►
Description
To learn more about Brains@Bay, visit our Meetup page: https://www.meetup.com/Brains-Bay/
Brains@Bay Meetups focus on how neuroscience can inspire us to create improved artificial intelligence and machine learning algorithms. In this Brains@Bay meetup, we focused on grid cells and how they act as an inspiration for machine learning architectures.
Numenta Senior Researcher Marcus Lewis started us off with a basic overview of grid cells and then dove into using grid cells and other cells to quickly form spatial memories.
Next, James Whittington, Postdoctoral Research Associate at University of Oxford talked about unifying space and relational memory in the hippocampal formation.
Finally, Kimberly Stachenfeld, Research Scientist at DeepMind, presented a relational view of grid cells in which grid cells represent geometry over graphs.
Professor Tim Behrens from University of Oxford joined us for the discussion.
Link to meetup: https://www.meetup.com/BraIns-Bay/events/278434338/
0:00 Introduction
2:02 Marcus Lewis: Grid cell intro + Quickly forming structured memories
34:32 James Whittington: Generalisation in the hippocampal formation
1:03:17 Kimberly Stachenfeld: Representation Learning with Grid Cells
1:46:16 Discussion + Q&A
- - - -
Numenta is leading the new era of machine intelligence. Our deep experience in theoretical neuroscience research has led to tremendous discoveries on how the brain works. We have developed a framework called the Thousand Brains Theory of Intelligence that will be fundamental to advancing the state of artificial intelligence and machine learning. By applying this theory to existing deep learning systems, we are addressing today’s bottlenecks while enabling tomorrow’s applications.
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A
Welcome
to
brainstorm
bay,
I'm
charmaine
from
dementa
and
today
we'll
be
focusing
on
the
topic
of
grid
cells.
So
just
for
a
little
bit
of
context.
First,
we
started
brains
at
bay
two
years
ago,
where
we
discussed
a
series
of
topics
at
the
intersection
of
neuroscience
and
machine
learning.
Our
main
goal
is
to
foster
the
study
and
development
of
machine
learning,
algorithms
that
are
inspired
by
neuroscience
research.
So
we're
super
excited
to
have
everyone
here
today.
A
Each
talk
will
be
around
30
minutes
and
then
there
will
be
a
q
a
session
towards
the
end,
so
for
all
our
attendees
feel
free
to
ask
your
questions
through
the
q
a
function
on
zoom
and
in
case
you
missed
some
of
this
meetup
we'll
be
recording
this
and
posting
this
on
our
youtube
channel.
I'll.
Add
the
link
to
our
meetup
page
once
the
video
is
uploaded.
B
C
B
But
I've
been
immersed
in
neuroscience
and
machine
learning
for
past
six
or
seven
years.
Our
second
speaker
is
going
to
be
james
whittington,
who
is
a
postdoctoral
research
associate
at
oxford
and
he
did
his
phd
at
oxford
where
he
worked
with
raphael
bogach
who
who
is
presented
at
one
of
these?
What
was
that
he's.
B
Excellent
hooray,
and
so
in
his
phd
he
was
co-advised
by
tim,
behrens
and
rafael
bogat
and
tim
is
here
today.
You
just
heard
him
speak
and
I
guess
I'll
say
well
I'll
I'll.
Do
a
little
more
intro
right
before
the
speeches
so
and
then
our
other
speaker
is
kim
stackenfeld
who's,
a
research
scientist
at
deepmind.
She
did
her
phd
at
princeton
where
she
worked
with
matt
bob
nick
and
sam
gershman,
and
so
I
will
go
ahead
and
get
started
with
my
talk,
which
is
really
more
of
a
two-part
talk.
B
So
yes,
I'm
doing
a
two-part
talk
and
since
I'm
basically
doing
two
talks
in
one,
I'm
probably
going
to
use
most
of
the
30
minutes.
So
questions
are
welcome
but
like
discussion,
more
discussion
topics
it'll
help
me
if
those
get
saved
till
the
end.
But
clarifying
questions
are
totally
fine,
so
the
first
half
of
this
is
going
to
be
an
introduction
to
good
cells.
B
Since
the
audience
for
this,
for
this
meeting
is
sort
of
a
mix
of
neuroscience
and
machine
learning
people,
and
maybe
just
people
who
are
generally
interested,
I'm
going
to
try
to
give
you
just
a
basic
intuition
of
grid
cells,
and
some
people
are
going
to
be
coming
here,
wondering
why
they
should
care
about
them,
especially
like
machine
learning.
People
wondering
why
what
what
ideas
do
good
cells
bring
to
the
table,
and
I'm
not
going
to
answer
that
question
at
least
not
immediately,
because
the
field's
still
figuring
out
what
these
cells
do
like,
what
their?
B
What
their
actual
role
in
the
brain
is.
That's
not
a
settled
question
and
in
the
second
half
of
the
talk,
I
will
adopt
one
point
of
view
on
that.
But
the
first
is
more
just
designed
to
like
give
you
the
raw
data
in
some
sense,
so
that
you
can
then
receive
a
series
of
views
from
me
and
the
other
speakers.
A
B
B
This
shouldn't
work,
so
this
rats
running
around
they're,
recording
a
neuron
and
each
of
these
white
diamonds
indicates
where
this
neuron
spiked
or
where
rather
where
the
rat
was
standing
when
the
neuron
spiked
and
they
sped
up
the
video,
you
can
even
see
a
person
dropping
food
and
they're
the
to
kind
of
force,
a
random
walk
of
the
animal.
So
so
yeah
the
rats
foraging
for
food
and
at
first
the
dots
might
have
seemed
a
little
bit
randomly
placed.
B
But
now
once
you've
seen
it
kind
of
averaged
over
time,
you
see
that
the
the
dots
are
not
random.
They,
the
the
spike
locations,
all
occurred
at
the
corners
of
this
sort
of
lattice
the
set
of
points
that
is
sort
of
equally
separated.
You
can
see
equilateral
triangles
roughly
or
you
can
see
a
hexagon
shape
here,
and
so
when.
C
B
When
the
when
the
mosers,
when
their
group
saw
this,
I
mean
these
neurons
didn't
have
a
name
and
they
saw
that
that
looks
like
a
grid,
and
so
they
called
them
grid
cells
and
I'm
going
to
move
into
a
more
cartoon
view
of
them.
First,
I'm
going
to
change
zoom
back
so
that
the
video
is
so
those
back
to
normal
and
so
to
give
more
of
a
cartoon
view
of
grid
cells.
B
Just
so,
we
can
talk
about
a
couple
of
the
properties,
a
single
good
cell
fires
at
multiple,
equally
spaced
locations
when
the
rat
is
at
multiple,
multiple,
equally
spaced
locations,
and
in
this
picture,
though,
we're
just
recording
one
neuron
and
and
we're
just
putting
like
one
electrode
into
one
part
of
the
inter
rhino
cortex
in
the
next
slide.
I'm
going
to
talk
about
if
you
record
different
points
along
the
enter
rhino
cortex,
what
you
see
and
what
you
see.
B
So
this
is
the
same
picture
but
multiple
electrodes,
actual
images
of
electrodes
just
kidding
and
and
what
you
see
when
you
record
these
neurons
as
you
get
grid
cells,
but
the
grids
are
different.
The
grids
are,
they
have
different
spatial
periods
or
scales,
we're
on
one
end
of
this
one
end
of
this,
this
of
the
anterinal
cortex,
the
dorsal
end
of
it.
B
A
second
interesting
thing
to
look
at
is,
if
you
look
at
it,
go
back
to
looking
at
just
one
electrode,
but
look
at
the
the
local
population
of
cells
and
and
what
the
other
cells
nearby
are
doing
and
what
you
find
there.
So
here
I'm
going
to
show
empirical
data
where
they
did
that
from
the
original
good
cell
paper,
where
they
record
three
different
neurons
at
a
single
electrode
and
the
empirical
data.
Here
on
the
left,
you
see
the
a
blue
cell,
the
blue
cells,
plotted
its
firing
locations
are
firing
in
a
grid.
B
Red
cell
is
firing
in
a
grid.
This
green
cell
is
firing
in
a
grid
and
they
all
in
the
sense
have
are
the
same
grid,
just
shifted
or
translated,
and
so
this
is
the
property
that
you
see
when
you
look
at
a
bunch
of
locally
clustered
neurons
they're,
the
they're,
the
the
same
grid,
but
translated,
and
you
get
the
sense
that
so
looking
at
looking
at
how
there's
still
some
like
uncovered
space.
B
A
way
a
valid
way
of
thinking
about
good
cells
is
that
a
population
of
cells
seems
to
be
tracking
the
location
of
a
rat
and
I'll
just
replace
this
with
a
different
cartoon
version
of
a
rhombus
just
so
I
can
make
the
point
a
little
bit
more
clearly,
so
you
have
it's
like
it's
like.
You,
have
a
bump
of
activity
that
moves
through
this
population,
regardless
of
how
they
work
from
the
outside.
This
is
seems
to
be
how
they
look.
B
So
the
term
for
this
set
of
cells
that
has
the
same
grid
is
a
grid
cell
module
and
and
also
you'll,
hear
people
talk
about
it
as
a
taurus
topologically.
This
is
a
taurus,
and
sometimes
when
people
hear
the
word
taurus,
they
picture
a
donut
which
is
right.
I
guess,
but
for
me
that
mental
image
doesn't
work
as
well
for.
C
C
B
Of
like
the
old
video
game
asteroids
where
the
ship
goes
off,
the
side
comes
on
the
other
side,
that's
kind
of
what's
going
on
here
with
the
bump,
except
it's
it's
a
promise,
a
shape
game
of
asteroids
that
a
bump
is
going
through
and
I'll
show
one
more
interesting
property
of
grid
cells.
That's
just
useful
for
making
sense
of
what
they're
doing
for
one
thing
after
a
wrap
comes
back
into
a
room
where
it's
been
before.
If
it
comes
back
to
an
environment,
it's
been.
E
B
Grid
cells
fire
consistently
with
where
they
did
before,
but
an
interesting
property
that
I
think
this.
This
was
an
elegant,
little
study
that
that
showed,
if
you'd,
if
you
now
put
the
rat
in
another
room,
that's
similar
but
sufficiently
different.
The
grid
cells
will
map
onto
the
room
differently,
but
they
do
maintain
their
like
their.
B
Their
relationship,
like
this
cell,
will
always
fire
next
to
this
cell,
the
the
the
cell
will
fire
next
to
this
cell,
but
but
you
get
this
translation
of
them
as
a
whole
and
in
the
study
they
had
to
change
the
colors
of
the
walls
and
they
had
to
change
the
odor
of
the
room
and
those
two
changes
combined
would
cause
the
grids
to
shift
like
this,
but
at
the
same
time
making
just
one
of
those
changes.
Often
wasn't
enough.
B
So
so
it
seems
like
the
system
is
doing
some
amount
of
generalization
or
reusing
of
its
representation
scheme.
But
some
amount
of
like
splitting
as
well,
and
as
I
mentioned,
the
relative
firing
of
the
cells
stays
constant.
It's
there's
always
you
get
this
sort
of
rhombus,
where
the
relationships
are
the
same
within
them
and
to
say
a
little
bit
more
about
good
cells,
though,
and
to
maybe
emphasize
their
importance.
Is
that
they're
not
just
used
for
representing
self-location.
B
It
seems
to
be
roughly
that,
no
matter
what
task
this
rat
is
engaged
in,
the
grid
cells
are
participating
somehow,
and
so
so.
In
this
first
set
of
studies,
I've
listed
that
literally
the
exact
same
grid
cells
are
are
found
to
do
other
things
like
a
rat's,
pulling
a
lever
listening
to
a
pitch
and
then
suddenly
the
cells
are
representing
the
pitch.
That
is
playing
and
second
there's
something
some
relation
to
them
tracking
time
as
well,
and
then
I'm
going
to
bring
up
this.
B
A
second
set
of
studies
where
the
the
distinction
I'll
make
here
is
that
this
is
another
case
where
grid
cells
are
being
used,
but
it
may
be
a
different
population
of
cells.
It's
not
clear
whether
it's
the
same
cells
being
used
for
these
are
a
different
one,
and
so
one
important
of
these
to
focus
on
is
first
of
all.
Primates
like
humans
and
non-humans
have
grid
cells,
but
a
lot
of
the
ones
that
have
been
found
track.
B
The
viewed
location
as
you
look
around
you
get
these
hexagonal
firing
fields
for
where
you're
looking
and
a
second
study
I'll
point
out.
So
some
of
these
people
are
here
today.
So
this
is
definitely
the
paper
that
wins
the
award
for
the
neuroscience
paper.
I've
talked
about
most
at
parties,
so
this
this
was
a
truly
surprising
thing.
B
So
if
humans
are
trained,
if
humans
practice
manipulating
a
virtual
bird
changing
its
neck
length
and
its
leg
length,
grid
cells
start
to
adapt
the
the
representation
scheme
of
responding
like
a
grid
in
this
space
of
neck
length
and
leg
length.
This
is
based
on
fmri
evidence,
but
it's
a
pretty
strong
signal
like
that.
Is
it's
a
pretty
it's
pretty
clear.
It
seems
to
be
what's
going
on
here
so
so
this
is
one
of
like
the
big
fun
results.
