►
Description
Marcus Lewis discusses and elaborates some of the ideas he explored and proposed on using grid cells as a prediction-enabling basis in a previous meeting. He first details the frameworks of the grid cell module and neural network. The team then asks questions on the technique used and evaluates the technique’s constraints and potentials. This is an informal continuation of the research meeting on December 21, 2020.
Dec 21's meeting: https://youtu.be/7tF-ofr7VUo
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A
So
so,
just
just
to
review
the
one
thing
I
did
that
may
have
seemed
a
little
weird.
It
was
kind
of
a
leap
I
made
was.
I
chose
to
simulate
these
biological
rings
using
these
pairs
of
units,
and,
like
my
the
philosophy
there
is,
that
artificial
units
and
neurons,
artificial,
sorry,
artificial
units
and
neural
networks
are
already
not
neurons,
they're
they're
already
non-biological,
it's
it's
not
like
they're.
If
you
draw
an
analog
between
neural
networks
and
biological
neural
networks,
it's
unlikely
that
that
the
right
connection
is
that
a
neuron
is
a
neuron
like.
B
Oh
there's,
a
set
of
neurons
that
are
actively
firing
in
in
succession
right.
It's
it's.
That
ring
is
consistent
with
a
bunch
of
neurons
and
they're
taking
turns
here.
I
can,
I
guess,
I'm
struggling
with
okay
yeah.
I
can
see
how
the
a
and
b
combined
mathematically
in
some
sense
would
link
to.
If
you
could
read
them
out
together,
they
would
lead
to
a
result,
but
you,
but
no
neuron
could
read
out
that
result
from
a
and
b
right.
It's
like.
A
B
A
A
Now
you
can
probably
see
my
cursor
and
I'm
just
going
to
show
it
like
yeah,
okay,
so
yeah,
okay,
so
yeah,
I
was
saying
a
c
unit
could
read
these
out,
and
here
I'm
I'm
in
artificial,
neural
network
land.
Yeah.
Talking
about
this,
the
c
could
could
read
about
any
point
on
this
ring
by
choosing
different
pairs
of
weights
on
the
a
and
b.
B
B
So
I'm
just
pointing
out
that
a
and
b
are
sort
of
the
the
basis
or
the
what
you.
If
you
take
a
and
b
you
can
create
a
set
of
ring
or
being
attractive
by
having
a
bunch
of
neurons
attached
to
a
and
b,
but
on
their
own
a
and
b
do
not
are
not
it's
not
a
ring.
Attractor
on
a
and
b
you
could
create
a
ring.
Attractor.
Maybe
is
that
right?
B
Okay,
so
so,
just
using
this,
I'm
just
saying
I
have
a
and
b
doesn't
mean
I
have
a
ring
attractor.
Yet
I
still
have
to
have
a
set
of
neurons
that
we're
going
to
read
from
a
b
and
then
the
next
question
is
when
we
talk
about
the
ring
there,
you
know
it's
a
simple
assumption:
that's
a
1d
grid
cell
module
is
that
right.
Is
that
that's
how
you're
thinking
of
it.
A
B
A
B
A
What
I
did
that
purely
because
this
lets
me
use
this
as
a
machine
learning
technique
that
lets
me,
I
can
th.
This
is.
This
is
something
I
can
do
as
an
unsupervised
learning
algorithm
where
here
we
have
two
two
sets
of
motivations.
I
want
to
find
a
good
unsupervised
learning
algorithm
that
does
useful
things
and
teaches
us
something
about
grid
cells.
A
B
A
C
A
B
A
Yeah,
okay-
and
I
can
do
everything
to
this-
I
can
do
path
integration
to
it.
I
can
perf
I
can
perform
updates
on
this
by
you
know,
multiplying
it
in
a
careful
way,
it
does
require
care
some
carefulness,
but
but
but
anything
this
can
do.
This
can
do
okay
and
so
okay.
The
the
second
point
I
made
here
is
so
I
I
you
know,
I
built
a
ring.
A
B
Can
you
go
through
this
network
again
see
we're
showing
we
have?
We
have
a
bunch
of
rings
and
they're
feeding
into
then
you
have
a
bunch
of
neurons
that
are
getting
pixels
and
they're
feeding
into
the
rings,
but
but
we're
learning
the
picture.
The
this
line,
these
these
these
neurons
are
actually
learning
to
be
filtered
right.
So
at
this
point
the
rings
aren't
even
contributing
this
you're
only
showing
it
you're
only
showing
information
go
from
those
from
those
neurons
to
the
rings.
C
A
A
B
Yeah,
I
I
obviously
you're
you're
you're
trying
to
learn
where
you're
learning
you're
learning
these
filters
on
these
in
this
rectangular
box
and
they're
on,
but
you're
also.
But
your
constraint
is
that
you
have
to
have
smooth
progression
in
the
ring
attractors.
Is
that
what
you
said
yeah,
but
but
that's
not
you're,
not
showing
the
feeding
back
into
the
into
the
filters
right?
It
must
be.
B
I
know
I
know,
but
this
doesn't
look
like
your
typical
network,
and
so
you
know
when
I
think
about
neurons.
