\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, 
Fenton et al. (2000a) 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}.  Firing field centers shifted during these trials,
stretching and shrinking the cognitive map.  Fenton et al. (2000b)
described this deformation with an {\em ad hoc} vector field equation.

We consider what sorts of neural network mechanisms might be capable
of accounting for their observations.  In an abstract, maximum
likelihood formulation, the rat's location is estimated by a conjoint
probability density function of landmark positions.  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.  If features are
chosen with appropriate care, the attractor network and maximum
likelihood models yield similar results, in accord with previous
demonstrations that recurrent neural networks can efficiently
implement maximum likelihood computations (Pouget et al., 1998; Deneve
et al., 2001).}

\rightheader{Deforming the Hippocampal Map}
\leftheader{Touretzky et al.}
\shorttitle{Deforming the Hippocampal Map}
\journal{Hippocampus}

\ifapamodeman{\note{
Number of words in Abstract: 182\\
Number of text pages: 35\\
Number of figures: 10\\
Number of tables: 2\\~\\
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{Introduction}

Many cells in the rat hippocampus show spike activity focused on times
when the rat is located at specific places in the environment.  Dozens
of studies over the past three decades have demonstrated that these
firing field locations can be controlled by visual landmarks.  The
studies have yielded enough quantitative information to enable
development of preliminary computational models of these ``place
cell'' responses.  Currently, however, the pool of candidate models is
very diverse, and there is not even consensus on the general form of a
correct model, much less the detailed values of parameters.  (See
Hippocampus 9:4, ``Special Issue on Place Cells,'' for examples and
references.) In the effort to narrow down the pool of viable models,
some of the most useful information has come from experiments that
have manipulated features of the environment and quantified the
resulting changes in spatial firing properties of hippocampal cells.
One set of particularly informative findings come from a recent study
by {}\citetext{fenton00jgp1} on the consequences of changing the angular
separation between two cue cards mounted on the walls of a cylindrical
arena.  The authors presented a mathematical model of the deformations
they observed in the ``cognitive map,'' but it was purely descriptive;
there was no claim that the hippocampus actually derived firing fields
this way.  The aim of the current study is to consider what sorts of
neural network mechanisms might be capable of accounting for their
observations.

After reviewing the Fenton et al. model, we develop a theoretically
justified account of the map deformation effect by constructing a new
model based on maximum likelihood estimation.  We then present a third
model, an attractor neural network, that approximates the behavior of
the maximum likelihood model as a computation that could be
implemented in the hippocampus.  Our results are in accord with
previous observations that recurrent neural networks can efficiently
implement maximum likelihood estimation
{}\cite{pouget98nc,deneve01nature}.

\section{The Experiment}

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 edge of the black card.  In probe trials the cards
were rotated to increase or decrease their separation by
{}\mdeg{25}. Firing field centers shifted systematically during these
trials, distorting the cognitive map.  In contrast, on trials where
one card was removed and the other rotated, the map did not distort,
but rotated with the remaining card, demonstrating its continuing
salience and a nearly ideal form of pattern completion.

{}\citetext{fenton00jgp2} presented a mathematical model of the
deformations they observed in their experiments.  The model described
how they believed firing field centers moved as a result of rotating
the cue cards or deleting a card: all cells were controlled by both
cards, but to varying degrees based on the distance of the field
center to each card.  This contrasts with previous models where cells
differ in their responses to environmental manipulations because they
receive input from different subsets of cues
{}\cite{shapiro97hc,hartley00hc}.

\section{Reformulated Vector Field Model}

The vector field model of {}\citetext{fenton00jgp2} had two parts: an
angular component that determined how firing field centers rotate
around the center of the arena, and a translational component that
corrected a problem with the prediction of the rotation equation when
the cards were moved together or apart.  We begin by presenting this
model in a slightly different formulation for improved clarity and
completeness.

Our variant 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 a
reference point on the cylinder wall half way between the closest
edges of the two cue cards.  If the cards rotate by opposite amounts
(e.g., {}\mdeg{+10} and {}\mdeg{-10}), their edges move closer
together or farther apart, but there is no change in the reference
point.  If the cards rotate by identical amounts, the reference point
rotates as well, so the change is undetectable by the model due to the
resetting of the head direction system.  If the cards rotate by
unequal amounts, after head direction reset the model will see only
their relative motion.  Finally, if one card is removed, the reference
point is defined to be in ``standard'' position with respect to the
remaining card, meaning {}\mdeg{45} counterclockwise from the left
edge of the black card or {}\mdeg{45} clockwise from the right edge of
the white card, and again, any rotation of the card is undetectable.

