\documentclass[man]{apa}

\input{psfig}     % Jeremy's version of psfig

\def\citetext{\citeA}
\newcommand{\calL}{{\cal L}}
\newcommand{\mdeg}[1]{#1$^\circ$}
\newcommand{\tup}[1]{\langle #1 \rangle}
\newcommand{\colvector}[2]{\left[\begin{array}{c}#1\\
#2\end{array}\right]}
\newcommand{\comment[1]}{}

\title{Deforming the Hippocampal Map}

\author{David S. Touretzky, Wendy E. Weisman, Mark C. Fuhs, 
William E. Skaggs, Andre A. Fenton, and Robert U. Muller}

\affiliation{$^1$Center for the Neural Basis of Cognition\\
$^2$Computer Science Department\\
Carnegie Mellon University\\
Pittsburgh, PA 15213-3891}

\abstract{To investigate conjoint stimulus control over place cells, 
{}\citetext{fenton00jgp1} recorded while rats foraged in a cylinder
with \mdeg{45} white and black cue cards on the wall; card centers
were \mdeg{135} apart. In probe trials the cards were rotated together
or apart by \mdeg{25}.  Place field centers shifted during these
trials, stretching and shrinking the cognitive map.
{}\citetext{fenton00jgp2} described this deformation with an ad hoc
vector field equation.

We present two other models of map deformation.  In a maximum
likelihood formulation, the rat's location is estimated by a conjoint
probability density function.  In an attractor neural network model,
recurrent connections produce a bump of activity over a 2D array of
cells; the bump's position is influenced by landmark features such as
distances or bearings.  The maximum likelihood and attractor network
models yield similar results, supporting previous conjectures that
maximum likelihood may be an appropriate framework for describing
hippocampal attractor dynamics.}

\rightheader{Deforming the Hippocampal Map}
\leftheader{Touretzky et al.}
\shorttitle{Deforming the Hippocampal Map}
\journal{Journal of Rodent Cognition}

\ifapamodeman{\note{
Number of words in Abstract: xxx\\
Number of text pages: \\
Number of figures: \\
Number of tables: \\~\\
Send correspondence to: \\
David S. Touretzky\\
phone: 412-268-7561\\
fax: 412-268-3608\\
e-mail: \sf{dst@cs.cmu.edu}}}{}

\begin{document}
\maketitle


\section{Place Field Distortions}

To investigate conjoint stimulus control over place cells,
\citetext{fenton00jgp1} recorded while rats foraged in a cylinder with
one white and one black cue card on the wall.  The cards each
subtended {}\mdeg{45} of arc, and their centers were {}\mdeg{135}
apart, leaving a gap of \mdeg{90} between the right edge of the white
card and the left of the black card.  In probe trials the cards were
rotated together or apart by {}\mdeg{25}. Firing field centers shifted
during these trials, stretching and shrinking the cognitive map.

\section{Vector Field Model}

{}\citetext{fenton00jgp2} presented a mathematical model of the
deformations they had observed in their experiments.  The model
described how they believe place field centers move as a result of
rotating the cue cards and/or deleting a card.  The approach was
presented as purely {\em ad hoc}: the equations simply describe the
desired vector field results, with no claim that the hippocampus
actually produces place fields this way.

The vector field equation had two parts: a rotational component that
determined how place field centers rotate around the center of the
arena, and a translational component that was added to correct a
problem with the result of the rotation equation when the cards were
moved together or apart.  We present the model here in a slightly
different formulation than {}\cite{fenton00jgp2} for improved clarity.

Our model assumes that upon entry into the cylinder during a probe
trial, the animal's head direction estimate is reset so that {\em
East} is the direction defined by the line from the arena center to
the spot on the cylinder wall that is equidistant from the edges of
the two cue cards.  (There are actually two such spots, on opposite
sides of the cylinder; the spot closer to the cards should be used.)
If only one card is present, then the reference spot is in its
``standard'' position with respect to the card: \mdeg{45}
counterclockwise from the left edge of the black card, or \mdeg{45}
clockwise from the right edge of the white card.  Note that if the
cards are rotated by opposite amounts, e.g., \mdeg{+10} and
{}\mdeg{-10}, then their edges move closer together or farther apart
with no change in the reference spot.  If the two cards are rotated by
identical amounts, or one card is deleted and the other rotated, the
reference spot rotates but, due to head direction reset, the rotation
is undetectable by the model.  If the cards move by unequal amounts,
the motion of the reference spot is cancelled by the head direction
reset, and the model sees only the relative motion of the two cards.

