11.2.1. Cognitive Maps as Spatial Structures

Thinking of cognitive structures in terms of cognitive maps has a long history in psychology and neuroscience. The view began as an explanation of maze-running abilities in rats and, over time, has developed and changed as it has been used to capture a range of activities, from semantic knowledge structures to the navigational expertise of London taxi drivers (Collins & Loftus, 1975; Maguire, Frackowiak, & Frith, 1997). Throughout, theorists have aimed to make connections between these abilities in experimental animals and humans, but they have offered subtly different accounts of why these maps exist, what’s essential to their structure, and how the extension from basic neural structure to broader human competences is characterized.

Tolman (1948) is often identified as originating the idea of a cognitive map in this literature.1 Tolman’s account of the cognitive map emerged from his work on maze running and spatial learning in rats—the dominant method and experimental framework in early twentieth-century psychology. For Tolman, cognitive maps were part of an argument that explaining the navigational abilities of rats required more cognitive, representational structure than was allowed for by the stimulus–response approach, which was dominant at the time. Specifically, Tolman documented rats’ ability to learn shortcuts in mazes—an ability inexplicable in terms of the animal’s learned association with particular places in the maze as individual stimuli. Tolman further observed that rats were capable of latent or non-reinforced learning. That is, rats that were simply allowed to explore mazes while fully fed—not receiving nor wanting any reinforcement for their exploration— were able to learn routes through the maze. In order to explain this behavior, Tolman (1948) argued, the rat needed to be credited with the possession of a cognitive map that provided a “field map of the environment” (p. 192). Although Tolman’s evidence was based on the maze-running behavior of rats, the cognitive maps that he posited to explain this behavior were intended to apply to a much wider range of cognitive creatures. Indeed, his 1948 paper was titled “Cognitive Maps in Rats and Men.” The paper even concludes with a few pages of speculation on how particular features of human personality and social organization may be explicable within this framework.

Tolman’s (1948) initial proposal was solidified into a theory of neural structure with the publication of O’Keefe and Nadel’s Hippocampus as a Cognitive Map (1978). For O’Keefe and Nadel, cognitive maps were not simply a general framework for thinking about mental structure; the cognitive map was posed as a theory of hippocampal function. It was the first such proposal, and a highly systematic one, which helps to explain both the initial excitement about the idea and its lasting influence. There are further differences between Tolman’s use of the term and use in O’Keefe and Nadel’s framework, the latter of which serves as the basis of “cognitive map theory” as it is now understood. First, O’Keefe and Nadel take the notion of a map far more literally than Tolman. The claim is not that the information processing of the hippocampus can be understood as map-like or spatial-ish, but rather that these cognitive maps are inherently spatial. In putting forward their theory, O’Keefe and Nadel make continued, explicit appeal to the Kantian idea of spatial structures as an organizing feature of cognition. These spatial maps are considered innate structures endemic to all cognitive creatures. Second, the extension of these maps to humans is not a metaphorical abstraction from the idea of a spatial map, but instead is characterized as an expansion of the kind of inputs that the spatial system can incorporate and process. This is best illustrated by their account of cognitive maps in humans: “the left hippocampus in humans functions in semantic mapping, while the right hippocampus retains the spatial mapping function seen in infra-humans. On this view, species differences in hippocampal function reflect changes in the inputs to the mapping system, rather than major changes in its mode of operation” (O’Keefe & Nadel, 1978, p. 3).

The centerpiece of cognitive map theory is the discovery of place cells (O’Keefe & Dostrovsky, 1971): neurons in the hippocampus that fire preferentially—that is, exhibiting a burst of action potentials—in response to a specific location in the organism’s environment. When a rat navigates a maze, for example, some place cells fire at the beginning of the maze, others at the first fork, still others at the second fork, and so on. These place cells are organized topographically, so that their collective firing pattern reflects the rat’s route. After the maze has been run, the pattern is rehearsed, establishing a “map” that allows the rat to navigate this environment more easily the next time it is encountered.