Head direction reset restricts the model to operating in the reference
frame defined by the cards.  There is no provision for an external
reference frame tied to the experimental chamber to influence place
cell firing, in agreement with the finding that when the two cards
were rotated together no influence of a room frame was seen.  The head
direction reset assumption 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}
\centerline{\psfig{file=coordsfig.eps}}
\caption{Calculation of the rotational displacement vector $D_{rot}$ for
a point $[x,y]$ when the cards are rotated by $2\alpha=-25^\circ$.
Dotted arcs show cards in their standard position, solid arcs show
rotated position.}
\label{fig:coordsfig}
\end{figure}

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

The next step in our formulation 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 firing fields influenced by this card
should rotate by $\alpha$; see Figure~\ref{fig:coordsfig}.  We
represent each field by the location of its center.  Assume the
cylinder is centered at the origin, and let $R$ be its radius.  A
field 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 rotational displacement vector for this specific
point would be:

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

Any point on the $x$-axis at a distance $r$ from the origin would have
the same rotational displacement vector $W_{R,0}$, 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 $\theta = \mathrm{atan}(y/x)$ from the positive
$x$-axis, the white card-dependent rotational displacement vector is
given by:

\begin{equation}
 W_{x,y} = r \cdot \left[ \begin{array}{cc}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{array} \right] \times \colvector{\cos\alpha-1}{\sin\alpha}
\label{eqn:Wrot}
\end{equation}

\noindent 
The black card-influenced rotational displacement vector $B_{x,y}$ 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_{x,y} + d_b W_{x,y}}{d_w + d_b}
\label{eqn:netrot}
\end{equation}

The result of this equation is shown in the leftmost plot in
Figure~\ref{fig:distortm25}.  This purely rotational displacement does
not accurately capture the distortion of relative field locations
observed by Fenton et al.  When the two cards move closer together,
all field centers are displaced slightly toward the cards; when the
cards move apart, field centers are displaced away from the cards.
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 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 the center plot of
Figure~\ref{fig:distortm25}.

The total displacement of a point is the sum of its rotational and
translational displacements, and is shown in the rightmost plot in
Figure~\ref{fig:distortm25}:

\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 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 fields will
rotate without distortion, i.e., they rotate in the room frame to
maintain their ``standard'' position relative to the remaining card.
The original formulation of the model accomplished this by assigning
an effectively infinite distance to the missing card, but this is
unnecessary in our revised scheme.  Defining the reference point to be
in ``standard position'' whenever a card is missing gives $\alpha=0$
for Equation~\ref{eqn:Wrot}.  The values for $d_w$ and $d_b$ in
Equation~\ref{eqn:nettrans} are thus unimportant, because $W_c$ and
$B_c$ are null vectors.  The model is already insensitive to rotations
of the cards and cylinder relative to the room frame, so setting the
reference point as if the missing cue card were in its standard
position relative to the remaining card poses no problem.

A rat could implement this operation without explicitly calculating a
``reference point'' by using the one visible cue card to reset its
head direction (HD) system.  \citetext{skaggs95nips} describe an
attractor-based model of the HD system where feature detectors tuned
to egocentric bearings of individual landmarks develop projections to
selected HD units as the animal explores the environment.  Once these
projections are established, they keep the system aligned with the
environment.  The nature of these attractor networks is such that when
two cards are present and have been moved together or apart by a
modest amount, so that they provide somewhat conflicting head
direction cues, the network will average their influence.  When only a
single card is present, it can control the alignment of the HD system.

\begin{figure}
\psfig{file=distortp25.eps,width=6in}
\caption{
Calculated displacement vectors when cards are rotated apart by
$25^\circ$: rotational displacement $D_\mathrm{rot}$ from
Equation~\protect{\ref{eqn:netrot}}, translational displacement
$D_\mathrm{trans}$ from Equation~\protect{\ref{eqn:nettrans}}, and
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 of 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 uses a probabilistic method to determine 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 to 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 firing field
map is distorted by cue card movement.

Let the probability distribution generated by a landmark observation
be a Gaussian function of distance.  If the rat perceives its distance
to landmark $i$ to be $v_i$, then the probability that its actual
distance to the landmark is $d_i$ is distributed across a Gaussian
annulus (in unobstructed space; the portion within the cylinder forms
an arc) with a peak at distance $v_i$ and a variance $\sigma^2$ which
we assume is proportional to $v_i^2$:

\begin{equation}
p(d_i | v_i) = \frac{1}{A_i}
 \exp \left[ \frac{-(d_i - v_i)^2}{v_i^2} \right]
\label{eqn:distprob}
\end{equation}

\noindent where $A_i$ is the area under the annulus.
Making $\sigma$ proportional to the perceived distance to the
landmark causes distributions to scale in accordance with Weber's
Law\footnote{Weber's Law, a fundamental precept of psychophysics,
states that the magnitude of the just noticeable difference in a
stimulus is proportional to the stimulus intensity.}.  Scaling the
width of each gaussian based on distance from the controlling landmark
was also used in the place field model of {}\citetext{okeefe96nature}.

Within the cylinder, the set of locations at distance $d_i$ from
landmark $i$ forms an arc centered on the landmark.  A single landmark
is thus 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 truly 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 fixed width.  Of course, the rat might not
assume that the width is fixed.  We will treat the landmarks as
independent for purposes of evidence combination, as a naive observer,
unaware that a card's two edges should move in unison, might do.  For
our maximum likelihood estimator, we therefore approximate the joint
probability distribution as:

\begin{equation}
p(x,y | v_1,\ldots,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 cell's firing rate distribution at
different arena locations.  Moving the cue cards alters the observed
combinations of $v_i$'s and thus changes the entire distribution.

