►
Description
Marcus Lewis reviews the paper “The Tolman-Eichenbaum Machine” by James Whittington, et al.. He first connects and compares the paper to the grid cell module in Numenta's “Locations in the Neocortex” paper. Marcus then gives a high-level summary of the paper and highlights two aspects - how grid cells and place cells interact, and how place cells can represent novel sensory-location pairs. The team then discusses the multiple grid cell modules and mechanisms presented in the paper.
“The Tolman-Eichenbaum Machine: Unifying Space and Relational Memory through Generalization in the Hippocampal Formation” by James Whittington, et al.: https://www.sciencedirect.com/science/article/pii/S009286742031388X
Papers mentioned:
“Locations in the Neocortex: A Theory of Sensory Recognition Using Cortical Grid Cells” by Jeff Hawkins, et al.: https://www.frontiersin.org/articles/10.3389/fncir.2019.00022/full
“What is a Cognitive Map? Organizing Knowledge for Flexible Behavior” by Timothy Behrens, et al. https://www.sciencedirect.com/science/article/pii/S0896627318308560
“A Stable Hippocampal Representation of a Space Requires its Direct Experience” by Clifford Kentros, et al.: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3167555/
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A
Versions
of
it
have
sort
of
been
out
there
for
the
past
a
couple
years,
but
but
it
all
kind
of
culminated
in
this
paper,
the
tolman
eikenbein
machine
and
so
yeah.
I'm
I'm
leaving
kind
of
leading
a
journal
club
here,
presenting
some
slides
extracting
some
ideas
from
it
that
are
particularly
interesting
to
us
and
then
just
kind
of
opening
it
for
discussion.
A
So
the
agenda
here
is:
first,
I'm
just
gonna
draw
connections
to
how
I
would
describe
this
paper
to
anyone
here,
because
we
can
describe
this
paper
very
succinctly
by
how
it
relates
to
the
paper.
We
did
that
we
released
in
2019
and
then
I'm
going
to
bring
up
three
highlights
from
the
paper.
The
first
one
is
the
thing
that
paper
just
like
hammers
on
repeatedly
the
main
point
of
the
paper.
So,
of
course
I
should
talk
about
this.
A
In
a
way,
that's
that's
interesting
and
also
worth
talking
about
so
to
start
off
with.
This
is
literally
roughly
what
I
said
to
the
team
about
this
paper.
How
we
describe
it.
We
can
describe
this
paper
quite
succinctly
based
on.
Like
our
past
experience,
I'd
say
it's
essentially
the
model
from
our
locations
in
the
neocortex
paper,
but
where
the
grid
cells
are
trained
and
where
the
grid
cells
can
learn.
A
D
A
I
say
all
that,
but
they
definitely
solved
a
harder
problem.
Training.
The
grid
cells
is
hard.
We
kind
of
hard-coded
our
good
cells,
so
in
a
sense
we
kind
of
knew
ours
was
going
to
work
before
we
even
simulated
it,
because
it
was,
it
was
sufficiently
simple
that
we
could
run
it
in
our
heads
and
say
yes,
that'll
work
and
the
simulations
were
just
kind
of
a
proof
of
concept
or
making
sure
we
weren't
fooling
ourselves,
whereas
this
paper
that
trains,
the
good
cells,
I
would
say,
there's
no
way
they
knew
it
would
work.
A
They
explored
the
wilderness.
They
they
went
deep
into
trying
to
get
this
to
work.
There's
no
way
they
just
knew
so
so
it
was.
It
was
a
more
difficult
simulation
test
than
ours
for
sure.
One
way,
ours
was
slightly
harder,
was
we
focus
on
learning,
multiple
environments
and
for
and
inferring
which
environment
the
agent
is
in,
whereas
they
focus
on
quickly
learning
and
predicting
within
a
single
environment?
A
A
B
A
A
To
generalizing
it
to
arbitrary
transition
graphs,
so,
rather
than
always
navigating
2d
space
or
3d
space,
it
might
navigate
a
family
tree
general.
I
don't
know
arbitrary
relation
graphs.
You
could
come
up
with
ways
where,
like
I
don't
know,
a
highly
advanced
version
of
this
might
be,
you
know
sense
of
humor,
maybe
maybe
a
sense
of
humor
is
somehow
one
of
these
complicated
graphs
like
you.
A
E
A
So
just
showing
the
iconography
or
visual
language
of
the
two,
both
of
us
these
tests,
the
left
paper,
tem,
stands
for
tool
and
I
can
bump
machine
this
paper
and
the
right
is
our
is
our
paper
from
a
couple
years
ago.
So
we
both
use
this
kind
of
iconography
of
arrows
and
simple,
like
symbolic
features
that
where
these
are
just
there's,
not
an
actual
motorcycle,
there's
just
standing
for
a
feature.
We
use
squares
and
circles
and
stuff
to
show
the
same
concept.
A
So
so
the
general
task
type
is
receiving
a
secret
sequence
of
actions
and
sensory
inputs
so
that
you
can
predict
subsequent
input
and
always
have
the
extra
the
extra
piece
of
of
distinguishing
which
object
you're
in
which
lets
you
predict
based
on
memory
but
but
taking
novel
sequences.
You've
never
seen
before
and
and
predict
what
you
sense.
So
so
it's
the
kind
of
this
is
the
kind
of
thing
where,
if
you've
seen
one
of
these
papers,
it
can
help.
You
understand
the
other
paper
and
I'm
so
I'm
just
drawing
these
distinctions.
A
This
type
of
iconography
I'll
show
one
more
correspondence.
Their
model
is
on
the
left.
Our
model
on
the
right
I'll
talk
about
ours.
First,
we
have
we
talk
about
these
two
cell
populations,
roughly
good
cells
and
place
cells.
They
receive
a
motor
input.
Then
they,
you
know,
then
they
predict
the
sensory
input.
Sensory
input
arrives
and
then
they
they
update
the
the
location.
Based
on
what
happened,
I
can
draw
these
same
numbers
onto
their
network,
where
one
two,
three
and
four,
those
that
same
general
sequence
of
things
happens.
A
So
it's
really
the
same
type
of
model,
but
once
again
where
their
grid
cells
are
trained.
We
kind
of
hard
coded
this
set
of
the
set
of
good
cells,
so
differences
in
mechanism,
and
I'm
going
to
talk
about
this
in
a
couple
of
slides,
is
in
our
model.
Even
though
we
didn't
fully
like
this,
we
settled
with
a
we
had
our
place,
cells
or
quote-unquote
place
cells,
delay
or
four
cells
read
out
from
multiple
grid
cell
modules
and
their
their
model
finds
a
way
to
avoid
this,
which
is
nice.
A
The
the
second
piece
is
we
used
to.
We
used
two
different
mechanisms
for
representing
and
learning
feature
location
pairs.
Ours
used
temporal
I'll
talk
about
it
later.
Ours
used
a
temporal
memory
thing,
theirs
uses
a
more
deterministic
mapping.
