explain
the
the
data
from
these
results
there.
I
just
wanted
to
lay
out
that
there
there
are
a
set
of
ways
that
you
can,
that
you
could
still
explain
this,
but
it
but
I,
but
I
will
say,
of
course,
that
the
it
would
have
been
great
for
our
theory.
B
A
Well
again,
I
think
the
savory
here
is:
if
we
don't,
if
we
think
of
those
we
think
of
the
blobs,
what
they're
calling
you
know,
the
the
grid
fields
are
really
the
intersection
of
ugly
fields
and
if
you
think
of
it
that
way,
then
maybe
you
don't
have
to
throw
anything
out.
I
mean
it's
yeah
right,
yeah,
all
right.
You
know
this.
This
whole
thing
about
this.
I
don't
remember
this
figure,
so
I
have
to
read
your
paper
again.
Sorry.
This
is
like
it's
like.
Oh,
it
looks
really
good.
C
C
C
Integration
just
works
in
2d,
it's
the
the
2d
module
nothing's
changed
about
the
2d
module,
it
does
path
integration,
but
the
way
you
map
the
the
3-dimensional
motion,
vector
down
to
the
2d
doesn't
and
what
I'm
asking
is
that
doesn't
really
have
to
be
a
linear
mapping.
It
could
be
some,
you
know
somewhat
distorted
kind
of
non-linear
thing
as
well.
D
B
C
A
C
A
It
would
that
means
that
that
that
that
there's,
a
sort
of
distortion
like
if
I
move
a
centimeter
and
my
my
finger
moves
a
centimeter
in
some
direction,
where
how
far
moves
in
the
object's
reference
frame
would
would
vary.
Depending
on
my
like
on
my
location,
I
may
be
like
well
that's
true
for
linear.
C
A
A
I
imagine
the
grid
cells
representing
some
sort
of
three-dimensional
grid
around
an
object,
and-
and
I
am-
and
I
imagine
that
grid
is
is-
is
it
doesn't
have
to
be
rectilinear,
but
it
has
to
be
regular,
like
the
the
spacing
between
equal
points
and
spacing
between
the
bars
and
the
grid
have
to
be
the
same.
It's
like,
I
can't
distort
it
otherwise,
path,
integration
wouldn't.
C
A
B
B
A
C
C
C
D
Could
I
give
you
a
thought
experiment
that
might
help?
Is
this
related
to
what
I
was
just
saying?
Yes,
so
imagine
you
as
an
adult
someone
gives
you
a
pair
of
glasses
that
distorts
the
world
so
that
everything's
non-linear
now,
but
they
hold
up
a
ruler
and
in
that
space
you
see
you.
You
somehow
know
that
the
graduations
are
still
in
some
kind
of
regular
spacing
with
respect
to
each
other.
Well,.
D
No,
no,
the
you
perceive
them
as
distorted,
but
physically
yeah,
it's
a
reference
yeah.
So
you
could
learn
how
to
navigate
that
space
over
time
to
understand
what
that
distortion
was.
As
long
as
you
have
a
consistent
feedback
now
for
humans.
We
don't
have
that
ruler,
but
we
have
the
physical
world
the
motion,
the
touch.
You
know
you
know
how
long
it
takes
me
to
do
something.
So
as
long
as
things
are
topologically
connected
as
they
were
in
the
real
world,
you
could
learn
this
non-linear
mapping
of
how
things
yeah.
D
C
D
You
don't
have
to
go
back
and
forth
is
the
point,
which
is
what
your
your
analysis
is.
It's
basically
as
long
as
your
feedback
is
similarly
distorted,
you
know
so
that
you
you
get
the
if
I
want
to
go
from.
If
I
want
to
move
my
hand
from
point
a
to
point
b,
you
know
all
I
need
is
the
instructions
to
get
me
from
there.
The
fact
that
my
eye
is
now
showing
something
weird
you
know
is
something
that
goes
away
in
time,
because
you
learn
what
the
real
distortion
is.
C
C
D
D
A
D
But
all
you
need
is
adjacency.
All
you
need
to
do
is
something
is
adjacent
to
something
else
and
then,
as
long
as
you
get
feedback
that
this
is
next
to
this,
you
can
it's
not
so
much.
You
know,
learning
an
arbitrary
thing.
That's
really
you
know
you,
that's
that!
That's
why
you
can
have
this
non-linear
space
and
it's
consistent,
yeah.
C
A
A
Out
how
to
get
from
point
a
to
point
b
in
some
space,
can
I
calculate
that
right
up
front
or
do
I
have
to
walk
through
sequence,
by
sequence,
step
by
step
to
get
from
point
a
to
point
b?
Meaning
can
I
literally
say
the
whole?
The
linearity
to
me
is
sort
of
this
and
the
whole
grid
cell
module
concept
says
hey.
