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Remembering the past and imagining the future:
a neural model of spatial memory and imagery
Patrick Byrne and
Department of Psychology, Neuroscience and Behvaiour, McMaster University; Neil Burgess,
Institute for Cognitive Neuroscience and Department of Anatomy, University College London
Suzanna Becker
Department of Psychology, Neuroscience and Behvaiour, McMaster University; Neil Burgess,
Institute for Cognitive Neuroscience and Department of Anatomy, University College London
Abstract
The neural mechanisms underlying spatial cognition are modelled, integrating neuronal, systems
and behavioural data, and addressing the relationships between long-term memory, short-term
memory and imagery, and between egocentric and allocentric and visual and idiothetic
representations. Long-term spatial memory is modeled as attractor dynamics within medial-
temporal allocentric representations, and short-term memory as egocentric parietal representations
driven by perception, retrieval and imagery, and modulated by directed attention. Both encoding
and retrieval/ imagery require translation between egocentric and allocentric representations,
mediated by posterior parietal and retrosplenial areas and utilizing head direction representations
in Papez’s circuit. Thus hippocampus effectively indexes information by real or imagined
location, while Papez’s circuit translates to imagery or from perception according to the direction
of view. Modulation of this translation by motor efference allows “spatial updating” of
representations, while prefrontal simulated motor efference allows mental exploration. The
alternating temporo-parietal flows of information are organized by the theta rhythm. Simulations
demonstrate the retrieval and updating of familiar spatial scenes, hemispatial neglect in memory,
and the effects on hippocampal place cell firing of lesioned head direction representations and of
conflicting visual and ideothetic inputs.
Keywords
navigation; path integration; representational neglect; hippocampus; computational model
Introduction
One of the most intriguing challenges in cognitive neuroscience is to understand how a
higher cognitive function such as memory arises from the action of neurons and synapses in
our brains. Such an understanding would serve to bridge between the neurophysiological
and behavioral levels of description via systems neuroscience, allowing for the
reinforcement of convergent information and the resolution of questions at one level of
description by inferences drawn from another. Moreover, a theory that bridges the cellular
and behavioural levels can lead to the development of experimental predictions from one
level to another, and improved ability to relate behavioral symptoms to their underlying
pathologies. In terms of developing such an understanding of memory,
spatial
memory
Correspondence concerning this article should be addressed to Sue Becker, Department of Psychology, Neuroscience and Behaviour,
McMaster University, Hamilton, Ontario, Canada L8S 4K1. E-mail: becker@mcmaster.ca.
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provides a good starting point due to the ability to use similar paradigms in humans and
other animals.
We are often faced with the challenging task of deciding how to act in the absence of
complete sensory information, for example, when navigating toward an unseen goal. To
solve such tasks, we must rely on internal representations of object locations within their
environment. Here we attempt to develop a model of the uses of these internal
representations in spatial memory, incorporating data from single unit recording, systems
neuroscience and behavioral studies, and describing how each relates to the other. Central
questions in the cognitive neuroscience of spatial memory concern the frames of reference
used for representations of location, e.g. whether they are egocentric (relative to parts of the
body) or allocentric (relative to the external environment), the durations over which different
representations are maintained, the uses they are put to, and how they interact with each
other. However, there is currently no clear consensus, with various investigators stressing
one or the other type of representation (e.g. cf. Poucet, 1993; Wang & Spelke, 2002). To
address these questions, we propose a general organizational structure for spatial memory
(see also Mou & McNamara, 2002; Burgess, 2006) encompassing encoding and retrieval of
spatial scenes as well as some aspects of spatial navigation, imagery and planning. We then
implement the key components of this structure in a neurophysiologically plausible
simulation, to provide a quantitative model relating behavior to the actions of networks of
neurons. We provide example simulations of four key test situations, showing that the model
can account for aspects of representational neglect, as well as spatial updating and mental
exploration in familiar environments, and place cell firing patterns seen in rats with lesions
to the head direction system and in normal rats navigating through environments that
unexpectedly change shape (Gothard, Skaggs & McNaughton, 1996). First we briefly
review some of the data at each of these levels of description that motivate the design of the
model.
Neuronal representations
Data from electrophysiological recordings in behaving animals provides perhaps the most
direct evidence of the nature of the representations at work in spatial cognition. We start
with the apparently allocentric representations associated with the mammalian medial
temporal lobe. View-invariant hippocampal “place cells” fire selectively for an animal’s
location in space (e.g. O’Keefe, 1976), but show little dependence on the animal’s
orientation during random, open field foraging. We refer to this representation as allocentric,
representing location relative to the environment, even though the location represented is
that of the animal itself. In a linear track place cells tend to be direction specific, however,
when the track environment is enriched with place-unique cues the place cells are much less
directionally selective (Battaglia, Sutherland & McNaughton, 2004). O’Keefe & Nadel
(1978) argue that this collection of place-selective neurons forms the basis of a cognitive
map and provides the rat’s internal allocentric representation of location within the
environment. Evidence for the existence of place cells has also been found in the
hippocampus in non-human primates (Matsumura et al., 1999; Ono, Nakamura, Nishijo &
Eifuku, 1993) and in humans (Ekstrom et al., 2003). The representation of the
complementary spatial information - orientation independent of location - has also been
found; “head direction cells” (see e.g. Taube, 1998) are found along an anatomical circuit
largely homologous to Papez’s circuit (Papez, 1937) leading from the mammillary bodies to
the presubiculum via the anterior thalamus. A representation related to place cells has also
been found in the parahippocampal and hippocampal region of both non-human (Rolls &
O’Mara, 1995) and human primates (Ekstrom et al., 2003): “view-cells”, which fire when an
animal is looking at a given location from a range of vantage points.
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The location of a place cell’s response depends on large extended local landmarks rather
than on discrete objects, while the orientation of the overall place and head direction
representations depend on landmarks at or beyond the reachable environment (see Barry et
al., 2006; Burgess & O’Keefe, 1996; Cressant, Muller & Poucet, 1997). Thus, the location
and shape of the firing fields of hippocampal place cells can be explained if it is assumed
that their firing is driven by the activity of a population of
boundary vector cells
(BVCs)
(Hartley, Burgess, Lever, Cacucci & O’Keefe, 2000; O’Keefe & Burgess, 1996). These
neurons, hypothesized to exist within parahippocampal cortex, show maximal firing when
an animal is at a given distance and allocentric direction from an environmental landmark or
boundary. The direct or indirect reciprocal connectivity of the hippocampal formation and
parahippocampal regions with each other and with the perirhinal cortex (for a review, see
Burgess et al., 1999), an area that is known to be important for object recognition (Davachi
& Goldman-Rakic, 2001; Murray & Bussey, 1999; Norman & Eacott, 2004), probably
allows for the positions and identities of landmarks visible at a particular location to be
bound to that location.
In parallel to the above allocentric representations, egocentric representations, which are
ubiquitous throughout the sensory, motor and parietal cortices, are clearly directly involved
in all aspects of spatial cognition. Sensory representations will be egocentric, reflecting the
reference frame of the receptor concerned (e.g. retinotopic in the case of visual input), while
motor output will reflect the reference frame appropriate for the part of the body to be
moved (see e.g. Georgopoulos, 1988). Coordinating these representations, the posterior
parietal cortices are heavily involved in sensorimotor mappings. Posterior parietal cortex is
known to contain neurons that respond to stimuli in multiple reference frames, especially
areas near or within the intraparietal sulcus. In particular, Galletti, Bataglini & Fattori (1995)
have found neurons in the anterior bank of the parieto-occipital sulcus (V6A) in
ventromedial parietal cortex, that represent the positions of visual stimuli in a craniotopic
reference frame. Also, area 7a contains neurons that exhibit egocentrically tuned responses
which are modulated by variables such as eye position and body orientation (Andersen,
Essick & Siegel, 1985; Snyder, Grieve, Brotchie & Andersen, 1998). Such coding can allow
transformation of locations between reference frames (Pouget & Sejnowski, 1997; Zipser &
Andersen, 1988). Furthermore, head direction selective neurons that exhibit responses tuned
to various different reference frames have been found in the posterior cotices of the rat
(Chen, Lin, Barnes & McNaughton, 1994). Such properties might allow for the
establishment of the angular relationship between different representational frames.
A number of single unit recording studies have shown that areas of primate posterior parietal
cortex, again in and around the intraparietal sulcus, contain neurons that exhibit firing
patterns modulated by various combinations of head position, velocity, acceleration and
visual stimuli (Andersen, Shenoy, Snyder, Bradley & Crowell, 1999; Bremmer, Klam,
Duhamel, Hamed & Graf, 2002; Klam & Graf, 2003). The nature of these interactions
appears to be complex, but Bremmer et al. suggest that this idiothetic modulation of parietal
neuron firing might be related to object tracking during self-motion. This argument is
indirectly supported by Duhamel, Colby & Goldberg, (1992) who have shown that eye
movements that bring the location of a previously flashed stimulus into the receptive field of
a parietal neuron elicit a response from that neuron even though the stimulus is no longer
present (see also Colby, 1999). Area 7a is the part of parietal cortex most strongly connected
with the medial temporal lobe, including efferent projections into the parahippocampus,
presubiculum, and CA1 (Ding, Van Hoesen & Rockland, 2000; Rockland & Van Hoesen,
1999; Suzuki & Amaral, 1994) and afferent connections from entorhinal cortex and CA1
(Clower, West, Lynch & Strick, 2001). In addition, single unit recordings from monkey
dorsolateral prefrontal and posterior parietal cortic es suggest that spatial working memory
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is, indeed, egocentric in nature (Chafee & Goldman-Rakic, 1998; Funahashi, Bruce &
Goldman-Rakic, 1989).
Finally, some hints of the temporal dynamics of neural processing during navigation come
from the observation that the theta rhythm (i.e. 4-12Hz) of the EEG invariably accompanies
voluntary displacement motion of the rat (O’Keefe and Nadel, 1978). In addition, the phase
of firing of place cells correlates strongly with the rat’s location within the firing field
(O’Keefe and Recce, 1993), and independently of firing rate or running speed (Huxter,
Burgess, O’Keefe, 2003). Recent results indicate a possible role for theta in human
navigation (Caplan et al., 2003; Kahana, Sekuler, Caplan, Kirschen & Madsen, 1999), and
several experiments indicate a role for theta phase (e.g., Pavlides, Greenstein, Grudman &
Winsom, 1988) in modulating hippocampal synaptic plasticity, and theta power (Sederberg
et al., 2003) or theta coherence between hippocampus and nearby neocortical areas (Fell et
al., 2003) in modulating encoding into memory .
Lesions, neuropsychology and functional neuroimaging
The medial temporal lobes, and hippocampus in particular, have long been know to be
crucial for long-term memory (Eichenbaum & Cohen, 1988; Scoville and Milner, 1957;
Squire, 1986), together with other elements of Papez’s circuit (Aggelton and Brown, 1999).
Within the spatial domain, neuropsychological studies have left little doubt that the medial
temporal lobe, particularly in the right hemisphere, is critical for remembering the locations
of several objects within a visual scene over a significant delay (Crane and Milner, 2005;
Piggott & Milner, 1993; Smith & Milner, 1989). Within a broader memory deficit,
hippocampal damage seems to specifically impair performance in tasks likely to require
allocentric representations of location or representations that can be flexibly accessed from
novel points of view, rather than being directly solved by use of egocentric representations.
For example, where locations must be remembered from a different point of view to
presentation, performance is impaired relative to location memory from the same view even
over short timescales (Abrahams, Pickering, Polkey & Morris, 1997; Holdstock et al., 2000;
King et al., 2002; Hartley et al., in press). More generally, accurate spatial navigation to an
unmarked goal location is impaired by hippocampal damage in rats (e.g., Jarrard, 1993;
Morris, Garrard, Rawlins & O’Keefe, 1982) and humans (Bohbot et al., 1998; Maguire,
Burke, Phillips & Staunton, 1996; Spiers et al., 2001). Human neuroimaging studies also
show involvement of the hippocampus in accurate navigation (Hartley, Maguire, Spiers &
Burgess 2003; Iaria et al, 2003; Maguire et al., 1998). Additionally, neuroimaging of the
perceptual processing of spatial scenes, including plain walled environments, implicates the
parahippocampal cortex (Epstein & Kanwisher, 1998), a region associated with landmark
recognition (Aguirre & D’Esposito, 1999) and navigation (Bohbot et al., 1998). See
Burgess, Maguire & O’Keefe (2002) for a review.
Human neuropsychology has long recognized the parietal lobes as playing a major role in
spatial cognition. Parietal damage leads to deficits in sensorimotor coordination such as
optic ataxia, deficits in spatial manipulation such as mental rotation and deficits in spatial
working memory (see e.g. Burgess, Jeffery & O’Keefe, 1999; Haarmeier, Thier, Repnow &
Petersen, 1997; Karnath, Dick & Konczak, 1997). Visual processing in the temporal and
parietal lobes has been generally characterized respectively in terms of the dorsal and ventral
‘what and where’ (Ungerleider and Mishkin, 1982) or ‘what and how’ (Goodale & Milner,
1992) processing streams. The parietal region in the dorsal stream is concerned with
representing the locations of stimuli in the various egocentric reference frames appropriate
to sensory perception and motor action, and translation between these frames to enable
sensori-motor coordination. In contrast, the occipital and temporal visual regions in the
ventral stream are concerned with visual perceptual processes related to object recognition,
see Neuronal representations above.
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Unilateral damage to parietal cortex (most often on the right) and surrounding areas
commonly results in the syndrome of hemispatial neglect: a reduced awareness of stimuli
and sensations on the contralateral side of space (‘perceptual neglect’). Of particular interest
here is the phenomenon of ‘representational neglect’ - a lack of awareness of the
contralateral side of internal representations derived from memory. In the classic
demonstration (Bisiach and Luzzatti, 1978), patients were asked to imagine the Piazza del
Duomo in Milan (with which they were very familiar) and to describe the scene from two
opposite points of view. Buildings to the left of the given point of view (e.g. facing the
Cathedral) were neglected, but those same buildings were described when given the opposite
point of view (e.g. facing away from the Cathedral), indicating intact long-term memory of
the entire Piazza, in spite of neglect of the left of each imagined scene. Perceptual and
representational neglect depend, at least in part, on different neural systems, and can be
dissociated, even within the same patient (Bechin, Basso & Della Sala, 2000). Interestingly
representational, but not perceptual, neglect is associated with impaired navigation to an
unmarked location (Guariglia, Piccardi, Iaria, Nico & Pizzamiglio, 2005). Consistent with
these findings of parietal involvement in imagery, neuroimaging experiments have shown
heightened activity within the precuneus (i.e. medial parietal cortex) during mental imagery
(e.g., Fletcher, Shallice, Frith, Frackowiak & Dolan, 1996) and visuospatial working
memory (e.g., Wallentin, Roepstorff, Glover, Burgess, 2006). Transcranial magnetic
stimulation and fMRI studies also indicate that areas surrounding the right intraparietal
sulcus, including areas 7a and 40, are essential in the generation and manipulation of
egocentric mental imagery (Formisano et al., 2002; Knauff, Kassubek, Mulack & Greenlee,
2000; Sack et al., 2002).