B
I,
in
my
mind
I
kind
of
organize
views
on
grid
cell
into
two
general
clusters
of
views
and
in
one
of
them,
one
of
them
it's
more
of
part
of
a
memory
system
and
in
these
and
these
types
of
models
by
the
way,
this
slide
does
not
list
all
of
the
theoretical
work
on
good
cells,
but
I
really
focus
on
the
the
high
level
ones,
though
the
the
types
of
models
that
are
using
grid
cells
to
solve
some
problem.
So
so
I'm
leaving
a
lot
of
theoretical
work
on
good
cells
out.
B
In
this
first
view,
there's
a
there's,
a
real
emphasis
on
learned
associations
and
these
types
of
model.
The
system
you
know,
updates
the
grid
cells
forms
an
association
updates.
The
good
cells
forms
an
association.
They
do
that
kind
of
thing,
as
a
second
class
of
models,
rather
than
treating
grid
cells.
As
like
this
blank
canvas
for
associating.
B
B
In
my
mind,
something
that's
common
to
both
of
these
views
is
that
grid
cells
participate
and
and
like
their
importance,
is
the
fact
that
they
can
be
updated.
Other
representation
schemes
might
optimize
for
other
things,
like
reconstruction
error,
the
grid
cells,
sort
of
optimize
for
being
able
to
be
updated,
and
so
the
rest
of
this
talk,
though
I'm
adopting
view
one
mainly
because
it's
it's
the
it's
the
where
most
of
my
work
has
been,
and
so
so,
but
I'm
still
very
intrigued
by
the
second
view.
B
It's
just
more
of
my
work
is
on
the
first
one,
and
so
I'm
jumping
into
the
second
part.
So
I'm
taking
off
one
hat
and
putting
on
a
different
hat.
Now,
I'm
suddenly
I'm
confident
of
good
cells
being
used
in
a
memory,
and
I'm
gonna
talk
about
that
to
you.
So
in
this
I
titled
this
quickly
forming
structured
memories.
B
So
I'm
putting
an
emphasis
on
the
quickness
here
and
and
I'll
tell
you
what
I
mean
there
so
in
the
rest
of
this
talk,
I'm
going
to
focus
on
two
operations
that
are
real,
that
are
important
for
mapping
and
so
the
first
of
these
operations
and
by
the
way
we've
been
working
on
these
for
over
the
years.
But
I'm
going
to
present
a
previous
version
of
them
and
then,
where
my
head's
at
right
now.
B
So
this
first
operation
is
that
of
learning
what
I'm
calling
here,
a
sensory
map
of.
So
if
it
you
can
imagine
if
a
rat
is
walking
around
and
it's
noting
what
it
senses
at
different
locations
and
it
associates
what
it
senses
with
those
locations.
That's
sort
of
learning
a
sensory
map
forming
a
map
of
this
object.
So
I
have
these
two
objects.
B
These
two
parts,
one
is
a
triangle:
one
is
a
pentagon
and
so
forming
different
maps
through
this
sort
of
brute
force
way
of
moving
around
and
learning
what
you
sense,
then
the
second
part-
and
this
is
where
the
speed
comes
in-
is
detecting
arrangements
of
maps.
So
imagine
a
rat,
it
has
learned
maps
of
these
two
objects
and
now
it's
able
to
say,
okay,
I
see
a
triangle.
I
see
a
pentagon
and
here's
the
arrangement
of
them
done.
B
So
this
is
a
problem.
We've
been
kind
of
focused
on
for
a
while
and
and
in
2019
we
had
a
couple
papers
that
laid
out.
Basically,
one
paper
focused
on
this
one
paper
focused
on
this
in
some
sense
and-
and
today
I'm
going
to
briefly
summarize
that
and
then
talk
about
an
alternate
version
and
and
what
what's
special
about
this
alternate
version
is
that
I'm
using
a
single
grid
cell
module
on
it?
B
So
it's
well
known
that
you
can
use
grid
cell
codes
for
representing
displacements,
but
a
kind
of
trippy
idea.
It
caught
us
off
guard
when
we
stumbled
on
it
is
that
you
can
also
do
displacements
between
maps
in
a
way
that
is
really
kind
of
surprising.
It's
sort
of
like
a
wormhole
between
two
maps,
so
you
detect
your
location
on
one
of
these
maps.
B
You
detect
your
location
on
another,
take
the
displacement
between
that
and,
and
you
get
something
that
really
does
represent
the
spatial
relationship
of
these
two
things,
and
so
this
this
was
a.
This
is,
this
is
a
cool
idea.
It's
still
a
definite
candidate,
but
I
want
to
point
out
things
that
we
don't
like
about
it
exactly.
One
is
that,
on
the
on
the
on
the
top
half,
it
doesn't
really
provide
much
opportunity
for
optimizations.
B
We
always
knew
it
from
the
start.
If
you
use
displacements
for
arranging
maps
well,
you're
only
using
displacements,
you're
not
you're,
not
giving
yourself
the
ability
to
rotate
them
or
scale
them
and
grid
cells.
As
far
as
I
can
tell,
don't
provide
much
up,
much
opportunity
or
much
affordance
for
scaling
or
rotation,
and
so
I'm
going
to
now
jump
into
an
interlude.
Some
some
inspiration
from
a
system
called
ratslam.
B
B
There
are
reasons
to
doubt
this
hypothesis,
for
example,
the
anatomy.
The
modules
are
anatomically,
separated
the
various
scales
of
the
grids
they
don't
occur
next
to
each
other,
and
it's
not
clear
any
neural
population
reads
out
from
multiple
modules
and
second,
the
the
paper
that
coined
the
term.
Gridson
module
really
came
to
the
conclusion
that
they're
truly
independent,
and
that
was
a
little
bit
of
a
controversial
claim.
B
Not
everyone
believes
it,
but
they
do
have
some
evidence
supporting
the
idea
that
they're
truly
independent,
that
they
don't
really
work
closely
together
and
so
to
lay
out
some
various
models
and
it's
sort
of
a
two
by
two
that
I
think
is
kind
of
useful.
These
two
operations,
I've
been
talking
about
learning
maps
and
arranging
them.
We
worked
on
through
these
two
papers
and
I'd
put
on
james's
work.
James
and
tim's
work
tolman
ikenbaum
machine
up
here,
though,
theirs
is
really
cool
because
it
generalizes
over
different
types
of
spaces.
B
It's
not
just
2d
space
and
wrestling
put
on
my
radar
that
this
2x2
has
a
whole
other
row,
and
and
yet,
though,
what
I
really
want
to
do
is
explore
this
quadrant.
I
have
a
system
that
can
arrange
maps,
but
do
it
all
with
one
module,
but
I
think
it's
good
to
take
a
page
out
of
the
rat
style
playbook
and
that
is
to
use
a
grid
cell
module
along
with
other
parts
to
to
solve
this
problem.
B
B
Meanwhile,
a
polar
coordinate,
like
representation
can
use
varying
resolution
like
by
default.
You
would
kind
of
expect
it
to,
and
that
gives
it
the
opportunity
to
extend
over
longer
distances
and
and
in
my
mind
this
is
appropriate
for
a
lot
of
things
like
navigation
vectors,
and
so
yes,
what
what
follows
is
I'm
going
to
speed
through
the
next
part
a
little
bit
but
by
design?
I
think
it's,
I'm
gonna
walk
you
through
a
system
that
uses
a
combination
of
these
and
but
I'm
really
gonna
kind
of
skim
through
the
system.
B
B
Sensory
information
arrives
and
it
activates
an
object,
vector
cell,
and
so
these
are
sort
of
like
polar
coordinates
that
are,
that
are
a
little
bit
more
appropriate
for
that.
Well,
you
can
get
further
away
and
things
will
work
as
as
you
would
want
them
to.
You,
can
optimize
it
with
things
like
scaling
and
so
so
to
me,
these
are
appropriate
for
that
type
of
operation.
B
Using
those
two
populations,
you
can
now
detect
the
location
of
the
object
on
the
grid
just
to
quickly
say
it.
I
mean
I'm
taking
this
vector
and
this
bump,
and
just
saying
this
is
where
the
object
is
now
roughly
adding
this
blue
vector
to
here
to
get
there
second
operation.
Oh
no,
I
do
I
should
point
this
out
is
like
this
was
an
accidental
realization,
but
these
cells
sort
of
resemble
grid
cells
and
primates
that
track
vision
and
attention.
B
So
I
could
argue
that
that,
in
fact,
all
three
of
these
are
empirically
observed,
cells
and
so
to
con,
continue
the
sort
of
lightning
talk
on
how
this
works.
I
think
the
second
important
point
to
talk
about
is
how
a
grid
cell
module
can
be
used
to
path
integrate.
Other
cell
types,
given
movement
information,
the
grid
cell
bump
of
activity,
can
move.
However,
this
bump
of
activity
does
not
move
because
the
object's
not
moving
and
through
that
these
two
combined
can
path,
integrate
the
object,
vector
cells.
B
So
so,
even
though
these
are
course
coded,
they
can
still
get
path
integrated
by
the
system.
They
can
still
get
updated
when
you
move.
A
third
operation
is
inferring
the
grid
cell
location
from
sensory
input.
So
this
this
gets
back
to
how
if
an
animal
comes
back
into
a
room,
its
grid
cells
are
re-anchored
to
the
same
way
they
were
before.
B
This
is
very
much
like
the
first
operation
just
switched
around
between
these
two
now.
The
second
thing,
the
second
set
of
operations-
I'm
gonna,
two,
two
total
more.
There
are
five
I've
showed
you
three,
the
the
second
set
of
operations
involved
arranging
objects.
So
what
happens
now
when
the
rat
attends
from
the
pentagon
to
the
to
the
triangle?
It
goes
from
looking
at
this
to
looking
at
this.
B
What
happens
is
those
are
going
to
activate
two
different
object,
vector
cells
and
the
displacement
between
those
optic
vector
cells
should
be
represented
somewhere,
and
so
this
red
cell
literally
represents
the
fact
that
this
is
diagonal
from
that
or
that
this
is
diagonal
from
that
and
once
you've
stored.
All
of
this,
you
can
now
do
prediction
as
you
attend
to
other
objects.
B
So
here
now
the
rat
is
attending
from
the
triangle
back
to
the
pentagon
and
using
everything
it's
learned
using
the
course
displacement
between
objects
using
the
location
of
this
object
on
the
grid,
which
is
fine-grained
but
ambiguous,
you
can
unambiguously
predict
which
object,
vector
cell
becomes
active,
which
is
then
usable.
It
can
then
be
used
to
predict
sensory
input
and
one
more
quick
slide
once
again,
just
going
through
this.
I'm
making
the
point
here
that
these
four
populations
really
are
all
necessary
together.
It's.
C
B
B
So
the
final
one,
I
would
say,
is
similarly
more
of
a
skimming
I'd,
say
the
the
main
content
I
wanted
to
get
across
to
you
is
this:
the
idea
of
of
a
module,
complementing
a
system
and
polar
coordinates
and
cartesian
coordinates
working
together,
and
so,
but
when
you,
let's
see
the
system
is
learning
these
arrangements
of
maps
and
the
way
that
we
think
this
happens
is
by
storing
them
in
a
graph
or
a
tree
and
and
in
a
graph
I'll.
Just
give
you
a
conceptual
version
of
what
I
mean
there.
B
So
what
you
need
is
some
sort
of
a
group
of
properties
that
are
associated
with
each
other.
What
information
and
where
information
and
an
association
between
those
memories
and
I'll
show
a
neural
diagram
in
this.
For
this
in
the
next
slide,
a
tree-like
one
is
very
similar.
So
so
here
just
make
sure
I
said
everything
associated
with
this
pair
of
memories
is
the
displacement
information
between
them
and
associated
with
each
node?
B
Is
what
and
where
information
a
tree
is
like
that,
but
where
you
have
a
sort
of
hierarchical
view
or
where
you
have
this
parent-child
relationship,
which
has
some
nice
properties
and
so
neural
mechanisms
for
this
I've
been
talking
about
these
four
populations.
If
you
can
add
another
population
to
learn
associations
to
associate
them
together,
then
it's
pretty
straightforward
to
to
learn
these
graphs
using
this.
This
association
approach
and
I've
roughly
placed
these
in
where
they'd
be
medial,
anterior,
cortex,
lateral
and
tyrannocortex,
and
the
so-called
engram
cells
and
hippocampus
and
nearing
the
end
of
this.
B
B
And
second,
second
takeaway
would
be
stepping
back
just
what
the
kind
of
substrate
the
computational
substrate
should
look
like.
It
seems
that,
having
a
set
of
independent
modules
that
quickly
learn
graph
and
tree
like
maps
is,
it
would
be
a
good
system,
and
in
this
system
it
would
associate
information
with
nodes
and
displacement
information
with
edges
and,
finally,
some
takeaways
for
neuroscience.
I
made
a
point
about
cartesian
and
polar
coordinates.
B
I'm
making
a
case
that
hippocampal
learning
is,
it
would
be
much
more
efficient
if
it
learns
arrangements
of
parts
rather
than
a
location
by
location
memory
of
what
is
sensed
that
doesn't
mean
this
is
right.
It
just
means
it
would
be
more
efficient.
So
it's
what
I
would
expect
to
find
in
the
hippocampus,
but
that's
just
based
on
thinking
about
how
I'd
build
it.
A
B
B
He
sorry,
I'm
really
fumbling
with
the
zoom
ui
here
yeah
a
few.
A
few
more
words
about
of
introduction
about
james
I'd,
say.
One
thing
that
I've
learned
from
james's
work
is
how
to
set
up
a
neural
system
where
everything's
probabilistic,
for
example,
if
I'm
modeling
grid
cells
and
I'm
taking
how
they,
how
they
receive
sensory
input
and
movement
input
and
how
how
you
kind
of
combine
those
in
a
in
a
principled
probabilistic
way.
