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Cognitive Science (2015) 1–35
Copyright ©2015 Cognitive Science Society, Inc. All rights reserved.
ISSN: 0364-0213 print / 1551-6709 online
DOI: 10.1111/cogs.12265
Concepts as Semantic Pointers: A Framework and
Computational Model
Peter Blouw, Eugene Solodkin, Paul Thagard, Chris Eliasmith
Center for Theoretical Neuroscience, University of Waterloo
Received 9 July 2013; received in revised form 24 February 2015; accepted 20 March 2015
Abstract
The reconciliation of theories of concepts based on prototypes, exemplars, and theory-like
structures is a longstanding problem in cognitive science. In response to this problem, researchers
have recently tended to adopt either hybrid theories that combine various kinds of representational
structure, or eliminative theories that replace concepts with a more finely grained taxonomy of
mental representations. In this paper, we describe an alternative approach involving a single class
of mental representations called “semantic pointers.” Semantic pointers are symbol-like representa-
tions that result from the compression and recursive binding of perceptual, lexical, and motor
representations, effectively integrating traditional connectionist and symbolic approaches. We pres-
ent a computational model using semantic pointers that replicates experimental data from categori-
zation studies involving each prior paradigm. We argue that a framework involving semantic
pointers can provide a unified account of conceptual phenomena, and we compare our framework
to existing alternatives in accounting for the scope, content, recursive combination, and neural
implementation of concepts.
Keywords: Concepts; Categorization; Neural computation; Semantics; Computational modeling;
Mental representation
1. Introduction
The study of concepts has played a central role in the advancement of recent theories
of cognitive function. Phenomena ranging from categorization to language use have been
profitably described in terms of conceptual processing (see Murphy, 2002, for a review),
and many influential descriptions of cognitive development have been produced on
the assumption that concepts are the basic representational entities that comprise our
Correspondence should be sent to Peter Blouw, Center for Theoretical Neuroscience, University of
Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada, N2L 3G1. E-mail: pblouw@uwaterloo.ca
knowledge of the world (e.g., Carey, 1985, 2009; Tenenbaum, Kemp, Griffiths, & Good-
man, 2011). However, despite the obvious importance of concepts to ongoing work in the
cognitive sciences, researchers have developed very different views regarding how
concepts are structured and represented in the brain.
A primary reason for this disagreement is that theorists working in different disciplines
have numerous and sometimes conflicting explanatory goals. Psychologists, for example,
typically wish to explain empirical data from experiments involving tasks like categoriza-
tion and concept learning (e.g., Lin & Murphy, 1997; Regehr & Brooks, 1993; Rips,
1989; Smith & Medin, 1981). Philosophers, on the other hand, typically wish to under-
stand the semantics and possession conditions of concepts (e.g., Fodor, 1998; Laurence &
Margolis, 1999; Peacocke, 1992; Prinz, 2002). Even when a set of explanatory goals is
agreed upon, the scope of the data to be accounted for is often too vast and disparate to
allow for the development of a unified theory (Murphy, 2002). Categorization phenomena
alone, for example, cannot be comprehensively explained using the individual resources
of prototype theories, exemplar theories, or theory theories of concepts (Rogers &
McClelland, 2004).
To deal with this impasse, researchers have most recently tended to adopt one of two
general strategies. The first strategy involves the proposal of “mixed” or “pluralistic”
models in which individual concepts correspond to a number of related or co-referring
representational structures that account for different phenomena (Laurence & Margolis,
1999; Murphy, 2002; Weiskopf, 2009). The second strategy, in contrast, involves elimi-
nating the term “concept” from the vocabulary of the cognitive sciences in favor of a tax-
onomy of more finely grained mental representations that each serve distinct functions
(Machery, 2009). The relative merits of these two approaches are the subject of ongoing
debate (Machery, 2010), but it is safe to say that neither view has achieved widespread
support.
In this paper, we propose an alternative, unifying solution to the current challenges in
concept research. Using methods for characterizing representational states in neural sys-
tems (Eliasmith, 2003, 2013; Eliasmith & Anderson, 2003), we describe concepts in
terms of processes involving a recently postulated class of mental representations called
“semantic pointers” (Eliasmith, 2013). Roughly speaking, semantic pointers are neurally
instantiated, symbol-like representations that can be transformed in numerous ways to
yield further representations that function to support cognitive processes like categoriza-
tion, inference, and language use. Notably, semantic pointers have been successfully used
to account for a range of perceptual, cognitive, and motor behaviors in what is currently
the world’s largest functional model of the human brain (Eliasmith et al., 2012). How-
ever, this past work does not explore the relevance of semantic pointers to conceptual
phenomena in detail. Our aim here, accordingly, is to show that a modeling framework
based on semantic pointers can offer a unified explanation of the kinds of phenomena that
concept theorists have traditionally been interested in. To support our claims, we describe
a biologically plausible spiking neuron model that processes semantic pointers to account
for data from categorization experiments that have been used to bolster three competing
accounts of concepts: prototype theory, exemplar theory, and theory theory.
2P. Blouw et al. / Cognitive Science (2015)
2. Criteria for a theory of concepts
Although the criteria by which theories of concepts are evaluated are often controver-
sial, it is widely acknowledged that certain cognitive functions are paradigmatically con-
ceptual functions. For example, language use, inference, and the formation of
propositional attitudes are just a few of many cognitive tasks uncontroversially defined in
terms of operations involving concepts. We accordingly take as our starting point the idea
that a theory of concepts ought to explain how these conceptual functions are
implemented. However, in addition to these functional explanations, certain theoretical
explanations should also be provided. For example, one ought to explain how concepts
can range in kind from the abstract to the ordinary, and explain how they can refer to
groups of objects in the world. With these considerations in mind, we propose the follow-
ing criteria as minimal requirements for a satisfactory account of conceptual processing
(cf. Barsalou, 1999; Fodor, 1998; Laurence & Margolis, 1999; Prinz, 2002):
1. Categorization
2. Recursive binding
3. Neural implementation
4. Scope
5. Content
There are, of course, other criteria one might choose, but we have selected these five
for the simple reason that they seem to capture properties common to a large number
of conceptual phenomena. Categorization tasks, for instance, are widely studied in the
literature on concepts (Murphy, 2002), and in some cases recruit background knowledge
in the form of inferences that relate object properties to category membership. Thus, it
is plausible that descriptions of additional conceptual processes involving inference and
language can be understood partly as more complex forms of categorization. We focus
exclusively on categorization effects in our simulations because of these and related
considerations.
1
An explanation of binding, too, is an important goal for any account of conceptual pro-
cessing: it underlies both the formation of compositional structures involving multiple
concepts (e.g., LARGE RED DOG) and the integration of multimodal representations of
category instances. Binding has attracted a great deal of attention amongst researchers
interested in the structure of mental representations (e.g., Jackendoff, 2002), and we thus
take it to be a somewhat uncontroversial constraint.
Regarding neural implementation, it is of course something of a platitude to say that
conceptual processes are neural processes. But since the nature of neural processes likely
constrains the types of functions that can be easily computed by a cognitive system (Elia-
smith & Anderson, 2003), it remains an open question whether the functions described
by any particular cognitive model are, in fact, neurally implementable. So, the adoption
of a neural implementation criterion suggests that, all else being equal, a demonstration
of the implementation of a particular model counts considerably in its favor.
P. Blouw et al. / Cognitive Science (2015) 3
As for the more theoretical criteria, scope refers to the broad range of different kinds
of concepts. There are concepts for perceivable objects (e.g., TABLE), abstractions (e.g.,
VIRTUE), theoretical posits (e.g., GENE), mathematical terms (e.g., SUM), and non-exis-
tent entities (e.g., CENTAUR), among other things (Prinz, 2002). A good theory should
be able account for these different classes of concepts, and it should also be consistent
with available evidence implicating particular neural systems and anatomical regions with
the processing of these classes. For instance, studies of neurological patients with seman-
tic deficits suggest that concepts for concrete and abstract entities are processed in distinct
neural systems (Shallice & Cooper, 2013).
Finally, the fact that concepts are about things means that they have content or mean-
ing. This content, in turn, can be roughly defined in terms of how a given concept charac-
terizes what it represents. An adequate theory must explain why a given concept refers to
some things and not others (i.e., provide an account of its extension), and it must also
explain why this concept describes these referents in some ways and not others (i.e.,
provide an account of its intension). The philosophical literature on the semantics and
individuation of mental representations provides the motivation for adopting this criterion
(e.g., Fodor, 1987, 1998; Peacocke, 1992; Prinz, 2002).
In summary, the first three criteria concern the nature and implementation of concep-
tual functions, while the last two criteria concern theoretical properties of the representa-
tions that enable these functions. Our framework is designed to meet these criteria,
although our discussion of the first three is meant to be more comprehensive, while our
discussion of the last two is meant to be more suggestive. To begin developing the frame-
work in detail, we first describe the principles of neural representation and computation
that motivate a number of our arguments.
3. Neural representation and computation
While it is widely accepted that mental representations are features of neural systems,
current approaches to cognitive modeling do not generally characterize representations in
highly detailed neural terms. Symbolic approaches, for instance, typically describe mental
phenomena in terms of computations defined over atomic representations structured by a
language-like syntax (e.g., Fodor, 1975); neural details are rarely, if ever, a consideration.