Behavioral and single unit studies indicate that memory for locations in general, and the
place cell representation of location in particular, is automatically updated by self-motion, a
process more generally known as ‘path integration’ or ‘spatial updating’ (see below). This
process may reflect an interaction between the parietal and hippocampal systems, as the
parietal cortex appears to be centrally involved in this process (Alyan & McNaughton, 1999;
Commins, Gemmel, Anderson, Gigg & O’Mara, 1999; Save, Guazzelli & Poucet, 2001;
Save & Moghaddam, 1996). For example, Save, Paz-Villagran, Alexinsky & Poucet (2005)
have shown that lesions to the associative parietal cortex of rats result in altered place cell
firing, suggesting that egocentric sensory information must travel through parietal cortex in
order to elicit appropriate place cell firing. This is consistent with a number of experiments
that demonstrate that mental exploration/navigation depends on posterior parietal and extra-
hippocampal medial temporal regions in primates, and on homologous regions in the rodent
brain (Ghaem et al., 1997; Pinto-Hamuy, Montero & Torrealba, 2004). The interaction
between parietal and medial temporal areas likely involves retrosplenial cortex, lesions of
which selectively disrupt path integration (Cooper, Manka, Mizumori, 2001), and the
parieto-occipital sulcus, which has been associated with topographical disorientation (Ino et
al., 2002) and cells coding for locations in space (Galletti et al., 1995).
Prefrontal regions, as well as parietal ones, are implicated in spatial working memory, with
parietal areas predominantly associated with storage and prefrontal areas with the
application of control processes such as active maintenance or planning (Shallice, 1988;
Levy and Goldman-Rakic 2000; Oliveri et al., 2001) using the posterior spatial
representations. Thus, fMRI studies have shown activation in both of these areas when
subjects were required to remember the locations of various objects for short periods of time
(Galati et al., 2000; Sala, Rämä & Courtney, 2003). Manipulations of working memory may
also involve making or planning eye movements in order to direct attention to spatial
locations in imagery. In support of this notion, voluntary eye movements disrupt spatial
working memory (Postle et al, 2006), while left hemispatial neglect patients show abnormal
eye movements, which deviate about 30 degrees rightward during visual search (Behrmann
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et al, 1997) as well as while at rest (Fruhmann-Berger and Karnath, 2006). Moreover,
adapting prisms that shift the neglected visual field toward the good side of space, which
would compensate for a rightward bias in gaze direction, ameliorate both perceptual and
representational neglect (Rode et al, 2001). Studies involving mental navigation and route
planning consistently find elevated activation in frontal regions, especially on the left side
(Ghaem et al., 1997; Ino et al., 2002; Maguire et al., 1998). For example, Maguire et al.
(1998) found additional activation in left prefrontal cortex associated with the planning of
detours when subjects were navigating in a familiar virtual town in which the most obvious
route had suddenly been blocked. This suggests that left prefrontal areas contribute to route
planning, perhaps guiding egocentric mental imagery within the temporoparietal systems
activated by the basic navigation condition.
Cognitive psychology
Given the electrophysiological and lesion evidence for parallel egocentric and allocentric
representations of location, we next consider converging evidence from cognitive
psychology in which one or the other or both may contribute to behavior. Simons & Wang
(1998; Wang & Simons, 1999) performed an elegant series of experiments in which subjects
were required to remember an array of objects presented on a circular table. During the
delay period preceding the memory test, the table would either remain stationary or rotate
through a fixed angle. At the same time, the subject would either remain stationary or walk
through the same angle around the table. Thus, the test stimuli could be aligned with the
studied view, with a rotated view consistent with the subject’s motion, with both, or with
neither. Subjects’ performance on a memory task (detecting which object had moved)
provided evidence for the use of both: 1) a visual-snapshot representation of the presented
array; and 2) an egocentric representation that is updated to accommodate self-motion; by
showing an advantage whenever the test array was aligned with either representation. The
latter “spatial updating” ability (Reiser, 1989) can be thought of as a generalization of path
integration, allowing an organism to keep track of several locations, including its origin of
motion, during real or imagined navigation in the absence of visual cues. The results suggest
that both of types of representation exist in the brain. Interestingly, evidence suggests that
allocentric representations of object locations (i.e., relative to visual landmarks external to
the array) are also employed in this type of experiment, as shown by a subsequent study
incorporating a rotatable landmark (Burgess, Spiers & Paleologou, 2004). Parallel influences
of egocentric and allocentric representations are also indicated by human search patterns
within deformable virtual reality environments (Hartley, Trinkler & Burgess, 2004). In these
experiments, the locus of search can be predicted by a model based on the firing of
hippocampal place cells, indicating allocentric processing of location. However, subjects
also tended to adopt the same orientation at retrieval as at encoding, indicating egocentric
processing of orientation.
Further evidence for the use of both egocentric and allocentric representations of space can
be found in reaction time data from a number of experiments involving the recognition/
recall of previously presented object configurations from novel viewpoints. Diwadkar &
McNamara (1997) had subjects learn the locations of objects on a desktop from a number of
viewpoints before taking part in a recognition test. When presented with a novel view of the
same or a different object configuration, subjects’ reaction time was found to vary linearly
with the angular distance between the observed view and the closest trained view. Related
results were found when blindfolded subjects had to point to where a given object would be
from a specific imagined viewpoint: accuracy and/or reaction time reflected the distance and
angle between the studied viewpoint and the imagined viewpoint (Reiser 1989; Easton and
Sholl, 1995; Shelton and McNamara 2001). These results are consistent with spatial
updating of an egocentric representation. However, the additional use of allocentric
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representations in these tasks is indicated by improved performance for viewpoints aligned
with the walls of the room or the sequence of learning (Mou & McNamara, 2002), or with
external landmarks (McNamara, Rump & Werner, 2003), and with the absence of a
relationship to distance or angle for objects configured into a regularly structured array
(Reiser, 1989; Easton and Sholl, 1995). In possibly related findings, Wang & Spelke (2000)
suggest that the high variance of the error in pointing to different objects after blindfolded
disorientation indicates independent egocentric representations for the location of each
object. In the same experiment, the lower variance in errors when pointing to features of the
testing room indicated a single coherent (allocentric) representation for the layout of the
room. Similarly, judgments of relative direction between objects from an imagined location
at a third object do not increase in variance with disorientation, indicating use of a more
coherent representation in this task than that used for egocentric pointing (Waller and
Hodgson, 2006). See Burgess (2006) for further discussion.
Theoretical analyses
It has been proposed (e.g. Milner, Paulignan, Dijkerman, Michel & Jeannerod, 1999) that
the relative contribution of egocentric and allocentric representations to spatial memory
depends on the timescale of the task concerned. Short-term retention of perceptual
information for the purpose of immediate action will be best served by egocentric
representations appropriate to the corresponding sensory and motor systems. By contrast,
long-term memory for locations will be best served by allocentric representations (i.e.
relative to stable landmarks) because the location and configuration of the body at retrieval
typically will be unrelated to that at encoding (see Burgess, Becker, King & O’Keefe, 2001
for further discussion). This observation is consistent with the evidence for the role of
parietal and prefrontal areas in supporting egocentric representations and short-term
memory, on the one hand, and the role of medial temporal lobe areas in supporting
allocentric representations and long-term memory, reviewed above.
For intermediate timescales (e.g. tens of seconds), it may be possible to relate the
configuration of the body at retrieval to that at encoding via the egocentric process of “path
integration” or “spatial updating” referred to above. Pierrot-Deseilligny, Müri, Rivaud-
Pechous, Gaymard & Ploner (2002) review evidence suggesting that spatial memory may
have at least three important timescales. For the first approximately 20 seconds, they claim
that a fronto-parietal spatial working memory system is the dominant mechanism, followed
for approximately five minutes by a medium-term, parahippocampally dependent memory
system, and finally by a hippocampally dependent long-term memory system which operates
only after delays of several minutes. Spatial scale might also be a factor in determining
which representations are used. For example, in mammals path integration becomes
unreliable over long or convoluted paths (see e.g. Etienne, Maurer & Seguinot, 1996), while
egocentric parietal and premotor representations may be preferentially recruited for
representations of locations in “peri-personal” space that can be directly acted upon (e.g.
Duhamel, Colby & Goldberg, 1998; Goodale & Milner, 1992; Graziano & Gross, 1993;
Ladavas, di Pellegrino, Farne & Zeloni, 1998).
Along the above lines, Mou, McNamara, Valiquette and Rump (2004) propose a transient
egocentric representation of object locations for immediate action and an allocentric
representation of the environment, including the subject’s own location, for actions
supported by information from long-term memory. On the basis of the experiments probing
memory for object location as a function of differences between the studied, imagined and
actual views, they argue that two types of spatial updating occur: spatial updating of
egocentric representations of object locations, and spatial updating of the subject’s own
location in the environmental representation. A related proposal suggests transient
egocentric representations of single objects in parallel with a more coherent enduring
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representation (Waller and Hodgson, 2006). For a discussion of the neural mechanisms
supporting the integration of self-motion and sensory information, see (Guazzelli, Bota &
Arbib, 2001; Redish, 1999).
In summary, evidence from psychology and neuroscience indicates that spatial cognition
involves multiple parallel frames of reference, with short-term/small-scale tasks more likely
to recruit egocentric representations and long-term/large-scale tasks more likely to recruit
additional allocentric representations. However, this proposed division of labour involving
different reference frames is neither absolute nor uncontroversial. Thus, Wang & Brockmole
(2003) have also argued that even long-term spatial memory is purely egocentric. They
found the current view to influence the ability of students to point to an occluded but very
familiar landmark on the campus. Conversely, even short-term memory can be shown to
depend on the hippocampus when the viewpoint is changed between study and test (King et
al., 2002; King et al., 2004; Hartley et al., in press), and on allocentric representations when
landmarks are parametrically manipulated (Burgess, Spiers & Paleologou, 2004), see
Burgess (2006) for further discussion.
The model: Overview
From the forgoing discussion, it appears that mammalian spatial memory can make use of
both egocentric and allocentric representations in parallel, depending on the nature of the
task. We now propose a model of spatial cognition that accounts for the interaction between
long- and short-term memory processes in encoding, retrieval, imagery and planning. The
model addresses data at multiple levels of analysis, from single unit recordings to large-scale
brain systems to behaviour, and the relative roles played by egocentric and allocentric
representations and by visual and idiothetic inputs. We first provide a brief overview of the
functional architecture of our model, with further details of its implementation given in the
next section and elaborated upon fully in the appendix.
In our model, long-term spatial memory formation involves the generation of allocentric
representations in the hippocampus and surrounding medial temporal lobe structures
(perirhinal and parahippocampal cortices). The hippocampal place cell representation is
driven by convergent inputs from the dorsal and ventral visual pathways. The ventral stream
input consists of object features in perirhinal cortex, while the dorsal stream input consists of
BVCs in parahippocampal cortex. These medial temporal lobe areas are all mutually
interconnected to permit pattern completion. When cued with a partial representation of a
place, such as a specific landmark, the model thereby automatically retrieves the full
representation of that place, comprising the location of the observer as well as the
surrounding landmarks and their visual appearance.
Both short-term spatial memory and imagery are modeled as egocentric representations of
locations in the precuneus which can be driven by perception or by re-construction from
long-term memory, see below. The neural activations within this medial parietal
representation can be modulated by directed attention, to capture the fact that one can attend
sequentially to the spatial locations of items in imagery just as in perception, presumably via
planned eye movements (see Postle et al, 2006). Both encoding and retrieval require
translation between the egocentric precuneus and allocentric parahippocampal
representations of landmarks. This occurs via a coordinate transformation mediated by
posterior parietal and retrosplenial cortices, reflecting the current head direction.
Retrieval from long-term memory, cued by knowledge of position and orientation relative to
one or more landmarks, corresponds to pattern completion of the parahippocampal
representation of the allocentric locations of landmarks around the subjects, via its
connections with the hippocampal and perirhinal representations. Thus, the medial temporal
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lobe acts as an attractor network within which a representation of the visual features,
distances and allocentric directions of landmarks can be retrieved, which is consistent with
perception from a single location (represented in the hippocampus). This representation is
translated into the egocentric precuneus representation, within which directed attention can
boost the activation of egocentrically defined locations of interest. Finally, the additional
activation can feed back to the parahippocampal representation, again via posterior parietal
translation, and thence to the perirhinal representation so as to activate the visual features of
the attended landmark.
Motor efference drives the ‘spatial updating’ of the egocentric representation of the
locations of landmarks. Specifically, modulation of the posterior parietal egocentric-
allocentric transformation by motor efference causes allocentric locations to be mapped to
the egocentric locations pertaining after the current segment of movement. The re-activation
of the BVCs by this shifted egocentric representation then updates the medial temporal
representation to be consistent with the parietal representation. The ‘bottom up’ (parietal to
temporal) and ‘top down’ (temporal to parietal) flows of information are temporally
organized into different phases of the theta rhythm. Additionally, the generation of mock
motor efference in prefrontal cortex allows mental exploration in imagery via mock spatial
updating.
A central component of our model is circuitry which transforms between different
representations of the space surrounding an animal. This proposed egocentric-allocentric
transformation suggests a solution to two puzzles regarding the functional anatomy of
memory and navigation. The first is the observation that Papez’s circuit (including the
mammillary bodies, anterior thalamus, retrosplenial cortex and fornix, as well as the
hippocampus) is both crucial for episodic recollection, which is impaired by lesions
anywhere along it (see e.g. Aggleton & Brown, 1999), and provides the neural basis for head
direction cells (Taube, 1998). A second, related puzzle is the ubiquitous involvement of
retrosplenial cortex and the anterior parieto-occipital sulcus in both navigation (reviewed in
Maguire, 2001) and memory (see e.g. Burgess, Maguire, et al. 2001). We propose (see also
Burgess, Becker et al., 2001; Burgess, Maguire et al., 2001) that the segment of Papez’s
circuit from the mammillary bodies to the hippocampal formation via the anterior thalamus
carries the head direction information needed to transform the allocentric directional tuning
of the BVC representation into an egocentric (head-centered) representation suitable for
mental imagery, and that the retrosplenial cortex/parieto-occipital sulcus may mediate or
buffer the stages of transformation between egocentric and allocentric representation (see
also Ino et al., 2002). A related proposal is that retrosplenial cortex serves to integrate
mnemonic and path-integrative information (Cooper & Mizumori, 2001), which maps onto
our own proposal given the assumption of allocentric long-term memory and egocentric
spatial updating.
The model: Architecture and dynamics
In this section, we discuss the architecture of our model and then describe the model
dynamics, and how spatial updating, mental exploration and learning are simulated. A
simplified version of our model with preliminary simulation results was described by Becker
& Burgess (2001). By lesioning the parietal region of the model, the authors were able to
simulate aspects of hemispatial neglect. The model presented here builds on this earlier
work by deriving in a more principled manner the neural circuits for allocentric
representation and allocentric-egocentric transformations, and augments this model with
parietal neural circuitry to support spatial updating and mental navigation. The architecture
of our model rests upon three key assumptions:
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1.
The parietal window hypothesis: An egocentric window provides exclusive access
into long-term spatial memory, in the service of mental imagery, planning and
navigation.
2.
Allocentric coding in the medial temporal lobe: Allocentric BVC representations
are constructed in the parahippocampal region, and project to hippocampal place
cells where long-term spatial memories are stored.
3.
Transformation circuit: Access by the parietal window into allocentrically stored
spatial representations is mediated by a transformation circuit; the same circuit also
operates in the inverse direction, such that the products of recall are mapped from
allocentric into egocentric representations of space.
The parietal window hypothesis
We hypothesize that a population of neurons maintains a head-centered, egocentric map of
space that can be driven either by bottom-up sensory input or by top-down inputs from long-
term memory. This map represents the locations of all landmarks/objects that are visible
from an animal’s current location in space, or from a location that the animal recalls from
previous experience. This neuronal population, assumed to exist within the posterior parietal
cortex, and very likely within the precuneus, will henceforth be referred to as the
parietal
window
. We claim that the contents of the parietal window are generated based upon some
combination of information from the senses (dorsal visual stream, for example) and from
allocentric long-term spatial memory, with the exact combination depending on the demands
of the current task. Manipulation of spatial information for the purposes of planning or
navigation, including spatial updating, occurs within the parietal window.
