Navigation and Acquisition of Spatial - IA de Investigación especializada en el MIT

Navigation and Acquisition of Spatial
Knowledge in a Virtual Maze

Sabine Gillner and Hanspeter A. Mallot
Max-Planck-Institut für biologische Kybernetik

Abstracto

n Spatial behavior in humans and animals includes a wide
variety of behavioral competences and makes use of a large
number of sensory cues. Here we studied the ability of human
subjects to search locations, to ªnd shortcuts and novel paths,
to estimate distances between remembered places, and to draw
sketch maps of the explored environment; these competences
are related to goal-independent memory of space, or cognitive
maps. Information on spatial relations was restricted to two
types: a visual motion sequence generated by simulated move-
ments in a virtual maze and the subject’s own movement
decisions deªning the path through the maze. Visual informa-
tion was local (es decir., no global landmarks or compass informa-

tion was provided). Other position and movement information
(vestibular or proprioceptive) was excluded. The amount of
visual information provided was varied over four experimental
condiciones. The results indicate that human subjects are able
to learn a virtual maze from sequences of local views and
movimientos. The information acquired is local, consisting of
recognized positions and movement decisions associated to
a ellos. Although simple associations of this type can be shown
to be present in some subjects, more complete conªgurational
knowledge is acquired as well. The results are discussed in a
view-based framework of navigation and the representation of
spatial knowledge by means of a view graph. norte

INTRODUCCIÓN

Spatial Memory and Cognitive Maps

All organisms capable of locomotion have to deal with
space and spatial relations within their environment.
Simple tasks like efªcient grazing and foraging, camino
integración, or systematic search can be achieved with-
out a mental representation of space, whereas more
advanced competences require the recognition of places
as well as knowledge of spatial relations, como el
distance and bearing of a goal, routes, or conªgurations
of places. en este documento, we address the problem of
exploration, path planning, and navigation in a virtual
laberinto (es decir., in an environment composed of streets and
junctions and with goals that are not generally visible
from the starting position). The knowledge or mental
representation required for this task is studied by be-
havioral experiments with human subjects navigating
in a virtual environment simulated on a computer
pantalla.

Mental representations of space are often called cog-
nitive maps. More speciªcally, there seem to be at least
three more or less independent ideas related to the
concept of a cognitive map:

1. Cognitive map as a spatial reasoning stage. Tol-
man’s original notion (Tolman, 1948) considers the abil-

ity to ªnd (or infer) novel shortcuts as crucial for the
presence of a cognitive map.

2. Cognitive map as a cue integration stage. Espacial
behavior rests on a fair number of different information
sources that are not easily combined. At the stage where
the integration occurs, all information has to be present
in a compatible way. This interaction stage may be called
a cognitive map (see Gallistel, 1990).

3. Cognitive map as goal-independent memory of
espacio. Information about spatial relations can be ac-
quired in neutral (unrewarded) situations and can be
used for goal-directed behaviors later (latent learning). En
contrast, routes are always headed toward a goal. Ver
O’Keefe and Nadel (1978) para una discusión detallada.

Claramente, the above deªnitions are not mutually exclusive
but simply highlight different aspects of cognitive maps.
In terms of the underlying mechanisms, the third notion
seems to allow the most clear-cut distinctions: If spatial
learning is achieved by a mere modiªcation of the
mechanism generating the behavior, it will be stereo-
typed, and we will not call this a cognitive map. Si,
sin embargo, a separate storage is involved that does not
itself produce behavior but is “loaded” into ºexible
mechanisms or referred to during planning, the term
appears to be appropriate. This distinction is akin to the
procedural versus declarative memory dichotomy as dis-
cussed by Squire (1987).

Revista de neurociencia cognitiva 10:4, páginas. 445–463

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Types of Spatial Memory

What types of spatial behavior can be achieved without
a cognitive map, and which ones cannot? We will split
the discussion of this question into three parts, related
to three basic navigational mechanisms: (1) path integra-
ción, (2) approaching recognized views (p.ej., “homing”),
y (3) route and graph memory.

Path Integration

In insect navigation, it has been shown that many impor-
tant tasks can be achieved by some kind of working
memory such as a continuously updated “home vector”
holding the egocentric coordinates of the starting posi-
tion of the current excursion (Wehner & Menzel, 1990).
The current position of some starting point in egocen-
tric coordinates can easily be computed by triangulation
(see Maurer & Séguinot, 1995, for review). Path integra-
tion has been studied in blind and blindfolded human
subjects by Loomis et al. (1993) and in sighted subjects
using virtual reality by May, Wartenberg, and Péruch
(1997). The representation required for path integration
is a simple buffer storing the two vector components
(Mittelstaedt & Mittelstaedt, 1972/73; see also Touretzky,
Redish, & Wan, 1993). Recientemente, McNaughton et al. (1996)
have proposed an alternative mechanism based on hip-
pocampal place cells. In all models, storage is achieved
by neuronal activity (rather than synaptic plasticity), eso
es, by some kind of working or short-term memory.

The memory involved in repetitions of a previously
traveled distance can be based on more elaborate
mechanisms as well. Recent results by Berthoz, Israël,
Georges-François, Grasso, and Tsuzuku (1995) indicate
eso, in humans, the repetition of short distances involves
not just a continuously updated vector buffer but uses a
stored velocity proªle. It is not clear, sin embargo, how this
result extends to longer routes.

An intriguing property of path integration is its close
relation to metric information. Although it is sometimes
assumed that the access to metric information requires
highly sophisticated cognitive maps, it appears that met-
ric is in fact one of the most basic properties of spatial
short-term memory.

Approaching Recognized Views

Recognizing and approaching views (local landmarks)
requires a long-term memory of the view or some of its
características. A strictly associative mechanism for this task has
been proposed by Barto and Sutton (1981). It actually
stores the required approach direction for every position
identiªed by its local position information. A more gen-
eral mechanism for homing that computes the approach
direction from the comparison of current and stored
views has been proposed by Cartwright and Collett
(1982). This scheme involves long-term memory of the

approached view, but not of the required movements,
which are computed. If only one view is to be ap-
proached (homing in a strict sense), memory can be
realized in a procedural and stereotyped way (p.ej., por
some sort of matched ªlter for the home view). Si, cómo-
alguna vez, the same machinery is to be used for many different
approach tasks, the appropriate target views would have
to be “loaded” into a comparison stage as needed. En el
meantime, they must be kept in some long-term memory.
The same argument applies to the somewhat more pow-
erful model by Benhamou, Bouvet, and Poucet (1995)
describing homing behavior in mammals. (See also Franz,
Schölkopf, Mallot & Bülthoff, 1998, for an alternative
implementation of this approach mechanism.)

Routes and Conªgurations

As the basic element of route memory and conªguration
memory, we consider an association of the form

(current view, (movement direction,
expected next view))

(1)

which is illustrated in Figure 1d. Associations between
views and movement decisions have been demonstrated,
Por ejemplo, in bees (Collett & Baron, 1995) and have
been used in the associative schemes of Barto and Sutton
(1981) and McNaughton and Morris (1987). When going
from one view to the next, navigation can initially follow
the movement direction associated with the present
vista. A scheme for robot navigation based on recognized
landmarks and movement behaviors associated with
them has been suggested by Kuipers and Byun (1991).
The additional information on what view to expect next
is required in order to switch to the appropriate ap-
proach behavior when arriving in the neighborhood
(“catchment area”) of that view. Alternativamente, stereotyped
approach behaviors for all known views could be active
in parallel. In this case, they would need to produce a
conªdence measure allowing the selection of the cor-
rect one.