The head direction reset assumption restricts the model to operating
in the reference frame defined by the cards; there can be no influence
of an external reference frame tied to the experimental chamber.  The
assumption's value is that it greatly simplifies the vector field
equations, ensuring that angular displacements of the cards with
respect to the reference point are always equal and opposite.  We will
therefore adopt the convention that the white card rotates by an angle
$\alpha$ and the black card by $-\alpha$.

\begin{figure}
\psfig{file=distortm25.eps,width=6in}
\caption{Calculated displacement vectors when cards are moved closer together by $25^\circ$.
(a) Rotational displacement $D_\mathrm{rot}$ from Equation~\protect{\ref{eqn:netrot}}.
(b) Translational displacement $D_\mathrm{trans}$ from Equation~\protect{\ref{eqn:nettrans}}.
(c) Total displacement $D_\mathrm{tot}$ from Equation~\protect{\ref{eqn:nettot}}.
}
\label{fig:distortm25}
\end{figure}

The next step in Fenton et al's procedure is to calculate displacement
vectors, in room coordinates, based on rotations of the individual cue
cards.  Ignoring the black card for the moment, if the white card
rotates by an angle $\alpha$, then all place fields should rotate by
$\alpha$.  Assume the cylinder is centered at the origin, and let $R$
be its radius.  A place field centered on the east edge of the
cylinder, at location $[R,0]$, would rotate to a new position
$[R\cos\alpha, R\sin\alpha]$.  Hence, the white card-dependent
rotatonal displacement vector for this specific point would be:

\begin{equation}
 W_0 = \colvector{R\cos\alpha}{R\sin\alpha}-\colvector{R}{0} = R\cdot\colvector{\cos\alpha-1}{\sin\alpha}
\end{equation}

Any place field centered on the $x$-axis at a distance $r$ from the
origin would have the same rotational displacement vector $W_0$,
scaled by $r/R$.  In the general case of an arbitrary point $[x,y]$ in
the cylinder, at a distance $r=\sqrt{x^2+y^2}$ from the origin and an
angular displacement $\theta = \mathrm{atan2}(y,x)$ from the positive
$x$-axis, the white card-dependent rotational displacement vector is
given by:

\begin{equation}
 W = r \cdot \left[ \begin{array}{cc}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{array} \right] \times \colvector{\cos\alpha-1}{\sin\alpha}
\end{equation}

\noindent 
The black card-influenced rotational displacement vector $B$ is
calculated analogously using $-\alpha$.  The overall rotational
displacement of a point $[x,y]$ is the average of its white and black
card rotational displacement vectors, weighted by relative distance to
the two cards, so that the nearer card has proportionally greater
influence.  If $d_w$ and $d_b$ are the distances from the point
$[x,y]$ to the centers of the white and black cards, respectively,
then the net rotational displacement is

\begin{equation}
D_\mathrm{rot} = \frac{d_w B + d_b W}{d_w + d_b}
\label{eqn:netrot}
\end{equation}

The result of this equation is shown in Figure~\ref{fig:distortm25}a.
This purely rotational displacement does not fully capture the
distortion of place field structure observed by Fenton and Muller.
When the two cards move closer together, all place field centers are
displaced slightly toward the cards (the map shrinks); when the cards
move apart, place field centers are displaced away from the cards (the
map stretches).  Fields close to a card show less translation, but
more rotation, than fields distant from either card.  Fenton et
al. added a translational term to their equation to reproduce this
effect.  Let $W_c$ be the rotational displacement vector denoting the
movement of the white card center from its standard position to its
current location.  Let $B_c$ be the rotational displacement vector for
the black card center.  The translational displacement of a point
$[x,y]$ at distances $d_w,d_b$ from the white and black cards,
respectively is defined as:

\begin{equation}
D_\mathrm{trans} = 
  \frac{W_c + B_c}{c_2 \cdot \left(\frac{1}{d_w} + \frac{1}{d_b}\right)}
\label{eqn:nettrans}
\end{equation}

{}\noindent where $c_2 = 83.4$~cm determines the dropoff of the
translational term as distance to the cards decreases.  Note that if
either $d_w$ or $d_b$ is small, the denominator of the equation will
be large, and the translational term will be attenuated relative to
the rotational term.  But at locations distant from both cards, the
translational term is significant. The contribution of this
translational displacement term is shown in
Figure~\ref{fig:distortm25}b.

The total displacement of a point $[x,y]$ is the sum of the rotational
and translational displacements, and is shown in
Figure~\ref{fig:distortm25}c:

\begin{equation}
D_\mathrm{tot} = D_\mathrm{rot} + D_\mathrm{trans}
\label{eqn:nettot}
\end{equation}

Figure~\ref{fig:distortp25} shows the rotational, translational, and
total displacement vectors when the cards are moved apart by
\mdeg{25}.  Once again, the translational component helps to overcome
the shortcomings of the rotational component at locations distant from
both of the cue cards.