Connectionist approaches, in contrast, characterize representations in sub-symbolic terms
using weighted connections between large numbers of individual processing nodes (e.g.,
Rogers & McClelland, 2004; Rumelhart & McClelland, 1986). These models, however,
only roughly correspond to the structure of the brain, and they leave out a number of
important details regarding the physiological properties, dynamical properties, and
connectivity of real neurons.
We favor an approach that describes representation and computation in terms of the
activities of large numbers of individually spiking neurons. More specifically, we adopt
the Neural Engineering Framework (NEF) developed by Eliasmith and Anderson (2003).
According to this framework, patterns of activity in spiking neurons can be characterized
4P. Blouw et al. / Cognitive Science (2015)
using mathematical objects such as vectors (i.e., sets of numerical values), which in turn
capture information about the world (via the tuning of the neurons to environmental stim-
uli).
2
By specifying sets of synaptic weights between two or more populations of
neurons, transformations of such vectors can be computed, including transformations that
bind together multiple vectors to embed complex hierarchical structures in a vector space.
It is for this reason that the NEF is occasionally characterized as a compiler that
translates algorithms defined over vectors into a language of neural spikes (see Eliasmith,
2013). For our purposes, there are two significant advantages to characterizing the behav-
ior of neural systems in this quantitative manner. First, there are well-established tech-
niques for translating various kinds of lexical, sensory, and motor representations into
vectors (Georgopoulos, Schwartz, & Kettner, 1986; Jones & Mewhort, 2007; Plate,
2003). Vectors accordingly have the necessary representational power to account for the
multimodal nature of conceptual representations (cf. Barsalou, 1999). Secondly, computa-
tions involving vectors can be used to implement powerful forms of recursive binding
(Gayler, 1998; Kanerva, 1994; Plate, 2003; Smolensky, 1990). Given the vast array of
contents associated with even the simplest concepts, an account of binding is likely essen-
tial for the development of a plausible model of conceptual processing.
Binding can be carried out in the NEF using a process called circular convolution
(Eliasmith, 2004, 2013). Leaving the mathematical details aside, circular convolution can
be thought of as a function that blends two input vectors into a single output vector of
the same dimensionality. Implementing this function is relatively straightforward: If two
“input” neural populations each representing a vector are connected to an intermediary
population that projects to an “output” population, one can use the NEF to solve for a set
of synaptic weights between the populations that will result in the output population
encoding a vector that is the convolution of the two input vectors. This process can be
repeated indefinitely, and it can also be reversed to recover an approximation of any one
vector bound into such a recursively generated structure.
3
Overall, the NEF has all of the
tools needed to describe highly complex syntactic operations involving the composition
and decomposition of a diverse range of neural representations. In combination, the NEF
principles of neural representation and computation allow for the description of a very
powerful kind of representation that Eliasmith (2013) refers to as a “semantic pointer.”
We use the notion of a semantic pointer as a starting point in developing an account of
concepts that can adequately satisfy all five of the criteria introduced in Section 2.
4. Semantic pointers
In its most basic form, a semantic pointer can be thought of as a compressed represen-
tation that captures summary information about a particular domain. Typically, such rep-
resentations derive from perceptual inputs. An image of an object in one’s visual field,
for instance, will initially be encoded as a pattern of activity in a very large population of
neurons. Through transformations of the sort described above, however, further layers of
neural populations produce increasingly abstract statistical summaries of the original
P. Blouw et al. / Cognitive Science (2015) 5
visual input (see Fig. 1). Eventually, a highly compressed representation of the input can
be produced. Such a characterization is consistent both with the decrease in the number
of neurons found in later hierarchical layers of the visual cortex, and with the develop-
ment of neurally inspired hierarchical statistical models for dimensionality reduction
(Hinton & Salakhutdinov, 2006; Serre, Oliva, & Poggio, 2007). Analogous representa-
tions can be generated in other modalities such as audition and sensation.
The reason compressed representations of this sort are called semantic pointers is
because they retain semantic information about the states they represent by virtue of
being non-arbitrarily related to these states through the compression process. The reason
why the representations are referred to as pointers is because they can be used to “point
to” or regenerate representations at lower levels in the compression network (Hinton &
Salakhutdinov, 2006). Moreover, any given semantic pointer can be manipulated indepen-
dently of the network that is used to generate it. A semantic pointer of a table percept,
for example, could be used in cognitive tasks related to tables without necessarily
prompting a reactivation of the richer perceptual representations at the bottom of the
relevant compression network.
The computational power of semantic pointers lies in their ability to be bound together
(using compression operations such as circular convolution) into highly structured repre-
sentations containing lexical, perceptual, and motor information from a variety of sources.
Importantly, such structured representations are themselves semantic pointers, because
(a) (b)
Fig. 1. Hierarchical populations of neurons used for the compression and decompression of perceptual data.
The number of nodes in each layer corresponds to the dimensionality of the representation. The low dimen-
sional semantic pointer at the top of the hierarchy in (a) is thus a compressed representation of a percept, and
the high dimensional representation at the bottom of the hierarchy in (b) amounts to a partial recovery of this
percept from the appropriate semantic pointer.
6P. Blouw et al. / Cognitive Science (2015)
they can point to and regenerate the subordinate representations from which they are
built. Consider again the toy example of a table. Using recursive binding of the sort
already described, which is a compression operation, semantic pointers for visual and tac-
tile images of tables could be combined, along with pointers for an auditory image of the
sound “table” and a visual image of the letters “t-a-b-l-e.” Additionally, various structures
corresponding to verbal information like “has a flat surface” or “used for eating meals”
might also be bound together. These structures would themselves be built out of other
semantic pointers, including compressed visual images of flat surfaces, meal settings, and
so on. Overall, the result of these numerous binding operations is a single representation
that captures relations among a wide range of table-related contents. And this single rep-
resentation can be transformed in numerous ways to re-access images of tables, verbal
information about tables, or motor commands commonly used to interact with tables.
At this point, it should be clear that semantic pointers are highly applicable to the
explanation of conceptual phenomena. They can account for symbolic processes, percep-
tual simulations, and a host of other functions all centered upon a single object class. In
other words, they can act as a summary representation of a category of things in the
world, which is precisely what a concept is often taken to be. Successful neural simula-
tions of conceptual tasks such as simple linguistic inference (Eliasmith, 2013), inductive
reasoning (Rasmussen & Eliasmith, 2011), and rule-based problem solving (Stewart &
Eliasmith, 2011) have all been produced using semantic pointers, along with a large-scale
brain model capable of executing a variety of cognitive functions (Eliasmith et al., 2012).
Based on these successful applications, we think that the notion of a semantic pointer
provides an ideal foundation for accounting for a wide range of conceptual phenomena.
5. Concepts as semantic pointers
It is tempting to claim that concepts just are semantic pointers. However, we avoid this
theoretical formulation for a simple reason: Semantic pointers cannot meet all of the
desired criteria when considered in isolation. Recall that a semantic pointer is simply a
vector encoded by the spiking activity in a population of neurons. This vector captures
relations between a wide range of other representations, and it can be transformed in vari-
ous ways to access these representations, but the vector itself does not possess anywhere
near the full semantic content of an ordinary concept. It is better, then, to think of a
semantic pointer as an entity that enables the occurrence of a concept rather than as an
entity that is equivalent to a concept.
On our account, concepts are best thought of dispositionally. To possess a concept is
to be able to activate various sequences of neural states that correspond to things like
visual and auditory simulations, expressions of natural language, and motor commands all
centered on a single category. The interrelated neural states that feature in these processes
result from the transformation of semantic pointers, and on any given occasion in which
a particular concept occurs, only a limited range of all possible transformations will be
carried out. In other words, the neural processes that comprise the occurrence of a given
P. Blouw et al. / Cognitive Science (2015) 7
concept are context and task dependent (Barsalou, 1999). Conceptual tasks involving ver-
bal reasoning will, for instance, activate different neural states than conceptual tasks
involving the categorization of tactile stimuli (see Fig. 2). Likewise, a cognitive task
involving the concept DOG might invoke a visual simulation of a large animal in one
individual while invoking a simulation of a smaller animal in another individual. Part of
the burden of elaborating on this theory is to give an account of the factors that influence
such contextual variability. Nonetheless, our claim is that these various neural states that
are constitutive of conceptual processing stem from a common point of origin: namely,
the transformation of a semantic pointer.
Given this description, it is important to consider the obvious similarities between our
view and recently developed “neo-empiricist” accounts that identify conceptual process-
ing with the partial re-activation of previously captured perceptual states (e.g., Barsalou,
1999; Barsalou, Simmons, Barbey, & Wilson, 2003; Barsalou, Santos, Simmons, &
Wilson, 2008; Prinz, 2002). Barsalou (1999), for example, identifies concepts with “simu-
lators” or organized systems of category-specific perceptual symbols that can be selec-
tively transferred into working memory. As one might expect, this transfer of symbols
into working memory is highly analogous to the transformation of a semantic pointer to
access detailed perceptual representations. Similarly, Prinz’s (2002) claim that concepts
are “proxytypes” or “perceptually derived representations that can be recruited by
Fig. 2. A simplified diagram of possible transformations of a semantic pointer for the concept DOG. Trans-
forming the semantic pointer can result in perceptual simulations, linguistic inferences, and the consideration
of related concepts. An occurrence of the concept DOG, in our account, is a process through which a set of
these possible transformations is realized. All the representations and transformations depicted in this diagram
are compatible with the principles of neural implementation described in Section 3.
