However, the size of V1 could introduce bias and change the dynamic interaction among the dorsal and ventral areas. As Grossberg [26] described, the architecture of the dorsal and ventral streams and the relationship between the two enables image recognition and determines appropriate actions. Furthermore, without bottom-up data, Grossberg's theory [13] describes how and why top-down interactions can lead to imagery with vo- litional modulation. Grossberg also suggested that imagery is not only an entity itself but also part of the complex dynamic system of seeing and thinking. In other words, in Grossberg's theoretical infrastructure, imagery is an important component in vision and thinking. Thus, imagery findings impact perception, and knowledge regarding perception can be applied to imagery. The vividness of an experience in the visual field, including light, color, and extra sensory information, draws human attention and necessarily activates both the ventral and dorsal streams. Within this system, we attempted to clarify the possible role of V1 through indirect measurements and suggested vividness as a possible bias guiding perception and imagery. Moreover, the novel aspect of this paper is the contribution and analysis of the role of V1. V1 is a high-level interaction region with the ventral and dorsal streams. An individual's V1 size may influence the individual's bias toward the dorsal or ventral stream through the reentrant loop. This would potentially contribute to reconcile why V1 activation during visual imagery is a ubiquitous find (e.g., in fMRI studies). Indeed, it should be expected that only imagery situations that induce extreme disequilibrium between dorsal and ventral processes *in certain individuals* (hence, usually a proportion of the sample) should reveal strong and clear evidence of V1 activation (or deactivation).

Kosslyn provided extensive independent supporting evidence demonstrating the importance of the dorsal and ventral pathways, as described in [40], and D'Angiulli and Reeves [28] described how vividness ratings reflect the strengths and weaknesses of the dorsal and ventral pathways. The vividness (or image strength) ratings and image latency could represent the combined action of the two pathways and their roles in both imagery and perception [6]. These two pathways are core mechanisms involved in image generation and high-level vision tasks that keep being rediscovered in current imagery research (see [19,41]), but no detailed model of their emerging dynamics has been previously proposed.

Although the role of the dorsal and ventral pathways has been described in the litera-
ture, the bias of V1 toward the dorsal or ventral pathway is another novel contribution of
the present study. Importantly, we do not make any assumptions about the bias toward the
dorsal or ventral streams, and we suggest that external bias, task demands, and individual
differences (including V1 differences) impact responses to perceptual stimuli. What we
call “voluntary” (for example, our unconscious bias) is a complex system of interactions.
Rather than voluntary, this intertwining nature is autonomous in attending, categorizing,
recognizing, and predicting objects, which is central to Grossberg’s ART and the idea of the
resonant brain. Our contribution emphasizes the primacy of the link among V1, the dorsal
and ventral pathways, and visual phenomenology [12].

4.3.2. Limitations and Future Research Directions

In this context, a limitation of this work, and the literature, is the lack of understanding of exactly how attention affects unconscious and conscious bias toward the ventral and dorsal pathways. Future research should therefore focus on conscious and unconscious bias and how attention is linked to V1 and the dorsal and ventral pathways. Overall, the interaction among the dorsal and ventral pathways, vividness, and the size of V1 could improve our understanding of how individual differences in perception (sometimes major differences with influential consequences) occur. This understanding might be useful in educational applications. In particular, imagery training is a key application for this research. For example, young children with less acuity (who are cognitively less attentive to external stimuli) could benefit from imagery training. In this paper, we found that image vividness affects the perception of external stimuli. Generating imagery could lead to stronger priming of perceptual processing. More importantly, training and practice could be determined according to individual characteristics and task types, as some children
may suffer unwanted interference and divided attention or distraction due to images that are too vivid. Similar applications could be considered for clinical, mental health, and psychotherapy settings, which routinely use mental imagery training and practice.

Some limitations of this paper include the relatively small sample size and the limited diversity of the participants. A larger, more diverse demographic would add more varied V1 size data as well as different imagery abilities. In light of these limitations, the interpretations and data are preliminary and should be considered an attempt to examine the impact of individual differences and phenomenology on dynamics that extends imagery research.

Our contribution should be considered a hypothetical extension of ART for future neurophysiological work on imagery and imagination. As mentioned, the present study is merely descriptive and limited to replicating Kosslyn's data and Grossberg's insights; however, connecting subjective and phenomenal results to actual empirical neural dynamics remains challenging. Ongoing investigations in our laboratory aim to extend the proposed framework using combined fMRI, EEG, and ERP techniques and deep learning simulations. We hope to elucidate the groundbreaking contributions of Grossberg's ART. The proposed approach might permit the prediction of phenomenology and subjectivity and individual differences starting from emerging complex properties of neural networks, in contrast to the large majority of research that, up to now, has proceeded in the opposite direction, going from insights driven from phenomenology and introspection to experimental phenomena and to cognitive and neural architectures (see [42,43]). The present proposed approach may considerably improve the current state-of-the-art in top-down attention, imagery, and consciousness research. In the most practical sense, the identification of the different possible scenarios of complex neural interactions between perception and imagery (as analyzed in Figures 6–8) and the reduction to a manageable set of system variables (as illustrated in Table 3) provide a testable and falsifiable computational working model of vividness which is amenable to further empirical and theoretical refinements, moving the bench from introspection to manipulation of objective variables within interacting complex systems, especially in terms validating implementations based on neural networks simulations. A far-reaching objective is that, through the advancement of multidisciplinary meta-theory, the present conceptualization of vividness might, in future research, be abstracted and generalized further, from the level of human domain to neuromorphic computing and neurorobotics to arrive at possible new forms of autonomous machine explainability (for more in-depth discussions see [2]).

**5. Conclusions**

In conclusion, we showed that concurrent imagery can exert interference, priming, or no effects on perception depending on the interaction between image vividness and individual differences in neuroanatomy, including dynamic complementary parallel processing in ventral and dorsal visual streams. Our results suggest that highly vivid images and large (ventrally dominant) V1 interfere with perception, whereas weak imagery and small V1 (dorsally dominant) prime perception. A possible interpretation is that the size of V1 affects the neural synchronization between the connectivity of this region and other dorsal and ventral regions according to reentrant feedback from the PCF, thereby also influencing the vividness of the perception. Hence, under our static vivid imagery condition, the interference effects are greater than those under the static nonvivid and dynamic vivid conditions and more than double the effect than that under the dynamic nonvivid imagery condition.

Overall, we conclude that in most situations, imagery enhances perception, possibly because mental images are often processed by ventral pathways. The ventral network is beneficial for perception during imagery tasks because this network carries information that is less likely to compete with working memory and the attentional systems necessary for accurate visual-spatial prediction in the where/how system. However, despite these findings, there is still large individual variation among individuals, which appears to be due to the vividness of the image and the relative cortical size of the V1 area.
Our interpretation of ART and related descriptive modeling based on ARTSCAN contributes to expanding Grossberg's comprehensive account of how and why individually experienced vividness may influence the differential use of the dorsal and ventral complementary visual processing pathways, resulting in the reported effects of imagery on concurrent perception.

**Author Contributions:** Conceptualization, A.D.; methodology, A.D.; validation, C.L. and D.M.B.; formal analysis, A.D. and D.M.B.; investigation, A.D.; data curation, A.D. and D.M.B.; writing—original draft preparation, A.D., C.L. and D.M.B.; writing—review and editing, A.D.; supervision, A.D.; project administration, A.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Natural Sciences and Engineering Research Council of Canada through the New Frontiers in Research Fund—Exploration Program (grant number NFRFE-2019-00503) to A.D.

**Institutional Review Board Statement:** The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Behavioural Research Ethics Board of Carleton University (protocol code 111569 27 November 2019).

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** The data presented in this study are available upon request from the corresponding author. The data are not publicly available because open data sharing was not included as an option for consent.

**Acknowledgments:** We thank Etienne Lefebvre for comments and editing the previous version of this manuscript. We thank Jennifer Goldberg and Leslie Lavoie for assisting with the data collection. We are indebted to Adam Reeves, whose two-decade discussions and guidance have shaped the empirical side of this research. This paper is also a celebration of his work and legacy. All remaining (inevitable) blunders are the responsibility of the first author. We thank the anonymous reviewers and the editors for helping us improve our manuscript.

**Conflicts of Interest:** The authors declare no conflicts of interest.

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Article

# Analyzing Biomedical Datasets with Symbolic Tree Adaptive Resonance Theory

Sasha Petrenko <fnref n="1" />,*, Daniel B. Hier <fnref n="1" />,<fnref n="2" />, Mary A. Bone <fnref n="3" />, Tayo Obafemi-Ajayi <fnref n="4" />, Erik J. Timpson <fnref n="5" />, William E. Marsh <fnref n="5" />, Michael Speight <fnref n="5" /> and Donald C. Wunsch II <fnref n="1" />

**Abstract:** Biomedical datasets distill many mechanisms of human diseases, linking diseases to genes and phenotypes (signs and symptoms of disease), genetic mutations to altered protein structures, and altered proteins to changes in molecular functions and biological processes. It is desirable to gain new insights from these data, especially with regard to the uncovering of hierarchical structures relating disease variants. However, analysis to this end has proven difficult due to the complexity of the connections between multi-categorical symbolic data. This article proposes symbolic tree adaptive resonance theory (START), with additional supervised, dual-vigilance (DV-START), and distributed dual-vigilance (DDV-START) formulations, for the clustering of multi-categorical symbolic data from biomedical datasets by demonstrating its utility in clustering variants of Charcot-Marie-Tooth disease using genomic, phenotypic, and proteomic data.

**Keywords:** adaptive resonance theory; biomedical data; categorical data; ontologies; knowledge graphs

## 1. Introduction

Precision medicine depends upon a detailed unraveling of the relationships between diseases, phenotypes, genes, and the underlying proteins and biological pathways [1–7]. The ready availability of protein, disease, gene, phenotype, and biological pathway ontologies makes it possible to construct purpose-specific datasets for studying human disease. These can take the form of symbolic relationships that can be organized into formal ontologies that are instantiated as knowledge graphs defining the permissible relationships between classes and the instances within them [8].

However, many elements in these disease–gene–protein datasets are formatted as categorical rather than numerical variables, bringing a unique challenge to machine learning algorithms. Although tools exist to analyze and visualize categorical data [9], the tools for clustering these datasets depend heavily on recasting categories into real-valued spaces, which is largely unavoidable due to the definition of the problem statement; all modalities of machine learning assume distance metrics or similarity measures of their feature spaces, whereas categorical data contain symbols that do not belong to ordered sets, and thus, do not inhabit metric spaces. An important design choice then when working with mixed or fully categorical data is how to recast categorical features into spaces with similarity measures [10]. This recasting, whether by one-hot encoding, ordinal encoding, or another encoding scheme, can bring its own deleterious consequences; one-hot encoding

**Citation:** Petrenko, S.; Hier, D.B.; Bone, M.A; Obafemi-Ajayi, T.; Timpson, E.J.; Marsh, W.E.; Speight, M.; Wunsch, D.C., II. Analyzing Biomedical Datasets with Symbolic Tree Adaptive Resonance Theory. *Information* **2024**, *15*, 125. https://doi.org/10.3390/info15030125

Academic Editors: Birgitta Dresp-Langley and Luiz Pessoa

Received: 29 January 2024

Revised: 9 February 2024

Accepted: 10 February 2024

Published: 23 February 2024

<figure><img src="image_78.png" /></figure>

**Copyright:** © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

<footnote n="1">¹ Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA; hierd@mst.edu (D.B.H.); dwunsch@mst.edu (D.C.E.II)</footnote>

<footnote n="2">² Department of Neurology and Rehabilitation, University of Illinois at Chicago, Chicago, IL 60607, USA</footnote>

<footnote n="3">³ Department of Science and Industry Systems, University of Southeastern Norway, 3616 Kongsberg, Norway; mary.bone@drmarybone.com</footnote>

<footnote n="4">⁴ Engineering Program, Missouri State University, Springfield, MO 65897, USA; tayoobafemiajayi@missouristate.edu</footnote>

<footnote n="5">5 Honeywell Federal Manufacturing & Technologies, Kansas City, MO 64147, USA; etimpson@kcnsc.doe.gov (E.J.T.); wmarsh@kcnsc.doe.gov (W.E.M.)</footnote>

<footnote>* Correspondence: petrenkos@mst.edu</footnote>
of categories can generate large sparse feature vectors due to many different categories, while ordinal encoding can introduce measures of proximity between categories that do not intrinsically exist. Meta-analyses of symbolic datasets may yield similarity meta-metrics that are useful for clustering [11,12], but these meta-metrics require domain knowledge of the categories in the dataset, limiting both their transferability to other datasets and applicability to streaming learning. While statistical machine learning algorithms can compensate for some of these input feature space shortcomings through sophisticated machinery that relies on a large dataset size and a high degree of feature cardinality, these methods naturally suffer in regimes with small categorical datasets. Furthermore, these encoding schemes and the machine learning algorithms do not gracefully extend to instances of hierarchical or nested attributes such as occur with the variably sized association of diseases with phenotypes, genes, and proteins.

Adaptive resonance theory (ART) algorithms principally belong to the class of incremental neurogenesis clustering/unsupervised [13] algorithms, with many additional variants for use in supervised learning [14,15], reinforcement learning [16,17], and even self-supervised and multimodal applications [18]. The design of these algorithms allows them to update existing categories or create new ones from the data alone in a stable, incremental, and lifelong manner. With the notable exception of the binary-valued ART1 algorithm, most of these algorithms work upon real-valued preprocessed feature datasets via the use of fuzzy feature membership [19–23]. In contrast, the Gram-ART algorithm was designed for the meta-optimization of genetic algorithms, and thus, is designed to work with variable-length symbolic datasets [24], but it too has its shortcomings when tackling the large numbers of terminal symbols encountered in medical disease datasets.

With these myriad design challenges in mind, this article describes the design of a new ART algorithm named symbolic tree adaptive resonance theory (START) for the clustering of variable-length symbolic statements. This formulation of START also includes both dual-vigilance (DV-START) and distributed dual-vigilance (DDV-START) variants [25,26] along with their supervised modifications. This article also outlines methods for casting categorical disease–gene biomedical datasets into symbolic datasets for both unsupervised clustering and supervised training where labels are available.

The changes in START compared to the Gram-ART algorithm summarize the novel
contributions of this article in addition to the use of this algorithm for the study of biomed-
ical disease-variant data. START extends Gram-ART as a novel approach to analyzing
biomedical disease-variant data in the following ways:

1. Both a match and activation function for the Gram-ART match rule.

2. Optimizations to the prototype-encoding scheme to mitigate memory complexity in grammars with large sets of terminal symbols.

3. A mechanism to grow prototype tree structures when novel production rule sets are encountered.

4. Both dual-vigilance and distributed dual-vigilance START variants [25,26].

5. A supervised modification for each unsupervised START variant.

This article is organized into the following sections: Section 2 provides a background of the literature pertinent to the formulation of START, while Section 3 describes the derivation and structure of START and its dual-vigilance and supervised variants. Section 4 outlines the datasets and experimental methodology utilized in the evaluation of START, including benchmark machine learning datasets and the target biomedical disease-variant datasets of the article, and Section 5 contains the results of these experiments. Section 6 discusses the experimental results and their biological plausibility, with Section 7 providing final conclusions on both START and the biomedical dataset analysis of the previous sections.

## 2. Background

2.1. *Adaptive Resonance Theory*

Adaptive resonance theory (ART) is a neurocognitive theory of how biological neural
networks for self-stable representations learn without catastrophic forgetting, online and
without supervision, through feedback and competitive dynamics [27–35]. Since its inception, a variety of machine learning models have been implemented using the theory as a basis [19,36–38]. Though these algorithms in large part belong to the class of incremental neurogenesis clustering/unsupervised algorithms, they have been adapted for applications in supervised, reinforcement, and even multimodal learning [19,39], tackling clustering issues from sample granularity [25,26] to distributed representations [40–42], pattern sequences [43], context recognition [44,45], and uncertainties [46–48]. Some algorithms based upon ART have even been combined with incremental cluster validity indices (ICVIs), metrics of clustering performance in the absence of supervised labels, to enable a variety of incremental, online, and multimodal clustering and biclustering applications [49–54]. ART algorithms are additionally well suited for lifelong learning (L2) applications because they are derived from theories on how biological neural networks address the stability-plasticity dilemma to mitigate catastrophic forgetting [55–57].

Nearly all ART formulations trade the explicit coarseness parameters of other clustering algorithms for a vigilance parameter ($\rho \in (0, 1)$), which behaves as a threshold of agreement between a sample and expectations to determine whether to update existing knowledge or to create new categories altogether, a process known as the ART match rule [41]. Furthermore, nearly all ART formulations are intrinsically prototype-based machine learning algorithms, meaning that categories are defined by representative prototypes in the sample feature space. This has two important consequences: ART algorithms theoretically have unlimited memory because new prototypes may always be instantiated, but they generally have no representational capacity in the sense of manifold learning, relying instead on the assumption that the feature space being used for clustering is sufficiently well separated. Samples in this feature space are provided in a feature representation layer $F1$, which is compared with a category representation layer $F2$ containing these prototypes through ART competitive dynamics that include a check against this vigilance parameter.