I
always
think
like
you're
showing
the
connection.
That's
the
only
way
it
moves
so
yeah.
So
there
is
a
back
prop
operation.
Yes,
that
that,
and
so
some
I
don't
understand
how
this
works
yet,
but
you're
saying
the
the
ring:
attractors
really
are
a
b
cells.
I
think
that
you're
saying
there's
a.
B
B
Oh,
that
was
not
obvious
from
the
drawing
I'm
sorry.
It
wasn't
obvious
to
me.
I
didn't
make
that
connection.
I
I
was
somehow
to
assume
that
the
a
b
cells
were
just
part
of
the
ring
attractor
and
you
just
you're
showing
us
a
ring
attractive,
but
the
a
b
cells
were
up
there
someplace,
but
no.
This
is
really
literally.
These
are
the
a
b
cells,
yeah
and
and
and
and
so
those
ring
attractors,
that's
those
are
just
those
are
actually
one
cell
in
the
same
ring.
Is
that
right?
B
So
this
remember,
I
said
you
had
to
have
10
sort
of
cells
that
were
reading
the
a
b
cells
to
make
a
ring
attractor.
So
these
these
are
like
you're
showing
three
of
those.
Is
that
right,
like
the
first
two
cells
lead
to
one
neuron
and
one
position
in
the
ring
and
the
next
two
position
of
the
ring,
and
so
on.
B
The
ab
cells
themselves
are
not
a
ring
to
have
a
ring.
You
have
to
have
ten
cells
reading
a
mouth
in
your
drawing.
You
know
you
have
to
have
ten
cells
that
are
looking
at
the
abs
and
so
okay.
So
so
so
these
three
rings
are
three
different
rings.
There's
ten
cells
in
each
ring
each
cell
is
learning
a
set
of
connections
to
it's
a
set
of
a
b
cells.
A
A
A
Demo
fail
it's
going
to
take
a
second,
so
the
fundamental
building
block
I'm
using
here
the
fun.
The
fundamental
thing
is:
these
abstract
rings,
I'm
going
to
remove
these
little
these
little
confusing
gray,
dots,
these
abstract
rings
that
have
a
phase
and
a
magnitude,
and
this
is
the
fundamental
thing
and
this
10
unit
this
10
unit
ring
right
here
is
just
one
way
of
implementing
this.
A
B
B
Are
no
additional
additions?
Okay,
okay,
so
you're
you're
you're,
trying
to
each
each
a
and
b
unit
is
looking
at
a
bunch
of
pixels
they're
trying
to
learn
how
to
respond
to
those
pixels
you
in
the
end,
you're
going
to
get
these
sort
of
gabor
like
filters
and
and
and
you're
saying
well.
What
is
the
constraint
that
I'm
applying
on
top
of
the
a
b
cells
to
that
would
cause
them
to
learn?
B
B
B
You
want
that
that
intersection,
the
dot
in
the
upper
right
hand
corner
you
want
that
to
be
smoothly
going
around
the
ring
and
and
I
don't
and
and
so
you
are
just-
is
it
like
okay,
you're
saying
here's
another
input
and
I'm
assuming
that
it's
that
the
dot
is
moved
and
therefore
what
weight
should
a
and
b
have
and
there's
another
input?
It's
in
the
dots
moved
again,
and
what
something
like
that?
Yes,.
A
B
A
B
A
B
B
A
I'd
say
what
the
ring
constraint
does
is:
is
it
gives
you
these
clusters
of
cells?
It
gives
you
these
clusters
of
orientations
and
scales
where
because
because
keep
in
mind,
as
you
already
know,
but
just
to
just
to
say
it
again,
these
two
filters
this
this
a
and
b
reflects
actually
like
in
the
in
this
interpretation.
It
reflects
a
cluster
of
ten
cells
with
different
translations
and,
and
so
the
clusterings
is
the
thing
that
the
ring
gets
you
and
and
without
the
rings
they're
they're.
B
B
All
right
now,
how
does
that
know?
Look
all
right,
so
we
haven't
gotten
there
yet.
So
this
is
that
wasn't
yet
presentation,
but
not
this
one,
this
one!
So
far
you
see.
Okay,
I
just
learned
how
to
use
this
sort
of
smooth,
smooth
movement
of
a
theoretical
sort
of
ring
type
of
thing
to
train
these
input
units
so,
but,
but
now,
how
does
that
now
lead
to
novel
environments.
A
C
A
B
A
B
And
then
so
all
right,
so
so
then,
how
do
let's
go
back
to
the
novel
environments?
Then
all
right
yeah?
Where
does
that
come
in?
I
know
I
said:
okay,
here's
a
object,
vector
cell
that
says:
oh,
I
have
some
boundaries,
often
diagonally
to
my
left
or
anywhere
along
that
line.
Okay,
fine!
So
that
could
fire
anywhere.
I
suppose
right.
Where
does
the
novelty
thing
come
into
this
thing?.
A
B
B
I
mean
it's
it
I
wouldn't
I
mean
I
missed
if
if,
if
the
cell
just
says,
I'm
going
to
respond
whenever
there's
an
object
off
at
this
diagonal
space
here.
