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B
B
A
A
D
B
D
A
ring
and
such
that
there's
a
pump
active
somewhere
in
that
ring
at
any
point
in
time
as
the
animal
performs
a
movement
in
any
3d
direction.
What
happens
in
that
ring,
and
so
I'm
not
worrying
about
modeling
cells
here,
but
you
can
imagine
it
as
a
bunch
of
little
cells
here
in
a
ball
of
activity
moving
through
them,
but
instead
I'm
just.
B
B
B
Yeah
I'm
just
going
back
to
sort
of
the
bigger
picture.
If
the
goal
this
is
to
say,
given
the
how
this
information
can
I
ask,
can
I
predict,
what's
actually
going
to
see
happen
in
orientation
cells,
but
the
second
part
of
the
problem
is
okay.
Now
that
I
have
that?
How,
when
I,
take
that
information
posted
the
gravity
vector
to
do
something?
How
would
I
literally.
D
It
took
a
while
for
me
to
come
up
with
the
right
update
rules,
so
first
I
had
to
come
up
with
what
something
wasn't.
One
way
to
do
this.
That
I
know
is
right,
even
though
it
now
how
the
biology
does
it
like
us
as
how
sign
of
observers?
How
can
we
find
the
ground
truth?
What
should
that
direction
update
be
so
in
their
system
like
quick
review
is
like
rotation,
yaw,
rotation
rotating
and
your
current
frame
always
causes
the
head
Direction
cells
to
to
update
in
your
rotation
direction
and
there's
a
half
day.
D
Rule
in
that
paper
is
that
as
the
current
plane
that
you're
standing
on
like
the
rat
walking
on
a
on
a
hill
or
a
sphere
here,
as
that
rotates
about
the
gravity
direction.
That
should
also
cause
this
too.
So
that's
just
quick
review
so
given
to
three
orientations
which
you
can
depict
as
being
on
services
of
a
sphere
while
pointing
in
a
certain
direction.
These
are
little
rats
the
way
to
to
the
way,
analytically,
less
is
outside
of
observers
to
tell
how
the
head
direction
should
update
from
here
to
here.
D
D
D
That
shouldn't
do
it
change
it.
So,
by
splitting
into
these
three
movements
it
becomes
trivial.
The
the
correct
head
direction.
Change
from
this
point
to
this
point
is
this
rotation
angle?
So
if
you
can
just
so,
if
you
solve
for
all
this,
you
can
find
it
analytically,
and
this
was
how
I
had
a
notion
of
correctness.
D
D
D
B
B
D
Yeah,
that's
also
at
the
second
little
circle
yeah.
So
there's
that
little
ghost
agent
right
there.
So
you
can
see
I
implemented
the
ghost
agent
I'd
solved.
All
this
I
solved
this
formula
and
then
then
I
did
a
lot
of
other
work
to
make
it
where
the
actual
bump
moves
smoothly
in
a
correct
way,
and
that's
just
a
yeah.
So
so
here
I'm
already
showing
it
working
and
then
I'm
gonna
say
how
it
worked
and
now
just
to
demonstrate
that
it
works
I'll.
D
D
Okay,
so
yeah,
that's
that's
my
big
demo
here
else
up
shank,
so
yeah
it
took
a
while
to
get
it
together
right
and
it
ended
up
being
quite
simple
on
the
end.
But
building
up
to
that
required
light,
it
was
like
I
was
building
a
lot
of
scaffolding
just
to
get
an
initial
version.
That's
working
that
is
giving
too
much
information
for
free
that
pulled
out.
Some
of
that
information
pulled.
C
B
This
is
Omega
and
then
the
new
axes
they're
at
sort
of
your
direction
and
so
mean
yes,
because
I've
been
making
this
you
know
inference.
Point
angular
in
France
is
similar
to
that.
I
got
a
linear
movement
entrance.
You
know
the
same
thing
years
and
so
I'm
just
trying
to
make
that
connection
here
as
well.