It's
very
likely
that
someday
I'm
going
to
borrow
james's
approach
to
that.
B
F
F
Roran
thinks
oh
right,
I'd
quite
like
that
cheese
again,
but
of
course
it
has
a
bit
of
a
question
of
how
it
goes
about
getting
that
cheese.
So
it
could
just
go
about
repeating
all
the
same
actions
it
took.
Of
course,
we
know
that's
a
little
bit
silly,
but
if
it
really
understood
something
about
how
the
space
works,
you
can
take
certain
shortcuts
or
you
just
need
to
always
wiggles
back
on
itself.
F
It
could
just
go
straight
to
the
cheese,
but
to
do
that,
I
really
need
to
understand
what
is
space,
the
relationships
of
space,
okay,
so
while
tolman
sort
of
put
rats
in
there
and
showed
that
they
really
did
take
these
shortcuts,
and
that
made
him
believe
that
these
rodents
actually
had
an
internal
model
internal
map
of
the
relationship
to
the
of
the
world.
He
didn't
really
know
what
the
neural
bases
were
behind.
F
There's
also
things
like
object,
vector
cells,
that
we
saw
that
always
fire
the
same
distance
and
direction
away
from
objects
or
border
cells
that
always
fire
borders
in
in
internal
cortex
and
again
in
hippocampus.
That's
the
the
green
bit
of
the
brain
there.
You
also
get
these
landmark
cells,
which
fire
distance
and
and
certain
orientations
away
from
objects,
but
not
for
every
object.
F
Okay,
so
while
a
little
bit
after
all,
the
sort
of
cognitive
maps
and
rodents
and
cheese
and
mazes
were
going
on,
the
hippocampus
was
also
implicated
very
deeply
in
this
notion
of
called
relational
memory.
So
if
I
told
you,
for
example,
that
jane
is
stronger
than
bob
and
bob
is
stronger
than
alice
and
alice
is
stronger
than
gina
and
so
on.
And
then,
if
I
came
across
an
art
came
along
and
asked
you
who's
stronger,
bob
or
gina.
F
It
turns
out,
if
you
legion,
if
you
these
parts
of
the
brain
like
hit
the
hippocampus,
these
rats
can
still
answer
these
single
pair.
Like
questions
like,
oh,
I
prefer
b
over
c,
but
they
cannot
correctly
answer.
I
prefer
b
over
d.
They
can't
chain
together
and
make
these
inferences.
These
transitive
inferences,
I
mean
again.
Similarly
similar
stories
found
in
what
problems
of
social
hierarchies
the
same
brain
regions
are
are
deeply
implicated.
There
too,
and
again,
we
just
again
have
sort
of
repeating,
but
I'll
run
through
it
quickly.
F
Those
place
cells
that
we
saw
in
space
only
fire
in
one
location
of
space
also
seem
to
fire
in
only
one
location
of
sound
frequency.
If
I
ask
a
a
rodent
to
hold
a
lever
until
a
certain
sound
comes
on
a
second
sound
frequency
appears,
that's
very,
very
non-spatial,
no
spatial
component
to
it
whatsoever.
These
hippocampal
cells
seem
to
be
like
place,
cells
from
sound
frequencies,
and
similarly,
you
also
find
these
multi-peak
cells,
which
would
be
a
little
bit
like
a
slice
of
selenius.
F
You
know
one
deep
slice
or
a
grid
cell
and
again
also
spoken
about.
If
you
get
participants
to
to
to
watch
a
bird
on
the
screen
where
its
neck
structures
or
its
leg
stretches,
but
unfortunately
those
stretches
are
like
taking
walking
through
some
2d
bird
space
participants.
Have
these
sort
of
hexadirectional
stick,
you
know
classic
signals
characteristic
of
grid
cells
found
in
internal
cortex.
F
F
This
all
seems
to
be
about
relational,
inferences
and
what's
going
on
there
and
so
we're
going
to
try
and
answer
these
questions
and
and
also
try
and
answer
why
on
earth
these
cells
look
look
the
way
they
do,
and
so
we're
gonna
start
doing
this
by
I'm
really
trying
to
think
about
what
is
the
actual
structure
of
the
problem
the
brain
is
trying
to
solve
in
these
particular
cases.
Okay,
so
we're
going
to
try
and
elucidate
that
now,
okay.
F
So
if
I
come
along
and
tell
you
that
bob
is
janice's
brother
and
alice's
janice's
child
I've,
given
you
two
bits
of
information,
but
you
actually
know
an
awful
lot
more
than
that.
You
also
know
that
bob
is
alice's
uncle
and
alice's,
bob's,
niece
and
etc,
etc,
etc.
You
know
that,
because
you
know
what
a
family
tree
is,
you
know
that
all
these
sort
of
relationships,
like
parent
plus
brother,
you
know
that's
the
same
as
an
uncle.
It's
sort
of
a
little
bit
like
path,
integration.
F
F
So
that's
not
all
that
we're
interested
in.
So
once
you,
you
know
we're
interested
in
more
than
just
one
family,
but
the
important
thing
is
once
you
know
what
a
family
tree
is
once
you
know
what
brother
and
sister
and
uncle
and
niece
and
parents
and
child
are,
you
can
take
that
knowledge
and-
and
you
can
apply
it
to
a
another
family.
So
when
I
come
across
jean
and
tim
and
elsie,
I
automatically
know
about
all
the
you
know
the
relationships
between
them.
F
I
I
have
a
sort
of
framework
for
understanding
this
new
family
and
I
can
generalize
this
knowledge
to
a
new
situation,
and
so
that's
true
for
family
trees.
But
it's
also
exactly
the
same
for
space.
Space
is
just
a
bunch
of
relationships
of
two-dimensional
sets
of
relationships
between
between
elements
of
the
world
okay.
F
F
That
sounds
like
a
bit
of
a
tricky
problem,
but
it's
made
an
awful
lot
easier
if
you've,
if
those
stimuli
are
drawn
from
you
know,
real
2d
graph
worlds
and
you've
seen
lots
of
these
stimuli
before
to
be
able
to
understand
what
2d
is.
So,
if
you're
able
to
understand
what
2d
is
you
can
abstract,
you
can
have
that
abstract
notion
of
you
know.
F
F
Okay,
of
course,
you
can
answer
what
the
light
bulb
is:
okay
and
again
exactly
the
same
thing's
true
for
family
trees,
okay,
so
this
sort
of
boils
this
problem
down
of
how
do
I
predict
things
in
the
world?
How
do
I
predict
sensory
observations
in
the
world
and
it
boils
it
down
to
two
really
key
components.
F
The
first
one
is:
can
I
sort
of
represent
location
in
some
abstract
space,
some
abstract
structure,
so
that
was
going
to
be?
Where
am
I
in
the
family
or
where
am
I
in
this
facial
graph
or
where
am
I
in
this
chain
of
transitive,
inferences
and
and
the
other
components
is
going
to
be
right
now
I've
got
this
abstract
thing.
How
can
I
link
this
abstract
space
to
the
real
world,
and
so
here,
for
example,
you've
got
the
lovely
family
of
tim
and
you
know
each
each
character.
F
There
takes
take
jupiter
location
in
this
family
graph,
tim's,
a
father
louise's
wife
alex
et
cetera,
et
cetera,
et
cetera.
So
it's
very
two
things.
Can
I
represent
this
abstract
structure
and
can
I
have
memories,
so
I
can
bind
together
specific
instances
to
places
in
this
in
this
family
tree
and
the
important
thing
about
having
a
memory
is.
F
F
So,
let's
start
with
the
memory,
because
that's
the
easier
bit
right,
so
people
think
about
memories
in
the
brain
as
these
use
of
hebbian
networks.
Okay,
so
what
they
do
is
they
take
a
sensory
representation,
here's
a
sensory
representation
of
a
motorbike
and
they
put
it
into
a
memory
network.
This
is
going
to
be
a
bit
like
a
hotfield
network
of
anyone's
I'm
familiar
with
that.
F
Okay,
that's
not
quite
what
we
want.
Unfortunately,
what
we're
really
interested
is
is
relational
memories,
that's
memories
of
observations
that
know
where
they
are
in
an
abstract
structure,
we're
not
just
interested
in
memories
or
observations
alone.
We
want
to
know
where
they
are
in
some
abstract
space.
F
We
want
to
know
that
this
motorbike
is
at
position
b.
Okay.
So
how
can
we
go
about
doing
that?
Well,
let's
just
take
from,
for
example,
that
we
that's
what
we
want
the
motorbike
at
b.
Let's
take,
for
example,
that
we
we
know
how
to
represent
this
motorbike
with
this
little
red
red
bunch
of
neurons
down
there.
So
we
do.
F
And
then
again,
we
can
just
do
this:
auto
associative,
heavy
and
learning
to
between
up
weight
weights
between
co-active,
neurons,
okay.
So
this
is
rather
good,
because
this
allows
us
to
play
a
bit
of
a
trick.
I
can
then
come
along
and
provide
a
partial
bit
of
input
of
this
memory.
But
now
the
trick
is
that
the
partial
bit
is
just
only
the
the
abstract
where
information
I
can
input
that
into
this
memory
network
and
then
retrieve
the
sensory
knowledge.
F
So,
just
like
I
put
a
partial
motorbike
in
before
and
retrieves
the
for
my
bike.
Now
I
can
put
the
partial
wear
information
and
retrieve
the
full.
What
and
where
information
or
exactly
the
same,
but
in
the
other
direction
I
can
put
in
just
a
motorbike
and
I
can
retrieve
the
wear
information-
that's
sort
of
rather
nice-
and
this
is
this-
is
what's
going
to
allow
us
to
to
make
all
these
sensory
predictions
that
I'll
show
you
in
a
moment.
F
Okay.
So
how
that's
you
know,
memory
sorted,
but
I
mean
hopefully
we'll
we'll
consolidate
that
knowledge
in
a
second.
But
the
question
is
like
we
made
a
bit
of
an
assumption
that
we
had
this
blue
representation
already.
So
what
we're
going
to
try
and
wonder
about
now
is
how
do
we
get
that
blue
representation
in
the
first
place?
Okay,
so
we're
going
to
do
this
by
training,
a
neural
network
and
let's,
let's
see
what
we
can
do
in
training.
F
Okay
again,
so,
let's
believe
that
we
have
some
sort
of
representation
of
where
this
red
x
is
and
that's
going
to
be
in
this
blue
blue
representation.
Here,
then,
I'm
going
to
come
along
and
say,
go
to
your
uncle
and
then
it's
going
to
transform
this
representation
just
by
some
linear
mapping
or
just
some
sort
of
mapping
to
a
new
representation,
just
a
set
of
weights
there,
and
these
are
going
to
really
be
the
important
weights.
F
The
weights
that
we're
interested
in,
because
those
weights
of
how
I
move
around
the
space
are
really
the
weights
that
define
what
the
space
is
to
find
the
family
tree
or
to
find
the
2d
space
whatever
it
is.
Okay,
then,
once
we've
transitioned
to
this
new
representation,
I
can
then
retrieve
the
memory
of
that
memory
network.
You
know
just
because
I
got
the
where
information
I
can
retrieve
the
full
memory
of
what
I
think
is
at
c,
and
in
this
case
I
predicted
it's
going
to
be
a
chair,
and
so
that's
wonderful.
F
When
I
open
my
eyes
and
see
it
see,
it
is
a
chair.
It's
brilliant!
I
predicted
that
correctly,
but
if
I
was
wrong,
if
I
wasn't
right,
then
that's
an
error,
that's
an
error.
I
can
use
to
do
error
back
propagation
and
sort
of
re-jig.
All
these
all
these
weights,
at
the
top
to
you
know,
predict
better
next
time.
Next
time
I
would
have
been
at
that
location
or
an
analogous
location.
F
F
Okay,
so
if
it
did
work,
I
would
show
it
this
whole
sequence
of
sensory
stimuli
and
it
should
be
able
to
put
it
what's
coming
next
and
in
particular,
let's
say,
for
example,
that
the
sequence
of
cement
sensory
stimuli
came
up.
Crumbs
came
from
this
sort
of
torturous
red
trajectory
on
this
hexagonal
world,
then
by
at
the
end
of
the
hexagon.
At
the
end
of
this,
this
trajectory
I've
only
ever.
I've
only
visited
all
the
nodes
of
this
graph.
F
I
actually
haven't
visited
an
awful
lot
of
the
edges,
but
once
I
visited
all
the
nodes,
it
should
really
know
everything
about
that
graph
because
it
already
hopefully
had
this
old
structural
knowledge
that
can
reuse.
The
old
has
the
old
knowledge
of
how
everything
relates
to
each
other
in
an
abstract
way,
but
now
I've
seen
all
the
particularities,
and
so
it
should.
It
should
get
perfect
prediction
after
it's
seen
all
of
these
nodes
and
that's
exactly
what
we
see
so
performance
is
in
line
with
the
number
of
nodes.
F
I've
seen
not
the
number
of
edges
that
I've
seen
okay
and
we
can
do
the
same
sort
of
stuff
in
these
worlds
of
transitive
influence.
I
should
say
all
the
worlds
are
actually
much
bigger
than
what
I'm
showing
here,
and
so
the
blue
line
is
right
at
the
very
beginning
of
training,
it's
got
no
idea.
