By: Anthony Aubel
The University of Edinburgh, Philosophy
April 11,2024
Abstract
The Predictive Processing (PP) framework offers a powerful model of perception and cognition that emphasizes error-minimization within a hierarchical generative model. However, extending PP to abstract concepts remains a challenging problem. Christian Michel's hybrid account (2023c) attempts to address this challenge by positing that concepts exist in different formats (exemplars, prototypes, theories) that mainly depend on context and task. Yet, his model leaves some representational mechanisms underspecified; particularly for prototypes and their interaction across hierarchical levels. To address this, I turn to Conceptual Space Theory (CST) that can model concepts as structured and multidimensional spaces, with prototypes as their central regions. In this dissertation, I plan to propose a plausible integration of CST into Michel's hybrid account where I will argue that CST's geometrically defined spaces provide the missing representational substrate for Michel's ‘higher-level nodes’ and that this joint approach can significantly expand its explanatory scope. I aim to detail how this integration refines the mechanisms that govern prototype representation including similarity judgments, and what is referred to as "vertical" processing of a concept within the PP hierarchy. This could in turn open some new avenues for modeling how prediction errors, precision weighting, and CST's geometric structure can jointly shape the representation and dynamics of abstract concepts. I will also discuss some implications of this CST-augmented model for processes such as categorization under uncertainty and conceptual learning, and highlight how it may address shortcomings in purely error-minimization based conceptual modeling. Finally, I will outline some limitations and potential directions for future research.
Introduction
Predictive Processing (PP) offers a powerful framework for understanding the brain's hierarchical processing of sensory input. However, challenges arise when extending PP to higher-order cognition. While sensory-grounded predictions and error minimization function well at lower levels, the abstract nature of concepts like prototypes seems to demand additional representational tools1. In his paper A hybrid account of concepts within the predictive processing paradigm (2023c), Michel provides an account that seeks to address this challenge. He suggests a neural network where higher nodes embody more abstract, prototype-like concepts. However, his model leaves two key areas underspecified: 1) the precise mechanisms governing interaction between nodes at different abstraction levels, and 2) how the graded nature of prototype membership and ‘fuzzy’ boundaries of concepts2 are captured. Hence, I claim that there is room to enhance the explanatory scope of his proposal by introducing some principled tools and constraints that deal specifically with modeling the dynamics of these types of formats.
I will attempt to make a case and argue that Conceptual Space Theory (CST), as proposed by Peter Gärdenfors (2000, 2014), offers the missing tools. CST models concepts as convex regions within multidimensional spaces, a structure with two key advantages. First, the convexity of regions aligns well with the concept of prototypes as central tendencies. Second, the geometric distances within CST spaces allow for fine-grained modeling of similarity relationships between instances and prototypes.
Here, I propose an integration of CST into Michel's hybrid account. I focus specifically on how prototypes can be represented using CST's spaces within the PP framework. The goal is to refine the mechanisms Michel proposes in his substantive work that may lead efforts to a more computationally explicit model. I will begin by reviewing Michel's hybrid account, highlighting his concept of ‘higher-level nodes’ as encoding prototypes (1). I will then introduce CST, focusing on its geometric structure, convexity, and similarity metrics (2). Then, I propose a detailed integration, suggesting CST spaces as the representational substrate for Michel's higher-level nodes (3). This allows us to explore how:
1) CST regions can refine Michel's “vertical” processing of a concept along different levels of abstraction, providing formal tools to model the interplay between levels within PP's hierarchy.
2) Geometric similarity relationships derived from CST could augment PP's reliance on mere prediction error, enhancing the model's handling of ambiguous concept boundaries.
I will then discuss some of the implications of this CST-Augmented Representation of Prototypes specifically with regards to categorization under uncertainty, conceptual change and learning, and speculations for neural implementation and some testable hypotheses (4). In (5), I will highlight the key contributions of this integration and discuss some limitations and open questions. I conclude with how this integration offers a more sophisticated account of prototype-based processes within the PP framework. By leveraging the strengths of both Michel's approach and the formal tools of CST, this integration has the potential to plausibly advance PP's ability to model conceptual cognition.
1 - Michel's Hybrid Account of Concepts within Predictive Processing
1.1 A brief overview of predictive processing
Predictive Processing (PP) offers a powerful perspective on brain function, emphasizing the proactive generation of sensory expectations and continuous model refinement (Clark, 2013, 2016; Hohwy, 2013; Friston, 2010; Sprevak, 2021). At its core, the PP framework involves a prediction-driven perception mechanism that views the brain not as simply a passive receiver of information but rather, actively constructing hypotheses about incoming sensory data (i.e. sight, sound, touch, etc.). Instead of solely analyzing features from the ground up, the idea is that the brain predicts what it expects to perceive. These predictions are constantly compared to actual inputs where they generate what are considered ‘prediction error’ signals. These signals are thought to drive the brain to refine its internal model that works to improve future predictions.