The network model also includes circuitry that can manipulate the contents of the parietal
window so as to allow for spatial updating or mental exploration. In the case of spatial
updating, this circuitry is activated by idiothetic information (proprioceptive cues signalling
the observer’s change in direction and location), whereas in the case of mental exploration,
it is activated by some mentally generated equivalent (e.g. imagined rotation and translation
during path planning). The former ability allows the model to maintain an internal
representation of its surroundings even with degraded or absent sensory input, while the
latter provides a means of recalling the locations of occluded landmarks and generating
navigational strategies for reaching them.
Allocentric representations in the medial temporal lobe
In contrast to the parietal window’s egocentric frame of reference, we postulate that an
allocentric frame of reference is employed in the medial temporal lobe. The model’s
egocentric reference frame has its origin bound to the observer’s location, with its y-axis
fixed along the observer’s heading direction. The model’s “allocentric reference frame” has
its origin bound to the observer’s location (in this sense, like place cell firing, it is not fully
allocentric), but its orientation is fixed relative to the external environment. Therefore, both
reference frames are similar in that they remain fixed with respect to the observer so long as
the observer undergoes translational motion only. However, when the observer’s head
rotates within the environment, while the egocentric frame rotates with it, the allocentric
frame remains stationary with respect to the environment. An example of an object in the
allocentric frame and its corresponding location in the egocentric frame is shown in figure 1.
Consider the situation depicted in figure 2 where an observer surrounded by six walls is
located at the position marked “X”, with a heading direction indicated by the arrow. If the
walls of this “two-room” environment are discretized uniformly into a set of “landmark
segments” (to simplify later calculations), then the egocentric frame positions of the
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segments viewable from “X” can be inferred readily. These positions are depicted by open
circles in the top panel of figure 3. Representation of this egocentric information by the
parietal window neurons is accomplished by first forming a one-to-one correspondence
between the set of neurons and a polar grid covering the egocentric reference frame. This
grid is depicted by the closed circles in the top panel of figure 3. Each neuron in the grid is
tuned to respond most strongly to an object or landmark at a particular direction and distance
relative to the organism’s head, which is at the origin of the grid. The neuron’s response
falls off exponentially for objects located further away from the neuron’s preferred distance
and direction (see the appendix for details). When multiple segments are present within a
neuron’s receptive field, they contribute additively to its firing rate, up to a maximum firing
rate of 1. The parietal window representation of the information depicted in the top panel of
figure 3 is shown in the bottom panel of the same figure, where the firing rate of each
neuron is plotted at the location of its corresponding grid point.
We assume that the observer in figure 2 aligns its allocentric frame such that the y-axis is
perpendicular to the wall labeled “1” and the x-axis is parallel to the same wall. The
locations of the landmark segments in this frame, which will not depend on the observer’s
heading direction, are depicted in the top panel of figure 4. By forming a one-to-one
correspondence between a set of neurons and a polar grid centered at the origin of the
allocentric reference frame, it becomes possible to represent the configuration of landmark
segments by the firing rates of this neural population. In analogy with the egocentric parietal
window neurons, each allocentric neuron in the grid is tuned to respond most strongly to an
object or landmark at a particular distance from the organism’s head, which is fixed to the
origin of the grid, and
allocentric
direction (relative to the fixed environment). Again, the
neuron’s response falls off exponentially for objects located further away from the neuron’s
preferred distance and direction. Note that these allocentrically tuned neurons are essentially
the same as the BVCs described in the introduction, and will be referred to as such from this
point on. The BVC representation of the information depicted in the top panel of figure 4 is
shown in the bottom panel of the same figure, where the firing rate of each neuron is plotted
at the location of its corresponding grid point. Although we assume these BVCs exist within
parahippocampal cortex, we note that cells with BVC-like responses have been found in the
subiculum (Barry et al., 2006; Sharp, 1999), an alternative location to the parahippocampal
cortex, but less consistent with neuroimaging results in humans showing parahippocampal
processing of spatial scenes including plain walled environments (Epstein & Kanwisher,
1998).
To form long-term memories for specific spatial locations, spatial input from BVCs and
visual input from the perirhinal layer are combined into a place cell representation. Although
in reality the hippocampal formation consists of multiple spatially selective regions (dentate
gyrus, CA3, CA1), for simplicity, our model hippocampus contains a single layer of
recurrently connected place cells. Their place preferences are arranged uniformly over a
Cartesian grid that covers the relevant allocentric space for a given environment (see figure
2). In particular, a one-to-one correspondence is formed between each of the model place
cells and the set of grid points so that a given place cell fires maximally when the model is
located at that cell’s corresponding grid point. These model hippocampal neurons are
reciprocally connected to the layer of BVCs and to a layer of perirhinal identity neurons,
thus allowing environmental geometry and landmark identities to be bound simultaneously
to a given “place”. In addition, the layer of BVCs is reciprocally connected to the layer of
perirhinal neurons, thereby allowing the association of landmark identities with allocentric
locations (see figure 6 for a schematic of the full model). The full reciprocal connectivity
between the three medial temporal lobe components of the model allows for the recall of a
landmark’s identity when attention is directed toward the parietal window representation of
that landmark’s location. This process of recall is described in the next section.
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Within our gross simplification of hippocampal circuitry, the model’s single layer of place
cells is most consistent with area CA3, an area that is heavily recurrently connected, and that
exhibits place selective firing. In our model, this recurrent connectivity allows for recall/
pattern completion, as it is often argued to do in CA3 (Brun et al., 2002; Nakazawa et al.,
2002). Another gross simplification in our model is the strictly spatial function of the
hippocampus. Although the hippocampus is known to be important in spatial memory, its
more general contribution to episodic memory is well established (for a review see Burgess,
Maguire & O’Keefe 2002).
Transformation circuit
The assumption in our model of egocentric access to allocentrically stored spatial
information has an important implication: there must be circuitry that transforms between
these representations. In order to be able to recall the locations and identities of
environmental boundaries relative to one’s own location
and
orientation, long-term
allocentric internal representations of space must be transformed into egocentric
representations. Conversely, in order for sensory input to cue such recall, or for it to enter
long-term allocentric storage in the first place, the inverse transformation from egocentric to
allocentric representation must be performed. That is, a visual stimulus at a retinocentrically
encoded location must be transformed into an allocentrically encoded location in order to
match against or store within spatial long-term memory. We assume that sensory
information is first transformed into the head-centered egocentric parietal window reference
frame and then to the allocentric BVC representation. The transformation from the parietal
window representation to the BVC representation, and its inverse, can be accomplished very
simply if absolute heading direction is known. Consider, for example, that you are facing
West (90 degrees in allocentric angular coordinates, where North is zero degrees) and there
is an object to your left (90 degrees in egocentric angular coordinates where straight ahead is
zero degrees); the object’s allocentric direction can be calculated simply by adding the
heading direction to the object’s egocentric direction to obtain 180 degrees; similarly if the
object is known to be located to the South (an allocentric angle of 180 degrees) then its
egocentric direction can be calculated by subtracting the heading direction from the object’s
allocentric direction. Thus, in our model the egocentric-allocentric transformations are
mediated by input from head direction cells which provide the necessary modulation of
firing rates by head direction (Snyder et al., 1998), and the same neural circuitry can then
perform the transformation in either direction. The computation is a bit more complicated
than a simple subtraction or addition of angles because angular directions are encoded across
populations of narrowly direction-tuned neurons; nonetheless, it can be accomplished in a
single layer of neurons whose activities are non-linearly modulated by head direction (c.f.
Pouget and Sejnowski, 1997). See figure 5 for a schematic of the full transformation circuit.
When an animal first enters a new environment, we assume that salient perceptual features
reliably orient the head direction system. We model the head direction system as a set of
neurons configured in a ring via lateral connections to behave as a one-dimensional
continuous attractor, as in previous models (e.g. Skaggs, Knierim, Kudrimoti &
McNaughton, 1995; Stringer, Trappenberg, Rolls & de Araujo, 2002; Zhang, 1996). The
continuous attractor property implies that the network will stabilize on a single bump of
activity corresponding to a single head direction, and this bump can move continuously
through 360 degrees to reflect self-motion or perceptual inputs. Moreover, the reliability of
the input mapping implies that if the animal returns to the same environment in the future,
the head direction system will be oriented in exactly the same fashion, and will exhibit the
same firing pattern as it did on the first exposure to the environment.
The egocentric-to-allocentric transformation is accomplished by a circuit that combines head
direction information with egocentric spatial input from the parietal window. The
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transformation circuit, assumed to be in retrosplenial cortex / intra-parietal sulcus, is
comprised of a set of N identical neural subpopulations, each tuned to a specific head
direction. Each sub-population encodes a rotated egocentric map consistent with the
direction of its preferred heading. Thus, connections between the parietal window and any
one of the transformation subpopulations are weighted such that a rotated version of the
egocentric spatial information contained in the parietal window is projected onto that
transformation sub-layer. In our model there are twenty such sub-layers corresponding to
evenly spaced allocentric directions. Each transformation sub-layer then projects an identical
copy of its activation pattern onto the layer of BVCs. By setting connections from the layer
of head direction cells to the transformation neurons such that only the sub-layer
corresponding to the current head direction is active, the transformation from egocentric to
allocentric coordinates is accomplished. See figures 5 and 6. In this way, when the animal’s
head rotates within the environment, head direction cell activity and parietal window activity
vary in time, but so long as the animal undergoes no translation, activity projected to BVC
neurons remains constant. The gating function of the head direction cells is accomplished
via a combination of direct excitation from the head direction cells to the appropriate
transformation sub-layer, and indirect uniform inhibition of all transformation layers by a
population of inhibitory interneurons driven by head direction cell activity. This circuitry
allows a localized bump of activity in the head direction layer to select the set of
transformation units corresponding to that head direction.
The egocentric-allocentric transformation results in a single viewpoint-independent
representation of each location in an environment. The allocentric representation consists of
a distributed pattern of activation across the boundary vector cell layer. To encode this
pattern as a distinct place memory, and to permit subsequent cued recall, this pattern can be
learned by an auto-associative memory system. A retrieval cue, such as incomplete
egocentric sensory or mentally generated spatial information, can then feed forward through
the transformation circuit and reactivate the correct allocentric representation of the model’s
real or imagined surroundings. Conversely, the place memory can generate a viewpoint-
specific mental image if we assume that the connections in the transformation circuit operate
with equal weights in both directions. The recalled allocentric representation can thereby be
converted back into egocentric mental imagery of the environment via the same neural
circuitry.
Model dynamics
Neurons in our model are rate-coded (i.e. their activations represent average neural firing
rates rather than individual spikes) and exhibit a continuous dynamics governed by “leaky-
integrator” equations. The complete mathematical details of the model, along with these
dynamical equations, can be found in the appendix. Here we present a more intuitive
description of the model’s overall behaviour. For now, the issue of biologically realistic
learning is ignored and it is assumed that the model has already learned about the spatial
environments it encounters. The actual
ad hoc
training procedure used to set the model
weights for this work will also be described briefly in a later section, with full details
presented in the appendix. In a later section, we also discuss general principles that might
underlie the learning of egocentric-allocentric transformations in biological systems.
At the highest level of dynamics, our model operates in alternating bottom-up and top-down
stages, each lasting for 15 arbitrary “time units”. This periodic alternation in dynamics is
based on modeling work by Hasselmo, Bodelón & Wyble (2002) who argue that the
hippocampal theta rhythm regulates the communication of this structure with interconnected
brain regions. In particular they argue that during troughs in the rhythm, the hippocampus
primarily
receives
input from surrounding structures, but that during peaks, it primarily
transmits information
to
these structures. We implement this alternating dynamics in our
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model both because of the evidence supporting its existence and because it allows the model
to account for more experimental data than it otherwise could. In particular, without these
distinct phases the model would have to engage in both bottom-up and top-down processing
at the same time. We have found that a functional version of such a model exhibits states
that strongly resist change in response to external inputs.
During the top-down phase, activity from the hippocampal layer feeds back to perirhinal
cortex and also to the parietal window via the BVC and transformation layers. In addition,
during this phase, the parietal window receives input from the senses, which we assume can
be down-regulated if the model is performing mental exploration or recall of a familiar
environment without actually changing its vantage point. See figures 5 and 6. During the
bottom-up phase, the activity of the window is “frozen” to the last pattern present during the
top-down phase. This activity pattern, which is the model’s current representation of the
geometry of egocentric space, is hypothetically maintained by a fronto-parietal short-term
memory system (which we do not model here), consistent with evidence presented earlier.
The frozen information from the parietal window feeds forward during the bottom-up phase
to the hippocampal layer along with information from perirhinal cortex, thus influencing the
current hippocampal attractor state. In principle, rigid freezing of the parietal window
representation during the bottom-up phase is not necessary, but such an approach eliminates
the need for additional neural circuitry in the model.
An animal would need to recall the details of an environment stored in long term memory
for two main reasons. First, there could be transient environmental conditions that impede
sensory input and thus leave the animal with little direct access to spatial information.
Second, the animal might need to remember what would be around it at an imagined
location for the purposes of planning. For the former case, we assume that the model has
enough sensory information to orient the head direction system. Although we only deal with
visual information here, the model could be extended easily to include other cues such as
vestibular input for this purpose as well. Once the head direction system is oriented, the
available but incomplete sensory input to the parietal window and perirhinal cortex can flow
to the hippocampus in a bottom-up phase and activate an attractor state for the complete
corresponding allocentric representation. During the next top-down phase, this attractor state
reconstructs the environmental geometric information in the parietal window. Once the
model has reconstructed the geometry of the environment, it must be able to identify the
boundaries/landmarks which surround it. This is assumed to occur via directed attention to a
spatial location. We simulate this in our model as extra activation (calculated from equation
A17, see Appendix) being directed to the area of interest in the parietal window. The
boundary within the focus of attention in the parietal window will generate a corresponding
focus of activation on its allocentric location within the BVC layer. The associative
pathways within the medial temporal lobe can then retrieve the object’s identity in the
perirhinal cortex.
As a concrete example of spatial attention, if the model is instructed (perhaps by some
prefrontal brain region controlling planned eye movements, not modeled here) to identify a
boundary to its egocentric left, then extra activation is directed to the parietal window
neurons that represent space to the egocentric left. This activation then flows through the
transformation circuit, to the BVC layer, and finally to the perirhinal layer. The extra
activation from the parietal window increases the firing rate of all perirhinal neurons
corresponding to boundary indentities that the model could encounter to its left when it has
the current heading direction. The correct boundary identity, consistent with the subject’s
current location, can then be disambiguated by allowing the top-down connections in the
model to operate at a low level (5% of the normal top-down value) even during a bottom-up
phase. In this way, the place cell activity can provide the requisite disambiguation. For
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consistency, we also allow bottom-up connections to operate at the same reduced level
during top-down phases.
In cases where an animal needs to recall the details of its surroundings from a particular
imagined point of view, we assume that the suggestion of (in the case of humans) or
memory of a highly salient environmental feature located at some point in the animal’s
egocentric space might be enough to orient the head direction system. The correct perirhinal
units could also be activated by this process, and activity corresponding to the location of
the feature could be sent to the parietal window. During the next bottom-up phase, the
processes of pattern completion and directed attention would then follow as described
above.
Spatial updating and mental exploration
The recall processes described in the previous section are useful only if an animal requires
stationary “snapshots” of an environment. However, a moving animal, often faced with
partially or fully occluded sensory information, requires an accurate, real-time
representation of its surroundings. Similarly, if an animal wishes to plan a route through a
familiar environment, the ability to perform mental exploration of the surrounding space
would be useful.
A key part of our overall theory is that parietally generated egocentric mental imagery can
be manipulated via real or mentally generated idiothetic information in order to accomplish
spatial updating or mental exploration in familiar environments. A detailed neural
mechanism for accomplishing such tasks in the case of pure short-term or working memory
has been described elsewhere (Byrne & Becker, 2004). Here we are concerned primarily
with the updating process applied to medial temporal lobe dependent long-term memory.