Chains of such association structures implement a
route memory. If different routes are to be learned that
share some common section, the decision at the cross-
roads requires more complicated memory. One way to
think of this memory is to store all possible connections

(current view, (movement direction 1,
expected next view)
.
.
.
(movement direction n,
expected next view))

(2)

and have a separate planning device select one of the
possible movements. A neural network theory for storing
the required information in the form of a labeled graph
has been presented by Schölkopf and Mallot (1995). Para

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Cifra 1. The graph approach to space representation. Top row (a–c): Place graphs. The nodes are places recognized irrespective of body ori-
entation, the links (arrows) between them carry allocentric direction information. Bottom row (d–f): View graphs. The nodes are recognizable
views or other positional information (es decir., depend on the observer’s viewing direction), and the arrows carry directional information relative
to gaze. Each view vi in parts e and f corresponds to a directed connection in parts b and c. From left to right, increasingly more complicated
spatial layouts are shown.

related approaches including hippocampal modeling,
see Muller, Stead, and Pach (1996), Prescott (1996) y
Touretzky and Redish (1996).

An Ecological View of Spatial Memory

en este documento, we will deal mostly with memory of routes
and conªgurations (es decir., relational knowledge of position
and the movements leading from one position to an-
otro). A further breakdown of this problem is given in
Mesa 1. The behavioral competences have been ar-
ranged in order of increasing complexity. Cognitive maps
may be unnecessary for the ªrst two but become in-
creasingly more relevant for the more complex tasks.
The list of sources of information usable in navigation
tasks is probably not complete. De nuevo, there are trivial
cases like pointers, which do not require any spatial
knowledge or map, as well as more complicated cues
that can only be interpreted correctly if map information

is available. Note that we have included “path integra-
tion” as a source of information. Simple path integration
does not require a cognitive map and can thus be con-
sidered a separate mechanism feeding into the map
module. Possible representations acquired by spatial
learning are listed in the third column of Table 1. As was
discussed earlier, the home vector is a form of working
memory. Associations and simple (nonbifurcating) chains
of associations can be learned in a stereotyped or pro-
cedural way. If the same knowledge is to be used in the
pursuit of different goals, a goal-independent, graphlike
memory is required. Finalmente, a topographic map with
coordinates and distances is the richest but rather un-
likely representation.

The View-Based Approach to Navigation

The problem, entonces, is to ªnd the minimum repre-
sentation required to explain an animal’s or human’s

Gillner and Mallot 447

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Mesa 1. An ecological view of cognitive maps. For explanation, see text.

behavioral competences in the presence of a certain
type of environmental information. This idea of eco-
nomic or parsimonious explanations of spatial behavior
is especially well developed for insect navigation.1 For
the type of knowledge studied here (es decir., the expectation
of the next snapshot generated from the current snap-
shot and the intended movement), the most simple ele-
ment is shown in Figure 1a and d and Equation 1. En
Figure 1a it is assumed that places are recognized irre-
spective of the observer’s direction of gaze. Intended
movements are then represented in a global coordinate
sistema (es decir., in relation to an additional system such as
global landmarks or path integration). These elements
can be combined into chains (Figure 1b) or graphs
(Figure 1c). In contrast to this “place-based” approach,
the view-based approach (lower row of Figure 1) como-
sumes that views, rather than places, are recognized and
movements are represented in egocentric coordinates
(es decir., without reference to an independent compass sys-
tema). This approach can be extended to chains and
graphs just like the place-based approach. For a mathe-
matical analysis of the resulting view graphs, see Schöl-
kopf and Mallot (1995).

Both the place- and the view-graph approaches are
local in the sense that bits and pieces of spatial informa-
tion can be accumulated without checking for global
consistencia. They focus on topological properties (estafa-
conectividad); metric relations can be added as labels to the
Enlaces. The main differences between the two approaches

son (1) that metric labels of the place graph have to be
allocentric (world-centered), whereas those of the view
graph are egocentric (observer-centered) y (2) eso
the place graph is planar and symmetric (conocimiento de
a connection implies how to return), whereas the view
graph is not.

It should be noted that the view-based approach to
navigation is closely related to view-based mechanisms
in direction-invariant object recognition (see Bülthoff,
Edelman, & Tarr, 1995). Places and objects can be repre-
sented by their respective views in quite similar ways.
The graph structure resulting for a maze with many
places is generally not planar (cf. Figure 1f), mientras que el
view graphs for object recognition are.

Behavioral Experiments in Virtual Reality

In the work reported in this paper, we chose interactive
computer graphics, or virtual reality (VR), as our experi-
mental method. Previous studies using virtual reality
have focussed on the transfer of knowledge between
different media used for acquisition and testing. Puede,
Péruch, and Savoyant (1995) and Tlauka and Wilson
(1996), Por ejemplo, have tested map-acquired knowl-
edge in a pointing task performed in virtual reality. Tong,
Marlin, and Frost (1995), using a VR bicycle, showed that
active exploration leads to better spatial knowledge than
passive stimulus presentation. Sketch maps produced
after exploration of various virtual environments have

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been studied by Billinghurst and Weghorst (1995). De-
sign principles for constructing easy-to-navigate virtual
environments have been studied by Darken and Sibert
(1996). In the present paper, we use virtual reality to
isolate the various cues used for the build-up of spatial
knowledge and to study the underlying mechanisms. El
advantages of virtual reality for this application are (1)
the high controllability of computer graphics stimuli and
(2) the easy access to behavioral data, such as the sub-
ject’s movement decisions.

Measuring Behavior

Navigation performance can be accessed most directly
by the paths or trajectories that the subjects take during
the exploration. In virtual reality experiments, egomo-
tion is very simple to record, because it is equivalent to
the course of the view port used for rendering the
computer graphics. en este documento, we present a number
of novel techniques for data evaluation that are par-
ticularly suited for the virtual reality experiments de-
scribed.

Stimulus Control

Plan of the Paper

When investigating the information sources used in navi-
gation, it is advantageous to be aware of the exact move-
ment trajectories of the subjects and the visual
information available along these trajectories. This can
easily be achieved with interactive computer graphic
(see “Methods” section). The various parameters of the
sensory input can be easily separated. For instance, en
our experiments, we varied the number of buildings
visible simultaneously in one view without changing the
illumination, etc.. In real-world experiments, such sepa-
rate stimulus conditions are much harder to realize.
Otro
is the
interesting experimental paradigm
modiªcation or exchange of various features of the en-
vironment after learning. Aginsky, harris, Rensink, y
Beusmans (1996) exchanged landmarks after training in
a route-learning task. The effects of landmark exchange
on navigation have been addressed by Gillner and Mallot
(1996).

The method also allows complete control over ves-
tibular and proprioceptive feedback. En nuestros experimentos,
both were completely absent, allowing the effects of
visual input to be studied in isolation.

The virtual reality setup and the procedure used in the
experiments are described in the “Methods” section at
the end of the paper. In the “Results” section, we present
subjects’ trajectories obtained during a search task, como
well as two derived measures for transfer of knowl-
edge between routes and for persistent associations of
views to particular movements. Además, distance es-
timates collected from the subjects after exploration are
compared to theoretical distances from various candi-
date representations. Finalmente, some examples of subjects’
sketch maps will be presented.

RESULTADOS

Exploration and Search

Actuación

Cifra 2 shows an example for the trajectory taken by a
single subject when searching view number 5 from start
vista 15. In the ªrst trial, the subject made a complete
turn in the starting position and then started the explo-
ration via view 17. At view 11, he performed a loop,

Cifra 2. Sample trajectories
for subject GPK (condición:
dark) searching the way from
vista 15 (comenzar) to view 5.
1a–1e: Search trials. In part 1e
the shortest path is found for
the first time completing the
tarea 15 fi
: Accumulated
trajectory from all five trials.
This plot appears again in
part 1 of Figure 3.