That's
interesting,
and
that's
the
final
thing
I'll
talk
about
in
this
presentation
and
now
another
thing
I
will
compliment
this
paper
on
a
lot
is
is
when.
A
To
models
that
train
grid
cells-
this
is,
when
I
say,
grid
cells.
I
mean
the
idealized
path,
integrating
ones
with
multiple
multiple
modules.
Theirs
are
very
good
and
importantly,
each
trained
module
has
a
complete
set
of
grid
cell
phases.
This
is
a
sentence
that
is
not
generally
true
of
papers
that
have
trained
good
cells,
at
least
from
what
I've
seen,
and
this
is
the
significant
thing
in
some
sense,
the
important
thing
about
grid
cells
isn't
their
multiple
fields.
It's
this.
A
It's
the
fact
that
they
come
in
a
set
of
grid
cells
and
instead
of
phases,
the
translations
of
the
same
grid,
and
this
this
paper
did
this,
so
they
they
like
nailed.
This
part,
it's
gonna
influence,
probably
the
way
I
model
things
in
the
future
or
it'll
it'll
it'll
yeah
it'll
definitely
put
some
influence
on
that
you're.
A
A
little
bit
it's
it's,
it
involves,
you
know,
training
and
network
here
I'll,
just
say
right
here:
it
involves
having
the
system
move
around
and
try
to
predict
its
input
and
they
use
back
propagation
to
train
a
set
of
weights
or
train
a
network
like
high
level
picture.
That
is
what
they
do
getting
more
into
the
details
of
how
they
do.
It
is
more
time
consuming
well.
E
Maybe
I
can
ask
you
this,
you
know
getting
the
different
phases
of
the
grid
cells
in
a
module
is
tricky
and
we've
read
several
papers
about.
You
know
how
parts
of
how
that
might
come
about,
and
I
was
been
thinking
about
that
myself
recently
and
I
talked
a
little
bit
about
that
last
week
or
maybe
on
monday.
D
E
I
probably
read
that
and
then
I
said
to
myself:
oh
then
it
may
be,
you
mean
you're,
giving
him
credit
for
doing
it,
but
I
never
get
too
excited
about
things
that
aren't
you
know
that
are
sort
of
made
to
work,
but
I
don't
think
it's
happening
in
biology.
So
I
guess
that's
the
key.
The
key
question
for
me
here
is
like
hey.
Can
I
learn
how
what
can
I
learn
from
this
model
about
how
biology
actually
does
this
and
they
cannot
that's
the
question?
Maybe
you
can
help.
A
Me
by
that
later
I
I'd
say
that
this
model
has
a
complicated
relationship
with
biological
plausibility,
because
in
some
cases
they
really
put
in
a
lot
of
constraints.
In
other
cases,
they
don't
so
much.
They
really
that
the
constraints
they
impose
are
like
what
are
place
cells
going
to
represent.
What
are
these
so-called
good
cells
going
to
represent,
and
then
then
they
allow
that
to
be
sort
of
a
neural
network.
That's
using
back
propagation.
C
E
A
A
They
apply
a
set
of
weights
to
the
good
cells,
but
the
way
they
decide
what
weights
to
apply
is
they
take
the
motor
command
of
what
movement
is
occurring,
pass
it
through
a
small,
deep
network
and
that
network
outputs,
a
set
of
weights
which
are
then
applied
to
the
grid
cells,
and
that
that
would
make
this
mechanism
would
make
a
certain
type
of
neuroscientist
cringe
like
it's.
It's
not
something.
C
E
A
E
E
So
that
means
you
must
think
something
is
biologically
relevant
or
not,
but
anyway
okay,
I
just
I
just
want
to
go
sense
for
it,
because,
because
when
I
now
it's
coming
back-
and
I
read
this
paper-
what
a
week
or
so
ago
and
read
a
whole
bunch
of
papers
since
and-
and
I
did
recall
like
thinking
of
I'm-
not
going
to
learn
anything
from
that
mechanism-
that's
why
you've
stated
it
nicely.
Man.
A
E
A
A
C
A
Yeah
I
mean
I,
I
definitely
so
I
looked
at
their
source
code.
Like
I
looked
at
how
a
lot
of
this
works
to
to
fill
in
the
blanks,
it
was
hard
to
get
it
from
the
paper,
not
to
say
that
that
was,
I
don't
know
it's
hard
to
communicate
a
model
and
papers
in
general.
So
for
me
I
I
had
to
look
at
the
code
and
like
so
so
in
order
to
understand
that
this
is
what
they
were
doing.
I
had
to
look
at
the
code.
A
The
the
the
big
idea
of
this
paper
is
something
they've
talked
about
before
and
a
couple
years
ago,
this
paper
came
out,
and
I
reviewed
it
here.
This
is
back
before
we
were
recording
these
and
I
praised
it.
A
lot
like
I
it
was
it
was
thought
provoking
to
me
and
and
the
general
idea
there
is
that
that
mediolater,
rhino
cortex
among
other
areas
learns
about
abstract
structure
that
occurs
in
multiple
contexts
and
when
I
say
abstract
structure
here,
for
example,
2d.
B
A
Is
a
form
of
abstract
structure?
It's
not
a
thing.
You
can
counter
directly
it's
a
thing.
That
is
a
type
of
structure
that
encounters
that
occurs
in
multiple
contexts,
and
so
one
thing
that
one
way
that's
influenced
me
is
it
made
me
wonder:
is
it
correct
to
think
of
an
oriental
cortex
as
a
space
processor,
or
is
it
a
something
else?
A
E
E
E
E
A
Yeah
yeah,
this
paper
uses
a
an
example
of
a
totally
different
type
of
task,
not
not
2d
navigation,
not
navigating
a
family
tree.
They
talk
about
harlow's
task,
which
is
I
mean
here,
I'm
talking
about
a
different
paper,
but
they
talk
about
the
task
of
of
you
know
a
monkey.
You
put
two
objects
in
front
of
it.
If
it
picks
one
up,
it
gets
a
reward
that
picks
up
the
other.
A
It
doesn't
and
and
repeatedly
if
it
just
keeps
picking
up
the
correct
object,
it's
going
to
keep
getting
the
reward
and
soon
monkeys
learn
this
rule,
and
and
after
through
various
trials,
the
objects
get
swapped
out.
Two
totally
random.
Other
objects
get
swept
in
a
different
one
gets
is
the
prize
object
and,
and
what
monkeys
learn
over
time
is
that
they
learn
the
rules
of
this.
That,
like
one
of
these
objects,
is
the
prize
one.
A
Once
I
find
it,
I
should
just
keep
grabbing
it,
and
so
there's
a
certain
structure
there
across
the
task,
and
this
paper
laid
out
that,
like
what
grid
cells
are
doing
and
what
a
network
is
doing,
that's
solving
that
problem
are
one
in
the
same.
You
you
give
it
one
task:
it
learns
grid
cells.
You
give
it
another
test.