For this case we assume that rotational and forward-translational egomotion signals act upon
the egocentric parietal window representation of space via different mechanisms. In the case
of rotation, the egomotion signal causes head direction cell activity to advance sequentially
through the head direction map, thus rotating the image that is projected into the parietal
window from the BVCs. This velocity-modulated updating of head direction is similar to the
model described by Stringer, Trappenberg, Rolls & de Araujo (2002). The potential for such
one dimensional continuous attractor networks to account for multiple aspects of the head
direction cell assembly has been investigated in detail by Conklin & Eliasmith (2005),
Goodridge & Touretzky (2000), Hahnloser (2003) and Redish, Elga & Touretzky (1996)
among others. However, a detailed summary of such work is beyond the scope of this paper.
For the case of forward translation, the egomotion signal gates the top-down connections
from the parietal transformation layer to the parietal window such that the “normal” top-
down weights connecting these regions are down-regulated, while a second, alternate set of
top-down weights are up-regulated. With no forward velocity signal, the normal top-down
connections perform reconstruction of a head-centered egocentric representation of the
model’s current spatial surroundings in the parietal window using information originating
from place cell activity. Once up-regulated by the velocity signal, the alternate set of top-
down connections performs an almost identical function, except that the representation of
space reconstructed in the parietal window is of the model’s current surroundings, but
shifted backwards slightly in the model’s egocentric space. When the next bottom-up phase
begins, the shifted spatial information, represented as parietal window activity, flows
through the transformation and BVC layers to activate place cells that correspond to the
location slightly ahead of the model’s current location. This process repeats itself during the
next top-down/bottom-up cycle until the velocity signal dissipates, resulting in a continuous
relocation of the model’s internal representation of its location in space. Further details of
this updating procedure can be found in the appendix.
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Learning in the model
The purpose of our model is to reproduce experimental data and generate novel predictions
of spatial behavior in adult animals, rather than to account for learning in a biologically
realistic manner. Hence, we use a simplistic Hebbian learning procedure that associates
together pre-specified activation patterns in each layer of the model, in order to train all of
the model connection strengths except for those involved with spatial updating/mental
exploration. The latter connection strengths are calculated as described in the appendix.
Briefly, learning for the remainder of the weights involves positioning the model at
numerous random locations and heading directions within an environment while, at each of
these locations, sequentially directing attention to each landmark segment viewable from the
current location. For each attending event at each location, appropriate activation patterns
are imposed upon the model layers and connection strengths between neurons are updated
via a simple correlational rule. Once training is complete, weights are normalized. A
detailed description of the training procedures is provided in the appendix.
It should be noted that the transformation circuitry in our model is only trained once, but the
medial temporal component is retrained on each unique environment in the simulations
reported here. Training on multiple environments with the relatively small-scale models
used here can result in a degradation of information when it travels through the
transformation circuitry, and activation of an incorrect hippocampal attractor state. This
problem could be addressed by including a greater number of model neurons in the
transformation layer. Additionally, a larger-scale version of the medial temporal lobe portion
of the model should, in principle, be capable of storing multiple environments in distinct
subsets of place cells (a possible role for the dentate gyrus and CA3 recurrent connections,
McNaughton and Morris 1987; Samsonovich and McNaughton, 1997). There is no reason to
expect that the simultaneous storage of attractor states corresponding to multiple
environments would affect any of the results we obtain from the model in this paper.
Simulation 1: Recall of landmarks and geometry in hemispatial neglect
Methods
In order to simulate representational neglect (see Introduction and Bisiach and Luzzatti,
1978), we first tested the ability of the intact model to recall environmental geometry and
landmark identity. This was accomplished by first training the medial temporal component
of the model on the simplified Cathedral Square depicted in the upper left panel of figure 7.
During training, the allocentric reference frame was taken to be aligned with this depiction
of the environment so that its y-axis would be perpendicular to the inward facing walls of
buildings 1 and 3, but parallel to the inward facing walls of buildings 2 and 4. In reality, it is
likely that the orientation of the allocentric reference frame within the environment would be
set by the head direction system alignment when the animal first experiences the
environment. Once training was complete, the model was cued to imagine itself facing the
cathedral in the trained environment by injecting appropriate activation into the head
direction, parietal window and perirhinal identity layers. Cueing activation for the parietal
window was calculated by applying equation A5 (in the appendix) to a discretized linear
boundary, representing the front of the cathedral, located directly in front of the model in the
egocentric reference frame. Similarly, cueing activation for the perirhinal neurons was
calculated from equation A3, with the cathedral (building identity 1) being the attended
landmark. Finally, it was assumed that the cathedral is sufficiently salient that cueing its
location relative to the subject is enough to orient the head direction system. Thus, activation
for the head direction layer was calculated from equation A6, with the heading-direction,
φ
,
set to zero, indicating perfect alignment between egocentric and allocentric reference
frames. The cueing activations were applied to the model for two full bottom-up/top-down
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cycles, after which they were down-regulated, and the retrieved attractor states in the head
direction system and hippocampal place cell layer maintained the model’s parietal window
representation of the imagined geometry of the environment.
In order to “ask” the model to identify the boundaries that would be visible from the current
viewpoint (see figure 7), we simulated the focus of attention along four different directions:
left, right, forward and backward. In each direction the corresponding activation calculated
from equation A17 was injected directly into the parietal window. During a subsequent
bottom-up phase this activation flowed forward through the transformation and
parahippocampal layers to activate the correct perirhinal identity neuron. For example, in the
case of rightward attention, the correct response would be perirhinal activity corresponding
to building 2, see Appendix for details.
Next, the model was cued to imagine itself in the square facing away from the cathedral.
This was accomplished by focusing attention on a boundary directly behind the model in the
parietal window, while simultaneously activating the perirhinal neurons representing the
visual features of the cathedral, and the allocentric head direction 180 degrees away from the
current egocentric frame
Once it was confirmed that the model could identify surrounding landmarks from different
viewpoints, hemispatial neglect was simulated by performing a random knock-out of 50% of
the parietal window neurons representing the left side of egocentric space, and then
repeating exactly the same procedures as just described for testing the intact model.
Results and discussion
The ability of the intact model to recall environmental geometry and landmark identity,
when cued that it was facing the cathedral, is shown in figure 7. The top four panels show
the activity in the various network layers averaged over one full cycle after the removal of
the cueing activity. Although the spatial resolution of the model’s representation of the
environment is coarse, the geometry represented in the parietal window is roughly correct.
The bottom panel of figure 7 shows the activity of perirhinal neurons at the end of a bottom-
up phase. Perirhinal activity is plotted with open circles for leftward attention, asterisks for
forward attention, crosses for rightward attention, and triangles for backward attention,
indicating that the model can identify all landmarks correctly. Performance of the intact
model when cued that it was facing away from the cathedral is shown in figure 8. The
resultant activities of the various network layers averaged over a full cycle after down-
regulation of cueing inputs are shown in the top four panels. Once again the model formed
the correct egocentric representation of spatial information in the parietal window and
directed attention resulted in the correct identification of the surrounding boundaries. For
example, when attention was directed to the egocentric right, the identity of building 4 was
activated in the perirhinal layer. Building 4 would be to the right of the model if it were
facing away from the cathedral.
Results of the simulations with the lesioned model, simulating hemispatial neglect, are
shown in figures 9 and 10, corresponding to figures 7 and 8 respectively. From these results
it is clear that the model could identify landmarks to its right, but not to its left, regardless of
its imagined heading direction. These simulation results are consistent with a central tenet of
our model, namely, that allocentric representations of space are formed in long-term
memory and are transformed into egocentric views as needed, in the service of memory
recall and imagery. Moreover, our model provides a mechanistic explanation for patterns of
deficits observed in perceptual and representational neglect patients, a previously perplexing
phenomenon in neuropsychology. Both the long-term memory representation and the
transformation mechanism are intact, whereas the egocentric representation projected from
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long-term memory, and/or the transformation mechanism itself, is faulty. This could arise in
patients either from a lesion to the pathway from the transformation circuit to the parietal
window (resulting in pure representational neglect) or a lesion to the parietal window itself
(resulting in both perceptual and representational neglect). Pure perceptual neglect in the
absence of representational neglect could arise from a lesion along the sensory or motor
pathways projecting into and out of posterior parietal cortex. Testing of these predictions
based on currently available data is difficult because of the extensive lesions suffered by
most patients suffering from unilateral neglect. For the case of perceptual neglect, recent
studies indicate that a disconnect between parietal cortex and prefrontal areas (Doricchi &
Tomaiuolo, 2003; Thiebaut de Schotten et al., 2005), or between parietal cortex and medial
temporal regions (Bird et al., 2006) is critical to a realization of the phenomenon. However,
we are unaware of any data that so clearly indicate which regions of the brain must be
damaged in order to induce pure representational neglect, the focus of the current set of
simulations.
Simulation 2: Spatial updating during physical and mental navigation
One of the key functions of the model is its ability to perform spatial updating of its internal
representations of location, given a motion signal. Spatial updating is critical for navigation
in the absence of perceptual input (path integration), as well as for mental imagery involving
viewpoint changes, and path planning. Spatial updating should allow relatively normal
navigation and place cell firing over short durations in the absence of perceptual input, as
well as accounting for data on spatial updating such as that of Wang & Brockmole (2003)
described in the introduction. In our model, path integration occurs outside of the
hippocampus, by updating the parietal egocentric representation. Further, the same
machinery accounts for the process of mental navigation by generating an imagined motor
signal in place of the efference/ proprioceptive/ vestibular signal generated by actual motion.
This should allow the model to address performance and reaction time data in tasks where
the subject is asked to respond from a different imagined viewpoint and/or location (e.g.
Diwadkar & McNamara, 1997; Easton & Scholl, 1995; Rieser, 1989; Shelton & McNamara,
2001), and to simulate some aspects of spatial planning.
Methods
In order to simulate spatial updating or mental navigation, the medial temporal component
of the model was trained on the “two-room” environment shown in the upper left panel of
figure 11, with the allocentric reference frame taken to be aligned with the vertical axis of
the environment as depicted. The training procedure and architecture for this component of
the model were identical to those used in the previous set of simulations, except that in
addition, within the parietal window, the velocity-gated translational weights given by
equation A9, and the rotational head direction weights, trained as described in the appendix,
now come into play.
After training was complete, the model was first cued to a location near to and directly
facing wall 1. Such cueing would be equivalent to asking the model to
imagine
itself facing
wall 1 in the two-room environment. This was accomplished as in the previous simulations
by injecting appropriate activations into perirhinal, head direction, and parietal window
neurons for two full cycles. Attention was then focused along four different directions,
leftward, rightward, forward and backward, to demonstrate that the model could identify the
surrounding landmarks from memory.
Next, we simulated spatial updating after several steps of imagined egomotion. The same
situation could arise during real navigation if an animal spontaneously loses sensory
information about its
real
surroundings (e.g. navigating in the dark). In either case, attractor
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states in the head direction system and in the hippocampal formation of our model are able
to maintain an internal representation of its real/imagined surroundings. Mental exploration
or spatial updating based on this self-sustaining internal representation was simulated in the
model by a series of eight egomotion steps. This egomotion, if assumed to be generated by
real idiothetic information, would correspond to spatial updating, or if generated by a mental
equivalent, would correspond to mental exploration. In the first step, to simulate making a
180 degree turn, a counter-clockwise rotational velocity signal lasting for 150 time units
gated the rotational head direction weights until the model’s egocentric representation of
space rotated by a full 180 degrees. In the second step, to simulate forward egomotion, a
translational velocity signal lasting 135 time units gated the transformation to parietal
window translational weights causing the model’s egocentric representation of the locations
of boundaries to translate backwards. Similarly, a further six egomotion steps were
performed to complete the simulation.
As a control, we compared spatial updating in imagined versus sensory-driven navigation.
Although the model’s ability to perform spatial updating/mental exploration on internally
maintained representations of space is of primary interest, it must also function in a
consistent way during real navigation through a familiar environment with intact sensory
information. Thus, we simulated the same situation as above but in the presence of accurate
sensory cues during the eight steps of egomotion. In this case, sensory information
corresponding to visible boundaries calculated from equation A5 was simultaneously
injected into the parietal window during egomotion.
Results and discussion
The ability of the model to retrieve the appropriate context in the two-room environment,
when asked to imagine itself facing wall 1, is shown in figure 11. Network activity averaged
over a full cycle after down-regulation of the cueing inputs can be seen in the top four panels
of figure 11. The results of the four directed attention events are shown in the bottom panel
of figure 11, indicating that the model could also identify the surrounding landmarks.
The performance of the model after several steps of imagined egomotion is shown in figures
12 and 13. Figure 12 shows activation in the various network layers averaged over one full
cycle following the first two egomotion steps. The remaining six steps brought the model’s
internal representation of space to that shown in figure 13, where it was nearby and facing
wall 2. Three directed attention events show that the model could correctly identify
surrounding boundaries from this new viewpoint (see bottom panel of figure 13).
In the case of sensory-driven navigation, the analogous results to figures 11-13 are shown in
figures 14-16 respectively. Results of the sensory-driven simulations after 8 steps of
egomotion are nearly indistinguishable from the corresponding results with imagined
egomotion.
The fact that an egocentric translational velocity signal causes spatial updating/mental
navigation to occur at a constant velocity will be discussed in more detail with respect to
Simulation 4 and in the General Discussion. Here we simply note that it is consistent with
the reasonably accurate (if scaled) correspondence between mental navigation times and
actual navigation times (see e.g. Ghaem et al., 1997; Kosslyn, 1980).
Simulation 3: Place cell firing with head direction cell lesions
In Simulations 1 and 2 we compared our model against behavioural data. The purpose of
simulations 3 and 4 was to evaluate the adequacy of our model in explaining and predicting
data at the level of single-unit recordings. For this third set of simulations the static model,
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i.e., in the absence of egomotion, is evaluated with respect to place cell firing after lesions to
the head direction system. In Simulation 4 the model is evaluated under conditions of cue
conflict between direct sensory and path-integrative inputs.
Calton et al. (2003) have shown that rats with lesions to the anterodorsal thalamic nuclei or
to the postsubiculum, two locations where head direction cells have been found, show
altered place cell firing characteristics when compared with intact animals. Although
variations in place cell firing properties between the two lesioned groups were seen, there
were a number of characteristics in common to both groups. Specifically, place cells in both
groups showed roughly normal in-field firing, but elevated out-of-field firing. Additionally,
this out-of-field firing showed dependence on heading-direction.
In order to understand how our model could address the results of Calton et al. (2003) it is
useful to return briefly to the description of how incoming sensory information activates the
correct place cell attractor states. Recall, we have assumed that incoming information about
environmental geometry first reaches the egocentric parietal window representation before
being transformed via the transformation layer into an allocentric BVC representation. The
BVC pattern, in conjunction with perirhinal activity, then activates the appropriate
hippocampal attractor state. This transformation relies upon a gating mechanism driven by
the head direction system that will be clearly disrupted if head direction cells are destroyed.
Thus, under normal circumstances, a given pattern of activity in the head direction system
allows only one transformation sub-layer to project activity onto the BVC layer. However, if
the former is damaged its gating function will be compromised, reducing the activity
received by the BVC layer from the correct transformation sub-layer and increasing the
activity from other sub-layers. Depending on the extent of the lesion to the head direction
system, the garbled BVC representation could still overlap significantly with that required to
activate the appropriate attractor state given the model’s current sensory information, or it
could be that the overlap is very small. In intermediate cases the correct hippocampal place
cells might receive enough activation to fire but other neurons might be driven past their
firing thresholds as well.