The discovery of grid cells further enriches our understanding of the maps created by the hippocampal system (Hafting et al., 2005). Grid cells are found in the medial entorhinal cortex and, in contrast to hippocampal place cells, fire at multiple regularly spaced locations in the environment. Seen over the rat’s trajectory, the spatial firing patterns of these cells provide a grid-like representation of the organism’s environment. Other cells select for additional elements of the map—for example, cells that track objects, landmarks, and other agents (Høydal et al., 2019); head direction cells that fire selectively based on the way the organism’s head is oriented relative to its route (Taube, Muller, & Rank, 1990); and those that encode information about the distance to borders and edges (Solstad et al., 2008).

In our brief survey of this work, we want to highlight two important features of this literature as we see it. First, even though work on cognitive maps and memory structure has been done mostly with rodents and has focused on low-level neural structure, the intent has always been to make claims about the role of such maps in cognitive creatures more general. That is, the aim was not simply to move away from overly simplistic stimulus– response models of nonhuman animal learning, but rather to think about the cognitive structures available to these nonhuman animals in terms of a framework that would encompass cognitive processes and cognitive creatures more generally. How the cross-species and beyond-spatial generalizations of the framework are envisioned differs across particular accounts and interpretations of cognitive maps.

Second, cognitive map theory remains influential and controversial. The framework continues to serve as a serious guide to inquiry into neural structure, especially with regards to the hippocampus (Redish, 1999; Bellmund et al., 2018). The view also serves as a steady target for alternative conceptions of neural structure and cognitive processing. Many of these criticisms involve claims and/or evidence that the information represented in these “maps” is nonspatial, including, for instance, findings of hippocampal cells encoding temporal context (Eichenbaum, 2014). That such interpretations of the content of cognitive maps are available is certain. Whether this sense of cognitive maps is an extension of the original framework or an objection to it is more contentious. Answering this question depends on asking, first, how the notion of “map” should be understood. We address this in section 11.3.

11.2.2. Nonspatial Cognitive Structures

Cognitive psychology has, since its beginnings, been interested in how humans and other cognitive creatures organize their vast amounts of knowledge so as to support efficient and effective search and retrieval. Although some of these cognitive structures are referred to as “maps,” in such cases the term is being stretched to nonstandard or metaphorical use. In this section, we’ll survey some of the foundational work on structures in cognitive science that do not seem to be map-like and do not primarily encode spatial content.

We’ll start with emergent conceptual structures (figure 11.1A and B). These are ways of organizing and relating information that emerge from amassing overlapping conceptual content and acquire their particular structures from patterns in the accumulated information. For example, this type of memory structure is often thought to support language comprehension and production. Adele Goldberg (2019) presents a view of language intended to explain the differences between the following kinds of minimal pair (p. 57):

Figure 11.1

Conceptual structures: A and B represent highly dimensional spaces, as described by Goldberg (2019), which combine semantic information with other dimensions such as syntax and phonetics with dark lines representing phrase frequency. We find “cried (more...)

1.

I’ll cry myself to sleep.

2.

I’ll cry myself asleep.

The former sentence is perfectly felicitous, whereas native speakers judge the latter to be odd. Perhaps sentence 2 is odd because it is novel or unusual, but as Goldberg notes, we are perfectly happy with unusual sentences like this (p. 76):

3. She’d smiled herself an upgrade.

Goldberg explains what is special about sentence 2 by appealing to the role of long-term memory organization. On her account, we encode much of the language we hear in a high-dimensional conceptual space that is structured by syntactic form, meaning (in context), phonetic features, and so on. Since there are systematic relationships between many of these features, over time, clusters emerge. Bits of language that are encountered more frequently are selectively strengthened, whereas the connection between a word or phrase and its initial context is weakened if it then fails to occur in similar contexts. We have only noisy, implicit access to this space. Thus, the problem with sentence 2 is that it is close to a stronger competitor, sentence 1. Conversely, sentence 3, while unusual, does not trigger us to recall a more common alternative formulation, and so we judge it to be felicitous.