What's
going
on,
but
after
I've
seen
10
20
30
of
these
sort
of
I
I'm
transitive
inference
graphs,
it's
sort
of
been
able
to
abstract
the
structure
of
what's
going
on
and
immediately
straight
away.
F
It
performs
really
well
in
new
worlds
and
that's
exactly
the
same
in
these
family.
Tree-Like
structures
as
well,
and
the
first
in
the
blue
bit
is
rubbish,
but
after
the
top
and
the
gold
immediately
only
need
to
see
something
once
and
then
it's
understood,
then
it
knows
what's
going
on
okay,
so
it
can
perform
these
relational
inferences
and
perform.
Those
transitive
inferences
like
like
what
like
what
I
showed
you
that
rodents
were
rodents
and
humans
can
do
okay,
so
it
works.
But
what
about
how?
F
This
network
is
actually
representing
the
world?
Okay,
so
I'm
going
to
first
just
take
this
model
tem
and
I'm
going
to
let
it
diffuse
about
randomly
in
in
space.
So
it's
just
going
to
wander
around
from
node
to
node,
trying
to
predict
what's
coming
up
next,
and
so,
if
we
have
a
look
and
peek
inside
these
neurons
in
this
abstract
that
blue
layer,
we
can
see
that
we
see
things
that
look
a
lot
like
grid
cells
and
there
are
grid
cells
of
different
frequencies.
F
That
grid
cells
have
different
phases
like
markers
of
talking
things
that
are
shifted
from
one
another,
and
you
also
get
things
that
look
like
band
cells
too.
So
that's
that's
rather
interesting
that
shouldn't
be
there
and
then,
if
I
also
look
inside
the
hippocampus,
this
sort
of
memory
bit
of
of
the
tomato
machine,
I
also
can
see
place
cells.
So
cells
are
just
fire
in
one
location.
F
F
Okay.
So
what
do
you
see?
You
see
these
things
that
look
like
object,
vector
cells?
That's
cells
that
fire
certain
angles
and
orientations
away
from
objects
and
secret
fights
this
one
fires,
always
five
pixels
to
the
north
and
objects
from
always
five
pixels
to
the
to
the
east
of
an
object
and
again
these
cells
generalize
across
worlds
and
that's
exactly
like
real
object,
vector
cells
and,
similarly
exactly
the
same
picture
as
seen
for
borders,
a
border
that
has
on
the
left
board
and
one
more
because
about
the
left
border,
the
other
world.
F
Okay.
So
what's
this
sort
of
saying
it's
sort
of
saying
this
whole
spectrum
of
cells
that
we
saw
and
then
grid
cells
or
direct
cells,
border
cells?
Well,
they
they
all
look
like
their
own
different
beasts.
They
all
actually
just
seem
to
be
representing
transitions
and
using
those
transitions
to
optimize
the
inferences
in
new
worlds
to
generalize
really,
okay
and
again,
if
we
peek
inside
hippocampus.
This
is
our
memory
bit.
We
see
these
landmark
cells,
these
cells,
that
fire
at
certain
distances,
distances
and
orientations
away
from
objects.
F
But
crucially,
they
do
not
do
it
for
every
object
in
the
environment,
only
a
subset
of
the
objects
from
the
environment
and
again
that's
exactly
like
landmark
cell
real
landmark
cells
in
in
rodent,
hippocampus
and
again
compare
that
with
the
internal
cells
which
which
fire
for
every
object.
We've
got
these
ones
on
the
right
that
really
generalize
across
everything
and
the
ones
on
the
left
that
are
specific
to
the
particular
world
that
we're
looking
at
okay.
F
F
Two
cells
that
fire
each
other,
your
grid
cells
fire
each
other
next
to
each
other
in
the
other
environments.
Okay,
but
what
do
we
see
inside
this
model?
We
see
exactly
that.
Here's
a
little
cell,
a
little
play
cell
fires
in
one
place
and
it
fires
a
new
place
or
a
big
play
cell,
which
falls
in
one
place
and
fires
a
new
place.
Okay,
let's
we've
got
that
in
time,
and
that
seems
to
also
that
well,
that
is
also
true
in
in
in
rodent,
hippocampus.
F
Okay
looks
pretty
random,
but
the
question
is:
is
it
really
so?
If
we
remember,
oh
sorry,
a
little
bit
faster,
I
was
expecting
so
if
we
remember
that
these
hippocampal
memory
cells
to
really
do
this
memory
properly,
we
have
to
make
it
conjunctive.
We
have
to
have
cells
that
combine
both
abstract.
Where
am
I
in
the
space
with?
What
am
I
seeing
right
now?
Is
each
cell
knows
a
bit
about?
F
Where
am
I
and
what
am
I
seeing,
and
that's
rather
interesting,
because
if
I
go
to
new
worlds,
the
sensory
things
might
all
be
totally
different.
You
know
my
house
is
probably
very
different
to
your
house,
but
the
structure
of
the
space
is
probably
pretty
similar.
You
know
corridor
and
your
daughter
named
rome
still
come
off
the
end
of
corridors,
and
you
know
there's
no
wormholes
or
teleports
or
anything
like
that.
Space
is
still
the
same.
F
F
Okay,
a
little
bit
more
precisely
over
what
that
might
look
like.
So,
although
a
place
cell
has
a
place
cell
will
remap
between
two
different
worlds,
when
it
remaps
it
will
be,
it
will
maintain
its
relationship
to
these
structure,
preserving
grid
cells.
So
so
here's
a
here's,
a
a
a
place
cell
that
fires
at
the
same
location
as
of
one
of
these
grid
cell
fires.
F
F
F
F
It
wasn't
just
that
it
was
running
in
a
loop
was
the
fact
that
after
it
completed
four
loops,
so
one
two
three
four
four
laps,
it
would
then
be
get
get
given
a
reward
and
so
do
lap
blah
blah
blah
reward,
blah
blah
blah
reward
that
that
reward
and
so
on-
and
that's
quite
interesting
because
now
there's
two
structures
at
play
here:
there's
both
the
structure
of
this
of
the
sensory
world
of
the
spatial
world.
I
I'm
on
a
loop,
then
there's
also
the
structure
of
the
task.
F
I
have
to
do
four
things
in
a
row,
four
laps
in
a
row
and
to
get
reward.
I've
got
a
simultaneous
thing
going
on.
Okay,
so
what
would
10
say
in
this
case?
Well,
10
would
say
right
if
you
want
to
be
predicting
your
immediate
sensory
experience.
You
better
have
cells
which
care
about
space,
because
you
know
when
I,
when
I
do
a
loop
of
space,
I
should
be
able
to
predict
something
that
I
that
I
saw
you
know
just
a
few
steps
before.
F
However,
I
also
needed
some
sort
of
representation
which
knows
about
for
lapners,
because
I
want
to
be
able
to
predict.
When
am
I
getting
that
reward
after
after
four
laps?
Okay,
so
I'll
get
these,
you
know
on
the
left,
I'll
get
cells
which
can
you
know
fire
in
the
same
place
every
lap,
but
then
I'll
also
get
cells
which
far
for
specific
laps,
that's
what
them
says.
F
Okay
and
chen
song
did
this
wonderful
experiment,
certainly
before
we
had
thought
thought
of
it.
We
really
thought
of
it
because
he
had
already
thought
of
it
and
he
saw
exactly
this.
He
saw
in
hippocampus
sorry,
he
saw
place
cells
which
fire
both
just
like
normal
place.
Cells
are
either
the
same
place
on
every
lap
or
you
see
at
the
top.
But
then
you
also
see
place
cells.
You
see
a
lap
one
cell,
you
see
a
lap.
F
F
Okay,
so
that's
rather
that's
rather
fun
as
well,
and
the
last
bits
that
I've
got
here
is
that
you
know
earlier.
We
saw
we
saw
this
remapping
in
space.
We
saw
that
10
was
saying
that
hippocampal
cells-
you
know
structures
preserve
some
when
in
into
animal
cells
when
when
they
when
they
they
move,
and
that
means
that
place
cells
should
also
inherit
a
bit
of
that
structural
preservation.
F
But
this
is
a
bit
of
a
more
of
a
funky
task,
because
now,
if
we
get
a
rodent
to
run
in
in
in
one
of
one
box
like
this,
do
four
laps
in
one
box
in
one
lap
and
then
move
it
into
a
new
box.
You
know
it
still
has
to
do
the
four
laps,
but
it
is
a
new
box.
What
tim
says?
It's
right:
the
cells
which
care
about
space
should
remap
because
space,
you
know
the
spatial
orient.
The
spatial
orientation
of
the
sensory
stuff
is
different.
F
Okay.
So,
in
summary,
I
I
try
to
convince
you
that
relational
reasoning
and
space
are
already
two
sides
of
the
of
the
same
same
coin
and
the
same
sort
of
fundamental
mechanisms
of
or
or
principles
of,
generalization
can
apply
to
both
I've
tried
to
say
that
relational
abstraction
structure
is
abstracted
and
represented
explicitly
in
in
the
brain,
where
entering
cells
are
just
a
basis
for
representing
transitions.
B
Well,
yeah
I'll,
say
a
few
more
awards
of
introduction,
as
as
kim
gets
set
up
so
yeah
james
stop
sharing
so
yeah.
So
I
one
thing
I
have
learned
from
kim
and
from
kim's
work
is
she
really
put
eigenvectors
on
my
radar?
Previously
they
were
a
thing
I
sort
of
learned
about
in
school
and
now
they're
a
thing
that
I'll
be
out
hiking
and
suddenly
I'm
thinking
about
eigenvectors.
So
thank
you
kim
for
that.
C
C
The
acquisition
of
new
information
look
like,
and
it's
a
bit
of
a
cliche
for
people
who
study
hippocampus
to
be
into
maps,
but
I
I
have
one
so
so
be
it.
This
is
an
old
drawing
of
a
map
I
really
like,
and
I
like
it,
because
it's
nicely
illustrative
of
many
of
the
challenges
of
representation.
Many
of
the
things
that
we've
talked
about
we'll
talk
about
more,
so
you
have
a
number
of
different
types
of
experiences,
bundled
up
into
this
map.
C
This
map
can
support
many
different
types
of
tasks,
whether
it
be
visual,
recognizability,
navigational
information,
information
indicating
the
different
functions
of
different
buildings
like
farms
and
churches.
You
also
have
this
hodgepodge
of
different
coordinate
systems
and
different
coordinate
systems,
and
the
importance
of
this
is
something
marcus
alluded
to.
You
have
egocentric
and
alice
centric
views,
in
particular
all
like
forced
into
this,
this
2d
drawing-
and
you
also
have
this
property,
where
certain
things
are
represented
with
a
high
degree
of
detail.
C
Probably
when
that
detail
is
relevant
and
other
details
are
neglected
or
entirely
absent,
for
instance,
I'd
be
pretty
confident
using
this
map
to
identify
the
church
in
the
town
square.
I
wouldn't
be
that
confident
using
it
to
navigate
that
country,
because
that's
almost
entirely
blank
only
roads
are
really
depicted.
C
So
I
think
it
strikes
this.
This
familiar
trade-off
between
richness
and
simplicity,
and
it's
a
it's
a
nice
illustration
of
some
of
these
principles.
It's
also
kind
of
fun,
because
it's
artsy
and
this
map-
I
think
it's
obvious
that
it
was
made
by
humans,
but
we
also
know
that
animals
seem
to
learn
and
represent
structure
too
and,
as
you've
heard
about
quite
a
bit
already,
we
think
that
place
cells
and
grid
cells
are
a
big
part
of
this.
We
think
of
these
in
in
map-like
terms
as
well.
C
So
one
kind
of
conventional
view
of
how
we
think
about
place,
cells
and
grid
cells
is
that
place.
Cells
are
representing
location
with
some
degree
of
specificity.
Some
some
degree
of
like
this
is
something
you
can
bind
to
that
will
keep
information
specific
to
this
one
episode
or
experience
that
you're
encountering,
whereas
we
think
maybe
grid
cells
are
representing
this
more
generalizable
metric
information
and,
of
course
this
is
something
that
james
talked
about
too
in
a
really
good
degree
of
detail
that
you
know
the
grid
cells
are
representing.
C
In
all
of
these
tasks,
we
see
hippocampal
activity,
that's
consistent
with
representing
your
location,
but
not
necessarily
in
a
task
that
has
the
same
kind
of
planar
structure,
and
I
should
mention
that
that
this
paper,
the
the
garvered
at
all
paper
from
2017,
is
also
from
tim's
lab.
So
if
you
want
to
ask
about
it
in
the
panel
he'll,
be
there
this
yeah.
C
If
you
have
a
representation
of
spatial
structures,
you
can
use
this
every
time.
You're
in
a
spatial
structure,
there's
a
lot
of
self-similarity
to
space.
Why
would
you
want
to
relearn
that
every
time
these
these
more
graph-like
tasks,
these
more
discrete
task
spaces,
are
in
a
way
a
lot
more
expressive
and
flexible?
You
can
learn
many
more
types
of
of
relational
spaces.
You
can
learn
more
ways.
Things
relate
to
each
other,
but
they're,
but
you
that's.
You
perhaps
lose
some
of
the
generalizability.
It's
not
that
you
have
this
self-similarity
anymore.
C
You
now
have
this
flexibility,
where
you
can
form
any
kinds
of
relations
to
each
other,
and
this
kind
of
view
is
more
consistent
with
the
idea
that
hippocampus
is
representing
relations
between
states.