The joint probability distribution $p(x,y|v_1,\ldots,v_4)$ need not be
computed directly.  It suffices to calculate $\calL_{xy}$, the log
likelihood, which has the advantage of eliminating the exponential
functions and replacing a product with a sum:

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

{}\noindent The maximum of the log likelihood will be found at the
same location as that of the underlying probability distribution.  To
find the location $[x^*,y^*]$ of the peak of the probability
distribution we eliminate constant terms and calculate:

\begin{equation}
[x^*,y^*] = \arg\!\max_{x,y} \sum_{i=1}^4 \frac{-(d_i^{xy}-v_i)^2}{v_i^2}
\label{eq:argmax}
\end{equation}

An interesting consequence of using Weber's law scaling is that nearer
landmarks have greater influence on the location of the peak.  To see
this, consider the one-dimensional case with landmarks at locations
$l_1$ and $l_2$ on the real line.  At any point $x$, the distances to
these landmarks are $d_1^x = x-l_1$ and $d_2^x = x-l_2$.  Suppose we
move the landmarks and try to estimate our position based on the observed
distances $v_1$ and $v_2$, using log likelihood:

\begin{equation}
\calL_{x} = -(d_1^x-v_1)^2/v_1^2 - (d_2^x-v_2)^2/v_2^2
\end{equation}

The local maximum, attained when $d\calL_x/dx = 0$, is 

\begin{equation}
x^* = \frac{(l_1+v_1) v_2^2 + (l_2+v_2) v_1^2}{v_1^2+v_2^2}
\end{equation}

\noindent which bears a strong resemblance to Equation~\ref{eqn:netrot},
substituting $l_1+v_1$ for $W_{x,y}$, $v_1^2$ for $d_w$, $l_2+v_2$ for
$B_{x,y}$, and $v_2^2$ for $d_b$.  Equation~\ref{eqn:netrot} was
chosen by Fenton et al. to give greater weight to nearby landmarks.

Unlike the vector field model, the maximum likelihood model operates
in room coordinates and makes no assumptions about head direction
reset.  When the two cards rotate by equal amounts, all points rotate
around the center by a corresponding amount.  When a card is removed,
we simply omit the corresponding terms from the sum; there is no
inconsistency among the remaining cues so the fields rotate without
distortion.  But when the cards move closer together or farther apart,
the four Gaussian arcs defined by the individual 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 calculated $v_i^\prime$
values for the rotated landmarks viewed from all points on the grid.
We then selected an evenly spaced subset of points from the interior
of the cylinder, and for each point $[x,y]$ representing a firing
field center, we found $[x^*,y^*]$, the peak of
$p(x,y|v_1^\prime,\ldots,v_4^\prime)$.  This was the grid location of
the cell's firing field center with the cards rotated.  We then drew a
vector from $[x,y]$ to $[x^*,y^*]$.

\begin{figure}
\begin{tabular}{cc}
(a)\psfig{file=bayes4dm25.eps,width=2.8in} &
(b)\psfig{file=bayes4am25.eps,width=2.8in} \\[4em]
(c)\psfig{file=bayes4bm25.eps,width=2.8in} &
(d)\psfig{file=bayes8adm25.eps,width=2.8in}
\end{tabular}

\caption{Maximum likelihood estimation of firing field movement 
when the cards are rotated 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 \citetext{fenton00jgp2}
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 in the eastern portion of the arena,
around the vertical line joining the left edge of the white card to
the right edge of the black card.  The vectors suddenly switch
direction in this region.  This is the result of
Equation~\ref{eq:argmax} choosing a single location for the maximum
value of $\calL_{xy}$.  Points on the line actually have two maxima,
one on either side.  As one moves off the line in either direction,
the symmetry is broken.  These unexpected results are a consequence of
the geometry of the arena (a concave, symmetric interior with
landmarks distributed asymmetrically around the edges) 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 them.  We explored this
possibility in a second model.  With four visible landmarks there are
six possible landmark pairs, but for simplicity we used just the four
pairs of circularly adjacent landmarks, since the additional pairs
would only provide redundant information.  (Only three angles are
actually independent; we use four to maintain symmetry in the model.)
Note that the retinal angle between a card's left and right edge
decreases with distance from the card, while the retinal angle between
edges of the white card and the black card is a function of both
distances to the cards and the card separation angle.

Let $u_{ij}$ be the rat's perceived angle between landmarks $i$ and
$j$, and let $a_{ij}^{xy}$ be the actual bearing difference 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 perceived angle between a pair of distinct
landmarks as:

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

{}\noindent where $K=\sqrt{2\pi\sigma_a^2}$ is a normalization
constant, and the variance $\sigma_a^2$ may either be fixed, or
proportional to $u_{ij}$.  Note: for circular values such as bearings,
a circular distribution should be used in place of the normal
distribution.  But here our feature detectors are really measuring the
{\em lengths} of the arcs between pairs of landmarks.  Although these
values are bounded by the circumference of the arena, they are not
periodic: an arc subtending a little more than {}\mdeg{0} of the
visual field is not similar to an arc subtending almost {}\mdeg{360}.
So a normal distribution is used.