This example helps us extract several key features of an emergent conceptual structure. Over time, in the case of language, certain features of language stand out as principal components. For instance, in English, sentences with the form of the double-object construction—for example, she (x) passed him (y) something (z)—almost always have the meaning that x causes y to receive z. This is a regularity that relates sentential form to semantic content, and crucially Goldberg (2019) argues that this regularity arises in memory without any need to learn an explicit rule or start off with innate knowledge. Instead, the emergent conceptual structure allows for all kinds of regularities between a wide variety of features to be learned over sufficient exposure. Such a structure must therefore be (1) highly dimensional in order to catch the relevant features, (b) associative in order to relate these features flexibly without a prior model, and (c) content addressable in order to utilize stored information to produce responses efficiently.

A second kind of nonspatial structure is a graphical model, a family of models within which we will focus on Bayesian networks (figure 11.1C). In the case we just considered, the regularity between the double-object construction and the type of causal-agential content was represented as a clustering or association. We neither represent the syntactic structure as dependent on the semantic content nor vice versa; the association is not specific enough to represent anything more than organization in terms of similarity. But in a Bayesian network, relationships between features are represented as a set of conditional (in)dependencies. For example, I might encode information about academic lectures as in figure 11.1C.

In this model, the nodes are random variables, and the edges represent dependencies (sets of local probability models). This allows us to assume that a node is independent of any other node, conditional on its parents: for instance, conditional on “conference quality,” “speaker style” is independent of “talk content”. Notice that in the above graph, an edge connects “conference quality” to “talk content,” but it seems unlikely that “conference quality” is a cause of “talk content.” A narrower class of these models, causal Bayesian networks, interprets dependence and independence causally. Consequently, laws of causality can be applied to the graph structures, such as transitivity, asymmetry, and nonreflexivity. As is, this graphical representation is completely equivalent to an enumeration of the local probability models. However, when treated as a representation structure, the graphical representation can have properties not shared by the set of local models. For instance, we might search the graphical representation with an algorithm designed specifically for search in a graph, which would produce different behavior than search over other forms of representing the same information (e.g., Dechter & Mateescu, 2007). Whether it is true that conceptual knowledge is in fact represented with, in addition to representable by, graphical models, this hypothesis provides an interesting model of nonspatial mental structures.

Glymour (2001) analyzes a wide swath of human cognition in terms of causal Bayesian networks. These representations have three functions on his account: control, prediction, and discovery. To varying degrees, these functions could be fulfilled just as well, no matter the format of the probabilistic and causal information. However, the graphical format is significant as soon as the thinker employing the models is not perfect. Graphical representations figure directly in heuristics and inductive biases, such as a preference for an explanation that appeals to fewer causes (Lombrozo, 2007). Graphical representations allow simple access to points of potential intervention (Gopnik et al., 2004). As we noted above, we can define distinctive algorithms for search over graphical representations, and both noise and lesions to the model will operate differently, depending on representational format.

Thus, our second class of nonspatial models, causal Bayesian networks, is used to represent all kinds of causal knowledge. These representations function to identify interventions, predict new outcomes, and enumerate new possible theories. Bayesian networks are well suited to performing these functions because they organize information according to principles (i.e., the principles of causation) that (1) apply to the entire domain and (2) align with our interests in manipulating the environment.

Emergent conceptual structures and causal Bayesian networks are both structures that have been posited as operant in memory. Neither of these structures is in any notable way spatial or especially suited for spatial information. Both of these structures are functional: they are thought to have certain features that map onto computational advantages for the thinkers who employ them. Emergent conceptual structures are more flexible than causal Bayesian networks, since the latter can only represent causal relationships, whereas the former can represent non-causal associations. Correspondingly, causal Bayesian networks can express a more complex set of relationships within the causal domain, differentiating cause and effect, and identifying potentially complex causal patterns. Emergent conceptual structures represent many features in exactly one relationship: similarity.