## 2.2. Gram-ART

Gram-ART is a clustering algorithm, based on ART learning dynamics, that defines its prototypes and input features as trees of parsed statements adhering to a formal grammar [24]. Originally designed to tackle the problem of comparing similarity between symbolic expressions for the meta-optimization of genetic algorithms, it is capable of accepting statements of an arbitrary length according to a user-defined context-free grammar (CFG) expressed in the Backus-Naur form (BNF). In the original formulation, Gram-ART samples are statements adhering to a CFG that are parsed into rooted syntax trees. These parsed samples are then compared according to ART learning rules to Gram-ART prototypes that are themselves rooted trees containing distributions of terminal symbols that are encountered at each node during learning. Gram-ART answers the questions of how to formulate prototype trees of varied shape, compute similarities of sample statements to prototypes of differing shapes, and update the terminal symbol distributions at each node during learning.

Gram-ART is the first ART algorithm capable of clustering inputs samples of arbitrary length, but it also inherits some problems from working with symbolic data. Terminal symbols under a grammar have no fuzzy membership or relation without an additional embedding scheme. Gram-ART tackles this by updating distributions of terminal symbols at each position along the rooted prototype trees during learning. However, this technique quickly grows in space and subsequent time complexity in grammars with sets of terminal symbols larger than the algebraic expressions that they were originally designed for.

# 3. Method

## 3.1. *START: Symbolic Tree Adaptive Resonance Theory*

This paper introduces a new formulation of the Gram-ART algorithm called START for the clustering of symbolic datasets. START is a prototype-based unsupervised clustering algorithm that when presented with a new sample utilizes ART dynamics to determine
whether to update an existing template or to instantiate a new one. START targets symbolic expressions adhering to a context-free grammar CFG(T, N, P, S) with a complete set of terminal symbols T, non-terminal symbols N, production rules P, and a statement entry point S. The prototypes of START are rooted trees containing learned distributions of the encountered terminal symbols at each node representing a non-terminal position, and symbolic statements are parsed into rooted constituency parse trees that are subsequently processed against these prototypes using ART learning dynamics. With such a formulation, the method is naturally extended to the clustering of purely categorical datasets of variable length sequences, such as in the myriad categorical fields of disease-gene-protein data.

### 3.1.1. Motivation

The realm of clustering, and indeed machine learning as a whole, requires a serious consideration and study of the various forms that data may take [10]. Datasets are often modeled as samples of the state space defined by some measuring device. Many samples of data are naturally real-valued, such as the readings from imaging sensors for the purposes of computer vision, while others are categorical in nature, such as descriptor labels of the color of an object (e.g., red, blue, yellow). Datasets may have one or more feature dimensions, and they may even be multimodal, containing a combination of real-valued and categorical data in each sample. A notion of the proximity of features is critical to machine learning algorithms that utilize similarity measures to model and interpret samples; metric spaces are defined as sets that can have such a similarity measure defining the distance between points in the set, and indeed even categorical features may sometimes have distance metrics if they have an ordering (e.g., low, medium, high), though they often only have a strict equivalence relation for comparing categories (e.g., red = red, red ≠ blue). The presence of a distance metric is especially important in unsupervised learning scenarios such as clustering where an algorithm has nothing available to model a dataset aside from the features themselves. As a consequence, the clustering of data with unordered categorical features is difficult, and many clustering algorithms are designed with the assumption that at least some ordered features exist in the data [10].

Nevertheless, purely categorical datasets such as those containing only label descriptors do exist, and it is desirable to cluster them to extract meaning and structure. It is even more challenging when such datasets contain a varying number of features for each sample; algorithms that tackle real-valued datasets of variable length such as time-series data utilize techniques like convolutions and pooling to turn a varying number of features at runtime into a fixed model size, but these techniques are ill-defined for purely categorical data, especially when individual symbols are sparsely populated throughout the dataset.

Purely categorical datasets of variable feature dimensions arise commonly in human-annotated datasets, such as those generated from medical research. Human-prescribed categories of diseases, their variants, and other ontological features can contain missing entries when data are missing or inapplicable, and categories can even be nested; for example, the presence or absence of the symptom of pain may be further qualified by pain in specific regions of the body or of varying intensity according to some pain scale.

One realm that specifically deals with categorical data of variable length is the study of languages [58,59]. Syntactically, sentences in a language are interpreted as statements that adhere to a formal grammar that determines the rules of what is or is not a valid statement in a given language. The study of language also applies in the design of lexers and parsers, which are used in computer science for the design of programming languages to structure valid symbolic statements of arbitrary length written by a programmer for compilation or interpretation. Parsers are especially important as a mechanism of applying the rules of a grammar to interpret strings of symbolic statements as syntax trees defining their structure. These grammar rules, however, do not define a notion of how similar or dissimilar two statements are, so a clustering algorithm working in this space must introduce a mechanism for comparing statement similarity.
Given that START shares the objective of Gram-ART to cluster variable-length symbolic expressions, the key design challenges of START's design are in how to formulate metrics of similarity between these symbolic expressions. In such a formulation, statements are collections of symbols sampled from unordered sets; individual symbols share no fuzzy membership, so similarity between symbols is dictated by strict equivalence in a set theoretic sense. Furthermore, though statements of equal length introduce a step-wise fuzziness when symbols in the same relative positions are identical, many datasets do not satisfy the assumption of equivalent non-terminal structure across all statements. In the pursuit of creating a clustering algorithm for variable-length symbolic datasets, START utilizes a prototype method as a proxy for direct comparison between statements, using ART-based competitive learning dynamics for determining when to update templates and when to instantiate new ones. As with all ART algorithms, START therefore inherits both the theoretically unlimited learning capacity of neurogenesis algorithms and the problems of category proliferation that they bring; though new prototypes can be instantiated for an arbitrary number of categories, this growing knowledge base incurs its own search time complexity [19,60].

### 3.1.2. START Algorithm

START shares the nomenclature of Gram-ART and other ART algorithms from its structure to its learning dynamics, so existing terminology is preferred where available. START also follows the procedure of most ART unsupervised clustering algorithms, with additional considerations for handling symbolic data. As in Gram-ART, START handles this symbolic data by working in the space of the syntactic trees representing the symbolic data as statements under a formal grammar. The shared notation of all START variants is listed in Table 1.

<table><caption>**Table 1.** Shared START notation. The learning dynamics of START and its variants follow the activation, competition, match, update, and initialization rules of unsupervised ART algorithms, so the notation here largely adheres to the elementary ART algorithm notation outlined in [19]. Dual-vigilance lower bound $\rho_{lb}$ and upper bound $\rho_{ub}$ follow the notation in DVFA [25] and DDVFA [26].</caption><tr><td><i>R</i>:</td><td>set of prototype nodes.</td></tr><tr><td><i>R</i>:</td><td>a single prototype node.</td></tr><tr><td><i>C</i>:</td><td>set of prototype node indices.</td></tr><tr><td><i>A</i>:</td><td>subset of active ART module node indices (<i>A</i> &sub; <i>C</i>).</td></tr><tr><td><i>p</i>:</td><td>START vigilance threshold, <i>p</i> &isin; (0, 1).</td></tr><tr><td><i>p</i><sub>lb</sub>:</td><td>dual-vigilance lower-bound vigilance threshold (<i>p</i><sub>ub</sub> &gt; <i>p</i><sub>lb</sub> &gt; 0).</td></tr><tr><td><i>p</i><sub>ub</sub>:</td><td>dual-vigilance upper-bound vigilance threshold (1 &gt; <i>p</i><sub>ub</sub> &gt; <i>p</i><sub>lb</sub>).</td></tr><tr><td><i>n</i>:</td><td>number of input dataset statements.</td></tr><tr><td><i>X</i>:</td><td>statements parsed as syntax trees with terminal metadata.</td></tr><tr><td><i>Parser</i>(·):</td><td>syntactic parsing algorithm taking a set of statements and a grammar and producing rooted constituency parse trees.</td></tr><tr><td><i>f</i><sub>T</sub>(·):</td><td>activation function.</td></tr><tr><td><i>f</i><sub>M</sub>(·):</td><td>match function.</td></tr><tr><td><i>f</i><sub>N</sub>(·):</td><td>node initialization function.</td></tr><tr><td><i>f</i><sub>L</sub>(·):</td><td>node weight update function.</td></tr><tr><td><i>f</i><sub>V</sub>(·):</td><td>the vigilance test function.</td></tr><tr><td><i>U</i>:</td><td>internal supervised category indices.</td></tr><tr><td><i>C</i>:</td><td>set of cluster indices.</td></tr></table>

A START module is initialized to contain the CFG(*T*, *N*, *P*, *S*) rules of the target symbolic dataset statements. This grammar can be inferred from an existing dataset of statements if all relevant symbols and production rules are represented in the dataset. Statements from the dataset are parsed according to the production rules of the grammar into rooted constituency parse trees, the basic unit of which is known in Gram-ART and START as a TreeNode. Each parsed statement tree is presented incrementally to the START module, and each sample either mutates an existing prototype or is used to instantiate an entirely new prototype [19]. Prototypes in START are themselves rooted trees with a
structure modified from the statement trees, the basic unit of which is known in Gram-ART as a ProtoNode. The stateful information of START TreeNodes and ProtoNodes can be seen in Tables 2 and 3, respectively.

<table><caption>**Table 2.** A simple UML diagram of the stateful information of one START TreeNode [24]. A symbol in a TreeNode in START is realized by either a terminal or non-terminal symbol at the syntax tree position of the node. A rooted tree of TreeNodes in this regard contains the minimum information necessary to describe the syntax tree of a statement parsed with a prescribed grammar.</caption><thead><tr><th>TreeNode</th></tr></thead><tbody><tr><td>Symbol: GrammarSymbol</td></tr><tr><td>Children: Vector{TreeNode}</td></tr></tbody></table>

<table><caption>**Table 3.** A simple UML diagram of the stateful information of one START ProtoNode, which is the basic element of the rooted trees constituting the prototypes of START [24]. A rooted tree of START ProtoNodes encodes only through the non-terminal positions of the syntax tree of a TreeNode tree. Each ProtoNode encodes a PMF of terminal symbols encountered at and below the non-terminal position of the ProtoNode itself, with instance counts of each terminal encoded for the renormalization of the PMF when learning occurs at the node itself.</caption><thead><tr><th>ProtoNode</th></tr></thead><tbody><tr><td>Symbol: NonTerminalGrammarSymbol</td></tr><tr><td>Distribution: Dictionary{TerminalGrammarSymbol, Float}</td></tr><tr><td>InstanceCount: Dictionary{TerminalGrammarSymbol, Integer}</td></tr><tr><td>Children: Vector{ProtoNode}</td></tr></tbody></table>

Here, START and Gram-ART differ on an important point in formulation: Gram-ART treats ProtoNodes and TreeNodes as modified dependency relation syntax trees where each node represents a terminal symbol, the children of which are the dependents of that symbol. This formulation is most apparent in the case of operators, such as in the algebraic statement $x + y$, where the operator terminal + would have branch dependents x and y. In START, however, ProtoNodes and TreeNodes are defined as relation parse trees with non-terminal symbols representing non-terminal positions and terminal symbols at the leaves of the rooted tree. The same algebraic statement $x + y$ is then treated in START as a relation parse tree with non-terminal symbols for the operation and its three branches represent the operator and its two arguments, with leaf nodes realizing the terminal symbols at these non-terminal positions (Figure 1).

In START, sample symbolic statements are preprocessed into parse trees via a syntactic parser such as an Earley parser according to the production rules *P* of the grammar written most generally in an extended Backus-Naur form (EBNF) [61,62]. These syntax trees can be interpreted as concrete constituency relation parse trees belonging to constituency grammars, also known as phrase structure grammars, where branches of a parse tree are all non-terminal symbols in the grammar, including the statement entry point, and leaf nodes are terminal symbols [58,59]. These parse trees are then converted to statement trees via an inclusion of metadata at each node indicating the symbol to be terminal or non-terminal. Prototypes in START are rooted trees containing probability mass functions (PMFs) of terminal symbols encountered at and below the position of each ProtoNode on the tree. In contrast with Gram-ART, these START prototypes do not contain terminal symbol leaves; instead, the nodes of the prototypes represent the non-terminal positions of the grammar production rules applied to the node's position on the tree, which reduces the effective size of each prototype tree while still encoding the occurrence of terminal symbols at and below those positions via their PMFs.
<figure><img src="image_79.png" /><figcaption>**Figure 1.** Comparison of the constituency relation parse trees of START (a) to the dependency parsing syntax trees of Gram-ART (b) for the simple algebraic statement $x + y$. START TreeNodes are full constituency relation parse trees containing terminal symbols at the leaves of the tree, while START ProtoNodes contain only non-terminal symbols at non-terminal positions on the parse tree. As in the Grammar Listing 1, non-terminal symbols are surrounded by arrows `<->` and terminal symbols are in single quotations. Here, `<oper>` denotes “operation,” `<op>` denotes “operator”, and `<arg₁>` and `<arg₂>` denote the two “arguments” of the operator.</figcaption></figure>

**Listing 1.** Formal grammar for parsing Charcot-Marie-Tooth disease-protein flat-file data. EBNF syntax is used for production rules with the exception of the regular expression symbol '+', which is used to denote one or more occurrences of the preceding symbol. Statements are composed of a series of one or more categorical attributes, all of which are listed in the non-terminal symbol `<attribute>`. When an attribute is missing or otherwise unknown for a CMT variant, then it is not included in the parsed syntax tree and handled accordingly by START. The production rules for two notable multi-category attributes, `<phenotype>` and `<biologic_process>`, are listed to demonstrate how statements formulated from CMT disease-variant entries illustrate how a gene can be associated with multiple phenotypes and biologic processes. Other multi-category attributes are not listed for brevity.

`(S) ::= <attribute> + ;`

`<attribute> ::= <num> | <gene_location> | <disease> | <disease_MIM> | <gene> | <gene_MIM> | <inheritance> + | <protein> | <uniprot> | <chromosome> | <chromosome_location> | <protein_class> + | <biologic_process> + | <molecular_function> + | <disease_involvement> + | <MW> | <domain> + | <motif> + | <protein_location> + | <length> | <disease_MIM2> | <phenotype> + | <weight_tag> | <length_tag> ;`

`<phenotype> ::= 'ataxia' | 'atrophy' | 'auditory' | 'autonomic' | 'behavior' | 'cognitive' | 'cranial_nerve' | 'deformity' | 'dystonia' | 'gait' | 'hyperkinesia' | 'hyperreflexia' | 'hypertonia' | 'hypertrophy' | 'hyporeflexia' | 'hypotonia' | 'muscle' | 'pain' | 'seizure' | 'sensory' | 'sleep' | 'speech' | 'tremor' | 'visual' | 'weakness';`

`<biologic_process> ::= 'Apoptosis' | 'Mitosis' | 'Lipid_metabolism' | 'Symport' | 'Ubl_conjugation_pathway' | 'Glycolysis' | 'Glucose_metabolism' | 'Ion_transport' | 'Unfolded_protein_response' | 'Cell_division' | 'DNA_repair' | 'Cell_adhesion' | 'Notch_signaling_pathway' | 'Protein_biosynthesis' | 'Stress_response' | 'Endocytosis' | 'Transcription' | 'Sodium_potassium_transport' | 'Transcription_regulation' | 'Fatty_acid_metabolism' | 'Host_virus_intersection' | 'Antiviral_defense' | 'Lipid_degradation' | 'Autophagy' | 'Sodium_transport' | 'Immunity' | 'none' | 'Protein_transport' | 'Nucleotide_biosynthesis' | 'Calcium_transport' | 'Transport' | 'Phagocytosis' | 'Inflammatory_response' | 'DNA damage' | 'Potassium_transport' | 'Carbohydrate_metabolism' | 'Cell_cycle' | 'Innate_immunity';`
### 3.1.3. Derivation of the START Match Rule

A fundamental characteristic of ART algorithms is the use of a match rule, whereby a process of bottom-up activations drive the evaluation of how much the input sample matches existing top-down categories [41]. Because of the origins of these algorithms lies in the analysis of the competitive dynamics of biological neural networks, these activation and match functions are frequently analogized with bottom-up prediction and top-down expectation, respectively.