It
learns
to
do
harlow's
task.
A
So
this
is
this
is
my
only
slide
for
this
idea
that
the
next
two
are
a
little
more
low
level
but
interesting
to
us,
so
something
nice
they
did
was
they
found
a
way
to
do
this
whole
model
where
they
don't
have
their
place
cells
reading
out
from
multiple
grid
cell
modules.
Here
their
model
uses
five
modules
and
I'll
go
ahead
and
bring
up
more
labeling
text.
A
So
so
there
are
cross
connections
at
the
place
cells,
but
the,
but
not
these
connections,
this
these
modules,
the
grid
cell
modules
and
medial
entry,
rhino
cortex,
are
only
projecting
to
and
from
this
these
place
cells
and
one
more
label
is
across
the
modules
the
spatial
scale
increases.
Of
course
they
let
the
network
decide
what
to
do
and
the
network
sort.
A
Chooses
to
use
different
spatial
skills,
they
have
to
nudge
it
a
little
bit.
Lateral,
lateral,
enter
rhino
cortex,
which
provides
the
sensory
input,
has
a
temporal
scale
increase,
and
what
I
mean
by
that
is
they
really
do
a
sort
of
smoothing
or
a
filtering
in
our
in
some
of
our
older
language.
We
would
have
called
it
union
pooling,
but
it's
it's.
E
I
want
to
make
sure
I
understand
what
you
just
said,
though,
maybe
I'm
confused
there's
two
ways.
I
can
look
at
this
diagram.
I
can
say:
oh,
these
are
like
columns
in
the
cortex
and
they
can
say
like
adjacent
columns
in
the
cortex
or
I
could
say
these
are
like
columns
of
the
cortex,
but
they're
hierarchically
arranged
columns
in
the
cortex
is
either
one
of
those
correct.
E
When
you
talk
about,
if
you're
talking
about
change
in
scale
going
from
left
to
right
here,
that
would
be
more
of
a
hierarchical
model
as
opposed
to
multiple
columns
at
a
single
level.
Voting.
Now,
if
you
look
at
you
look
at
the
place
cells,
you
know
projecting
back
and
forth
to
place
cells.
You
can
easily
imagine
that's
the
voting
between
columns,
but
that's
not
what
you're
saying
here.
I
think.
A
D
E
Well,
that's
that
that
would
be
interesting
because
that
is
in
some
sense,
like
voting
across
regions
or
roading
from
scale
which
we
haven't
really
dealt
with
at
all.
We've
done
voting
within
a
region.
You
know
which
I'm
sure
it's
happening,
but
but
we
don't
really
have
a
good
theory
about
voting
between
or
what
we
don't
have
a
good
theory
about
our
hierarchy
in
our
model
right,
we
don't,
we
understand
it
less
than
we
used
to
understand.
E
I
think
so
so
this
that's
very
interesting
about
this-
is
that
if
this
is
sort
of
a
key
to
saying
okay,
we've
done
voting
across
columns
in
a
single
scale.
This
could
be
voting
across
columns
in
multiple
scales
across
modules,
which
I
think
I
really
want
to
understand,
because
we
don't
understand
that
right
now
in
our
models.
If
that's
what
it
is,
yeah.
A
A
E
Did
you
did
it
change
your
opinion?
I
mean
that's
a
big
part
of
our
thinking
in
our
model
is
I'm
just.
I
want
to
double
check.
Should
I
be
concerned
about
that.
Is
that
something
like
hey?
Maybe
we
got
that
wrong
and
we
got
it
right
and
they
just
didn't
pay
attention
that
this
particular
problem
here,
I'm
under
the
assumption
that
we
have
a
single.
E
So
there
has
to
be
this,
but
it's
it's
not
a
feature.
It's
not
just
a
feature
pair.
Yes,
you
know
it's
not
just
like
a
a
location
feature
pair.
It's
it's
a
whole
object.
So
can
I
would
it
be
correct
assume
that
that
that
would
would
you
personally?
Would
you
feel
like
hey?
That's
still
there,
they
just
didn't
talk
about
it
or
do
you
think
they
somehow
made
it
obviated
the
need
for
it.
A
A
D
B
C
E
Yeah
again
well,
okay,
so
so
everything
I've
done
up
to
now
has
been
pretty
much
grounded
in
the
belief
that
you
do
have
that
it's
an
observation
as
much
as
anything
so
so
you're.
What
you
learned
here
doesn't
shake
you
that
doesn't
say
like.
Oh,
they
showed
that's
not
necessary,
and
you
know
no,
it's
not
like
that.
You're
saying
he
just
didn't,
do
it,
it
doesn't
that
doesn't
change
how
I
feel
about
it.
E
A
grid
cell
module,
for
example,
is
by
using
this
hierarchy
of
scale.
Is
that
I
think
that's
what
this
picture
implies,
yeah,
which
is
a
very
interesting
idea,
the
both
the
things
could
be
happening.
You
know,
if
you
think
about
the
hierarchical
projections
in
the
cortex
they're,
not
the
same
I'd,
be
like
from
layer
three
to
layer,
four,
going
up
a
hierarchy,
not
not
not
even
going
directly
to
the
thalamus,
but
just
like
from.
E
Therefore,
the
layers
redirect
on
the
other
street
and
therefore
direct
and
portable
hierarchy
like
v1
and
v2
in
that
case,
that
could
be
the
arrows
here,
and
that
could
be
something
that
this
could
explain
that
perhaps
so,
I
think
that's
an
interesting,
very
interesting
idea.
So
I'll,
let
you
continue
on
terminology.
D
A
A
D
A
D
E
But
yeah
in
our
work,
though,
when
we
talked
about
having
multiple
modules.
One
of
the
objections
I
had
to
it
was
is
that
there
only
seemed
to
be
one
module
that
in
in
the
internal
cortex
grid
cells,
nbc
grid
cells,
there
only
seemed
to
be
one
module
at
each
scale,
and
and
so
when
you
mixed
in
our
model,
if
you
were
trying
to
read
out
for
multiple
modules,
I
would
always
argue
that
a
cell,
that's
reading
getting
input
from
multiple
scales
would
be
fooled
because
some
would
be
stable.
E
Some
would
be
changing
those
inputs
and,
if
you
may
recall,
I
argue
like
I
don't
think,
that's
going
to
work,
and
so
we
try
to
find
ways
that
we
have
multiple
modules
that
differ
by
orientation
or
slight
variations
in
scale.
Slight
variations
of
scale
and
and
different
orientations
would
require
or
solve
the
multiple
module
problem.
E
E
So
this
they
may
have
a
way
of
getting
to
work
with,
because
they're,
not
no
no
cell.
Here,
like
it's
your
opponent,
pointing
at
you,
but
look
at
all
the
blocks:
labeled
mec,
there's
no
single
cell,
that's
looking
at
inputs
from
those
different
blocks
and
which
is
the
objection
I
had
to
having
multiple
scales.
So
this
is.