The \citetext{fenton00jgp2} model produces a good match to the
experimental data of Fenton and Muller on cards moving together or
apart by \mdeg{25}.  Furthermore, when one card is removed and the
other card rotated, the model correctly predicts that place fields
will rotate without distortion.  The original formulation of the model
accomplished this by assigning an effectively infinite distance to the
missing card, but this is unnecessary, as the same result can be
achieved by defining the ``reference point'' in such a way that
$\alpha=0$ whenever a card is missing.  Since the model is already
insensitive to rotations of the cylinder relative to the room frame, this
reference point assumption does no harm.

\begin{figure}
\psfig{file=distortp25.eps,width=6in}
\caption{Calculated displacement vectors when cards are moved apart by $25^\circ$.
(a) Rotational displacement $D_\mathrm{rot}$ from Equation~\protect{\ref{eqn:netrot}}.
(b) Translational displacement $D_\mathrm{trans}$ from Equation~\protect{\ref{eqn:nettrans}}.
(c) Total displacement $D_\mathrm{tot}$ from Equation~\protect{\ref{eqn:nettot}}.
}
\label{fig:distortp25}
\end{figure}


\section{Maximum Likelihood Model}

Our first alternative to the purely descriptive approach exemplified
by the vector field equations (\ref{eqn:netrot}--\ref{eqn:nettot}) is
to look for a theoretically justified account of map deformation.  Let
us assume the rat takes a probabilistic approach to determining its
position, with each landmark an independent source of position
information.  In this framework, each landmark observation generates a
probability distribution for the animal's current location.  When the
cue cards are in the standard configuration, all evidence sources
should agree (in the absence of noise).  But when the cards move
closer together or farther apart, the evidence becomes inconsistent.
A reasonable response in this situation is to take all the evidence
into account and choose the peak of the combined probability
distribution as the animal's most likely location.  This {\em maximum
likelihood} approach offers a probability-theoretic justification for
why the place field map stretches and shrinks with cue card movement.

Let the probability distribution generated by a landmark observation
be a Gaussian function of distance.  If the rat's observed distance to
landmark $i$ is $v_i$, then the probability that its actual distance
from the landmark is $d_i$ is distributed according to a Gaussian with
mean $v_i$, and variance $\sigma^2$ which we assume is proportional
to $v_i^2$:

\begin{equation}
p(d_i | v_i) = \frac{1}{\sqrt{2\pi v_i^2}}
 \exp \left[ \frac{-(v_i - d_i)^2}{2 v_i^2} \right]
\end{equation}

Within the cylinder, the set of locations at distance $d_i$ from
landmark $i$ forms an arc centered on the landmark.  Since any of
these locations is equally probable, $p(x,y|d_i)$ is a positive
constant if $d_i = d_i^{xy}$, the actual distance from location
$[x,y]$ to landmark $i$ with the cards in standard configuration.
Hence $p(x,y|v_i) \sim p(d_i^{xy}|v_i)$.

A single landmark is insufficient to uniquely determine location, but
a pair of landmarks is usually adequate.  We assume that the two edges
of each cue card are both utilized as landmarks, so there are normally
four landmarks visible.  When one card is deleted two landmarks are
still available.

If landmarks are independent evidence sources, then they can be
combined by multiplying their probability distributions.  The two cue
cards would be independent if cards could overlap.  Even when this is
prohibited, the cards are nearly independent.  But the four cue card
edges are not independent, since if we know the location of the left
edge of a card, the right edge can only be in one location, determined
by the card's width.  Nonetheless, we will treat the landmarks as
independent for purposes of evidence combination.  The joint
probability distribution is thus:

\begin{equation}
p(x,y | v_1,v_2,v_3,v_4) = \prod_{i=1}^4 p(x,y | v_i)
\end{equation}

The product of several Gaussian arcs that intersect at a single point
is, roughly, a Gaussian bump.  Thus, the Gaussian firing rate
distribution of a place cell whose field center is at distances $v_1$
through $v_4$ from the four landmarks when the cards are in standard
position can be viewed as an estimate of $p(x,y)$.  Moving the rat
(changing $x$ and $y$) samples the distribution at different arena
locations.  Moving the cue cards alters the observed combinations of
$v_i$'s and thus changes the entire distribution.  We assume the
variance $\sigma^2$ of each distribution is proportional to the
perceived distance to that landmark, so that distributions scale in
accordance with Weber's law.