8P. Blouw et al. / Cognitive Science (2015)
working memory to represent a category” (p. 149) shares much with our notion of con-
cepts as processes corresponding, in part, to perceptual simulations.
There are two significant differences between the semantic pointer framework and
these neo-empiricist accounts. For one, the semantic pointer theory is consistent with the
existence of amodal representations. A semantic pointer corresponding to a lexical term,
for example, can contain amodal statistical information regarding the co-occurrence of
related terms in certain contexts (Eliasmith, 2013). Access to such information in the
absence of perceptual representations can explain response times during certain tasks that
require the verification of matches between words and properties (e.g., Solomon & Barsa-
lou, 2004). Moreover, given the controversial nature of the arguments and evidence com-
monly cited in support of strictly empiricist accounts (Machery, 2007), and the existence
of evidence suggesting that concrete and abstract concepts are processed in at least par-
tially separable systems (Shallice & Cooper, 2013), we see the neutrality of the semantic
pointer framework on this issue as a virtue.
Second, the view we propose offers a distinct account of how concepts are actually
represented. Neo-empiricist accounts generally identify concepts with either (1) simula-
tors (i.e., organized systems of perceptual symbols) or (2) simulations (i.e., temporary
representations in working memory). The problem with the first option is that it is un-
derspecified with respect to how simulators carry out their functions (Dennett & Viger,
1999). The semantic pointer framework addresses this shortcoming by identifying
concept occurrences with neural processes and by mechanistically describing these
processes in terms of independently motivated principles of neural computation and a
model-based implementation, described below. The problem with the second option is
that it entails that each instance of a specific simulation corresponds to a unique con-
cept. If so, then one can have multiple concepts that denote a single category, and one
would rarely elicit the same concept twice, since the production of identical simulations
across time is unlikely. Without some explanation of why such simulations form dis-
tinct concepts (or why they are related if they are not in fact distinct concepts), the
theory is left vague and imprecise. Our account avoids this problem by unifying diverse
occurrences of a single concept through the postulation of a common underlying neural
mechanism.
Further similarities are also evident between the semantic pointer framework and con-
nectionist models that are trained through gradient descent to associate various modality-
specific representations (e.g., percepts, verbal descriptions, etc.) via mediating, amodal
semantic representations (Rogers et al., 2004; Roy & Pentland, 2002; Rumelhart & McC-
lelland, 1986). Two key features differentiate our approach. The first is the account of
representational binding that we provide. Connectionist models that learn to associate
representations have no means by which to bind representations recursively, and thus no
means by which to account for compound concepts or syntactically structured representa-
tions. Second, a form of localist encoding is often used wherein individual processing
nodes are taken to represent unique linguistic predicates and perceptual features. This
encoding violates the implementation criterion, because the nature of the correspondence
between these nodes and the neural substrate is left unspecified.
P. Blouw et al. / Cognitive Science (2015) 9
For all of these reasons, we propose that our account offers a promising new approach
to satisfying the criteria introduced in Section 2. However, before returning to an evalua-
tion of the theory with respect to these criteria, we first describe a semantic pointer-based
model that is able to account for important features of the prototype, exemplar, and the-
ory-theory accounts of concepts. While this computational model is not comprehensive, it
provides a good initial demonstration of the potential for a semantic pointer-based frame-
work to offer a unified explanation of conceptual phenomena.
6. Model description
We focus our modeling efforts on three paradigmatic categorization studies. The first
study, conducted by Posner and Keele (1968), suggests that when subjects learn to cate-
gorize patterns of dots generated through the disturbance of various prototype patterns,
they abstract and utilize information about the relevant prototype. The second study, con-
ducted by Regehr and Brooks (1993), demonstrates that similarity-based categorization
strategies can override more analytic strategies under certain circumstances. The third
study, conducted by Lin and Murphy (1997), investigates the effects of background
knowledge on categorization decisions involving identical visual stimuli given distinct
functional descriptions. Together, these studies catalogue a diverse range of categorization
effects.
We propose a single model architecture that does not change when accounting for any
of the different effects. Functionally, the model receives vectors corresponding to com-
pressed natural images as input and produces vectors corresponding to motor responses
as output. All intermediate processing is implemented using approximately 300,000 simu-
lated leaky-integrate-and-fire (LIF) neurons, and 128 dimensional vectors are used in
every simulation. Details regarding how the LIF neurons are used to encode, decode, and
transform vectors can be found in Part A of the Supplemental Materials.
At a more specific level, the model architecture involves a working memory system,
an action selection system, and two further subsystems that implement perceptual and
inferential evaluations of input stimuli (see Fig. 3). Working memory stores semantic
pointers that encode visual exemplars and soft rules that define category membership.
The action selection system controls how information is extracted from these semantic
pointers and used to manipulate input stimuli in a task-dependent manner. Anatomically,
the action selection system is mapped on to portions of basal ganglia and thalamus, while
the other subsystems are mapped on to cortex. These anatomical mappings are largely
motivated by other work (Eliasmith, 2013; Eliasmith et al., 2012) and for present pur-
poses are best viewed as plausible assumptions. The functionality of the model is our pri-
mary concern, and in the simulations reported below, we do not use the model to account
for neural data in a way that would independently justify the mappings in question.
During each experiment, the model is presented with a visual input that cues the
current experimental task, followed by a stimulus to categorize. Based on the cue, the
action selection system initiates processes that decompress a semantic pointer to either
10 P. Blouw et al. / Cognitive Science (2015)
(a) compare the stimulus to previously learned exemplars or (b) apply a set of rules that
define category membership.
To provide more formal details, the working memory system contains neural popula-
tions whose activities represent semantic pointers corresponding to concepts learned dur-
ing the training phase of each experiment. In the case of experiments involving
perceptual categorization, these semantic pointers will take on the following mathematical
description:
SP ¼E1~LabelE1þþEn~LabelEnð1Þ
where Eiis a training exemplar (i.e., a semantic pointer generated through the compres-
sion of an image percept), and LableEidenotes a vector representing the category label
for Ei. The symbol ~denotes the operation of circular convolution in all equations. The
implementation, use, and learnability of this operation in spiking networks are described
in Stewart et al. (2011). In the case of experiments involving background knowledge and
rule-based categorization, an analogous mathematical description is applied:
SP ¼R1~K1þþRn~Knð2Þ
where Riis a vector indexing a particular rule, and Kiis a representation of the contents
of this rule. By sequentially retrieving and applying these rules, the model is able to infer
the degree to which a given input stimulus is consistent with the category description
the rules encode. Overall, (1) and (2) provide representation schemes for the semantic
Fig. 3. Functional architecture of the model. Thick black lines indicate connections between subsystems,
and thin lines with arrows indicate connections that allow the action selection system to monitor and modify
representational states in these subsystems. During each run, the model is presented with a vector indicating
the current task, followed by a vector corresponding to a compressed image of a stimulus. The task vector
triggers actions that decompress a semantic pointer stored in working memory to obtain perceptual or inferen-
tial information that enables categorization of the stimulus. Categorization judgments are routed through the
motor buffer as output. The visual and motor compression hierarchies are not modeled in the simulations
below (cf. Eliasmith et al., 2012), but the former is taken to compress images into semantic pointers, while
the latter is taken to decompress semantic pointers into motor activities. Subsystems contain internal compo-
nents that perform simple processing and routing tasks as described in the main text.
P. Blouw et al. / Cognitive Science (2015) 11
pointers used in our model, but it is important to note that these schemes are chosen to
accommodate specific categorization experiments and are not reflective of the representa-
tional power of our framework.
The details of the action selection system are more complicated. A number of incom-
ing connections allow the system to monitor representational states in other neural
populations in the model, and a small subset of these states are associated with actions
the system performs to control information flow. The system consists of a collection of
neural populations anatomically mapped to basal ganglia, and the input populations corre-
sponding to the striatum encode the similarity (i.e., dot product) between each monitored
representational state and the states that trigger particular actions. The output globus palli-
dus internus populations connect to the thalamus, which connects back to the rest of the
model, resulting in the execution of those actions that correspond to the highest encoded
similarity measure at a given time. Details about the implementation and biological plau-
sibility of this basal ganglia model of action selection can be found in Stewart, Choo, and
Eliasmith (2010). Again, though, our concern is with exploiting the functionality of this
system rather than with using it to account for neural data directly.
The systems that perform perceptual and inferential evaluation are best illustrated
through example. In each experimental condition, the model is first presented with a
vector indicating the current task, followed by a vector corresponding to a visual stimulus
to be categorized. When the task vector is passed through the visual buffer, it triggers an
action that updates a working memory representation of task context, which in turn deter-
mines how the stimulus vector will be processed. In the case of a perceptual categoriza-
tion task, this context representation triggers an action that compares the input stimulus
to a decompressed semantic pointer in the subsystem labeled “Perceptual Evaluation” in
Fig. 3. Mathematically, the subsequent output of the perceptual evaluation system can be
given the following description:
Output ¼SP~Stimulus1ð3Þ
where SP is a semantic pointer of the sort described by (1), and Stimulus1is the
pseudo-inverse
4
of the input vector being categorized. This output is routed to the motor
buffer via an action triggered by the working memory representation of the current task
context. Fig. 4 illustrates this process unfolding as the model performs a task drawn from
Posner and Keele (1968).