This
is
using
a
sort
of
indirect
method
of
doing
it.
A
Also
one
more
thing
about
voting
a
conversation
we've
had
in
the
past
that
that
you've
often
agreed
with
me
on
or
we've
often
agreed
on,
is
that
is
the
voting
on
object.
Identity
is
not
that
powerful
compared
to.
If
you
could
vote
on,
you
know
an
object
at
a
pose.
For
example,
if
you
can
vote
on
something
a
little
more
informative,
it
makes
the
system
much
more
powerful.
A
E
E
I
mean,
I
think,
I
think
our
we,
our
our
thinking,
has
evolved.
Clearly
I
mean
we
don't
even
think
it's
features
at
locations
anymore.
We
think
it's
displacements
right,
so
we
haven't
gotten
into
that,
but
but
I
think
you're
right
I
mean
it's
interesting.
You
can
really
do
a
lot
of
introspection
here.
You
look
at
something
and
I
have
the
stable
representation
of
this.
This
outdoor
chair,
I'm
looking
at
right
now
and
it's
it's
the
chair
at
a
pose
right.
I
don't
just
see
the
chair.
E
B
E
Now
it's
it's
the
pose
that
it's
described
in
somehow
it's
going
to
be
described
in
the
displacement
cell,
but
but
we'll
get
to
that
again,
so
I
think
we're
in
agreement
with
that.
So
so,
if
their
model
actually
shows
how
you
get
to
pose
here,
that's
being
voted
on
across
scale.
That's
a
very
interesting
idea
and
I'd
like
to
understand
that,
because
that
solves
some
problems
that
we
haven't
been
able
to
solve.
E
A
So
final
highlight
is
going
to
dive
into
what
interaction
is
occurring
here.
This
medial,
anterior
angle,
which
is
the
location,
lateral
internal
cortex,
which
is
sensory
information
and
how
the
place
cells
are
determined
from
that
which
is
analogous
to
in
our
model.
It's
our
location,
layer,
layer,
six,
which
would
be
this
our
input,
which
would
be
kind
of
the
thalamus
really
and
and
how
it
determines
which
layer
four
cells
are
active.
E
E
A
E
E
E
I
just
I
had
heard
that
before
yeah-
I
don't
I
don't
remember
I
was
discussing
that
before,
which
is
which
is
very
interesting.
I
find
it
so
then
I
could
say
then
you
could
say
the
following.
You
could
say:
okay,
you've
got
your
sort
of
grid
structure
in
mbc.
You've
got
your
spatial
pattern
in
lc
and
what
place
cells
are
really
doing
is
essentially
saying.
Okay,
what
feature
is
that?
What
location
and
call
that
a
play
cell.
A
E
I
don't
I
just
don't
remember
us
discussing
that
before,
so
that's
that
that
opens
up
a
whole
different
set
of
possibilities
for
me
to
think
about
lbc.
That
way.
In
that
case,
lac
might
be.
You
know,
I
just
said
the
input
from
the
thalamus,
I'm
not
sure
if
that's
really
what
it
is
or
if
it
could
be
the
well.
You
know
it
recently,
I'm
sorry
to
go
off
topic,
but
this
I'm
just
working
on
this
all
the
time
right
now.
So
I
we're
talking
about
how
you
know
your
your.
E
Both
your
motor
commands
and
your
spatial
pattern
on,
say
the
retina
or
your
finger
has
to
go
through
a
reference
frame.
Transform
has
to
be
transformed
by
orientation,
and
so
I
argued
that
was
what's
going
on
in
the
thalamus
before
it's
projected
the
layer
four,
so
maybe
lec
is
providing
that
role
here.
Could
that
be
somehow
this
again
we're
in
that
we're
not
in
the
cortex
or
up
in
the
you
know,
part
of
the
brain?
A
Possibly
because,
like
one
thing,
you
can
one
thing
one
type
of
cell
you'll
find
in
lac.
Is
they
respond
if
the
animal
is
next
to
a
an
object,
but
they
don't
have
to
be
facing
the
object
they
they
can.
It's.
They
don't
have
to
be
looking
at
the
object.
So
that
seems
a
little
bit
like
a
reference
frame.
Transformation.
D
A
I
that's
the
best
thing
I
can
it
it's
been
it's
been
literally
years
since
I've
looked
at
that.
Okay,
let's
just
keep
going
then,
okay,
so
yes,
the
what
this
mechanism
is
so
their
different
feature,
location
pair
approach,
their
hippocampus,
they
place
cells,
their
place
cell
population
has
one
place
cell
for
every
mediological,
cortex,
lateral
and
tyrannocortex
cell
pair,
which
they
visualize
down
here.
A
So
the
lateral
and
torino
cortex
like
a
temporally,
filtered
sensory
input
and
their
medial
entrance
cortex,
which
is
a
set
of
grid
cells
which
they
visualize
as
one
d
here,
but
could
be
visualized
otherwise,
for
every
pair
of
those
cells,
they
have
a
place
cell,
and
this
is
visually
evocative
of
mini
columns
with
sensory
input,
deciding
which,
which
many
columns
receive
input
so
in
the
next
slide
I'll
show
that
a
little
more
how
inference
occurs
but
yeah
this
is
this.
Is
this?
A
Is
a
core
part
of
of
this
you're
going
to
need
to
see
the
next
slide
to
really
grok
it?
I
think
so
I'll
move
on
to
that,
but
we
can
come
back
to
this,
and
so
they
have
a.
This
is
three
modules.
Three
three
modules:
where
and
I've
I've
shown
this
as
a
I've,
basically
taken
their
figure
here
and
stretched
it
out
like
this
and
I've
also
intentionally
made.
It
resemble
arm
models
with
many
columns
because
it
at
least
can
be
visualized
this
way.
D
A
Remembered
feature
location
pair
so
so
what
happens
here
is
the
lateral
into
rhino
feature
information
selects
a
set
of
conjunctive
cell
conjunctive
place,
conjunctive
location,
sensory
cells,
aka
place
cells
and
gives
them
excitatory
input,
but
this
happens
in
each
module
and
so
different
pairs
are
different.
Pairs
of
so-called
mini
columns
receive
excitatory
input.
A
Then
an
attractor
state
is
found.
A
memorized
set
of
cells
that
has
been
active
in
the
past
before
is
found
in
this
using
recurrent
connections
using
you
could
think
of
it
as
any
kind
of
auto-associative
memory,
any
kind
of
almost
like
a
hopfield
network
where
this
pattern
has
been
memorized.
This
pattern
has
been
learned
by
synapses.
It's
an
attractor
state
of
the
network,
so
that
if
this
set
of
columns
ever
becomes
active,
it
will
then,
after
a
couple
of
iterations,
resolve
to
this,
and
from
that
you
wouldn't
beat
out
which
graded
cells
should
be
active.