Well,
of
course,
you'll
respond
to
novel
environments
too.
So
what
what's?
What's
what's
surprising
about
that?
It
just
says:
I
see
something
over
there,
it'd
be
like
saying
I'm
looking
at
a
picture
and
there's
some
pixels.
You
know
lines
in
the
picture
and
I
say:
well:
here's
a
novel
image
and
there's
a
diagonal
line
off
in
this
part
of
the
image.
B
A
B
B
Obviously,
if
I
go
to
a
new
spot-
and
I
just
observe
I'll
form-
another
representation
yep-
these
are
the
objects.
There
are
different
positions
around.
They
have
a
different
image,
but
you're
saying
I
could
move
to
that
new
spot
and
and
not
observe
and
yet
predict
what
my
my
image
would
be.
Yes,
all
right.
So
where
does
movement
come
into
this
this
into
this
figure
here.
B
Now,
that's
not
is
that
built
into
this
network.
We
don't
there's
no
movement
vectors
here.
There's
no
movement
input
at
all
in
this
thing.
Right,
so
is
that
a
separate
system
that
says
oh
we're
just
going
to
assume
that
we
have
some
way
of
taking
movement
to
move
ourselves
around
these
rings
is
that
that
yeah.
C
B
B
A
A
C
A
A
Yes,
because
the
it's
it's
objective,
like
its
objective,
is
literally
that
it
wants
the
the
distance
from
here
to
here
to
be
the
same
distances
from
here
to
here
and
the
same
as
from
here
to
here,
and
so,
if
you
scramble
the
inputs,
it's
going
to
be
just
a
bunch
of
I'm
using
the
word
distance.
But
like
the
the
phase,
the
the
the
change
in
phase
should
be
the
constant
within
this
module
for
for
a
sequence
of
inputs.
A
B
Now
I
I'm
going
to
move
to
a
new
spot
in
the
room,
and-
and
I
want
I
I
I
want
to
be
able
to
calculate
what
the
new
object
vector
cells
will
be
in
that
new
location
room
as
opposed
to
just
observing
it.
So
that's
a
prediction:
I'm
saying:
okay,
I'm
going
to
move
over
here
and
I
move
from
one
corner
to
room
to
the
other
corner.
B
What
will
be
my
representation
of
the
room
and
object
vector
cells?
Now
we
can
do
this
right.
I
can
mentally
do
this.
I
can
walk
in
the
room.
The
new
room
look
around
sort
of
understand
the
room.
Then
I
could
just
imagine
what
what
what
would
be
my
perception
of
the
room
from
a
different
from
a
different
corner.
I
can
do
that.
I
can
mentally
do
that
exercise.
B
So
this
is
a
way
of
saying
here's,
a
a
potential
mechanism
for
how
you
might
do
that,
how
you
might
say,
okay,
form
a
representation
for
one
part
of
the
room.
Now
now
move
or
pretend
to
move
or
think
you're
going
to
move
to
another
part
and
predict
what
the
representation
will
be
from
that
part
of
the
room,
and
this
is
the
mechanism
we'll
do
that,
and
now
is
that
correct?
Yes,
okay,
now
the
way
I've!
B
We
assign
different
objects,
different
displacements
in
the
in
this
reference
frame,
and-
and
now
I
know
my-
I
move
to
a
new
location
in
the
reference
frame
and
then
from
that
new
location,
I
can
build
up
a
set
of
object,
vector
cells,
building
my
model
of
the
room
and
it's
a
bit
hand
waving,
but
that's
how
I
would
have
thought
about
it.
But
you're
saying
here
this
is
this
is
perhaps
a
very
sort
of
different
mechanism
achieve
the
same
results
that
that's
somehow
much
simpler.
A
Yeah
that's
correct
there.
There
there
are
two
ways
of
approaching
the
problem,
and
I-
and
I
even
I
I
feel
like
this-
would
be
too
big
of
a
topic,
but
I
can
actually
draw
a
link
between.
I
can
draw
a
direct
connection
between
how
this
solves
that
problem
and
how
the
just
how
the
displacement
cells
version
solves
the
problem
and
how
they're
actually
kind
of
the
same.
I
I
have
a
unifying
theory
in
my
head
about
why?
How
those
two
are
the
same?
B
Yeah,
you
know,
I
think
that
that'd
be
very
useful.
You
know,
as
I've
been
thinking
about
these
issues,
it's
it's
really
complex,
of
course,
and
it's
all
these
steps
you
have
to
go
to
to
solve
these
problems.
You
know
like.
Oh
I'm
dealing
with
you
know.
Movements
are
going
to
be
egocentric
reference
frame
and
we
have
changes.
You
know
an
orientation
and
that
from
what
I
observe
is
in
this
egocentric
frame,
but
I
have
to
turn
into
these
allocentric.
B
You
know:
objects,
cells,
object,
vector
cells,
just
all
these
steps
involved
that
are
very
complex
and
and-
and
so
it's
enticing
to
say-
oh
maybe
there's
a
shortcut
to
achieve
the
whole
thing
right.