\comment{
von~Mises distribution~\cite{evans00}, giving a probability density
function proportional to $\exp(\beta cos(a_{ij}^xy-u_{ij}))$, where
$\beta$ is a ``mixing parameter'' that controls the sharpness of the
tuning curve.  But the difference is insignificant for our purposes,
and the normal is easier to compute.}

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 change fairly slowly with position, producing a shallow
gradient with broad peaks.  When the two cards move in a non-rigid
fashion, the location of a firing field peak 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.  Therefore angle-based
features are also not ideal for determining position in the cylinder.

The single landmark feature that produced the best firing 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{taube90a}.  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{taube90b}.  Therefore, to
compute allocentric bearings in our maximum likelihood model, we
assume that the animal's heading reference ({\em East}) is defined as
mid-way between the white and black cue cards, just as in our vector
field model.  Moving the cards together or apart by equal but opposite
amounts leaves the reference point unchanged, and thus does not affect
the alignment of the head direction system, although individual
landmark bearings will of course shift.  Rotating the cards by
identical amounts rotates the heading reference as well, so the rat
does not notice any bearing change in that situation either.

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, firing
field centers can rotate but will 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}b).  Combining the log 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 set $\sigma_a$
to unity in this case.  The locations of peak firing were estimated
by:

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

Similar results were obtained using distance plus allocentric bearing.

\begin{figure}
\begin{tabular}{cc}
(a)\psfig{file=bayes4dp25.eps,width=2.8in} &
(b)\psfig{file=bayes4ap25.eps,width=2.8in} \\[4em]
(c)\psfig{file=bayes4bp25.eps,width=2.8in} &
(d)\psfig{file=bayes8adp25.eps,width=2.8in}
\end{tabular}

\caption{Maximum likelihood estimation of firing field movement
when the cards are rotated 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 various versions of the
maximum likelihood model when the cards are moved apart by \mdeg{25}.
Once again, a combination of distance and angle features produces the
best results.  Using this combination, the mean horizontal
displacement of firing fields was $+6.56$~cm when the cards were moved
together, vs.  $-7.32$~cm when they were moved apart.  These values
are somewhat larger than the predictions reported by
\citetext{fenton00jgp2}: $+5.59$~cm for cards together and $-5.95$~cm
for cards apart.  They are also larger than our own numerical
simulations of Equation~\ref{eqn:nettot} using a grid with 1~cm
resolution ($+4.71$~cm together, $-5.09$~cm apart).  But in all three
simulations, the magnitude of the displacement in the cards-together
case is approximately 90\% of that in the cards-apart case.  In the
maximum likelihood model, the larger displacement values can be
attributed to the particular choice of feature values selected (angle
plus distance) and their relative weightings.

\comment{*** Note: I calculated the horizontal displacement by
setting gridstep=1 and calling bayes\_common, then taking the mean of
the non-NaN values of xd.}

\section{Attractor Bumps}

Dynamical systems, or ``attractor bump'' networks, are a popular
approach to modeling aspects of hippocampal place cells
{}\cite{samsonovich97jns}, and have been widely adopted by hippocampal
modelers {}\cite{redish98nc,redish99book,doboli00nc,kali00jns}.
One-dimensional attractor models have been used to model the head
direction system {}\cite{zhang96jns,redish96network,goodridge00jnp},
orientation tuning in visual cortex {}\cite{benyishai97}, and the
oculomotor system {}\cite{seung96}.  Two-dimensional attractor
networks have been proposed as models of hippocampus, superior
colliculus {}\cite{droulez91,pouget02nature} and motor cortex
{}\cite{lukashin96a}.  Here we examine the ability of an attractor
network to function as a deformable map, producing the stretching and
shrinking effects observed in the Fenton and Muller two-card
experiment.

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 firing 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
mechanism for finding the peak of an input signal projected onto the
grid.

To produce visual control of firing fields, the external input may be
taken from a collection of visual feature detectors tuned to landmark
distances and/or bearings.  To model the two-card experiment, 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 with firing field centered at $[x,y]$ received an excitatory
connection from feature detector $F_{i,j}$ or $G_{i,j}$ if the
distance from $[x,y]$ to the $i$th landmark was approximately $r_j$,
or the allocentric bearing of landmark $i$ viewed from $[x,y]$ was
approximately $\phi_j$.

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 rat's perceived distance from its present location to the
$i$th landmark be $v_i$.  Those distance detectors $F_{i,j}$ whose
preferred distance value is close to $v_i$ 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.

The estimate would be exact if the external input to the place cell
grid were a circularly symmetric 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 centered 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 of the input 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.  We
therefore added a feedforward inhibition term $I_{FD}$ 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.  The
amount of this feedforward inhibition was calculated iteratively,
using the procedure in the Appendix.  Biologically, it may be
considered analogous to excitatory projections onto interneurons that
in turn make inhibitory projections onto place cells.

The feedfoward inhibition mechanism is separate from the recurrent
inhibition component of the attractor network itself.  Recurrent
inhibition within the attractor network is important for producing a
stable bump of the correct {\em shape}.  The feedforward inhibition
term ensures that the bump forms in the correct {\em place}.

Another difficulty with the projection from feature detectors to place
cells 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; the feedforward
inhibition assures that the location of the peak of the input is more
important than its overall shape.