Considering these two structures leaves us with a few key takeaways. First, even for models of abstract, domain-bridging, and perhaps distinctively human knowledge, cognitive structures are still thought to be tailored to particular functions. Second, there seems to be a trade-off between the generality of a representation (i.e., the kinds of features it could in principle represent) and its inferential power (i.e., the conclusions that can be derived from the connections among representational subunits). When data and processing limitations are held fixed, we could either utilize a structure with more flexible (and hence weak) connections or with less flexible (but more inferentially generative) connecting links. This idea is fairly intuitive, following from a more general connection between flexibility and informativeness. In the case of emergent conceptual structures, we saw an advantage of flexibility at work: Goldberg’s (2019) model allows speakers not just to track semantic or syntactic patterns separately, but also to combine all the information we have about a string of language and thereby to learn patterns of association that crosscut traditional linguistic categories. Causal Bayesian networks displayed one of the advantages of inferential power: by representing causal structures in graphs, we made the task of determining points of intervention vastly easier. These cases offer helpful comparisons for considering how to manage these trade-offs in characterizing the functions of spatial structures that serve as the basis for cognitive maps.

11.3.1. Spatial-Isomorphic Information

Spatial-isomorphic content is a kind of content that is structured according to dimensions that functionally correspond to spatial dimensions. By functional correspondence, we mean that the regularities, limitations, and inference patterns that we commonly apply to spatial dimensions will for the most part apply to these nonspatial dimensions. For example, (Euclidian) spatial distance is symmetric: if my office is ten feet from the coffee machine, then the coffee machine is ten feet from my office.5 Spatialisomorphic content, since its dimensions functionally correspond to spatial dimensions, will tend to have a “distance-like” measure that is symmetric in the same way. It seems reasonable that, were we to have a dedicated cognitive mapping system for dealing with spatial content, this system might also be used for dealing with spatial-isomorphic content.

Constantinescu, O’Reilly, and Behrens (2016) offer some preliminary evidence that some of the core processes for representing spatial maps can be used for spatial-isomorphic content. Unlike related work on spatial representations of nonspatial information (e.g., Tavares et al., 2015), the authors went beyond neurally co-locating spatial and nonspatial activity. Instead, they focused on a signature of spatial representation: coding of a space into a hexagonal lattice, such that the rate of cell firing corresponds to the orientation of movement relative to the orientation of the lattice. Because strongest firing occurs at 60° increments in orientation, the 360° of phase space are divided into six identical regions, giving rise to the lattice’s hexagonal symmetry.6 The authors looked for this hexagonal symmetry as a mark of what are sometimes called human grid cells. Unlike the grid cells discussed in section 11.2.1, these neurons are not thought to be restricted to regions of the medial temporal lobe but instead are thought to occur throughout (some of) the brain regions that also form the default mode network, including the ventromedial prefrontal and posterior cingulate cortex. Still, previous work has associated these more distributed cells with spatial representations (e.g., Doeller, Barry, & Burgess, 2010). Rather than a strictly spatial task, Constantinescu and colleagues (2016) taught participants a pattern of association between the appearance of a bird and a set of symbolic cues. The bird figure varied according to neck height and leg lengths, which allowed for a representation of possible bird figures in a two-dimensional space structured by these two features. The bird-cue relationships were chosen so that each cue picked out a single region of this “bird space.” The authors indeed found hexagonally symmetric responses (measured in fMRI) in a variety of default mode brain regions that seemed to correspond to hexagonal, grid-like representations of “bird space.”

The bird space used in this study was spatial-isomorphic, since it was structured according to two dimensions (neck height and leg length) that could be used to carve up a feature space with several space-like functional dimensions: it was a two-dimensional Euclidean space, with distance and orientation operating just as they would in a real space. Intuitively, the bird space is space-like in that it articulates a “conceptual space,” but also space-like in that neck height and leg length are themselves literally spatial dimensions. However, the design of this study allows Constantinescu and colleagues (2016) to differentiate between these two spatial aspects of the stimulus: because the bird space and the bird’s features in regular space are two distinct spaces, moving through each would produce different patterns of symmetrical activation. Since the stimuli were carefully chosen to avoid passing through phase space in the same way, the observed symmetries in fMRI signal should not reflect the bird’s position in visual space.