Methods
A realistic simulation of the effects of lesions to the head direction cells in our model is not
possible because of the use of a single inhibitory interneuron that causes each head direction
cell to inhibit all transformation sub-layers equally. A more realistic circuit would employ a
population of inhibitory interneurons that were connected randomly within the constraint
that they would achieve the same gating function (in combination with excitatory head
direction connections to the transformation layer). We did not employ such a population
because, given the unnatural training methods used, it would have behaved like a single unit
anyway. With a more natural configuration, partial lesions to the head direction system
would result in reduced excitation to the selected transformation sub-layer and decreased
inhibition to random regions of the overall transformation layer. To simulate the equivalent
effect in our model, for each lesioned head direction, the excitatory head direction input to
the corresponding transformation sub-layer was reduced, while the inhibitory input to a
random selection of other transformation sub-layers was decreased. See Appendix for
details.
Since the lesioning procedure does not involve the medial temporal structures, the latter
region was trained once on the “box” environment shown in figure 17. The model was then
localized at numerous positions within the environment by injecting appropriate egocentric
sensory information from all of the environmental boundaries into the parietal window
neurons. At each location the sensory input was maintained for one top-down/bottom-up
cycle and the activity of a selected place cell was recorded and averaged over the bottom-up
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cycle. This procedure was performed for two simulated head directions, one of which
corresponded to perfect alignment between egocentric and allocentric representations, and
another that corresponded to perfect anti-alignment between the two representations.
Results and discussion
The average firing rates for a model place cell recorded when the lesioned model was
localized at numerous locations within a rectangular sub-region of the “box” environment
are depicted in figures 17 and 18. In figure 17 these rates correspond to the aligned heading
direction, while in figure 18 the results correspond to the anti-aligned simulation condition.
Clearly, the firing field of the model neuron varied with simulated head direction, and
moreover, its peak-firing location for either head direction did not correspond to the location
where the cell would have attained its maximal firing rate in the non-lesioned model
(marked with an ‘X’ in both figures). In addition, for the aligned condition, the cell
exhibited a firing maximum in one location, but with an additional area of elevated firing
near ‘X’. These data are qualitatively similar to the data shown in figure 4B of Calton et al.
(2003).
Our model makes two unique predictions regarding the outcome of experiments similar to
those of Calton et al. (2003). First, a place cell that has a pre-lesion preference for a location
about which there is a high degree of rotational symmetry (e.g. the center of a cylinder)
should maintain its place preference post-lesion. Conversely, place cells that show pre-lesion
preferences for locations of low rotational symmetry should tend to show shifts in their
preferred locations after a lesion. An example of this latter effect is seen clearly in the
simulation presented in figures 17 and 18. Second, the relative firing rates for place cells
when measured at locations of high rotational symmetry should demonstrate little
dependence on head direction after a lesion. For example, if cell A demonstrates a high post-
lesion firing rate at the center of a cylinder for a given head direction, and if cell B
demonstrates a low firing rate at that location and head direction, then for all other head
directions, cells A and B should show similar relative firing rates at that location.
Conversely, the relative firing rates for place cells when measured at locations with lower
levels of rotational symmetry should exhibit higher levels of head direction dependence after
a lesion.
In order to understand these predictions, one only needs to note that each transformation
sub-layer contains a representation of the same egocentric space, but rotated about the
origin. Therefore, if the egocentric parietal window representation shows a reasonable
degree of rotational symmetry at a given location, then allowing extra regions of the overall
transformation layer to project to the BVCs will not have a large effect on the resultant
geometric information represented there, regardless of head direction. Hence, a place cell
that fires maximally/minimally at such a location before a head direction system lesion
would still receive high/low levels of stimulation there after a lesion; moreover, because of
the rotational symmetry, it will do so for all head directions.
Simulation 4: Place cell firing with conflicting visual and path-integrative
inputs
The basis of the medial temporal component of our model was derived from a simple feed-
forward model of place cell firing (Hartley et al., 2000; O’Keefe & Burgess, 1996) driven by
input from BVCs. This earlier model included a number of simplifications, one of which
was that BVCs and therefore place cell firing rates were independent of firing history.
However, memory in general, and path integration in particular, make important
contributions to place cell firing, in addition to immediate sensory perception such as vision,
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olfaction etc. For example, place cells can continue to fire normally in the dark (O’Keefe,
1976); path integration, distant visual cues and multi-modal local cues can be pitted against
each other to control the orientation of place cell firing (Jeffery, Donnett, Burgess &
O’Keefe, 1997; Jeffery & O’Keefe, 1999); and congenitally blind rats show normal place
fields once they have explored the polarizing environmental cues (Save, Cressant, Thinus-
Blanc & Poucet, 1998).
Here we have coupled the medial temporal model to a parietal system capable of spatial
updating. An obvious test of this extended model is to see if it can capture the joint effects of
path integration and sensory perception on place cell firing, thereby extending the simple
feed-forward place cell model. Another line of evidence for the differential contributions of
path integration and sensory perception to place cell firing comes from Gothard et al.
(1996), who examined the activity of hippocampal place cells in rats running along a linear
track. By varying the track length during recording sessions they were able to pit sensory
and locomotor cues against each other. In our final set of simulations, we sought to compare
the performance of the model to Gothard et al.’s data.
Gothard et al. (1996) trained rats to run back and forth along a narrow, elevated track with
food cups at either end. One food cup was fixed directly to one end of the track while the
other was fixed to the floor of a sliding box that could be in any one of five locations (
box1
thru
box5
), thereby changing the overall track length (see the left panel of figure 19). Rats
were habituated to the apparatus in the maximum length, or
box1
state, for three to five days
prior to recording. During a recording session, an animal was placed in the box at one of the
five positions and allowed to run to the fixed food cup (outbound journey). The box was
then moved to a new position before the rat turned around to make the return journey
(inbound journey). Most cells fired prederentially in one direction of running, consistent
with previous experiments on linear tracks (McNaughton, Barnes & O’Keefe, 1983;
O’Keefe & Reece, 1993). The firing profile for each cell was calculated separately for all
types of journey (e.g.,
box1-out, box2-out, box1-in, box2-in
, etc.), and was compared with
the corresponding
box1
profile. Specifically, the amount by which the peak firing location
for a given cell was shifted from its preferred location in the
box1
condition was plotted
against the corresponding shift of the box relative to its
box1
position, see figure 19. This
measure is sensitive to whether the place field shifts with the movable box, or remains at a
fixed location relative to stationary cues, but note that deformations in firing field shape
occurred, such as bimodal fields, as well as simple shifts. By fitting a regression line to the
data for a given cell across box positions, a
displacement slope
, normalized to range
between 0 and 1, was calculated. A slope of 0 corresponds to firing peaked at the same
location relative to the fixed food cup in all conditions, while a slope of 1 corresponds to
peak firing at the same location relative to the movable box, regardless of its position. Thus
the movable box
controls
the location of firing fields with a large displacement slope, while
the fixed food cup and other room cues control the location of fields with small
displacement slopes.
Gothard et al.’s (1996) displacement slope results for inbound and outbound selective
neurons are shown in figure 20 along with some sample firing fields. Neurons that fired near
the box or the cup in the original configuration continued to fire near the box or cup in the
other configurations. Similarly, cells that fired in between the two cups did so in all
configurations, except on the shortest journeys when they did not fire at all. However, for
most of the distance traveled on a given journey, place cell firing appeared to be
predominantly controlled by the landmark which the animal was moving away from. For
outbound journeys, firing peaked near to the box in the
box1
configuration have
displacement slopes around 1, and this value gradually decreases to zero for neurons with
peak firing positions farther away from the box. However, the slope value remains above 0.5
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for peak firing locations much more than halfway down the track from the box. This
additional influence of the cue the rat is running from is also clearly evident for the inbound
journeys where most neurons, excepting those with peak firing very close to the box, are
controlled by the cup, showing displacement slopes close to zero.
The BVC model of place cell firing (O’Keefe & Burgess, 1996; Hartley et al., 2000)
predicts much of Gothard et al.’s pattern of data, e.g., that the location of maximal firing will
tend to remain a fixed distance from the nearer of the two boundaries and how the fields
stretch, develop sub-peaks, reduce in firing rate and disappear when the component BVCs
fail to coincide in one or other new configuration. However, the increased influence of the
boundary behind the rat compared to the one in front is not captured by this model (also
noted in O’Keefe & Burgess, 1996). These results appear to require an interaction between
BVC’s responsive to the inconsistent visual cues and path-integrative locomotor information
(see also Redish et al., 2000), consistent with the idea that both path-integrative and
perceptual inputs are required to determine the hippocampal representation of location
(O’Keefe & Nadel, 1978). Here we investigate the behavior of the model, which now
includes both BVCs and motion-related spatial updating, in the Gothard et al. paradigm.
We model initial place cell firing when the animal is placed at either end of the apparatus as
consistent with the place cell firing for that location within the full-length track. This
assumption is reasonable given that the majority of local cues available at either location are
consistent with this representation. These cues consist of the three box walls for the box and
all the other room cues at the fixed food cup. Upon leaving the start position for a given
trial, input from both locomotion-related updating and from visual cues combine to update
the animal’s internal representation of its position. Within the
full-length
track (
box1
)
condition of Gothard et al.’s (1996) experiment, neuronal activity follows a “normal”
continuous trajectory through the set of states representing all intermediate locations within
the full-length track and terminating with the state corresponding to the destination end of
the track. At each stage the perceptual input from both ends of the track is consistent with
the internally updated input from the previous step. In the remaining conditions (
box2-box5
)
the visible landmark ahead is closer to the rat than would be consistent with the motion-
updated representation; this causes previously unimodal place fields to reduce in peak
activity and to deform, showing a compromise between firing at a fixed distance from both
ends of the track. At the start of an outbound journey, the cues behind the rat
and
the
idiothetically updated internal representation predominantly control place cell firing, but as
the rat proceeds along the track there is an increasing influence of the nearer than expected
destination end; at some point past the midpoint of the track there will be a transition in the
cues controlling place cell firing, from the cues behind the rat to the cues in front of the rat.
For the shortest track conditions, some place cells with fields near to the “transition point”
may not fire at all, having roughly equal inputs from both ends on the full length track,
which entirely fail to overlap on the short track. In this case the inferred location of the rat
will jump from one reference frame to the other, rather than making a smooth transition.
Before describing our simulations of Gothard et al.’s (1996) experiment in detail, we note
one further piece of data. The preceding explanation predicts that if sensory information
about the nearer-than-expected destination end of the track is degraded, then the internally
updated representation of landmark positions should take precedence in the control of place
cell firing for an even longer portion of the journey. Consistent with this, when rats
performed Gothard et al.’s linear track task in darkness, it was found that the cue from
which the rat was running maintained control over place cell firing for a greater portion of
the journey than it did in the light (Gothard et al., 2001).
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Methods
To simulate the key aspects of the linear track environment of Gothard et al. (1996), we
trained our model on a symmetric environment consisting of two “boxes” that open towards
each other, as in the lower left/middle panels of figure 21. Due to the absence of surrounding
room cues, either box can be considered the movable box. In this way we were able to
perform one set of simulations representing both outbound and inbound journeys. Medial
temporal and parietal connections were set in the same manner as for the previous
simulations. Before performing actual simulations of the Gothard et al. (1996) data, the
forward translational velocity of the place cell representation under application of an
egocentric velocity signal had to be calibrated. This was accomplished by applying the
velocity signal after cuing the model to localize itself near box 1, facing box 2 (see figure
21) until place cell firing indicated localization near box 2. The model’s representation of its
own location within the environment was calculated at any given instant by averaging the
coordinates associated with maximally active place cells. By fitting a regression line to the
roughly linear position-time data (see the rightmost panel of figure 21), a velocity of 0.044
space units per time unit was found. Such a simulation would correspond to the model
mentally exploring this familiar environment, or performing spatial updating during actual
locomotion in the absence of visual cues.
In the next step of the simulation the model was cued to a location two units away from box
1 along the direction towards box 2, while facing box 2. To simulate a shortened track,
sensory input corresponding to box 2 was applied directly to the parietal window layer at
either 0, 2, 4, 6, or 7 units closer to the egocentric origin than what would be consistent with
the model’s learned representation for that location (see the top and bottom panels at the left
of figure 22 for an example). For our initial set of simulations, sensory information
corresponding to box 1 was not applied because we assumed that this landmark did not have
the salience of the target landmark and a rat’s field of view is only approximately 300
degrees. Locomotion was simulated by turning on the forward velocity signal
(corresponding to a velocity of 0.044 space units/time unit) and moving the sensory input
corresponding to box 2 towards the origin of the parietal window coordinate system at the
same speed. When this sensory input came within one unit of the origin, its movement was
stopped, the velocity signal was turned off and the model was allowed to relax for 50 time
steps before sensory input was down-regulated.
During locomotion, the rat’s head tends to bob up and down, so that it might well receive
visual information from box 1. With this in mind we performed a second set of simulations
identical to those just described, but with input representing box 1 also being applied to the
parietal window component of the model. For these simulations, the additional input
representing box 1 was initially configured so as to represent this landmark at 2 units behind
the animal. During simulated locomotion, this “sensory” input was moved through the
parietal window coordinate system at the same speed and in the same direction as the input
representing box 2.
Finally, we performed simulations identical to those above, but with weakened overall
connection strengths for the connections terminating on the BVC layer (see Table 1 for
parameter values). The motivation for this was that a smaller proportion of space was filled
with landmark segments in the linear-track environment than in the previous two
environments. This was found to result in a very low resolution representation of space due
to reduced lateral inhibition in the BVC, transformation and parietal window layers.
However, results for both sets of simulations (with and without weakened parameters) are
qualitatively similar, except for one difference as discussed below. Furthermore, a more
realistic simulation in which the BVC and parietal window layers covered a more extensive
region of space would have allowed for the inclusion of distal landmarks (room walls, etc.).
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Such inclusion would have generated increased lateral inhibition and a sharper
representation of space without the need for altering any connection strengths.
Results and discussion
Results for the
6-unit-closer
trial with no box 1 sensory information are shown in figure 22.
Of particular interest is the fact that the maximum velocity of the place cell activity was
0.058 space units/time unit, or about 32% faster than when no inconsistent sensory input
was present (see the rightmost panel of figure 22). Therefore, as with the data reported by
Gothard et al. (1996), place cell activity was initially under the control of the nearest
landmark, but during locomotion it “caught-up” to what it should have been had it been
primarily under the influence of the target landmark (box 2).
In addition to recording the trajectory of place cell activity, the activity of eleven cells,
representing equally-spaced locations within the environment, were recorded. If the
simulation trials are considered as outward journeys, then we can plot the firing profiles in a
way similar to that used by Gothard et al. (1996) to calculate displacement slopes. In figure
23 the firing profiles for four of the eleven recorded place cells in the condition with no box
1 sensory information are shown along with displacement slopes for all eleven in both
conditions. The same information is plotted in figure 24 for the weak BVC input
simulations. For the weak BVC input condition, place cell activity of the navigating model
in the shortest track length trial hopped from one representation of location within the
longest environment to another, resulting in a complete lack of firing from one of the four
selected cells. Given the symmetry of our environment, displacement slope data can be
determined for inward journeys by transforming the data for outward journeys as follows:
(1)
where
DS
(
x
) is the displacement slope for a neuron with peak firing position,
x
, in the
box1
condition, and
x
is normalized to range between 0 (at the movable box) and 1 (at the fixed
food cup). The transformed curves are shown in the lower right panel of 23 and 24. Notice
that both sets of simulation-generated displacement slopes show patterns consistent with
Gothard et al.’s results. In particular, the landmark that the animal is moving away from
maintains considerable control over place cell firing until the target landmark is nearly
reached. For the normal BVC input conditions this effect is similar whether or not we
assume the animal has access to sensory information from both box 1 and 2. For the weak
BVC input simulations we obtain a stronger effect if we assume the model has sensory input
from both boxes.