\section{Model Details}

Parameter values are given in Table~\ref{tab:params}.  The activity of
the feature detector $F_{i,j}$, tuned to distance $r_j$, in response
to perceived distance $v_i$ from the $i$th landmark, is:

\begin{equation}
F_{i,j}(v_i) = \exp\left( - \frac{(v_i-r_j)^2}{2\sigma_d^2} \right)
\label{eq:F}
\end{equation}

The activity of feature detector $G_{i,j}$, tuned to bearing $\phi_j$,
in response to perceived bearing $b_i$ from the $i$th landmark, is
calculated (using circular subtraction) as:

\begin{equation}
G_{i,j}(b_i) = \exp\left( -\frac{[b_i-\phi_j]^2}{2\sigma_b^2} \right)
\label{eq:G}
\end{equation}

There are 43 distance and 120 allocentric bearing feature detectors
per landmark, giving 652 feature detectors total.  Let $FD_k^{xy}$
denote the activity of the $k$th feature detector (of either type $F$
or type $G$) when the simulated rat is at location $[x,y]$ with the
cue cards in standard position.  Let $i$ be the index of the place
cell with firing field centered at $[x_i,y_i]$.  The strengths of the
connections from feature detectors to the $i$th place cell,
$w^{EF}_{ik}$, are set equal to the feature detector activations when
the rat is at the location that is to be the place field's center:
$w^{EF}_{ik} = FD_k^{x_i,y_i}$.

\begin{table}
\begin{tabular}{lr}
{\bf Parameter} & {\bf Value} \\
Number of place cells & 2025 ($45\times 45$) \\
Number of landmarks (card edges) & 4 \\[1em]

Distance FDs per landmark & 43 \\
Distance values $r_j$ & $2j$ cm, $0\leq j<43$ \\
Distance std. dev. $\sigma_d$ & 2 cm \\[1em]

Bearing FDs per landmark & 120 \\
Bearing values $\phi_j$ & $3j$ deg., $0\leq j<120$ \\
Bearing std. dev. $\sigma_b$ & $6^\circ$ \\[1em]

Feature detector projection $w^{EF}_{ik}$ & per Eqs.~\ref{eq:F}, \ref{eq:G}, and $w^{EF}_{ik} = FD_k^{x_i,y_i}$ \\
Feed-forward inhibition $I_{FD}$ & per App. A \\
Recurrent excitation $w^{EE}_{ij}$ & per Eq.~\ref{eq:wEE} \\
Recurrent excitation scale factor $k_{EE}$ & 0.1125 \\
Recurrent excitation std. dev. $\sigma_{EE}$ & 3.1818 \\
Recurrent inhibition $w^{EI}$ & $-0.35$ \\
Inhibitory interneuron drive $w^{IE}$ & $0.12$ \\
Interneuron self-inhibition $w^{II}$ & $-1.6$ \\[1em]

Initial place cell drive $S_i(0)$ & $6\left[\sum_k w^{EF}_{ik} FD_k - I_{FD}\right]_+$ \\
Initial interneuron drive $SI(0)$ & 3.5 \\
Time constants $\tau_E$, $\tau_I$ & 0.0015, 0.004 \\
Time step $\Delta{}t$ & 0.001
\end{tabular}
\caption{Attractor model parameters.}
\label{tab:params}
\end{table}

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 38
units in diameter, centered on the grid origin.  Each grid unit
therefore covered a surface of 4~cm$^2$.  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 the feature
detectors.  Shunting inhibition was used because it improves the
stability of the attractor \cite{kali00jns}.  The feature detector
input included a previously mentioned feedforward inhibition
term. $I_{FD}$, and was thresholded at zero.

\begin{equation}
V_i(t+\Delta{}t) = \sum_j w^{EE}_{ij} S_j(t) + 
	w^{EI} SI(t) V_i(t) + \left[ \sum_k w^{EF}_{ik} FD_k - I_{FD} \right]_+
\label{eq:Vi}
\end{equation}

$V_i(t)$ is analogous to the cell's membrane potential.  The cell's
firing rate $F_i(t)$ is equal to this value thresholded at zero.  The
integral of firing rate over time is the cell's synaptic drive
$S_i(t)$, the influence it exerts on other cells, governed by a time
constant $\tau_E$ \cite{pinto96}:

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

\begin{equation}
S_i(t+\Delta{}t) = S_i(t) + (-S_i(t)+F_i(t)) \frac{\Delta{}t}{\tau_E}
\end{equation}

A global inhibitory unit receives excitation from all place cells and
makes recurrent inhibitory projections back to them and to itself.
The equations for this inhibitory unit are:

\begin{equation}
VI(t+\Delta{}t) = w^{IE} \sum_j S_j(t) + w^{II} SI(t)
\end{equation}

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

\begin{equation}
SI(t+\Delta{}t) = SI(t) + (-SI(t)+FI(t)) \frac{\Delta{}t}{\tau_I}
\end{equation}

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

\begin{equation}
w^{EE}_{ij} = k_{EE} \,\, \exp \left( -d_{ij}^2/\sigma_{EE}^2 \right)
\label{eq:wEE}
\end{equation}

For each of the four landmarks there were 43 distance-based Gaussian
feature detectors, tuned to even distance values from 0 to 84~cm.
$\sigma_d$ 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 $24/(24+d)$,
where $d$ is the distance in cm to the landmark, so the relative
weights from distance-based feature detectors varied from 1.0 down to
0.22.  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.  Figure~\ref{fig:FDprojection}
shows typical activation patterns that distance and bearing-based
feature detectors transmit to the place cell grid.