C
E
E
B
A
E
C
C
A
E
Told
well,
I
think
that
I
think
the
point
you
make,
though
markers
is
really
important
one
in
our
in
our
mind,
you
you,
you
should
be
able
to
understand
these
with,
in
some
sense,
without
the
scalar
that
the
scalar
is
really
helpful,
but
there's
a
structural
element
to
this.
That
is
not
built
purely
on
scalar
activations.
E
It's
built
on
the
it's
the
intersection
of
these
cell
populations
right.
So
I
think
it's
important.
I
mean
if
you
take
a
typical,
deep
neural
network,
the
seller,
you
know-
and
you
have
a
dense
activation
state.
Well,
that's
all
you
have
is
scalar
activation
right,
but
as
soon
as
you
start
having
sparse
representations,
then
you
have
a
population
encoding,
it
doesn't
mean
you
don't
have
scalar
representations,
but
the
population
coding
is
so
in
some
sense
dominant.
A
E
E
We
were
dealing
with
right,
resolving
that
ambiguity
and
and
we
got
to
work,
but
we
weren't
totally
happy
with
how
we
got
to
work,
because
you
know
the
the
unions
can
get
too
big
and
then
the
whole
thing
falls
apart,
and
so
you
know
it
didn't
bother
me
too
deeply,
because
there's
various
ways
you
can
imagine
getting
around
that
problem,
how
much
all
kids
right
now,
but
but
but
that
was
one
of
the
key
problems
we
were
dealing
with.
It's
this
ambiguity.
You
touch
something
with
your
finger
and
you
don't.
E
A
So
what
I'll
say
is
they?
I've
read
their
model?
I've
read
the
code
and
my
understanding
is
like
in
my
mental
model,
it's
going
to
work
a
whole
lot
like
ours.
What
we
call
path,
integration
of
unions,
their
their
model,
is
going
to
do
the
exact
same
thing.
It's
going
to
activate
multiple
blue
cells.
You
move
it's
going
to
path,
integrate
them
all,
and
it's
going
to
have
all
the
same
pitfalls
as
ours
as
well.
A
A
B
E
Edge
could
be
an
edge
on
my
finger
could
be
part
of
a
hundred
different
objects,
but
a
curved
edge,
for
example,
but
a
curved
edge
on
the
coffee
cup
could
only
in
a
few
places
where
it
could
be
so,
but
we
still
haven't
gotten
to
the
issue
yet
of
the
core
thing
we're
gonna
talk
about,
which
is
how
they
get
away
with
one
module
right.
We
haven't.
We
haven't
addressed
that
yet.
A
E
A
E
E
So
the
specific
pairing,
as
shown
here,
let's
say
the
left-
module
of
a
sensory
input
in
a
location
well
that
location
that
blew
up
that
blue
triangle
on
the
bottom
left-hand
corner
here
could
represent
a
location
in
a
hundred
different
objects.
Again,
maybe
they're
not
dealing
without
different
objects.
Maybe
that's
the
key
here.
If
that
location
could
be
on
100
different
objects,
then
you
know
it's
just
it's
just
it's!
It's
not
specific
to
a
particular
object.
Then
then
you
have
this.
C
I
mean,
I
guess,
also
they're
not
really
trying
to
address,
or
at
least
not
explicitly,
trying
to
address
something
like
object
recognition
here.
It's
more
about
learning
the
structure
of
quote-unquote,
an
object
which
is
here
in
environment,
so
that
you
can
predict
how
it
changes
or
kind
of
where
you
are.
E
D
C
That's
where
both
the
resetting
of
the
heavy
and
weights
between
environments
and
also,
as
far
as
I
understand
it's,
it's
different.
It's
completely
different
networks
that
learn
the
family
tree
structure
versus
the
2d
graph
structure,
for
example.
So,
even
though,
like
there
might
be
a
an
elegant
solution,
at
least
at
this
point,
they
haven't
specifically
addressed
kind
of
how
how
to
infer,
for
example,
what
kind
of
structural
situation
you're
in
or
or
how
to
yeah
have
maybe
the
representational
capacity
to
deal
with
preserving
weights
across
environments.
A
Yeah
definitely
one
of
the
moving
pieces
here
is
that
there
are
some
problems
they
chose
not
to
solve
yet
and
yeah,
like
the
things
neil's
just
pointed
out
where
they
actually
use
different
sets
of
weights
for
different
environments
even
and
and
looking
in
their
code.
I
can
see
that
they've
tried
to
deal
with
that
other
ways.
They've
tried
to
use
like
essentially
masks
where
you
mask
out
certain
sets
of
weights
for
different
environments
when.
E
E
It's
got
a
set
of
many
columns
in
it
and,
in
my
mind,
the
space.
The
motor
sensor,
motor
space,
that's
represented
by
that
column,
is
fixed
aft,
it's
learned,
but
it
doesn't
change
dynamically.
That
is
that
column
basically
says
everything.
I'm
going
to
learn
is
going
to
be
built
on
the
same
sensory
motor
space.
I
can't
I
can't
have
a
hexagon.
I
can't
have
a
two-dimensional,
you
know
or
square.
E
They
have
pictures
in
their
paper,
like
you
know
the
square
grids
and
hexagonal
grids
and
whatever
a
single
column's
not
going
to
swap
back
and
forth
between
those
it's
going
to
it's
going
to
learn
some
representational
space
and
it's
going
to
try
to
put
every
damn
thing.
It
learns
into
that
representational
space
and
what
I
think
you're
telling
me
here
is
that
they
can.
They
can
swap
those
out
that
is
they
can
they
can
go
from
a
hexagonal
space
to
a
square
space
dynamically.
Is
that
what
you're
saying.
D
E
A
One
one
thing
like
one:
one
piece
of
evidence
that
they
might
bring
up
is
is
also
an
it's
a
different
paper
from
tank
slab
than
we
usually
bring
up.
It's
one
with
a
rat,
putting
its
hand
on
a
lever
pulling
down
the
lever
here.
A
musical
pitch
plays
the
pitch,
goes
up
and
up
and
grid
cells
seem
to
represent
the
location
in
that
sequence
and
that
pitch
sequence,
but
this
same
cell
well
recorded
if
the
animal
is
moving
around
in
2d
is
a
standard
good
cell,
and
so
it.
E
I
guess
I'm
trying
to
differentiate
the
the
metric
shape
of
the
space
versus
what
the
space
is
being
used
to
represent.
Therefore
you
know
yeah,
you
know
I
could
say
imagine
this.
I
can
say
I
have
three
dimensions
x,
y
and
z,
and
I
have
movements.
I
know
how
to
move
and
path
integrate
in
those
three
dimensions:
x,
y
z
and
I'm
a
column.
E
That's
what
I've
learned,
I'm
gonna
now
everything
I'm
gonna
learn
has
to
fit
within
that
I
mean
I
can
I
learned
that
space
and
it
could
slowly
morph,
but
but
the
moment
the
moment
basis.
If
something
else
comes
along,
I
have
to
fit
it
into
that
space.