The use of hexagonal coding itself, if the authors are correct, suggests a second kind of isomorphism. Hexagonal coding is thought to be optimal for spatial representation in particular. Mathis, Stemmler, and Herz (2015), for example, present an optimal model that ranks hexagonal coding highest for spatial resolution in the two-dimensional plane. In ordinary space, we don’t normally privilege one dimension over another—that is, the north– south axis is not in general more informative than the east–west axis. This allows us to value spatial resolution uniformly across the plane. But we do typically privilege those two axes over the up–down axis in navigation. These two features must be assumed in order to show the hexagonal lattice is optimal in the spatial domain. Neither feature needs to obtain in conceptual space. For instance, resolution in the bird neck-height dimension may be more valuable than information in the bird leg-length dimension. Were this to be true, the hexagonal symmetries observed by Constantinescu and colleagues (2016) would reflect a suboptimal representation. And so we can conclude that the use of a hexagonal symmetry code either reflects (a) a genuine isomorphism between the conceptual space and real space, or (b) a representational choice that favors spatial-isomorphism over customization to the optimal division of conceptual space.

Another kind of spatial isomorphism centers on temporal rather than conceptual structure. Researchers commonly motivate the division of a temporal sequence into parts by analogy with the division of a spatial layout into parts. For instance, Zacks and Swallow (2007) write:

For quite a while, psychologists have known that in order to recognize or understand an object people often segment it into its spatial parts (e.g., Biederman, 1987). A new body of research has shown that just as segmenting in space is important for understanding objects, segmenting in time is important for understanding events (p. 83).

This literature on event segmentation asks how and why we draw boundaries between events. While Zacks and Swallow take the process of segmentation to be somewhat automatic, DuBrow and colleagues (2017) present contrasting evidence suggesting that segmentation can be active, abrupt, and driven by top-down goals.

Is this use of space merely a helpful metaphor, or is event structure genuinely spatial-isomorphic? One genuine isomorphism comes from the local structure of both representations—that is, a ubiquitous feature of cognitive maps is their locality. While I have a clear idea of how things in my apartment complex are oriented, and a good idea of how things in the Philadelphia Museum of Art are oriented, I do not necessarily have a joint map that neatly connects the two. Kuipers (1982, 2007) views this as a key starting assumption of cognitive maps even in machines: breaking a map into smaller, local maps allows the agent to remain noncommittal about global connections. This locality of representation seems to hold for temporal segmentation as well. Upon hearing a story, I might build a temporal “map” of the events of my friend’s adventure last week without forming any particular representation of how the details of the events she is describing fit into a temporal sequence of my own schedule last week. Locality naturally arises from the use of schemas. Baldassano, Hasson, and Norman (2018) found that temporal boundaries in event schemas across different kinds of events had a common neural signature, provided they shared an abstract schematic structure—that is, schematic representations impose a local structure relative to the device of the schema itself (e.g., from when you enter a restaurant to when you pay the check). Anchoring event segmentation in local (temporal) structure, then, creates an abstract isomorphism with spatial maps, which are anchored in local (spatial) structures.

We could point to a long tradition locating isomorphisms between space and time, tracing at least back to Kant (1781/1787, A33/B49–50). The strength of this tradition, however, is a double-edged sword. The abundance of spatial language used in our everyday talk about time makes it hard to conceive genuinely of the capacity to represent space and the capacity to represent time as distinct. The question of isomorphism between space and time may, from this perspective, be ill formed if the two capacities are more than accidentally linked to one another.

In summary, one way to extend the core notion of a cognitive map to nonspatial information is to treat the nonspatial information as spatialisomorphic. These expansions are most efficient in cases where the nonspatial domain has significant regularities that mirror regularities that compose our representations of space, such as a symmetrical distance measure, roughly equal value assigned to discriminability among the dimensions on a two-dimensional plane, and a representation of related “spaces” that can be composed locally and independently.