This
eichenbaum
and
cohen
view
that
james
talked
about,
or
maybe
that
hippocampus
is
learning
a
transition
model
between
states
and
it
can
iterate
over
this
in
order
to
produce
reasoning.
C
So
one
of
the
things
that
we're
interested
in
doing
is
trying
to
reconcile
these
views
a
bit,
get
the
the
good
parts
of
geometry,
this
generalizability
and
reusability,
as
well
as
the
the
flexibility
and
expressiveness
that
that
that
graphs
give
us-
and
this
is
a
topic
that
james
also
talked
about
a
lot
a
lot
of
the
ideas
we've
talked
about-
have
a
lot
of
overlap
which
isn't
a
coincidence.
I've
just
been
like
really
influenced
by
them
and
talked
to
tim
and
james
a
lot.
C
The
view
that
we're
going
to
take
for
this
perspective
is
a
reinforcement,
learning
view,
basically
we're
going
to
anchor
our
our
desiderata
of
what
we
want
in
a
representation
with
the
reinforcement,
learning
framework,
we're
going
to
say,
what's
a
good
representation
to
support,
efficient
reinforcement,
learning
and
the
reinforcement
learning
problem.
Is
this
one?
Given
some
task?
You
want
to
learn
a
policy
to
maximize
your
expected
cumulative
reward
or
value
over
future
states
by
trial
and
error.
C
The
reason
we're
using
the
reinforcement,
learning
framework
here
is
it's
super
general
and
kind
of
minimally
specified.
You
can
apply
it
to
lots
of
different
types
of
problems.
You
can
apply
to
things
like
chess,
where
your
state
is
the
configuration
of
pieces
on
the
board.
You
want
to
maximize
your
probability
of
winning.
C
You
could
apply
it
to
something
really
open-ended
like
cooking,
where
your
state
is
whether
you're
chopping
vegetables
or
chopping
bread,
and
you
want
to
or
if
you're,
stirring
or
or
chopping,
and
you
want
to
maximize
your
probability
of
eating
a
delicious
meal
as
soon
as
possible,
and
then
you
can
apply
it.
Of
course.
C
This
familiar
example
of
navigation,
where,
if
you're
this
rat,
you
want
to
maximize
the
number
of
delicious
morsels
you
encounter,
while
navigating
this
very
complicated
maze,
the
problem
with
reinforcement,
learning
the
challenge
we
face
is
that
this
can
be
very
hard
and
actually
complicated.
Real
world
type
problems
like
cooking,
even
even
things
like
navigation,
and
one
of
the
reasons
for
this
is
that
reinforcement
learning
throws
away
a
lot
of
information
if
you're
only
learning
about
reward,
and
you
only
get
fruit
loops
every
once.
C
In
a
while,
then,
you've
thrown
away
a
lot
of
the
information
about
the
environment
that
wasn't
directly
linked
with
reward
information
has
to
prove
its
value
by
a
very
high
threshold,
if
you're
using
reinforcement,
learning
and
then
the
other
thing,
that's
a
challenge
in
all
types
of
learning.
Is
that
there's
just
a
lot
of
states
that
you
want
to
learn
about?
It
can
be
hard
to
explore
them
efficiently
and
you're
learning
in
a
very
high
dimensional
space.
So
you
have
this.
C
So
what
do
we
need?
In
rl
representation?
This
question's
been
asked
a
lot
in
machine
learning
and
we
can
break
this
down
into
the
two
challenges
I
described
for
reinforcement
learning,
the
first
being
that
if
you're
only
learning
about
reward
you're
throwing
away
a
lot
of
information,
so
what
additional
structure
should
we
be
learning?
C
C
So
this
is
just
like
a
safe
way
to
anchor
the
learning
process.
According
to
the
the
logic
of
predictive
representations,
this
has
been
the
focus
of
a
lot
of
work.
I
think
my
list
stops
at
2018,
but
it
shouldn't
there's
been
a
lot
of
really
nice
work
in
machine
learning,
on
using
predictive
representations
as
a
representation
to
support,
reinforcement,
learning
and
learning
in
general.
C
So
this
is
kind
of
the
logic
of
why
compression
is
an
important
part
of
forming
a
representation.
That's
useful
for
reinforcement
learning
instead
of
listing
individual
works
here.
I
I
really
just
listed
like
areas
of
machine
learning
that
take
a
different
perspective
on
compression.
It's
like
a
very
ubiquitous
and
widely
applied
concept.
So
there's
there's
just
a
lot
of
stuff
in
that
space.
C
The
low
dimensional
structure
among
these
predictions
they're
trying
to
summarize
this
predictive
information
as
succinctly
as
possible,
and
I
should
say
that
one
of
the
reasons
that
that
we,
that
we
kind
of
define
things
in
these
really
broad
ways
that
we
talk
about
things
in
terms
of
prediction
and
compression
rather
than
immediately
going
to
a
specific
algorithm,
is
that
this
means
that
we're
talking
about
our
representations
in
terms
of
objectives.
We
should
have
for
them,
and
this
is
really
easy
to
apply
to
deep
learning.
C
But
for
now
we're
going
to
go
through
a
much
simpler
manifestation
of
these
ideas
using
linear
algebra,
so
we're
going
to
start
with
a
predictive
representation
called
a
successor
representation.
The
rows
of
this
matrix
are
possible.
Starting
states,
you
could
have.
The
columns
are
possible
states,
you
could
visit,
and
every
entry
in
this
matrix
captures
the
expected
number
of
times
you're
going
to
visit
some
state
j
given
a
starting
state.
C
I
so
basically,
this
matrix
captures
captures
how
often
you're
going
to
visit
every
state
and
we're
going
to
assume
a
discretized
linear
environment
that
our
rat
is
wandering
around
it's
just
a
linear
track
and
it's
taking
a
random
walk.
So
there's
no
directional
biases
or
anything
interesting
like
that.
C
So
we
can
compress
this
matrix
using
eigen,
decomposition
or
more
generally
singular
value
decomposition,
and
that
involves
rewriting
this
matrix
in
terms
of
the
product
of
three
matrices.
The
first
is
a
matrix
of
eigenvectors
of
this
matrix,
which
is
why
marcus
now
thinks
of
them
on
his
long
hikes.
C
The
second
matrix
is
the
matrix
of
eigenvalues
that
are
it's
a
diagonal
matrix
that
weights
each
of
these
eigenvectors
by
some
amount
and
the
last
matrix
is
the
matrix
of
eigenvectors
transposed
it's
just
the
same
as
the
first
but
flipped
on
the
side,
and
we,
if
we
have
10
states
in
our
starting
environments,
we'll
have
10
eigenvectors.
However,
if
we
truncate
these
eigenvectors
removing
the
ones
that
have
the
smallest
eigenvalue,
then
we
can
reconstruct
this
matrix,
but
with
a
lower
resolution.
C
C
So
we
represent
this
here
as
just
truncating,
and
only
keeping
six
or
four
or
three
or
two
eigenvectors,
but
you
can
also
think
of
this
as
weighting
these
eigenvectors,
so
that
you're
downwarding,
the
high
frequency
components
these
these
eigenvectors
that
have
very
small
eigenvalues
and
upgrading
the
the
low
frequency,
coarser
ones.
And
this
view
of
like
waiting
and
filtering
them
will
come
up
later,
so
these
eigenvectors
were
a
model
of
how
grid
cells
could
be
generated
that
that
we
proposed
in
this
2017
paper.
C
We
use
this
to
simulate
some
of
the
ways
in
which
eigenvectors
change
in
irregularly,
geometric
environments,
so
in
environments,
where
you
don't
necessarily
have
a
square,
but
you
have
a
trapezoid
or
a
hexagon
or
circles.
You
see
different
shapes
of
grid
cells
and
we
could
capture
some
aspects
of
this
same
thing
with
how
grid
cells
fragments
in
a
hairpin
maze
or
how
they
change
in
multi-compartment
environments.
C
I
also
should
say
that
the
this
work
is
related
to
a
bunch
of
other
approaches
in
the
grid
cell
world
these.
What
what
many
of
these
works
have
in
common
is
that
they
all
anchor
grid
cells
to
the
transition
statistics
of
space.
They
show
that
this
these
grid-like
patterns
can
emerge
from
a
particular
statistical
view
of
space
and
don't
necessarily
require
some
like
innate
encoding
of
navigation.
You'll
see
james
models
is
in
there.
C
There's
a
bunch
of
ways
in
which
these
eigenvectors
are
like
differ
from
what
you
actually
see
in
grid
cells,
namely
like
they're,
a
little
more
czech
or
more
than
they
are
hexagonal,
and,
and
this
work
by
sorceratal
explains
like
the
different
constraints
you
can
use
to
determine
whether
or
not
that's
the
case.
Dortico
also
covered
that
in
some
early
work.
C
So
one
of
the
things
that
I
mentioned
was
nice
about
about
compression
is
that
it
can
capture
the
structure
of
your
environment
and
in
a
in
a
succinct
way.
So
this
manifests
differently
in
different
types
of
environments.
If
we
look
at
the
top
left,
you
can
see
that
in
environments
with
clusters,
the
first
the
first
few
eigenvectors
here
is
just
shown.
C
The
first
non-trivial
eigenvector
will
partition
your
your
graph
environment
into
different
clusters,
and
this
is
used
in
spectral
clustering
techniques
in
if
you've
learned
about
some
of
those
in
your
machine
learning
courses.
Another
thing
that
you
can
do
with
these
is
capture
low,
dimensional
representations.
C
Another
important
aspect
of
these
representations
is
that
you
can
organize
information
according
to
the
spatial
scale,
so
the
first
few
eigenvectors
will
change
very
slowly
across
your
environment.
If
we
look
at
the
top
row,
then
we
basically
reconstruct
the
the
fourier
basis
for
a
line.
We
have
sinusoids
that
change
at
different
frequencies.
If
this
were
a
circular
environment,
we'd
have
sines
and
cosines
for
this
linear
one.
We
have
only
cosines
for
2d
planar
environments.
We
get
essentially
the
analogous
thing
we
get
2d
fourier
components
that
vary
both
along
the
x
and
the
y
axis.
C
But
if
we
start
to
look
in
these
environments
that
don't
have
euclidean
geometry
anymore,
that
aren't
linear
or
planar,
then
we
see
that
you
also
get
this
range
of
scales.
But
it's
adjusted
for
the
particular
graph
structure
that
you
have
so
you
kind
of
get
like
different
peaks
and
troughs
in
this
triangular
environment,
but
the
peaks
and
troughs
are
forced
onto
the
structure
of
the
graph.
C
You
have
the
same
thing
in
hippocampus
too,
where
the
very
dorsal
cells
that
you
can
record
the
ones
that
are
recorded
most
commonly
because
they're
right
at
the
top
of
hippocampus
that
are
that's
easy
to
get
to
these
ones,
have
very
small
place
fields,
and
you
can
see
this
in
the
bottom
left-hand
corner
here,
as
you
record
more
and
more
eventually
towards
the
bottom
of
the
hippocampus,
the
the
lower
part
of
this
red
c
shape.
You
get
larger
and
larger
place
fields.
C
So
if
we
simulate
a
place
cell
that
has
a
very
small
field,
then
this
is
going
to
lead
to
a
very
precise
representation
of
space
and,
as
I
mentioned
early
on,
when
we're
reconstructing
a
predictive
map
from
many
different
grid
cells
or
many
eigenvectors,
we're
going
to
need
a
lot
of
them
to
get
the
map
with
precision
and
the
fewer
we
use
the
more
corsa
map
we
get.
So
in
order
to
get
this
small
place
field
that
has
a
lot
of
precision
we'll.
C
As
our
simulated
place,
cells
get
coarser
and
coarser.
We
need
fewer
and
fewer
grid
cells
to
stimulate
them.
We
can
down
weight
the
high
frequency
components
to
a
large
degree,
because
they're
not
contributing
very
much
when
we
have
a
course
representation
of
space,
and
this
is
because
this
eigenvector
basis
organizes
information
according
to
changing
spatial
scales,
and
you
have
the
very
large
spatial
scales
early
on
which
you
can
recombine
to
generate
very
coarse
representations.
C
If
you
want
a
more
precise
representation,
you
used,
you
need
to
use
a
larger
number
of
them,
so
a
nice
property
of
this
of
this
model
is
that
you
can
use
the
same
eigenvector
grid
cell
population
to
support
a
wide
range
of
place
field
sizes,
and
this
is
consistent
with
the
fact
that
you
get
the
same
grid
cell
populations
projecting
to
different
regions
of
hippocampus.
That
seems
to
be
encoding
information
with
different
scales.
So
that's
really
nice.
This
is
also
anchored
to
relational
logic.
C
These
are
learned
from
the
transition
structure,
even
though
right
now,
they're
depicted
over
space.
So
it
starts
giving
us
hypotheses
about
multi-scale
representations
in
these
relational
environments,
and
one
word
that
I'll
I'll
use
for
this
again.
So
I
should
introduce
now,
for
this
kind
of
re-weighting
and
recombining
of
eigenvectors
to
form
maps
with
different
scales
is
spectral
modulation
and
this
kind
of
structural
modulation
that
I've
showed
is
useful
for
generating
maps
with
different
sizes.
This
is
great
when
you
don't
know
a
priori.
C
C
We
see
a
lot
of
activity
in
hippocampus
that
appears
to
be
sampling
from
a
representation
of
your
of
your
environment
from
a
cognitive
map
I'll
go
into
some
of
this
in
a
little
more
detail
later.