Gram-ART utilizes an activation function, while START introduces separate activation and match functions. The distinction between the two lies in the normalization scheme of the activation and match functions; for example, in ART1 the match function (Equation (2)) is the activation function (Equation (1)) normalized by the size of the input [19].

$$T_j = \|x \cap w_j\|_1 \qquad (1)$$

$$M_j = \frac{\|y^{(F_1)}\|_1}{\|x\|_1} = \frac{\|x \cap w_j\|_1}{\|x\|_1} \qquad (2)$$

FuzzyART replaces the binary intersection with the fuzzy intersection in both equations and normalizes the activation by the magnitude of the weight vector [19]. When evaluated at a single node, an input terminal symbol can be interpreted as a one-hot binary vector encoding at the terminal symbol position, so the magnitude of the membership of sample $x$ in weight $w_j$ is indeed the fuzzy intersection $\|x \cap w_j\|_1$. This is computed in START for the terminal distribution of each ProtoNode climbing up from the aligned leaf representing the terminal symbol. In statements with many branches arising from non-trivial production rules, this means the evaluation of the activation at each ProtoNode for potentially multiple terminal descendants.

The activation is then normalized by the size of the input pattern, which can be realized in multiple manners requiring a design decision; with the rooted tree definition of parsed input statements, the size of the input pattern could be interpreted as the number of nodes in the parsed statement, the number of terminal symbols in the unparsed statement, or a more complex function of the number of terminals that could be realized beneath the non-terminal position of the node in question according to the production rules of the grammar of the sample. For simplicity, the remainder of this study utilizes the length of the unparsed statement itself as a normalizing factor, having the effect of discounting the disproportional contributions to the match value of increasingly longer statements. In grammars where statements are of equal length, such as in the processing of tables with single-category data, each decision trivially scales the required vigilance values to satisfy the vigilance criterion.

The remainder of the match rule follows the activation, competition, match, and vigilance test of unsupervised ART algorithms, as can be seen in Algorithm 1, with the exception of the dual-vigilance variants of START, which can be seen in Section 3.1.5 and Algorithm 2.

### 3.1.4. Derivation of the Weight Update

When a prototype is selected for learning according to the START match rule, the input TreeNode and selected ProtoNode are root-aligned and compared, similar to in the activation and match processes. The terminal symbols contributing to the activation and match functions of the winning prototype are used for updating the PMF at each non-terminal symbol position at each ProtoNode up the prototype tree. The instance count of the observed terminal symbol is incremented, and the PMF update is weighted by the instance count of each terminal of the distribution to renormalize. In Equation (3), the weight value $w$ of the PMF indexed at terminal $T$ in node $i$ is updated with instance count $N$ and a Kronecker delta $\delta_T$ that is satisfied if the terminal symbol $x$ being evaluated is equivalent to the PMF index $T$ (Equation (4)).
$$w_i^T = \frac{w_i^T * N + \delta_{Tx}}{N + 1} \qquad (3)$$

$$\delta_{Tx} = \begin{cases} 1 & \text{if } T = x \\ 0 & \text{otherwise} \end{cases} \qquad (4)$$

**Algorithm 1:** START algorithm. A set of symbolic statements under a formal context-free grammar are parsed into their syntax trees. Prototypes are defined as learning dynamics otherwise follow the activation, competition, match, update, and initialization rules of unsupervised ART algorithms [19]. ART dynamics notation here largely follow the elementary ART algorithm outlined in [19]. Inference during classification follows the same match rule dynamics without the instantiation of new categories; in the case of complete mismatch, either an "unknown" label or the best matching unit (the category that maximizes the match criterion) may be returned. Please see Table 1 for full notation

**Data:** Symbolic statements **S**; CFG grammar **G** with terminal symbols **T**, non-terminal symbols **N**, production rules **P**, and statement entry symbol **S**.

**Result:** Cluster labels **Y** ∈ $\mathbb{N}^n$

<table><tr><td>/* Parse statements into constituency parse trees</td><td>*/</td></tr><tr><td>1 X &larr; Parser(S, G)</td><td></td></tr><tr><td>/* Iteration over parsed statement trees</td><td>*/</td></tr><tr><td>2 foreach x &isin; X do</td><td></td></tr><tr><td>3 /* Compute activations for all nodes</td><td>*/</td></tr><tr><td>T<sub>j</sub> &larr; f<sub>T</sub>(x, R<sub>j</sub>), &forall;j &isin; C</td><td></td></tr><tr><td>4 /* Perform WTA competition for active nodes</td><td>*/</td></tr><tr><td>5 J &larr; arg max<sub>j &isin; A</sub> (T<sub>j</sub>)</td><td></td></tr><tr><td>/* Compute match for the winning category</td><td>*/</td></tr><tr><td>6 M &larr; f<sub>M</sub>(x, R<sub>J</sub>)</td><td></td></tr><tr><td>/* Vigilance test</td><td>*/</td></tr><tr><td>7 if M &gt; &rho; then</td><td></td></tr><tr><td>8 /* Update category</td><td>*/</td></tr><tr><td>9 R<sub>J</sub> &larr; f<sub>L</sub>(x, R<sub>J</sub>)</td><td></td></tr><tr><td>/* Deactivate category</td><td>*/</td></tr><tr><td>10 &Lambda; &larr; &Lambda; - {I}</td><td></td></tr><tr><td>11 if &Lambda; &ne; &emptyset; then</td><td></td></tr><tr><td>12 /* Continue match search</td><td>*/</td></tr><tr><td>13 Goto Line 4</td><td></td></tr><tr><td>14 else</td><td></td></tr><tr><td>15 /* Create and initialize new category</td><td>*/</td></tr><tr><td>16 K &larr; ||C||<sub>1</sub> + 1</td><td></td></tr><tr><td>17 R<sub>K</sub> &larr; f<sub>N</sub>(x, G)</td><td></td></tr></table>

If no prototype satisfies the vigilance criterion, a new one is instantiated. START prototypes do not encode all combinations of non-terminal production evaluations during instantiation, as this would quickly combinatorially explode towards the Catalan number of the non-terminal production rules, and it could be infinite in some recursive grammars. Instead, prototypes are instantiated as structural clones of the input TreeNode without the inclusion of the terminal symbols at their leaves. This design decision is made to mitigate the time and memory complexity of the ProtoNode evaluation given that the non-terminal node preceding
a terminal leaf already encodes all of the instances that the terminal symbol encounters. The new structural clone prototype is then trained upon the input sample, updating the PMFs of each ProtoNode for the first time. In the case that an existing winning prototype does not contain the input TreeNode as a structural subset (i.e., it is missing a non-terminal production rule path describing the parsed TreeNode), these new non-terminal paths are instantiated on the winning prototype and updated as usual.

**Algorithm 2:** Dual-Vigilance START algorithm. This algorithm combines Algorithm 1 with the dual-vigilance procedure of DVFA [25]. The vigilance test is split into a cascade of two vigilance checks for the current match candidate node. Passing the upper vigilance check updates the current category node, while passing only the lower vigilance check creates a new category node belonging to the same cluster label. Failing to pass both vigilance checks results in the instantiation of a new category node belonging to an incrementally new cluster label. Please see Table 1 for full notation

**Data:** Symbolic statements S; CFG grammar G with terminal symbols T, non-terminal symbols N, production rules P, and statement entry symbol S.

**Result:** Cluster labels $Y \in \mathbb{N}^n$

/* Parse statements into constituency parse trees */

1 X $\leftarrow$ Parser(S, G)

/* Iteration over parsed statement trees */

2 **foreach** x $\in$ X **do**

/* Compute activations for all nodes */

3 $T_j \leftarrow f_T(x, \mathcal{R}_j)$, $\forall j \in \mathcal{C}$

/* Perform WTA competition for active nodes */

4 $J \leftarrow \arg\max_{j \in \Lambda} (T_j)$

/* Compute match for the winning category */

5 $M \leftarrow f_M(x, \mathcal{R}_J)$

/* Dual-vigilance tests */

6 **if** $M > \rho_{lb}$ **then**

/* Update current category */

7 $\mathcal{R}_J \leftarrow f_L(x, \mathcal{R}_J)$

8 **else if** $M > \rho_{lb}$ **then**

/* Create a new category within the same cluster */

9 $K \leftarrow \|C\|_1 + 1$

10 $\mathcal{L}_K \leftarrow \mathcal{L}_J$

11 $\mathcal{R}_K \leftarrow f_N(x, G)$

12 **else**

/* Deactivate category */

13 $\Lambda \leftarrow \Lambda - \{J\}$

14 **if** $\Lambda \neq \emptyset$ **then**

/* Continue match search */

15 Goto Line 4

16 **else**

/* Create and initialize new category and cluster */

17 $K \leftarrow \|C\|_1 + 1$

18 $\mathcal{L}_K \leftarrow \max(\mathcal{L}) + 1$

19 $\mathcal{R}_K \leftarrow f_N(x, G)$
### 3.1.5. Dual-Vigilance and Distributed Dual-Vigilance START

The FuzzyART algorithm provides a foundation for how to adapt ART learning rules to real-valued datasets [19]. Like most ART modules, FuzzyART utilizes the ART match rule evaluated at a single threshold value that is either the vigilance hyperparameter $\rho$ or a function thereof. Dual-vigilance FuzzyART (DVFA) utilizes instead two vigilance parameters for the match rule evaluation, a lower bound $\rho_{lb}$ and upper bound $\rho_{ub}$, which separates prototypes in a many-to-one mapping from categories to clusters and introduces the ability to compensate for differing granularity both within and between clusters [25]. Distributed dual-vigilance FuzzyART (DDVFA) advances this idea by representing entire clusters with FuzzyART modules governed by a global FuzzyART module, compensating for even varying granularity within different clusters and enabling the ability to learn arbitrary cluster shapes [26]. Each node in the global F2 layer competes for assignment of a provided sample through modified activation and match linkage methods, defining the relevant proximity measures of the sample to an entire F2 FuzzyART module node.

The principles of dual-vigilance and distributed dual-vigilance are extended here for START. In the dual-vigilance formulation (DV-START), the same cascading technique as in DVFA is used for determining category-cluster assignments through upper- and lower-bound vigilance hyperparameters during the ART match evaluation:

1. $M_J > \rho_{ub}$: if the current match candidate satisfies the upper vigilance threshold, then the winning category is updated according to the START weight update rules.

2. $\rho_{ub} > M_J > \rho_{lb}$: if the current match candidate only satisfies the lower vigilance threshold but not the upper, then a new category prototype is instantiated that belongs to the same cluster as the winning node.

3. $\rho_{lb} > M_J$: if the current match candidate does not satisfy even the lower-bound vigilance threshold, then the normal mismatch procedure is followed, where a new category is instantiated belonging to an entirely new cluster.

In the distributed dual-vigilance formulation (DDV-START), additional modifications are made to accommodate the rooted tree structures of the prototypes. DDVFA utilizes a global FuzzyART module that represents nodes themselves as FuzzyART modules [26]. The basic units of DDV-START are the rooted ProtoNode trees, but global module dynamics are not restricted to their use; because the global module of DDV-START is largely agnostic to the formulation of the input samples, the global module may be approximated as a FuzzyART module coordinating the learning of its START F2 nodes. With the exception of the centroid linkage method, which in DDVFA is defined as a function of local FuzzyART weights, all other linkage methods from DDVFA can be utilized in DDV-START; by independently defining the activation and match values for each ProtoNode within an F2 START module, the global values can be compared using the hierarchical agglomerative clustering (HAC) methods of DDVFA, as can be seen in Table 4.

### 3.1.6. Supervised Variants

Most ART algorithms are designed as unsupervised clustering algorithms with variants and compositions of the elementary ART module motif providing supervised and reinforcement learning variants [19]. ARTMAP is a formulation of ART, comprised of two elementary ART modules and an inter-ART map field, that enables multidimensional mapping between two feature fields [63]. A simplified version of FuzzyARTMAP, where the second module $ART_B$ is replaced with vectors representing class labels, provides a basic procedure for adapting unsupervised ART modules to simple supervised ARTMAP variants [64]. Though START is designed as an unsupervised clustering algorithm, it utilizes these supervised modifications for evaluation on the supervised machine learning benchmark datasets in Section 4.2 and in the Supplementary Materials of this paper. Algorithm 3 outlines this procedure of mapping the internal category representation labels to supervised labels for any START variant.
Because these supervised variants are derived from a procedure to modify an ART module to a simplified ARTMAP variant, their naming follows the same notation (e.g., START to Simplified STARTMAP).

<table><caption>**Table 4.** Distributed dual-vigilance START activation and match linkage methods where hierarchical agglomerative clustering (HAC) functions and distributed dual-vigilance notation are shared with DDVFA [26]. Global activation $T_i^g$ and match $M_i^g$ functions are defined via the generic function $h_i^g$ for the global F2 node index $i$ as a function of inner node indices $j = 1...k$, where $k$ is the number of $F_2$ nodes in the local START module $i$. Each HAC method then is a "function of functions" evaluated at each F2 node in the global module to determine either the match or activation value in the global module match rule dynamics.</caption><thead><tr><th>HAC Method</th><th>$h_i^g$</th></tr></thead><tbody><tr><td>Single</td><td>$\max_j(f_j^i)$</td></tr><tr><td>Complete</td><td>$\min_j(f_j^i)$</td></tr><tr><td>Median</td><td>$\text{median}(f_j^i)$</td></tr><tr><td>Average</td><td>$\frac{1}{k_i}\sum_{j=1}^{k_i}f_j^i$</td></tr><tr><td>Weighted <sup>1</sup></td><td>$\sum_{j=1}^{k_i}p_jf_j^i$</td></tr></tbody></table>

### 3.1.7. Summary of START Variants

The previous sections have outlined three unsupervised algorithms for the clustering of categorical data of varying feature dimensionality: START, its dual-vigilance variant DV-START, and its distributed dual-vigilance variant DDV-START. The core START algorithm is outlined for clustering this categorical data using the incremental learning and update rules of ART algorithms with a single vigilance value; if a category match is found, that prototype is updated according to the ART match rule, and if there is instead a complete mismatch, a new category is instantiated. The dual-vigilance (DV-START) and distributed dual-vigilance (DDV-START) variants of this core algorithm follow as extensions of the algorithm through modifications of the prototype update method in a similar manner as FuzzyART is extended to DVFA and DDVFA [25,26]. In both dual-vigilance variants, two vigilance values (upper and lower) are instead used to determine how a single update should proceed, allowing for differing inter- and intra-cluster granularities. DV-START utilizes an internal category-cluster map for determining if a single prototype is updated, if a prototype is updated in an existing cluster, or if an entirely new cluster is instantiated. DDV-START, on the other hand, distinguishes between global and local nodes, where global nodes are themselves START modules and local nodes are their prototypes; this distinction necessitates the use of hierarchical agglomerative clustering (HAC) functions to determine the distance measures between START modules when evaluating the match and activation values of a sample, and the upper and lower vigilance values are used to determine which global and local nodes to update or instantiate.

<footnote n="1">¹ $p_j = \frac{n_j^i}{n_i^g}$, where $n_j^i$ is the number of samples (i.e., instance count) encoded by $j$ of the local START module at global F2 index $i$ and $n_i^g = \sum_j n_j^i$.</footnote>
**Algorithm 3:** Simplified supervised modification for all START variants (e.g., Simplified STARTMAP). The variation between START variants is captured in the evaluation of the vigilance test as a function $f_V$; if some node satisfies the match rule of the START variant, the sample is said to fall within the vigilance region of the prototype [19]. Complete mismatch instead occurs when no vigilance test is satisfied, and the prototype initialization procedure of the START variant is triggered. Inference after training is run through to the vigilance test procedure, reporting the supervised label mapping to the winning internal node category. In the case of complete mismatch, where no nodes satisfy the vigilance test of a supplied inference sample, either the supervised label mapping to the best matching unit (i.e., the node with the highest match value) or a custom mismatch signal may be reported depending on the desired application. Please see Table 1 for full notation

**Data:** Symbolic statements **S**; supervisory labels **Ω**; CFG grammar **G** with terminal symbols **T**, non-terminal symbols **N**, production rules **P**, and statement entry symbol **S**.