11.3.2. Abstractions over Spatial Information

Another way to extend cognitive map theory—integrating work on neurallevel spatial maps and cognitive-level structure—is to explore ways in which the neural systems that support cognitive maps can process and represent abstractions from spatial information. Here, we consider two kinds of spatial abstraction: (1) a structure where the abstraction itself is still isomorphic to the lower-order representation of space, and (2) abstractions over space that are no longer spatial-isomorphic.

Michael Hasselmo (2011) has used cognitive map theory—with its place, grid, and head direction cells—to build a map-based account of episodic memory. In keeping with key themes of cognitive map theory, Hasselmo’s theory is derived largely from work with rats but is meant to provide an account of episodic memory that can scale to humans. His book-length articulation of the view is entitled How We Remember, and the “we” here encompasses all mammals with similar hippocampal structure. Critical to Hasselmo’s particular version of cognitive map theory is the idea that the hippocampus and surrounding structures are a phase-coding mechanism, where the map-making activity of place and grid cells is integrated into maps of the environment at multiple spatial and temporal scales—that is, the hippocampus produces a series of cognitive maps, in Rescorla’s (2009) loose sense, representing the environment in more or less detail, as a function of the scale imposed by the cells from which they are activated. Together, these maps represent a particular event or experience, serving as the content of an episodic memory. To support the idea of multiple maps, Hasselmo incorporates neurons from the entire hippocampus into his model rather than focusing primarily on the dorsal portions of the hippocampus, as is common in much of the literature.

Hasselmo argues that neurons across the hippocampus share the mapping function. The differences between dorsal and ventral neurons are a matter of the size of their receptive fields not their general function. As one moves across the hippocampus, from dorsal to ventral, the receptive field size of the neurons increases. This increase in receptive field size results in a comparable increase in the scalar proportion of the map. Maps featuring place cells with the smallest receptive fields represent the organism’s immediate surroundings in detail, whereas larger maps are produced by place cells with larger receptive fields, situating the experience within its (increasingly broad) spatial and temporal context. Importantly, the broadest “maps” may remain spatial in only the loosest or most metaphorical sense, situating the event within a social, conceptual, or experiential context.

The result is a mechanism that produces representations rich enough to support episodic remembering. The existence of multiple maps allows for a single episode to be recorded at several “scales of experience” (Hasselmo, 2008), capturing the episode as occurring not only at a particular place and time but as associated with various objects and events. For example, consider my episodic memory of walking from my campus office to the university library to return a book this morning. On Hasselmo’s view, the representation of this episode is a conjoined set of maps of the event at different scales of experience. We can think of the smallest-scale map of the event in terms of traditional cognitive map approaches—as an allocentric map of the campus, along with my route from my office to the library. But other maps associated with this episode will represent this event at different spatial, temporal, and contextual scales. The more abstract spatial maps may represent campus with relation to the part of town, city, state or continent in which I live. More abstract temporal maps will represent my route through this map as part of my schedule for the day, or schedule for the week, or activities characteristic of this time in the academic year. Further contextual maps will also be available, where the items represented in the map situate the landmarks along the route on different contextual scales—for example this trip to the library as a stage in a particular research project, trees along this route at this time of the year, campus construction at this time, and so on.

Hasselmo’s model proposes that the cognitive map system can process increasingly abstract characterizations of space and time that can then serve as the content for more elaborate and higher-order episodic memories. However, his hierarchical picture would seem to preserve some degree of structural similarity between levels. On the other hand, Behrens and colleagues (2018) also provide an account of abstraction from first-order spatial information in the form of eigenvectors corresponding to transformations between firstorder spatial (and nonspatial) environments. Unlike a hierarchical, nested representation of experience, an eigenvector is an abstraction that does not share a structure with its first-order counterparts. Eigenvectors fall into a broader class discussed by Behrens and colleagues (2018), including inductive biases and factorizations. These are all features applying to a set of environments or state spaces that aid in learning but seem to require additional representational resources.