But
this
is
the
the
motivation
for
thinking
about
hippocampal
activity
in
terms
of
forming
a
distribution
over
space,
so
these
different
lines
show
different
weights
that
you
can
have
on
eigenvalues
to
produce
different
distributions,
and
so
we
see
eigenvalues
along
the
x-axis
and
a
weight
of
that
eigenvector
is
depicted
along
the
y-axis.
C
So
these
blue
lines
are
not
just
different
random
walks
that
have
different
widths
associated
with
them.
They
are
walks
that
have
a
different
type
of
statistic:
they're,
more
heavy-tailed
and
they're
called
levy
flights.
They're
also
called
super
diffusion
because
they're
like
diffusion,
but
but
with
bigger
steps.
It's
super
the.
If
you
plot
this
in
linear
coordinates,
you
can
see
that
a
levee
walk
looks
locally
a
lot
like
a
normal
distribution.
It
sort
of
has
the
same
bell
curve
shape.
However,
it
has
a
much
heavier
tail.
C
The
the
the
blue
part
of
the
line
is
above
the
red
part
of
the
line,
as
you
start
to
look
at
more
extreme
values
and
if
you
sample
from
this
style
of
taking
step
sizes,
then
what
you
get
is
something
called
a
levee
flight
where
you
are
following
something
that
looks
like
a
random
walk
for
a
little
while,
but
then
every
once
in
a
while.
You
take
a
huge
step
to
get
to
some
other
part
of
the
state
space.
C
And
so
this
is,
this
is
work
I
I
forgot
to
mention
this
is
work
with
dan
mcnamee
matt
boffinik
and
sam
grishman.
Dan
mcnamee
was
the
first
author
on
this
and
one
thing:
sam
grishman
realized,
as
we
were
like
making
these
math
models
is
that
this
is
actually
connected
to
a
huge
literature
of
levy
flights
on
in
the
animal
foraging
literature,
and
these
these
styles
of
exploring
environments
are
seen
in.
Like
all
of
these
different
kinds
of
exploration
patterns,
you
can
see
them
in
albatrosses
searching
for
prey.
C
You
can
see
them
in
foraging
spider,
monkeys,
you
see
them
in
humans
looking
at
a
visual
scene,
if
this
is
for
humans
that
are
looking
at
fractal
patterns,
that
they've
never
seen
before.
The
reason
they're
hypothesized
to
be
so
ubiquitous
is
that
they
have
good
coverage
time,
whereas
random
walks
tend
to
dither
they
tend
to
spend
too
much
time
in
the
same
area
and
not
move
on
to
other
areas
fast
enough.
The
large
jumps
that
you
occasionally
take
with
a
levee
walk
cause
you
to
avoid
repeating
the
same
locations
again
and
again.
C
One
nice
thing
about
this
formulation
is
because
we've
anchored
it
to
these
eigenvectors,
which
are
learned
from
the
transition
structure,
is
we
have
a
notion
of
levy
flight
that
can
be
applied
to
environments
that
are
not
necessarily
euclidean
that
are
that
are
maybe
relationally
specified,
and
this
is
because
we
have
this
notion
of
of
big
jump.
That's
anchored
to
something
that's
specific
to
the
actual
geometry
of
your
environment
and
not
just
euclidean
geometry
in
general,
so
in
in
this
example,
we
have
a
random
walk
in
a
four-room
environment.
C
You
have
a
notion
of
big
jump
that
will
take
these
doorways
into
account.
So,
in
this
case,
you
now
get
an
agent
that
takes
a
big
jump
that
propels
it
through
a
doorway,
frees
itself
from
the
room
and
it
can
explore
this
environment
a
lot
faster.
So
I
think
that's
really.
The
main
upside
of
of
defining
these
in
relationally
anchored
terms
is
you're
using
geometry,
that's
specific
to
the
actual
environment
that
you're
in
rather
than
a
more
general
general
geometry
that
could
be
refined
further.
C
So
one
thing
that
dan
also
realized
is
that
we
can
use
this
model
to
explain
different
types
of
sampling,
statistics
that
are
observed
in
different
cognitive
states.
So
one
example
of
this:
this
activity
that
is
thought
to
be
sampling
from
hippocampus.
Is
this
spontaneous
reactivation
you
see
in
hippocampal
data?
So
what
spontaneous
reactivation
is
is
this?
C
However,
sometimes
when
the
animal
is
at
rest
or
just
kind
of
hanging
out
or
grooming
itself
or
eating
reward,
then
you
see
this
spontaneous
reactivation,
where
distal
states
that
the
animal's
not
currently
at
are
active
in
hippocampus,
it's
very
like
evocative.
It
looks
like
the
animal's
imagining
things
that
it
did
before,
but
isn't
doing
now.
C
Interestingly,
you
also
see
this
kind
of
activity
during
sleep,
where
you
see
the
spontaneous
reactivation
of
states
that
the
animal
has
visited,
perhaps
in
the
the
last
day,
but
the
animal
is
of
course
unconscious
and
not
currently
experiencing
them.
If
this
is
hypothesized
to
relate
to
consolidation,
to
the
kind
of
strengthening
of
memories
that
that
is
hypothesized
to
happen,
while
people
are
asleep,
but
of
course
I
think
it's
not
really
known
for
sure.
C
So
there
was
this
data
from
stella
at
all,
which
showed
that
the
statistics
of
spontaneous
reactivation
during
sleep
were
different
from
the
statistics
of
experience
when
the
animal
was
awake,
so
the
the
experience
was
more
consistent
with
super
diffusion
or
these
levee
flights.
This
isn't
surprising,
because
this
is
the
the
animal
foraging
pattern
that
we're
familiar
with,
but
the
pattern
that
you
saw
during
the
spontaneous
reactivation
while
the
animal
was
asleep,
looked
more
like
diffusion,
so
this
is
cool
for
a
couple
of
reasons.
C
One
of
them
is
that
it
seems
like
this.
Reactivation
is
not
just
recapitulating
what
happened
during
the
day.
It's
not
just
sampling
from
experience,
because
it's
got
different
statistics
associated
with
it,
and
it's
also
interesting
because
it
suggests
that
the
same
architecture
can
generate
samples
that
have
different
statistics
that
are
varied
in
this
particular
systematic
way
where
you
can
turn
super
diffusion
into
diffusion.
This
is
something
we
can
do
with
spectral.
Modulation,
so
that's
that's
kind
of
exciting.
That's
like
one
way
you
could
generate
this.
C
C
So
basically
you
can
re-weight
the
eigenvectors
and
recombine
them
to
form
a
sampling
pattern
where
the
consecutive
states
you
sample
are
unlikely
to
overlap
with
each
other,
and
this
is
consistent
with
the
kind
of
activity
you
see
during
hippocampus,
as
the
animal
is
approaching
a
decision
point.
You
see
this.
This
sub
second
cycling
between
representations
of
possible
futures.
C
It's
like
left
right
left
right
left
right,
and
this
is
consistent
with
sampling
states
that
don't
overlap
with
each
other
very
much,
because
your
different
futures
are
unlikely
to
have
overlap
with
each
other.
So
this
is
also
something
that
could
potentially
be
generated
with
spectral,
modulation
and
and
that's
kind
of
neat.
C
So,
as
I
mentioned,
if
you're
exploring
an
environment-
and
you
want
to
cover
it
as
fast
as
possible,
leveewalk
is
your
guy.
That's
going
to
be
the
fastest
way
to
cover
the
environment,
and
that's
shown
by
this
blue
line.
Up
top
random
walks
are
much
less
efficient,
and
so
are
these
minimally
autocorrelated
sampling
modes,
the
environment.
We're
simulating
is
on
by
the
way,
is
this
multi-cluster
environment
below.
C
However,
you're
not
actually
subject
to
the
constraint
where
you
have
to
walk
among
locations,
this
minimum
autocorrelation
sampling
mode
is
really
good
for
that,
because
that's
basically
saying
let's
sample
as
many
non-overlapping
states
as
possible
as
quickly
as
possible.
If
you're
doing
this
in
mental
simulation,
you
don't
actually
have
to
traverse
between
these
locations.
So
this
minimum
autocorrelation
is
ideal.
If
you
actually
have
to
walk
between
these
locations,
you'll
spend
a
lot
of
time,
walking
back
and
forth,
and
a
levy
flight
is
better
for
this.
C
So
this
work
was
recently
published
in
nature
neuroscience.
If
you're
interested
in
more,
you
should
check
it
out.
There's
more
simulations,
there's
much
denser
math,
all
things.
People,
like,
I
think,
the
one
of
the
the
key
insights
of
this
work,
as
I
mentioned,
is
that
you
can
use
these
these
eigenvector
grid
cells
to
do
something
that
is
easy
to
define
for
euclidean
geometries
to
to
sample
in
these
different
ways
across
space,
and
you
can
translate
it
to
relational
environments
this
this,
like
these.
C
So
in
this
case
it
permits
us
to
have
a
multi-scale
representation
of
space
and
define
step
size,
which
is
a
key
property
of
these
distributions
for
these
different
types
of
environments.
So
I,
on
that
note.
I
made
this
slide
during
marcus's
talk,
because
I
thought
this
dichotomy
is
really
lovely
and
I
wanted
to
go
further
with
it.
C
This
idea
that
there's
one
a
view
that
you
could
use
grid
cells
for
for
structured
memory
as
like
a
scaffolding
for
acquiring
new
associations
where
you're
learning
associations
between
like
a
map
and
and
things
that
are
bound
to
that
map.
The
second
view
is
that
grid
cells,
encode
information
directly.
You
have
like
a
a.
E
C
Of
basis,
you
have
grid
cells
that
are
capturing
statistics
of
your
environment.
Certainly,
these
eigenvectors
are
used
for
like
learning
clusters
and
representing
the
underlying
low
dimensional
structure,
so
you
can
think
of
them
in
that
way,
but
I
think
one
thing
I'd
want
to
point
out
as
a
connection
between
them
is
that
these
these
eigenvectors,
these
representations
that
that
are
derived
from
trying
to
succinctly
capture
the
structure
of
a
particular
environment,
or
also
a
really
nice
basis
for
binding
things
as
well.
C
You
can
take
a
lot
of
the
the
geometric
euclidean
ideas
and
you
can
apply
them
to
relational
spaces
if
you
have
a
representation,
that's
anchored
to
the
to
the
specific
relational
statistics
of
space,
one
of
the
reasons
so
powerful
in
our
models
of
spatial
reasoning.
Is
they
capture
these
geometric
principles,
which
aren't
entirely
absent
in
these
other
types
of
environments?
You
just
need
to
use
something
that
captures
that
particular
geometry,
so
I
think
this
this
is
more
most
succinctly.
C
Summarized
is
what
you
use
to
encode
information
is
possibly
what
you
should
use
to
scaffold
the
rapid
acquisition
of
memory.
This
is
whatever
is
representing.
The
underlying
structure
of
your
environment
is
capturing
all
the
statistics
you
already
know
and
want
to
take
for
granted
and
don't
want
to
relearn.
So
rapid
memory
is
all
about
not
relearning
things
you
already
know,
and
so
this
is
kind
of
this
is
the
main
idea.
C
So
this
is
an
illustration
of
a
dragon
that
was
in
one
of
davinci's
sketchbooks,
and
he
accompanied
it
by
this,
like
really
delightful
recipe
for
generating
dragons,
you
take
for
its
head,
that
of
a
mastiff
or
a
hound
eyes
of
a
cat
ears
of
a
porcupine,
etc.
All
of
these
different
animal
parts-
you
you
separate
them
from
their
natural
context
and
you
recombine
them
according
to
the
the
the
statistics
of
how
these
parts
should
relate
to
each
other.
You
know
noses
and
faces,
relate
to
each
other
in
a
systematic
way.
C
You
could
take
a
particular
nose
and
a
particularly
different
face
and
recombine
them
using
this
kind
of
relational
logic.
So
I
think
relations
are
are
not
the
only
language
of
compositionality
you
can
do
geometric,
compositionality
and
and
generalization
you
could
use
programmatic,
but
relational
are
really
powerful
and
we
have
this
geometric
way
of
describing
them
that,
I
think,
is
really
nice.
C
So
on
that
note,
I
want
to
thank
my
collaborators
and
various
mentors
dan
mcnamee,
who
did
the
spectral
modulation
sampling
work,
which
you
should
check
out
in
nature:
neuroscience
jesse
gertz
who's,
a
grad
student
at
ucl,
who's
work.
I
didn't
talk
about
today,
but
is
working
on
really
cool
stuff
and
a
number
of
professors
and
colleagues
in
deep
mind
that
I've
worked
with.
C
C
A
A
A
B
B
Okay,
cool,
I
think
I
think
we're
all
here,
so
I
mean
yeah.
I
guess
I'll
bring
up
just
a
couple
empirical
things
about
good
cells
that
I
just
want
to
see
if
people
have
thoughts
on
things
that
puzzle
me-
and
I
just
wanted
to
see
what
people
think
so,
these
I'm
actually
going
to
use
some
deleted.
Some
slides
that
I
deleted
from
my
presentation
as
I
was
cutting
it
down
to
size.
B
So,
first
of
all,
one
of
the
things
that
puzzles
me
about
good
cells
is
is
distortions,
and
here
I'm
showing
a
picture
of
of
a
2x2
meter
box
and
like
if
you,
if
you
really
start
to
look
closely
at
the
fields,
the
the
firing
fields
of
the
cells
over
here.
The
scale
seems
to
be
like
larger
than
it
is
over.
D
C
C
I
mean
I've
heard
other
models
too.