\begin{figure}
\begin{tabular}{ll}
\psfig{file=distance-arcs-0-10.eps,width=2.8in} &
\psfig{file=bearing-cones-11-7.eps,width=2.8in}
\end{tabular}
\caption{Feature detector activity projected onto the place cell grid
with the cue cards in standard position.  Left: distance-based
detectors produce gaussian arcs centered on the four card edges.
Right: allocentric bearing-based detectors produce gaussian cones
emanating from the card edges.}
\label{fig:FDprojection}
\end{figure}

\section{Results from the Attractor Model}

\begin{figure}
\centerline{\psfig{file=att-m25.eps,width=4in}}
\caption{Map distortion in the attractor model with the cards rotated closer together
by \mdeg{25}.
}
\label{fig:attm25}
\end{figure}

\begin{figure}
\centerline{\psfig{file=att-p25.eps,width=4in}}
\caption{Map distortion in the attractor model with the cards rotated 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 maximum likelihood
models, though a little less smooth, due in part to the lower
resolution of the grid.  The mean horizontal movement of firing fields
in this model, measured on a grid of 1129 points, was $+6.70$~cm for
the cards together case, and $-6.98$~cm for the cards apart case.
These values are comparable to those of the maximum likelihood model
and larger than the predictions of the vector field model.  The
cards-together value is 96\% of the cards-apart value.

\comment{Horizontal movement was measured by running the simulation,
setting the grid spacing to 1x1, and then computing
mean(Vxy(:,1)).}

\begin{table}
\begin{tabular}{llll}
Statistic & Standard & Cards Apart & Cards Together \\
\underline{Measured} & \underline{Configuration} & \underline{by \mdeg{25}} & \underline{by \mdeg{25}} \\
mean peak activation level & 1.5374 & 1.3852 & 1.3517 \\
standard deviation & 0.0889 & 0.0739 & 0.0860 \\
maximum peak activation & 1.9733 & 1.7993 & 1.7554 \\
minimum peak activation & 1.3252 & 0.9707 & 0.9430 \\
\end{tabular}
\caption{Distribution of peak activation levels (in dimensionless units)
for all 1129 place cells within the arena in the attractor network
model, with the cue cards in standard position or rotated apart or
together by \mdeg{25}.}
\label{tab:peakrate}.
\end{table}

The attractor network model also reproduces an independent finding of
\citetext{fenton00jgp1}, a reduction in peak firing rates when the
cards were rotated together or apart, as shown in
Table~\ref{tab:peakrate}.  Peak activation levels in dimensionless
units for three cells are shown in Figure~\ref{fig:fields}.  The mean
peak activation (over all 1129 cells) declined by 10\% in the cards
apart condition, and by 12\% in the cards together condition.  Fenton
et al. report reductions in the mean centroid firing rate, i.e., the
firing rate of a cell in the centroid pixel of its firing field, of
36\% (apart) and 35\% (together), and reduction in the mean in-field
firing rate of 21\% (apart) and 15\% (together).

The reduction in peak activation in the model is a consequence of the
feature detector input becoming defocused when the cards are moved out
of standard position.  The Gaussian arcs from the various feature
detectors no longer intersect perfectly at a single point, so place
cells at the center of the stimulus bump receive less total input than
before.

\begin{figure}
\begin{tabular}{c@{}c@{}c}
\psfig{file=att-p25-0-0.eps,width=2in} &
\psfig{file=att-n-0-0.eps,width=2in} &
\psfig{file=att-m25-0-0.eps,width=2in}\\[0.94in]
\psfig{file=att-p25-10-14.eps,width=2in} &
\psfig{file=att-n-10-14.eps,width=2in} &
\psfig{file=att-m25-10-14.eps,width=2in}\\[0.94in]
\psfig{file=att-p25-18-0.eps,width=2in} &
\psfig{file=att-n-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) rotated apart by \mdeg{25}, (center) in standard position,  and
(right) rotated together by \mdeg{25}.  The top row shows a 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}

\subsection{Comparison of Distance Weightings}

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
(leftmost plots in Figures~\ref{fig:distortm25} and
\ref{fig:distortp25}.)  A similar effect is obtained in the maximum
likelihood model by giving distance-based landmark features a standard
deviation $\sigma_d$ proportional to the perceived distance $v_i$
(Equation~\ref{eqn:distprob}), 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}, entorhinal cortex cells are
tuned to both distance and egocentric bearing to walls, and the
distance tuning is again sharper for closer walls.  It is also sharper
for walls behind the animal versus those 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 thus likely
to have a more accurate distance estimate based on path integration.)
The models described here use allocentric bearing, not relative
bearing, so they do not distinguish landmarks ahead of vs. behind the
animal.

The boundary vector cells (BVCs) of \citetext{hartley00hc} use a
response function that is a {\em product} of two Gaussians, one tuned
to distance and one to allocentric bearing.  BVC firing rates are
calculated by integrating this response function over all points on
the surfaces surrounding the rat.  We could instead take the firing
rate to be the product of the Gaussian response functions at the
specific bearing and distance of one point landmark.  Compare this to
our maximum likelihood and attractor network models which compute the
{\em sum} of distance-based and bearing-based tuning functions.  The
log of a product being equal to the sum of the logs, the two
approaches appear similar, and we would expect similar map deformation
results.