The other key aspect of PP is the proposed hierarchical generative model (HGM). Predictions are thought to occur at multiple levels within this hierarchical network. Lower levels are seen as dealing with more fine-grained, sensory details, while higher levels encode broader, more abstract patterns. ‘Top-down’ predictions act as constraints on lower-level expectations forming a rich interconnected system. This model is also considered probabilistic; meaning it represents beliefs about the world in terms of some probabilities, not absolute certainties.
The idea of ‘precision weighting’ plays a critical role in PP framework. It is assumed that the brain needs to assess the reliability of incoming information to avoid overreacting to every minor discrepancy. Precision weighting could be seen as assigning a level of 'importance' to errors that allows the system to selectively adjust its model based on what it determines are the most salient differences between predictions and external environment. Importantly, for some of our discussions to follow, this mechanism is crucial for understanding how context shapes our perception and responses.
Michel posits a neurocognitive basis for PP that claims the dynamics emerge from networks of what are called 'prediction units' (or nodes). It is understood that each node combines a representation of a possible state of the world with a mechanism for detecting errors. Precision weighting functions by modulating the strength of error signals, determining which ones should drive significant model updates.
1.2 Why higher cognition is a problem for PP
While PP provides a compelling framework for lower-level perceptual processes and it is grounded in action-guided cognition, the theory nevertheless encounters significant challenges when trying to account for the complexities of higher conceptual thought. For instance, one core issue is the compositional nature of concepts; how concepts recombine to produce new concepts. This ability of our brain to flexibly combine concepts in new ways and create representations that the system has never directly encountered, often poses a significant problem for PP; especially given its primary emphasis on minimizing sensory prediction error (Lake et al., 2017). Also, the grounding of PP in sensory modalities makes it difficult to explain abstract concepts that lack direct sensory referents (Barsalou, 2008; Dove, 2016). Concepts like "justice" or "causality" may rely on complex internal states and are heavily influenced by social constructs. These characteristics are considered ‘higher-order’ relationships that aren't easily reducible to the prediction and correction of sensory input. Moreover, PP, as currently formulated seems to lack the internal representational structures that allows it to mirror the way concepts are organized based on features like: similarity, typicality, and hierarchical relationships within its framework. Such limitations highlight that while PP presents valuable insights, it likely requires augmentation to provide a more comprehensive account of higher-order conceptual thought.
In his work A Hybrid Account of Concepts Within the Predictive Processing Paradigm (2023c), Michel attempts to address this challenge by proposing a flexible model where concepts can shift between different representational formats (outlined below). He draws upon some empirical observations that suggest concepts are used in various ways depending on the task and context. Specifically, he identifies three distinct formats:
1) Exemplars: Exemplar-based representations encode specific instances of a concept, tied closely to sensory-motor details. For example, your representation of the concept "dog" might include exemplars derived from encounters with your pet Labrador or a neighbor's Dachshund.
2) Prototypes: Prototypes capture the central tendency or 'average' of a category. They represent what is most typical for a concept, such as the image of a dog that might come to mind if asked to simply picture "dog" without a specific instance in view.
3) Theories: Theory-based representations are considered the most abstract. They encompass knowledge of causal and explanatory relationships between concepts, drawing heavily on background knowledge and long-term memory. One’s concept of "dog" may likely include theory-like knowledge about dogs being mammals, domesticated animals, etc.
To account for this representational flexibility, Michel proposes a neural foundation based on a hierarchical network of nodes (Barsalou 2017; Kiefer & Pulvermüller, 2012). The following aspects in his model are noteworthy for our purposes here:
1) Nodes and Levels of Abstraction: Each node corresponds to a population of neurons, and nodes are distributed throughout the cortical hierarchy. Lower-level nodes encode sensory-motor details, while those higher in the hierarchy represent more abstract, schematic information.
2) Vertical Processing for Exemplars to Prototypes: Michel argues that the activation patterns across this network, moving between levels of abstraction, underlie the shift between the three representational formats. Focused activation of lower-level nodes corresponds to exemplar-like representations. In contrast, the activation of higher-level nodes, encoding more generalized information, would give rise to prototype-like representations.
3) Precision Weighting for Context: A core strength of this model lies in its alignment with PP's precision weighting mechanism. This mechanism allows the brain to adjust the relative influence of top-down predictions and bottom-up sensory error signals based on their reliability within the current context (Hohwy, 2013). Michel argues that precision weighting can modulate which levels in the concept network are most active, thus shaping the current representational format. This precision weighting mechanism is a powerful aspect of Michel's model that enables flexible, context-dependent activation of different representational formats. However, the specific computations governing the interaction between precision weighting and the graded, fuzzy boundaries of prototype regions remain underspecified. This is a key area where integrating insights from CST could refine Michel's account, as will be explored in Section 3.
1.3 Limitations and CST's Potential
While Michel's account provides a valuable step towards integrating concepts within the PP framework, I believe there are aspects related to prototype representation and interaction across levels of abstraction that remain underspecified and in need of further development. Specifically, I see limitations in the following regards:
1) Prototype Structure: While the idea of higher-level nodes is compelling, the specific mechanisms of how prototypes, with their characteristic graded membership and fuzzy boundaries, are represented remain unclear.