**Result:** Cluster labels $\mathbf{Y} \in \mathbb{N}^n$

/* New supervised prototype initialization procedure taking
supervised label $\omega$ */

1 **Function** initialization($\omega$):

/* Increment the count of unique internal categories */
2 $K \leftarrow \|C\|_1 + 1$

/* Initialize a new prototype according the START variant with
the new internal category label $K$ */
3 $\mathcal{R}_K \leftarrow f_N(\mathbf{x}, \mathbf{G})$

/* Map the supervised label the new internal category */
4 $\mathcal{U}_K \leftarrow \omega$

/* Parse statements into syntax trees */
5 $\mathbf{X} \leftarrow \text{Parser}(\mathbf{S}, \mathbf{G})$

/* Iteration over parsed statement trees with supervised labels */
6 **foreach** $\mathbf{x}, \omega \in \mathbf{X}, \Omega$ **do**

/* Instantiate a new prototype with the supervised label if the
label is entirely novel */
7 **if** $\omega \notin \mathcal{U}$ **then**

initialization($\omega$)

/* Run the vigilance test specific to the START variant */
/* Update winning node $J$ if it correctly predicts label $\omega$ */
/* Run START update procedure */

$V_J = f_V(\mathcal{R}, \mathbf{x})$

$f_L(\mathcal{R}_J, \mathbf{x})$

/* Otherwise, initialize a new category */
initialization($\omega$)

Furthermore, these three unsupervised algorithms are extended to their own supervised variants using the procedure of Simplified FuzzyARTMAP to map internal category labels to supervised labels, and their nomenclature follows the same procedure (e.g., START to Simplified STARTMAP) [64]. Table 5 arranges the resulting six variants and their names in a table according to their learning modality and vigilance formulation.
<table><caption>**Table 5.** A summary of the START variants and their abbreviations. Three vigilance formulations are developed, starting with a core START algorithm and extending it with dual-vigilance and distributed dual-vigilance variants (Section 3.1.5). These three variants are intrinsically incremental, unsupervised clustering algorithms, but a supervised procedure in the vein of Simplified FuzzyARTMAP (summarized in Section 3.1.6) generates a supervised variant for each of these three algorithms as well.</caption><thead><tr><td>Vigilance Formulation</td><td>Unsupervised</td><td>Supervised</td></tr></thead><tbody><tr><td>Single-Vigilance</td><td>START</td><td>Simplified STARTMAP</td></tr><tr><td>Dual-Vigilance</td><td>DV-START</td><td>Simplified DV-STARTMAP</td></tr><tr><td>Distributed Dual-Vigilance</td><td>DDV-START</td><td>Simplified DDV-STARTMAP</td></tr></tbody></table>

### 3.1.8. Comparison of START Variants

Similar to the FuzzyART variants that they are inspired by, the six variants of START each have their own advantages and drawbacks according to the machine learning context at hand. Each algorithm is designed for learning upon purely categorical datasets where each sample may have a variable length; as a result, the use of these algorithms necessitates the design of a parser that may take such a dataset and transform it into a series of statements and their corresponding relation parse trees according to a context free grammar (CFG) that describes that dataset, which may be expressed in a series of production rules in an extended Backus-Naur form (EBNF). When the entirety of the dataset is available, the CFG and its production rules may be immediately inferred from the data itself.

ART algorithms such as START are designed to completely learn upon a single sample at a time, which makes them suitable for streaming clustering applications. Because of its formulation, START tracks only distributions of symbols that it has encountered, without requiring full knowledge of the populations or distributions of symbols in advance, so new symbols may be added naturally in a streaming clustering context.

The unsupervised variants of START are naturally suited to symbolic clustering problems, and the supervised variants may be used in both supervised and multimodal contexts because the supervised modification is exterior to the weight update and instantiation process. When supervised labels are available, the label map is populated as a many-to-one mapping of internal categories to supervised labels, and when supervised labels are not available, updates to the label map correspond to updates to the internal labels. When no supervised labels are available in this scenario, the label map is populated as a one-to-one mapping of internal labels to supervised labels and is equivalent to running in the original unsupervised mode.

The selection of which vigilance formulation to use, however, is more nuanced; the original single-vigilance START formulation only has one hyperparameter to tune according to the application at hand, whereas DV-START and DDV-START have two. Furthermore, the use of dual-vigilance variants has a trade-off of variable cluster granularity versus computational and memory complexity; DDV-START, for example, is capable of capturing arbitrary cluster shapes, but this comes at the cost of the potential for prototype proliferation and the added computation necessary to compare global nodes. On the other hand, START is suited for capturing more globular clusters with fewer computations at the risk of excessive category proliferation when cluster densities vary. These considerations also apply to the supervised formulations of each variant, making the selection of which START variant to use dependent upon both the availability of supervised labels and the availability of *a priori* knowledge of the statistics of the dataset in question.

### 3.1.9. Comparison with Existing Methods

START is most directly comparable with Gram-ART for two important reasons: Gram-ART is the first and indeed only, prior to START, ART-based categorical data clustering algorithm, and the design of START uses Gram-ART as a basis with important modifications.
tions. Details on the design differences between START and Gram-ART can be seen in the Supplementary Materials section of this paper.

The related Cascade ARTMAP handles symbolic data rather than real-valued or binary input patterns, but it is designed to handle if-then rule-based knowledge datasets rather than the variable-length categorical data targeted by START [65].

## 4. Evaluation

START is evaluated here both on existing benchmark machine learning datasets with known labels (outlined in Section 4.2) and on a custom biomedical dataset (outlined in Sections 4.3 and Appendix A).

### 4.1. Software Implementation

The START algorithm and the experiments outlined in this paper are implemented in a version-archived software repository [66]. In this repository, the START algorithm is implemented in the Julia scientific programming language [67] and utilizes the Lerche.jl package for implementing parsers [62] and AdaptiveResonance.jl for ART post-processing and analysis tools [39]. Clustering result analysis was also performed on the CMT dataset using the Python SHAP library (detailed in Sections 4.5 and 5.3) [68]. Visualizations of the SHAP analysis and additional post-processing were performed with the Orange data mining toolbox [69] and the IBM SPSS toolbox.

All algorithms and tool dependencies are implemented in serial without parallel or GPU acceleration. Individual experiments involve parsing and clustering the dataset in question, and they are run on the scale of seconds with large vigilance parameter values and minutes with small vigilance parameter values when run with the single-thread performance of a desktop Ryzen 9 3950X CPU. This variation is a consequence of the variable number of categories instantiated, where small vigilance parameter values tend to over-partition the data into many categories and large vigilance parameter values tend to generalize the data as belonging to a small number of categories. These individual experiment iterations themselves were run in parallel on a university computing cluster for hyperparameter sweeps and for gathering performance statistics.

### 4.2. Benchmark Datasets

Purely categorical machine learning benchmark datasets are not as widespread and well-studied as real-valued benchmark datasets, and the START algorithm and its variants are not designed to handle real-valued data without modification. Therefore, START and its variants are evaluated on a combination of both real-valued clustering datasets and purely categorical datasets with caveats.

Gram-ART is originally verified upon a discretized version of the UCI Iris dataset, the UCI Mushroom dataset, and the UCI Unix User dataset [24,70-72]. For comparison, START is evaluated upon the following open-source machine learning benchmark datasets with existing labels: a set of real-valued clustering benchmark datasets [73,74], the categorical UCI Mushroom dataset [71], and a categorical lung cancer patient dataset [75]. Because benchmark datasets such as the Iris dataset's elements are real-valued, each feature is range-normalized and binned into a set of terminal symbols representing each bin.

Both the written procedures for accommodating real-valued benchmark datasets for evaluation and the results of all real-valued and categorical benchmark evaluations can be viewed in the Supplementary Material of this paper.

### 4.3. Charcot-Marie-Tooth Disease Dataset

To test the ability of START to cluster rows in a complex dataset with various multi-category fields of varying length, we created a test dataset based on Charcot-Marie-Tooth disease (CMT). CMT, also known as hereditary motor and sensory neuropathy, is one of the most common neurogenetic diseases, with a population prevalence of 1 in 2500 [76]. As a starting point, we began with 81 variants of CMT in the Online Mendelian Inheritance of
Man (OMIM) phylogenetic series. A known genetic mutation characterizes each variant. The protein associated with the mutation is known in all but three variants. For each CMT variant, we added a row to a flat file with the following columns: variant name, OMIM number, gene, gene location, chromosome, mode of inheritance, phenotype, protein name, UniProtKB number, protein location, biological process in which the protein participates, protein molecular function, protein length, and protein weight. External data sources were identified to populate the dataset (Table 6), including the Online Inheritance in Man (OMIM), the Human Phenotype Ontology (HPO), UniProtKB, and the Human Protein Reference Database (HPRD) [77–80]. The final dataset had 81 rows and 17 columns, as shown in (Table 6). Seven columns were multi-categorical. Gene number (OMIM), phenotype number (HPO), protein number (UniProtKB), and variant number (OMIM) were not used in the clustering.

Example production rules resulting from the interpretation of this dataset as statements sampled from a grammar can be found in Appendix A. The following clustering methodology and analysis was performed on this CMT dataset using the original START unsupervised variant.

<table><caption>**Table 6.** Table of features and their characteristics in CMT flat file. Protein numbers were from UniProtKB [79]. Variant and gene numbers were from OMIM [77]. The phenotype numbers were from HPO [1,81]. Since genes, proteins, and diseases have multiple names, the names were normalized to the standard form. Most of the features were categorical, and some were multi-categorical. The features were formatted as integers or strings of variable or fixed length.</caption><thead><tr><td>Feature</td><td>Type</td><td>Format</td><td>Length</td><td>Multi-Category</td></tr></thead><tbody><tr><td>variant name</td><td>categorical</td><td>string</td><td>variable</td><td>no</td></tr><tr><td>variant number</td><td>categorical</td><td>string</td><td>fixed</td><td>no</td></tr><tr><td>gene name</td><td>categorical</td><td>string</td><td>variable</td><td>no</td></tr><tr><td>gene number</td><td>categorical</td><td>integer</td><td>fixed</td><td>no</td></tr><tr><td>protein name</td><td>categorical</td><td>string</td><td>variable</td><td>no</td></tr><tr><td>protein number</td><td>categorical</td><td>string</td><td>fixed</td><td>no</td></tr><tr><td>protein length</td><td>numerical</td><td>integer</td><td>variable</td><td>no</td></tr><tr><td>protein weight</td><td>numerical</td><td>integer</td><td>variable</td><td>no</td></tr><tr><td>protein location</td><td>categorical</td><td>string</td><td>variable</td><td>yes</td></tr><tr><td>protein molecular function</td><td>categorical</td><td>string</td><td>variable</td><td>yes</td></tr><tr><td>protein biological process</td><td>categorical</td><td>string</td><td>variable</td><td>yes</td></tr><tr><td>protein class</td><td>categorical</td><td>string</td><td>variable</td><td>yes</td></tr><tr><td>mode of inheritance</td><td>categorical</td><td>string</td><td>variable</td><td>yes</td></tr><tr><td>phenotype</td><td>categorical</td><td>string</td><td>variable</td><td>yes</td></tr><tr><td>phenotype number</td><td>categorical</td><td>string</td><td>variable</td><td>yes</td></tr><tr><td>chromosome</td><td>categorical</td><td>string</td><td>variable</td><td>no</td></tr><tr><td>chromosome location</td><td>categorical</td><td>string</td><td>variable</td><td>no</td></tr><tr><td>chromosome location</td><td>categorical</td><td>string</td><td>variable</td><td>no</td></tr></tbody></table>

Each row of the flat file was interpreted as a statement of symbols corresponding to each column entry. In this manner, each statement was of variable length due to some rows missing entries while other entries contained more than one element. These statements of sequential symbols were then used to infer the grammar and production rules of the dataset; a statement could have one or more attributes (e.g., names of the columns containing data entries), which themselves could have one or more terminal symbols, to reflect how a
disease variant could be associated with multiple different phenotypes, biologic processes, etc. The resulting grammar production rules seeded a parser that was used to process each statement into a parse tree. These trees were then interpreted as START TreeNodes (Figure 1) for clustering according to the prototype instantiation, comparison, and update procedures of the START algorithm (Figure 2, Algorithm 1), clustering in a single pass and updating weights or instantiating new prototypes at each incremental sample presentation. A hyperparameter sweep of the vigilance parameter with statistics generated by shuffled presentation order was performed to determine the most meaningful vigilance parameter selection for subsequent cluster analysis (Section 5.1).

<figure><img src="image_80.png" /><figcaption>**Figure 2.** A set of figures demonstrating the evaluation and update of a START prototype on a new sample. (a (left)) demonstrates a START prototype as a rooted tree of ProtoNodes instantiated on the algebraic statement $x + y$ (a). ProtoNodes are labeled by a non-terminal symbol, and they contain a probability mass function (PMF) of the terminal symbols generated both by that non-terminal and by any descendant non-terminals, where $p(x,y)$ is shorthand for the PMF of the set of outcomes $S = \{x_1, x_2, \dots, x_n\}$ that gives $p(x_1, x_2, \dots, x_n) = P(X = x|x \in S)$. (b (center)) demonstrates the relation parse tree of a new algebraic statement $x + m \cdot n$. The rooted trees of the prototype and parsed statement are aligned and compared as a graph intersection at the non-terminal positions. The START match rule (Section 3.1.3) then determines the activation and match values of this graph intersection as a function of the PMFs at each non-terminal position and the terminal symbols at the leaf nodes of the sample, and the hypothetical prototype of (a) is selected from a pool of other candidate prototypes. (c (right)) demonstrates the prototype after update, accommodating the new non-terminal symbol positions of the sample and updating the PMFs at each non-terminal position according to the START weight update rule (Equations (3) and (4)).</figcaption></figure>

## 4.4. Cluster Feature Means and Heat Maps

After clustering by START, a cluster membership (between 1 and 9) was assigned to each row. Multi-categorical features (see Table 6) were flattened into individual features by one-hot encoding. Feature means for each cluster were calculated using the AGGREGATE procedure from SPSS (version 29.0, IBM). The features were visualized using heat maps from Orange 3.35 [69]. For the heat maps, raw feature means were used for the categorical variables, and normalized feature means (in the interval [0, 1]) were used for the numerical variables (see Table 6).
4.5. SHAP Values

SHAP summary values were calculated using the method of Lundberg et al. [68]. START cluster membership was added to the flattened feature array (see above). The cluster configuration was fitted to the HistGradientBoostingClassifier (scikit-learn). The shap TreeExplainer and the shap summary plot procedures were used to compute SHAP values and create the SHAP summary plot.

5. Results

5.1. *Selection of Cluster Configuration for the CMT Dataset*

The vigilance parameter ρ was varied between 0.0 and 1.0 in a Monte Carlo of shuffled
sample presentation order (Figure 3). To minimize the size of the largest cluster and mini-
mine the number of clusters with one member, ρ = 0.6 was selected, yielding nine clusters
(Figure 4).

<figure><img src="image_81.png" /><figcaption>**Figure 3.** Effect of vigilance parameter ρ on number of clusters. A Monte Carlo of shuffled sample presentation order was run to generate 1σ intervals of the results at each vigilance parameter value. As ρ was increased from 0.0 to 1.0, the maximum cluster size decreased, the number of clusters increased, and the number of singleton clusters increased. A value of ρ = 0.6 (yellow dashed line) was selected to yield 9 clusters with only two singleton clusters. Larger ρ values gave too many singleton clusters, and smaller ones put too many cases into one cluster.</figcaption></figure>

5.2. *Cluster Characterization by Feature Composition*

We used heat maps to visualize the features that characterized each cluster. The clusters differed in mode of inheritance, protein localization within the cell, protein participation in biological processes, protein length, molecular weight, motifs and domains in amino acid chains, phenotype, and protein molecular function (Figures 5–12). The heat maps were used to create a narrative summary of each cluster’s most important feature characteristics (Table 7).

