\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 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 (Deneve
et al., 2002) that maximum likelihood may be an appropriate framework
for describing attractor network behavior.}

\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: 160\\
Number of text pages: \\
Number of figures: 10\\
Number of tables: 1\\~\\
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{Firing 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 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}.

The vector field approach of {}\citetext{fenton00jgp2} was presented
as purely {\em ad hoc}: the equations simply described the desired
results, with no claim that the hippocampus actually derived firing
fields this way.  After reviewing this 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
support previous conjectures \cite{deneve02nature} that maximum
likelihood may be an appropriate framework for describing attractor
network behavior.

\section{Vector Field Model}

The vector field equation 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 than {}\citetext{fenton00jgp2} 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
head direction reset.  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.

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, on 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 $25^\circ$.
(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 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 $\theta = \mathrm{atan2}(y,x)$ 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 Figure~\ref{fig:distortm25}a.
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
Figure~\ref{fig:distortm25}b.

The total displacement of a point is the sum of its 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 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 zero vectors.  The model is already insensitive to rotations
of the cards and cylinder relative to the room frame, so treating a
missing cue card as if it were present in standard position does not
pose a problem.

\begin{figure}
\psfig{file=distortp25.eps,width=6in}
\caption{Calculated displacement vectors when cards are rotated 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 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.  One reasonable
resolution 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]
\end{equation}

\noindent where $A_i$ is the area under the annulus.

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 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.  Nonetheless we will treat the
landmarks as independent for purposes of evidence combination, as a
naive observer, unaware that a card's two edges must move in unison,
might do.  The joint probability distribution is thus:

\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.  We
assume the standard deviation $\sigma$ of each distribution is
proportional to the perceived distance to that landmark, so that
distributions scale in accordance with Weber's Law\footnote{Weber's
Law, a fundamental precept of psychophysics, states that the magnitude
of 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}.

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 \log A_i - \sum_{i=1}^4 \frac{(d_i^{xy} - v_i)^2}{v_i^2}
\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 bounding the arena, with $l_1<l_2$.
At any point $x$ within the arena, the distances to these landmarks
are $d_1^x = x-l_1$ and $d_2^x = l_2-x$.  Suppose we move the
landmarks and try to estimate our position based on observed distances
$v_1$ and $v_2$:

\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}.

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 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}
\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 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 right edge of the white card to
the left 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 located only at 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.  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}{\sqrt{2\pi\sigma_a^2}}
  \exp \left[\frac{-(a_{ij}^{xy} - u_{ij})^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 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 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}
\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 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.

\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,doboli00nc,kali00jns}.
One-dimensional attractor models have been used to model the head
direction system {}\cite{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.

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 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 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 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 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} F_j(t) + 
	w^{EI} FI(t) V_i(t) + \sum_k w^{EF}_{ik} 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} 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 $w^{EE}_{ij}$ between cells
$i$ and $j$ on the torus are a Gaussian function of the distance
$d_{ij}$ between them, measured in cm:

\begin{equation}
w^{EE}_{ij} = 0.1125 \,\, \exp \left( -d_{ij}^2/40.5 \right)
\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=3in} &
\psfig{file=bearing-cones-11-7.eps,width=3in}
\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.

\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}{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) rotated apart by \mdeg{25}, 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 Models}

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 maximum likelihood model by giving distance-based
landmark features a standard deviation $\sigma_d$ proportional to the
perceived 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 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 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 maximum likelihood model and our attractor network model
compute the 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.  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, our attractor network model
utilizes recurrent connections and network dynamics to produce roughly
gaussian shaped firing 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 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 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 maximum likelihood 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 effect of the distance weighting function
is similar to that of increasing the variance in the maximum
likelihood model, in that it reduces the feature detector's
contribution to the gradient of the activation.

The good agreement between the maximum likelihood and attractor
network models supports previous suggestions that attractor networks
can function as maximum likelihood reasoners \cite{deneve02nature},
integrating evidence from multiple sources and cleaning up noise in
the input.

\subsection{Strength of Recurrent Connections}

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}

The map distortion effects visualized in Figures~\ref{fig:attm25} and
\ref{fig:attp25} as shrinking and stretching
of the grid of firing field centers do 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
(Figure~\ref{fig:distortfig}.)  When the rat is at location
$c_i^\prime$, by definition cell $i$ is firing at its maximum rate
with the cards rotated.  The firing field center of cell $i$ is closer
to cell $j$'s firing field center (now at $c_j^\prime$) than when the
rat was at $c_i$ with the cards in standard position.  Hence with the
rat at $c_i^\prime$, cell $j$ will be firing at a rate that is closer
to its peak rate than when the cards were in standard position.  Due
to recurrent excitation and inhibition, a change in one cell's firing
rate affects the behavior of other cells with overlapping firing
fields.  The effects will be seen as changes in peak firing rates and
in the steepness of the gradient of firing rate change with distance
from the center.  The shapes of firing fields of cells in a recurrent
network must therefore change somewhat when the locations of their
centers move relative to each other.

When recurrent connections are too strong relative to the external
input, that input can only influence where on the grid the activity
bump appears, but not its shape.  Place field centers could shift
uniformly in response to card manipulations, but map distortions
caused by relative motion would not be possible.

\subsection{Threshold for Remapping}

{}\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, 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 model 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 vs. distal cues, here the two discordant cues are of the
same type and presumably equal in salience.  We therefore 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.  In attractor models with strong recurrent
connections, a spatially localized stimulus applied to one flank of
the bump will cause the bump to shift in that direction, but the same
input applied far from the bump will be suppressed 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.

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

\end{document}