5. (cid:229)

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Gillner and Mallot 449

turning 60(cid:176) to face a street and back again. He then
proceeded to view 6, where he performed the same
search behavior. At view 12, the trial was stopped be-
cause the subject deviated from the shortest path by
more than one segment. In the second trial (part 1b in
Cifra 2), he ªnds the goal, though not the shortest
possible path. Curiosamente, the third trial is an exact
replication of the second one. The ªrst time he ªnds the
shortest way is trial 5 (part 1e), which thus terminates
the exploration of that route.

The cumulative trajectory shown in the lower right
panel of Figure 2 appears again in the upper left panel
of Figure 3. The other panels in this ªgure show the
cumulative trajectories for the other routes performed
subsequently in the sequence indicated by the number
in the upper left of each panel. Paths 1 a través de 4 son
excursions, 5, 7, 9, y 11 are returns, y 6, 8, 10, y
12 are novel routes. En general, there is a tendency for lower
error rates in the search tasks performed later. That is to
decir, there is a transfer of knowledge obtained in earlier
searches to the later searches. The decrease of errors is
not monotonic, aunque. Nota, sin embargo, that the three last
routes were found in just ªve trials. In some subjects, No
such decrease of the error rate is found.

Error Rates

Errors were deªned locally as decisions that do not
reduce the distance of the goal. Each movement decision
equals clicking the mouse buttons twice (cf. Cifra 12
in the “Methods” section). Distance to the goal is meas-
ured as the minimum number of decisions needed to
reach it (“decision distance”). De este modo, if a subject enters a
street leading away from the goal, the return from that
street will be counted as a correct decision even though
the current position is not part of the shortest path. En
cases where the correct decision is a 60(cid:176) turn left fol-
lowed by a “go,” the 120(cid:176) turn left would leave the
decision distance to the goal unchanged. This decision
(and the mirror-symmetric case) is also counted as an
error.

Average error rates for each path type are shown in
Cifra 4. For each viewing condition (1 a través de 4; ver
“Methods” section), the excursions, returns, and novel
paths were lumped into groups of four. As mentioned
arriba, the excursions were preformed ªrst, y el
novel and return paths were performed alternatingly,
starting with a return in one group of subjects and
starting with a novel route in a second group. Los datos

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Cifra 3. Traveling frequencies for each view transition for the 12 paths, subject GPK (viewing condition: 1; sequence condition: returns first).
Top row: excursions, fila del medio: returns, fila inferior: novel paths. The overall number of errors decreases at later stages of exploration.

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Volumen 10, Número 4

transfer occurs, the route explored ªrst should have
higher error rates in both cases. We deªne

ER,1 –

ER,2 + EN,1 –
ER,1 + EN,1

EN,2

(3)

If error rates do not depend on position, t will be zero;
if everything is learned already when exploring the ªrst
route, ER,2 and EN,2 will be zero and t evaluates to 1.
Statistical signiªcance of transfer is tested by comparing
the various error rates with the t test.

4 routes ·

For this evaluation, the subjects from viewing condi-
ciones 1 y 2 were pooled because there were no sig-
niªcant differences between the respective error rates
(three-way ANOVA 2 conditions ·
2 género,
F(1, 36) = 0.014, pag = 0.9075). If we take the average over
todo 40 subjects, no signiªcant effect of transfer is found.
Si, sin embargo, only the 20 subjects with the lowest overall
error rate are considered, a transfer effect with t = 0.4
is found (see Figure 5). In this case, 11 subjects were
from the returns-ªrst condition and 9 subjects from the
novel-ªrst condition. The result indicates that the good
navigators show signiªcant transfer learning even from
one route to the next. Transfer across more steps of the
exploration procedure is not visible in this evaluation,
which does not mean that we exclude such a transfer.

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Cifra 5. Average error rates for novel paths and returns in the
novel-first and returns-first sequence conditions. All subjects from
viewing conditions 1 y 2 with an overall error rate below the me-
dian were included in this plot. • novel routes; + returns. For both
returns and novel routes, error rate drops when other routes are ex-
plored before. The transfer coefficient (Ecuación 3) is t = 0.4.

Gillner and Mallot 451

Cifra 4. Total number of wrong movements performed in the dif-
ferent path types (excursion, novedoso, return). Numbers are averaged
encima 20 subjects, error bars are standard deviations. 1 a través de 4:
viewing conditions.

show a learning effect in the sense that excursions take
more errors than the later paths. They also show a clear
effect of condition: Higher visibility results in lower error
tarifas. This general relation does not hold for the com-
parison of conditions 1 y 2, sin embargo, which differ in
the visibility of the neighboring places.