5.3. *Identifying Features that Contributed the Most to Cluster Configuration*

We used SHAP [68] to find the features that drove the cluster configuration. The SHAP summary plot (Figure 12) showed that protein length, chromosome number (autosomes 1 – 22 and X and Y), mode of inheritance (autosomal recessive and autosomal dominant), protein localization in the cell (cytoplasm and plasma membrane), and phenotype (hypertonia, auditory and cognitive) contributed the most to cluster formation.
<figure><img src="image_82.png" /><figcaption>**Figure 4.** With $\rho = 0.6$, clustering by START yielded nine clusters from 81 variants of CMT. Each cluster is a different color on the heat map. Order of clusters on heat map is 7, 9, 8, 2, 1, 6, 5, 3, 4, with ordering by Euclidean distance between cluster centroids [69]. The largest cluster is 4 (dark green), with 53 members. Singleton clusters are 9 (white) and 6 (pea green). A shortened variant name is shown in the right margin. Dejerine–Sottas disease appears four times in the heat map because it is caused by four distinct mutations in the MPZ, PMP22, PRX, and EGR2 genes.</figcaption></figure>
<table><caption>**Table 7.** Summary of features that characterize CMT clusters. k is the cluster number and N is the count of members in each cluster. Phenotype Plus lists signs and symptoms in addition to weakness, atrophy, deformities, sensory loss, and hyporeflexia that characterize most cases of CMT. AD is autosomal dominant inheritance; AR is autosomal recessive; XLR is X-linked recessive. TM is the transmembrane protein domain. GNRF is the guanine nucleotide-releasing factor. Note that some of the characteristics identified by the SHAP analysis, including cognitive, hypertonia, auditory, plasma membrane, autosomal recessive, and autosomal dominant (Figure 12), recur in this summary table.</caption><thead><tr><th>k</th><th>N</th><th>Process</th><th>Function</th><th>Location</th><th>Domain</th><th>Inherit</th><th>Phenotype Plus</th></tr></thead><tbody><tr><td>1</td><td>6</td><td>apoptosis</td><td>hydrolase</td><td></td><td></td><td>AD</td><td>auditory, visual</td></tr><tr><td>2</td><td>3</td><td></td><td></td><td>cytoplasm</td><td></td><td>AD</td><td>hypertonia</td></tr><tr><td>3</td><td>7</td><td>protein synthesis</td><td>transferase</td><td></td><td></td><td>AD, AR</td><td></td></tr><tr><td>4</td><td>53</td><td></td><td></td><td>plasma membrane</td><td>TM</td><td>AD,AR</td><td></td></tr><tr><td>5</td><td>4</td><td></td><td></td><td>plasma membrane</td><td>TM</td><td>AD</td><td>cognitive, auditory</td></tr><tr><td>6</td><td>1</td><td>immunity transcription</td><td>transferase</td><td>plasma membrane</td><td></td><td>AD</td><td>cognitive, ataxia, seizure, hypertonia, speech, hyperreflexia</td></tr><tr><td>7</td><td>4</td><td>transcription</td><td>DNA binding</td><td>plasma membrane</td><td></td><td>AD, AR</td><td>cognitive, hypotonia</td></tr><tr><td></td><td></td><td></td><td>transferase</td><td></td><td></td><td></td><td></td></tr><tr><td>8</td><td>2</td><td>autophagy apoptosis</td><td>hydrolase GNRF</td><td>nucleus</td><td></td><td>AR</td><td>cognitive, auditory, hypertonia</td></tr><tr><td>9</td><td>1</td><td></td><td>transferase</td><td>mitochondrion</td><td>TM</td><td>XLR</td><td>cognitive, auditory</td></tr></tbody></table>

<figure><img src="image_83.png" /><figcaption>**Figure 5.** Heat map of molecular function for proteins in CMT clusters. Kinase function is associated with cluster 9, hydrolase function with clusters 1 and 8, DNA binding with cluster 7, activator function with cluster 7, and transferase function with cluster 9.</figcaption></figure>
<figure><img src="image_84.png" /><figcaption>**Figure 6.** Heat map of biological process for proteins by CMT cluster. Cluster 1 is apoptosis, cluster 8 is autophagy and apoptosis, cluster 3 is protein synthesis, cluster 6 is transcription and immunity, and cluster 7 is UBL protein conjugation and transcription.</figcaption></figure>

<figure><img src="image_85.png" /><figcaption>**Figure 7.** Heat map of protein locations by CMT cluster. Cluster 2 is cytoplasm, clusters 5, 6, and 7 are plasma membrane, cluster 8 is nucleus, and cluster 9 is mitochondrion.</figcaption></figure>
<figure><img src="image_86.png" /><figcaption>**Figure 8.** Heat map showing protein motifs and domains by CMT cluster. Motifs and domains are characteristics of configurations of the amino acid chains that make up proteins and are often associated with a specific function. Note the over-representation of the transmembrane (TM) domains in clusters 5, 6, and 9 (red arrow). The CC motif is found in most proteins except for cluster 7.</figcaption></figure>

<figure><img src="image_87.png" /><figcaption>**Figure 9.** Heat map of molecular weights and amino acid chain lengths for proteins for CMT clusters.</figcaption></figure>
<figure><img src="image_88.png" /><figcaption>**Figure 10.** Phenotype scores for each of the nine clusters for the 81 variants of CMT. Scores have been normalized to the interval [0, 1], where 1 indicates 100% and 0 indicates 0%. Note, as expected, that gait, atrophy, deformity, hyporeflexia, weakness, and sensory loss are common features in most cases (red bracket). Cluster 6 with one case and cluster 9 with one case are different because they manifest auditory and cognitive symptoms (cluster 9) or ataxia, cognitive, hyperreflexia, hypertonia, seizures, and speech symptoms (cluster 6). Cluster 6 is also of interest because it lacks weakness and atrophy, two of the core symptoms of CMT. Cluster 2 (3 cases) is also interesting because subjects have hypertonia. Cluster 4, with 53 cases, is the most common pattern and shows a typical phenotype of gait, atrophy, deformity, hyporeflexia, weakness, and sensory symptoms, which is characteristic of CMT.</figcaption></figure>

<figure><img src="image_89.png" /><figcaption>**Figure 11.** Modes of inheritance for the nine CMT clusters. Cluster 8 is largely autosomal recessive. Cluster 9 is X-linked recessive. Clusters 5, 6, and 7 are autosomal dominant inheritance.</figcaption></figure>
<figure><img src="image_90.png" /><figcaption>**Figure 12.** SHAP cluster summary plot for the 9 clusters derived from CMT dataset with $ρ = 0.6$. The SHAP plot shows which features contributed the most to the cluster configuration by cluster. Important features are protein length, chromosome, mode of inheritance (autosomal dominant and recessive), protein location (cytoplasm and plasma membrane), and certain phenotypes (auditory, cognitive, and hypertonia). The domain expert rated these features as highly biologically plausible. SHAP plots were created using the method of Lundberg et al. [68].</figcaption></figure>

# 6. Discussion

## 6.1. Feasibility of Clustering Multi-Categorical Biomedical Data with START

START demonstrates several important capabilities that make it particularly useful for the clustering of multi-categorical data. Firstly, it directly represents the categorical data without an intermediate encoding representation and all the problems introduced therein; categorical data by definition does not define distance metrics or fuzzy membership between categories and feature dimensions. The problem is circumvented here by the definition of prototype parse trees tracking the distributions of symbols from learned statements using the ART match and learning rules.

Secondly, it naturally compensates for data points with missing elements entries in its fields; rather than requiring a special encoding scheme for missing fields or removing data points altogether, START can represent missing fields as unused non-terminal positions when representing multi-categorical datasets as statements containing one or more attributes, which has the effect of penalizing the degree to which samples with missing features match existing prototypes while still accommodating prototypes of varying sizes.

Thirdly, and as a consequence of the previous point, START can handle symbolic data of varying length when interpreted as statements under a grammar; in fact, this paper demonstrates an analysis of multi-categorical datasets of depth 2 due to the nature of the CMT data available, but categorical datasets of arbitrary depth can be analyzed with START when treating categories as themselves non-terminal symbols with production rules mapping to other sets of categories. This can be interpreted as processing hierarchical symbolic databases where individual fields can themselves link to other symbolic database tables.
## 6.2. Biological Interest and Plausibility of Derived Clusters

When the START vigilance parameter was set to $\rho = 0.6$ (Figure 3), we obtained nine clusters (Table 7). Cluster 4, the largest, had 53 members, and clusters 6 and 9 were singleton clusters. The fact that cluster 4 is large is not surprising since most cases of CMT are similar and have similar core symptoms of weakness, sensory loss, hyporeflexia, orthopedic abnormalities, atrophy, and gait abnormalities in common [76]. Although it is usual to differentiate clinically between axonal forms (involving the neuron axon) and demyelinating forms (involving the myelin sheath of the axon) of CMT, it is not surprising that we did not find axonal and demyelinating clusters of CMT since we did not input electromyographic data into the clustering algorithm. The finding of small clusters of CMT variants with auditory, hypertonic, or cognitive phenotypes is interesting and plausible biologically and is consistent with clinical observations.

The clusters differed in inheritance (Table 7) in biologically plausible ways and consistent with clinical practice. Since each variant of CMT was due to a gene mutation and since each gene coded for a unique protein, protein weight, protein length, protein configuration (motifs and domains), protein involvement in biological processes, protein molecular function, and protein locations could be examined for each CMT cluster and compared to the observed phenotype (Figures 5–12, and Table 7). Although these observations are intriguing, they do not offer a precise path to connect protein function, location, and process to the neurological phenotype in CMT. As an example of explainable AI [82], the SHAP plots in Figure 12 provide biologically plausible explanations for how START relied on certain features to form clusters.

## 6.3. Limitations

One limitation of this work is that START is used to cluster a small biomedical dataset without ground truth labeling. Although the diagnosis of each row (CMT disease variant) is known, cluster membership for the dataset as a whole is unknown. As a result, this work cannot contain an analysis of either truth in cluster membership and structure or performance of START with respect to such a ground truth. The reader is referred to the Supplementary Materials of this article for a study of the START algorithm on various other machine learning benchmark datasets, including fully symbolic and real-valued datasets including details of the procedure necessary to adapt the START algorithm to real-valued data.

Another limitation of this work is that all available features are used as inputs to the START clustering algorithm. A separate study is warranted to study how withholding some of the features as meta-features would allow potentially interesting cluster composition analyses.

Furthermore, the results of clustering and analyzing the CMT dataset with START has limited generalizability to other diseases datasets. It is seldom expected of clustering models or indeed machine learning models as a whole to generalize to distinct domains from which they were trained, outside of research areas such as lifelong machine learning that tackle this specific issue [83]. Nevertheless, some transferability should be expected to related datasets of the class of neurogenetic diseases that CMT belongs to, but this transferability is limited in two ways: by the format of the selected data and the START methodology itself. Biomedical data are themselves notoriously multifaceted, and the creation of generalizing models in this field depends on the narrowed problem statement and subsequent dataset at hand. For example, this work considers the clustering of data with categorical membership in specifically phenotype-gene relationships; with respect to the OMIM elements, each entry in the CMT dataset subsumes a variable number of clinical studies that generate the datapoints, discarding the qualitative aspects of the various individuals studied and the other clinical features that are inconsistently included between entries. This is a consequence not of the quality of the OMIM database but rather of the project of aggregating vastly disparate clinical data.

In addition to this, an ontological level of granularity is necessarily selected by the researcher when working with any clinical features; for example, a full hierarchy of pain
may be studied as symptoms of a disease with respect to an individual patient from the location, duration/periodicity, and subjective intensity according to some pain scale, etc. Taking the location of pain as an example, some studies describe the location of pain with a varying degree of specificity; in the hierarchy of location on the human body, should one hypothetical study citing simply pain and another citing leg pain be considered the same by virtue of belonging on the same hierarchy or different by virtue of being two different points on that hierarchy? If neither is true, then what is the distance metric along this hierarchy until two points are considered distinct? Even clinical definitions of the term pain itself vary and are subject to debate [84]. As a result, simplifying assumptions must be made for any learned model of clinical data, such as the ontological subsumption of the phenotype of pain in this study, and the interpretation of clustering results and its usefulness for the treatment of a disease is tied to the selection of ontological granularity.

Lastly, the methodology of the START algorithm successfully tackles the clustering of the resulting variable-size categorical datasets, but it does so at the level of the symbols themselves without forming a feature-transformed intermediate representation of them. Additional similar phenotype-gene datasets could be clustered with an existing model trained on this CMT dataset, but relationships between them would only be found if they shared exactly the same symbols, such as gene locations and disease phenotypes. Completely disparate datasets with no shared symbols may indeed be clustered by the same START model after appropriate modifications to its parser's grammar, but this would be functionally equivalent to clustering with two separate START models; this benefit of START is also to its detriment, as if there are no shared symbols between data points, they are treated as having no shared features for computing similarities.

## 6.4. *Future Work*

This paper demonstrates that START can work with data from a knowledge graph or ontology when flattened into a rectangular file, even with missing or nested elements. Alternatively, knowledge graphs and ontologies can be converted into triplets as subject-object-predicate triplets, which retains the underlying graph architecture. In the future, we plan to determine whether START can successfully cluster these triplets derived from knowledge graphs into meaningful clusters.

Additional future work includes an evaluation of START clustering on large multi-categorical datasets with a known ground truth cluster membership and further experiments on datasets in which some features are withheld from input and retained as meta-features for post-clustering analysis.

## 7. Conclusions

This work introduces the START algorithm for the clustering of symbolic data with arbitrary length statements. This work also introduces dual-vigilance and distributed dual-vigilance variants of START along with a supervised modification for each. Because START is designed for symbolic datasets, it is naturally suited for the clustering of both categorical and multi-categorical datasets where each sample feature may realize multiple values. This multi-categorical clustering capability is demonstrated on a curated biomedical dataset of Charcot-Marie-Tooth disease variants and their disease-gene attributes, such as disease phenotypes and protein molecular functions.

For a dataset such as the CMT dataset used here, START is useful as a tool for studying structural relationships between disease variants for guiding future clinical research or in the formulation of useful models of those diseases via the hierarchies, clusters, and outliers identified during the clustering process; for example, distinct gene locations with shared phenotypes between two groups of disease variants may illustrate to a researcher some other shared molecular mechanism for future research, which could guide drug research in a data-driven manner.
**Supplementary Materials:** The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/info15030125/s1, Listing S1: Discretized Iris dataset grammar illustrating the symbolic binning procedure of real-valued data used to evaluate START and Gram-ART; Table S1: Hyperparameters for each START variant during supervised train/test evaluation. Table S2: Performance statistics of the supervised implementations of each START variant derived in the original paper on a set of benchmark real-valued and categorical machine learning datasets.

**Author Contributions:** Conceptualization, D.B.H., S.P. and D.C.W.II; methodology, S.P. and D.B.H.; software, S.P. and D.B.H.; validation, S.P.; formal analysis, S.P. and D.B.H.; investigation, S.P. and D.B.H.; resources, D.B.H. and D.C.W.II; data curation, D.B.H.; writing—original draft preparation, S.P. and D.B.H.; writing—review and editing, S.P., D.B.H., T.O.-A., M.A.B., E.J.T., W.E.M., M.S. and D.C.W.II; visualization, S.P. and D.B.H.; supervision, D.C.W.II; project administration, D.C.W.II and E.J.T.; funding acquisition, S.P. and D.C.W.II. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is funded by the Department of Energy's Kansas City National Security Campus, operated by Honeywell Federal Manufacturing & Technologies, LLC, under contract number DE-NA0002839.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Disease-protein datasets are gathered from the openly available Online Mendelian Inheritance in Man (OMIM) knowledge base [85]. All data, code, and documentation related to the experiments outlined in this paper are contained in a version-archived repository [66].

**Conflicts of Interest:** The authors declare no conflicts of interest.

## Abbreviations

The following abbreviations are used in this manuscript:

<table><tr><td>ART</td><td>Adaptive resonance theory</td></tr><tr><td>BNF</td><td>Backus-Naur form</td></tr><tr><td>CFG</td><td>Context-free grammar</td></tr><tr><td>CMT</td><td>Charcot-Marie-Tooth disease</td></tr><tr><td>DDVFA</td><td>Distributed dual-vigilance FuzzyART</td></tr><tr><td>DDV-START</td><td>Distributed dual-vigilance symbolic tree adaptive resonance theory</td></tr><tr><td>DVFA</td><td>Dual-vigilance FuzzyART</td></tr><tr><td>DV-START</td><td>Dual-vigilance symbolic tree adaptive resonance theory</td></tr><tr><td>EBNF</td><td>Extended Backus-Naur form</td></tr><tr><td>F1</td><td>ART Feature input layer (field 1)</td></tr><tr><td>F2</td><td>ART Category representation layer (field 2)</td></tr><tr><td>HAC</td><td>Hierarchical agglomerative clustering</td></tr><tr><td>L2</td><td>Lifelong learning</td></tr><tr><td>ML</td><td>Machine learning</td></tr><tr><td>PMF</td><td>Probability mass function</td></tr><tr><td>START</td><td>Symbolic tree ART</td></tr><tr><td>WTA</td><td>Winner-take-all</td></tr></table>

## Appendix A. Charcot-Marie-Tooth Dataset Grammar

An analysis of the Charcot-Marie-Tooth (CMT) dataset *a posteriori* demonstrates the process used in this article for interpreting tabular multi-categorical data as statements sampled from a context-free grammar that can be expressed as a set of EBNF production rules, which can be seen in Grammar Listing 1. Gene-protein disease data are gathered for 81 variants of CMT with categorical attributes (Table 6). Categories such as phenotype are subsumed where hierarchically relevant to reduce attribute feature dimensionality (e.g., variants of "pain" symptomology are subsumed to one feature belonging to the "phenotype" attribute). This process results in a 81-row flat-file dataset of features with
multi-categorical attributes represented as piped entries for each disease variant, including attributes with missing entries.

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Review

# From Information to Knowledge: A Role for Knowledge Networks in Decision Making and Action Selection

Jagmeet S. Kanwal

Department of Neurology, Georgetown University Medical Center, Washington, DC 20057, USA;
kanwalj@georgetown.edu

**Simple Summary:** This perspective article examines the differences between memory, information, and knowledge. It is proposed that the creation of knowledge is not simply the extraction of information or sequencing and storage of memories, but its contextualization that offers a point of advantage for survival within a decision-making framework; outside these contexts there are no useful memories and therefore no relevant knowledge. A constellation of neural networks spread across multiple brain regions must work together to grow knowledge over time. Knowledge must be stored in such a way that it can be accessed by multiple tokens, or handles, at any time. The emergence of knowledge networks underlies the evolution of complex brains that can predict outcomes, induce imagination and expand knowledge. Attention and sleep play important roles in creating and protecting knowledge.