The conjoint distribution $p(x,y)$ need not be computed directly.  It
is more efficient to calculate $\calL_{xy}$, the log likelihood, whose
peak will be at the same location as that of $p(x,y)$:

\begin{equation}
\begin{array}{rcl}
\calL_{xy} &=& \log p(x,y | v_1,v_2,v_3,v_4) \\
 &=& \log \prod_{i=1}^4 p(x,y | v_i) \\
 &=& \sum_{i=1}^4 \log p(x,y | v_i) \\
 &=& -\frac{1}{2}\sum_{i=1}^4 \log 2 \pi v_i^2
 - \sum_{i=1}^4 \frac{(v_i - d_i^{xy})^2}{2 v_i^2}
\end{array}
\end{equation}

{}\noindent To find the peak of this distribution we eliminate
constant terms and calculate:

\begin{equation}
(x^*,y^*) = \arg\min_{x,y} \sum_{i=1}^4 \frac{(v_i-d_i^{xy})^2}{2 v_i^2}
\end{equation}

Unlike the vector field model, this model operates in room coordinates
and makes no assumptions about head direction reset.  When the two
cards rotate by equal amounts, the model produces a vector field in
which all points rotate around the center by a corresponding amount.
For a missing card, we omit the corresponding terms from the sum;
there is no inconsistency among the remaining cues so the fields
rotate without deforming.  But when the cards move closer together or
farther apart, the Gaussian arcs defined by the landmarks shift
relative to each other, and the map undergoes stretching and
shrinking.

We implemented this model using a grid of points spaced 1~cm apart.
The 76~cm diameter cylinder contained 4,513 of these points.  To plot
the vector field with the cards rotated, we selected an evenly spaced
subset of points from the interior of the cylinder.  For each point
$[x,y]$ we calculated the $v_i$ values for the landmarks viewed from
that location, and then evaluated $p(x',y'|v_1\ldots{}v_4)$ at every
point on the grid.  We picked the most probable value and plotted a
vector from $[x,y]$ to $[x',y']$.

\begin{figure}
\begin{tabular}{cc}
\psfig{file=bayes4dm25.eps,width=3.25in} &
\psfig{file=bayes4am25.eps,width=3.25in} \\[4em]
\psfig{file=bayes4bm25.eps,width=3.25in} &
\psfig{file=bayes8adm25.eps,width=3.25in}
\end{tabular}

\caption{Maximum likelihood estimation of place field movement when the cards
are moved closer together by \mdeg{25}, using (a) distance to
landmarks, (b) angles between pairs of landmarks, (c) allocentric
bearings to landmarks, or (d) a combination of distance and angle
information.}
\label{fig:bayesm25}
\end{figure}

As shown in Figure~\ref{fig:bayesm25}a, the results of this
formulation do not always match that of the Fenton et al. model.
Specifically, when the two cards move closer together by {}\mdeg{25},
most of the vectors point roughly east, but there is a region around
the arena center where the vectors are noticeably attenuated.  This is
because, at the center of the arena, the distance to all four
landmarks is equal to the arena radius, and this distance is unchanged
by cue card rotation.  Another striking feature of the plot is the
sharp discontinuity near the eastern edge of the arena, around the
vertical line joining the right edge of the white card to the left
edge of the black card.  The vectors suddenly switch direction in this
region.  These unexpected results are a consequence of the geometry of
the arena and our assumption that distance from landmarks is the rat's
sole source of evidence for its position.

A plausible alternative hypothesis is that rats determine their
position based on angles between pairs of landmarks, i.e., the
difference in their relative bearings, or equivalently, the retinal
angle subtended by a line connecting the two landmarks.  With four
visible landmarks there are six possible landmark pairs, but we chose
just the four pairs of circularly adjacent landmarks.  The additional
pairs would only provide redundant information.  Note that the
retinal angle between a card's left and right edge decreases with
distance to the card, while the retinal angle between one edge of the
white card and the opposite edge of the black card is a function of
both the distance to the cards and the card separation angle.

Let $u_{ij}$ be the observed value of the angle between landmarks $i$
and $j$, and let $a_{ij}^{xy}$ be the angle between these landmarks,
viewed from location $[x,y]$ with the cards in standard position.  We
can define a Gaussian probability distribution for position based on
the angle between a pair of distinct landmarks as:

\begin{equation}
p(x,y | u_{ij}) = \frac{1}{\sqrt{2\pi\sigma_a^2}}
  \exp \left(\frac{-[u_{ij} - a_{ij}^{xy}]^2}{2\sigma_a^2} \right)
\end{equation}

Once again, we make the simplifying assumption that evidence sources
are independent, so that the overall probability distribution is the
product of the probability functions for the four landmark pairs
$u_{12}$, $u_{23}$, $u_{34}$, and $u_{41}$.  The result,
Figure~\ref{fig:bayesm25}b, shows good behavior in the vicinity of the
cue cards but anomalous results in other parts of the cylinder.  This
plot was made with $\sigma_a$ proportional to $u_{ij}$; using a
constant value produced slightly less satisfactory results.  In either
case, at locations far from either cue card the angles between
landmarks changle fairly slowly with position, producing a shallow
gradient with broad peaks.  When the two cards move in a non-rigid
fashion, the location of the absolute peak for a place cell in the
western half of the cylinder can shift by a large amount even though
the change in magnitude of the probability values is small.  Thus,
angle-based features are also not ideal for determining position in
the cylinder.