3 path types ·

A three-way analysis of variance (ANOVA, 4 condi-
tions ·
2 genders) of error rate as the
dependent variable reveals signiªcant effects of condi-
ción (F(3, 72) = 17.31, pag < 10-4) and path type (F(2, 144) = 60.65, p < 10-4) but not of gender (F(1, 72) = 0.22, n.s.). Additionally we found an interaction of con- dition and path type (F(6, 144) = 2.66, p = 0.018). The error rates of novel paths are slightly higher than those of the returns (see Figure 4). This effect is not signiªcant, however. Transfer In our procedure, learning occurs on two time scales. During each of the 12 tasks, a route is learned as illus- trated in the example in Figure 2. When switching from one route to the next, part of the knowledge acquired in the earlier routes might be transferred to the new ones. To test this, we deªne a transfer coefªcient t in the following way: Let R and N denote two routes, for instance the ªrst return and novel path, respectively. Our group of sub- jects is divided into two subgroups, one of which ex- plores R ªrst and N second, whereas the second group explores N and then R. As can be seen from Figure 15, four such pairs of routes have been tested. We accumu- late the data from these four tested pairs of returns and novel paths: ER,1 Errors in returns in the returns-ªrst condition EN,1 Errors in novel paths in the novel-ªrst condition ER,2 Errors in returns in the novel-ªrst condition EN,2 Errors in novel paths in the returns-ªrst condition Thus, ER,1 and EN,2 refer to the ªrst group of subjects (returns-ªrst condition) and EN,1, ER,2 to the second. If The transfer-coefªcient t averages the transfer in both directions. In order to look at direction-speciªc transfer effects, let us consider the two subject groups separately. In Figure 5, this would amount to connecting the left open dot with right ªlled dot, etc. It turns out that there is a much stronger improvement in the novel-ªrst con- dition (t test: t = 6.13, FG = 16, p < 0.001) but no improvement in the returns-ªrst condition (t = 0.19, FG = 20, n.s.). Persistence An inspection of Figure 2 shows that the subject re- peated the ªrst route that led to the goal (trial 2) exactly in the following trial. Similarly, it can be seen from Fig- ure 3 that in almost all cases where the subject started from view 15, the ªrst movement decision was LL even though RR would have been just as good. These and similar observations from other subjects lead to the conjecture that at least some movement decisions reºect simple, ªxed associations between the current view and some motion that is performed whenever the view oc- curs. In order to test this in more detail, we analyzed the return statistics of the decision sequences. ˛ Let mh,u {LL, LG, LR, RL, RG, RR} denote the move- ment decision taken at the h th encounter of view u (see Figure 12 for possible movement decisions). We are in- terested in cases where the movement chosen at the h th encounter of view u is the same as that taken at the 1th encounter. More generally, we count the cases 1 and move- {LL, LG, LR, RL, RG, RR}): - where movement j is taken at encounter h - ment i at encounter h Fi,j:= #{(h, u 1,u = j}. = i, mh (i,j ˛ )|mh,u (4) - -1,u ˛ It is important to note two points: First, the two encoun- ters h and h 1 do not occur in subsequent time steps {RL, LR}). Rather, long sequences of other (unless mh views may occur in between. Second, the frequency Fi,j is accumulated over all views. Thus we are looking for an average persistence rate rather than for a view- speciªc one. In the experiments, each search task is repeated until the subject ªnds the shortest possible path. This proce- dure can in itself produce repetition rates above chance if parts of the path are created correctly several times. To exclude this type of error, we restrict our analysis to repetitions where both decisions were false in the sense that they did not lead to an approach to the goal (local deªnition of errors). Finally, we dropped the cases involv- ing the decisions LR and RL because these are quite rare. Example data from individual subjects are shown in Figure 6. The numbers on the diagonal correspond to cases where the same decision was chosen in two sub- sequent encounters even though the decision was false in both cases. From these matrices, we can estimate average movement transition probabilities pij := P(mh, . = D o w n l o a d e d l l / / / / / j t t f / i t . : / / f r o m D o h w t n t p o a : d / e / d m i f r t o p m r h c . p s i l d v i r e e c r t c . m h a i e r d . u c o o c m n / j a o r t c i c n e / - a p d r t 1 i 0 c l 4 e 4 - 4 p 5 d 1 f 9 / 3 1 1 0 8 / 3 4 9 / 0 4 8 4 9 5 8 / 9 1 2 7 9 5 9 8 8 3 5 8 6 2 1 8 / 6 0 1 8 9 p 8 d 9 b 2 y 9 9 g 8 u e 5 s 6 t 2 o 8 n 6 0 1 7 . p S d e f p e b m y b e g r u 2 e 0 s 2 t 3 o n t . f / Figure 6. Examples of return statistics for selected subjects for the four viewing conditions. In the subjects shown for the first three conditions, hypothesis 1 could be rejected in all cases (i.e., persist- ence rate l was significantly different from zero). For condition 4, where error rates were generally very low, hypothesis 1 could not be rejected for any subject. i | mh ,. = j ); averaging is performed with respect to the -1 different views involved. A simple statistical model for these transition probabilities is 1 8 M a y 2 0 2 1 pij = l + pi pi if i = j if i „ j (5) ,0 £ l £ 1 and l where l pi = 1. It states that there is a bias l for the repetition of the movement chosen at the previous encounter. Other than that, the decisions at + S 4 i =1 452 Journal of Cognitive Neuroscience Volume 10, Number 4 h - (cid:236) (cid:237) (cid:238) = 0, true Judged Distances subsequent encounters are independent. If l independence is obtained. This model was ªtted to the data by a maximum likelihood procedure, that is, by minimizing 4 c 2 := (cid:229) i = 1 4 (cid:229) j = 1 (Fij - pijFi(cid:215))2 pijFi(cid:215) (6) where Fi(cid:215) denotes the marginal frequencies S 4 Fij. If j=1 Fi(cid:215) = 0 (empty columns in the matrices of Figure 6), the corresponding terms were deleted from the above sum. The analysis could be applied to data from 67 out of 80 subjects. For the remaining 13 subjects, the number of total errors was too low to ªt the model. Ten of these had been tested in viewing condition 4, where the error rates were lowest. Goodness of ªt was tested with the c 2 test; choosing a signiªcance level of 5%, the best- ªtting model could not be rejected in any of the 67 subjects. Figure 7 shows the histogram for the best-ªtting persisting rates. In order to get an impression of the conªdence inter- vals for l , we repeated the analysis with ªxed l = 0 in Equation 5. By testing goodness of ªt for this model with the c 2 test, 18 cases could be rejected on the 10% level, 9 of which could be rejected on the 1% level as well. The 18 cases are highlighted in Figure 7. Here, persist- ence rate is signiªcantly different from zero. Average persistence rate over all subjects was 0.33, indicating that about one-third of the decisions were based on persistence. A regression analysis of persistence rate l with the overall number of errors for each subject did not reveal a signiªcant correlation. Figure 7. Histogram of best-fitting persistence values l . Dark col- umns: Data from n = 67 subjects where the analysis could be ap- plied (mean = 0.329, s = 0.242). Light columns: Data from n = 18 subjects where l was significantly different from zero. Data are cu- mulated from viewing conditions 1 through 4. Analysis of Variance Following the exploration phase, subjects were asked to rate the distances between 20 pairs of views (see Table 6) on an ordinal scale from 0 to 4. A 20 · 4 · 2 ANOVA on ranks as dependent variable and view pair, viewing condition, and instruction as independent variables re- veals a signiªcant effect of view pair (F(19, 1368) = 38.7, p < 10-4) but no effect of viewing condition (F(3, 72) = 1.63, p = 0.19) or instruction (F(1, 72) = 1.39, p = 0.24). In addition, a signiªcant interaction of view pair and viewing condition was found (F(57, 1368) = 1.74, p = 0.0007). Thus, the instruction (“distance” versus “airline distance”) did not inºuence the result. In the following, we discuss the two signiªcant effects separately. Judged and True Distance Depending on the type of representation acquired, dif- ferent distance estimates could be expected (Figure 8): 1. Walking distance. This is the length of the mini- mum path connecting the two views. Because all seg- ments have the same length in our model, it is equivalent to the number of streets traveled or the number of “Go”-decisions taken, that is, to the graph distance in the place graphs (top row of Figure 1). 