**Abstract:** The brain receives information via sensory inputs through the peripheral nervous system and stores a small subset as memories within the central nervous system. Short-term, working memory is present in the hippocampus whereas long-term memories are distributed within neural networks throughout the brain. Elegant studies on the mechanisms for memory storage and the neuroeconomic formulation of human decision making have been recognized with Nobel Prizes in Physiology or Medicine and in Economics, respectively. There is a wide gap, however, in our understanding of how memories of disparate bits of information translate into "knowledge", and the neural mechanisms by which knowledge is used to make decisions. I propose that the conceptualization of a "knowledge network" for the creation, storage and recall of knowledge is critical to start bridging this gap. Knowledge creation involves value-driven contextualization of memories through cross-validation via certainty-seeking behaviors, including rumination or reflection. Knowledge recall, like memory, may occur via oscillatory activity that dynamically links multiple networks. These networks may show correlated activity and interactivity despite their presence within widely separated regions of the nervous system, including the brainstem, spinal cord and gut. The hippocampal-amygdala complex together with the entorhinal and prefrontal cortices are likely components of multiple knowledge networks since they participate in the contextual recall of memories and action selection. Sleep and reflection processes and attentional mechanisms mediated by the habenula are expected to play a key role in knowledge creation and consolidation. Unlike a straightforward test of memory, determining the loci and mechanisms for the storage and recall of knowledge requires the implementation of a naturalistic decision-making paradigm. By formalizing a neuroscientific concept of knowledge networks, we can experimentally test their functionality by recording large-scale neural activity during decision making in awake, naturally behaving animals. These types of studies are difficult but important also for advancing knowledge-driven as opposed to big data-driven models of artificial intelligence. A knowledge network-driven understanding of brain function may have practical implications in other spheres, such as education and the treatment of mental disorders.

**Keywords:** neural network; learning and memory; attention; adaptive action; knowledge acquisition; education; artificial intelligence; decision making; goal-directed behavior; sleep

**Citation:** Kanwal, J.S. From Information to Knowledge: A Role for Knowledge Networks in Decision Making and Action Selection. *Information* **2024**, *15*, 487. https://doi.org/10.3390/info15080487

Academic Editors: Birgitta Dresp-Langley and Luiz Pessoa

Received: 7 June 2024
Revised: 5 August 2024
Accepted: 9 August 2024
Published: 15 August 2024

<figure><img src="image_91.png" /></figure>

**Copyright:** © 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
# 1. Introduction

## 1.1. Motivation

Some humans are very good at memorizing facts and others have uncanny imag-
inative abilities that can be translated into beautiful works of art. Still others are very
knowledgeable, i.e., they have a deep understanding in a particular information domain
that can lead to insights on solving nontrivial problems. The term "knowledge" can be
construed to represent a working model of a particular aspect of the real world. Acquiring
this knowledge takes time and is built on processes involving cross-validation and justi-
fication in an attempt to arrive at the "truth". It is not merely a memory gained through
sensory experience or through repeated association of a stimulus with a reward or punish-
ment. Though typically applied to human domains of information, knowledge is equally
important, however, for any organism to gain insights from their social and environmental
interactions to make informed decisions.

In this perspective article, I extend neuroscientific findings on memory acquisition, storage and recall to propose a neurological framework within which to define knowledge as a wide-area neural network. Such a network can be referred to as a knowledge network (KN). KNs participate in creating and storing knowledge and have several key properties. These properties include interconnectivity between multiple brain regions, such as fronto-cortical cognitive and limbic emotive structures. KNs are considered to be highly dynamic networks, portions of which link up transiently depending upon the context, an explicit query or a physiological state or drive. KNs also need to transiently link up with attentional brain circuits to extract current information either from an incoming sensory stream or from memory. KNs eventually direct their output to decision-making and action-selection networks. Catastrophic events, such as "9-11" for humans in the US, can transiently crash KNs, causing confusion and the need to create a new real-world model.

Speech perception, reading and language can be considered as types of development-
tally entrained, sapient KNs that acquire cross-validation via explicit instruction and are
designed to rapidly gain new knowledge [1-6]. Acquiring a navigational map equates to
developing a KN of one's surroundings within which to make quick decisions about escape
from predators, foraging and to find mates. A navigational KN undergoes cross-validation
and certainty or truth-seeking via repeated, direct interactions with the environment and
has been partially elucidated for navigation in rodents [7,8]. KNs are important for decision
making and survival. A navigational KN must be reliable for making split-second decisions
for survival. This becomes obvious when observing a rabbit swerving in seemingly random
directions to dodge a wild cat or a dog, and still able to find its rabbit hole while running at
top speed on a rough terrain.

Below, I first briefly describe the philosophical idea and usage of the term "knowledge"
and show how it relates to well-defined concepts of information and memory at the
phenomenological level. I then elaborate on some of the neuroscientific attributes of
knowledge and propose the building blocks and mechanisms for knowledge acquisition,
storage and recall. Furthermore, I conceptualize the formation of KNs in an equation form
and show where they fit in our current understanding of the evolution of the nervous
system. Finally, building on recent studies on neural mechanisms for attention and sleep
for memory formation and consolidation, I clarify their role in acquiring knowledge that
gets embedded within KNs.

## 1.2. From Information to Knowledge

Ever since the formulation of information theory [9,10], a lot has been written on infor-
mation processing from informational and neuroscientific perspectives [10–14]. Knowledge,
however, remains less well defined from both a neurological and a computational perspec-
tive. Neither are there any mathematical definitions of knowledge or a knowledge theory.
Lynn et al. [15,16] have recently provided a formulation of the perceived information
(cross-entropy) within a communication system as being the sum of the large amount of
information produced (having high entropy) and the efficiency of the observer's representa-
tion of that information (low divergence from expectations). The brain's ability to store and retrieve information, i.e., learning, memory and recall, is of fundamental importance for the elaboration of goal-directed behaviors that frequently involve decision making [17]. This requires querying the brain and extracting bits of relevant information simultaneously from various loci to direct and question decision-making and goal-setting tasks. We may term this contextualized information set as "knowledge". Neural networks that are required to perform these functions have not been clearly identified either in the neurobiological or computational domains. In this brief review, I provide a nonformal overview of the differences between information, memory and knowledge from a neurocognitive perspective. I propose a conceptual framework for describing KNs as a dynamic grouping of realistic neural networks different from those needed to extract information and create a memory.

A theory for information processing was first formulated to quantify the information load that the human mind can encode, store and retrieve over the short term, emphasizing capacity limits without external aids or techniques [18]. Its formulation was based on experimental data generated by others from the presentation of auditory, taste and visual stimuli to human subjects. Further mathematical elaboration was focused on the coding and decoding of signals for transmission and reception [19]. Work on cognitive and neural mechanisms for long-term storage capacity and the contextual linkage with existing information required animal experimentation using sophisticated techniques that were not available at that time. One way to do so is to integrate relevant information bits as synaptic connections whose strength or "weight" within local and global networks can be continuously modified. The configuration and patterns of activity generated by this network, when queried by inputs (in the form of either real or virtual stimuli), function as a model of the world in which an organism exists. This model can be equated to what we term as "knowledge" and the neural networks that participate in creating, consolidating and modifying knowledge can be referred to as knowledge networks or KNs. KNs incorporate the core memory networks described by others [20,21], but also include cross-validation (for truth seeking), organizational (sequencing and timing), contextualizing and attentional networks [22]. Parts of these nonmemory networks remain less well-defined anatomically but must come online via their interconnections during query (recall) and reflection. In this context, it is noteworthy that even simple memories need to be reactivated before they can be modified or erased, emphasizing the inbuilt labile nature of a KN [23–25].

1.2.1. Feature Extraction

Behavioral and cognitive psychology can go only so far in providing mechanistic insights into the way brain networks accomplish the task of developing and implementing a knowledge-based model of the world for decision making and action selection. For that, we need to take a step back and first understand how information is created from the sensory environment. A necessary step in the process of information creation via sensory perception is feature extraction. Naturalistic signals or stimuli in the environment are complex and typically have multiple elements and parameters by which they can be specified. From these, the brain uses parallel-hierarchical processing to extract features of varying levels of complexity [26]. Feature extraction is perhaps one of the most universal and useful aspects of sensory processing regardless of stimulus modality. Some of these features contain information that contributes to stimulus discrimination and object recognition [26–31]. A defining feature of an object does not solely depend on the physical properties of an object but also is the outcome of neural processing governed by the receptive fields of higher-order neurons [32–35]. A large amount of research on sensory processing and cognition has focused on feature identity and extraction and this is also a central theme of deep learning in artificial neural networks [13,36,37]. It should be noted that, at a more abstract level, cognition also involves terms like thinking, knowing, understanding, reasoning, judging and problem-solving. This review does not attempt to tackle all of these concepts from a neuroscientific perspective, though in some cases they may be intimately connected to and rely on knowledge embedded within KNs. An understanding of how neural networks
learn, recognize and problem-solve has led to major advances in artificial intelligence (AI) models for face, handwriting and voice recognition—problems that were seemingly impossible to solve by conventional programming and engineering approaches [38–40].

Restating the argument made above more succinctly, a physical entity is intrinsically recognized as a feature depending on its relevance for a particular function or goal, such as for food and mate recognition or selection for reproduction [41,42]. A feature may consist of simpler elements that can be easily synthesized and experimentally tested, such as orientation columns in the visual cortex [34,43]. From a behavioral perspective, simple elements within a feature are referred to as information-bearing elements or IBEs [44,45]. They contain information that is critical for computing a feature and its variants. Non-IBEs within a signal can usually vary without affecting the perception of a particular feature but may carry other types of information. Furthermore, an IBE can be quantitatively defined based on any number of physical parameters. Those parameters that are behaviorally meaningful or communicate information to an organism (low surprise), and by design are extracted and encoded by the relevant sensory system are referred to as information-bearing parameters or IBPs [44,45]. IBEs and IBPs were first defined to provide a framework for identifying and studying the acoustic elements in the pulse-echo combinations in the vocalizations produced by bats for echolocation [45].

In the neuroethological studies of bats, empirical studies yielded putative IBEs that could be meaningfully attributed to the substructures or components of the acoustic structures of the naturalistic pulse-echo signals used for echolocation. These readily identifiable components, such as constant-frequency (CF) tones with harmonics, and frequency-modulated (FM) sweeps, were shown to play a critical role in the computation of important acoustic features, such as relative target velocity and distance, and wingbeat frequency and the amplitude of insects. Relative target velocity, wingbeat frequency and amplitude are IBEs for a bat tracking a target, such as an insect [46]. In our example, the IBPs would be the values of relative target velocity as a function of the Doppler-shifted frequency (60.6 to 62.3 kHz in mustached bats), with the wingbeat frequency resulting in an amplitude modulation rate of 50 to 500 Hz in the range of 48.2 ± 10.7 dB SPL, corresponding to particular insects that are a good (nontoxic) food source [47–49]. Key elements or putative IBEs and IBPs are also present in the sounds produced for social communication in bats and other species [50–52]. For vision, IBEs may correspond to the combination of a particular shape, color and/or texture of an object, and for the olfactory system the combination of a mixture of odorants that can be used to identify a flower [53–56]. IBEs and IBPs can be determined by performing behavioral studies and estimated by determining receptive fields of neurons.

Neurophysiological and computational studies show that the IBEs are represented in the responses of neurons in many brain regions [57–61]. Cortical neurons and those present within hidden layers of multilayered, artificial neural networks exhibit high activation levels to either identifiable or abstract, low-dimensional representations corresponding to the principal components of the physical measures of an object [62,63]. They constitute IBE-like representations within real neurons and networks. Studies on the visual system of nonhuman primates identified a representation of IBE-like basis functions in the response of cortical neurons [64]. These basis functions do not necessarily represent easily identifiable elements within a naturalistic stimulus. Rather, they are statistical formulations of the real elements that are extracted as useful information. Features represent the building blocks of objects, and objects make up a scene [65,66] (Figure 1). Clearly, association between features is critical for computations that guide decision making and flight behavior, such as insect-tracking and capture, and the recognition of a food source. However, the memory of a feature that may help to identify a target or one that could be useful in identifying a social call does not in itself constitute knowledge.
<figure><img src="image_92.png" /></figure>

**Figure 1.** The parallel-hierarchical processing of sensory inputs leads to the extraction of information from features and objects defined by the information-bearing elements or IBEs. The physical proximity and/or temporal coherence of extracted features create objects, and that of objects within a scene creates perceptual associations or memories. Most of this processing is accomplished via ascending lemniscal pathways in the brain. Thalamocortical loops facilitate egocentric selection by neurons tuned to the parameters of an incoming stimulus, and signal amplification occurs via descending projections (see arrows) [67].

1.2.2. What Is Knowledge?

To be meaningful, information must exist in the form of knowledge. The study of
knowledge borders the domains of educational science, cognitive science and philoso-
phy [68,69]. Knowledge is mainly acquired from sensory inputs to the brain, though even
at the very first step of sensory transduction of various forms of energy into electric or
neural energy, knowledge acts like a filter, biasing what reaches the neural networks deep
within the brain. In other words, the creation of new knowledge and learning is itself
influenced by existing knowledge. Thus, knowledge is much more individualized than
information. Two people may be given identical information, but the knowledge it creates
in each of their brains may be quite different.

It is useful to also consider the meaning of knowledge from a philosophical perspective since a concept of knowledge has existed from a long time before a conception of mind and brain. Wikipedia defines human knowledge as “an awareness of facts, a familiarity with individuals and situations, or a practical skill” [70]. This type of definition of knowledge exists more within the domain of philosophy rather than neuroscience or information science. From an epistemological perspective, propositional or declarative knowledge has been proposed to have three essential features or components. These features are individually necessary and jointly sufficient for achieving a state of declarative knowledge [71]. Knowledge of facts, also called propositional knowledge, has been characterized as true belief that is distinct from opinion or guesswork by virtue of justification. Belief involves various levels of certainty about an observation or thought [72]. A low level of certainty of an event or a relationship may be equated to imagination by others, whereas an elevated level of certainty indicates a strong belief in a perception. If a belief happens to accurately represent reality, then it is taken to be true. The process of testing is important for obtaining evidence to justify that a belief is true. The process of justification protects against a lucky guess becoming knowledge. This suggests that KNs must involve neuronal processes that test for the level of certainty and establish certainty or belief in a perception go through a process of justifying that belief and check for the presence of evidence in memory circuits or find a way to seek that evidence. This makes knowledge, unlike memories, more robust and less prone to forgetting. In neural terms, this means that knowledge about an issue may have a fractured, distributed representation so that knowledge can be evoked in many different situations. In other words, there are multiple entry points to activating KNs or their subsets and their activation can impact decision making in many related circumstances. How exactly certainty is tested and justified at the neural level remains unclear (however, see [73]).

A KN's representation of knowledge, i.e., a species-specific model of the world, may guide decision making in many circumstances, e.g., the strategy an animal adopts to find food or to interact with conspecifics to facilitate various aspects of social behavior. Neural networks that represent knowledge are more difficult to define and usually require simultaneous recordings of the activity of neurons in multiple brain regions in awake-behaving animals exhibiting naturalistic behaviors. For example, neurobehavioral studies of a laboratory rat, mouse or a primate species maintained or placed within a small cage
or apparatus where the animal has limited degrees of freedom of movement and decision
making are not well-suited to explore their KNs. A naturally enriched environment offers a
much greater opportunity for an organism to create and use KNs for action selection. Thus,
the use of neuroethological approaches to study animals under natural or semi-natural
environments has a greater potential to identify KNs and the neural mechanisms underlying
knowledge processing [74–76]. With a few exceptions, such as social vocalizations in the
house mouse and bats, and spatial navigation studies in rats and bats, neuroethological
and neuro-ecological studies are easier to perform in invertebrate species. Advanced
invertebrate brains likely contain rudimentary KNs, supporting limited reflection and
justification processes that are difficult to extract. The technology to examine neural network
activity at a large scale in vertebrate species has recently become available, however, and its
application to study KNs via appropriately designed experiments should be feasible [77,78].

Knowledge of the environment in which an organism functions is critical for accom-
plishing multiple activities that are essential for survival. These include territorial, social
and foraging forays on a regular basis. Figure 2 captures the periodic, behavior-driven
incorporation of epistemological aspects (e.g., reinforcement, trust and belief) and mech-
anisms (associative memories) that define knowledge in general, within a navigational
framework. During navigation, a cognitive route map within networks (e.g., using place
and grid cells) is automatically created by an animal's behavior, and brain mechanisms
(oscillations, excitation, inhibition, facilitation and spike-time-dependent plasticity) embed
context-driven network properties within a navigational KN. Place cells fire maximally at
a particular location based on multimodal sensory cues, whereas grid cells compute the
vector from the starting to goal location. A cognitive map, typically created by strategic,
free-exploration and probabilistic associations, is a neural model of the external spatial
world which represents the distances and directions between locations.