The authors argue for a common set of abstractive capacities operating over both spatial and nonspatial representation, which would utilize higher-order features to drive first-order prediction and planning. Presumably, whatever representational resources would be needed to supplement first-order maps with these higher-order features must be integrated tightly with the first-order maps themselves. Behrens and colleagues (2018) provide a few suggestions, but how this integration might work is still very much an open question.

11.3.3. Embedding Nonspatial Information in a Spatial Format

A third way to expand spatial representations would be to keep the spatial structure intact and embed nonspatial information within that structure. The most prominent example of memory success, the method of loci (MoL), can be understood as organizing information in this way. In the MoL, subjects typically memorize a list of unstructured items such as random words or phone numbers by imagining these items in a structured environment such as a familiar childhood walk through the neighborhood. The MoL thus appears to be a way of using a useful feature of spatial memory to store nonspatial content. The explicit process in the MoL involves two stages: (1) a strategy for encoding items through visualization, and (2) a strategy for retrieving items through a parallel visualization. For example, I could encode a list by imagining a walk through my childhood home, and then later recall the items by imagining walking through the home again and picking up the items.

Questions about how and why mnemonic structure works have received very little attention from memory theorists and scientists. The scant evidence that exists is, however, suggestive and intriguing, and invites us to ask more detailed questions about memory structure. Both the testimony of memory champions and some preliminary studies of expert performance reveal that success in the use of mnemonics does not require any particular level of intelligence or distinct cognitive skills. Instead, the key to success using mnemonic techniques is simply practice (Ericsson, 2003; Wilding & Valentine, 1997). These behavioral reports are complemented by neuroimaging studies indicating that those who use these techniques regularly differ from controls in functional but not structural brain features (Maguire et al., 2003; Raz et al., 2009). A recent and intriguing paper showed that after only six weeks of training, cognitively matched novices exhibited the same functional changes seen in memory champions (Dresler et al., 2017). Similarly, Yoon, Ericsson, and Donatelli (2018) have just shown that a person trained to increase their digit span to more than 100 digits thirty years ago has retained many of the associated skills, despite undertaking no training in the meantime.

In the majority of these cases, the items to be memorized do not have an interesting nonspatial structure themselves (e.g., digit span, presentation order of a shuffled deck of cards). However, looking more closely at the history of this mnemonic technique reveals that it has also been used for semantically structured information. In De Oratore, Cicero recommended the technique for memorizing speeches, lyric poems, and the like. The

Classicist Minchin (2001, p. x) argued that Homer used this method to compose and perform the Iliad and Odyssey. In the Middle Ages, it was common for monks to use this technique to memorize the Bible’s 150 Psalms, as well as long passages from other scholarly texts (Carruthers, 2008). This is continued in some forms of contemporary use, as when students use such techniques to help them remember conceptual blocks of information (Kerr & Neisser, 1983; Roediger, 1980; Wang & Thomas, 2000). The ability to achieve additional mnemonic gains by translating information that is already semantically or conceptually structured into a spatial format suggests that there is something about spatial structure in particular that is useful for memory retrieval.

We have surveyed three ways in which spatial maps might be “scaled up” to accommodate nonspatial information. First, spatial structures might be repurposed to represent nonspatial content with a suitably isomorphic structure: for instance, a conceptual “space” can be modeled as a two-dimensional Euclidean plane. Second, spatial structures might be used to represent abstractions over spatial information, whether in the form of higher-order but still spatially structured representations, or with summary statistics that are to some degree represented jointly with traditional maps. Third, nonspatial information might be embedded in a spatial map, overriding or supplementing nonspatial structure with an exogenous spatial structure.

Taking stock of these results in light of the two traditions discussed in section 11.2, we can draw two key conclusions. First, following on from Tolman’s original essay, the idea of seeing how far the basic components of the cognitive map can be stretched to fit nonspatial content is still very much an open research question. Second, the trade-off between flexibility and inferential power that we observed in the case of conceptual representations characterizes the question of expanding the cognitive map as well. The more we stretch the notion, the more we reduce inferential power in favor of flexibility.