2) Interplay of Levels: The precise computations involved in the "vertical" processing between nodes at different levels of the hierarchy need further elaboration. This is crucial to understanding how PP's error minimization and precision weighting mechanisms contribute to a subtle representation of a single concept across levels of abstraction.
These gaps motivate the integration of CST. As explored in the next section, I claim that CST offers a geometric framework that naturally addresses these shortcomings, suggesting ways to refine and computationally ground Michel's hybrid account of concepts.
2 - Conceptual Space Theory: A Foundation for Modeling Prototypes
2.1 Conceptual Spaces and its Core Features
Conceptual Space Theory (CST), as proposed by Peter Gärdenfors (2000, 2014), offers a geometrically structured framework that is particularly well-suited for representing the prototype-like nature of many of our conceptual categories. At its core, CST proposes that concepts can be modeled as multidimensional spaces, with each dimension representing a fundamental attribute or quality relevant to distinguishing instances of that concept. To make this concrete, let’s consider presenting the key features that that we’ll focus on using an example: the concept of "fruit":
Quality Dimensions: In CST, conceptual spaces are constructed from a set of quality dimensions, which are the basic building blocks of conceptual representations (Gärdenfors, 2000). These dimensions correspond to the perceptual, cognitive, or affective attributes that are relevant for distinguishing between instances of a concept. For example, the concept of "apple" might be represented in a space with dimensions such as color, shape, size, and taste. Each point in this multidimensional space could correspond to a specific instance or exemplar of the concept. Quality dimensions provide a natural way to capture the continuous and graded nature of many conceptual attributes (Gärdenfors, 2014). By representing concepts as regions in a space defined by these dimensions, CST can account for those instances of a concept that vary along multiple attributes while still being recognized as members of the same category. Relevant dimensions might include sweetness, color, size, shape, texture, and seed type. Each instance of a fruit (e.g., an apple, a mango) can be located as a point within this multidimensional space based on its values on these dimensions.
Convex Regions and Prototypes: In CST, concepts are represented as convex regions within the conceptual space (Gärdenfors, 2000). A region is considered convex if, for any two points within the region, all points between them are also contained in the region. This convexity constraint has important implications for the structure of concepts and their relationships. Firstly, convex regions naturally give rise to prototype effects (Gärdenfors, 2000; Rosch, 1978). The prototype of a concept corresponds to the central point of the convex region, which is the point that minimizes the total distance to all other points within the region. Instances closer to the prototype are considered more typical or representative of the concept than instances near the boundaries of the region. Secondly, the convexity constraint enables CST to capture the idea of graded category membership (Gärdenfors, 2014). Instances near the center of a convex region are clear members of the corresponding concept, while instances near the boundaries are less typical and may be borderline cases. This graded structure aligns well with empirical findings on the fuzzy nature of many human concepts (Hampton, 2007; Rosch, 1973). In our "fruit" example, an apple might be closer to this prototype than a less typical fruit like a starfruit.
Graded Membership and Fuzzy Boundaries: This convexity property is critical for modeling graded membership, addressing a limitation in simpler models. Instances near the center of the region are considered clear category members, while those on the periphery are less typical and may constitute borderline cases. This captures the intuition that a starfruit, though technically a fruit, is intuitively a less central member of the category than an apple—a fuzziness difficult to model with rigid rules or purely predictive networks.
CST also offers a subtle understanding of similarity, a basis for processes like categorization and inference. Unlike PP which is primarily driven by prediction error, CST uses distances within a geometric space. Instances that are closer together are considered more similar. CST also offers multiple distance metrics (Euclidean, Manhattan, Mahalanobis, etc.). Choosing the appropriate metric is vital here, as it determines how differences along different dimensions are weighed in judging similarity. This flexibility allows CST to capture the context-sensitive nature of similarity judgments For example, when sorting fruits as a cook, color and ripeness dimensions might be most heavily weighted. But, for say a botanist, seed type and growth patterns might take more precedence3.
The choice of distance metric has important implications for capturing context-dependent similarities. For example, the Euclidean metric is invariant to rotations of the space and may be most appropriate for integral dimensions, while the Manhattan metric respects the orientations of the axes and may better suit spaces with separable dimensions (Gärdenfors, 2000). The ability to flexibly apply different distance metrics enables CST to naturally model the context-sensitivity of similarity judgments in a way that prediction error minimization alone does not obviously afford. This context-sensitivity is an important facet of human concept use that a CST-enhanced PP model may better capture.
2.2 Neural Realization and Population Coding
CST's geometric emphasis doesn't preclude a neuroscientific grounding. Research suggests a potential neural implementation based on population coding (Gärdenfors, 2000; Reimann et al., 2017). In this view, each neuron acts as a 'basis function,' responding most strongly to a particular point in the conceptual space and decreasing its activation in proportion to distance from that ideal point. The collective activity across a population of neurons thus effectively codes for a specific instance within the conceptual space. This distributed coding scheme resonates with neural organization throughout the brain, including the hierarchical networks found in PP models. Neuroimaging studies lend support, finding brain regions that respond differentially based on the similarity structure of conceptual categories (Edelman et al., 1998; Simmons et al., 2007), aligning with CST-like representations.