Training animals on stereotypic actions, such as nose-pokes, within a constrained
environment is important for examining the physiology and pharmacology of specific local
circuits involved in either aversive or reward memories. Chronic neural recordings during
free exploration behavior, such as for navigation, offer a more open-ended approach to
finding key neuronal and network properties [79,80]. These types of studies led to the
discovery of place neurons within the hippocampus and of grid cells within the entorhinal
cortex (EC), important findings that were recognized with a Nobel prize [81]. In the last
decade, research on networks and neurons underlying navigation continues to be a source
of groundbreaking new findings, such as of ring and toroidal structures, border cells, stripe
cells and head-direction cells in rats and free-flying bats [82,83] (Figure 3A). We still need to
understand how navigational knowledge is generated, contextualized, and recalled within
different contexts.

Within a KN concept, the understanding of navigation can be further advanced by
adopting techniques of chronic recordings of positional information and neural activity
from multiple brain regions while animals engage in navigation within different contexts
(Figure 3B). Activity-dependent mapping, e.g., using c-FOS and ZENK as markers of early
gene expression, has been used in songbirds to identify neurons for song discrimination and
production [84–87]. The use of two-photon imaging and neural activity markers, such as
CaMPARI, in freely behaving animals can also facilitate the identification of sensorimotor
networks that are co-activated, and show how they engage a core KN to accomplish
different phases of navigation [88]. These approaches can also explain how contextual
queries are initiated and how they might reconfigure particular networks for multiple
functions, e.g., shifting the excitatory-inhibitory balance in core networks and engaging
new networks. This is important to know because the brain evolves as a whole, and
while reductionist approaches are useful for identifying gene and receptor function and
pharmacology, they also take us away from achieving a holistic understanding of the brain.
<figure><img src="image_93.png" /><figcaption>**Figure 2.** A schematic showing behavior-driven flow of information to extract object- and scene-specific cues for creating navigational knowledge. Associative memories are typically created via valence-driven idiothetic cues, or show statistically significant coincidence of occurrence in the form of allothetic cues. KNs are expected to play an important role in top-down modulation for sensory selection by sustaining attention at various levels of sensory processing and may be modified by reward- and aversion-driven associative memory mechanisms.</figcaption></figure>

In common terms, knowledge is often understood as an awareness of facts (declarative knowledge) or as practical skills (procedural knowledge) and may also mean familiarity with objects or situations. The same terms have been used to categorize memories. Episodic memories require the activation of action-related networks that are associated with an activity or an event. For a long time, it was debated whether memories are stored within specific areas in the brain or distributed as information bits throughout the brain. Current data support the latter scenario [89,90]. If so, then, as in a computer, one also needs a directory of sorts to point to the address of each information bit. This means that to access a bit of information, one first needs to look up the registered address and then utilize the information in memory. This could mean first accessing another memory to locate the directory to in turn locate a bit of pertinent information. Theoretically, this sequencing of addresses and memories could lock up the neural access within an infinite loop. While loops are important components of information processing, they do not represent an efficient way to store information. Also, it is energetically costly to not only “remember” a certain bit of information but then also go to a location to recall where that information is stored. This further complicates recall by the number of instances or bits of information that need to be retrieved. In real life, one can equate this to finding a user manual for an electronic device, such as a television set. If one wants to obtain some information on the location or function of a port or connector, then it will be best if one can either look directly at that device or look up relevant information in a user manual sitting next to the television set rather than first locating and retrieving it from a different location. Moreover, this process involves
the need for additional memory storage that can greatly increase the cost of obtaining information by a factor proportional to the number of bits accessed, slowing down recall. It may also trigger an infinite loop making the system crash. Alternately, directory information may be stored at a central location and automatically channeled to the source of the query. How this is accomplished within the brain without the involvement of an outside agent, as in the case of an individual using their brain to access the memory and physically locomoting to retrieve it, is still unclear.

<figure><img src="image_94.png" /><figcaption>**Figure 3.** (A). Networks and cell types discovered for navigation in rodents, bats and primates, including humans. Lines connect to the brain structures where they are located and arrows show the directionality in which networks and one cell type carries the receptive field of another. (B). Examples of research questions that can be addressed using a KN-driven approach to provide a foundational understanding of the neuroscience of knowledge. These approaches can help us to define the physiological properties, configuration, extent and plasticity of KNs. MEC: medial entorhinal cortex.</figcaption></figure>

1.2.3. Memory vs. Knowledge

Memory remains a vital component of knowledge, and considerable progress has been made towards understanding memory formation, consolidation and recall. Short-term memories are first transiently stored within the electrophysiologic activity of neurons and networks [91]. Some memories are then consolidated within neural networks throughout the brain via the modification of dendritic spines and synapses that are only just beginning to be delineated [92–95]. Over the long term, the relevant networks somehow become activated to access a specific memory each time a decision is to be made. It is as if the brain has the capacity to “dial-in” to different networks within a knowledge domain to extract information.

As a real-world example, to communicate with others to share and/or obtain information, one may access a digital phonebook and click on a number to be dialed. These actions require energy so that there is a cost–benefit ratio not only for movement but also of network usage within the brain. Many mental disorders, such as depression and schizophrenia, can be considered, respectively, as a high perceived cost of action and decision- making going array. These disorders can be triggered by chronic anxiety and stressful states that release hormones in the bloodstream via the activation of the hypothalamus and other endocrine organs, such as the adrenal glands [96]. Chronic release of these hormones
can result in changes in the wiring of the brain [97–100]. It can also disrupt metabolic processes and deteriorate telomeres or the endcaps of chromosomes [101–103]. Knowledge not only lowers the basal state of energetic cost by providing access to previously obtained information, i.e., via experience, but also allows an organism to make decisions and take action in the interest of lowering energy consumption over the long term and increasing the probability of survival. But, what is knowledge in contrast to information and memory, and how is it created and stored in the brain? Semantic or conceptual knowledge has been defined and studied only loosely from a neurobiological perspective [69,104], and studied in a very limited way from a computational perspective [17]. Computationally, structured knowledge is thought to reside in long-term memory as a distributed activity pattern and accessed via partial retrieving cues. According to a proposed model, sequences of memories can be potentially stored as single attractors within recurrent neural networks [17,105]. To gain a more tangible sense of KNs, let us now consider brain structures that are well known to play a role in memory mechanisms.

1.3. *Brain Structures Involved in Memory Formation and Recall*

Memories, such as those used for face recognition, may be purely experiential [106], or
they may be created via association with either reward or punishment [107]. Experimentally,
memory formation can be studied in animals only via either reward or fear associations, i.e.,
within an emotive context. While an emotive context is a powerful mechanism of memory
formation given the direct relationship of emotions with survival, not all memories are
formed in this way. Associations can happen within any two temporally bound sensory
inputs, provided those associations occur at a statistically significant level over other
chance associations.

The hippocampus, a seahorse-shaped brain structure, stores information transiently until it is contextualized and stabilized, and then channeled to a neural network at a specific location for long term storage. In this scenario, memory may be accessed by reactivating the memory in a context-driven manner. In other words, context may activate a specific section of the “phonebook” to automatically “dial-in” to a particular network to access a specific memory and/or to transform it, given a new set of contextualized inputs. In the phonebook analogy, the same brain structure, the hippocampus, cannot however function both for long-term storage and as a pointer to the correct location of a phone number entry since the information represented by each phone number may change with time. We are, however, concerned less here with details of information storage and more about what that information represents and what it accomplishes for an organism’s survival.

The hippocampi can be divided into a dorsal and a ventral subregion [108]. Whereas the ventral hippocampus receives information from the basal nucleus of the amygdala (BA), the dorsal hippocampus sends information to the basolateral nucleus of the amygdala, which also receives sensory information from the cortex and thalamus [109–111]. The reciprocal connectivity of the ventral hippocampus with the amygdala plays a central role in creating contextual memories, as summarized in Figure 4. The hippocampus binds together item and context information related to a study event. It receives information from the amygdala, an almond-shaped brain structure consisting of multiple nuclei, as well as the perirhinal and parahippocampal, including entorhinal, cortices. These structures transfer information to the hippocampus, respectively, about an event from the emotive value stream, and the “what” and “where” streams. The BA also projects to both the medial and lateral regions of the entorhinal cortex (EC). The BA receives projections back from the CA1 region and the subiculum as well as from the lateral EC (LEC). The LA sends excitatory output to the inferior colliculus (IC) that inhibits both the medial and lateral nuclei in the central amygdala (CeA). The LA also sends excitatory output to the BA, which in turn can excite both medial and lateral nuclei in the CeA. The basolateral amygdala (BLA) has reciprocal connectivity with the prefrontal cortex [112]. Theta and gamma oscillatory activity within the hippocampus plays an important role in creating and retrieving navigational knowledge. Theta-gamma phase coupling encodes navigation-
related functions, and spike timing within theta oscillations is important for memory consolidation and recall (Figure 3). With respect to navigation, e.g., during free exploration in a familiar environment (Figure 2), and during memory recall, CA1 pyramidal neurons respond most effectively to CA3 input [113]. Eventually, all information is transmitted to the central nucleus of the amygdala for action-related decisions.

<figure><img src="image_95.png" /><figcaption>**Figure 4.** Schematic showing the localization of oscillatory activity (top-left) in the ventral hippocampus within a navigational context (see Figure 2) and for memory functions, and its complex interconnectivity with the amygdala for context-dependent associative learning. The lateral amygdala (LH) receives processed sensory inputs from the cortex and thalamus. These inputs contain information about features and objects in the sensory landscape. The subiculum, presubiculum and parasubiculum are extensions of the CA1, CA2 and CA3 regions of the hippocampus and they all receive outputs from the basal amygdala (BA). The basolateral amygdala (BLA) has reciprocal connections with the prefrontal cortex. Projections to the hypothalamus trigger hormonal changes, and those to the periaqueductal gray in the brainstem control respiration, heart rate and vocalization.</figcaption></figure>

Finally, the prefrontal cortex (PFC), particularly the dorsolateral PFC, is heavily involved in working memory functions. It is also clearly important for maintaining and updating information, as well as for emotional regulation and motor control. The PFC, in general, plays a central role in orchestrating complex cognitive processes, such as executive functions, which include planning, decision making, problem solving, and controlling attention [114,115]. Therefore, KNs are expected to be intimately connected with and continuously interact with the PFC, though knowledge itself is likely distributed in wide-area networks extending to higher-order sensory cortices as well as diencephalic structures, such as the habenula.

Episodic memory for navigational and other tasks involves the conversion of sensory inputs into working memory. A working memory activity pattern in the hippocampus must be transferred to long-term memory stores, such as in the neocortex, to process the next set of events or sensory inputs. By definition, navigation is a sequential event. Therefore, it is not surprising that the hippocampus plays an important role in both functions and has been studied as such. Building on the adaptive resonance theory (ART) family of neural models, Grossberg and colleagues have advanced hypotheses to explain both the
working memory and navigational functions of the hippocampus [116,117]. Their model stresses the importance of mass action-induced theta rhythms [118] in both mapping and extracting the information of an organism's spatial environment for navigation. As indicated earlier (see also Figure 3A), time- and distance-encoding cells, together with head-direction cells, project to grid cells in the entorhinal cortex (EC), which, in turn, project to place cells in the hippocampus [7,116]. In this scheme, the theta period represents a temporal metric for sequence learning [119] and also allows activity in widespread hippocampal and neocortical networks to be temporally coordinated [116]. The idea is that theta activity results in either gamma or beta oscillations, depending on match (resonance) and mismatch (reset) between expected vs. actual inputs. Altogether, the KN for navigation in mammals contains idio- and allothetic sensory, working memory, long-term memory, emotive-limbic, and action-selection modules. These reside in interconnected anatomical structures, namely, the thalamus, sensory and parietal cortex, hippocampus, subiculum and EC, amygdala, as well as the prefrontal cortex and subthalamic nucleus for action selection and the motor cortex/basal ganglia for motor output—for details see Figure 1 in Bermudez-Contreras et al. [120]. The retrosplenial cortex is a key structure that receives both head-direction and allocentric information. Most recently, Rolls and colleagues have expanded their quantitative theory of hippocampal function for short-term memory storage and recall without invoking mass action in the form of oscillations [121,122]. They include the orbitofrontal cortex as a source of reward-related input that is used to bind multimodal information within the hippocampal circuitry and invoke “concept” and “spatial view” cells within the hippocampus to explain goal-directed navigational behavior [121,123].

As with mirror neurons in Broca's area [124], it is becoming clear that the initial terminology developed to identify neurons and networks that conduct a specific function can usually explain a general class of functions. Thus, a part of the navigational network can also contribute to other cognitive processes based on associative thought [121]. This stresses the multifunctional nature of cortical, and possibly subcortical, neurons and networks [125–129]. Recent findings that expand the interconnectivity of a working memory module to its application to a task-directed navigation network are a prime example of the need to establish a viable and reliable KN where cross-validation can occur on a daily basis as an organism roams its environment under different environmental conditions and physiological states. Similar cross-validation and certainty-seeking processes can also occur via reflection-driven fronto-cortical activity.

## 2. Proposition

### 2.1. *Knowledge Equation*

The brain is essentially a complex and highly plastic network of neurons. Therefore, we presume that knowledge is stored as a state of synaptic interconnectivity within neural networks, partially as what we term as a memory of some information. To be meaningful (for survival), new information must be contextualized, linked up with existing information and tested for consistency or coherence before being stored as knowledge. In philosophical terms, knowledge of “facts” is often defined as true belief that is distinct from opinion or guesswork by virtue of justification. The process of justification is what delineates knowledge from a piece of factual information or a memory of it. The generation of knowledge involves the summation of cross-validated information over time and potentially across multiple timescales.

In this section, I propose a mathematical formulation of knowledge and of KNs. The motivation to do so is two-fold. First, it should allow those working in the domain of AI to incorporate a concept of KNs and develop new processes that go beyond using brute force, big data approaches to train artificial neural networks. Once established, the idea of KNs will allow systems and robots to learn from their interactions with the environment as humans and other animals do. Second, a mathematical conceptualization of KNs together with enabling technologies, in turn, can stimulate a more rigorous and comprehensive understanding of human cognition as well as predict the effects of neurological disorders,
aiding in the development of diagnostic tools and treatments. I hope this will enable the simulation of knowledge-driven brain networks, such as for a better understanding of navigation, language learning and education in general (e.g., incorporating the role of emotional, motivational and movement networks to generate trust and belief). This will allow researchers to test hypotheses and explore the effects of different variables included in the mathematical formulation. Additional details of the mathematical extensions for these applications are beyond the scope of this review, but I hope will be picked up by others working in their respective fields.

First, let us try to formulate, in rigorous terms, the concept of knowledge residing within KNs. For knowledge to remain relevant, forgetting is as important as incorporating additional information. This is in fact critical for planning and efficient decision making. The state of knowledge at any time therefore can be given by a knowledge equation, where the information is updated by integrating new with existing information, and outdated or irrelevant information is deleted via forgetting to keep knowledge viable. Later, we will discuss how and when the forgetting happens in neurobiological terms.

To mathematize the idea that knowledge is a result of integrating new incoming information while forgetting old information, we can model knowledge as a dynamic system that changes with time. Some variables relevant to such a model are as follows:

1. $K(t)$: the cumulative knowledge at time "t";

2. $G(t)$: The rate of incoming new information at time "t". $G(t)$ is modeled as a function of time, depending on how information is received. For instance, it could be constant, exponentially growing or influenced by other factors like attention or exposure;

3. $L(t)$: The rate of loss or forgetting old information at time "t". This can be transient, triggered by distraction or change permanently via neuronal or synaptic degradation, as in Alzheimer's disease;

4. $\alpha$: the integration rate constant, which determines how efficiently new information is integrated into the current knowledge state;

5. $\beta$: the forgetting rate constant, which determines how quickly old information is forgotten.

$\alpha$ and $\beta$ are constants that quantify the efficiency of integration and the rate of loss of information via forgetting, respectively. In terms of memory, these can be determined either empirically from learning curves or theoretically based on the context.

The change in the state of knowledge over time, $dK(t)/dt$, can be expressed as a first-order linear differential equation, where

$$ \frac{dK(t)}{dt} = \alpha G(t) - \beta K(t) \quad (1) $$

where $\alpha G(t)$ represents the gain in information or contribution of new information to knowledge, and, within the context of memory, $\beta K(t)$ represents the loss of information due to forgetting.