In the case of a knowledge-based categorization task, the working memory representa-
tion of the task context triggers actions that carry out a number of inferences, each of
which decompresses the same semantic pointer in a slightly different way. Decompres-
sion in this context involves extracting a representation associated with a particular rule
from the semantic pointer, and then comparing this extracted representation to an input
stimulus by computing a dot product. More specifically, instead of convolving a semantic
pointer with the pseudo-inverse of the stimulus vector, the pseudo-inverse of a rule vec-
tor, Ri, is applied: KiSP~R1
i. Because circular convolution is only approximately
reversible, the inferential evaluation subsystem contains a simple associative memory that
12 P. Blouw et al. / Cognitive Science (2015)
“cleans up” each Ki. Then, the dot product between the input stimulus and this clean ver-
sion of Kiis computed and added to a scalar value that measures the “coherence” of the
stimulus with the category knowledge encoded by the semantic pointer. This running
coherence measure is implemented by a neural population with recurrent connections that
function to integrate input from a neural population that computes the dot product. The
output of the inferential evaluation system at the conclusion of processing thus takes on
the following mathematical description:
Output ¼X
i
ðSP~R1
iÞStimulus ð4Þ
After the sequence of inferences is completed, the final state of the task context represen-
tation triggers an action that routes the output of the inferential evaluation system to the
motor buffer. Fig. 5 illustrates this process unfolding as the model performs a task drawn
from Lin and Murphy (1997).
We take this model to be unified in the following sense: All representations are
semantic pointers, the model structure does not change across tasks, and the set of
Fig. 4. An example run of the model performing a perceptual categorization task. Each plot depicts the sim-
ilarity over time between the representational state in a particular neural population of the model and a set of
known representational states. Below each plot is a spike raster depicting the activity in a subset of the neu-
rons encoding each representational state. The vertical axis indicates the value of the dot product used to
measure similarity, while the horizontal axis indicates time. The labels on the plot for the visual buffer, for
instance, indicate that its representational state is initially similar to a vector indicating that a task from Pos-
ner and Keele (1968) is to be performed. The state then changes as the novel visual stimulus is presented,
and varying degrees of similarity with known items are indicated. As the model concludes its processing, the
representational state in the motor system indicates that the model has chosen “Label A” to categorize the
stimulus.
P. Blouw et al. / Cognitive Science (2015) 13
actions that the action selection system can perform also does not change across tasks.
It could be argued that because we have defined different actions for different task
contexts, we have really proposed a hybrid model of some kind. However, this is
analogous to arguing that a calculator does not provide a unified implementation of
arithmetic. Merely changing the flow of information through the device based on input
(e.g., which operation button is pushed) does not make the device implement a
hybrid account of arithmetic. Representation, structure, and processing steps are held
constant.
Similarly, in our model, changing the transformations performed by the action selec-
tion system is akin to manually swapping out the operation button on a calculator. We
have chosen the present implementation to minimize both the complexity of the model
and its run time. But, critically, changing the transformations performed by the action
selection system only changes the control of information flow in the model—it does not
change the nature of the representations used, the structure of the model, or the overall
process by which stimuli are categorized.
Fig. 5. An example run of the model performing an inferential categorization task. Again, each colored plot
depicts the similarity over time between the representational state in a component of the model and a set of
known representational states. In this case, the plot for the visual buffer indicates that its representational state
is initially similar to the vector triggering a task from Lin and Murphy (1997). This task vector initiates a
sequence of actions that decompress a semantic pointer stored in memory to extract a number of different rules
(as labeled) for categorizing the stimulus. The effects of these rules is to assign particular weights to particular
features (as shown by the values in the plot labeled “Applications”), resulting in changes to the representation
that tracks the “coherence” of the stimulus with the knowledge encoded by the rules. The “Coherence” and
“Applications” plots correspond to subsystems of the Inferential Evaluation system in Fig. 3.
14 P. Blouw et al. / Cognitive Science (2015)
7. Simulations
7.1. Prototype theory: Experiment 1
The first study we simulate is Experiment 3 of Posner and Keele’s (1968) examination
of dot pattern classification. The experiment was designed to investigate whether subjects
abstract information about category prototypes when they are only trained to classify
patterns that are generated by distorting the prototypes. In the training phase of the
experiment, 30 subjects are taught through corrective feedback to categorize a set of 12
slides, each of which depicts a distinct arrangement of nine dots placed within a 30 9
30 matrix. The slides divide into three categories, and the four slides in each category are
generated by randomly distorting a single “prototype” dot pattern that is not present in
the training set. A distortion rule that specifies the distances each dot is moved from its
starting point is used to generate the four training slides associated with each category-
defining prototype. Training is considered complete when subjects are able to achieve
two consecutive classifications of all 12 slides without error.
After completing of the training phase, 32 subjects are placed in a transfer phase and
asked to classify a set of 24 slides without feedback. These slides consist of six old pat-
terns (two per prototype) from the training phase, six new patterns (two per prototype)
generated using the distortion rule from the training phase, six new patterns (two per
prototype) generated using a weaker distortion rule, the three prototypes, and three com-
pletely random patterns. Results from the transfer phase indicate that the training patterns
and prototypes are categorized best of all and equally well, while the new low-level
distortion patterns and new high-level distortion patterns are categorized progressively
less accurately.
To model this experiment, we first assume that all of the visual stimuli are compressed
into semantic pointers using neural transformations of the sort described in Section 3.
Thus, the dot patterns are presented to the model as individual vectors encoded into neu-
ral spike patterns. The three prototypes are constructed through a slightly constrained
form of random vector generation to ensure a certain degree of similarity, before being
normalized to unit length.
5
All vectors are 128 dimensions. To generate both the training
stimuli and the transfer stimuli, the following equation is used:
Stimulus ¼Prototype þNkð0;rIÞð5Þ
where krefers to the dimensionality of the vector, rrefers to the level of distortion, and
Irefers to a k9kidentity matrix. In plain language, a stimulus vector is constructed by
adding a random number drawn from the normal distribution with standard deviation rto
each element of the relevant prototype vector. The value of ris used to approximate the
distortions applied to the prototype patterns by Posner and Keele. Note that the low and
high distortion rules are exact ratios of one another, so a single rvalue is sufficient to
describe both.
6
P. Blouw et al. / Cognitive Science (2015) 15
To run a trial of the experiment, an instance of the model is created with a seman-
tic pointer encoding 12 labeled training images as per (1) provided as direct input
into the working memory. Vectors corresponding to the task context (i.e., “Posner”)
and the test stimulus (e.g., “AT1” –Prototype A, Training Item 1) are then sequen-
tially provided as input to the visual buffer. The task vector triggers an action which
updates a task context representation in working memory to indicate that perceptual
evaluation should be performed; this representation then triggers a further update to
the task context representation, which results in the output of the perceptual evaluation
system being routed to the motor buffer. Fig. 4 illustrates this process in detail for a
single trial of the experiment. Each trial corresponds to 450 ms of simulated process-
ing time.
To replicate Posner and Keele’s experiment in detail, we use 32 random seeds to gen-
erate a unique instance of the model for each of the 32 experimental subjects. The seeds
fix the random number generator used to set various neuron parameters in the model
(e.g., maximum firing rates, preferred stimulus vector, etc.) and allow identical instances
of the model to be recreated across trials. To run the complete experiment, each model
instance is tested on a set of test stimuli using independent trials. The same test stimuli
are used across all trials involving a particular model instance, and an overall total of
21 932 =672 trials are conducted.
7
Results are obtained by tallying the proportion of
errors the model makes in each stimulus category, and averaging this proportion over the
32 model instances.
This procedure is used in all subsequent experiments, and the same model instances
are used across experiments to ensure that all results are strictly due to changes in rele-
vant parameter values.
To evaluate the model, we found a best fit between the free parameter rand the data
reported by Posner and Keele. We perform 11 complete experiments at rvalues ranging
from 0.05 to 0.15, and Fig. 6 plots categorization error as function of rfor each stimulus
condition. These results indicate that the model generalizes Posner and Keele’s finding
across a range of stimulus distortion values: Categorization accuracy is highest for the
training patterns and prototypes, and it gets progressively worse for low-level and high-
level distortion patterns.
Further examination of these results indicates that a rvalue of 0.1 minimizes the root
mean squared difference between the model results and the data. Confidence intervals of
95%for both model data and the human data are computed based on the percentages of
positive and negative categorization judgments in each stimulus category.
It is helpful to conclude with a brief summary of these results. The model’s perfor-
mance in this experiment can be largely attributed to the mathematical structure of the
semantic pointers and the way in which they are processed. To explain, each randomly
generated prototype vector can be thought of as a point in 128 dimensional space, and
each stimulus can be thought of as another point randomly displaced from a prototype by
an amount specified by r.Asris increased, these displacements grow larger in both the
low and high distortion conditions, and the probability of a given stimulus being located
in a region of space more closely associated with an incorrect prototype (and hence an
16 P. Blouw et al. / Cognitive Science (2015)
incorrect category label) increases. It is therefore not surprising that the general pattern of
results displayed in Fig. 6 is obtained.
The model does, however, perform better than humans on the training patterns and
prototype patterns. Two remarks can help clarify the significance of this discrepancy.