You
know,
I'm
not
sure,
that's
what
you're
saying,
but
it's
what
it
sounds
like
there
might
be
a
much
simpler
way
of
achieving
these
results
and
that's
really
appealing.
B
But
I
I
I
don't
yet
sort
of
have
a
deep
understanding
of
in
the
way
that
I
have
a
deeper
understanding
of
grid
cells
in
grid
cell
modules.
It's
sort
of
like
okay,
you've
said
all
these
things.
It
kind
of
makes
sense.
There's
a
lot
of
abstractions
or
some
diff.
You
know
I
haven't
internalized
yet
and
and
somehow
it
just
magically
works.
It's
just
like.
Okay,
that's
really
great!
I
don't
understand
it.
I
mean
I,
you
walk
through
it,
but
I
don't
deeply
understand
it.
B
I
don't
I
couldn't
explain
it
to
somebody,
but
I
want
to
make
sure
that
I'm
understanding
what
you're
proposing
so
the
alternate
way
of
at
least
going
from
one
allocentric
representation
to
another
allocytic
representation
based
on
some
sort
of
probably
allocentric
movement,
that
just
that
basically
converts
object,
vector
cells
to
object,
vector
cells,
and
so
I
could
you
can
think
of
it.
This
way,
giving
a
set
of
object,
vector
cells
plus
a
allocentric
movement.
What's
my
next
object?
Vector
cells,
yeah?
B
Okay,
I
think,
if
you
phrase
the
problem
up
front,
that
I
mean
that's,
a
subset
of
everything
has
to
happen
right.
It's
clearly
a
subset.
How
do
we
get
the
house
of
representatives?
How
do
we
get
these
object
vectors
now?
So
all
these
things
are
like
okay,
but
but
at
least
you
could
just
if
you'd
phrased
it
that
way,
it
would
help.
For
me,
at
least,
you
know
it's
like
hey,
it
reminds
me
of
we
saw
whose
paper
it
was.
We
did
a
review
of
a
paper.
B
Remember
I
did
it
and
at
the
end
it
was
this
big.
There
was
a
figure
that
occupied
an
entire
page
and
they
were
showing
how
you
could
do
a
mapping
between
egocentric
and
allocentric,
vector
cells
with
orientation,
and
you
might
remember
it's
like
a
big
grid
so
down
the
left
side
was
like
the
egocentric
object,
cells
and
the
the
horizontal
axis
was
object,
allocent
objects
like
the
cells,
and
then
they
had
orientation
coming
from
the
side
and
they
routed.
B
They
said
he
could
have
a
rounding
mechanism
and
if
anyone
remembers
that
they
figured,
I
could
find
the
paper,
but
anyway
they
were
trying
to
convert
between
egocentric
and
allocentric
vector
cells,
and
this
was
a
paper
they're,
arguing
that
there
was
that
there
was
complementary,
egocentric,
vector
cells
and
as
well
as
object,
allocator
cells,
and
so
this
that
was
like
a
transform-
and
I
was
like.
Oh
that's
something
like
that-
has
to
happen
here.
This
is
another
type
of
transform.
B
This
is
saying:
okay,
I've
got
a
bunch
of
allocentric
rectal
cells
and
I
have
an
allocytic
movement
and
I'm
going
to
get
another
set
of
I'll
subject:
ventricular
cells
to
represent
my
new
movement.
I
suppose
this
would
work
if
I
was
doing
it
egocentric
too
right.
I
could
have
an
egocentric
method
of
cells,
an
egocentric
movement,
and
this
mechanism
would
also
maybe
give
me
egocentric
the
new
egocentric
vector
cells,
okay
and
then,
and
now
I'm
now,
I'm
going
to
ask
the
question:
what
is
the
fundamental
core
idea
here
that
allows
that
to
happen?
B
A
B
B
A
A
B
B
A
Though,
if
if
all
of
the
magnitudes
were
the
same,
then
these
types
of
filters,
you
would
you'd
have
to
add
them
all
up
like
and
describing
what
this
rep
represents.
You
add
up
all
these
filters.
The
same
amount
you
weight
this
filter
by
one
you
ate
this
filter
by
one
add
them
all
up,
and
what
you're
gonna
get
is
a
a
single.
B
B
A
B
B
A
Because
because
the
goal
of
these
cells
is
to
work
together
to
reconstruct
the
image
not
to
not
to
single-handedly,
do
it
and
and
so
reconstructing
their
that,
like
hey,
there's
an
object
here
there
and
there,
but
not
there
requires
a
population
working
together
where
they
can
each
where
some
of
them
can
be
kinda
on.
Some
are
responding
vigorously
well,.
B
A
B
Maybe
that
does
make
sense.
I
guess,
if
I
look
at
one
of
these
filters,
the
filters
have
multiple
bands
in
them
right
at
least
some
of
them
do
so
if,
in
some
sense
each
filter
is,
is
not
on
its
own,
going
to
tell
you
exactly
what's
out
there
right,
multiple
cases,
so
these
are
not.
These
are
not
true
object,
vector
cells
because
correct
if
they
were
object,
vector
cells,
it
would
be
no
it's
it's
object.