In summary, our model performs in a manner consistent with the Gothard et al., (1996) data.
In a subsequent experiment, the influence of the cue the rat is running from was seen to last
for a constant time through the run rather than for a constant distance (Redish et al., 2000).
This indicates either a time-limited usefulness for path integration (see e.g. Etienne, Maurer,
& Seguinot,, 1996), or (as argued for in Redish et al., 2000) some temporal inertia in place
cell firing possibly due to attractor dynamics (which can be seen under other experimental
circumstances, e.g. Wills, Lever, Cacucci, Burgess & O’Keefe, 2005). Simulations
comparing time and distance in this way were not performed (we used constant velocity),
and remain for future work.
Finally, we compared our full model to a model lacking path integration. By considering
only the part of the full model consisting of the BVCs, the place cells, and the feed-forward
connections from the BVC to place cell layer, we were able to verify that the simple BVC
explanation of Gothard et al.’s (1996) results does not produce the noted asymmetry.
Specifically, we simulated navigation along each track length by providing direct input to
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the BVC neurons corresponding to the box 1 and box 2 landmarks, and then translated this
input through the BVC coordinate system at 0.044 space units/time unit. In this way BVCs,
and hence place cells, were driven directly by sensory input and the model’s current
representation of space was not affected by previous representations of space or idiothetic
information. Displacement slope curves for these simulations were calculated as above and
plotted in the lower two panels of Figure 23 and 24. Notice that these curves are
approximately symmetric about the mid-point of the full-length track. Thus the simple BVC
model, in which distances to boundaries in allocentric directions are the only concern, is
insufficient to produce the dependence on running direction noted in Gothard et al. (1996),
O’Keefe & Burgess (1996), or Redish et al. (2000).
In the current model perceptual inputs and motion-related updating combine to influence the
animal’s internal representation of location, and the operation of this mechanism seems to be
consistent with the relevant existing data from place cell recording. The functional
architecture of the current model was largely informed by thinking about imagery and
planning in human spatial memory, however the simulations reported here indicate that it is
also able to explain data at the single-unit level of description.
General Discussion
We have outlined a model of the neural mechanisms underlying spatial cognition, focusing
on long-term and short-term spatial memory and imagery, egocentric and allocentric
representations, visual and ideothetic information, and the interactions between them. We
proposed specific mechanisms by which long-term spatial memory results from attractor
dynamics within a set of medial temporal allocentric representations, while short-term
memory results from egocentric parietal representations driven by perception, retrieval and
imagery, and can be investigated by directed attention. However, perhaps our main novel
contribution is to propose specific mechanisms by which these systems interact. Thus we
propose that encoding and retrieval require translation between the egocentric and
allocentric representations, which occurs via a coordinate transformation in posterior parietal
and retrosplenial cortices, and reflects the current head direction. In our model, the
hippocampus effectively indexes information by real or imagined location, allowing re-
construction of the set of visual textures and distances and allocentric directions of
landmarks consistent with being at a single location (see also King et al., 2004). In turn,
Papez’s circuit translates this representation into an egocentric representation suitable for
imagery according to the direction of view (and also translates from egocentric perception
during encoding of the allocentric representation). For partially related models, see Becker
and Burgess (2001), Burgess et al., (2001), Recce and Harris, (1996) and Redish (1999). We
further propose that modulation of the allocentric to egocentric translation by motor
efference allows “spatial updating” of egocentric parietal representations, which in turn can
feedback to cause updating of the medial temporal representations. Finally, the generation of
mock motor efference (e.g. representing planned eye movements) in prefrontal cortex allows
mental exploration in imagery, making a potential contribution to spatial planning. The
temporal coordination of the alternating interaction of the temporal and parietal regions was
assumed to be provided by the theta rhythm.
For concreteness, and to demonstrate the actual ability of the theory to bridge between
single-neuron and systems neuroscience and behavioural data, we implemented it as a fully
specified neural network simulation for the case of long-term, hippocampally dependent,
spatial memory and its interaction with short-term working memory and imagery. Our
simulations provide straightforward explanations for a number of experimental results. The
first provides a neural implementation of the idea that representational neglect results from a
damaged egocentric window into an intact long-term spatial memory system (see also
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Baddeley & Leiberman, 1980). From the model architecture we are able to suggest that
unilateral lesions to the precuneus, retrosplenial cortex, parietal area 7a, areas connecting 7a
or retrosplenial cortex with parahippocampal gyrus, or combinations of these areas have the
potential to generate representational neglect. However, currently available patient data
makes this prediction difficult to test. The second simulation provides a neural
implementation of self-motion related spatial updating of object locations in memory and of
imagined navigation and route planning. The third shows that our interpretation of the role
of head direction in memory is consistent with the effects of lesions to the head direction
system on single unit responses in the hippocampus. With this interpretation we are also able
to make two simple predictions about the outcomes of similar experiments, thus allowing
the translation component of our model to be tested directly. The final simulation shows that
our proposed mechanism for integrating sensory information and self-motion also provides
an explanation for single unit responses in situations of conflicting sensory and ideothetic
information (Gothard et al. 1996). In the following, we discuss the implications, predictions
and limitations of the model with respect to the wider literature on the neural bases of spatial
cognition and memory more generally.
Temporo-parietal interactions, planning and imagery
Our specific model of the temporo-parietal interaction has some straightforward
implications for functional anatomy. Thus, it explains why Papez’s (mammillo-anterior-
thalamic-medial temporal) circuit is required for ‘episodic’ recollection into rich visuo-
spatial imagery (Aggleton & Brown, 1999), and also provides the head direction signal in
rats (Taube, 1998). It also suggests a functional role for retrosplenial cortex and intraparietal
sulcus, which are well positioned to integrate or buffer the translation between egocentric
and allocentric representations (Burgess, Becker et al., 2001), or correspondingly path
integrative and mnemonic information (Cooper & Mizumori, 2001). Cooper & Mizumori
(2001) and Maguire (2001) provide evidence that lesions to the retrosplenial cortex, an area
interconnected with parietal and medial temporal regions (Kobayashi & Amaral, 2003;
Wyss & Groen, 1992), do indeed impair the navigation of rats and humans under such
circumstances. In humans, the intimate link between spatial imagery and navigation is made
clear by the correlation of impairments in these two faculties following unilateral damage
(Guariglia et al., 2005). Finally, our model proposes a role for the theta rhythm in
coordinating the flow of information between medial temporal and parietal components of
the model. Thus, “top-down” activation from medial temporal to parietal areas occurs at one
phase of theta, while “bottom-up” activation from parietal to medial temporal areas occurs at
the opposite phase of theta. A related proposal relates hippocampal encoding and retrieval to
opposing phases of theta (e.g., Hasselmo et al., 2002), corresponding to our bottom-up and
top-down phases respectively. In our model, spatial updating occurs over repeated top-down
and bottom-up cycles as each (top-down) translation from allocentric to egocentric
representations maps to locations adjusted for the subject’s velocity and then passes
(bottom-up) back to update the allocentric representation.
In order to plan routes through complex environments, the brain must make use of long-term
memories of the layout of those environments. Route planning also requires the ability to
perform mental navigation: to imagine moving in a given direction, and the consequences of
that action. Thus the task in our second set of simulations, involving mentally generating a
velocity signal or “mock motor efference”, could be viewed as mental exploration of a
familiar environment. This exploration would be useful for path planning and many other
tasks. For example, this may be how people accomplish the task of Wang & Brockmole
(2003) described in the introduction. Recall that in this task subjects were led along a path
through a familiar environment and asked to point to occluded landmarks at various
predetermined times. It was found that when subjects could not accurately point to a given
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landmark, they often could do so if allowed to walk to some point further along the path
from which the landmark was still occluded. Within the framework of our model, subjects
may have been mentally navigating from their current location to a location from which the
occluded landmark was visible. By integrating the direction of the mentally generated
velocity signal, a pointing direction could be generated. However, if the mental path was too
long or complex, then the calculation would be swamped by cumulative error. In physically
moving further along the path, subjects may have been simplifying the task by reducing the
amount of mental navigation required.
Within the framework of route planning, a final prediction of the model presented here is
that damage to connections between parietal and medial temporal cortices would impair the
ability of an organism to navigate to occluded landmarks in familiar environments. This is
because, without access to long-term spatial memory, the parietally supported egocentric
window would only have access to short-term memory and direct sensory information,
rendering the organism unable to mentally explore the familiar environment beyond regions
very recently encountered. Equally, we might expect to see increased theta coherence
between temporal and parietal regions as a function of this type of actual, or mental,
navigation.
Differences between spatial updating and path integration in temporal and parietal
cortices
Path integration can be defined as the ability of an organism to keep track of its current
location relative to its starting point as it moves around, on the basis of idiothetic
information alone, while spatial updating refers to the ability to also keep track of other
locations within the environment, again using idiothetic information alone (for examples,
see Etienne et al., 1998; Loomis et al., 1993; Mittelstaedt & Mittelstaedt, 2001;
Morrongiello, Timney, Humphrey, Anderson & Skory, 1995). However, either process
could operate either by individually updating the required egocentric location(s) relative to
oneself, or by updating an allocentric representation of one’s own location relative to the
environment. Both types of updating are probably available in parallel, with the former
suitable for small numbers of locations and short movements and the latter for updating
multiple locations and longer movements, when perceptual support from the environment is
unavailable. Thus spatial updating over short timescales and small movements (e.g. less than
135 degree rotation) in unfamiliar environments appears to operate on transient egocentric
parietal representations, showing independent accumulation of errors in the locations of
different objects (Wang and Spelke, 2000; Waller and Hodgson, 2006). By contrast, spatial
updating over longer durations or movements or in very familiar environments appears to
operate on a coarser but enduring allocentric representation (Mou, McNamara, Rump &
Xiao, in press; Waller & Hodgson, 2006). See Burgess (2006) for further discussion.
Corresponding to these two types of spatial updating, separate models have been proposed
for the mechanisms within each (temporal or parietal) region. Byrne & Becker (2004)
propose a purely parietal mechanism for motion-related updating of the egocentric locations
in the parietal window, which would be consistent with single unit recording and effects of
lesions within this region (see Introduction). On the other hand, strictly medial temporal
mechanisms have been proposed for updating the location of the subject relative to the
environment (see e.g. O’Keefe & Nadel, 1978; Redish, Rosenzweig, Bohanick,
McNaughton & Barnes, 2000; Samsonovich & McNaughton, 1997; Howard, Fotedar,
Datey, & Hasselmo, 2005). These latter models are supported by the recently discovered
‘grid cells’ in entorhinal cortex (Hafting et al., 2005), which appear well-suited to this task,
with the hippocampus potentially required when path integration has to be tied to
environmental locations (O’Keefe & Burgess, 2005; McNaughton et al., 2006). See
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Whishaw & Brooks (1999) and Save, Guazzelli, & Poucet (2001) for related discussion of
the hippocampal contribution to path integration.
Our model primarily concerns the interaction of parietal and medial temporal
representations, and assumes a single spatial updating mechanism derived as an extension of
this interaction. Our second set of simulations provides a detailed mechanism by which the
parietal cortex might make use of stored spatial representations in the medial temporal lobe
to provide egocentric representations of an arbitrary number of locations within a familiar
environment, and to update these locations following real or imagined self motion. Other
tasks (such as pointing to a recently seen object, or imagery for objects or actions as opposed
to environmental layout) will be purely parietal, and are not addressed by our model. Even
within tasks that depend on both regions, such as those simulated, our model will not capture
the finer distinctions between spatial updating driven more strongly by one region than the
other. Similarly, we do not distinguish the processing of discrete objects, likely more
strongly represented in parietal areas, from the processing of extended boundaries, likely
key to driving the hippocampal representation. The BVC representation used provides the
appropriate dependence of hippocampal representations on environmental geometry, but
probably does not correspond so well to some aspects of egocentric parietal representations.
The provenance of the model
We have presented a working model of spatial cognition, without really addressing how the
brain might have ‘learned’ such a solution. While a number of models of hippocampal
learning have been presented (see e.g. Becker, 2005), principles underlying the learning of
egocentric-allocentric transformations have not been firmly established. In recent work, we
have attempted to elucidate more biologically realistic principles upon which such learning
could be based (Byrne & Becker, submitted). Specifically, we have proposed two relatively
simple learning principles that, when applied to a transformation circuit similar to the one
presented here, reliably result in the generation of allocentric representations of space. The
first principle is that of
minimum reconstruction error
. That is, for a given heading direction,
the representation produced at the medial temporal lobe level should, through top-down
connections, be able to reproduce the corresponding egocentric input. The second principle
is the
maximization of temporal inertia
in medial temporal representations. This is motivated
by empirical evidence that both hippocampal pyramidal cells (Redish, McNaughton &
Barnes, 2000) and, under certain circumstances, superficial (Klink & Alonso, 1997) and
deep layer (Egorov, Hamam, Fransén, Hasselmo & Alonso, 2002) entorhinal cells exhibit a
resistance to rapid changes in firing rate. We speculate that spatial representations that vary
as little as possible in time should maximize accuracy and precision in storage, as well as
allowing more rapid spatial updating or mental exploration, because the medial temporal
representations would have to vary less rapidly to keep up with the retrieval demands. We
have tested the utility of these learning principles in two very different models, one trained
by direct minimization of a cost function using steepest descent learning, and one consisting
of a coupled network of restricted Boltzmann machines trained sequentially by contrastive
Hebbian learning (Hinton, 2002; Hinton et al, 2006). Both models were able to learn
allocentric representations of space at the medial temporal lobe output layer, and generate
good reconstructions of the egocentric input layer.
Implications beyond spatial memory
Although we have concentrated on the role of the hippocampus in spatial memory, this
structure is also known to be important in the maintenance of more general episodic
memories (for recent reviews, see e.g. Burgess, Maguire & O’Keefe, 2002; Eichenbaum,
2001; for models see Howard et al., 2005; Marr, 1971; McClelland, McNaughton &
O’Reilly, 1995; McNaughton & Morris, 1987; Treves & Rolls, 1992; Becker, 2005; among
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many others). In our model, hippocampal place cells bind the outputs of various BVCs and
visual feature units together to form an allocentric map of an environment. The attractor
dynamics of the medial temporal system then performs retrieval by allowing only those
conjunctions of visual feature, distance and allocentric direction that are consistent with
being in a single location (represented in the hippocampus). This information is then rotated,
with the aid of Papez’s circuit, to form an egocentric parietal image corresponding to a
specific direction of view, for conscious inspection. Our model is highly consistent with the
pattern of fMRI activation in retrieving the spatial context of an event (Burgess, Maguire et
al., 2001; King et al., 2005). Having defined this functional anatomy in the context of spatial
memory, we suspect similar processing occurs much more generally during any detailed
mental imagery for environmental layouts derived from long-term knowledge. This would
be consistent with reports of deficits in detailed imagery for novel or future events in
amnesic patients (Klein, Loftus & Khilstrom, 2002; Hassabis, Kumaran, Vann, Maguire,
submitted; but see also Bayley, Gold, Hopkins, Squire, 2005), and similar patterns of
activation for thinking about past and future events (Okuda et al., 2003; Addis, Wong &
Schacter, submitted). This function might relate to characterizations of episodic or
autobiographical memory in terms of retrieval of rich contextual information or feelings of
“re-experiencing”, as distinct from the imagery for simple objects and actions which is
preserved in amnesia (e.g., Rosenbaum, McKinnon, Levine, Moscovitch, 2004).