Let’s see how these attributes can support Michel’s proposed model.
2.3 Key Strengths for Augmenting Michel's Model
CST's emphasis on geometric structure has the potential to refine Michel's proposal in several ways:
1) Prototypes with Structure: CST's convex regions directly address Michel's underspecified "higher-level nodes" for prototype modeling.
2) Similarity beyond Prediction Error: CST's distance metrics offer an alternative to PP's prediction-error-based similarity calculations. This may help model human cognition in cases of high ambiguity or when prediction error signals alone are insufficient.
3) Context-Sensitivity and Precision Weighting: Dynamically weighting dimensions, central to CST, mirrors the flexibility of PP's precision weighting mechanism. This synergy may provide a richer account of how context shapes conceptual processing.
The next section will delve into the specifics of how we propose to integrate CST with Michel's model, focusing on enhancing the representation and processing of prototypes within the PP architecture.
3 - Integrating Conceptual Space Theory into Michel's Hybrid Account
The preceding sections introduced Michel's hybrid account, emphasizing its strengths in accommodating the context-dependent nature of conceptual representations, and CST's power in modeling prototypes with their graded structure and geometric similarity relations. This section proposes a detailed integration of these frameworks to produce a more robust and computationally explicit model of concept representation within the PP paradigm.
3.1 Conceptual Spaces as the Substrate for Michel's Account of Higher Nodes
I propose that CST's multidimensional spaces provide the missing representational substrate for the higher-level nodes posited in Michel's account. His proposal hinges on the idea of concepts as hierarchical networks of nodes in the PP framework. As he states:
"C1. Conceptual representations are realized as extended networks of nodes: A concept is neurally realized as the activation of a set of neuron assemblies (nodes) in the form of a distributed network that can cover different brain areas, from higher cortical areas down to lower-level sensorimotor ones." (Michel, 2023, p. 1354)
We can attempt to directly enrich this idea with CST by providing a principled geometry for these nodes, with prototypes as convex regions in a multidimensional space. We can build on Michel’s description using a modified version of it; we could say:
"C1-CST. Conceptual representations are realized as hierarchical networks of nodes, with each node corresponding to a convex region in a conceptual space. Prototypes could represent the central regions in these spaces, with exemplars as points within the regions."
This modified CST-augmented characterization maintains Michel's key insights about the distributed and hierarchical nature of conceptual networks while adding a precise representational substrate for the nodes. But by grounding the nodes in the geometric structure of conceptual spaces, we can capture the graded, similarity-based nature of prototype representations that is central to many theories of concepts (e.g., Rosch, 1978; Gärdenfors, 2000).
As an example, let’s consider the concept of the "dog". Michel suggests a hierarchical network with higher nodes embodying more abstract, dog-related features. We can take this further. Consider a space for "Dog". The higher reaches of this network instantiate a conceptual space with dimensions like size, typical behaviors, domestication status, etc. Prototypical "dog-ness" is then modeled as the central region within this space.This CST-based view of nodes provides a natural way to accommodate the context-sensitive flexibility of conceptual processing that Michel emphasizes. As we will discuss in more detail below, the salience of dimensions in the conceptual spaces can be modulated by the PP precision-weighting mechanism, allowing for dynamic, context-dependent reshaping of the prototype regions.
3.2 Exemplars as points within the space
In the CST-augmented PP framework, exemplars can be understood as specific points within the multidimensional conceptual space defined by the prototype region. Each exemplar could be characterized by its unique combination of values along the relevant dimensions that determine its location within the space.
For instance, consider again the "dog" conceptual space. A particular dog breed, such as a Labrador Retriever, would be represented as a point within this space, with specific values for dimensions like size (large), coat color (typically yellow, black, or chocolate), temperament (friendly, outgoing), and so on. Another breed, like a Chihuahua, would occupy a different point in the space, with contrasting values along these dimensions (small size, varied coat colors, more energetic temperament).
By embedding these exemplars within the prototype's conceptual space, we can capture our intuition that some exemplars are more central or typical than others. Exemplars that lie close to the center of the prototype region, like a Golden Retriever for the "dog" concept, are considered highly representative of the category. In contrast, exemplars that fall near the periphery of the space, like a hairless Chihuahua, are seen as more atypical or borderline cases.
This CST-based view of exemplars aligns well with Michel's emphasis on the graded structure of concepts. As he notes: "The tokening of the same concept on different occasions can reach into lower levels of the hierarchy to different degrees" (Michel, 2023, p. 1354). In our integrated model, this "reaching into lower levels" can be understood as the selective activation of specific exemplar points within the prototype's conceptual space, depending on the current context and task demands.