First, the difference between the mean error rates in each stimulus category is roughly
10%, which corresponds to roughly one additional error in each category for every two
participants in the experiment. Given that over 180 training stimuli are categorized in
each experiment, this difference is actually quite small. Second, since the prototypes are
randomly generated unit vectors, they can be quite dissimilar, which reduces the likeli-
hood that miscategorization occurs, since the stimuli associated with each prototype are
more likely to lie in disjoint regions of the vector space. Enforcing a minimum similarity
value between the prototypes reduces the discrepancy observable in Fig. 7, but also adds
a further free parameter to the model (we do not set this parameter explicitly in the
results reported—see endnote 6). Adjusting the model on the basis of these factors would
likely reduce the performance discrepancies observed here.
7.2. Exemplar theory: Experiment 2
While prototype theories and exemplar theories have traditionally been developed as
competing explanations of the same phenomena, there are a number of experimental
results that suggest that exemplar representations play a unique role in conceptual pro-
cessing (Murphy, 2002). In order to account for such effects, we model an experiment
(1C) by Regehr and Brooks (1993) that is designed to test the relative importance of
Fig. 6. Modeled error percentages for each stimulus category with varying degrees of stimulus distortion.
The parameter ris varied across 11 simulations to generate different levels of stimuli distortion, as per (5).
The results indicate that larger values of rcorrespond to proportionally more errors on low- and high-
distortion stimuli. Errors on training and prototype stimuli, by comparison, are not significantly altered with
increased values of r. Error bars indicate 95% confidence intervals. In accordance with Posner and Keele’s
data, the model categorizes training items and prototypes equally well, and it makes progressively more
errors on low- and high-distortion stimuli. This general pattern of results holds across a range of stimuli
distortion levels, thereby generalizing Posner and Keele’s observations across a range of stimuli types.
P. Blouw et al. / Cognitive Science (2015) 17
analytic feature matching and more holistic measures of stimulus similarity in the forma-
tion of categorization judgments. During the experiment, 32 subjects are trained to cate-
gorize drawings of imaginary creatures. Each creature possesses a unique feature set
defined over five binary dimensions and belongs to one of two categories on the basis of
this feature set.
8
The two categories are referred to as the “Builder” category and the
“Digger” category. In order to be a Builder, a creature has to possess at least two of three
specific features. Otherwise, a creature is a Digger. For each subject in the experiment,
one of the following four rules is used to specify which features can be used to identify a
Builder (p. 99):
1. Long legs, angular body, and spots.
2. Short legs, long neck, and spots.
3. Six legs, angular body, and spots.
4. Two legs, long neck, and spots.
The use of these separate rules is intended to balance the extent to which a given fea-
ture dimension is relevant to determining category membership (note, however, that the
spots vs. no spots dimension is always relevant; Regehr & Brooks, 1993, p. 99).
Importantly, the perceptual character of each feature can vary across the drawings. For
example, a “long neck” feature might have various curves in one drawing while being
comparatively straight in another. The presence or absence of these sorts of perceptual
differences across analytically identical drawings is used to define two experimental con-
ditions. In the “composite” condition, analytically equivalent features are perceptually
Fig. 7. Comparison of modeled and observed error percentages on a categorization task from Experiment 3,
Day 1 of Posner and Keele (1968). Model results are produced through simulations employing the architec-
ture described in Fig. 3. A rvalue of 0.1 is used to generate the low distortion stimuli, and a rlevel of
0.197.7/5 is used to generate the high distortion and training stimuli, as per the protocol of Posner and Ke-
ele. The value of ris fit to minimize the root mean squared difference between the model results and Posner
and Keele’s reported results. Error bars indicate 95% confidence intervals. Like human subjects, the model
categorizes the training patterns and prototypes best of all, and it makes progressively more errors on the
low- and high-distortion patterns.
18 P. Blouw et al. / Cognitive Science (2015)
equivalent. In the “individuated” condition, analytically equivalent features are perceptu-
ally distinct. By comparing categorization performance across these two conditions, ana-
lytic structure and perceptual similarity can be assessed for their relative importance in
the formation of categorization decisions.
In the training phase of the experiment, each of the 32 subjects is taught through cor-
rective feedback to classify a set of eight figures in accordance with one of the four rules
just described. Half of the subjects are placed in the composite condition and the other
half are placed in the individuated condition; each condition has its own set of eight
training drawings. In the transfer phase, the subjects are asked to categorize a total of 16
drawings without feedback, eight of which are the training exemplars, and eight of which
are new drawings. All of the new drawings are paired with a “twin” from the training set
that differs on only one dimension (namely, the presence or absence of spots; twins are
accordingly quite perceptually similar to another). New figures belonging in the same cat-
egory as their twins are deemed “good transfer” (GT) items, while new figures belonging
in the opposite category as their twins are deemed “bad transfer” (BT) items. The percep-
tual similarities between twin items can thus suggest either correct or incorrect categori-
zation decisions: For the GT items, perceptual similarity is suggestive of the correct
decision, while in the case of the BT items, perceptual similarity is suggestive of the
incorrect decision.
Results from the experiment indicate that subjects in the composite condition make
roughly the same percentage of categorization errors on training, GT, and BT items in
the transfer phase. In the individuated condition, however, a significantly greater propor-
tion of errors are reported for BT items.
To model the experiment, we use similar methods to those employed in the prototype
simulation. We assume that the stimuli are converted into semantic pointers via a com-
pression process, and that the structure of each semantic pointer conforms to the follow-
ing mathematical description:
Stimulus ¼X
FFeatures
DimensionF~ValueFð6Þ
where DimensionFand ValueFare randomly generated vectors used to define the compo-
nents of the analytic structure of each stimulus. For example, a possible dimension-value
pair is SPOTS ~YES. To increase feature individuation and approximate the difference
between the composite and individuated experimental conditions, a Gaussian disturbance
of variable magnitude is applied to each feature value:
Stimulus ¼X
FFeatures
DimensionF~ðValueFþNkð0;rIÞÞ ð7Þ
where, again, krefers to the dimensionality of the vector, rrefers to the standard devia-
tion of the Gaussian distribution, and Irefers to a k9kidentity matrix.
P. Blouw et al. / Cognitive Science (2015) 19
Each experimental trial is conducted using the same method employed in the Posner
and Keele simulation. A semantic pointer encoding a set of eight labeled training stimuli
is provided as direct input to working memory, and the visual buffer is sequentially pro-
vided with both a task vector and stimulus vector. The task vector initiates the same
sequence of actions described in Fig. 4, so the change in the model’s performance is only
due to the use of different test stimuli and different semantic pointers. Each trial again
corresponds to 450 ms of simulated processing time, and the model’s categorization judg-
ment is determined by evaluating the representational state in the model’s motor system.
To assess the impact of feature individuation on categorization performance, we per-
form 15 experiments using stimuli generated with rvalues ranging from 0.01 to 0.15.
Each experiment involves testing 16 stimuli on the first 16 instances of the model, for a
total of 16 916 =256 trials. Fig. 8 plots categorization error as function of rfor each
stimulus condition. Model evaluation was performed by fitting the free parameter rto the
data reported by Regehr and Brooks. We observe that a rvalue of 0.02 minimizes the
root mean squared difference between the model results and the data in the composite
condition. Likewise, a rvalue of 0.1 minimizes the root mean squared difference
between the model results and the data in the individuated condition. Using different val-
ues of rto account for the different stimulus conditions is quite reasonable, because a
low rvalue corresponds to comparatively small differences between analytically equiva-
lent features on distinct stimuli, while a high rvalue corresponds to comparatively large
differences between such features. A direct comparison of model results and experimental
data for both the composite and individuated conditions is reported in Fig. 9. Confidence
intervals are computed as before.
For an intuitive explanation of these results, it is useful to again think of each stimulus
as a point in high-dimensional space. The labeled training stimuli define regions in the
space that are associated with one of the two category labels. When no feature individua-
tion is present, differences in analytic structure are the only differences that exist between
the two classes of stimuli. This is important because it means that each labeled training
exemplar signals that a particular analytic structure is diagnostic of membership in a par-
ticular category. When a novel stimulus is mapped to the high-dimensional space, one can
think of each training exemplar stored in memory as “voting” on the category membership
of the new stimulus on the basis of points of structural overlap. For example, if a training
exemplar possesses a “long neck” feature and is labeled a Builder, then the presence of
this feature in the test stimulus would result in this training exemplar supplying one vote
in favor of categorizing the test stimulus as a Builder. The balance of all such votes deter-
mines the resulting categorization judgment (subject to some noise given that the ran-
domly generated vectors used in stimulus construction described in (6) are not guaranteed
to be orthogonal). Overall, when no feature individuation is present, the votes produced
by each memorized exemplar are sensitive only to analytic structure, which means that
the change in analytic structure between GT and BT items results in a change in the num-
ber of votes a test stimulus gets for each category. This sensitivity to analytic structure
explains why the model is less prone to BT error in the low feature individuation condi-
tion—the model notices the change in structure between BT items and their training pairs.
20 P. Blouw et al. / Cognitive Science (2015)
As stimuli individuation is increased, however, the votes supplied by each training
stimulus become less sensitive to analytic structure. To see why, recall that feature indi-
viduation produces features that are analytically equivalent but perceptually distinct. This
means that the votes supplied by a particular training exemplar only apply to test stimuli
with perceptually similar features, since analytically equivalent features might be repre-
sented by highly distinct vectors due to the distortions used to approximate feature indi-
viduation. In this case, if a training exemplar has a highly individuated “long neck”
feature, then the exemplar does not produce votes in favor of long necks in general. It
only produces votes in favor of particular kinds of long necks. This behavior translates
into increased error on BT items for the following reason. The counterpart to a BT item
in the training set contains identical features on all but one dimension (Spot vs. No
Spots). This means that as feature individuation is increased, the training counterpart sup-
plies an increasingly large proportion of the votes that are relevant to categorizing the BT
test stimulus. And because the training counterpart is in the opposite category as the BT
test stimulus, these votes translate into an increased likelihood for categorization error. It
is accordingly not surprising that the model performs progressively worse on BT items as
the value of ris increased over a range of experiments in Fig. 8.