For simplicity, our simulations concerned a single familiar environment. However, retrieval
from the best matching of several familiar environments could be mediated, as described by
our model, by distinct subsets of place cells (McNaughton and Morris, 1987; Samsonovich
and McNaughton, 1997) providing a distinct attractor representation of each environment
(Wills et al., 2005). In this way, the hippocampus might be described as providing the spatial
context appropriate to recollection (O’Keefe and Nadel, 1978), explaining its role, for
example, in context-dependent fear conditioning, but not fear conditioning itself (Phillips
and LeDoux, 1992; Kim and Fanselow, 1992). An interesting prediction here is that two
situations can be identified in having different “contexts” requiring hippocampal
disambiguation, if they illicit “remapped” (Muller, 1996) patterns of place cell firing, as
occurs rapidly with dramatic multi-modal changes (Wills et al., 2005) or more slowly with
unimodal changes (Lever et al., 2002).
Of course, hippocampal neurons are probably not limited to the spatial functions we have
focused on here. For example, rat CA1 and CA3 pyramidal neurons can also respond to
various non-spatial cues (see e.g. Huxter, Burgess & O’Keefe, 2003; Young, Fox &
Eichenbaum, 1994). This ability to connect non-spatial and spatial information may allow
the association of location within an environment to various other elements of experience,
i.e. providing a spatio-temporal context to support context-dependent episodic memory more
generally (see e.g. chapters 14 & 15 in O’Keefe & Nadel, 1978). We also note that the
ability to perform spatial updating of the imagined viewpoint may both aid the process of
search during episodic retrieval and the binding of places into remembered trajectories, or
sequences, in memory for more extended dynamic episodes (see also Jensen & Lisman,
1996; Levy, 1996; Wallenstein, Eichenbaum & Hasselmo, 1998; Howard et al., 2005).
Howard et al.’s temporal context model (TCM) of memory for lists of items provides an
example of how such association across time might occur. The TCM works by associating
items to a slowly varying context representation containing history-dependent information
relating to the items themselves. Howard et al. note that this model is broadly compatible
with a spatial function for the medial temporal lobe in providing a mechanism for path
integration by representing the recent history of movements. In our model, the medial
temporal lobe could be thought of as providing the spatial context of events by representing
the actual surrounding spatial scene. Generation of more general representations of context,
such a temporal contexts, would be one way in which our model might be extended to
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include the involvement of the medial temporal lobe in memories for trajectories through
space, or in non-spatial memory.
Finally, while we have concentrated on spatial memory, the question of how long-term
memory and short-term or working memory interact is equally pertinent to non-spatial
memory. For example, although much has been learned about both long-term and working
memory for verbal stimuli, the interaction of these two systems is a topic of much current
interest (e.g. Baddeley, 2000; Burgess & Hitch, 2005). By staying within the spatial domain
where there is much data at the single-unit level, we have provided a detailed model of one
form of the interaction between long-term medial temporal and short-term parietal systems.
However, our proposals for the functional roles and interactions of the regions in question
should generalize to the generation of dynamic visuo-spatial imagery from stored verbal
knowledge. Given the slight lateralization of visuo-spatial processing to the right
hemisphere (e.g. Piggott & Milner, 1993; Smith & Milner, 1989; reviewed in Burgess et al.,
2002), we would hope that some of the mechanisms considered here might generalize to the
interaction of left medial temporal lobe long-term memory systems for narrative (e.g. Frisk
& Milner, 1990) and parietal short-term memory systems for verbal working memory.
Acknowledgments
We thank John O’Keefe, Tom Hartley and Lynn Nadel for useful discussions, and Allen Cheung for pilot
simulations. NB is supported by the MRC and Wellcome Trust, UK, and SB is supported by NSERC, Canada.
Code for the model presented herein, along with detailed comments can be found at: http://psycserv.mcmaster.ca/
beckerlab/ByrneBeckerBurgessModel/
Appendix: Mathematical details
In presenting the mathematical details of the training procedure for the model, each
component (medial temporal, transformation, etc.) will be considered separately. Following
this, the dynamical equations governing the model’s behaviour during simulation will be
presented.
Medial Temporal component
Before the model was trained on a particular environment the landmarks/boundaries of that
environment were discretized by overlaying them on a Cartesian grid with a linear
dimension of approximately 3 grid points/unit length. Any grid point that fell within half a
lattice spacing of a boundary was then marked as a
landmark segment
. This set of landmark
segments, examples of which have been presented in figures 3 and 4, constituted the training
data for the current environment. Training proceeded by positioning the model at random
locations within the environment while, at each location, sequentially directing attention to
each landmark segment that was potentially viewable from that location. For each of these
attending events at each location, appropriate firing rates were imposed on all neurons in the
medial temporal layers and connection strengths between neurons were incremented via a
Hebbian learning rule. The procedure for calculating the firing rates during the training
phase will now be considered.
For the hippocampal layer, a one-to-one correspondence was established between the model
neurons and the points on a Cartesian grid, such that each neuron fired maximally at its
preferred location. The grid points were spaced with linear density of 2 grid points/unit
length covering the relevant allocentric space for each of the environments simulated (see
figure 2 for an example). When the model was located at the location with coordinates (
x,
y
), the firing rate of the
i
th
hippocampal neuron was calculated via
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(A1)
where (
x
i
, y
i
) are the coordinates of that neuron’s preferred location. Next, for the BVC
layer, a one-to-one correspondence between the set of BVCs and a radial grid centered at the
model’s current location and covering allocentric space (see figure 4) was formed. For all
environments, this grid had a radial resolution of 1 grid point/unit length to a maximum of
16 units and an angular resolution of 51/2π grid points/rad. The contribution of a landmark
segment with allocentric coordinates (
r,θ
a
) to the firing rate of the
i
th
BVC neuron was
calculated via
(A2)
where are the allocentric co-ordinates of that neuron’s corresponding grid point, and
σ
θ
and
σ
r
are chosen to have values of (0.005)
1/2
and (0.1)
1/2
respectively. The total firing
rate of the
i
th
BVC neuron was obtained by summing equation A2 to a maximum value of 1
over all landmark segments viewable from the current location. The particular values chosen
for
σ
θ
and
σ
r
allow for reasonable spatial resolution with the model architecture; however,
the exact values of these parameters are not critical. In fact, with a sufficiently high number
of neurons covering space, the only constraint on these values would be the desired spatial
resolution of the model. It should be noted that the above definition of BVCs simplifies that
of Hartley et al. (2000) and O’Keefe & Burgess (1996) for which the sharpness of the
distance-tuning decreased with the preferred distance
r
i
of the cell. However, a similar effect
of increased influence for nearby verses distant boundaries is achieved due to the increased
angle subtended by a nearby boundary, which therefore controls the firing of a larger
proportion of the BVC population (see Barry et al., 2006). Finally, boundary/landmark
identity neurons were modeled by associating each perirhinal neuron with an environmental
landmark identity. Thus, the firing rate of the
i
th
perirhinal neuron is given by
(A3)
where,
C
PR
is set to one.
Once firing rates for a given training step (attending event) were imposed upon all medial
temporal layers, the model weights were updated via the Hebbian learning rule
(A4)
where
α
and
β
are layer labels chosen from {BVC, H, PR}, and is the weight
connecting the
j
th
neuron in layer
β
to the
i
th
neuron in layer
α
at training step
t
. After the
completion of the training session, each neuron’s vector of incoming weights from each
other layer was normalized to sum to unity. Each hippocampal neuron’s vector of incoming
weights on recurrent connections was normalized by dividing by its maximum incoming
recurrent weight. Note that no learning rate parameter was required in Equation A4 because
of the weight normalization after learning.
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Parietal component
The parietal component of the model, including the parietal window, the transformation
layer, the head direction system, and the connections within/between these regions and
from/to the BVC layer, was trained separately from the medial temporal component because
the former needed training only once. For each training step a heading direction,
φ
, was
randomly chosen from the set of heading directions, , corresponding to the set of
transformation sub-layers. Next, a linear boundary of random location and orientation in
allocentric space was discretized in the same way as landmark boundaries were in the
medial temporal training procedure described above. The length of this linear boundary was
chosen proportional to the distance between its midpoint and the allocentric origin in order
to sample sparsely distributed neurons distant from the origin as frequently as densely
distributed neurons near the origin. BVC firing rates were then calculated for the discretized
boundary using equation A2 and identically imposed upon the BVC layer and the
transformation sub-layer corresponding to the randomly chosen rotation angle,
φ
. By
rotating the linear boundary through
φ
about the allocentric origin, the egocentric positions
of the individual landmark segments for this boundary were then found. As with the BVC
layer, firing rates of the parietal window neurons in the presence of the boundary were found
by first forming a one-to-one correspondence between the set of parietal window neurons
and a radial grid centered at the model’s current location and covering
egocentric
space (see
figure 3). The contribution of a single landmark segment with egocentric coordinates (
r,θ
e
)to
the firing rate of the
i
th
such neuron was calculated via
(A5)
where are the egocentric co-ordinates of that neuron’s corresponding grid point,
C
PW
is set to one, and
σ
θ
and
σ
r
are chosen as in equation A2. The total firing rate of the
i
th
parietal window neuron was calculated by summing equation A5 to a maximum value of 1
over all landmark segments viewable from the current location. Finally, the head direction
layer is a one-dimensional continuous attractor (e.g. Skaggs, Knierim, Kudrimoti &
McNaughton, 1995; Stringer, Trappenberg et al., 2002; Zhang, 1996) composed of 100
neurons uniformly covering 360 degrees of angular head direction space, with the firing rate
of the
i
th
such neuron calculated via
(A6)
where
φ
i
is the preferred heading direction of that neuron, and where
C
HD
is set to 1.
Once firing rates were imposed on each layer for a given head direction and linear boundary,
all connection weights were incremented according to equation A4. After 400,000 such
training iterations, the vector of incoming weights for each parietal neuron from each other
layer was normalized to sum to unity. Weights from the transformation layer to the PW were
clipped so that the smallest 30% were set to zero. This was done so that the weight matrices
became sparse, a manipulation which decreased required simulation time considerably. For
normalization purposes, all transformation sub-layers were taken as part of the same layer.
The vector of weights on incoming recurrent connections for each head direction neuron was
normalized by dividing by the maximum incident weight value for that neuron. Although all
weights in the parietal component of the model were trained on a discrete set of 20
transformation angles, the model was found to interpolate accurately between these values.
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Velocity integration
In order to maintain a localized packet of self-sustaining activity, the head direction system
must have a set of recurrent excitatory connections, each originating from a particular head
direction cell representing and terminating on another cell that represents a nearby or equal
direction. Overall, connections from any given head direction cell must be balanced in such
a way that that cell’s activity equally excites neurons representing directions to either side of
the current direction. The training procedure described in the previous section results in the
formation of just such a set of weights. An applied angular velocity signal can move an
activity bump around in this network in a continuous fashion by modulating an appropriately
formed second set of self-excitatory connections (Zhang, 1996). Any connection in this set
also originates from a cell representing a particular direction and terminates on another cell
that represents a nearby direction, but these “rotational” connections are asymmetric so that
activity in the presynaptic head direction cell preferentially excites cells corresponding to
nearby directions that are to one side of the current direction. In principal the angular
velocity of the shift is proportional to the size of the asymmetric component (Zhang, 1996),
however for simplicity, we simulate rotations of fixed velocity, with an angular velocity
signal that simply gates the use of a fixed set of “rotational” connections in either sense
(clockwise or anti-clockwise). We achieved such a weight distribution by moving a bump of
activity around the head direction neurons at a constant velocity in order to simulate
rotational egomotion. During this simulated rotation, the velocity-gated weights on recurrent
connections within the head direction layer were updated by the trace Hebbian learning rule
given by
(A7)
where is the velocity-gated weight from the
j
th
to the
i
th
head direction neuron at
training step
t
, where is given by
(A8)
and where Δ
t
=0.05 time units. After training, the velocity-gated head direction weights were
normalized in the same way as the non-velocity-gated recurrent head direction weights. A
similar model of the head direction cell ensemble has been described in detail by Stringer,
Trappenberg et al. (2002).
Translation, which can occur in parallel with rotation in our model, is accomplished by
introducing a second set of velocity-gated “translational” weights from the transformation
sub-layers to the parietal window. The original “static” set of weights is responsible for
projecting a rotated image of BVC activity onto the parietal window during top-down phases
and becomes inactive during translational motion. Instead, the translational set of weights
projects a similar rotated image onto parietal window neurons, but it is displaced by a small
amount in egocentric space. This is accomplished by setting the translational weights as
(A9)
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where are the maximal firing coordinates of the
i
th
parietal window
neuron in the egocentric map, and is the static weight connecting the
j
th
neuron in
the
n
th
transformation sub-layer and the
k
th
neuron in the parietal window layer. Although
σ
in this equation could be set to a constant, we found that with our limited resolution for
landmark representation at larger distances, a more practical form was given by
(A10)
Since feedback connections propagate the displaced parietal window activity resulting from
the up regulated weights of equation A9 back to the place cell layer during bottom-up
phases, BVC and place cell firing shifts to reflect the new parietal window activity. This, in
turn, results in a further shifting of the activity projected back onto the parietal window in
the next top-down phase. Thus, translation of both the egocentric and allocentric
representations of space continues until the velocity signal is removed and the original static
weights are up-regulated again. As with the rotational connections, we simulate only a single
speed of motion. A more complete model might simulate different speeds of translation,
using a number of different sets of connections from the transformation layer to the parietal
window, each corresponding to a slightly different displacement, and each gated by separate
signals for the corresponding speeds. Alternatively it might titrate the influence of static and
translational weights according to speed of movement. However, due to their intense
computational requirements, we have not explored these more detailed models here.
Dynamics
During simulations, all neurons are in our model were of the “leaky-integrator” variety and
all dynamical equations were integrated using the simple Euler method with a time step of
0.05 units. For the medial temporal part of the model {PR, BVC, and H} we have
(A11)
where A
α
is the activation vector for layer
α
, W
α, β
is the weight matrix connecting layer
β
to layer
α,φ
α,β
is a scalar representing the overall strength of the connection from layer
β
to
layer
α
,
δ
is the Kronecker delta function (unity for equal arguments, zero otherwise),
represents an inhibitory bath of interneurons to which all neurons in a given layer are
reciprocally connected with equal weight, is a square matrix with all elements equal to
one, and I
PR
is an externally applied source of input (see below) representing direct lower
level input into the perirhinal layer. Bottom-up/top-down dynamics are governed by the
χ
functions, of which
χ
H,
β
(
t
) and
χ
BVC,
TR
n
(
t
) are 1 during a bottom-up phase and 0.05
during a top-down phase,
χ
α
,H
(
t
) is 1 during a top-down phase and 0.05 during a bottom-up
phase, and the remaining
χ
’s in equation A11 are 1 always. The length of each of the
bottom-up/top-down phases is 15 time units. Finally, the firing rate of the
i
th
neuron in
layer
α
is given by a sigmoid function of its activation, as follows
(A12)
where
ν
α
acts as a threshold. Exact numerical values for all unspecified parameters are
presented in table 1.
Byrne and Becker
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The dynamics of the parietal window and head direction layers are given by
(A13)
and
(A14)
respectively, while the dynamics of the
i
th
neuron in the
n
th
transformation sub-layer are
given by
(A15)
where W
ν
×TR
n
and W
ω
×HD
are the “translational” transformation layer to parietal window
weights and the “rotational” recurrent head direction weights respectively, where
χ
TR,α
(
t
)
is 1 for
α
= BVC during a top-down phase or for
α
= PW during a bottom-up phase, and
0.05 otherwise, and where 1 is a vector of ones. Finally, the dynamics of the inhibitory
interneuron are given by
(16)
Parameters in the model were chosen so that the fourth term on the right hand side of
equation A15 was a constant for all head direction cell activity packets maintained in our
simulations, either by attractor dynamics or injected current. This constant was equal to the
maximum value of W
TR
n
,HD
. Therefore, the fourth term on the right hand side of equation
A15 could have been eliminated by simply subtracting a constant from W
TR
n
,HD
so that
their maximum value was zero. With such a simplification, the model could be interpreted
as having only inhibitory direct connections from HD tothe transformation layer, without
any inhibitory interneurons. Note also that all neurons in the model interact with their
connected neighbours in an identical fashion. Apparent differences in the form of the above
dynamical equations are superficial and reflect the fact that the various network layers have
unique patterns of connectivity with their neighbors.