We can think of the contexts as shaping the geometry of the spaces. Here, precision weighting mechanisms that are central to PP, now directly can influence the shape of the conceptual space itself. Thinking of dogs as pets, for instance, might emphasize dimensions of companionship and breed, while say a veterinarian context could increase the salience of health-related dimensions.
3.3 Mechanisms for Vertical Processing and Representation
The integration of CST allows us to refine Michel's notion of "vertical" processing between lower, sensory-grounded nodes in the hierarchy and those abstract spaces at higher levels. To consider the dimension weighting through precision weighting, we can look at Michel's emphasis on the context-sensitive nature of conceptual processing. He states:
"C3. Context-sensitive and flexible conceptual processing: On different occasions different parts of the network of a concept are activated in a task- and context-sensitive manner. The tokening of the same concept on different occasions can reach into lower levels of the hierarchy to different degrees." (Michel, 2023, p. 1354)
CST naturally accommodates this flexibility through the differential weighting of dimensions in the conceptual space. We can reformulate C3 as follows:
"C3-CST. Context-sensitive and flexible conceptual processing arises from the selective activation of nodes in the conceptual network, with the salience of dimensions in the CST spaces modulated by the PP precision-weighting mechanism to reflect current context and task demands."
In this view, precision-weighting acts to prioritize certain dimensions over others, effectively reshaping the conceptual space to suit the current situation. For instance, in a context where fine-grained color discrimination is crucial (say, choosing a paint color), the "color" dimension of the relevant conceptual space would be ‘up-weighted’, increasing its influence on processing. But in contrast, in a context where color may be less relevant (like identifying objects in a dimly lit room), the "color" dimension would be ‘down-weighted’ that could allow other features to dominate.
This CST-based perspective can provide a plausible computational mechanism for the context-sensitivity that Michel highlights. We can see that by linking precision-weighting to the selective activation of dimensions in conceptual spaces, one can more precisely model how the salience of different features shifts dynamically to match task demands.
Let’s consider how prediction errors may play a role in altering the prototype region's geometry. One key aspect of vertical processing in the integrated CST-PP framework is the way prediction errors can drive updates to the geometry of prototype regions over time. As Michel notes, "all of the nodes are interlocked and have an influence on the overall state of the information package associated with the concept" (Michel, 2023, p. 1371). CST allows us to cash out this idea in terms of the dynamic reshaping of conceptual spaces:
"CST-PP Processing Dynamics: The holistic, interlocked processing of conceptual networks in PP is driven by prediction error minimization. Errors generated by outlier exemplars can propagate up the hierarchy that can induce gradual changes to the shape and extent of prototype regions in CST spaces. This allows the conceptual system to flexibly adapt its representations to fit the structure of the environment."
For example, encountering a novel, atypical exemplar of a category (like a platypus as an instance of "mammal") would generate large prediction errors, since it deviates strongly from the current prototype region. We can imagine these errors then propagating upward and inducing incremental adjustments to the geometry of the "mammal" space which could better accommodate this rather unusual creature. Over time, with repeated exposure to diverse exemplars, the prototype region would again expand and reshape itself to minimize prediction errors, leading to a more comprehensive and well-calibrated representation of the category.
So we can envision how both of these mechanisms likely playing key roles as follows:
1) Dimension Weighting and Abstraction: As specific exemplars activate lower-level nodes, this signal propagates upwards. Through precision weighting, relevant dimensions become active within the higher-level conceptual space. If I see a Golden Retriever, features like "golden fur" and "large size" activate sensory nodes; these then increase the weight of the relevant dimensions within the "dog" concept space. This focuses the processing on a region of the space corresponding to these features.
2) Prediction Errors Within the Space: The PP framework's error-minimization principle can operate within the CST space. To make the prototype updating process more concrete, consider again our dog example but this time in the instance of encountering a novel breed of dog, say a Chihuahua, for the first time. Given its small size and unusual features, it would likely generate significant prediction errors when processed through one's existing 'dog' prototype network. One would expect that these errors would propagate up the hierarchy, inducing some form of plasticity in the higher-level CST space that represents the prototype. Over time, with repeated exposure to Chihuahuas and similar small breeds, the geometry of the prototype region would shift to accommodate this new cluster of dog exemplars. The dimensions of size and shape would be expanded or reshaped, updating the prototype to better minimize prediction error in future encounters. In this way, the prototype could serve as a flexible, high-level prior that is continuously refined via interaction with the world, as Michel emphasizes in his account.
3.4 Advantages of CST-Augmented Representation of Prototypes
The integration of CST into Michel's PP-based account of concepts offers several key advantages that address limitations in the original proposal:
1) Structure for Prototypes: CST's convex regions provide a formal model of prototypes as the central tendency of a multidimensional space. This specifies how they are represented and how distance from the center models the graded membership of exemplars.
2) Similarity beyond Error Signals: While PP primarily relies on prediction error to compare representations, CST's geometric structure offers an alternative notion of similarity based on distance within the space. This allows for a richer account of how the PP system might handle cases where prediction errors are not the primary driver of conceptual similarity judgments, such as in highly ambiguous or abstract domains.