2. Decision distance. This is the minimum number of decisions (mouse clicks) required to travel from one view to the other. It is also the number of views encoun- tered and thus the length of a chain of view-movement associations or the graph distance in the view graph. Using the conventions of Figure 12, we take the unit to be a view-to-view transition (i.e., two subsequent mouse clicks). 3. Metric or euclidian distance. Metric distance can be measured in meters in the three-dimensional model underlying the simulation. Figure 9 shows the average distance ratings from all 80 subjects as functions of each of the three possible dis- tance measures. Judged distance increases with true dis- tance, indicating that subjects have in fact learned some of the distance relations. This dependence of the ratings on the actual distance between the views of the pair accounts at least in part for the effect of view pair found in the ANOVA. However, it is not obvious from the data presented in Figure 9 which of the three theoretical measures is closer to the subject’s sense of distance. Correlation with the data is highest for decision distance, whereas standard deviations are smallest for walking distance. One reason for this poor discrimination lies in the fact that all three theoretical measures are closely correlated to each other. Clearer distinctions between the three theoretical dis- Gillner and Mallot 453 D o w n l o a d e d l l / / / / / j t t f / i t . : / / f r o m D o h w t n t p o a : d / e / d m i f r t o p m r h c . p s i l d v i r e e c r t c . m h a i e r d . u c o o c m n / j a o r t c i c n e / - a p d r t 1 i 0 c l 4 e 4 - 4 p 5 d 1 f 9 / 3 1 1 0 8 / 3 4 9 / 0 4 8 4 9 5 8 / 9 1 2 7 9 5 9 8 8 3 5 8 6 2 1 8 / 6 0 1 8 9 p 8 d 9 b 2 y 9 9 g 8 u e 5 s 6 t 2 o 8 n 6 0 1 7 . p S d e f p e b m y b e g r u 2 e 0 s 2 t 3 o n / t f . 1 8 M a y 2 0 2 1 Figure 8. Possible distance measures. a. Walking distance, b. Decision distance, c. Metric distance. For further explana- tion see text. D o w n l o a d e d b and b ﬁ tance measures can be achieved by selecting discriminat- ing view pairs. For example, the decision distance of view pairs a ﬁ a is often different, whereas walking and metric distances of both directions are of course the same. Table 2 shows the results for the four such cases tested in our experiments. The ratings do not depend on direction (i.e., they do not reºect the deci- sion distance). This result was also obtained when sepa- rately comparing the ratings from the different viewing conditions. For the comparison of metric and walking distance, we pooled the ratings from both directions of each view pair, which were shown to be equal in Table 2. The results from discriminating cases are shown in Table 3. The ªrst two rows compare view pairs with equal metric distance and different walking distance. Here, the ratings differ signiªcantly and are in agreement with walking distance. The next two rows of Table 3 show the reverse case (i.e., equal walking distance but different metric distance). Here again, a signiªcant difference is found, which, however, does not agree with metric distance: pair 5 « 19 is rated closer than pair 15 « 19, in contradiction to the metric distances. Thus, the differ- ence between these two ratings does not indicate an inºuence of metric distance. In the last row of the table, a possible alternative explanation is illustrated. Here, a signiªcant difference between two pairs with equal walking and metric distance is found. The pair involving view 15 (“home”) is rated further apart. This might indi- cate a perceptive expansion of the area around view 15, which would also explain the ratings found in rows 3 and 4 of Table 3. However, further experiments are needed to clarify this point. The effects illustrated in Table 3 do not depend on viewing condition, even though the signiªcances are weaker when analyzing the four groups separately. Interaction of View Pair and Condition In the ANOVA including all ratings (Analysis of Variance section), no effect of viewing condition was found, indi- cating that average ratings were the same in all condi- l l / / / / / j t t f / i t . : / / f r o m D o h w t n t p o a : d / e / d m i f r t o p m r h c . p s i l d v i r e e c r t c . m h a i e r d . u c o o c m n / j a o r t c i c n e / - a p d r t 1 i 0 c l 4 e 4 - 4 p 5 d 1 f 9 / 3 1 1 0 8 / 3 4 9 / 0 4 8 4 9 5 8 / 9 1 2 7 9 5 9 8 8 3 5 8 6 2 1 8 / 6 0 1 8 9 p 8 d 9 b 2 y 9 9 g 8 u e 5 s 6 t 2 o 8 n 6 0 1 7 . p S d e f p e b m y b e g r u 2 e 0 s 2 t 3 o n f t / . 1 8 M a y 2 0 2 1 Figure 9. Average distance ratings from all subjects and all viewing conditions (n = 80) plotted as a function of a. walking distance, b. deci- sion distance, and c. metric distance. r: Pearson correlation (ranks treated as numbers). Error bars are standard deviations. 454 Journal of Cognitive Neuroscience Volume 10, Number 4 Table 2. Distance rating in view pairs with different decision distance and equal walking and metric distance (different directions on the same path). dw: walking distance (number of segments). dd: decision distance (number of double mouse clicks required). p significance of difference as obtained from the Mann-Whitney U-test (n = 80). Errors are standard deviations. No significant differences are found. Pair 7ﬁ 10 10ﬁ 7 5ﬁ 15 15ﬁ 5 2ﬁ 15 15ﬁ 2 19ﬁ 15 15ﬁ 19 dw dd 1 1 2 2 3 3 4 4 1 3 2 4 3 5 4 6 Rating 1.48– 1.36 1.51– 1.09 2.33– 1.01 2.43– 0.98 3.24– 0.72 3.33– 0.73 3.14– 0.96 3.29– 0.83 p 0.39 0.31 0.22 0.22 tions. More interestingly, however, a signiªcant interac- tion between viewing condition and view pair could be demonstrated. One possible explanation of this interac- tion is that in one viewing condition, ratings correlate more strongly with true distance than in another view- ing condition. To test this possibility, we calculated Pear- son (product moment) correlations individually for each viewing condition (Table 4). The correlation is smallest Table 3. Distance rating in view pairs discriminating for walking and metric distance. dw: walking distance (number of segments). dm: metric distance (meters). p significance of difference as obtained from the Mann-Whitney U-test (n = 160). Errors are standard deviations. For explanation see text. Pair dw dm(m) 2 « 15 « 5 « 15 « 2 « 2 « 15 « 5 « 2 « 5 « 13 19 15 19 5 15 19 19 13 15 2 4 2 4 3 3 4 4 2 2 173 173 173 173 200 265 173 300 173 173 Rating 1.96– 1.12 3.21– 0.90 2.38– 1.00 3.21– 0.90 2.74– 0.91 3.28– 0.72 3.21– 0.90 2.80– 1.15 1.96– 1.12 2.38– 1.00 p <10–5 <10–5 <10–5 0.001 0.0005 Table 4. Pearson correlation r of distance ratings with walking distance in the four viewing conditions. Condition Correlation, r 1 0.99 2 0.92 3 0.94 4 0.89 in condition 4 (open environment) and highest in con- dition 1 (dark). Additionally, the interaction might be due to condition-dependent rating differences of view pairs with equal true distances; such dependencies have not been found, however (see Judged and True Distance section). Sketch Maps As a ªnal part of the experiments, subjects were asked to draw a map of the explored maze. Three subjects refused to draw a map (i.e., 77 maps have been col- lected). Each row of Figure 10 shows examples from one viewing condition, a good navigator (few errors in ex- ploration phase) on the left side and a poor navigator (many errors in exploration phase) on the right side. In each viewing condition, subjects were ranked according to the number of errors that occurred during the explo- ration phase. The best navigator is ranked 1, the poorest is ranked 20. The position of view 15 (often labeled “MP,” “Institut,” or “Schild” by the subjects) is marked with a circle. It has been chosen as the start of the drawing by 74 out of 77 subjects. In Figure 10 all maps have been oriented in roughly the same way. Good navigators often produce sketch maps that are topologically or even metrically correct and contain identiªable objects. Subject CBK, for example, drew a perfectly correct map except for four missing objects whose locations are included. An equally good map had been drawn by two other subjects, whose maps are not included in the ªgure. A frequent deªcit of maps are omissions or additions of places. For example, subject LIS drew a good map with the rightmost place missing. Subject GBC, on the other hand, included two nonexisting places in a map with otherwise correct connectivity: one in the lower left and one in the spiral part on the right side. This map (GBC) also shows another interesting feature: The regu- lar Y junctions (120(cid:176) ) are represented as T junctions. This locally feasible assumption leads to global problems such as nonexistent intersections. In her drawing, GBC solved this problem by rolling the right branch to a spiral. T junctions were found in 11 out of the 77 sketch maps. In most maps (43 out of 77) three streets meet at each place. Examples of four- and ªve-way junctions appear in the map of VOJ. The number of structurally correct places Np (i.e., junctions meeting nonortho- identiªable three-way gonally) has been determined for each sketch map. The Gillner and Mallot 455 D o w n l o a d e d l l / / / / / j f / t t i t . : / / f r o m D o h w t n t p o a : d / e / d m i f r t o p m r h c . p s i l d v i r e e c r t c . m h a i e r d . u c o o c m n / j a o r t c i c n e / - a p d r t 1 i 0 c l 4 e 4 - 4 p 5 d 1 f 9 / 3 1 1 0 8 / 3 4 9 / 0 4 8 4 9 5 8 / 9 1 2 7 9 5 9 8 8 3 5 8 6 2 1 8 / 6 0 1 8 9 p 8 d 9 b 2 y 9 9 g 8 u e 5 s 6 t 2 o 8 n 6 0 1 7 . p S d e f p e b m y b e g r u 2 e 0 s 2 t 3 o n t f / . 