Vector
cells
are
just
it's
here,
it's
not
like
here
or
here
or
here.
So
these
are
it.
B
Then
I
can't
really.
I
can
I
see
these
are
like
object,
vector
cells
that
are
better
horses
or
the
ring
ring
attractors,
or
something
like
that,
and
so
so
so
then,
how
would
I
actually
get?
Because
when
we
look
at
object,
vector
cells,
they
don't
appear
to
be
like
that
right
object,
vector
cells,
don't
they
they're,
not
they're,
not
like
grid
cells,
they're
more
like
play
cells,
they
they
just
they
they
activate,
but
they
don't
activate
multiple
locations.
B
B
B
It's
the
same
with
grid
cells
like
which
will
say
I'm
at
this
location,
this
location,
this
location-
I
know
where
I
am,
but
if
I
have
a
bunch
of
grid
cell
modules,
then
I
can
say
oh
yeah,
I
know
where
I
am.
Is
that
a
good
analogy?
Yes,
it's
like
okay,
do
we
have
any
evidence
that
the
brain
actually
has
repeating
object?
A
B
B
A
B
C
I
wonder
if
it'd
be
interesting
to
have
a
visualization,
where
you
literally
created
those
1d
modules
up
top
and
then
showed
the
animal
kind
of
moving
around
at
the
bottom.
You
know
that
thing
and
see
how
that's
changing,
and
you
would
see
those
things
going
around
and
around
in
a
in
a
circle
I
mean
the
grid
cells
would
be
the
actual
dots
in
the
ring
module
up
top
right
right.
Each
of
those
would
be
a
1d
grid
cell.
B
B
I
think
you
know
market,
I
feel
like
it
really
onto
something
here,
but
I
I
I
I
sometimes
it's
happened
in
the
past.
I
feel
like
there's
a
there's,
a
some
sort
of
mismatch
between
what
you're
thinking
about
it,
the
way
I'm
thinking
about
it
and-
and
so
I'm
and
I
I'm
not
I'm
saying
it's
just-
I
need
to
bridge
that
and
I'm
not
putting
any
it's
just
my
issues,
your
issue,
perhaps
both
maybe
all
my
issues,
I'm
just
trying
to
so
I
keep
asking
these
questions
I'm
trying
to
like
get
a
better
interpretation.
A
A
C
B
B
I
guess
I'm
asking
is
that
what
this
is
suggesting
again,
I'm
trying.
I
have
this
idea,
how
great
cells
work
right
right.
You
know
and
in
some
sense
the
way
the
way
we
think
about
grid
solid.
You
can
think
about
completely
almost
completely
independent
of
sensory
input.
I
mean
once
they
get
anchored.
They
basically
path
integrate.
B
Of
course,
the
path
integration
is
very
noisy,
so
they
have
to
get
re-anchored
all
the
time,
but
in
an
ideal
world
you
can
imagine
that
they
don't
need
to
get
rid
if
it
was
perfectly
accurate,
but
then
we
so
that's
like
one
view
of
the
world.
Okay.
Yes,
these.
I
have
a
bunch
of
grizzle
modules,
updated
by
movement
put
in
a
new
movement.
I
know
my
new
location.
We
have
several
reasons.
We
know
this,
nothing
doesn't
work,
one
is
it
they
have
to
get
re-anchored
all
the
time
two
is.
B
The
grid
cells
are
actually
distorted.
There's
a
bunch
of
reasons.
This
doesn't
work,
but
we
I
still
in
the
back
of
my
mind,
this
idea
that
that's
how
it
basically
works
and
then
where
the
system
is
compensating
somehow
for
noise
and
distortions,
but
think
of
it
as
a
purely
motor
driven
update
of
your
location.
B
B
But
the
ultimate
view
is
it's.
You
know.
It's
more
sensory
different
than
the
grid
cells
are
sort
of
a
local
path,
integration
phenomena
that
works
in
some
on
some
places,
but
not
everywhere,
and
therefore
your
your
reference
frame
is
sort
of
it's
discontinuous,
or
at
least
it's
not.
It
could
be
distorted.
Something
like
that.
So
I'm
wondering
if
you're,
if
if
this
is
sort
of
coming
out
from
that
way,
like
you're
saying
here,
I
have
these.
A
Yeah,
I'm
drawing
a
link
to
sensory
input.
A
way
I
would
say
is
so
you
at
the
beginning
of
everything
you
just
said.
You
said
you
said
that,
like
the
anc,
the
grid
cells
are
anchored
at
the
beginning
and
then,
from
that
point
forward
you
can
move
and
update
them
in
an
ideal
world.
You
could
just
update
them
endlessly
and
everything
will
stay
in
sync
here.
I'm
saying
that
same
thing,
you,
you
anchor
the
grid
cells
at
the
beginning,
but
I'm
no
longer
anchoring
them
randomly.
B
A
B
All
right,
that's
a
okay!
That's
a
really
big
thing.
It
gets
back
to
my
my
conversation
earlier
about,
like
oh
I'm
in
the
novel
room.
Can
I
predict
my
objective
cells
would
be
at
this
point
in
the
room
this
this
so
the
the
crux
of
this
is
not
maybe
I'm
going
to
stake
this
and
see
if
you
agree
with
it.