B
B
B
D
So
this
so
an
angular
velocity
can
be
specified
as
the
direction
of
this
and
it's
its
magnitude,
and
so
you
conveniently
you
can
actually
express
an
angular
velocity
as
a
single
vector
where
the
direction
the
vectors
pointing
is
the
axis
and
the
length
of
the
vector
is,
is
the
velocity
itself,
which
ends
up
working
out
nicely
for
this,
but
I
I'll
just
leave
it
at
that.
So
now,
just
to
just
restate
the
task
in
terms
of
these,
this
notation,
given
a
movement
and
direction
of
gravity,
given
movement
and
direction
of
gravity
figure
out.
D
What
is
the
update
to
to
the
head
Direction
cells,
so
I
tried
a
bunch
of
ways
of
specifying
this
and
finally,
the
way
that
worked
was
to
now
think
of
rotation.
Now
I
just
laid
out
one
way
to
think
about
angular
velocity,
but
another
another
way
that
people
specify
rotations
is.
This
is
basically
Euler
did
everything.
This
is
called
the
Euler
access.
D
This
is
called
Euler
angles,
and
he
just
you,
saw
all
these
things,
and
so
another
way
to
specify
rotation
is
with
this
series
of
Euler
angles
and
I've
shown
one
example
of
how
that
can
be
set
up
and
the
one
in
the
example.
This
is
the
one
that's
relevant
to
us.
You
can
specify
a
rotation
using
three
angles
where
you're
doing
a
rotation,
and
this
is
like
a
gyroscope
whatever
you
want
to
call
it
rotating
about
this
axis
rotating
about
this
axis
and
rotating
about
the
center
one.
D
If
you
set
it,
be
something
the
right
way.
There
are
a
number
of
ways
you
can
do
it.
You
can
reach
any
orientation
by
by
turning
these
three
by
performing
these
three
rotations,
and
so
just
it
just
like
envision
this
for
a
second,
if
you,
if
you
picture
a
constant
angular
velocity,
if
you
picture
this
in
agent
just
kind
of
rotating
about
this
axis
smoothly
at
a
constant
rate,
so
now
picture
this
agent
doing
the
same
thing.
This
is
rotating
smoothly
at
a
constant
rate.
These
are
different
actions
right,
yeah,
well,.
B
D
That
some
points
like
these
are
going
to
be
updating
quickly
and
other.
Quite
sir,
and
in
this
picture
this
a
solution
are
like
the
with
the
initial
scaffolding
was
that
the
correct
update
to
the
head
Direction
cells?
Is
this
angle
Plus
this
angle,
where
the
middle
one
is
basically
capturing
rotations,
I.
D
B
D
B
B
B
So
as
I
move,
if
I
think
about
just
my
change
relative
to
my
body's
reference
frame,
this
yah,
then
it's
just
a
simple
change,
but
my
change
relative
to
the
ground
effect
is
much
more
complex.
Yes,
exactly
so
that's
going
to
require
these
two
dimensions
are
specify
that
those
are
really
they
talk
about
you'd,
imagine
yeah
and.
D
Then
one
one
other
nice
thing
here
is
one
thing.
Awkward
with
these
angles
is:
that
is
that
from
for
most
cases
there
is
a
one-to-one
mapping
like
there's
a
one-to-one
here.
I'll
just
say
it.
This
way,
suppose
you're
standing
on
the
North
Pole
and
you
rotate
there's
an
ambiguity
of
here.
You
represent
that
by
rotating
this
or
by
rotating
this,
so
either
it
can
be
correct.
D
The
thing
I
wanted
to
point
out
here
is
that
ambiguous
solved
by
the
fact
that
head
Direction
cells
represent
the
some
of
them
such
that
I'm,
the
beauty
just
so
here's
yeah,
so
that
there's
a
there's
something
computationally
elegant
about
representing
the
some
of
these
disasters.