1 8 M a y 2 0 2 1 D o w n l o a d e d l l / / / / / j f / t t i t . : / / f r o m D o h w t n t p o a : d / e / d m i f r t o p m r h c . p s i l d v i r e e c r t c . m h a i e r d . u c o o c m n / j a o r t c i c n e / - a p d r t 1 i 0 c l 4 e 4 - 4 p 5 d 1 f 9 / 3 1 1 0 8 / 3 4 9 / 0 4 8 4 9 5 8 / 9 1 2 7 9 5 9 8 8 3 5 8 6 2 1 8 / 6 0 1 8 9 p 8 d 9 b 2 y 9 9 g 8 u e 5 s 6 t 2 o 8 n 6 0 1 7 . p S d e f p e b m y b e g r u 2 e 0 s 2 t 3 o n t . / f 1 8 M a y 2 0 2 1 Figure 10. Sample sketch maps from eight subjects. Cond: viewing condition. Error-rank: subject’s rank in terms of navigation errors during the exploration phase. Rank 1 indicates lowest number of errors in the respective condition group, and rank 20 indicates highest error num- ber. For further explanation see text. 456 Journal of Cognitive Neuroscience Volume 10, Number 4 “place error,” |Np - 7|, is taken as a measure of map quality. It correlates moderately with the navigation er- rors in the exploration phase (Table 5), indicating that good navigators tend to include the correct number of places in their sketch maps. An ANOVA over the viewing conditions revealed a signiªcant dependence of Np on condition (F(3, 76) = 3.18, p = 0.029). As can be seen from Table 5, this relation is not monotonic. Place error is high in conditions 2 and 4 and low in conditions 1 and 3. Figure 10 also illustrates two kinds of global errors. Subject SFC drew a map with largely correct connectiv- ity that is basically a mirror image of the correct map. The lower left place (with two trucks, identiªed by the word “Laster”), however, has been replaced from its cor- rect position at the upper end of the drawing without any changes to its local structure. Altogether, three mir- ror-inverted maps were drawn. Another example of global errors is the map of subject QFM, who invented closed hexagonal loops. Most maps distinguish places and objects (67 out of 77). In the remaining cases, each junction or place is identiªed by just one object, resulting in a structure resembling a view graph (EMN). Subject TER has a re- duced number of connections, so that the map consists mainly of isolated objects. DISCUSSION Navigation in Virtual Environments The results presented in this paper clearly show that spatial relations can be learned from exploration in a virtual environment even under rather restricted view- ing conditions. Here, we brieºy summarize the most important ªndings: conditions. Bringing the objects closer to the places in condition 3 and removing the occluding hedges in con- dition 4 are reminiscent of zooming out the whole scene with a wide-ªeld lens. May et al. (1997) showed that this zooming does not improve path-integration performance in a triangle-completion task. This discrepancy may char- acterize a difference between path integration and land- mark navigation. Alternatively, it may be due to the marked errors in perspective associated with zooming. The comparison between conditions 1 and 2 (night and day) does not show an improvement in error rates. This is surprising because more information is available in condition 2 (objects at the far end of the streets become visible). This ªnding may be related to the fact that the local structure of the maze becomes more complicated in condition 2, where six objects are visible from each place. The correlation of distance estimates with walking distance in the maze decreases from viewing condition 1 to viewing condition 4 (Interaction of View Pair and Condition section). We take this as evidence that less information is stored in the open environment where navigation need not rely on memory as strongly as in the other conditions. This interpretation is also in line with the observation that sketch maps from condition 4 are not better than those from condition 1. In conclusion, it appears that the amount of knowledge acquired is de- termined not by its availability but by the different needs in the four conditions. Irrespective of this difference of correlation between the viewing conditions, the analysis of discriminating view pairs shows that walking distance is the theoretical distance measure closest to the subject’s ratings. This may indicate that the structure of the representation acquired from all four conditions is the same. Effect of Viewing Condition Transfer and Latent Learning The four viewing conditions differ in the amount of information available to the subjects. Not surprisingly, the number of errors during the search phase decreases as more information is provided. This is in spite of the fact that the ªeld of view was the same in all four Table 5. Average number of structurally correct places Np in the sketch maps and the correlation of |Np - number of errors made in the exploration phase. r coefª- cient of correlation. n = 20. 7| with the Condition 1 2 3 4 Np – s 6.05– 3.70 3.10– 3.24 5.75– 3.21 3.95– 3.45 r 0.61 0.48 0.47 0.35 The overall number of errors was smaller for the later search tasks. For the 50% best subjects, this effect was already clearly visible for the comparison of one search task with the next (Figure 5). If subjects simply learned a set of independent routes (e.g., by reinforcement learn- ing), each search would be a new task and no such transfer would be expected. The knowledge being trans- ferred from one route to the next is not just a route memory but involves the recombination of route seg- ments; this is to say, it is of the conªguration type. Its acquisition is akin to latent learning, because knowledge obtained during one search can be employed later in other, unrelated search tasks. As can be seen from Figure 5, transfer was strong from the novel to the return paths but not the other way around. One possible explanation of this ªnding is that the novel paths are more difªcult than the returns. When considering the shortest possible paths, the novel paths involve 14 different views, 8 of which also occur in the Gillner and Mallot 457 D o w n l o a d e d l l / / / / / j f / t t i t . : / / f r o m D o h w t n t p o a : d / e / d m i f r t o p m r h c . p s i l d v i r e e c r t c . m h a i e r d . u c o o c m n / j a o r t c i c n e / - a p d r t 1 i 0 c l 4 e 4 - 4 p 5 d 1 f 9 / 3 1 1 0 8 / 3 4 9 / 0 4 8 4 9 5 8 / 9 1 2 7 9 5 9 8 8 3 5 8 6 2 1 8 / 6 0 1 8 9 p 8 d 9 b 2 y 9 9 g 8 u e 5 s 6 t 2 o 8 n 6 0 1 7 . p S d e f p e b m y b e g r u 2 e 0 s 2 t 3 o n . / f t 1 8 M a y 2 0 2 1 returns. The returns involve only 9 different views (i.e., almost all of their views are already known from the novel paths). The only view not occurring in the novel paths is the ªnal goal of the returns, view 15. The transfer asymmetry may thus be due to the fact that the novel routes contained more information about the returns than vice versa. The occurrence of transfer from one route to another is also evidence for the presence of goal-independent memory of space (i.e., a cognitive map). Persistence Along with these arguments for conªguration knowl- edge, evidence for simpler types of spatial learning was also found. The persistence rates presented in Figure 7 indicate that at least some of the subjects based a con- siderable part of their movement decisions on simple associations of views with movements. This strategy is efªcient for learning nonintersecting routes but will lead to errors for views at a crossroads where the correct motion decision depends on the current goal. We specu- late that the persistence rate will decrease if longer training sequences are used. Distance Measures Judged distance agrees best with the walking distance in the maze, rather than with metric or decision distance. This result is based on the rather small