The
crux
of
this
is
something
you
just
said
that
grid
cells
are
not
randomly
anchored,
which
is
an
assumption.
A
B
All
right,
that's
that
may
be
the
simplest
explanation.
What
you're
proposing
here
and
it's
a
big
one,
because
we
had
we
and
you're
saying
I
can
do
that
without
having
learned.
Well,
I
still
didn't.
I
still
have
to
you.
I
guess
I
don't
have
to
learn.
I
don't
have
to
learn
the
room.
I
just
have
to
see
that
the
positions
of
objects
are
similar,
even
though
they're
they're,
somehow
the
same
in
the
room's
reference
frame
even
on
the
different
parts
here.
So
that's
it.
That
would
be.
B
You
know,
that's
that,
let's
just
rephrase
it
again
up
to
now,
we've
assumed
the
grid
cells
randomly
randomly
anchored
and
then
you
have
to,
but
and
then
you
build
a
model
up
on
top
of
that,
and
this
model
says
no:
they
don't
randomly
anchor
they're,
somehow
they're
going
to
be
based
on
observation
and
and
if
I
were
dropped
into
the
same
room
in
a
different
location.
I
would
I
would
even
though
I
have
different
observations,
I
would
I
would
anchor
them
correctly
and
be
compatible.
A
A
B
When
they
have
to
agree,
I
mean
I
mean
I
somehow
yeah
somehow
I
have
to
form
a
representation
yeah
all
right,
all
right.
Obviously,
if
I'm
dropped
in
some
place-
and
I
have
a
bunch
of
object,
vector
cells
and
and
another
place,
a
bunch
of
objects
like
themselves.
If
I
was
in
different
orientations-
and
I
didn't
know
that
that
the
whole
thing
would
be
messed
up
right.
Yes,
so
how
does
that
get
resolved?.
B
B
Integration
with
orientation
like
I
can
close
my
eyes
and
I
can
rotate
my
body
a
certain
amount
and
I
have
a
sense
of
how
far
I've
rotated
I
can
predict
what
I'm
gonna
see
and
and
so,
and
so
we
have
this
path,
integration,
there's
an
anchoring
component
to
it,
it's
very
analogous
to
grid
cells,
except
that
the
we're
instead
of
unfolding
our
taurus
or
our
our
ring
literally
over
in
long
distances.
We
we
wrap
around
on
itself,
so
360
degrees.
B
You
know
back
to
where
I
started
from,
and
yet
so
the
question,
then,
is
because
you
know
how
does
it
get
anchored
and-
and
you
know
this
idea-
that
you
pick
these
long
distance
things
to
look
for,
there's
some
evidence
for
that,
but
it
still
seems
very
weird
to
me.
It
just
doesn't
seem
how
sufficient
somehow
it
feels
like
almost
like.
Somehow
you
have
to
infer
what
your
orientation
is
and
not
just
base
it
on
some
simple
observation
like
there
could
be
many
places
where
it's
not
obvious.
What
the
correct
orientation
is.
B
B
B
We
may
never
randomly
pick
anchoring
points
in
modules
that
we're
going
to
do
it
on
observations.
We're
going
to
do
that,
whether
it's
an
orientation
module
we're
going
to
do
that,
whether
it's
a
good
cell
module
that
somehow
observations
are
going
to
choose
what
our
anchoring
point
is
and
somehow
you've
magically
made
it
work
so
that
in
a
room
with
different
objects,
the
room
with
objects.
I
would
pick
the
copacetic
anchoring
points
to
different
locations
of
the
room
based
on
organization,
and
I
hope
to
think
about
how
you
did
that.
A
Yeah,
that's
that's
the
motivation,
so
the
last
bullet
point
here,
the
the
the
point
I
wanted
to
make.
First
I'll,
just
read
it
and
then
say
what
I
mean.
So
this
was
a
simple
linear
mapping.
It's
a
single
set
of
weights,
and
so
a
multi-layer
network
or
recurrent
network
will
be
more
advanced
it'll.
It
will
use
population
codes
and
won't
be
the.
It
won't
necessarily
be
these
simple
sine
waves.
A
A
But
it
won't
be
constrained
to
it
and
and
and
like
right
right
now.
Each
of
these
rings
is
sort
of
independent
of
all
the
others.
In
the
sense
that,
like
you,
can
describe
what
this
ring
responds
to,
you
can
respond.
You
can
describe
what
this
ring
responds
to
independently
of
each
other,
but
once
you
add,
multiple
layers
of
processing,
they're
kind
of
co-linked,
they're
kind
of
you
can't
really
describe
what
just
one
responds
to
in
isolation,
and
that's
that
might
not
help
this
presentation,
because
I'm
making
things
more
confusing.
A
A
B
So
the
reason
I
asked
this
I
said
this
many
times
ago,
but
you
know
forward,
but
this
conversion
from
one
reference
frame
to
another
has
to
occur
and
I
believe
it
has
to
occur
in
every
corner
of
a
column
all
right,
because
it's
going
to
occur
anywhere.