In addition to calculating neuronal firing rates for training purposes, equations A3, A5, and
A6 were also used to calculate the cueing/sensory or mentally generated inputs I
PR
, I
PW
,
and I
HD
. For this purpose,
C
PR
,
C
PW
,
C
HD
,
σ
φ
, and
σ
r
were set to 60, 60, 40, (0.01)
1/2
and
(0.1)
1/2
respectively. When the weak BVC terminating weights were used in simulation 4,
C
PW
was increased to 100 during calculation of sensory input. Again, the exact values of the
listed parameters were not critical, but were found to generate localization quickly. In fact, a
relatively wide range of parameter values would have produced qualitatively similar results.
Finally, after the model has been cued to “imagine” itself in a certain location and
orientation, or during mental exploration/spatial updating, attention can be directed in any
egocentric direction in order to identify surrounding landmarks. To simulate focused
attention in the direction,
ψ
, an input given by
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(17)
was applied directly to neurons in the parietal window layer, where
σ
A
was set to for all
attending events except during the identification of building 1 in simulation one. In the latter
case, an increased value of was used for
σ
A
(this stronger attention signal would have
resulted in the correct identification of the remaining buildings as well and would not have
affected any of the results presented here). The value
C
PW
was set to 40 for our simulations.
Simulation of head direction cell lesions
Input from the head direction cell system to transformation neurons was recorded for all
head directions by storing the combined value of the third and fourth terms of the right hand
side of equation A15 in a vector, I
HDrec
(
φ
). Each element of this vector corresponds to one
transformation layer neuron and is a function of the head direction,
φ
. Thus, the third and
fourth terms of the right hand side of equation A15 could be replaced by I
HDrec
(
φ
) during
simulation. For a given value of
φ
, all values of I
HDrec
(
φ
) are less than or equal to zero, with
only elements corresponding to transformation layer neurons in the “selected” sub-layer
being close to zero. All other values are strongly negative, reflecting the gating function of
the head direction system.
In order to simulate a head direction cell lesion for a “realistic” model in which inhibition
for gating is accomplished via a large
population
of inhibitory interneurons, a two-part
modification of I
HDrec
was employed. First, all values of I
HDrec
greater than a cut-off of
33% larger than the minimum value were set to the cut-off (the average minimum value was
-96, so the cut-off was -64). This modification was intended to simulate the loss of direct
excitation to the “selected” transformation sub-layer. Second, random regions of each
transformation sub-layer were selected (see below) and the I
HDrec
elements corresponding to
those neurons were increased in value to the level of the cut-off. The exact random
transformation layer regions selected for this manipulation varied with head direction. This
modification was intended to simulate the loss of inhibition resulting from lowered levels of
stimulation to the inhibitory neuron population.
In selecting random regions of the transformation layer for reduced inhibition, a one-to-one
correspondence between the neurons in each transformation sub-layer and a radial grid was
formed (as above in the training section). A circle with randomly located center and a radius
of 7.5 units was formed for each sub-layer and all neurons corresponding to grid points
within the circle were selected for reduced inhibition. These circular regions were randomly
re-selected for each head direction.
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Figure 1.
Top: Egocentric reference frame in which the observer is always at the origin, facing along
the positive y-axis. A triangular landmark sits in front and to the left of the observer in this
frame. Bottom: The same situation as above, but depicted in the allocentrically iligned
reference frame. In this frame the observer is always at the origin, but the direction of the y-
axis is fixed to the external environment instead of the observer’s heading direction. With
the heading direction depicted (approximately 45 degrees away from the positive y-axis in
the counterclockwise direction), the triangular landmark lies directly on the positive y-axis
and is rotated 45 degrees in the counter-clockwise direction.
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Figure 2.
Map of the “two-room” environment used in the second set of simulations. Solid rectangles
represent environmental boundaries/landmarks. Each grid point corresponds to a maximal
firing location for one hippocampal place cell. The ‘X’ represents the model’s current
location and the arrow its heading direction.
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Figure 3.
Top: Egocentric reference frame. Each grid point corresponds to the preferred boundary/
landmark location of a parietal window neuron, which fires maximally when a landmark
segment is located at that grid point’s coordinates. The landmark segments for the
discretized “two-room” environment, as viewed from position ‘X’ in figure 2, are also
shown. The landmark segment at egocentric direction,
θ
e
, is indicated by the dashed arrow.
Finally, the model’s heading direction, which is
always
the same in egocentric space, is
indicated by the solid arrow. Bottom: Activation of parietal window neurons corresponding
to the landmark segment configuration. The firing rate of each neuron is plotted at that
neuron’s corresponding grid point, with lighter shades indicating higher firing rate.
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Figure 4.
Top: Allocentric reference frame. Each grid point corresponds to the preferred boundary/
landmark location of a BVC, which fires maximally when a landmark segment is located at
that grid point’s coordinates. The landmark segments for the discretized “two-room”
environment, as viewed from position ‘X’ in figure 2, are also shown. The dashed vector
points to the same landmark segment highlighted in figure 3. In this map it is located at the
same distance from the model, but its direction,
θ
a
, is equal to
θ
e
plus the model’s current
heading direction. Finally, the model’s heading direction within the allocentric reference
frame is indicated by the solid arrow. Bottom: Activation of BVCs corresponding to the
landmark segment configuration. The firing rate of each neuron is plotted at that neuron’s
corresponding grid point, with lighter color indicating higher firing rate.
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Figure 5.
Top panel: Transformation circuit in bottom-up mode. A representation of the egocentric
positions of all viewable landmark segments is shown in the parietal window (PW). Rotated
representations are projected onto the various transformation sub-layers, which are inhibited
by current head direction (HD) activity via a population of inhibitory interneurons (I). One
transformation sub-layer receives direct excitation from the HD system, thus allowing its
representation to project forward to the BVCs. Bottom panel: Transformation circuit in top-
down mode. The allocentric BVC representation of the environment is projected identically
onto each of the transformation sublayers. Each of these identical representations would be
rotated through different angles by the transformation to PW weights, but excitation and
inhibition from the head direction system allows only the correct sub-layer to maintain
sufficient activity to drive PW neurons.
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Figure 6.
Schematic of the model. Each box or oval represents a set of neurons in a different brain
region. Thin, solid arrows represent full bottom-up interconnectivity between the neurons in
the connected regions, while the broken arrows represent full top-down interconnectivity.
Thick, solid arrows represent full connectivity, which is unaffected by the bottom-up/top-
down cycling. The thick dashed line from the inhibitory interneuron population (I)
represents inhibition that is unaffected by the bottom-up/top-down phases. A given
perirhinal (PR) neuron fires maximally when the model attends to a landmark segment with
a particular identity. Hippocampal neurons are associated with a Cartesian grid covering
allocentric space such that a given neuron fires maximally when the model is localized at its
corresponding grid point. BVCs or parietal window (PW) neurons are associated with a
polar grid covering allocentric/egocentric space. A given BVC/PW neuron fires maximally
when a landmark segment is a certain distance and allocentric/egocentric direction away
from the model. A given HD neuron fires maximally for a given head direction. The
transformation layer neurons are responsible for transforming allocentric BVC
representations of space into egocentric PW representations. A second set of top-down
weights (curved-dashed arrow) from the transformation layer to PW are gated by egocentric
velocity signals to allow for spatial updating/mental exploration.
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Figure 7.
Top four panels: Activation in the various model layers averaged over a full cycle after it
was cued to face the Cathedral (building 1). Top left panel: Environmental boundaries are
represented by gray walls superimposed upon the hippocampal place cell representation.
Here, the firing rates of
all
hippocampal place cells are presented, with each shown at its
corresponding grid point within the environment. Middle left panel: The HD activity peak
indicates that the model was facing “forward” relative to the stored allocentric map.
Therefore, PW activity (middle right panel), which is the model’s representation of its
surrounding egocentric space, was highly similar to parahippocampal (PH) BVC activity
(upper right panel), which corresponds to the model’s allocentric representation of space.
The various symbols superimposed upon the egocentric PW representation indicate the
directions in which attention was directed. Bottom panel: Activation in PR identity neurons
at the end of the first bottom-up phase after attention is directed in the PW. For example,
when attention is directed to the egocentric right (+ symbols), PR neuron 2, which
corresponds to boundary/building 2, is the most active identity neuron.
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Figure 8.
Top four panels: Activation in the various model layers averaged over one full cycle after it
was cued to face away from the Cathedral. The HD activity peak indicates that the model
was facing “backwards” relative to the stored allocentric map. Therefore, PW activity is
rotated 180 degrees relative to BVC activity. The various symbols superimposed upon the
egocentric PW representation indicate the directions in which attention was directed. Bottom
panel: Activation in PR neurons at the end of the first bottom-up phase after attention is
directed in the PW.
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Figure 9.
Top four panels: Activation in the various model layers averaged over one full cycle after
the lesioned model was cued to face the Cathedral. Bottom panel: Activation in PR neurons
at the end of the first bottom-up phase after attention is directed in the PW.
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Figure 10.
Top four panels: Activation in the various model layers averaged over one full cycle after
the lesioned model was cued to face away from the Cathedral. Bottom panel: Activation in
PR neurons at the end of the first bottom-up phase after attention is directed in the PW.
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Figure 11.
Top four panels: Activation in the various model layers averaged over one full cycle after it
was cued to localize itself in the “two-room” environment facing wall 1. Environmental
boundaries are represented by gray walls superimposed upon the hippocampal
representation. The various symbols superimposed upon the PW representation indicate the
directions in which attention was sequentially directed. Bottom panel: Activation in PR
neurons for the various attention conditions at the end of the first bottom-up phase after
attention is directed in the PW.
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Figure 12.
Activation in the various model layers averaged over one full cycle after the application of
the rotational velocity signal for 150 time units followed by a forward translational velocity
signal for 135 time units.
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Figure 13.
Top four panels: Activation in the various model layers averaged over one full cycle at the
end of the eight step sequence of egomotion. Bottom panel: Activation in PR neurons for the
various attention conditions at the end of the first bottom-up phase after attention is directed
in the PW.
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Figure 14.
Similar to the top four panels of figure 11 except that results are for the simulation in which
sensory information about the environment is being continuously input to the PW
representation throughout the duration of the simulation.
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Figure 15.
Similar to the top four panels of figure 12 except that results are for the simulation in which
sensory information about the environment is being continuously input to the PW
representation throughout the duration of the simulation.
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Figure 16.
Similar to the top four panels of figure 13 except that results are for the simulation in which
sensory information about the environment is being continuously input to the PW
representation throughout the duration of the simulation.
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Figure 17.
Activity of a
single
place cell recorded from the model with a simulated head direction cell
lesion. Recordings were made when the model was localized at numerous points within the
dashed rectangle. In this simulation the model’s head direction was consistent with perfect
alignment between parietal window and BVC representations of space. Note also that the
recorded cell would fire maximally at the ‘X’ for all head directions in the non-lesioned
model.
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Figure 18.
Activity of the same place cell as shown in figure 17, but in this simulation the model’s head
direction was consistent with perfect anti-alignment between parietal window and BVC
representations of space. Note also that the recorded cell would fire maximally at the ‘X’ for
all head directions in the non-lesioned model.
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Figure 19.
Adapted from Gothard et al. (1996). Left: Linear track apparatus used by Gothard et al.
Upper middle: Rat on outward journey from box to fixed cup for the five different box
positions. Upper right: Hypothetical average firing patterns for a place cell in each of the
five outward conditions plotted against relative position along the track in the
box1
condition (0 is the position of the box in the
box1
condition while 1 is the position of the
fixed cup). The slanted dashed line is the regression line used to calculate displacement
slope, which is 1.0 for this cell since it fires near the box in all conditions. The vertical
dashed line shows the location of peak firing on the
box1
-out trials. Lower middle: Rat on
inward journey from fixed cup to the box for the five different box positions. Lower right:
Hypothetical average firing patterns for a place cell in each of the five inward conditions
plotted against relative position along the track in the
box1
condition. This cell fires near the
fixed cup in all conditions, giving a displacement slope of 0.0.
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Figure 20.
From Gothard et al. (1996), used with permission. Upper left: Averaged firing profiles of
four outward selective neurons in each condition. Small rectangles represent the movable
box. Upper right: Displacement slopes for multiple outward selective cells plotted against
their peak firing positions in the
box1
condition. Positions are relative to full track length
with 0 representing the box position in the
box1
condition and 1 representing the position of
the fixed food cup. Lower left/right: Equivalent results for inward selective cells.
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Figure 21.
Left top/bottom: Activation in PW/hippocampal neurons near the beginning of a top-down
phase after the model was cued to localize itself 2 units away from box 1 facing box 2.
Environmental boundaries are represented by gray walls superimposed on the hippocampal
representation. Middle top/bottom: Activation in PW/hippocampal layer near the beginning
of a bottom-up phase after application of forward velocity signal. Right: The model’s
representation of its location within the environment as a function of time.
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Figure 22.
Left top/bottom: Activation in PW/hippocampal neurons near the beginning of a top-down
phase after the model was cued to localize itself 2 units away from box 1 facing box 2.
Additional activation has been applied directly to PW neurons representing box 2 at a
position 6 units closer to the origin than expected. Environmental boundaries are represented
by gray walls superimposed on the hippocampal representation. Middle top/bottom:
Activation in PW/hippocampal layer near the beginning of a top-down phase after the model
comes within 1 unit of box 2. At this point the velocity signal is switched off, and the
sensory input ceases to move. Right: The model’s representation of its location within the
environment as a function of time.
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Figure 23.
Top: Activity from four of eleven selected model place cells (maximal firing coordinates for
the selected cells: and
y
i
= 0.25 for all
i
) in five simulated conditions
(
box1
thru
box5
without box 1 sensory input) plotted against relative position in the longest
track-length condition (
box1
condition). Rectangles represent box 1 and 2. Bottom left/right:
Displacement slopes calculated from the eleven sampled model place cells during outward/
inward journeys. Squares/triangles represent results from full-model simulations with only
box 2 (squares) or box 1 and 2 (triangles) sensory input. Circles represent results from the
simple BVC explanation. The dashed line is what would be expected if landmarks exerted
control over place cell firing in direct proportion to their proximity to the animal.
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Figure 24.
Identical to figure 23 except results are for the simulations with weakened BVC input
parameters. Note the hopping behaviour of place cell activity in the shortest track-length
condition.
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Table 1
Model Parameters
Parameter Value
ν
α
5 (50 for the inhibitory interneuron)
ϕ
inh
H
2.1
ϕ
inh
PR
9
ϕ
inf
BVC
0.2
ϕ
inh
HD
6
ϕ
inh
TR
0.1
ϕ
inh
PW
0.1
φ
H
21
φ
H,BVC
140
φ
H,PR
25
φ
BVC,H
900
a
φ
BVC,PR
1
φ
PR,H
6000
φ
PR,BVC
75
φ
TR,BVC
54
φ
TR,PW
63
φ
BVC,TR
900
b
φ
PW,TR
880
φ
HD
15
φ
TR,HD
85
φ
TR,I
90
φ
I,HD
10
φ
ω
×HD
2
φ
ν
×TR
φ
PW,TR
a
decreased to 150 for weakened BVC input simulation on linear track
b
decreased to 540 for weakened BVC input simulation on linear track
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