That
I
think
are
basically
the
one
that
tim
mentioned
first,
that
just
like,
depending
on
what
you're
anchoring
your
representation
to
you,
have
different
certainty
and
and
resolution
at
different
places.
If
you're
like
trying
to
infer
your
allocentric
location
relative
to
something
that's
over
there,
it's
going
to
get
noisier
and
maybe
larger
as
you
get
further
away
from
it,
and
that
might
be
fine,
because
you've
got
a
whole
bunch
of
good
cells
that
all
have
different
different
types
of
uncertainty
that
they're
they're
juggling.
D
C
D
Yeah
exactly
lisa
jacomo
one
shows
that
the
rewards,
the
grids
do
not
move,
and
it's
really
telling
I
mean
this.
This
is
a
clear
example
of
what
kim
and
I
are
saying
so
in
the
in
the
behavior
in
the
in
the
tits
very
paper:
they're
just
running
a
set
of
predefined
transitions
all
day,
and
that
will
warp
the
grids,
but
in
the
tacoma
paper
they
have
context.
C
D
B
D
D
C
I
think
that's
that's,
maybe
a
bit
of
innovation
yeah,
I
I
don't
I
I
don't
know
I
mean
I
thought
it
was
cool.
I
I
I
it's
in
keeping
with
a
lot
of
the
things
that
I
think
are
like
interesting
to
model
hippocampus.
It
moves
it
in
this
meta-learning
direction
which
like
for
a
while
I've
wanted
to
be
done
and
think
this
is
a
really
nice
way
to
do
it.
I
think
yeah.
That's
that's!
Probably
my
if
I
had
to
to
pick
something
really
really
aggressive.
D
I
so
again,
I
think
they're
great
and
they've
moved
everyone
forward.
I
also
I
I
so
I
think
that
the
fact
that
they
have
to
you
have
to
the
fact
that
you
have
to
experience
the
whole
world
before
learning.
All
the
eigenvectors
means
that
the
means
that
the
transition
means
that
you
lose
some
of
the
benefits
of
all
the
all
the
so
the
there
needs
to
be
some
way
of
inferring,
what's
going
on
vectors,
rather
rather
than
learning
them.
D
D
Some
of
them
try
well
try
to
model
like
states
and
and
values
or
predictions
which,
which
enable
values
and
others
try
to
just
model
sequences
and
capture
the
statistics
of
sequences,
and
so
I
I
wonder
whether
whether
you
could
say
that
hippocampus
doesn't
care
about
reinforcement,
learning
at
all,
leave
that
to
straight
term,
but
hippocampus
sort
of
tries
to
copy
the
sequences
that
stratum
is
building
or
something
like
that.
It's
like
generalizing
or
making
yeah.
C
Yeah
I
do.
I
do
think
this
is
a
really.
This
is
a
really
interesting
question
to
discuss.
I
guess
I
think
I
don't
really
know
I
mean
the
the
way
that
I
have
like
liked
using
reinforcement
learning
is,
I
think
it's
a
nice
way
to
anchor
like
a
formalization
of
kind
of
autonomous
learning
like
it
has
some
of
the
it
has
some
of
the
child.
C
It
sets
up
the
challenges
of
learning
nicely
like
if
you're,
if
you
have
a
specific
notion
of
relevance
which
is
equal
to
your
reward,
then
is
that
enough
to
generate
a
learning
process
and
like,
if
not
what's
missing
from
it,
and
I
think
that's
that's
like
really
nice,
but
I
don't
know
to
what
extent
like
I
mean,
none
of
the
representation
learning
logic
really
requires
reinforcement,
learning
to
be
downstream
of
it.
It's
just
like,
if
you
have
some
uncertainty
about
what
you're
going
to
be
learning
later
on.
C
That
has
an
effect
on
hippocampus,
like
in
terms
of
stabilizing
memories
or
or
changing
like
which
sequences
are
are
generated
during
spontaneous
reactivation,
but,
like
this
whole,
like
cognitive
map
logic,
the
whole
idea
of
it
is
that
you
are
representing
something
that
is
like
capturing
information.
That
is
not
just
related
to
reward
like.
If
you
have
this
episodic
logic,
where
it's
trying
to
represent
episodic
memories,
the
relevance
of
which
you
don't
yet
know
it's
trying
to
capture
details
from
which
you're
you
haven't
yet
generalized.
C
The
whole
idea
of
it
is,
you
should
be
learning
things
that
aren't
necessarily
have
that
haven't
necessarily
proved
their
relevance
yet
and
representing
a
map.
Similarly,
that
you
should
be
able
to
like
generalize
to
new
things
you
haven't
learned
yet.
So
it's
almost
like
you
could
think
of
it
as
like
anti-all,
in
a
way
because
it's
representing
the
stuff
that,
like
we
wouldn't
have
thought,
reinforcement
learning
would
be
really
like
cauterized.
B
So
I
guess
one
one
yeah
one
one
question
I
have
with
the
eigenvector
one
and
as
I'm
trying
to
learn
to
think
about
it.
Like
one
statement
I
made
and
and
in
the
slide
you
screenshotted
was
that
the
thing
we
all
agree
on,
maybe
is
that
the
grid
cells
are
used
for
and-
and
you
know,
similar
cells
are
used
for
representations
that
you
can
update
and-
and
I
found-
and
I
found
that
I
find
that
with
your
work.
B
Sometimes
that's
the
case,
and
sometimes
it
seems
less
the
case
when
you're
with
your
sequence
generation.
It
absolutely
seems
the
case
with
the
with
other
stuff.
It
seems
more,
like
grid
cells
are
performing
a
change
of
basis
on
something
else
and
then
they're
just
they're
doing
that,
so
that
they
provide
some
useful
information.
You've
you've
changed
the
basis,
but
they're
not
there
to
be
updated,
and
I'm
curious.
If
my
logic
there
is
right,
like
am
I
right
that
some
that
you're,
not
as
focused
on
updates
in
some
of
your
work.
C
C
That,
I
think,
is
more
that
they're
that
they're
there
to
be
updated,
like
I
think
the
if
I
had
to
like
make
a
like
a
guess,
that's
consistent
with
this
work
on
why
they're
there
it
would
be
like
to
represent
structure
in
a
way
that
supports
some
efficient
process
downstream,
as
as
best
as
I
could
model
at
the
time-
and
I
think
the
the
ability
to
be
updated
is
sometimes
part
of
that.
But
it's
like
it's
subservient
to
that
more
more,
like
main
orbit
of
concern.
Does
that?
B
B
Like
recently,
the
moser
lab
had
a
had
data
where
they
used
neuropixels
recorded
many
many
neurons
and
and
recorded
a
single
grid
cell
module,
and
they
reached
the
conclusion
that
it
forms
a
a
taurus
like
the
the
rhombus
I
was
showing
and
they
they
even
reached
the
conclusion
that
it's
like
always
using
that
rhombus
like
whether
the
animal's
sleeping
or
if
it's
or
or
if
the
grid
is
distorted,
it
still
sort
of
forms
a
rhombus.
Honestly,
I'm
not
sure.
B
I
totally
believe
that
you
could
come
up
with
other
explanations
where
in
sleep
it
was
actually
the
average
of
a
taurus
with
something
else.
I
don't
know
if
it's
actually
true,
but
I'm
curious,
like
does
that
fit
with
really
you
know
kind
of
both
of
both
of
your
models,
I
think
involve
a
lot
of
switching
of
what
the
entering
cortex
is
doing.
If
you
move
away
from
2d
space,
if
you
move
right,
move
to
family
trees,
I'm
not
sure
you'd
expect
these
tauruses
anymore
and
similar
with
similar
with
kim's
work.
D
I
can
maybe
mayo
james
can
take
that
first,
then
kim
so
so
well
I'll
go
first,
and
so
so
I
think
so.
The
first
question
your
first
answer
questions
we
don't
know
yet.
So
I
agree
with
you
that
the
moses
data
is
exciting,
but
not
at
all
compelling
in
the
use
of
the
word
all
like
we
don't
know
what
happens
in
a
task
that
isn't
spatial
in
those
in
those
things.
D
So
hopefully
so,
and
so
here.
So
that's
the
first
question
I
agree
with
you,
we
don't
know
yet.
The
second
question
is
what
would
tim
do
first
and
then
I
and
then
I
think
kim
can
answer
after
by
thinks
he
might
give
a
similar
answer.
So
vanilla
10
right
now
in
a
task,
that's
embedded
in
space
where
they
still
have
to
do
a
spatial
re
spatial
part
of
the
task,
but
I
have
to
do
something
non-spatial
in
space.
Remember
it's
a
mouse!
Everything
it
does
is
all
it
does.
D
D
D
That's
what
vanilla
10
will
do
right
now.
If
you
there's
a
we've,
now
got
hierarchical
versions
of
10,
which
impose
these
kinds
of
interesting,
factorizations
and
and
we'll
learn
much
much
faster
because,
obviously,
for
it,
learning
to
factorize
would
require
a
lot
of
data,
but
you
can
obviously
impose
these
kinds
of
things
say
most
of
your.
D
If,
if
you
can
split
your
problem
into
various
factors,
then
you
can
do
much
more
rapid,
generalizations,
and
so
james
is
working
on
that
at
the
moment,
with
with
and
and
got
some
interesting
kind
of
results
with
with
some
a
master
student
of
his
or
else.
Maybe
james
should
have
answered
that
question,
but
but
but
yeah
so
it's
gonna,
be
I
don't
know
so.
D
I'm
gonna
be
interested
to
know
what
happens
to
these
non-spatial
task
elements
in
on
the
tourists,
but
I
don't
think
that
you
can
say
that
these
ideas
wouldn't
predict
grid
cells
in
those
things,
because
a
mouse's
task
is
almost
always
embedded
in
space.
I
guess
kim
is
going
to
say
something
similar
to
that.
C
C
I
think
one
thing
that
this
also
reminds
me
of
that's
maybe
worth
mentioning
is
there's
this
work
on
like
default
representations
that
and
nathaniel
dave
worked
on
that
that
talk
about
like
on
how
you
could
relate
a
default
policy
and
an
efficient
representation
of
that
default
policy
with
representations
that
are
more
task.
Specific
might
be
interesting
if,
if
we
think
about
these
more
toroidal
representations
as
as
belonging
to
a
default,
because
so
many
tasks
are
naturally
embedded
in
space.
F
I
was
also
I
mean
I
guess
the
question
you're
wondering
is,
if
you're
doing
a
task
which
has
no
spatial
elements
in
it,
do
do
your
do
you
utilize,
reuse
and
trying
representations
in
some
way.
I
need
a
good
sales
change
to
represent
that
task,
or
do
you
does
another?
You
know
the
brains,
much
bigger
than
entering
sex
and
just
says
some
other
parts
of
the
brain
represent
that
task,
while
entering
a
horchata
sits
around
that.
Do
it
does
nothing?
I
don't
know
at
all.
F
D
D
Out
of-
and
so
I
was
really
interested
in
in
this
idea-
that
you
could
just
use,
grids
and
effectively
sort
of
store,
all
the
pairs
of
locations,
all
the
pa
and
and
so
in
in,
and
it
seemed
like
you
were
what
you
were
storing
them
with
is
the
is
some
memory
in
hippocampus
of
some
of
the
object
vector
cell
code
is.
Is
that
what
you're
I
couldn't.
B
D
Displacement
between
obvious
vector
cells,
measured
at
any
one
time
where
the
displacements
are
measured
by
the
grids
which
you
obviously
can
just
read
off
the
displacements
of
the
two
different
grid
codes,
so
it
I
had
to
sort
of
a
number
of
so
it's
an
interesting
thing.
This
it's
an
interesting
thing,
because
that
same
system
does
solve
compositional
problems
as
well
as
and
scene
construction
problems
as
well
as
navigation
problems,
and
so
it's
clearly
amazing,
if
you
can
do
that
with
exactly
the
same
representations.
D
I
think
the
first
thing
that
I
would
question
is
that
means
that
you
have
to
effectively
sequence
out
all
pairs,
because
you
can't
mean
you
comprehend
two
things
at
once
in
the
grids
right,
so
you're
gonna
have
to
sequence
out
all
pairs
or
you're
gonna
have
to
know
which
sets
of
pairs
to
sequence
out
to
to
give
you
a
robust
enough
set
of
pairwise
things.
To
make
a
map.
Is
that
right.
B
Yeah,
so
I
guess
there
are
two
parts
where
pairs
come
in
one
is
that,
like
just
make
sure
making
sure
I'm
covering
everything,
object,
vector
cells,
you
want
to
you
want
something
that
can
detect
the
displacement
between
them.
So
that's
kind
of
a
pairing
approach,
but
I
think
what
you're
asking
is,
like
the
learned
thing,
the
learned
data
structure
in
the
end.
D
B
D
D
It
has
to
be
some
much
much
richer,
relational
structure
than
whenever.
I
think
about
this.
I
think
that
it's
that
it's
the
kind
of
relational
structure
you
want.
There
is
the
kind
that
you
want
in
language
like
like
a
really
rich
relational
structure.
To
allow
you
to
do
compositional
reasoning,
I
I
would
have
guessed
in
the
same
way
that
you
know
the
statistical
relationship
between
words.
You
also
do
between
concepts.
B
Yeah
the
way
I
at
least
in
my
current
mindset,
the
way
I'd
go
about
getting
more
of
that
flexibility
of
what
can
be
learned
like
heads
are
above
bodies
or
whatever
you
know.