number of “dis- criminating view pairs” where the theoretical measures disagree. It holds irrespective of the instruction (“dis- tance” versus “airline distance”). The analysis of distance ratings presented here rests on the assumption that dis- tances in different parts of the maze can be compared. As was discussed with respect to Table 3, this need not be the case. Rather, the perceived distances in the vicin- ity of well-known places (home) might be increased. Subject Differences Subjects differed strongly in terms of the number of errors made when searching a goal as well as in the quality of their distance estimates. However, no cluster- ing in different groups can be obtained from our data. In particular, no signiªcant gender differences were found. View-Based Navigation In viewing conditions 1 and 2, subjects had to rely on local views as their only position information. Their per- formance and the transfer learning is therefore view- based in an obvious sense. However, this result does not exclude the possibility that some more complicated rep- resentation of space is constructed from the local view information. Here we summarize the evidence against such a representation (i.e., evidence for a view-based mechanism of navigation). Returns Aren’t Easy After having learned the four excursions, the returns to the starting point along the very same paths are almost as difªcult as novel paths (Figure 4). The advantage on the order of just one error per search task is not sig- niªcant (F(1, 76) = 2.860, p = 0.095). If the subjects acquired a place-based representation of space, it would be the same for excursions and returns, because the corresponding place graphs are symmetric (see Figure 1). In this case, we would therefore expect that returns should be much easier and more reliable than novel paths. The weak difference between the number of er- rors occurring in returns and novel paths seems to indi- cate that this is not the case. It is rather more in line with a view-based mechanism. because the views occurring along the return path are as different from the original views as are any other views in the maze. Recognition and Action The average persistence rate of 32.9% (Figure 7) indi- cates that direct associations of views to movement decisions can be learned. As was pointed out in the introduction, the association pair of view and motion decision is the basic element of a view-based memory of space. Local Information Combined to a Graph? If the representation is in fact view-based, a graph struc- ture is the only representation we can think of that would account for the transfer and planning behavior observed. Independent evidence for a graphlike repre- sentation comes mainly from the sketch maps: as was point out in the Sketch Maps section, maps are often locally correct but globally inconsistent. Also, places with correct local connectivity have been translocated to er- roneous positions. Connectivity can be correct even though metric properties of the sketch maps, such as angles and lengths, are grossly mistaken. The distance estimates do not reºect the decision distance, which is the graph distance of the view graph, but correlate better with walking distance (i.e., the graph distance of the place graph). It therefore appears that we cannot decide between the view- and place-graph repre- sentations at this point. In ongoing work (Mallot & Gill- ner, 1997), this question is addressed with additional experiments. CONCLUSION In our view, the most important result of this study is the fact that conªguration knowledge can be acquired 458 Journal of Cognitive Neuroscience Volume 10, Number 4 D o w n l o a d e d l l / / / / / j f / t t i t . : / / f r o m D o h w t n t p o a : d / e / d m i f r t o p m r h c . p s i l d v i r e e c r t c . m h a i e r d . u c o o c m n / j a o r t c i c n e / - a p d r t 1 i 0 c l 4 e 4 - 4 p 5 d 1 f 9 / 3 1 1 0 8 / 3 4 9 / 0 4 8 4 9 5 8 / 9 1 2 7 9 5 9 8 8 3 5 8 6 2 1 8 / 6 0 1 8 9 p 8 d 9 b 2 y 9 9 g 8 u e 5 s 6 t 2 o 8 n 6 0 1 7 . p S d e f p e b m y b e g r u 2 e 0 s 2 t 3 o n . f / t 1 8 M a y 2 0 2 1 in virtual environments. This is in spite of the fact that the subjects did not actually move but were interacting with a computer graphics simulation. With respect to the high controllability of visual input, this result may well make virtual reality a valuable addition to more realistic ªeld studies, where stimulus control is often a problem. As a novel experimental opportunity, we are presently using object transpositions after learning, which could be done in real environment only with great difªculties (Gillner & Mallot, 1996). With respect to our starting point (i.e., view-based navigation), we think that three conclusions can be drawn: 1. Views sufªce. Map learning is possible if only local (i.e., view information) is provided. In this sense, naviga- tion can be view-based. 2. Graph versus view from above. The representation contains local elements (i.e., a place or view with one or several movement decisions and the respective outcome associated with it). These local elements need not be globally consistent, and they need not combine into a metric survey map. Rather, a graphlike representation is sufªcient to account for our results. 3. Places versus views. It is not clear from our data whether the nodes of this graph are places or views. We have not found evidence that the local views are com- bined into a representation of space independent of the orientation of the viewer. So, a view-based representation seems more likely at this point. METHODS The Virtual Maze (Hexatown) The virtual town was constructed using Medit software and animated with a frame rate of 36 Hz on a SGI Onyx RealityEngine2 using IRIX Performer software. A sche- matic map of the town appears in Figure 11. It is built on a hexagonal raster with a distance between two places of 100 m. At each junction, one object, normally a building, was located in each of the 120(cid:176) angles be- tween the streets; so each place consisted of three ob- jects. In the places with less than three incoming streets, dead ends were added instead, ending with a barrier at about 50 m. The hexagonal layout was chosen to make all junctions look alike. In contrast, in Cartesian grids (city-block raster), long corridors are visible at all times and the possible decisions at a junction are highly un- equal: going straight to a visible target or turning to something not presently visible. The whole town was surrounded by a distant circular mountain ridge that showed no salient features. The mountains were con- structed from a small model that was repeated peri- odically every 20(cid:176) . Subjects could move about the town using a com- puter mouse. In order to have controlled visual input and not to distract subject’s attention too much, move- ments were restricted in the following way. Subjects could move along the street on an invisible rail right in the middle of each street. This movement was initiated by hitting the middle mouse button and was then car- ried out with a predeªned velocity proªle without fur- interact. The ther possibilities for the subject to translation took 8.4 sec with a fast acceleration to the maximum speed of 17 m/sec and a slow deceleration. The movement ended at the next junction, in front of the object facing the incoming street. Sixty-degree turns could be performed similarly by pressing the left or right mouse button. Again, the simulated movement was “bal- listic” (i.e., following a predeªned velocity proªle). Turns took 1.7 sec with a maximum speed of 70(cid:176) /sec and symmetric acceleration and deceleration. Figure 12 shows the movement decisions that subjects could choose from. Each transition between two views is mediated by two movement decisions. When facing an object (e.g., the one marked “a” in Figure 12), 60(cid:176) turns left or right (marked “L”, “R”) can be performed; they will lead to a view down a street. If this is not a dead end, three decisions are possible: the middle mouse button triggers a translation down the street (marked “G” for go), whereas the left and right buttons lead to 60(cid:176) turns. If the street is a dead end, turns are the only possible decision. In any case, the second movement will end in front of another object. An aerial view of Hexatown is shown in Figure 13. It gives an impression of the objects used. The spacing and position of the trees correspond to viewing conditions 1 and 2 (see below). D o w n l o a d e d l l / / / / / j f / t t i t . : / / f r o m D o h w t n t p o a : d / e / d m i f r t o p m r h c . p s i l d v i r e e c r t c . m h a i e r d . u c o o c m n / j a o r t c i c n e / - a p d r t 1 i 0 c l 4 e 4 - 4 p 5 d 1 f 9 / 3 1 1 0 8 / 3 4 9 / 0 4 8 4 9 5 8 / 9 1 2 7 9 5 9 8 8 3 5 8 6 2 1 8 / 6 0 1 8 9 p 8 d 9 b 2 y 9 9 g 8 u e 5 s 6 t 2 o 8 n 6 0 1 7 . p S d e f p e b m y b e g r u 2 e 0 s 2 t 3 o n / t f . 