3) Dynamic Interaction Across Levels: The proposed mechanisms for "vertical" processing elaborate on how PP mechanisms can directly shape the geometry of the concept space (and vice versa). This interplay between prediction errors, precision weighting, and CST's geometric properties opens up new avenues for computational modeling of this interaction.
Taken together, these advantages stem from the close integration of key PP principles with the representational structures and mechanisms of CST. By grounding Michel's insights about the hierarchical, context-sensitive nature of conceptual processing in the specific geometries of conceptual spaces, we can develop a more precise and explanatorily powerful model of prototype representation.
There is also growing evidence that many aspects of human perception and categorization can be modeled by assuming an underlying spatial structure (Gärdenfors, 2000; Goldstone, 1994; Shepard, 1987). The geometric structures in CST allow for a more intuitive modeling of concepts through the use of metric distances and directions in a conceptual space (as discussed). This helps enable representation of complex relationships between concepts (i.e. hierarchical structures and family resemblances), in a straightforward and visually comprehensible way. For instance, if we consider the conceptual space for "color", we could position "red" closer to "orange" than to say "blue” which is helpful in our intuitive understanding of similarity. Hierarchical relationships can also be represented with broader concepts ("animal") encompassing more specific ones ("dog") as subregions within the space.
3.5 Empirical evidence and Neural Implementation speculation
While further research would certainly be needed, let me propose some potential neural implementations that may be plausible based on very interesting findings from cognitive neuroscience and psychophysical studies.
3.5.1 Neurobiological plausibility of conceptual spaces
The geometric nature of CST seems to align well with findings from cognitive neuroscience and suggests a potential biological basis for its posited theoretical principles.
Grid Cell Parallels: The discovery of grid cells within the entorhinal cortex, with their regular firing patterns mapping onto spatial locations (Moser et al., 2008) suggests the brain might use similar coding for abstract spaces. The neurons are observed to fire in a regular hexagonal pattern during spatial navigation (Hafting et al., 2005) which can provide a compelling mechanism. Research indicates that grid cell activity extends beyond just physical spaces, exhibiting grid-like representations within conceptual domains (e.g., visual properties of birds in Bellmund et al., 2018). This suggests that grid cells might serve as a general coding mechanism for organizing information along continuous dimensions, which is a core feature of CST's quality dimensions. Other recent work by (Constantinescu et al., 2016; Bao et al., 2019) showing the extension of grid-like patterns to non-spatial concepts also strengthens this plausibility of the brain utilizing similar computational principles for CST-like conceptual spaces.
Distributed Coding within Hierarchy: Furthermore, the concept of population coding, where groups of neurons represent various perceptual and conceptual features, supports CST's representational structures. For instance, studies demonstrate that populations of neurons can become tuned to increasingly abstract features in higher-order brain areas (Kriegeskorte & Kievit, 2013). This mirrors the hierarchical abstraction process posited by CST. The hierarchical PP network likely remains crucial. CST regions in higher levels could be instantiated across populations of neurons, with their collective firing patterns encoding locations within the space, just at a more conceptual than sensory level.
Taken together, these findings suggest that the brain might employ neural coding strategies well-suited to representing the geometric relationships central to CST. While grid cells offer direct support for spatial coding of concepts, population coding provides the flexibility needed for complex, abstract features and the hierarchical organization of conceptual spaces.
3.5.2 Psychophysical plausibility of Conceptual spaces
The geometric emphasis of CST dovetails with several findings from psychology about how humans perceive similarities and differences. This provides some empirical support to the theory's foundational principles. In Conceptual Spaces: The Geometry of Thought, Gärdenfors shows that people's similarity judgments often display geometric properties (Gärdenfors, 2000). This could mean that our intuitive sense of how similar or different concepts are correlates with spatial relationships within theoretical-based conceptual spaces. Furthermore, other evidence from (Bokeria et al., 2021) also suggests that our conceptual knowledge is spatially organized and influences tasks such as semantic categorization. More sophisticated studies using techniques like multidimensional scaling (MDS) also reveal that subjects implicitly organize concepts based on perceived similarities, and that these relationships exhibit geometric properties that can be modeled within CST's framework.
Let us consider again the example of colors. We intuitively tend to perceive certain colors as more similar than others (The color red, for example, is closer to orange than blue). Gärdenfors points to studies that reveal these similarity judgments can be mapped onto a circular arrangement. This reflects geometric properties like distance (closer colors on the circle are more similar) and dimensionality (the circle implies underlying dimensions of hue and saturation). This type of evidence gained from behavioral experiments, demonstrates that our intuitive sense of similarity isn't merely arbitrary; but that it tends to align with the geometric representational principles core to CST.