The single landmark feature that produced the best place field
deformation pattern was allocentric bearing, i.e., bearing to the
landmark with respect to some external coordinate system independent
of present heading.  The result is shown in
Figure~\ref{fig:bayesm25}c.  Neurons in the rodent head direction
system have been shown to encode the animal's heading with respect to
the environment {}\cite{sharpHD}; this suggests that rats are capable
of computing allocentric bearings of landmarks.  Behavioral
experiments in gerbils also indicate that rodents can use allocentric
bearing information to disambiguate landmarks \cite{collett86}.
Futhermore, in a familiar environment the alignment of the rodent head
direction system is known to be controlled by visual landmarks
{}\cite{taube}.  Therefore, to compute allocentric bearings in our
model, we assume that the animal's heading reference ({\em East}) is
defined as mid-way between the white and black cue cards.  Moving the
cards together or apart by equal amounts does not affect the alignment
of the head direction system, although individual landmark bearings
will of course shift.  Rotating the cards in the same direction by
equal amounts rotates the heading reference as well, so the rat does
not notice any bearing change in that situation.

In the standard cue configuration, the reference point is half a card
width clockwise from the right edge of the white card, and half a card
width counterclockwise from the left edge of the black card.  When one
card is deleted, we assume that {\em East} is in its standard position
relative to the remaining card.  Hence, with one card present, place
fields can rotate but not deform.

The overall best result was obtained by combining distance with either
the angle or bearing based features.  The distance-based probability
function (Figure~\ref{fig:bayesm25}a) has a stronger gradient in the
western half of the cylinder than the angle-based function
(Figure~\ref{fig:bayesm25}a).  Combining the likelihoods by simple
addition allows each to compensate for the other's shortcomings.  As
shown in Figure~\ref{fig:bayesm25}d, the resulting vector fields are
very close in appearance to those of Fenton et al.  We held $\sigma_a$
constant in this case.  Place field centers were estimated by:

\begin{equation}
(x^*,y^*) = \arg\min_{x,y} \left( \sum_{i=1}^4 \frac{(v_i-d_i^{xy})^2}{v_i^2} +
  \sum_{i,j} (u_{ij}-a_{ij}^{xy})^2 \right)
\end{equation}

Similar results were obtained using distance plus allocentric bearing.

\begin{figure}
\begin{tabular}{cc}
\psfig{file=bayes4dp25.eps,width=3.25in} &
\psfig{file=bayes4ap25.eps,width=3.25in} \\[4em]
\psfig{file=bayes4bp25.eps,width=3.25in} &
\psfig{file=bayes8adp25.eps,width=3.25in}
\end{tabular}

\caption{Maximum likelihood estimation of place field movement when the cards
are moved apart by \mdeg{25}, using (a) distance to landmarks, (b)
angles between pairs of landmarks, (c) allocentric bearings to
landmarks, or (d) a combination of distance and angle information.}
\label{fig:bayesp25}
\end{figure}

Figure~\ref{fig:bayesp25} shows the output of the model when the cards
are moved apart by \mdeg{25}.  Once again, a combination of distance
and angle features produces the best results.

\section{Attractor Bumps}

Dynamical systems, or ``attractor bump'' networks, are a popular
approach to modeling aspects of hippocampal place cells
{}\cite{zhang96jns,samsonovich97jns}, and have been widely adopted by
hippocampal modelers {}\cite{redish98nc,redish99book,minai,kali00jns}.
One-dimensional attractor models have been used to model the head
direction system {}\cite{redish96network,goodridge00jnp}, orientation
tuning in visual cortex {}\cite{benyishai}, and the oculomotor system
{}\cite{tank}.  Two-dimensional attractor networks have been proposed
as models of superior colliculus and motor cortex.  Here we examine
the ability of an attractor network to function as a deformable map,
producing the stretching and shrinking effects observed by Fenton and
Muller.