It's
kind
of
the
multiple
places,
the
cortex,
so
it's
going
to
occur
everywhere
in
the
cortex.
So
this
makes
sense
so
v1
has
to
somehow
make
some
movement,
maybe
not
complete,
but
some
movement
from
a
pure
egocentric.
Perhaps
reference
frame
to
a
less
egocentric
reference
right.
B
Maybe
it
makes
it
all
in
one
fell
swoop,
which
is
the
way
I've
been
thinking
about
it.
Maybe
there's
some
sort
of
intermediate
phases
it
takes.
Maybe
I
don't
know,
but
at
least
it
has
to
be
some
amount
of
ego
to
allow
it
has
to
be
occurring
in
every
column
and
maybe
a
complete
eagle
down
the
course
in
every
column.
That
would
be
nice.
A
B
I'm
totally
confined
to
the
idea
that
each
column
does
a
reference
frame
transform.
So,
for
example,
you
know
in
a
you
know
a
where
column
you
might
be
doing
a
transform
from
a
fingers
reference
frame
to
a
to
a
hand,
reference
frame
or
you
might
be
doing
a
ref.
You
might
be
doing
a
reference
frame
from
your
eyes
orientation
in
your
head
to
your
eyes
orientation
to
your
body.
B
So
it's
less
about
ego
and
aloe,
it's
more
about
there's
a
reference
frame
transformation
that
occurs
in
every
column.
I'm
pretty
wedded
to
that
that
that
one's
I
feel
strongly
about
it.
The
aloe
ego
thing
is:
is
it
can
be
applied
to
lots
of
different
things,
but
a
reference
frame
transformation
I
think,
has
to
occur
everywhere
and
so
again
I
hope
those
examples
were
clear
that
you
know
you
could
be
doing
between
multiple
different
ego-centric
reference
rooms.
B
And
other
people
have
written
about
that.
You
know
like
they
just
point
out
that
when
you,
you
know
when
you
move
your
muscles
of
your
arm
and
your
fingers,
it's
gazillion
transformations
that
have
to
be
done,
and
you
know,
and
so
just
to
to
know
where
every
part
is
relative.
Every
other
point
of
just
to
know
where
your
finger
is
relative
to
your
head
right.
You
know,
that's
that's
to
know
where
my
fingers
well,
it's
with
my
nose
requires
my
fingernails
into
my
hand.
My
hand
melts
my
arm.
B
You
know
all
these
things
so
something
like
that
so
yeah,
I'm
wedded
to
that.
I
do
so
anyway.
The
point
what
I'm
getting
at
here
is
okay,
so
I
I
think,
that's
a
to
me
a
really
key
problem
here.
How
you
do
that
and-
and
I
just
was
wanting
to
know
if,
if
you
had
addressed
it
here-
you
hadn't
yeah,
but
I
think
I
think
we
have
to
somehow
do
it
so
I'll.
Think
about
that.
If
I
can
sort
of
think
about
what
you're
suggesting
here.
D
Fundamentally,
it
doesn't
change
all
the
informa
most.
The
information
ceiling
information
is
in
the
phase
part
of
the
of
the
image,
and
so
when
I
I
see
these
guys
contributing,
you
know
these
axes
in
quadrature
to
that
that
phase
information
you
know
contains
a
lot
of
interesting
information
and
you're,
showing
examples
of
things.
Pairs,
you're,
saying,
might
not
be
a
linear
mapping,
but
the
fact
that
you're,
combining
these
things
in
a
way
that
is
indicative
of
like
a
phased
relationship,
means
that
you
can
embed
a
lot
of
information.
That's
that's
important.
D
It's
a
question
of
you
know:
does
the
brain
find
this
encoding
way
of
using
this
encoding
of
information?
Importantly,
I
think
that
what
you're
what
you're
trending
toward
is
is
something
that's
important.
How
do
you
take
these
1d
things
and
and
get
you
know
these
these
these
relationships,
these
phase
relationships
to
things?
So
I
I
I
find
it.
You
know
evocative
because
of
what
I
see
in
fourier
transforms.
A
D
A
A
One
one
thing
that
seems
like
that
this
is
lacking
as
the
the
fact
that
the
grid
cells
are
set
from
the
instantaneous
input
that
this
this
system
is
not
gonna
handle
ambiguity.
Really
it's
gonna.
If
you
give
the
system
identical
input
at
two
times,
it's
gonna
activate
the
identical
output,
and
so
so,
something
that
does
incorporate
time
into
inference
is
going
to
it's
a
big
to-do
item.
So
that's
that's
a
hard
task,
that's
in
front
of
me
and
I'm
deciding
whether
that's
the
right
next
step.
A
Another
another
task
is
like
the
this.
Third
third
bullet
point
is
the
most
confusing
part
of
this
because
I
haven't,
I
haven't
tried
it
out
and
and
figuring
out
how
to
like
fl
flesh
out
this
third
bullet
point
and
make
it
into
something
that
I
can
explain
to
everyone
and
and
demo
it
I
it's
sort
of
this
is
a
big
unknown
and
I
think,
exploring
the
space
is
also
important
one.
I
guess
one
thing
on
the
topic
of
what
to
do.