Importantly, the geometric modeling of similarity judgments extends beyond just simple perceptual domains. There are studies that also have shown people organize more abstract concepts that have emotional basis or involve social relationships and interactions in ways that reflect spatial relationships that seem to be consistent with CST's core principles. This type of behavioral evidence can plausibly support CST's core idea that concepts are organized within multidimensional spaces based on similarity, suggesting that the intuitive geometric way we think about concepts may underlie our fundamental cognitive processes and representational structures.
In the next section, I will discuss how this integrated model enhances understanding of categorization, learning, and other processes central to human concept use.
4 - Implications of the CST-Augmented Representation of Prototypes
Integrating CST into Michel's hybrid account, with its focus on prototype representation, can have far-reaching implications for the PP framework's ability to model core aspects of higher-order cognition. Here, I will delve into these implications and consider some testable hypotheses, along with some possible objections that may go towards guiding future research.
4.1 Categorization: Beyond Errors and into Geometric Similarity
Traditional PP models often tend to present categorization as a direct matching of inputs to predictions. But this could undermine the subtleties associated with categorization in the real world. We can see how CST can fine-tune and therefore offer a more powerful way to address this complexity. For instance, let’s consider a blurry photograph where we find it difficult to determine whether we are seeing a cat or a small dog. Even with significant uncertainty, if features like ear shape and body proportions are at least partially discernible, their relative positions within a ‘CST-like’ conceptual space could strongly influence our categorization decision. This can introduce geometric similarity as a critical factor alongside the PP's focus on prediction-error minimization.
In addition, CST challenges this notion of rigid category boundaries that is often found in earlier PP models. Within CST, an atypical example – say a slightly unusual cat – may fall near the periphery of the "cat" prototype region and still be categorized correctly. This seems to align with empirical evidence demonstrating human flexibility in categorization and the significant impact of context on how we assign categories (Barsalou, 1983).
4.2 Reshaping and Learning: From Error Signals to Evolving Spaces
We can also consider how this CST-PP integration can offer a more dynamic perspective on conceptual representations, with implications for both conceptual change and learning. Atypical exemplars, like the instance of encountering a Chihuahua for the first time, can initially generate prediction errors within the PP framework. But the CST augmentation goes beyond just registering these errors. Instead, it can provide a mechanism for adaptation: for instance, the reshaping of the prototype region itself within the conceptual space. This also aligns with the PP's focus on model updating, while offering a geometrically grounded way to understand the nature of this change.
In addition, CST allows us to model subtle and experience-driven shifts in prototypes. Again, take for example an instance where exposure to multiple unusual dog breeds might gradually expand or refine the "dog" prototype region. This type of incremental learning could also align well with research on statistical learning, where humans are able to demonstrate the ability to extract category structure implicitly through repeated exposure (Knowlton & Squire, 1993).
4.3 Some Testable Hypotheses for proposed CST-PP Integration
The integration of CST and PP may offer exciting avenues for exploring the neural underpinnings of conceptual representation and in effect generate some concrete testable hypotheses. We may consider for example that if CST-like spaces exist at higher levels of the PP hierarchy, we should be able to identify some neural populations whose activity shifts in specific ways during categorization tasks. In conditions under uncertainty or when presented with exemplars that force conceptual change, we would expect the patterns of activity within associated brain regions to mirror the hypothesized reshaping of the abstract conceptual space. This aligns, for instance, with some evidence that shows the anterior temporal lobe's (ATL) is sensitive to conceptual similarity (Binney et al., 2016).
Moreover, the shared emphasis on precision weighting within both CST and PP could also offer some opportunities for direct experimental investigation of context effects. Here, we can imagine that by manipulating participants' expectations (priming them to think of dogs as "pets" vs. "working animals"), one could explore whether this alters how neural populations representing the "dog" concept are engaged in classifying some of the more ambiguous cases that we discussed earlier. A key prediction of the integrated model would be that this sort of contextual priming would predictably shift the likelihood of categorizing borderline exemplars that are in line with the altered precision weighting.
Naturally, claiming a theoretical model of this complexity raises potential objections and addressing such concerns will be important if our aim is to continue developing and refining this proposed integration. I will therefore examine some of these theoretical questions and possible objections in the following section.
4.4 Theoretical Challenges and Refinements
One potential objection we may anticipate may be that, while the CST integration can handle relatively concrete concepts (e.g., animals, objects), it will still struggle to scale meaningfully to the complexity of highly abstract concepts (e.g., "democracy," or "justice"). Such concepts become notoriously challenging to track and often involve complex relationships between other concepts, where simple geometric distances may be insufficient. But we are not claiming here that CST alone is the full solution; but we could envision a type of multi-layered extension where the relationships between CST-based conceptual spaces themselves become elements of a higher-order space. A framework like this would retain the core strengths of CST and geometric representations while potentially also allowing for the necessary structural complexity. Research on conceptual metaphors (Lakoff & Johnson, 1980) might offer useful insights here, as it suggests even abstract domains are grounded in more embodied ways of thinking.