We begin with a population of place cells arranged as a 2D grid.  Let
each cell have strong excitatory connections to the cells nearby,
weaker excitatory connections to cells somewhat further away, and
inhibitory connections to all the remaining cells.  With appropriate
parameter settings, a network organized this way will have an infinite
number of stable states, each consisting of a ``bump'' of activity
localized to some region of the grid.  Such a state is analogous to
the population activity observed in the hippocampus, because when the
rat is at a particular location, the place cell whose field is
centered closest to the rat's location will be firing at its maximum
rate, while cells whose place fields just overlap with the rat's
location will fire at lesser rates, and cells whose fields are far
from the rat's location will be quiescent.  As the rat moves through
the environment, the activity pattern over the place cell population
shifts to reflect this.

If the units comprising an attractor network are initialized with
random activity levels, the network will settle into a stable state
with a well-formed bump at a random location.  However, if a
smoothly-varying external input is applied to some region of the grid,
the bump will tend to form in the region of maximal external input.
The attractor network can thus be regarded as a parallel, distributed
method for finding the peak of an input signal projected onto a 2D
grid.

To produce visual control of place fields, the external input may be
taken from a collection of visual feature detectors tuned to landmark
distances and/or bearings.  To model the Fenton et al. task, we
created a separate set of feature detectors for each landmark.  For
the $i$th landmark there was a set of distance detectors $F_{i,j}$
tuned to various distances $r_j$, and a set of bearing detectors
$G_{i,j}$ tuned to various allocentric bearings $\phi_j$.  The place
cell at grid location $[x,y]$ received an excitatory connection from
feature detector $F_{i,j}$ or $G_{i,j}$ if the distance $d_{xy}^i$
from the place field center to the $i$th landmark was approximately
$r_j$, or the allocentric bearing $a_{xy}^i$ of landmark $i$ was
approximately $\phi_j$.

To demonstrate visual control of place fields once the feature
detectors have been wired up to the place cells, consider the rat
entering the environment at the start of a trial.  Let the perceived
distance from the rat's present location to the $i$th landmark be
$v_i$.  Those distance detectors $F_{i,j}$ for which $r_j$ is close in
value to the perceived distance $v_i$ will be active, and will supply
excitation to the appropriate subset of place cells.  A similar
situation holds for bearing detectors $G_{i,j}$.  A bump of activity
will then form over the place cells with its peak centered at roughly
the location receiving the greatest amount of feature detector input.
The simulated rat has thereby estimated its position in the arena.

This approximation is exact if the external input to the place cell
grid takes the shape of a gaussian bump.  However, the projections
from individual feature detectors to place cells form arcs, not bumps.
For example, a distance-based feature detector $F_{i,j}$ will project
to the arc of cells at distance $r_j$ from landmark $i$.  If several
arcs cross at a single point at roughly equal angles, the
resulting pattern of external input will look bump-like.  But this
condition does not always hold.  Angle-based feature detectors produce
very broad arcs in the western half of the arena, and the geometry of
the arena and cue cards constrains all arcs to be nearly coincident in
that region.  Bearing-based feature detectors were used instead
because they do not suffer this problem.  Another problem is that in
the eastern half of the arena, when the cards are moved, distance arcs
that once overlapped now merely pass close by, producing elongated
patterns of external input that are far from bump-like.  These effects
were not a problem for the maximum likelihood model because it only
looked at the peak of the input distribution; the overall shape was
ignored.  But the attractor network is sensitive to this shape, and
thus requires some refinement of its input features to assure that the
peak of the external input is close to the center of the input
distribution.  To correct for this, we added a feedforward inhibition
term from the feature detectors to the place cells that was strong
enough to cancel any individual arc or intersection of a few arcs, but
not the intersection of many arcs.  This ``decluttered'' the external
input signal, producing a stimulus that was more focused and bump-like.

Another difficulty arises close to the arena walls.  The 2D attractor
grid extends beyond the arena boundaries, but since the animal cannot
experience the environment beyond the walls, feature detectors were
not wired up to place cells lying outside the cylinder.  When the
simulated animal is at a point along the wall, the attractor bump
should be centered on a place cell right at the wall.  This cell will
excite (and receive excitation from) its nearby neighbors, both those
closer to the arena center and those further away (hence outside the
wall).  However, only the cells within the wall receive external input
from the feature detectors; the cells outside the wall do not.  Thus
there is a danger that the bump may form some distance short of the
wall, since that is where the center of mass of the external input
lies.  To minimize this we use relatively weak weights from the
feature detectors to the place cell grid, so that the attractor
dynamics dominate, and adjust the feedforward inhibition so that the
location of the peak of the input is more important than the overall
shape.