Finally, it is worth remarking on the fact that the model performs better than humans
on GT stimuli with minimal feature individuation. This is likely because Regehr and
Brooks’ additive feature rules are designed such that GT items occasionally (and uniquely)
possess none of the features that signal membership in the opposite category. To explain,
each three-feature rule can be used to divide all of the possible features into sets that are
either indicative of a Builder, indicative of a Digger, or diagnostically neutral. Most stim-
uli, upon examination, possess one feature that is diagnostic of the category they do not
Fig. 8. Modeled error percentages in each stimulus category with varying degrees of stimulus distortion.
The parameter ris varied across 15 simulations to generate different levels of stimulus feature individuation,
as per (7). The results indicate that larger values of rcorrespond to proportionally more errors on bad trans-
fer stimuli, and proportionally fewer errors on training and good transfer stimuli. Error bars indicate 95%
confidence intervals. These results are consistent with the observation that highly individuated stimuli are
much more likely to be misclassified when they have a twin stimulus in the opposite category (i.e., in the BT
condition).
P. Blouw et al. / Cognitive Science (2015) 21
belong to. Half of the GT items, however, possess no features that are diagnostic of the
category they do not belong to (see Regehr & Brooks, 1993, p. 102). As a result, these GT
items are somewhat more likely to reside in a region of the vector space that is associated
with the correct categorization judgment. It is quite possible that people are not very
attuned to these subtle differences among the stimuli, especially given Regehr and Brooks’
observation that many subjects report relying on only one or two features to arrive at a
classification judgment. The model has no attentional mechanism that allows it place
unequal priority on features, which provides a potential explanation for the more accurate
performance we observe in the GT condition. Again, adjusting the model on the basis of
such factors could improve the model-data fits we observe.
7.3. Theory theory: Experiment 3
In addition to the ongoing development of prototype and exemplar views, the idea that
concepts are structured like intuitive theories of the categories they denote has become an
increasingly popular target of research (Keil, 1989; Murphy & Medin, 1985; Rogers &
Fig. 9. Comparison of modeled and observed error percentages for composite stimuli and individuated stim-
uli in Experiment 1C of Regehr and Brooks (1993). The stimuli in the composite condition are generated
using a rvalue of 0.02, and the stimuli in the individuated condition are generated using a rvalue of 0.1.
These levels were selected to minimize the root mean squared difference between the model results and the
data in each condition. Bad transfer (BT) items have a twin in the training set that belongs in the opposite
category while differing on only one feature dimension. Good transfer (GT) items have a comparable twin
item in the training set that belongs in the same category.
22 P. Blouw et al. / Cognitive Science (2015)
McClelland, 2004). The basic insight prompting this development is that individuals pos-
sess beliefs about things like causal relations, essences, and ontological distinctions that
seem to influence how they use concepts (Keil, 1989; Murphy & Medin, 1985; Prinz,
2002; Rogers & McClelland, 2004). For example, the reason why BIRD denotes a coher-
ent category and groups entities in the way that it does is because many of the features
shared by most birds (such as flight, wings, feathers, and hollow bones) are related to one
another via a set of one or more explanations: Birds can fly because they have wings,
feathers, along with hollow bones; and birds fly because doing so helps them gather food
and avoid predators (Murphy, 2002; Rogers & McClelland, 2004). In categorization tasks,
effects of this sort manifest themselves when subjects use explanatory inferences to match
an object to a category.
To provide an account of simple effects of this sort, we model Experiment 2 of Lin
and Murphy (1997) in which two groups of subjects are given distinct functional descrip-
tions of artificial categories and then asked to categorize identical sets of images. The
results indicate that subjects are attentive to different features of the images depending on
the category description they receive. Thus, background knowledge is shown to have an
effect on subjects’ performance in an image-based categorization task.
In the training phase of the experiment, 20 subjects are divided evenly into one of two
groups (A and B). Each group is given a different interpretative description of a set of
three training examples, each of which is comprised of four distinct features, for eight
different categories. The category descriptions are devised so that of the four features
present on each training example, one is characterized as functionally critical, two are
characterized as functionally optional, and one is characterized as functionally irrelevant.
So, in the case of the examples pictured in Fig. 10, participants in Group A are given the
following description:
Quinese hunters use tuks to catch Bondu, a type of animal that people like to eat in
the Quine country. To catch a Bondu with a tuk, grab the tuk at its handle (3). Once a
Bondu is spotted, throw the loop (1) over the Bondu’s neck and quickly pull the string
(4) at the end to tighten the loop. The cover (2) in front of the handle protects your
hand from being bitten or scratched by the animal. (p. 1156)
Participants in Group B, in contrast, are given this description:
Quinese people use tuks to spray pesticides. The triangular shaped bottle (2) contains
the pesticides. When (3) is unscrewed, the pesticides flow out through the hose (4).
The loop (1) is used to hang the tuk on the wall. (p. 1156)
Once a given subject learns the training examples and category descriptions for all
eight categories, the subject is required to recall descriptive information about each cate-
gory, and to answer questions about how best to take care of the items in each category.
After the subject completes this recall process without error, they are allowed to proceed
to the transfer phase of the experiment.
P. Blouw et al. / Cognitive Science (2015) 23
During the transfer phase, each subject is asked to categorize a set of new items as
quickly as possible while maintaining accuracy. First, a category label (e.g., “Tuk”) is
presented on a computer screen for 1 s, after which an image appears. The subject then
provides a Yes/No judgment before moving on to the next item. Importantly, the images
that are presented after each category label vary in their consistency with the description
of the category provided during the training phase. Four types of images are used. First,
there are “Prototypes,” which contain all four features mentioned in the category descrip-
tions. Second, there are “Consistent A” items, which lack a feature that is functionally
optional for the subjects in Group A but functionally critical for the subjects in Group B.
Third, there are “Consistent B” items, which lack a feature that is functionally optional
for subjects in Group B but functionally critical for subjects in Group A. Finally, there
are “Control” items, which lack features that are functionally critical for subjects in both
Group A and Group B.
The 20 subjects are each tested on three images per image-type for all eight of the cat-
egories. The images are presented in random order, and each subject undergoes a total of
96 trials (i.e., 3 9498). Lin and Murphy’s results indicate that the prototype items eli-
cit very high proportions of positive judgments, while the items that are consistent and
inconsistent with a given subject’s category knowledge elicit progressively fewer positive
judgments. The control items elicit very few positive judgments.
To model this experiment, we assume that each simulated participant has learned a
semantic pointer that encodes a category description as a set of simple rules. These rules
function to determine whether a particular feature is important for belonging to the cate-
gory under consideration. For example, one of the rules learned by a participant in Group
A might be “if the item is a Tuk, then it should have a loop.” The semantic pointers
encoding these rules are structured in accordance with (2) as follows:
SP ¼R1~F1þR2~F2þR3~F3þR4~F4ð8Þ
Fig. 10. Training examples reproduced from Lin and Murphy (1997). The numbers identify features of the
stimuli that are given distinct functional descriptions (see text) across two experimental conditions. Once
given these descriptions and the training examples, the subjects are subsequently placed in a transfer phase
and asked to categorize a set of similar stimuli.
24 P. Blouw et al. / Cognitive Science (2015)
where each Fiis a semantic pointer encoding a representation of one of the features
defining the category (e.g., a loop or a handle). There is accordingly one rule for each of
four category-defining features. To apply a rule Ri, the action selection system performs
an action that extracts Fiby decompressing the semantic pointer in the model’s inferen-
tial evaluation subsystem. This decompression operation is performed by convolving the
semantic pointer with the pseudo-inverse of Riand cleaning up to a scaled version of the
result. The cleanup process can be symbolized as a transition SP~R1
i!rFi, where ris
scalar value that defines the strength of the rule. Finally, the dot product between rFiand
the input stimulus is computed and added to a score. The value of this score after all of
the rules are applied determines whether or not a positive or negative categorization
judgment is routed to the motor system as output.
We do not model the process by which the values of rare learned. Rather, we assign
a probability distribution to the strength of each rule, and sample from this distribution
when generating each instance of the model. The distributions are calculated by assuming
that on any given trial, the model should positively categorize each stimulus type with a
probability equal to the proportion of positive human responses recorded by Lin and Mur-
phy (1997) for this type. Two free parameters, a constant standard deviation rand the
mean l1of distribution for the first rule, are fit to best approximate Lin and Murphy’s
results. The mathematics underlying this statistical technique are described in detail in
Part B of the Supplemental Materials.
Fitting the parameters rand l1yields a threshold that we use to gate the output of the
inferential subsystem. If the score is over the threshold, the output of the subsystem is a
vector that corresponds to a positive categorization judgement. Otherwise, the output of
the subsystem is a vector that corresponds to a negative categorization judgement. During
each trial of the experiment, a task vector is provided as input to the visual buffer, which
in turn initiates a sequence of actions that selectively decompress the semantic pointer
from working memory to incrementally evaluate the stimulus. A complete illustration of
this process can be found in Fig. 5.