What
I'm
saying
so
now
getting
from
here
to
like
a
final,
simple
formula,
took
time
and
I'm
just
going
to
gloss
over
that,
so
solving
for
how
these
are
changing
over
time
and
then
just
like,
and
then
just
like
pummeling
those
formulas
and
to
being
simpler
in
step.
One
simpler.
D
D
D
B
B
D
What
one
thing
is
like?
Okay,
this
all
simplified
I,
went
through
this
elaborate
process
of
setting
this
up
in
this
popped
up.
Oh
there's
a
chance
that,
by
just
looking
at
this
formula
for
a
while,
I'll
realize
oh
here's
a
much
simpler
way.
I
could
have
done
it
and
that
will
cause
me
to
realize
something.
Maybe
I'm
missing
something
else.
Yes,
so
I,
don't.
D
This,
like
1
plus
G,
sub
Z
on
the
and
the
denominator,
I,
don't
know
someone
who's
experienced
with,
like
mechanics,
might
look
at
that
and
say
like:
oh
that's,
the
dot
product
is
the
cross
product.
There's
something
like
that.
Something
really
basic
so
just
to
bring
this
back
to
I
wrote
this
musing
section
just
thought
unless
to
bring
this
back
to
the
sort
of
unifying
this
with
grid
cells.
B
D
Very
singular:
it's
unnatural
for
head
Direction
cells,
another
type
of
matrix
velocity-
and,
in
this
case
it's
a
two
by
two
years
around
by
three,
but
I
should
actually
2
by
3.
If
this
is
a
3d
print
Selvin,
but
anyway,
I'm
at
the
stage
of
still
processing
this
one.
One
thing
that
I
want
to
know
is:
okay,
I've
only
tested
when
the
animal
is
right-side-up,
not
when
the
animals
sat
down
and
it
was
and
here's
where
here's,
what
we're
done.
This
formula
should
work
when
it's
upside
down
the.
D
A
B
B
D
So,
basically,
from
like
you
know,
last
night
at
6
p.m.
until
right
now,
I
feel
like
I've,
been
grabbing
more
and
more
low-hanging
fruit,
because
this
all
clicked
into
place
and
this
all
simplified
and
so
you're
catching
me.
Like
a
snapshot
of
me,
you
know
I'm
still
grasping
it.
The
fruit.
Now
that
I've
poured
a
bunch
of.
B
D
Yeah,
so
the
both
in
both
cases,
grid
cells,
head
Direction
cells,
were
applying
some
sort
of
projection
matrix
to
the
velocity
and
I
mean
this
is
just
the.
This
is
just
the
thing
from
the
paper
with
Neela.
Where
is,
and
mine
too,
by
three,
for
example,
of
history.
The
idea
that
you're
applying
some
sort
of
matrix
to
the
velocity
getting
together
I'm
just
trying
to
put
these
next
to
each
other
and
see
if
any,
if
anything
comes
to
mind,
I
still.
B
And
so
I
just
the
same
idea.
Oh
we
take
on
the
orientation
cells
models.
We
can
treat
the
representations
in
three
two
orientations
by
having
two
protection
project
and
that
doesn't
seem
to
be
what
the
brain
is
doing
correctly.
This
paper
and
and
there's
evidence
suggesting
that
we
don't
really
have
multiple
orientation
that
are
updating
differently
and
there's.
No,
we
actually
don't
even
have
any
evidence.
There's
multiple
clicks
on
modules
that
are
off
taking
differently,
like
active
projections
in
3d
space,
so
there's.
C
B
B
Mind
dimensional
orientation
cells,
but
it's
going
to
work
because
we
have
the
gravity
vector:
okay,
now,
really
that
kind
of
leads
open,
where's
the
gravity
vector
come
from.
You
can
imagine
that
no
gravity,
but
we
can't
assume
that
from
for
finding
our
orientation
to
objects
in
the
world,
there's.