A
Next
is
like
a
lot
of
this:
I'm
I'm
sort
of
living
in
machine
learning
land
still
like
the
fact
that
I
represented
all
this
using
these
two
units
use
back
prop
use.
This
loss
function
interpret
this
as
a
bunch
of
rings,
so
so
one
question
on
what
to
do
next
or
one
one
point
of
view:
is
that,
like
I'm
kind
of
doing
it
in
the
machine
learning
world?
A
Well,
while
still
like
mainta,
well
maintaining
a
grip
on
neuroscience,
and
so
what
so,
I
I
I'm
bringing
that
up,
because
you
could
come
in
with
the
perspective
of
like
no
it's
time
to
dive
deeper
into
the
neuroscience
and
you
know,
take
a
focus
start
implementing
these
rings.
Instead
of
these
and
right
now,
I'm
more
on
the
side
of.
I
think
I
can
get
more
done
by
living
in
the
machine
learning
world
and
analyzing
this
technique
from
that
end.
So
that's
that's
more
just
a
discussion
topic
on
the
on
what
to
do
next.
C
B
C
Yeah,
as
I
say,
it
seems
like
the
the
real
kind
of
the
core.
Powerful
idea
here
is
that
novel
environments,
together
with
locations,
can
be
represented
using
this.
You
know
some
encoding
scheme
like
this
right
and
it's
not
clear
to
me
that
you've
fully
shown
that
can
happen
well
and
it
seems
like
whatever
you
can
do
to
really
show
that
working
well
would
be
great.
You
know
whether
I'm
not
sure
whether
you
really
need
multi-layer
networks
or
not
to
do
it,
but
and
then
from
there
being
able
to
show
the
predictive
nature
of
this.
C
B
C
B
B
I
know
what
that
means
now
I
just
just
I
just
throw
out
an
observation,
marcus
that
think
about
what
I've
I
used
to
when
we
were
in
the
office.
I
used
to
talk
about
this
I'd
say:
oh
I
I
walk
into
a
conference
room.
I've
never
been
in
before,
and
I
quickly
look
around
and
and
now,
if
I
I
and
I've
just
what
I
said
earlier,
I
could
then
close
my
eyes.
B
B
My
eyes
are
closed
and
I
know
where,
where
I
am
in
the
room,
but
I
can
also
visualize
what's
in
the
room,
what
my
viewpoint
would
be
from
this
point
so
clearly
I've
built
some
sort
of
model
of
the
room
to
allow
me
to
do
that.
But
this
idea
that
that
the
observations
initially
are
are
do
not
lead
to
random
grid
cell
assignments.
B
It's
very
that's
the
key
thing
I
think
you
said
here
and
and
that's
part
of
this
solving
this
problem,
but
the
observation
I
wanted
to
make
here
here's
the
objection
I
want
to
become
so
I
get
off
track
when
I
walk
into
the
room,
I
just
don't
open
my
eyes
and
take
a
glance
and
close
them.
I
literally
have
to
attend
the
different
objects
in
the
room.
I
I
think
I
have
to
do
that.
I
can't
just
like
take
a
picture
and
go
bing
open
shut.
B
I
have
to
say:
oh
there's
a
chair,
there's
this
and
as
I
do
this,
I'm
somehow
I
see
where
these
things
are
relative
to
me.
I
clearly
form
a
representation
where
the
chair
is
relative
to
me
and
where
the
table
is
relative
to
me
and
where
the
coffee
pot
is
relative
to
me,
and
so
this
this
it's
not
a
snapshot.
It's
a
I'm!
I'm
building
up
something
by
looking
at
multiple
things
in
their
position
and
almost
to
me
and-
and
my
point
is
it's
not
like-
I
just
have
an
image-
that's
a
flash.
B
It's
like.
I
have
to
go
through
a
series
of
observations
of
the
different
components
in
the
room
to
to
get
to
that
representation.
That
allows
me
to
do
this,
and-
and
so
somehow
you
might
think
about
that
issue.
Here
too,
I
I'm
just
pointing
out,
like
you
just
said
something
a
moment
ago
where
you
say
well,
this
is
like
a
single
image,
and
from
that
I
can
do
all
this.
I
in
the
biology-
I
don't
think
that's
the
case
in
biology.
B
I
think
we
have
to
somehow
construct
something
through
multiple
observations
that
allows
me
to
that
that
that
somehow
allows
me
to
anchor
my
grid
cells
or
something
like
that.
I
don't
know
anyway,
I'm
going
to
shut
up
I'm
just
rambling
here.
A
big
take
away
again
for
me
is
no
random
good
selling
ring.
C
B
It's
a
very
powerful
idea
and
in
hindsight
it
almost
seems
like
it
has
to
be
true.
You
know
it's
like.
Oh
that's
pretty
likely,
true,
not
that
you
mentioned
it.
It's
like
oh
yeah.
That
almost
has
to
be
true.
So
I
like
that
the
striking
is
like
oh
yeah.
Okay,
that's
a
good
idea.
Now.
You've
got
a
mechanism
here.
I
don't
quite
understand
you
don't
think
about
it.