To run a complete experiment, we create 20 instances of the model and test each
instance on the same set of stimuli. For the sake of improving run-time, we only test
each model instance on one stimulus per transfer type per category, for a total of
498920 =640 trials. The stimulus vector in each trial is generated by adding
together a set of vectors corresponding to each feature present in the stimulus type
under consideration. For example, a consistent stimulus from the Tuk category would
encoded as:
Stimulus ¼1F1þ0F2þ1F3þ1F4
where F1;F2;F3,andF4correspond to the loop, guard, handle, and string that are refer-
enced in the category description for Tuks. Vectors corresponding to features are randomly
generated for each of the eight different categories used in the experiment.
Fig. 11 reports the results of this experiment, which indicate that the model’s performance
is quite comparable to the human performances reported in Lin and Murphy (1997).
P. Blouw et al. / Cognitive Science (2015) 25
Overall, these results indicate that the model is able to incrementally apply knowledge
encoded in a set of rules to perform rule-based stimulus categorization. When a stimulus
accords with these rules, the model is very likely to judge that the stimulus belongs to
the category under consideration. When a stimulus accords with the rules to a lesser
degree, the model is less likely to judge that the stimulus belongs to the category under
consideration. As such, the model is appropriately sensitive to how consistent a given
stimulus is with respect to a simple knowledge base encoded into a set of rules. We take
these results to be a very preliminary demonstration of the model’s ability to account for
simple rule-based knowledge effects during categorization. A discussion of possible
extensions and improvements to knowledge-based conceptual processing is presented in
the next section.
7.4. Model summary
We have provided a unified process model and applied it to three experimental results
covering three different theories of concepts. A few remarks can help clarify what we
take to be the significance of these simulations.
First, they demonstrate that semantic pointers can be manipulated and decompressed in
qualitatively distinct ways. In a perceptual task, a semantic pointer is used to match the
stimulus to a category label in a one-shot process that essentially amounts to pattern
recognition. In an inferential task, by comparison, a semantic pointer is used to facilitate
a series of computations that perform an incremental, rule-based analysis of the stimulus.
Second, the simulations generalize the empirical findings reported in the studies we
Fig. 11. Comparison of modeled and observed positive categorization judgments for an experimental task
from Lin and Murphy (1997). The downward trend across the conditions indicates that as the test stimuli
become more inconsistent with the relevant functional descriptions, both human subjects and the model make
fewer positive category membership judgments. Error bars indicate 95% confidence intervals. The rules
encoded into each semantic pointer specify the degree to which each stimulus feature is consistent with a
functional description of a particular category. As such, the model’s rule applications involve applying
knowledge of these functional descriptions to categorize stimuli.
26 P. Blouw et al. / Cognitive Science (2015)
model. By varying a single parameter r, we are able to replicate the patterns of percep-
tual categorization reported in these studies across a wide range of new stimuli. The
model can therefore be used to generate novel predictions about human behavior. Third,
during every experimental trial, a single semantic pointer is used to encode all of the cat-
egory information required to arrive at a classification judgment. The model thus demon-
strates how semantic pointers can offer a unified framework for studying conceptual
phenomena.
One potential criticism of our model is that it fails to genuinely account for the type of
phenomena that knowledge-based approaches to concepts are designed to handle. These
approaches emphasize the role of causal and explanatory reasoning during categorization,
and since the rules we apply in Experiment 3 essentially perform weighted feature com-
parisons, it seems doubtful that any such reasoning is occurring. However, rules suffice
for providing explanations in the form of inferences, and they can express causal regulari-
ties when they are tied to sensory-motor manipulations (Thagard, 2012). To explain, on
any appropriate understanding of reasoning, inferences can be characterized as transitions
between mental states. Such transitions can be naturally captured in the form of rules or
actions, and highly complex forms of reasoning involve highly complex rules and actions,
while simpler forms of reasoning involve simpler rules and actions. On this understand-
ing, it is quite apparent that our model is performing a simple form of inference. So what
is really at issue is the degree rather than kind of effect being exhibited by the model.
And if it is conceded that the model captures knowledge effects in kind, then this
criticism loses much of its force.
9
Moreover, it is simply not true that feature comparisons are inconsistent with the pres-
ence of rule-based categorization. Consider, for instance, Rips’s (1989) classic study in
which participants are asked to decide whether an object 3 inches in diameter is more
likely to be a pizza or a quarter. Since quarters have a fixed diameter, respondents typi-
cally answer that the object is more likely to be a pizza, even though it is more similar to
a quarter. This result is often taken to indicate that there are categorization processes
involving rules that can override more typical processes involving assessments of similar-
ity. Notice, though, that the relevant background knowledge about quarters can only be
applied through a feature comparison: One must assess the diameter of the stimulus
object and compare it to the known diameter of a quarter. As such, it is entirely plausible
that effects of this sort could be accounted for using our model. A set of soft rules about
quarters could be encoded into a semantic pointer as per Eq. (2), and one of these rules
could function to very strongly penalize the coherence score of any stimulus with a suffi-
ciently large diameter. Accounting for more sophisticated effects is, of course, an impor-
tant goal for future work, but given that models of complex reasoning tasks such as the
Tower of Hanoi puzzle have been implemented using semantic pointers (Stewart & Elia-
smith, 2011), our claim to have provided a good starting point for handling knowledge
effects is a reasonable one.
Another potential criticism of the model is that all of the explanatory insight it provides
is due to the mathematical structure of the semantic pointers. While good categorization
performance can be obtained using vector-based models that abstract away from neural
P. Blouw et al. / Cognitive Science (2015) 27
implementation (e.g., Knapp & Anderson, 1984), such models do not describe temporal
dynamics of the sort that are present in our simulations. These dynamics can be used to
make rough predictions about (a) differences in reaction times across tasks and (b) the
temporal and anatomical localization of neural activity during task performance. For
example, our model predicts that comparatively more neural activity would be observed in
basal ganglia and thalamus during the performance of a task from Experiment 3 than dur-
ing the performance of a task from Experiment 1 or 2, since more actions are performed.
Similarly, the model predicts that an onset of increased motor system activity should occur
later in Experiment 3 than in Experiment 1 or 2, since the action that performs motor rout-
ing is delayed as a result of the longer sequence of actions initiated by the task vector. In
general, the model predicts that tasks that involve more operations on a semantic pointer
should take longer than tasks that involve fewer operations, all else being equal.
10
One final concern is that the model currently predicts no difference in reaction times
(RTs) across conditions involving the same experimental task. In Experiment 3, for exam-
ple, the time it takes for the model to generate a categorization judgment does not depend
on the type of the stimulus being categorized (i.e., prototype and control stimuli are catego-
rized in approximately the same amount of time). This behavior conflicts with Lin and
Murphy’s observation (p. 1160) of quicker response times for stimuli that are consistent
with a category description in comparison to stimuli that are inconsistent with a category
description. However, it is important to note that these RT differences could be due to
factors we do not explicitly model. For example, it might be that the application of a rule
encoded in a semantic pointer also results in processes that prime particular motor behav-
iors. In our model, prototype stimuli tend to achieve above-threshold coherence scores more
quickly than other stimuli, even though the categorization judgment reflecting this is not
routed through the motor buffer until all four rules have been decoded from the semantic
pointer stored in memory. If motor priming occurs in proportion to the value of the coher-
ence score, it might be possible to explain these RT differences. It is also worth noting that
in the case of Regehr and Brook’s results, no significant RT differences are observed across
the stimulus conditions in experiment 1C (see p. 100). Overall, while accounting for differ-
ences across conditions in a single task is open to further exploration within our modeling
framework, it is nonetheless true that the framework currently makes interesting predictions
concerning the temporal and anatomical localization of neural activities across different
tasks. We take this latter point to be the main insight offered by our modeling framework,
and recognize the need for further study of reaction time data within this framework.
8. General discussion
It is worth reflecting on a few more of the general properties of the semantic pointer
framework. As described, the framework offers a fairly straightforward strategy for
accounting for a variety of conceptual functions. The first step in this strategy is to
hypothesize the structure of the semantic pointers underlying some phenomenon of inter-
est. The next step is to hypothesize a set of mechanisms that manipulate, compress, and
28 P. Blouw et al. / Cognitive Science (2015)
decompress these semantic pointers to bring the phenomenon about. Recent work sug-
gests that this strategy can be used to motivate a novel cognitive architecture (Eliasmith,
2013). Recent articles have also used semantic pointers to explain priming, intentions,
emotions, creativity, and consciousness (Schr€
oder & Thagard, 2013; Schr€
oder, Stewart, &
Thagard, 2014; Thagard & Schr€
oder, 2014; Thagard & Stewart, 2011; Thagard & Stew-
art, 2014).
In keeping with this breadth of application, the semantic pointer framework plausi-
bly satisfies our five criteria for a theory of concepts. With respect to categorization,
we are able to account for an important selection of experimental results and derive
predictions about related results using our model. Moreover, the inferential and per-
ceptual evaluations carried out by our model indicate that it has the capacity to
explain categorization behavior involving both rules and memory in a unified manner
(cf. Sloman, 1996; Smith, Patalano, & Jonides, 1998). Extensions involving additional
categorization phenomena are also possible. For example, Eliasmith (2013) describes
simulations in which semantic pointers are used to classify images of hand-written
digits with human-level accuracy. Similarly, Hunsberger, Blouw, Bergstra and Elia-
smith (2013) achieve human-level categorization performance in the tasks described in
Experiments 1 and 2 while using a hierarchical visual network that takes raw images
of stimuli as input.