However, BVCs are not place cells.  Hartley et al.'s place cells
compute a thresholded linear combination of two or more BVCs, an extra
layer of processing our model lacks.  This appears to be necessary to
derive spatially compact place fields when BVC's use entire walls as
landmarks.  In contrast, our attractor network model utilizes
recurrent connections and network dynamics to produce its roughly
gaussian shaped firing fields, from inputs with quite varied shapes.

One drawback of using only points as landmarks is that our model does
not produce crescent-shaped firing fields along cylinder walls as the
Hartley et al. model does.  Such fields have been reported by
{}\citetext{muller87a}.  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 maximum likelihood model, distance-based feature
detectors in the attractor network model require a fixed variance,
because otherwise, feature detectors tuned to large distance values
would supply excitation to broad swaths of place cells, which would
deform the shape of the input governing the location of the attractor
bump.  The maximum likelihood model was unaffected by broad input
excitation because it simply picked the single point of maximum
activation 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.  {}\citetext{pouget98nc} observed that their recurrent
network best approximated a maximum likelihood estimator when the
input and output encodings were identical.  In our case, that means
the afferent projections to place cells should approximate the shape
of a stable activity bump.  The feed-forward inhibition mechanism
discussed previously helps to cut off the tails of an elongated input
pattern, but is not sufficient in itself to assure a well-conditioned
input.  Therefore, in order to reintroduce the effect of Weber's law
scaling on a distant landmark's contribution to the activity gradient,
we adopted the inverse distance weighting for distance-based feature
detectors.  

Olypher et al. recently examined the amount of location information in
place cell spikes as rats foraged randomly in the same arena
configuration as the one modeled here.  They found that local
information measures in the center of a place cell's firing field were
higher when that field was close to a cue card than when it was far
from either card {}\cite{olypher03jnm}.  If the feature detectors
driving these place cells also carry greater information content when
tuned to nearby landmarks, this might provide another justication for
weighting nearby features more highly.

\subsection{Firing Field Deformations}

The strength of the recurrent connections relative to feature detector
afferents determines how the attractor network model responds to
inconsistent cues.  Recurrent connections provide both excitation and
inhibition, making firing fields robust against changes in total input
and largely preserving their shapes when the cards are rotated.  If
the recurrent connections are too weak, so that the activity bump with
the cards in standard configuration is mainly a result of feature
detector input, the reduction in input when a cue card is removed
could cause the attractor bump to collapse.  And the change in the
input distribution when cards are moved together or apart by a
significant amount could cause gross distortions in firing field
shapes.

\begin{figure}
\centerline{\psfig{file=distortfig.eps}}
\caption{Place cells $i$ and $j$ have firing field centers $c_i$ and $c_j$
that move to $c_i^\prime$ and $c_j^\prime$ when the cards are rotated.
As a consequence of map distortion, when the rat is at $c_i^\prime$
with the cards rotated, cell $j$ will be firing at a higher rate than
when the rat was at $c_i$ with the cards in standard position.}
\label{fig:distortfig}
\end{figure}

On the other hand, if recurrent connections are too strong relative to
the external input, that input can only influence where on the grid
the activity bump appears, not its shape.  But even with a rigid bump
shape, the shapes of individual firing fields can still change, since
these depend on how the bump moves over the map.  In fact, the map
distortions visualized in Figures~\ref{fig:attm25} and
{}\ref{fig:attp25} as shrinking and stretching of the grid of firing
field centers necessarily entail some change in firing field shapes.
To see this, consider two place cells $i$ and $j$ whose firing field
centers $c_i$ and $c_j$ move closer together, to locations
$c_i^\prime$ and $c_j^\prime$, when the cards are rotated together
(Figure~\ref{fig:distortfig}.)  Assuming the shape of the activity
bump remains rigid due to strong recurrent connections, the firing
rate of cell $j$ when the rat is at location $c_i^\prime$ in the
rotated environment would be the same as its firing rate when the rat
is at $c_i$ in the standard environment.  But since $c_i^\prime$ is
closer to cell $j$'s firing field center $c_j^\prime$ then $c_i$ is to
$c_j$, cell $j$'s firing field has changed.

\subsection{Predictions About Larger Card Movements}

{}\citetext{knierim02jns} argues that recurrent connections must be
weak relative to external inputs because in a double cue rotation task
where local and distal cues rotated in opposite directions by
substantial amounts, some cells followed the local cues, some followed
the distal cues, some developed split firing fields, and some remapped
entirely.