\section{Model Details}

The attractor network was implemented as a $45\times{}45$ grid of
cells, toroidally connected to eliminate edge effects.  This assures
that all cells have the same number of neighbors, so that in the
absence of external input, the attractor bump has a uniform shape
everywhere on the grid.  The arena was defined as a circular region 39
units in diameter, centered on the grid, so each grid unit covered
roughly 1~cm$^2$ of the cylinder.  The attractor bump was roughly 17
units in diameter, so a bump located at one edge of the cylinder would
have minimal effect on cells at the opposite edge via wrap-around on
the torus.

Each place cell's activation $V_i(t)$ was computed as the sum of
recurrent excitation from other place cells, a global shunting
inhibition term, and the external input received from feature
detectors.  Shunting inhibition was used because it improves the
stability of the network \cite{kali00jns}.

\begin{equation}
V_i(t+1) = \sum_j w^{EE}_{ij} \cdot F_j(t) + 
	w^{EI} \cdot FI(t) \cdot V_i(t) + \sum_k w^{EF}_{ik} \cdot FD_k(t)
\end{equation}

The integral of activation over time is the cell's synaptic drive
$S_i(t)$, governed by a time constant $\tau_E$ \cite{pinto96}.  The
cell's firing rate $F_i(t)$ is proportional to the synaptic drive
thresholded at zero.

\begin{equation}
\tau_E \frac{dS_i(t)}{dt} = -S_i(t) + V_i(t)
\end{equation}

\begin{equation}
F_i(t) = \left[ S_i(t) \right]_+
\end{equation}

A global inhibitory unit receives excitation from all place cells and
makes inhibitory projections back to them and to itself:

\begin{equation}
VI(t) = w^{IE} \sum_j F_j(t) + w^{II} \cdot FI(t)
\end{equation}

\begin{equation}
\tau_I \frac{dSI(t)}{dt} = -SI(t) + VI(t) 
\end{equation}

\begin{equation}
FI(t) = \left[ SI(t) \right]_+
\end{equation}

The strengths of the recurrent connections $w^{EE}_{ij}$ are a function of the
Euclidean distance $d$ between cells $i$ and $j$ on the torus:

\begin{equation}
w^{EE}(d) = 0.1135 \exp \left( -d^2/10.124 \right)
\end{equation}

For each of the four landmarks there were 43 distance-based Gaussian
feature detectors, tuned to integral distance values from 0 to 42.
$\sigma_d$ was $2$~cm, and the distance feature detectors were
weighted relative to each other to place more emphasis on detectors
tuned to small distances.  The weighting function was $24/(24+d)$.  In
addition there were 120 allocentric bearing feature detectors tuned to
bearing values from \mdeg{0} to \mdeg{357} in increments of \mdeg{3}.
$\sigma_b$ was {}\mdeg{6}, and all the bearing detectors had uniform
weights.

\section{Results from the Attractor Model}

\begin{figure}
\psfig{file=att-m25.eps,width=6in}
\caption{Map distortion in the attractor model with the cards moved closer together
by \mdeg{25}.
}
\label{fig:attm25}
\end{figure}

\begin{figure}
\psfig{file=att-p25.eps,width=6in}
\caption{Map distortion in the attractor model with the cards moved farther apart
by \mdeg{25}.
}
\label{fig:attp25}
\end{figure}

Figures~\ref{fig:attm25} and \ref{fig:attp25} show the map distortion
patterns when the cue cards are moved together or apart by \mdeg{25}.
The results are similar to the vector field and Bayesian models,
though a little less smooth, due in part to the lower resolution of
the grid.

\begin{table}
\begin{tabular}{llll}
Firing rate & Standard & Cards Apart & Cards Together \\
\underline{Statistic} & \underline{Configuration} & \underline{by \mdeg{25}} & \underline{by \mdeg{25}} \\
mean & 1.5374 & 1.3852 & 1.3517 \\
std. dev. & 0.0889 & 0.0739 & 0.0860 \\
min & 1.3252 & 0.9707 & 0.9430 \\
max & 1.9733 & 1.7993 & 1.7554 \\
\end{tabular}
\caption{Distribution of peak firing rates for all 1129 place cells within the
arena in the attractor network model, with the cue cards in standard
position or moved apart or together by \mdeg{25}.}
\label{tab:peakrate}.
\end{table}

The model also reproduces the reduction in peak firing rate observed
by Fenton et al. when the cards were rotated together or apart, as
shown in Table~\ref{tab:peakrate}.  Peak firing rates for three cells
are shown in Figure~\ref{fig:fields}.  This reduction in peak firing
rate is a consequence of the feature detector input becoming defocused
when the cards are moved out of standard position, so that the place
cell at the center of the stimulus bump receives less total input than
before.