To achieve recursive binding, we use convolution to define richly structured semantic
pointers of the sort described by the representation schemes in Eqs. (1) and (2). These
schemes can also be modified to account for the formation of simple natural language
expressions. For example, Eliasmith (2013) demonstrates that simple sentences can be
encoded using semantic pointers that bind representations of words to representations of
the grammatical roles they occupy, and Stewart, Choo and Eliasmith (2014) suggest a
means of parsing simple natural language sentences with this same architecture. Investi-
gating the use of semantic pointers in natural language processing tasks is an important
topic for further study.
The neural implementation criterion is satisfied through our use of LIF neurons and
biologically plausible patterns of connectivity between anatomical areas such as basal
ganglia, thalamus, and cortex. Additionally, since perceptual and inferential processing
are performed in distinct subsystems of the model, the model is consistent with evidence
indicating the abstract and concrete concepts are processed in distinct neural systems
(Shallice & Cooper, 2013). This said, there are aspects of our neural implementation that
require further investigation. For example, there is no direct evidence that the brain
makes use of a convolution operation during conceptual processing. But given the numer-
ous explanatory advantages accrued by postulating such an operation, we think it consti-
tutes a reasonable working assumption.
With respect to scope, the different semantic pointers used in our model encode differ-
ent kinds of representations, and it is straightforward to generalize these differences to
account for highly complex and abstract concepts. For example, Eliasmith (2013)
describes techniques for manipulating semantic pointers that include several hundred
bound elements taken from an adult sized vocabulary, while remaining within known
P. Blouw et al. / Cognitive Science (2015) 29
anatomical constraints. In a similar vein, Crawford, Gingerich and Eliasmith (2013) have
demonstrated a scalable encoding of the entire WordNet graph that employs semantic
pointers.
A bit more needs to be said about representational content. Previous work suggests
that the NEF is compatible with what is known as a two-factor theory of semantics
(Eliasmith, 2000, 2003). Two-factor theories describe the content of a mental represen-
tation in terms of both its external causes (e.g., the stimuli that drive activity in a pop-
ulation of neurons) and its computational role (e.g., the subsequent effect the neural
population has on other populations). So, in the case of a neural population represent-
ing, say, an auditory image, the encoding of the stimulus into neural spikes would
specify the causal factor,
11
and the decoding of the spikes (to recover the signal being
passed on to other neurons) would specify the computational factor. Together, these
two factors define the content of the representation, and act, roughly, to pick out its
extension and intension. For a semantic pointer built through the compression of
numerous representations, the relevant factors would be underwritten by spiking patterns
in other neural populations. The activity in these populations might, in turn, be more
directly driven by perceptual stimuli, which would causally contribute to the content in
the constructed semantic pointer. Of course, the theoretical details need to be fleshed
out, but the general strategy of tracing functional relations among patterns of neural
activity provides a principled method for identifying the content of arbitrarily complex
semantic pointers.
Overall, while much work remains to be done to scale up the semantic pointer
framework to account for more sophisticated conceptual phenomena, it has clear advan-
tages over existing approaches. First, it is neurocomputationally specified to a degree
that surpasses most, if not all, other accounts. Second, it offers a principled unification
of a range of categorization phenomena. Third, it offers a mechanistic description of
concept binding and the formation of natural language expressions. On a more philo-
sophical front, the semantic pointer framework has tools to give an account for the
wide range of different kinds of concepts and the semantic content of concepts. Discus-
sions of semantics and scope are often ignored or bracketed in the psychological litera-
ture, while philosophers, albeit with a few exceptions (Prinz, 2002), have paid
relatively little attention to empirical research on conceptual processing. Finally, we
suggest that the primary contribution of this framework is that it provides a general
representational scheme and biologically plausible mechanisms that can be used to
implement conceptual functions often thought to be fundamentally distinct in kind (e.g.,
Machery, 2009). There are clear limitations to the scope of the phenomena that we
have modeled here, but it should be apparent that semantic pointers provide a single
representational format capable of describing all of the main kinds of conceptual pro-
cessing. Applying this representational format to a wider range of phenomena is an
important avenue for future work.
30 P. Blouw et al. / Cognitive Science (2015)
9. Conclusion
To return to our introductory remarks, we think that semantic pointers offer a promis-
ing solution to the problems framing contemporary research on concepts. Pluralism is
avoided because the framework does not require the existence of multiple co-referring
representational structures to account for category knowledge. In all of our simulations,
one semantic pointer (in tandem with processing machinery for decompression etc.) suf-
fices to explain all of the cognitive processes involved in using a concept to perform cate-
gorization. Since a single semantic pointer can comprehensively support conceptual
processing in this manner, it makes little sense to claim that the term “concept” picks out
a set of unrelated representations and processes. We suggest that if our approach is at all
persuasive, concepts are here to stay.
Acknowledgments
We thank Terry Stewart for assistance with the simulations and helpful discussions on
a range of topics. For other helpful suggestions, we thank the members of the University
of Waterloo’s Computational Neuroscience Research Group and three anonymous review-
ers, whose comments have helped improve this paper in a number of ways. This research
was supported by the Social Sciences and Humanties Research Council of Canada, along
with the Natural Sciences and Engineering Research Council of Canada.
Notes
1. There are three other reasons for focusing on categorization simulations exclu-
sively. First, space restrictions prohibit the modeling of numerous conceptual phe-
nomena in one article. Second, categorization is a paradigmatic conceptual task and
is therefore of interest to a wide range of researchers of different disciplinary per-
suasions. Third, a majority of the existing empirical research on concepts has
focused on the study of categorization, and there is accordingly a rich store of data
to which we can compare our results. Model-data comparisons are less feasible for
tasks that have been of limited research interest.
2. The NEF defines mental representations in terms of both the encoding of stimuli
into patterns of neural spikes and the decoding of sets of spike trains into the phys-
ical variables they represent (Eliasmith, 2003). To give a very simple example, two
regions in the brainstem called the nuclei prepositus hypoglossi (NPH) and the vos-
tral medial vestibular nucleus (VN) contain neurons with tuning curves that plot a
relation between horizontal eye position and spiking activity (Eliasmith & Ander-
son, 2003, pp. 44–49). Accordingly, neurons in NPH and VN collectively “encode”
a measurement of eye position into a pattern of neural spikes. The decoding
P. Blouw et al. / Cognitive Science (2015) 31
procedure involves assigning an optimal weight (either a scalar or a vector depend-
ing on the dimensionality of the decoded representation) to the responses of each
neuron, and summing all such weighted responses over the relevant population and
over time. In NPH and VN, the result of this sum is an estimation of the position
of the eye. For a more detailed mathematical definition of these encoding and
decoding relations, see Part A of the Supplemental Materials.
3. It is also worth noting that a single binding network of this sort can compute the
circulation convolution of any two input vectors, and that the NEF has the
resources to explain how the weights that implement such a binding network can
be learned through the use of a biologically plausible Hebbian learning rule (Stew-
art, Bekolay, & Eliasmith, 2011).
4. Obtaining the pseudo-inverse of a vector is a simple linear transformation, and it
can thus be computed on a connection between neural populations. Convolving a
semantic pointer with the pseudo-inverse of a vector extracts an approximation of
any item bound to that vector in the semantic pointer.
5. Specifically, the vector for the first prototype is randomly generated, while vectors
for the second and third prototypes are sums of a randomly generated vector and
the first vector.
6. In Posner and Keele’s (1968) paper, the training and high-distortion stimuli are pro-
duced with a 7.7-bit distortion rule, and the low-distortion patterns are produced
with a 5-bit distortion rule. Because the distortion values are multiples of one
another, we fit the model using a “base distortion level” equal to sigma and gener-
ate the stimuli patterns in the training and high conditions using a standard devia-
tion equal to 7.7/5 times sigma.
7. To reduce the model’s runtime, no trails involving the completely random stimuli
are conducted, since there is no measure of correctness for these stimuli.
8. The complete set of feature dimensions and values is as follows: body (angular or
round); legs (long or short); number of legs (2 or 6); spots (yes or no); neck (long
or short).
9. One further point worth making here is that it is important to distinguish between
concept acquisition and concept use. In the case of the Lin and Murphy experi-
ment, it is during the process of concept acquisition that knowledge of causal rela-
tions between the constriction of a loop and the killing of a Bondu is used to
prioritize the presence of a loop on a Tuk. During the use of a concept to perform
categorization, these causal relations need not be directly considered. Since we are
not proposing to give a comprehensive account of concept acquisition, omitting
these causal inferences from our model is reasonable.
10. There is nothing particularly significant about our choice of a 450 ms simulation
window other than the fact that this window provides sufficient time for all three
kinds of categorization task to complete while minimizing the overall amount of
simulation time. Furthermore, there is no implicit prediction in our work that
reaction times should be on the order of 450 ms, because the model excludes a
32 P. Blouw et al. / Cognitive Science (2015)
considerable amount of perceptual and motor processing that needs to be incorpo-
rated into exact estimates of reaction times.
11. This is an oversimplification. Statistical dependencies among various triggers of
neural activity are used to specify the relevant causal factor to avoid problems
associated with misrepresentation. See Eliasmith (2000) for details.
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