I
have
to
fit
into
three-dimensional
space
and
the
movements
I
have
there's.
No.
I
can't
change
myself
to
represent
something
completely
different
very
quickly.
It
would
be
a
very
slow
process.
A
E
Would
totally
do
that
right
and
v1
the
space
represented
by
v1
and
v2?
I
mean
essentially
every
every
cortical
column,
in
my
mind,
has
to
learn
its
own
space
and
and
depending
on
what
its
inputs
and
its
motor
behaviors
are
primarily
what
its
motor
behaviors
are.
Then
that
determines
the
space
and
so
they'd
be
different.
You
know,
column
and
s1
is
going
to
be
different
than
column
d1,
but
but
yeah,
okay,
that's
fine,
but
there
wouldn't
be.
I
wouldn't
be
swapping
them
out
in
the
individual
comment.
E
C
E
C
E
E
E
Is
that
that
the
the
inputs,
the
the
state
of
the
mod
of
the
module
on
the
left,
will
be
changing
much
more
rapidly
than
the
state
of
the
module
on
the
right?
If
I,
at
least
in
my
thinking,
that's
what
happens
at
least
that's
what
I
think
you
know.
That's
the
way
I've
been
thinking
about
this,
and
so
it
it's
it's
interesting.
What
is
an
attractive
state
when
some
of
the
elements
are
moving
and
some
of
them
are
changing
and
some
of
them
aren't?
E
A
E
Yeah,
okay,
I
mean
I'm
really
I'm
trying
to
make
this
work
because
I
like
the
idea,
I
said
like.
Oh
that's
a
really
brilliant
idea.
We
could
we
could.
We
could
somehow
find
this
voting
across
scale
across
regions
or
across
hierarchy,
and
I'm
trying
to
I'm
thinking
about
trying
to
make
it
work
and
I'm
struggling
a
little
bit
so
it'd
be
nice.
A
A
C
E
Within
a
particular
module
I
say:
here's
a
feature
red
triangles
and,
and
there
is
where
it's
going
to
be
yeah,
you
don't
have
a
choice.
It's
going
to
be
that
green
triangle
location,
but
that
doesn't
seem
possible.
I
mean
it
doesn't
impossible
to
interpret
the
blue,
the
blue
triangles
and
the
blue
triangle
as
as
a
as
a
specific
location,
because
because
in
any
particular
environment
it
it
just,
I
can't
do
that.
I
mean
I
it's
like
or
or
either
that
or
the
location
is
not
a
metric
location
and
you
don't
do
path
integration.
A
E
Well,
if
it's
hardwired,
I
don't
see
how
it
works.
I
don't
see
how
you
can
do
path
integration
to
the
target
right,
if
it's
not
hardwired.
Well
it
should
then
it's
that
this
is
just
the
associative
linking
between
feature
and
location,
or
you
know
the
observation
and
the
metric
space,
but
if
it's
hardwired
then
it
can't
be
that
then
it's
it's!
It's
just.
I
can't
do
path
integration,
because
because
the
I
can't
say
that
the
next
blue
triangle
down,
if
I
move,
move
north
step,
then
then
it
and
then
it's
going
to.
E
But
I
don't
see
how
you
can,
if
the,
if
the,
if
the
small
red
triangles
are
not
learned
if
they're,
just
if
they're,
if
that's
just
a
hard-coded
intersection
between
blue
and
big
red,
then
then
I
can't
it
just
can't
work.
I
mean
it'd
be
like
saying
that
that's
sort
of
like
dictating
at
certain
location,
you're,
going
to
see
this
feature
in
the
next
location.
You're
always
going
to
see
this
feature.
There's.
E
E
Yeah
but
but
but
this
intersection
is
unique,
we
must
not
be
communicating
properly
if
the
little
red
triangles
are
not
learned
that
they're
just
like
there's
this
mechanistic
intersection,
then
then
they
represent
a
spatial
feature
at
a
location,
but
it's
not
learned.
If
it
can't
be
learned,
then
I
can't
use
it
to
learn
the
actual
structure
of
an
object.
It
has
to
be
learned.
A
A
B
E
E
We're
you're
relying
on
the
scale
if
we're
relying
on
the
scale
and
activation
of
these
little
red
triangles.
Maybe
that's
what
you're
saying
perhaps
like
like
the
intersection
between
the
two
is
fixed,
but
the
scale
of
activation
of
the
little
red
triangles
is
not
fixed.
Is
that
maybe
what
you're
saying.
C
A
E
E
A
E
So
yeah
all
right,
so
so,
okay.
That
was
really
helpful.
You
just
said
a
moment
ago,
so
I'm
going
to
now
paraphrase
this
from
again
from
my
thinking
here.
It
helps
me.
I
don't
want
to
dominate
the
conversation,
but
it
helps
get
through
these
things.
We
relied
on
multiple
grid
cell
modules
to
achieve
sort
of
a
unique
representation
of
an
object
space
for
an
object,
and
we
didn't
want
to
go
to
different
scales
because
we
didn't
see
how
it
worked.
E
So
we
we
hoped
that
there
would
be
multiple
modules,
a
similar
scale
orientation
slightly
different
scale,
but
so
on
here,
they're,
basically
going
to
rely
on
multiple
modules
again,
but
they're
going
to
rely
on
multiple
modules
across
scale,
which
we've
already
identified.
We
think
is
a
problem
with.
We
don't
understand
how
it
could
work.
E
Yet
maybe
we
could,
but
at
the
moment
we
don't
understand
how
it
could
and
but
but
that's,
okay,
but
and
and
they're
voting,
not
at
the
grid
saw
level,
but
at
the
the
the
modules
that
we're
we're,
reforming,
we're
looking
across
are
more
more
like
play
cell
modules,
and
so
in
that
regard
an
individual
place
cell.
Any
one
of
these
modules
is
not
unique
to
that
object,
which
we
know
is
true
anyway,
but
across
modules.
E
Yeah,
but
I
think
the
scale
issue
the
change
of
scale
across
the
module
just
finger
through
my
head
right
now,
it's
still
a
real
big
problem.
I
don't
I
don't
see
how
you
get
around
that,
but
but
I
think
the
idea
that
thinking
about
grid
cell
modules
I
mean
place
on
modules,
either
voting
or
play
cell
modules,
combining
together
for
uniqueness,
is,
is
an
interesting
idea
that
I
don't
think
we've
talked
about
a
lot.
A
At
all
yeah-
and
I
should
then
shout
out
to
them
that
they
part
of
you
know
what
took
them
a
lot
of
time
on
this
paper
is.
They
went
and
then
searched
neural
data
from
two
experiments.
They
made
the
prediction
that
they
would
find
that
grid
cells
are
correlated
with
specific
place
cells
across
multiple
environments
and
they
got
a
decent
result
like
a
nice
little
correlation
there.
It's
not
it's
not
for
sure,
but
it's
there.
There
seems
to
be
a
correlation
between
good
sales
and
play
cells
and
cross
across
environments.