The cue card manipulations studied here are sufficiently subtle that
they do not trigger remapping.  Moreover, in our models all cells are
influenced by both cue cards; place fields do not dissociate into two
sets, one following the white card and one the black card, when the
cards move relative to each other.  But moving the cards by a greater
amount must eventually exceed some threshold beyond which the vector
field transformation of \citetext{fenton00jgp2} no longer applies.
Note that rotating the cards apart by {}\mdeg{90} is equivalent to a
mirror image reflection of the arena, where the white and black cards
swap places.  Unlike in the double cue rotation experiments
\cite{tanila97hc,knierim02jns,brown02jnp}, where the discordance is
between local cues on a track vs. distal cues on the walls, here the
discordant cues are of the same type and presumably equal in salience.
We would therefore not expect to see place fields dissociate, as such
an effect has never been reported with homogeneous cues.  Instead, we
predict that at some critical amount of card rotation between 25 and
90 degrees, either a complete remapping will occur, or one card will
lose its influence over firing fields, and the fields will rotate with
the other card without deforming.

Our current attractor-based model produces neither of these effects.
It cannot undergo remapping because there is only one map stored in
the recurrent connections.  Neither can it reject the influence of one
cue card when the inputs from the two cards are so far apart that they
do not overlap.  A spatially localized stimulus applied far from the
flank of an attractor bump should be ignored due to recurrent
inhibition {}\cite{redish99book}.  However, the feature detectors in
our attractor model of the cylinder have broad projections (see
Figure~\ref{fig:FDprojection}), so rotating the cue cards still
produces inputs that intersect with and influence the bump location.
Thus, for the rat to ignore one card when the separation is increased
to 90 degrees, some attentional mechanism not considered here would
have to come into play.

\subsection{Summary}

We have argued that the maximum likelihood formalism gives a
satisfying theoretical account of map deformation, in which evidence
from multiple independent cues is combined arithmetically to derive a
probability distribution for the location of the animal.  We assume
that the activity of hippocampal place cells reflects this probability
distribution.  The good agreement between the attractor network and
maximum likelihood models supports previous observations that
attractor networks can efficiently implement maximum likelihood
reasoning {}\cite{pouget98nc,deneve01nature}.  If rats make near
optimal use of available landmarks, they may do so via a network of
this sort.

However, the success of both the maximum likelihood and the attractor
network models was found to depend on the choice of feature detectors
used, since the distribution of landmarks is not symmetric.  For the
maximum likelihood model the best choice was an equally-weighted
combination of distance and angle features.  For the attractor network
model, a combination of distance and allocentric bearing features gave
the best performance.  These models and the vector field model all
predict larger average horizontal displacements of firing field
centers in the cards-apart case than in the cards-together case, a
natural consequence of the landmark geometry.

The maximum likelihood and attractor network models predict larger
horizontal displacements than the vector field model in both the
cards-together and cards-apart cases, but this might be remedied by
further tweaking of the feature detectors and their relative
weightings.  The vector field model in turn predicts a significantly
larger displacement in the cards-together case ($+3.80$~cm) than was
observed experimentally by Fenton et al. ($+1.80$~cm).  None of the
models can explain this discrepancy.  

Another area where all the models disagree with the experimental
observations is the displacement of fields at the 9 o'clock position,
i.e., close to the wall at the western edge of the arena.  Fenton et
al.  report almost no horizontal displacement of firing fields in this
region, while the models predict substantial displacements.  A
potential explanation for this result is that the rat is using
distance from the arena wall as another source of location
information.  In most locations this is a less salient cue than the
cue cards, but at the 9 o'clock position the cards are maximally
distant and the wall is close, so it may receive greater weight and
serve to ``anchor'' the firing field.  Note that adding distance from
the wall as another type of landmark feature, combined with
allocentric bearing information, would allow us to derive
crescent-shaped place fields.

Finally, these models raise the question: what is the threshold for
cue card motion beyond which the hippocampus either remaps or ignores
one of the cards?  We are currently investigating a Bayesian model
selection account of remapping that may shed light on this question
{}\cite{fuhs03ns}.

\section*{Acknowledgments}

The authors thank two anonymous referees for helpful comments on the
manuscript.

\clearpage
\section*{Appendix.}

The feedforward inhibition term $I_{FD}$ produces a more focused and
bump-like afferent input from the feature detectors to the place
cells.  It is calculated iteratively by making a large initial guess
for $I_{FD}$ and then repeatedly reducing it by 10\% until at least 10
grid cells receive feature detector activation of at least 0.4.  The
quantity $B_i$ below denotes the feature detector afferent input for
place cell $i$, and $C_i$ is this afferent input with feedforward
inhibition and thresholding applied.  $C_i$ is equal to the second
summation in Equation~\ref{eq:Vi}.  The variables $i$ and $k$ range
over place cells and feature detectors, respectively.  The expression
{\em count}$_i(p)$ counts the number of place cells satisfying
predicate $p$.\\[1em]

\begin{enumerate}
\item Calculate $ \displaystyle B_i = \sum_k w^{EF}_{ik} FD_k $ for each place cell $i$.

\item Set initial guess for the inhibition term $I_{FD}$ to $\displaystyle \max_i(B_i) - 0.5$

\item Define $C_i(I_{FD})$ as $\left[ B_i - I_{FD} \right]_+$ for each place cell $i$.

\item {\bf while} $\,\,$ {\em count}$_i(C_i(I_{FD}) > 0.4) \,\, < \,\, 10 \,\, $ {\bf do}
	 $I_{FD} \leftarrow 0.9\, I_{FD}$

\item {\bf return} $I_{FD}$

\end{enumerate}

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

\end{document}