I
can
get
from
any
point
to
any
point
calculate
the
vector
to
get
me
there
because
I
consume
everything
is
regular
everything
all
these
spacings.
A
All
the
big
cell
modules
are
regular
and
all
this
stuff.
If
there
was
distortions
in
it,
it
seems
to
me
well,
I
might
still
be
able
to
get
there,
but
I
have
to
sort
of
wander
through.
You
know
the
distorted
fields
in
moment
to
moment
and
say.
Well,
if
I
go
forward
now,
it's
going
to
take
me
here
and
I'll
go
for
it.
It's
going
to
be
there
so
anyway.
A
A
I
don't
know,
I
think
I
think
I
don't
know
I
would
agree
with
that.
I
mean
oh
yeah,
I
don't
know
hey,
I
haven't
we're
getting
late
here.
Well,
I
don't
want
to
shut
down
the
conversation,
but
I
do
think
there's
a
lot
of
good
ideas
came
up
here
and
I
would
I'm
gonna
go
back
and
read
the
ilia,
marco
and
marcus
paper
again,
because
you
guys
got
some
new
figures
in
there
that
I
didn't
see
well.
B
C
There's
no
neuroscience
in
it:
it's
about
bias
and
machine
learning
and
some
of
the
societal
problems
and
how
we
go
about
address
it
as
a
community,
but
it's
a
really
really
well
done
talk.
It's
amazingly
well
done!
Okay,
I'll!
Listen
to
it!
It
gives
me
ideas
of
how
we
might
want
to
have
a
video
about
the
the
thousand
race
theory.
We.
B
So
yeah
there's
this
paper
from.
Oh
sorry,
there's
a
talk
by
here
just
to
make
sure
I
get
the
the
name
totally
right
because
I
haven't,
I
haven't
read
his
work
before,
but
but
he
works
with
them
with
moser
dater,
benjamin
dunn,
and
he
he
gave
a
talk
that
showed
some
new
some
new
data
from
the
moser
lab,
and
but
the
first
thing
I
just
want
to
emphasize
is
what
neuropixel
allows
them
to
do.
B
They
can
they
can
in
a
single
rat
record,
124
grid
cells
in
a
single
grid
cell
module
and
and
and
get
get
those
maps
from
get
get
the
rate
maps
for
124
cells
concurrently,
and
so
this,
on
its
own,
is
just
like
a
discussion.
Topic
is
like.
First
of
all,
I
didn't
even
know
that
I
I
wasn't
confident
that
a
good
cell
module
even
had
124
cells
that
are
sufficiently
grid-like
to
be
considered
good
cells.
I
mean
we
didn't.
B
D
A
A
Up:
okay,
okay!
Well,
that's
not
okay,
but
that's
a
different
definition
of
modules
than
some
other
people
use,
because
some
people,
they
would
say.
Oh
you
know,
there's
these
like.
I
think
some
cells
don't
show
the
phase
procession
or
something
like
that
and
those
are
in
different
layers
and
other
cells,
and
but
they
all
have
the
same.
Spacing.
B
Yeah
so,
like
I
mean
this
is
a
whole,
I
don't
know
that's
a
whole
discussion
topic
of
its
own.
We
have.
We
have
some
new
data
on
on
how
many
ballpark
figure,
how
many
grids
ours
cells
are
in
a
module
of
course.
What
do
I
mean
by
module
yeah
it's
across
different
layers,
but
even
just
like
looking
at
this,
you
see
some
diversity
how
some
of
them
are,
like.
B
A
A
B
So
I
really
didn't
try
to
do
a
good
job
of
summarizing
this
talk
here
other
I
wanted
to
get
the
neuropixel
point
of
par
a
neuropixel
idea
across,
and
I
just
wanted
to
echo
there
that
they
have
some
data
that
says
that
good
cell
modules
seem
to
kind
of
stay
coherent
and
then
distort
together.
I
could
do
a
better
job
of
actually.
B
A
A
A
B
B
B
That,
like
the
main
thing,
I
want
the
most
interesting
thing
I
was
like.
Oh,
I
need
to
share
this
nero
pixel
thing
and
by
the
way,
this
idea
of
grid
cell
modules,
kind
of
distorting
together
is
also
a
nice
observation
that
they
kind
of
have
some
data
backing
up.
Okay,
that's
that's
the
that's
the
summary
that
I'm
giving
yeah.
A
B
All
right
and
and
connecting
it
back
to
the
the
other
topic,
one
of
the
big
questions
of
grid
cells
is:
is
it
fundamentally
a
2d
sheet
that
is
being
stretched
over
different
things?
Whether
that
thing
be?
You
know,
wagon,
wheel,
mazes
or
3d
space,
and
the
thing
I
just
showed
kind
of
it
didn't
contradict
the
results
you
showed,
but
there's
certainly
not
a
unifying
theory
right
now
that
that
incorporates
those.