\begin{figure}
\begin{tabular}{ccc}
\psfig{file=att-n-0-0.eps,width=2in} &
\psfig{file=att-p25-0-0.eps,width=2in} &
\psfig{file=att-m25-0-0.eps,width=2in}\\[0.95in]
\psfig{file=att-n-10-14.eps,width=2in} &
\psfig{file=att-p25-10-14.eps,width=2in} &
\psfig{file=att-m25-10-14.eps,width=2in}\\[0.95in]
\psfig{file=att-n-18-0.eps,width=2in} &
\psfig{file=att-p25-18-0.eps,width=2in} &
\psfig{file=att-m25-18-0.eps,width=2in}
\end{tabular}
\caption{Firing fields of three place cells with cue cards
(left) in standard position, (center) moved apart by \mdeg{25}, and
(right) moved together by \mdeg{25}.  The top row shows cell whose
field, when the cards are in standard position, is located at the
center of the arena.  The cell in the middle row has a field near the
black cue card, and the cell in the bottom row has a field at the west
edge of the arena.}
\label{fig:fields}
\end{figure}

Figure~\ref{fig:fields} also shows that the shapes of firing fields
distort along the direction of cue card motion.  The least distortion
is observed at the center of the arena, where only the card bearings
change, not the distances.



\section{Discussion}

In the vector field model, a cue card's influence in the rotational
component (Equation~\ref{eqn:netrot} is weighted inversely by distance
in order to achieve the desired map distortion effect
(Figures~\ref{fig:distortm25}a and \ref{fig:distortp25}a.)  A similar
effect is obtained in the Bayesian model by giving distance-based
landmark features a standard deviation $\sigma_d$ proportional to the
observed distance $v_i$, on the assumption that such perceptual
measurements should obey Weber's law.  The scaled variance gives the
evidence from a closer landmark a steeper gradient than that from more
distant landmarks, hence the position estimate shows greater influence
by the closer card.

In the model of \citetext{kali00jns}, EC cells are tuned to both
distance and egocentric bearing to walls.  The distance tuning is
sharper for closer walls, and also for walls behind the animal rather
than ahead of it.  Kali and Dayan justify this by assuming that if the
animal is headed away from a wall, then it has been close to the wall
recently and is likely to have a more accurate distance estimate based
on path integration.

The boundary vector cells (BVCs) of \citetext{hartley00hc} use a
product of two Gaussians, one tuned to distance and one to allocentric
bearing.  Our Bayesian model and our attractor network model, both
based on log likelihood, compute the sum of distance-based and
bearing-based Gaussians.  The log of a product being equal to the sum
of the logs, the two approaches appear similar.  However, Hartley et
al.'s place cells compute a thresholded linear combination of two or
more boundary vector cells, an extra layer of processing our model
lacks.  This appears to be necessary to derive spatially compact place
fields, because BVC's use entire walls as landmarks, whereas our
models use only point landmarks.  Furthermore, the attractor network
model utilizes recurrent connections and network dynamics to produce
roughly gaussian shaped place fields from inputs with more varied
shapes, while the Hartley et al. model produces these shapes in a
strictly feed-forward manner.  One drawback of the attractor approach
is that the network does not produce crescent-shaped place fields
along cylinder walls, as the Hartley et al. model does.  Such fields
have been reported by {}\citetext{muller87}.  Their existence suggests
that place cells -- at least those associated with boundaries --
should be tightly sensory-bound.  Place cells not associated with a
boundary are presumably driven by distal landmarks and recurrent
excitation, making them robust against landmark deletion.

Unlike in our Bayesian model, distance-based feature detectors in the
attractor network model have a fixed variance, but the strength of
their projections to place cells is weighted inversely by distance.
This was necessary because active feature detectors with large
variances would supply excitation to broad swaths of place cells,
which deforms the input governing the location and shape of the
attractor bump.  The Bayesian model was unaffected by broad excitation
because it simply picked the single point of maximum input as the
animal's most likely location.  But the attractor model relies on
network dynamics to settle the activity bump over approximately the
peak of the input, and is therefore less tolerant of inputs that
depart significantly from a compact, roughly symmetric, unimodal
shape.  The feed-forward inhibition mechanism discussed previously
helps to cut off the tails of an elongated input pattern, but is not
sufficient in itself.  The empirically-determined weighting function
for the distance-based feature detectors was $24/(24+d)$, where $d$ is
the distance in cm, so the relative weights from distance-based
feature detectors varied from 1.0 down to 0.22.


\subsection{Loose Ends}

- We used strong attractor weights but Knierim argues they should be weak.
But they're not THAT strong, since field shapes distort.  Are they strong at all?


What next?

- Extend to O'Keefe rectangular arena?  What would the landmarks be?

- directionality issue:  compare with Kali \& Dayan

- remapping with card reversal?  

\bibliography{/afs/cs/usr/dst/Spatial/dst}

\end{document}