That's.
B
A
Well,
I've
on
one
side
less
where
I
left,
where
I'm
just
elaborating
that
at
this
point
or
putting
a
little
words
on
what
I
just
said
so
with
multiple
modules,
the
place
cells
can
represent
novel
feature,
location
pairs,
which
is
slightly
different
from
what
we
just
we're
talking
about.
I
mean
it's
not
it
doesn't
contradict
it,
but
I'm
not
saying
exactly
what
we
just
said.
A
It's
interesting
that
this
system
can
represent
novel
feature
location
pairs
without
it
is
this,
it's
the
same
good
cell
trick,
as
we've
always
talked
about
with
multiple
modules.
You
can
represent
multiple
things.
Well,
you
can
represent
novel
things,
but
now
it's
at
the
place
cell
level,
and
so
so
that's
just
that.
That's
just
a
nice
idea
and
in
our
model
I'll
read
the
final
bullet
point.
A
We
can
only
represent
a
new
feature
location
pair
by
forming
new
grid
cell
to
place
cell
connections,
whereas
theirs
they
can
represent
any
pair
they
want
and
they
don't
have
to
learn
it
right
away.
They
can
trust
that
in
the
future,
if
this
feature
location
pair
occurs,
the
same
rep,
the
same
set
of
cells
is
going
to
activate.
A
They
don't
have
to
do
any
initial
learning,
whereas
our
temporal
memory
kind
of
does
because
it
generates
these
random
patterns,
and
it
will
only
ever
recover
those
random
patterns
if
it
forms
synapses
to
to
resurrect
them.
So
there's
a
there's,
a
nice
idea
there,
that's
the
other
side
of
the
coin.
That
of
what
I
was
saying
in
the
previous
slide.
E
I
still
don't
quite
get
this
last
bullet.
I
still
don't
really
understand
how
and
and
how
path
integration
works
in
that.
In
that
case,
we
bounced
around
a
little
bit.
But
if
you're
not
forming
new
connections,
then-
and
you
said
they
have
empirical
evidence
for
this-
that
the
same
grid
cells
partner
with
the
same
place
cells.
E
So
maybe
this
maybe
that's
why
it
is,
but
it
does
seem
to
me
to
make
it
hard
to
imagine
how
path
integration
occurs
in
that
case,
because
if
the
placeholders
are
determined
by
sensory
input,
then
then
the
grid
cells
are
can
be
determined
by
central
input
and
therefore
you
do
not
you're
not
able
to
use
them
as
a
metric
space
in
path
integrated.
So
we've
talked
about
that
a
bit
already,
but
you
still
don't
see
how
it's
going
to
work
and
it's
by
the
way.
A
Right,
I,
let's
see
the
the
connection
I
can
draw
between
what
they're
saying
and
what
I'm
saying
here
this
this
middle
bullet
point.
They
have
their
place
cells
being
deterministically
determined
by
it's
a
anyway
turnover
phrase
determined
by
the
grid
cells
and
the
sensory
cells.
Whereas
what
I've
been
saying
is
grid
cells
are
determined,
deterministically
determined
by
the
sensory
input
has
been
what
I've
been
saying.
So
that's
one
connection
between
these,
where
we're
using
similar
words,
but
talking
about
different
things.
A
Their
work
does
not
require
any
kind
of
distortions
or
they
they
wouldn't
predict
any
kind
of
distortions
of
this
they're
they're.
The
two
populations
are
totally
independent
like
this.
This
can
path
integrate
whatever
it
wants,
and
the
fact
that
the
place
cells
can
be
computed
from
the
input
and
the
grid
cells
doesn't
really
influence
the
metric-ness
of
it.
E
E
Yeah
but
but
then,
but
then
we
can't
say:
there's
this
one
to
one
mapping
right.
It's
it's.
I
mean
this.
The
argument
here
is
that
there's
this
sort
of
pre-wired
deterministic
mapping
and
it's
not
learned,
and
that
then
leaves
in
my
mind
the
problems.
But
then,
if
we
say
oh
well,
it's
it's
got.
It's
got
this
permanent
mapping,
but
we
can
just
remap
it
when
we
need
to
for
another
environment.
Well,
then,.
B
Entirely
so
the
way
that
people
think
about
this-
I
remember
a
conversation
I
had
at
cosine
last
year,
kind
of
interesting
that
this
is
like
almost
exact
one
year
ago,
since
we're
about
to
enter
the
next
cosine
2021.
Now
was
that
how
did
he
put
it?
He
said
that,
like
that
these
weak
correlations,
you
called
them
between
grizzles
and
play
cells,
may
underlie
sort
of
a
hidden,
attractor
landscape,
so
not
entirely
free
to
put
these
wherever
you
want,
and
that's
what
what's,
this
name
kentros
also
has
shown.
B
I
think
I've,
like
briefly
talked
about
cliff
kendros,
talked
about
his
work.
He's
been
like
trying
to
like
destabilize
these
grid
surplus
cell
maps
and
analyze,
how
one
re
recovers,
the
others
or
or
doesn't,
and
so
it's
it's
messy
for
us
modelers,
but
it's
a
mixed
picture
because,
on
the
one
hand
we
we
like
sort
of
like
the
the
stable
phase
relationship
between
grid
cells,
for
example,
and
that
there
is
a
link
between
grid
cells
and
play
cells.
That
seems
to
be
reliable.
B
Yet,
on
the
other
hand,
when
you
destabilize
them,
there
are
multiple
different
places
in
which
it
like
they
might
re-materialize
so
to
speak,
so
so
you're
not
free.
These
mappings
are
not
entirely
free,
they're
constrained
somehow,
but
on
the
other
hand,
they
tend
to
be
stable,
a
lot
in
the
same
environment.
E
E
Now,
if,
in
the
middle
of
the
room,
there
was
a,
there
was
a
some
obstacles,
it
looked
like
the
corner
kind
of
very
similar
to
the
corner,
but
just
it's
not
in
the
corner.
It's
in
the
middle
of
the
room.
It's
like
two
walls
coming
together
in
the
middle
of
the
room,
well,
that
same
place
cell
may
fire
at
that
corner
in
that
in
the
middle
of
the
room.
When
I
approach
that
that
that
interior
feature
that
looks
kind
of
like
the
the
one
in
the
corner,
environmental.
E
Yeah
so,
but
what
you're
saying
I
want
to
make
sure
what
you
I
think
you
just
said
is
that
binding
between
oh,
this
is
kind
of
a
corner
that
looks
like
this
and
therefore
my
play
cell
should
fire
that
mapping
would
be
different
in
a
different
environment.
Is
that
what
you're
saying
like
if
I
had
another
environment
that
I
perceived
as
different
and
I
ran
into
that
similar
corner
in
the
middle
yeah.
E
E
B
B
E
Yeah,
but
I
guess
but
the
point
is
that
when
I
go
in
a
different
environment,
I'm
just
I'm
just
trying
to
reiterate
what
you're
saying
I'm
not
making
a
point.