Another objection we could foresee is that a pure PP proponent could argue that prediction errors carry more weight and that it’s ultimately the error signal (and its precision weighting) that will drive major conceptual change, not the geometric refinement of CST spaces. But while I have claimed CST could offer an alternative way to think about similarity, the model I proposed still fundamentally relies on prediction-error minimization within the PP framework; It does not deny the importance of prediction error. Rather, it offers a model with two interacting mechanisms that can expand the explanatory scope of the framework. Of course carefully designed experimental paradigms that can specifically test this integration could potentially disentangle the relative contributions of geometric similarity vs. error magnitude in categorization tasks and shed more light on this interplay. Further, the CST augmentation may help explain context-dependent categorization where the same objective error takes on different meanings depending on a concept's position within the broader scope of the conceptual space.
It's important to recognize that these theoretical refinements must ultimately be tested against empirical data. The testable hypotheses outlined earlier could provide a roadmap for a more extended research program that will either validate and strengthen this integrated model or reveal the need for any significant revisions. In the following section, I will briefly summarize and highlight some of the key contributions that I believe could be addressed by this integration along with its limitations and future directions.
5 - Key contributions and some Open Questions
The integration of Conceptual Space Theory (CST) into Michel's hybrid account of concepts may have the potential to significantly advance the PP framework's ability in modeling abstract concepts. By providing a formal representational structure for prototypes and their interaction within the PP hierarchy, I have discussed how this integrated model can address some key limitations in the current PP framework. This exploration has revealed several core insights that emerged from this integrated model which I outline as follows:
1) Structured Prototypes: we discussed how the convex regions of CST provide a geometrically grounded account of prototypes that can directly address some lack of specificity in Michel's original proposal which I had identified. This formalization, I think, is important for modeling the graded membership and the category fuzziness which play central roles in human cognition.
2) Similarity and Flexibility: with CST's notion of distances within a conceptual space, we discussed how it offers an alternative measure of similarity that goes beyond PP's primary reliance on prediction error. This suggested that we can have a more subtle way for the PP system to handle conceptual similarity judgments in cases where ambiguity or top-down predictions are weak or unreliable.
3) Dynamic Interaction Across Levels: I proposed mechanisms for CST-guided "vertical" processing within the PP hierarchy that could enhance our understanding of how lower-level sensory details and abstract concepts may influence each other. We also considered how precision weighting, which is central to PP, now can have the potential to directly shape the geometry of abstract conceptual spaces which could provide a computationally tractable model for their dynamic nature. In effect, the proposed CST-guided interactions across hierarchical levels offers a basis for understanding how both 'bottom-up' input and 'top-down' predictions could continuously shape and reshape our conceptual understanding of the world.
This integrated model offers a promising starting point, but several aspects pose significant challenges and naturally leave open questions that warrant further investigation. I have primarily focused my attention on the prototype level in this thesis, but we must also address how CST can model relationships between concepts themselves. It is crucial when theory-like knowledge, as Michel discusses in his work, can also interact with exemplar and prototype representations within the PP hierarchy. The sheer complexity and scaling is also a big challenge. For instance, how well does this model scale to highly abstract concepts or those with complex relationships within the conceptual space? Will extensions be needed to accommodate the full breadth of conceptual knowledge? Understanding how specific memories or causal knowledge interact with the established prototype regions proposed within the hierarchical PP model presents an important theoretical challenge.
6 - Conclusion
In this dissertation, my aim has been to explore the challenges of modeling abstract conceptual representations within the PP framework. In attempting to address these challenges, I proposed that elements from CST be integrated into Michel's hybrid account as an alternative approach that can plausibly enhance prototype representations and their interactions within PP's hierarchical model. By grounding Michel's higher-level nodes in the geometric framework of CST, I have hoped for this integration to offer some key contributions. My suggestions here have attempted to formalize prototypes through CST's convex regions and provide a computationally grounded model that may address some of the limitations that I have identified in Michel's original formulation. This formalization, I think, is important for understanding the graded membership and the inherently fuzzy boundaries of concepts that were discussed.
With this type of integration, I have also aimed to rethink similarity within PP. We have seen that while PP heavily relies on prediction error, CST may introduce the idea of distances within conceptual spaces as an alternative mechanism in modeling efforts. This approach may help open new paths for the PP system to model similarity judgments, especially under conditions of uncertainty or high abstraction. Furthermore, I have proposed how interactions that may be guided by CST across hierarchical levels could shape abstract concept representation within PP itself, with precision weighting directly influencing conceptual space geometry. This dynamic interplay, I believe, can enhance the model's capacity to account for context-sensitivity and conceptual change.
Another realization in this exploration was that this integrated model may also have implications for understanding core cognitive processes like categorization under uncertainty and incremental concept learning. It was emphasized that if testable hypotheses can be generated at both behavioral and neural levels, this proposal could have the potential to offer some ground for future research. Of course, limitations remain, especially with regard to the incorporation of theory-like representations and also the scaling of this model to more complex/abstract concepts. Nonetheless, my intention with this proposed integration has been to make an attempt to mark a step in expanding the PP framework's ability to capture the full flexibility and subtlety of human conceptual knowledge.
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