1 8 M a y 2 0 2 1 Figure 11. Street map of the virtual maze with 7 places num- bered 0 through 6 and 21 views numbered 0 through 20. The ring around each place indicates the hedges used in viewing conditions 1 through 3 to occlude distant places. Gillner and Mallot 459 arrangement was such that when entering the circular hedge in conditions 1 and 2, the buildings to the left and right were already outside the observer’s ªeld of view (60(cid:176) ). Thus, the three buildings at one junction could never be seen together in these conditions. In condition 3, the buildings were at a distance of only 15 m from the junction. In this case, all three buildings were seen at once when entering the place. The town was illuminated from the bright sky, except in condition 1, where an exploration at night was simulated. Here, illumination was as with a torch or the headlights of a car and reached about 60 m. Thus, the building at the far end of a street was not visible in this condition. A summary of the viewing conditions and variants of Hexatown is given in Figure 14. Procedure Figure 12. Possible movement decisions when facing the view marked a. L: turn left 60(cid:176) . R: turn right 60(cid:176) . G: go ahead to next place. We used four stimulus conditions with varying de- grees of visibility of the environment. In conditions 1, 2, and 3, a circular hedge or row of trees was placed around each junction with an opening for each of the three streets (or dead ends) connected to that junction. This hedge looked the same for all junctions and pre- vented subjects from seeing the objects at more distant junctions. In conditions 1, 2, and 4, the objects were placed 22 m away from the center of the junction. The Experiments were performed using a standard 19-in SGI monitor. Subjects were seated comfortably in front of the screen and no chin rest was used. They moved their heads in a range of about 40 to 60 cm in front of the screen, which results in a viewing angle of about 35 to 50(cid:176) . The experiment was performed in three phases. In the exploration phase, subjects found themselves facing some view v1. They were then presented with a target D o w n l o a d e d l l / / / / / j t t f / i t . : / / f r o m D o h w t n t p o a : d / e / d m i f r t o p m r h c . p s i l d v i r e e c r t c . m h a i e r d . u c o o c m n / j a o r t c i c n e / - a p d r t 1 i 0 c l 4 e 4 - 4 p 5 d 1 f 9 / 3 1 1 0 8 / 3 4 9 / 0 4 8 4 9 5 8 / 9 1 2 7 9 5 9 8 8 3 5 8 6 2 1 8 / 6 0 1 8 9 p 8 d 9 b 2 y 9 9 g 8 u e 5 s 6 t 2 o 8 n 6 0 1 7 . p S d e f p e b m y b e g r u 2 e 0 s 2 t 3 o n f t / . 1 8 M a y 2 0 2 1 Figure 13. Aerial view of Hexatown. Orientation as in Figure 11. The white rectangle in the left foreground is view 15, used as “home” posi- tion in our experiments. The aerial view was not available to the subjects. Object models are courtesy of Silicon Graphics, Inc., and Prof. F. Leberl, Graz. 460 Journal of Cognitive Neuroscience Volume 10, Number 4 view v2 printed out on a sheet of paper and asked to ªnd this view in the virtual town (task v1 ﬁ v2). When they found the view, feedback was given in the form of a little sign appearing on the screen. If they got lost in the maze (i.e., if they deviated from the shortest possible path by more than one segment), the trial was stopped and another sign announced the failure. This feature of the procedure was included in order not to discourage subjects by long unsuccessful searches. In pilot experi- ments with free exploration (no goals speciªed), learn- ing was much harder. The number of terminations varied from 9.9 times per subject in viewing condition 1 to 1.5 in viewing condition 4; terminations occurred almost exclusively during the excursion phase (see below). If a trial was terminated, or if the way found was not the shortest possible way, the subject was relocated to the starting point and a new trial for the same goal started. The sequence was terminated when the shortest possi- ble way, that is, the way involving the minimal number of decisions (mouse clicks), was found. The whole explo- ration phase contained 12 such search tasks, or ways to be found. The ªrst four ways were excursions from view 15, which served as a “home” position. View 15 showed a poster wall saying “Max-Planck-Institut für biologische Kybernetik.” The following eight searches were either returns to home or novel paths not touching on view 15. The return and novel path tasks were presented alternatingly in two sequence conditions: in the returns- ªrst condition, the ªrst task was a return, whereas in the novel-ªrst condition, the sequence started with a novel path. In both conditions, the four excursions were per- formed prior to both returns and novel paths (Figure 15). In the distance estimation phase, following immedi- ately after the exploration, subjects were presented with pairs of views on the screen. The ªrst view was shown without time limit; after hitting the spacebar, the second view was presented again without time limit. Subjects were asked to rate the distance between the two views on an integer scale ranging from 0 to 4, where 0 meant “very close” and 4 “very far apart.” One of two instruc- tions was used: In the ªrst, subjects were asked to rate according to distance (“Abstand”); in the second, they were told to estimate the airline (metric) distance (“Luftlinie”). After hitting the appropriate number button, the next view pair was presented. Ranking data from a total of 20 view pairs was collected from each subject (Table 6). D o w n l o a d e d l l / / / / / j f / t t i t . : / / f r o m D o h w t n t p o a : d / e / d m i f r t o p m r h c . p s i l d v i r e e c r t c . m h a i e r d . u c o o c m n / j a o r t c i c n e / - a p d r t 1 i 0 c l 4 e 4 - 4 p 5 d 1 f 9 / 3 1 1 0 8 / 3 4 9 / 0 4 8 4 9 5 8 / 9 1 2 7 9 5 9 8 8 3 5 8 6 2 1 8 / 6 0 1 8 9 p 8 d 9 b 2 y 9 9 g 8 u e 5 s 6 t 2 o 8 n 6 0 1 7 . p S d e f p e b m y b e g r u 2 e 0 s 2 t 3 o n t / . f 1 8 M a y 2 0 2 1 Figure 14. Viewing conditions used in the experiments. For each condition, three views ar shown. The views in the top row occur when look- ing from a place (no. 5) into a street. The views in the middle row can be seen during the motion along a street, in this example from place 5 to place 3. The views in the bottom row show an object (view 11) as seen from the corresponding junction (place 3). Object models are cour- tesy of Silicon Graphics, Inc., and Prof. F. Leberl, Graz. Gillner and Mallot 461 In the sketching phase, subjects were asked to draw a sketch map of the virtual town on a sheet of paper. The experiment was run on 80 paid volunteers, 40 male and 40 female, aged 15 to 38. Twenty subjects (10 male, 10 female) took part in each of the four viewing conditions (Figure 14). Within each viewing condition, the group of subjects was split equally (10 to 10) be- tween the two sequence conditions (returns ªrst and novel ªrst, Figure 15), as well as the two instructions for the distance estimation (“distance” and “airline dis- tance”). Acknowledgment This work was supported by the Deutsche Forschungsgemein- schaft, Schwerpunktprogramm Raumkognition. We are grateful to Silicon Graphics Inc. and Prof. F. Leberl (University of Graz) for providing VR-models used in the experiments. We are also grateful to Tordis Philipps for help with the experiments and to Heinrich H. Bülthoff, Bernhard Schölkopf, and Hendrik-Jan van Veen for comments on prior versions of the manuscript. The writing of the ªnal text was supported by a fellowship from the Institute of Advanced Study Berlin to HAM. Reprint requests should be sent to H. A. Mallot, Max-Planck- Institut für biologische Kybernetik, Spemannstr. 38, D-72076 Tübingen, Germany, or via e-mail: hanspeter.mallot@ tuebingen.mpg.de. Notes 1. Bennett (1996) has driven the minimalistic view to the point of denying the very existence of declarative memory of spatial relations. Although this may be true for insects (Wehner & Menzel, 1990), we argue that such memory does exist in humans. The view-graph model is a simple model of declarative memory of spatial conªgurations that builds on features of insect spatial memory. 2. Additional material on Hexatown is available http://www.kyb.tuebingen.mpg.de/links/hexatown.html. from REFERENCES Aginsky, V., Harris, C., Rensink, R., & Beusmans, J. (1996). 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