My
real
answer
is,
I
don't
know
how
to
use
grid
cells
for
that,
the
these
displacement
graphs
that
I'm
talking
about
like
the
object,
vector
cell
graphs.
In
a
sense,
I
think
I
can
do
it
with
in
my
mental
model.
B
It
works
with
that
and
associating
information
with
edges
when,
when
we
come
to
these
more
flexible
versions,
it's
more
rather
than
using
these
very
discreet
or
very
rigid
displacements,
not
spatial
displacements,
but
allow
it
to
be
more
of
like
a
learned
thing
like
a
learned
relationship,
a
learned,
quote-unquote
displacement
that
really
is
more
hand-wavy.
That's
that's
how
I
begin
to
address
this
is
so
thinking
about
graphs
where
the
edges
are
learned
and
the
edges
are
really
kind
of
flexible
and
what
they
can
mean.
So
that's
that's
how
I
begin
to
address
that.
C
B
E
Yeah,
if
I
can
speak
up,
I
hope
you
don't
mind.
I'm
somewhat
familiar
with
marcus's
idea
and
we've
been
we've
been
working
on
the
idea
that
in
cortex
throughout
the
cortex,
the
same
basic
mechanisms
are
being
used,
modeling
everything
in
the
world.
This
is
a
chemical,
complex
models
and
it's
specifically
the
model
mortgage
proposal.
I've
been
trying
to
map
this
onto
cortical
anatomies
and
the
specific
types
of
cells
in
physiology.
E
You
see
there
and
if
I,
if
I
understood
your
question
tim,
I
mean
I
may
not
have
so,
but
I
think
you're
asking:
how
do
we
keep
it
apart?
The
separate
underlying
structure,
from
the
specific
knowledge
about
a
memory
of
that
particular
environment
and
in
the
details
of
marcus's
model,
there
are
different
cell
types
representing
these
things.
The
nodes,
the
nodes
of
his
graph,
are
not
singular.
They're,
multiple
cell
types
in
those
nodes
and
as
I've
been
working
through
this
it
it
feels
like
you.
E
Can
you
can
infer
just
by
structure
or
you
can
infer
by
specific
instances,
but
I
think
it's
compatible.
I
think
what
marcus
has
proposed
is
compatible
with
that
idea.
If
I
understood
your
question
correctly
and
I
may
not
have-
and
it
gets
into
the
details
of
the
model,
but
I
agree
with
you
that
that
you
have
to
be
able
to
which
I
thought
some
of
the
speakers
elegantly
explained.
E
I
don't
think
that's.
I
think.
That's
that's
something
really
tripped
us
up
for
a
long
time,
because
we,
we
assume
our
models
of
the
world,
assume
that
we
don't.
If
we
think
about
the
neocortex,
we
don't
want
to
assume
that
it's
even
three-dimensional
space.
We
we
rather
assume
that
the
cortex
figures
out
the
the
dimensionality,
the
things
it's
modeling
and
but
let's
assume
it's
3d
and
and
we
never
really
got
a
good
way
of
doing
that
with
with
grid
cells.
It
just
never
really
really
worked
very
well.
E
We
went
through
all
sorts
of
generations
to
try
to
make
it
work,
but
I
think
the
idea
that
marcus
is
proposing,
which
is
really
the
modeling,
is
done
in
sort
of
the
the
polar
coordinates,
the
object,
vector
cell
types
of
cells.
That
is
amenable
to
many
dimensions.
You
you,
don't
that
you,
you
might
think
a
rat
might
believe
in
the
two-dimensional
space
but
the
the
actual
mechanism,
but
I
think
that
underlying
that
could
be
any
dimensional.
D
And
and
just
yeah
and
those
that's
why
you
understand
those
polar
coordinates
is
that
is
that
different
from
it
are
you
imagining
that
that's
an
egocentric
coordinate
space
so
that
you
can
so
you
have
the
whole
parietal
level
to
play
with,
or
are
you
I
mean
is
I
was
surprised
I
thought
you
were
going
to.
I
didn't
think
you
were
going
to
say,
object,
vector
cells
when
you
said
them,
because
they're
still
allo
centric
and
in
math.
E
B
E
They
are
another
thing,
but
you
also
have
other
representations
they're
not
allowed
so
anyway,
I
think
going
back
to
this
very
interesting
question
with
grid
cells
seems
to
be
two-dimensional
and
and
the
world
is
not
two-dimensional,
and
so
we
have
this
problem.
I
think
one
of
the
nice
things
that
marcus
came
up
with
is-
and
I
don't
think,
we've
worked
it
through.
I
don't
maybe
market
that
yet,
but
I
haven't
understood
it
yet
in
detail.
Is
that
the
the
the
grid
cells?
Are
there
really
one
way
to
think
about
it?
E
They're
there
for
path,
integration
they're
there
to
help
the
assist
the
model.
The
model
is
not
in
the
grid
cells.
The
models
is
in
the
op
and
the
vector
cells,
and-
and
yet
you
really
can't
do
path
integration
well
with
those.
So
you
can,
then
you,
then
you
have
this
subsystem.
That's
really
the
grid
cell,
which
are
sort
of
saying
hey.
I
can
assist
you
in
path.
Integration
market
share
this
sort
of
three-way
thing.
I
call
the
flux
capacitor,
which
says
you
know,
maybe
a
two-dimensional
path.
B
E
E
Yeah
yeah,
so
so
maybe
that's
right
right,
but
but
we
don't
know
so,
but
I
think
the
nice
thing
is
that
the
market's
proposed
it
separates
out
these
two
worlds.
You
can
say:
okay,
there's
a
path,
integration
component
and
then
there's
a
model
component
and
the
path
integration
could
be.
3D
could
be
and
there's
a
lot.
E
Things
about
grid
cells
that
bother
me,
you
know,
sometimes
they
they
realist,
reliably,
don't
fire
in
certain
situations,
certain
positions
at
certain
locations.
We
don't
have
an
explanation
for
that,
so
there's
weird
stuff
going
on.
We
just
don't
know.
Obviously
you
guys
are
the
experts
on
that,
but
I
think
the
idea
of
separating
out
path
integration
is
a
separate
from
the
modeling
component
and
microsoft.
I
don't
want
to
speak
to
you
if
I
got
that's
wrong,
correct
me,
but
that
seems
to
be
a
really
interesting
idea
which
to
me
was
like.
B
E
That
is
not
your
story,
but
that
that
tells
you
to
key
back
into
the
into
the
path
integration.
It's
not
the
it's,
not
the
location
of
the
thing,
we're
storing.
It's
not
like
what
the
the
basic
idea
is:
you're,
storing
these
relative
positions
of
of
objects
in
we've,
we've
kind
of
known
that
had
to
happen
for
a
long
time,
but
we
didn't
have
a
mechanism
that
really
we
proposed
one
in
our
2019
paper,
but
it
yeah,
but
it
wasn't,
it
had
holes,
they
had
problems
didn't
work.
This
is
the
displacement.
C
C
A
D
A
D
F
D
D
It's
trying
to
learn
to
make
the
same
representation
every
time
it
comes
back
to
the
same
place,
so
it
can
retrieve
the
correct
memory
as
path
integration
and
it's
trying
to
learn
to
make
representations
that
are
very
different
from
each
other
at
different
places.
So
it
has
an
efficient,
so
it
can
access
that
memory
efficiently.
D
C
D
The
physical
location,
the
question
is:
why
why
isn't
it?
Why
isn't
it
distributed
throughout
what
what's,
I
think,
you're
you're
being
asked
what
we're
being
asked?
Why
isn't
every
cortical
module
like
in
this
like
how
what
we
think
of
as
the
hippocampus
enter?
I
know
that
which
I
think
is
what
jeff
jeff's
model
suggests
and
would
be
amazing
if
true,
I'm
willing
asked.
Why
isn't
that
true,
so
kim?
Why
isn't
that
true.
D
C
Slightly
harder
question
I
mean,
I
think
the
I
don't
know
I
mean
the
the
classical
theory
that
I
think
most
relates
to
this
question
is
the
complementary
learning
systems
idea
that,
like
hippocampus,
is
specialized
in
a
particular
type
of
fast
rapid
acquisition
of
new
memories?
Cortex
is
doing
a
a
slower
more,
like
generalizable
form
of
learning
that
it's
learning
patterns
more
gradually
and
not
trying
to
rapidly
acquire.
E
It
could
be
that
could
be.
You
know.
One
thing
you
see
along
the
hippocampus,
you
see
a
lot
of
these,
so
silent,
synapses
that
are
and
the
parametral
cells
here
are
huge
numbers,
30,
000
synapses
on
them,
and
that
would
be
very
suitable
for
very
fast
learning
where
the
cortex
you
wouldn't
want
to
do
that,
so
it
could
be
the
same
base.
I
believe
it's.
The
same.
Basic
mechanisms
are
working
everywhere,
but
you
wouldn't
have
this
as
fast
as
learning
in
the
cortex.
E
D
We
should
also
quickly
return
to
kim's
initial
point
about
what
makes
a
memory,
and
what
do
you
want
to
remember
fast,
because
there
is
like
a
fundamental
difference
between
something
that
is,
that
knows
about
a
part
of
what's
happening
now,
as
in
a
cortical
column
versus
something
that
can
make
a
memory
that
can
go
and
reactivate
all
of
cortex
to
recreate
the
experience
of
now
and
hippocampus
really
historically
has
been
about
episodic
memory,
not
about
navigation.
That's
how
we
found
out
about
it
and
the
reason
people
are
excited.
D
The
reason
it's
interested
in
episodic
memory
is,
is
it
can?
It
can
remember
a
whole
episode,
and
that's
for
that
set
that
reason
that
that
kim
was
saying
earlier.
Is
it
gets
this
extraordinary
influx
of
connections
which
which
come
from
all
parts
of
cortex
and
are
combined
together
with
these
very
long
axons
which
go
all
the
way
up
hippocampus
and
combine
things
that
seem
to
be
that
you
could
never
bind
together
in
cortex
right?
You
could
never
buy
like
if
the
situation
now
I
need
to
bind
together.
D
My
schema
of
understanding
a
zoom
call
with
the
fact
that
I'm
on
jeff
hawkins's
buddy
over
there,
with
the
with
with
of
xyz
and
and
that
all
has
to
make
one
memory,
and
then
you
can
sequence
out
those
experiences,
and
then
you
have
to
partition
those
you
have
to
partition
the
relevant
parts
of
that
experience
to
into
the
relevant
cortical
regions
for
learning
and-
and
so
there
is
something
different
about
hippocampa,
not
about
hippocampus
from
a
cortical
column,
in
that
it
has
access
to
the
whole
experience
which
called
column.
Doesn't.
E
Experiment,
I
think
that's.
I
think
that
the
argument
is
it's
the
same
basic
mechanisms
and-
and
I
just
don't
disagree-
that
the
hippocampus
is
in
this
amazing
place
to
do
you
know
these
things.
You
know
it's
like
something
like.
I
have
knowledge
about
my
computer
and
where
every
key
is
and
where
the
ports
are,
and
so
is
that
stored
in
the
hippocampus
complex
was
that
stored
in
the
in
the
neural
cortex
right
I
have
knowledge
about
mathematics.
E
D
E
So,
that's
all!
I
don't
think
I
think
there's
just
it's.
You
know
the
the
interesting
it's
now
looking
like,
there's
similar
types
of
mechanisms
that
exist
in
birds
and
they're,
blob
sections,
and
things
like
that.
So
I
think
the
question
of
the
the
audience
member
is,
you
know
why
don't
we
see
similar
structures
throughout
the
brain?
I
think
actually
we're
going
to
see
them
throughout,
not
just
the
mammalian
brain.
You
can
see
them
throughout
other
any
animal
who
can
sort
of
do
any
kind
of
path,
integration,
any
kind
of
knowledge.
E
D
D
D
E
D
D
E
Right-
and
I
think
I
think
your
your
program-
the
paper
out
of
china,
which
said
that
I
I
agree-
that's
a
problem
with
that,
if
that's
what
you're
arguing
yeah,
so
I'm
not
going
to
put
much
weight
in
that.
I
think
it's
interesting
to
find
out
how
that
paper
came
about
what
did
they
actually
measure?
You
know
what
are
the
issues
there,
but
I
was
defined
by
the
two.
I
wouldn't
expect
that
that
you
wouldn't
expect
to
see
the
same
thing.
E
E
So,
let's
just
dismiss
that
I
mean
I
mean
the
the
beautiful
paper
that
you
know
you
guys
worked
on
the
fmr
experiments
with
with
humans.
Just
amazing.
You
know
that
was
showing
prefrontal
cortex
and
grid
like
structures,
and
so
I
think
we'll
see
over
time.
But
I
I'd
be
very
surprised
if
we
don't
see
a
continuing
theme
of
these
basic
mechanisms
throughout
the
cortex
it
it
just
makes
sense.
The
brain
has
to
model
things,
and
this
is
a
modeling
system
and
it's
unlikely
to
be
different.
A
Okay,
I'm
aware
of
the
time,
so
I
guess
we'll
wrap
this
up
so
yeah.
Thank
you
for
such
a
productive
and
interesting
conversation.
Thank
you
kim
james
and
tim
for
sacrificing
your
lunch
and
dinner
time
to
join
us
and
also
marcus
for
presenting
for
the
audience
we
weren't
able
to
go
through
that
many
questions.
So
if
our
panelists
have
some
time,
maybe
we
can
address
them
via
email
and
we'll
post
them
on
our
websites
and
yeah.