I'm
just
trying
to
make
sure
I
understand
the
point
in
a
different
room
that
that
pairing
wouldn't
necessarily
be
more
likely.
It
would
be
like
hey.
I
might.
I
might
find
the
same
corner
looking
thing
resulting
in
completely
different
play
cell,
which
which,
if
that
were
the
case-
and
it
tells
me
that
the
play
cell
is
surely
not
being
driven
purely
by
the
sensory
input.
E
The
place
cell
is
already
taking
advantage
of
the
fact
that
there's
a
it
knows
what
environment
I'm
in.
So
it's
like
saying
yeah.
This
is
this
is
an
a
corner
and
I'm
in
room
b.
Therefore,
every
time
I
see
an
acorn
in
room
b,
I
should
become
active,
but
if
I'm,
if
I'm
in
a
different
room,
well,
who
knows
something
else
right
now
right,
I'm
saying
all
that
to
say
yeah
did
I
get
that
right.
B
D
E
B
I
didn't
understand
that
I
might
want
to
like
briefly
point
out
that
I
hope
to
build
like
a
spiking
implementation
of
the
tome
and
ikenbaum
machine
as
part
of
like
an
upcoming
research
project
with
a
computational
neuroscience
group,
and
so
as
part
of
that,
I
want
to
like
fully
understand
the
the
implementation
that
they
published,
and
so
I'm
taking
it
apart
with
three
colleagues
of
mine,
so
we're
actually
going
through
the
methods
in
in
you
know
really
small
detail.
I
actually
want
to
run
the
original
code.
B
A
C
E
E
Because,
if
yeah,
because,
if
you're
trying
to
do
some
sort
of
stable
attractor
network,
some
of
those
modules
will
be
changing
rapidly.
Some
won't
because
they're
running
at
different
scales
and
and
it's
hard
for
me
to
imagine
what
it
means
to
have
an
attractive
network
when
some
parts
are
not
or
some
parts
are
moving
and
other
parts
are
not
moving.
E
Some
parts
are
changing
and
approaching
not
changing
and-
and
so
we've
always
avoided
trying
to
vote
across
scales
because
it
or
look
at
modules
across
scales,
because
because
of
that
issue,
so
the
question
is
we
didn't
get
a
resolution
on
that
here?
How
they're
handling
that
in
this
model?
Why
does
it
work.
D
E
One
way,
one
way
it
might,
they
might
got
around
it
if
you
don't
consider
the
system
being
dynamic.
If
you're
just
saying,
let's
take
a
snapshot
in
time,
then
it
works
right.
You
can
say,
take
a
snapshot
on
time
and
in
some
sense
that's
how
people
thought
about
the
multiple
gridsole
module
at
different
scales
issues
at
a
snapshot
in
time
you
have
a
unique
representation,
but
if
you,
if
you
flow,
if
you
watch
the
flow
pattern
through
time,
then
you
have
this.
These
varying
scales
of
change
and
then
the
attractor
doesn't
work
anymore.
E
B
B
B
If
you
go
a
little
bit
further
down
the
gradient,
you're
gonna
have
like
two
that
are
stacked
on
top
of
each
other
and
then,
if
you
go
even
further,
you
might
have
four
or
five
different
scales
of
of
grid
cells
like
layered
on
top
of
each
other,
yeah
and
people
are
debating.
You
know
whether
seven
or
eight
different
scales,
and
so.
B
E
E
I'm
not
claiming
that
these
connections-
no,
no,
I
know
you're
not
claiming
anything.
I
just
I'm
just
I'm,
not
I'm
not
trying
to
trip
you
up.
There's
one
thing
be
precise
about
this,
because
I
think
we've
also
read
that
the
evidence
that
that
cells
actually
collect
information
across
those
different
scales
is
is
not
there
or
they
haven't,
found
it
yet
or
it's
marginal
or
something
like
that.
E
So
anatomically
they're
stuck
like
that.
But
if
an
individual
cell
doesn't
make
that
connection,
then
maybe
it
doesn't
happen
anyway.
I
think
it's
still
a
very
interesting
problem
and
it
gets
it's
just
one
of
the
dimensions
of
this.
I
think
we're
all
everyone.
This
paper
we're
reviewing
my
work.
The
markets
are
working
with
you
they're
all,
there's
a
couple
of
key
issues
here,
we're
all
trying
to
resolve.
E
We
may
think
about
the
problems
differently,
but
there's
several
things:
how
do
you
get
uniqueness?
How
does
how
does
sensory
relate
to
spatial
representation?
Like
you
know,
what
is
the
uniqueness
of
the
space
code,
the
place
sales
versus
the
grid
code,
and
so
on?
You
know,
I
think
we're
moving
markets
has
been
arguing
moving
away
from
having
these
purely
metric
spaces.
This
paper,
we're
viewing
today,
seems
to
me
metric
spaces
are
going
to
be
really
rocky.
It
might
be
wrong.
B
Yeah,
but
the
more
important
thing
is
that
it
re-strengthens
numenta's
sort
of
you
know
like
statement
that
you
can
do
this
relational
learning
and
that,
since
it
holds
true
for
for
all
the
different
nodes
that
you
can
like
do
much
faster
inference,
which
was
the
point
that
the
cortical
learning
framework
also
makes.
I
thought
it
was
like
again
like
too
bad.
They
didn't
cite
you
guys,
but
on
the
other
hand,
you
know
maybe
tolman
and
eikenbein,
like
deserved
a
shout
out,
it's
least
controversial
to
name.
You
know.
C
B
E
I
sure
we're
probably
running
out
of
time
here,
but
I
just
here
we
are
right.
Thank
you.
We
have
a
little
bit.
Oh
no,
it's
yeah.
We
could
do
it.
I
just
want
to
point
out
fly.
I
don't
know
if
you've
been
paying
anything,
but
at
least
I've
recently
been
talking
about
really
inspired
by
the
papers.
You
reviewed
when
you're
here
about
the
the
you
know
to
generate
a
grid
cell.
You
need
a
bunch
of
one-dimensional
movement,
vector
things
and.
E
E
Been
working
on
that
idea
right
and
we
have
multiple
cells
in
the
minicom
that
are
all
oscillating
at
the
same
frequency,
but
but
offset
by
their
phase,
and
so
now,
if
you,
if
you
think
of
that
as
like
a
1d
grid
cell
module
and
any
individual
column,
has
lots
of
those,
then
you
have
a
unique
code
right
within
the
column.
It's
it's
got
some
other
issues,
but
so
I
think.
B
No,
it's
not
settled,
I
mean
in
the
modeling
literature,
there's
still
people
you
know
doing
all
you
know.
I
mean
the
nobody
is
contesting
they're
like
the
inhibitory
surround
idea
of
the
can
models
which
work
really
great,
like
for
stability
and
they're
awesome
for
lots
of
different
things.
But
but
of
course
this
is
about
the
update
mechanism
right
yeah.