B
B
B
It
probably
is
it
sort
of
like
we
have
one
to
DV
and
somehow
that
2d
grid
cell
module,
apparently
and
with
someone
with
some,
maybe
something
else
it's
going
to
just
like
here
we
had
something
else:
the
complete
solution
is
not
the
simple,
elegant,
simple,
more
complicated
and
and
so
probably
the
bleach
solution
on
the
location.
Salado
say
it's
going
to
be:
not
the
simple
one
either
become
something
like
this,
so
this
is
all
the
way
of
saying.
Okay,
I
accept
this.
B
B
B
B
B
C
C
B
Like
it's
like
you
know,
you
sent
something
and
you're
not
only
determining
your
location,
you
don't
know
your
location,
but
they
don't.
Even
with
the
object,
you're
honest,
you
do,
training
both
at
the
same
time.
Well
now
you're
sensing
something
you
have
to
you-
have
to
determine
and
you're
in
Charlie's
out
all
these
variables.
At
the
same
time,
you
try
to
resolve
your
your
grounding,
vector
or
:
your
orientation
and
your
occasion
an
object
and-
and
these
are
obviously
all
integrated
together.
They
come
with
all
these
things.
B
B
And
we
know
that
the
orientation
comes.
We
know
that
the
features
which
look
like
orientation
in
the
court
we
go
they're,
not
the
entire
all
the
layers
as
far
as
we
know,
and
so
it's
sort
of
like
just
sensory
input-
that's
coming
up.
Let's
just
call
it
sensory
input,
the
Centurions
up
coming
in
through,
like
a
spatial
polar
now,
you
got
the
many
common
representations,
it's
impacting
every
layer,
every
layer
faces
again.
B
And
you're
trying
to
so
it
seems
like
they're,
not
really
separate
problems,
or
probably
we
can.
We
can
break
them.
Apart
of
a
separate
conceptual
issues,
but
seems
like
a
solution,
that's
all
them.
At
the
same
time,
we
referring
all
somebody
probably
affirm
that
you
know
I
reach
my
hand
into
the
into
a
black
box
and
I
feel
something
I
can't
lose
one
sensation.
I,
don't
know
that
so.
B
C
B
So
if
we
said
okay,
this
is
orientation
the
original
models
there's
going
to
be
the
sensory
input
and
all
these
things
are
interacting
with
the
way
BBC
the
equivalents
can
we
can
we
resolve
how
all
this
gets
resolved
through
that
mechanism?
And
that's
that's.
The
three
doesn't
like
the
open
goal,
so
we're
different.
This
is
sort
of
the
find
the
problem
and
part
of
the
solution.
B
D
For
inferring
the
orientation
for
that
there's
a
very
good
chance
that
we
use
some
heuristic
to
use
a
common
heuristic
across
environments
across
objects
to
anchor
our
that
direction.
Cells
become
a
head
to
toe
or
similar
like
when
you
look
at
an
object.
You
kind
of
search
for
the
the
main
axis
of
the
object.
B
B
B
D
D
B
But
that
implies
not
if,
if
you're
saying
it
doesn't
matter
in
different
rooms,
a
lot
of
that
same
basic
shape,
so
the
good
self
seen
them
it's
different
yeah.
It's
also
you
much
the
same
yeah.
So
that
was
that's
a
big
clue.
Yeah
that's
been
replicated
the
recent
one
for
Kia
Hardcastle
and
it's
been
replicated.
B
B
B
B
B
May
not
be
true,
it
looks
more
like
it's
not
true,
so
you
know
so
that's
to
me,
like
the
big
that
to
me
like
that
I
should
I
were
to
identify
the
next
big
problem.
That's
going
to
be
more
likely
to
break
open
this
whole
thing.
That
would
be
it.
How
could
I
get
results
to
work
with
the
single
module?
What
what
additional
information?
What
I
mean
just
like
here,
you've
got
orientation
to
work
with
a
single
orientation
module
yeah
introduced
a
vector.