If we assume $G(t)$ for memory networks, we have to set up the initial condition where $M(0) = M_0$, and $K(0)$ is the initial knowledge at time $t = 0$. Then, we arrive at a form of the equation for memory:

$$ M(t) = e^{-\beta t}(\alpha \int_{0}^{t} I(s)e^{\beta s} ds + M_0) \quad (\text{Memory equation}) $$

where the integral term, $\int_{0}^{t} I(s)e^{\beta s} ds + K_0$, represents the cumulative effect of incoming new information over time, adjusted for the forgetting rate. "s" represents a process reducing the contribution of memory modules' contribution weight. It should be noted, however, that knowledge is less prone to forgetting, and this equation applies strictly to the memory components of knowledge.
Assuming, for simplicity, that $G(t)$ is a constant $G_0$, we can find the explicit form of $K(t)$ by solving the differential equation

$$ \frac{dK(t)}{dt} = \alpha G_0 - \beta K(t) \qquad (2) $$

The general solution, using an integrating factor, $\mu(t)$, where $\mu(t) = e^{-\beta t}$ (an exponential nature of information decay over time, assuming $G(t)$ is not constant), is

$$ K(t) = \alpha G_0 / \beta + (K_0 - \alpha G_0 / \beta) \dots (\text{Knowledge equation}) $$

* The term $\alpha G_0 / \beta$ represents the steady-state knowledge level when the rate of integrating new information and the rate of forgetting are balanced.

* The term $(K_0 - \alpha G_0 / \beta)$ represents the transient behavior of knowledge over time, showing how it approaches the steady-state level. The rate at which $K(t)$ approaches the steady-state is governed by $\beta$.

Overall, the Knowledge Equation captures the dynamic nature of knowledge accumulation and decay over time. It provides a mathematical framework for understanding the dynamics of knowledge as a function of new information integration and forgetting. This model can be adapted or extended based on specific contexts or additional complexities in the real-world scenarios being modeled. The variables and constants used are intuitive and can be adjusted based on empirical data or specific scenarios to reflect different learning and forgetting processes.

## 2.2. Knowledge Networks

Within a neural network framework, it is presumed that knowledge is embedded within a network that, when activated, will produce a set of outputs different from what it might in its nascent or initial state. A network carries information by its activity and the properties of the neurons within the network, but the knowledge it imparts depends on the context within which it is queried. The same network can produce different sets of outputs depending on how it is queried, that is, how other inputs activate this network. Thus, a generic information network can be a storehouse of multiple knowledge sets. This conception of a knowledge network is supported by the multifunctional nature of many neurons and networks [127–130].

Locally re-afferent and looping circuits are much more likely to be a component of KNs that store category-specific object knowledge [131,132]. An example of this type of network is shown in Figure 5A. That means that KNs will likely have neurons that show either tonic firing or bursting as the information loops between different networks as a representation of knowledge. This indeed appears to be the case when recording single-unit activity from neurons in the frontal cortex in response to complex, naturalistic sounds [60,62,133,134]. In contrast, neural responses to the same sounds from the primary auditory cortex are phasic and time-locked to stimulus onset. Also, because of their dynamic nature, the representation of knowledge within neural networks requires an activity component that is difficult to define a priori and is best studied using naturalistic stimuli [135]. These properties are not yet incorporated within artificially trained, deep learning networks, but in time hopefully AI will utilize an integration between multiple networks, each of which is trained for a specific task but relies on inputs or knowledge from other networks for justification. The knowledge could be related to ethics and other such constraints that can be independently updated. Presumably, the next generation of artificially intelligent agents will rely on knowledge-based models rather than generative language ones that are driven largely by statistical features. KNs have been alluded to previously from the viewpoint of social networks. Within this context, “KNs” are collections of individuals and teams who come together across organizational, spatial and disciplinary boundaries to invent and share a body of knowledge [136]. From an educational perspective, a network that provides knowledge to an organism is more representative of what we think of as “understanding”.
Therefore, KNs need to be continuously updated, are multifunctional and are more highly distributed than an information-extracting or a specific-memory network.

<figure><img src="image_96.png" /><figcaption>**Figure 5.** Knowledge network construction and distinction. (A) A generic multilayered network for signal extraction or associative learning, including recurrence (looping back arrows on green circles) that can be either excitatory or inhibitory, and feedforward (arrows) and feedback (filled circles) connections. The green, blue and orange circles belong to hidden layers of the network. (B) A series of modular networks showing the bottom-up flow of information to extract IBEs, features and objects from the sensory landscape (see Figure 1). (C) A putative knowledge network consisting of multiple distributed memory modules, predictive and attentional, and multimodal cross-validating networks as the basic components. This type of network contains multimodal representations, and has long-distance connections with inputs that can modify as well as query the network via sensory and cephalic triggers. Information is expected to flow bidirectionally (gray arrows) between these</figcaption></figure>
networks for decision making and action selection. The entire network can be triggered by either query or contextual signals. Transient coupling between networks (dashed arrows) can occur via oscillatory activity (see text for details). Physical dissociation via synaptic degradation or temporal unbinding over time is also possible via the de-correlation of neural activity. Prediction mismatch triggers large mismatch negativity-evoked potentials, leading to error signals and potential re-learning that may result in a reconfiguration of the knowledge network.

KNs can be considered as both valence-driven and referential. A simple example of a contextualized KN involves circuits within both the hippocampus and the amygdala. For example, if one walks into a dark alley, the hippocampus provides spatial information, whereas the amygdala provides valence information from either past experiences or from “instructional” inputs. This “knowledge” then guides one’s decision about a choice to enter the alley, or once there to leave it as soon as possible. If this knowledge is based on experience, then the hippocampus plays a role in embedding this in memory within the entorhinal cortex where valence appears to be represented within spatial coordinates. Figure 5B shows neural activity flowing through a sequence of generic information networks where each network holds a specific memory or bit of information. A KN would constitute a cluster of smaller networks that are typically accessible via a query or contextual input (Figure 5C). Intuition is a form of intrinsic knowledge that is spontaneously activated, and is less under the control of conscious inquiry and rational analysis. The components of intuitive KNs are expected to be present within the enteric nervous system and subcortical brain structures, such as the reticular formation.

Neural networks filter, process, code and decode signals as information either in the form of sensory, recurrent or feedback signals [137–139]. KNs, as formulated here, do not necessarily perform any of the low-level functions, but they can modulate, gate and facilitate activity within neural networks, including memory modules. KNs store the information as knowledge that can later prove useful to the organism. Hence, unless genetically encoded, KNs are created over time, taking into account the consequences of prior action and information processing. KNs tend to be distributed globally and need to be dynamically interconnected, especially when a decision needs to be made and/or an action taken. Whole brain networks have previously been constructed using graph theory [140,141]. Thus, KNs and the neurons within them can be thought of as being multifunctional. This is not a requirement for purely sensory neurons, though they may also process multiple types of signals. Information can reside within static networks. Knowledge, however, requires information to be stored in contextualized networks (Figures 2 and 5C). Since the context can change with every instance, knowledge cannot be represented by a static network.

## 2.3. Mathematical Formulation of KNs

As with knowledge, the state of a KN must also evolve with time. One can consider a KN to be a sum of multiple transient networks that are updated by new external (allothetic) and internal (idiothetic) sensory inputs and states. This may be represented symbolically as

$$KN_t = NN1_t + NN2_{t+1} + NN3_{t+2} + \dots NNx_{t+x}. \dots (Network equation)$$

where KN = knowledge network, NN = refers to a neural network and t = time at which the KN and NNs are accessed; NN’s can be memory, cross-validation, context or organizational networks that are transiently linked during knowledge creation and access.

To mathematize the concept of KNs rigorously, we need to model the dynamics of these networks, their interactions and how they contribute to the overall knowledge state [142]. First, we make the following assumptions:

1. The activity of neural networks, including memory modules, can be described by continuous functions of time.

2. The connection weights between neural networks and memory modules are constant over time.
3. The contribution weights of neural networks and memory modules to the knowledge network are dynamic and vary with time.

The following components, including variables listed, are an integral part of model formulation:

1. Neural Networks and Modules:

* $N_j(t)$: the activity level of the i-th NN at time t;

* $M_j(t)$: the activity level of the j-th memory module at time t;

* $w_{ij}$: the connection weight between the i-th NN, and the j-th memory module.

2. Knowledge Networks (KNs):

* $K(t)$: total knowledge at time t;

* $\alpha_i(t)$: the weight of the i-th NN's contribution to the KN at time t;

* $\beta_j(t)$: the weight of the j-th memory module's contribution to the KN at time t.

3. Dynamics and Interactions:

* $G(t)$: the rate of gain of new information at time t;

* $F(t)$: the rate of forgetting old information at time t.

Furthermore, a mathematical model of KN has the following properties:

1. Neural Network Dynamics: The activity level of each NN, $Ni(t)$, represents the dynamic activity levels of different neural circuits that contribute to knowledge. This can be influenced by incoming information and interaction with memory modules.

2. Memory Module Dynamics: The activity level of each memory module $Mj(t)$ represents the activity levels of memory storage systems that interact with neural networks, which can be influenced by the activity of a neural network.

3. Connection Weights, ($w_{ij}$), represent the strength of interaction between neural networks and memory modules.

4. Contribution Weights Dynamics: the weights $\alpha_i(t)$ and $\beta_j(t)$ represent the dynamic importance of each neural network and memory module to the KN.

5. Total Knowledge Dynamics: The total knowledge at time t, after adjusting weights, is a function of the contributions from neural networks and memory modules. The final solution for total knowledge within a KN at time t, can be modeled as

$$K(t) = \sum_{i} \alpha_{ij}(t) Ni(t) + \sum_{j} \beta_{j}(t) M_{j}(t) \quad (\text{KN knowledge equation})$$

A model incorporating the above features captures the dynamic nature of knowledge networks, integrating new information, and the forgetting process, with contributions from both neural networks and memory modules.

Using partial differential equations (PDEs) can make the model more realistic by capturing the spatial and temporal dynamics of KNs, reflecting how knowledge is dynamically structured and represented in the brain [143]. Specifically, sparse connectivity contributes efficiency, robustness and flexibility to the KN model [144]. Efficiency reduces the computational and energetic load on the brain by minimizing unnecessary connections. Robustness enhances the brain's ability to isolate damage or dysfunction, as fewer connections mean that problems in one area are less likely to propagate widely. Flexibility allows the brain to adapt and reorganize more easily in response to learning and new experiences, as specific pathways can be strengthened or weakened without affecting the entire network. The mathematization of a PDE model is beyond the scope of this review, but PDE equations allow one to model how knowledge, neural activity and memory activity evolve not only over time but also across different regions of the brain. Hence, as noted earlier, to test these types of models, it is necessary to record neural activity over time and across different regions of the brain in actively behaving animals.

## 2.4. Evolution of Knowledge Networks

The identification and quantification of naturalistic, unimodal stimuli and binary motor behaviors have played an important role in advancing neuroethological approaches
to understanding neural organization and mechanisms. Although such approaches will continue to be of benefit, given the present-day technologies and those that will be available in the near-future, it is timely to consider neural organization within more complex, integrative multimodal frameworks. It is time to address the fundamental integrative unit of brain organization and potentially its evolution. This unit almost always has to be multimodal because organisms function and evolve within a multimodal environment even though sometimes one sensory system may play a dominant role, e.g., the auditory system in bats and dolphins, and the olfactory system in rodents. Dogs have excellent olfactory and auditory capabilities. Even within a single sensory modality, neurons can be multifunctional, switching their role depending on the context, such as for echolocation vs. communication in bats [129,145]. Similarly, memory networks may be transformed with context via neuromodulators, such as dopamine and oxytocin, as happens for pair bonding vs. maternal bonding in prairie voles [146]. The conceptualization of KNs offers a more robust, naturalistic and adaptive way that may govern not only the organization of brain networks but also its evolutions as such. A KN-based model emphasizes the acquisition and processing of information to gain knowledge as a real-world model and the primary and urgently needed goal for survival of a newborn. A KN must consist of and engage with multiple and multimodal networks for decision making and action selection. As the brain matures, specific memories may be lost or become inaccessible having served their function of creating knowledge and this knowledge is both essential and sufficient for an organism's survival in its adopted environment and ecological niche. Thus, one way to think about the evolution of the brain is to consider that it was gradually configured over millennia towards a dynamic, knowledge-directed system to enable information storage and processing over increasing timeframes.

KNs incorporate distant memories and the present context for building predictive models of the social and physical environment. In humans, knowledge can sustain motivation and goal-directed action over a timeframe of several years. A long-term goal stored within a separate network can periodically query KNs and act as a planning and “decision center” for action selection as needed. A query can be triggered by either a sensory cue in the environment (a physical change or a conspecific interaction) or an internally generated, imaginative signal. It may also be triggered by the physiological state of the body, such as sleep, arousal and hunger, driving the organism to retrieve information based on knowledge about the location of a food source and activate a behavioral algorithm that will bring that organism to its food source [147–149]. Thus, a KN may connect with components of sensory, attention, goal-setting/planning and motor networks that work together for decision making and action selection (Figures 2 and 6).

Foraging strategies and many other adaptive behaviors depend on the complexity and size of the information-processing and storage system that an organism is endowed with. Single-celled organisms and simple multicellular organisms, such as sponges and coelenterates, respond to stimuli by orienting and translational movements that are both directed and random [150–152]. These organisms require only a sensor and a motor element to react reflexively in a preprogrammed manner with a built-in, limited range of tolerance for environmental change (Figure 7). They do not require a knowledge network. Alternatively, in more complex organisms, particularly vertebrates as well as some invertebrate groups, such as cephalopods with a well-organized brain, a goal-driven process may involve querying many networks that can extract knowledge before triggering a sequential decision set that ultimately gets converted into a sequence of actions [153].
<figure><img src="image_97.png" /><figcaption>**Figure 6.** A line diagram depicting the relationship between sensory inputs, motor control and behavior within a sense of self (idiothetic) in both physical (time and space) and physiological terms. The reciprocal interconnectivity of the various functions emphasizes the importance of and role of various neural states and mechanisms that constitute components of KNs. The behavior or motor output is typically triggered in response to a query generated by signals from the internal or the external environment. Physiological states, such as hunger, drive an organism to attend to contextual or allothetic cues in its environment and activate central motor programs in the brain to trigger behavioral actions. These in turn become the source of sensory inputs that allow an organism to monitor its movement and location within its relevant region of space. Some of this feedback leads to contextual learning through motivational cues related to reward and fear. Direct idiothetic feedback (diagonal arrow) can modulate coordinated and directed motor activity.</figcaption></figure>

In vertebrates, action sequences are embedded within brain structures, such as the basal ganglia and motor (including pre-motor) cortex for bodily movement, and the amygdala for autonomic functions and physiological activity of the internal organs [154,155]. Thus, neural networks are proposed as having gradually evolved from simple, non-overlapping reactive nerve nets, dedicated to specific functions of neuroids in sponges, to hydra-like organisms, more complex, bilateral brains in worms and to a multilobed, distributed central nervous system in insects and cephalopods. These advancements led to the emergence of primitive to advanced KNs present within proactive and imaginative brains, respectively (Figure 7). Some knowledge elements become embedded within neural networks through genetic encoding, especially in invertebrates and lower vertebrates [156,157]. Recent findings on the organization of the lamprey brain support this possibility [158,159]. Eventually, increases in complexity and connectivity led to the development of imaginative brains where scenarios could be simply imagined with or without follow-up with action selection [160]. Here, KNs became critical for providing top-down information via predictive computations [16,161–164]. Predictive networks enable “fact/error-checking” and the justification for decision making and action selection without reference to incoming sensory information [165].
<figure><img src="image_98.png" /></figure>

**Figure 7.** Flow-chart showing the evolution of organisms with or without "brains" (shaded boxes) within the animal kingdom. Early forms of life (over 540 million years ago) did not require knowledge to adapt to their environment. Their action selection was directly determined by sensors and effectors operating at either molecular, cellular or simple reflex levels. Later, as organisms and their brains became more complex, just-in-time information availability was supplanted by KNs that allowed a brain to be proactive in terms of evaluating its environment, goal-setting and action selection. Some molluscan species with large brains, such as the octopus, and protochordates are expected to have rudimentary KNs, whereas the neurally advanced species, such as cetaceans and primates, have the capacity of knowledge abstraction and manipulation. Humans are assumed to have the most advanced KNs capable of symbolic representation through cultural evolution and education.

**3. Discussion**

I have proposed here that a paradigm shift in our thinking is required to approach neurobiological research from a more comprehensive perspective. Despite numerous studies on learning and memory, we have barely elucidated the networks in which knowledge resides. Navigational systems represent an important frontier in this regard (see Figure 2). Below I discuss what we know about mechanisms for knowledge acquisition and, capitalizing on studies of memory consolidation, the important role of attention and sleep in the conversion of memory stores and contextual information into knowledge. As discussed later, an interdisciplinary approach promises to enhance the effectiveness of educational strategies and the sophistication of AI technologies, leading to better outcomes in both fields.
3.1. *Knowledge Acquisition and Action Selection*

Let us first examine in a little more detail the steps of information acquisition and processing within putative KNs. Exteroceptors, such as in eyes and ears, are designed to feed information about the external environment, whereas interoceptors, such as proprioceptors and carbon dioxide sensors, monitor and transmit information about the internal state of the body. These are a part of the peripheral division of the nervous system. These receptors send inputs to various parts of the brain where the inputs are consolidated and evaluated based upon experience and stored as “memory”. Because sensory systems are adaptive, to some extent sensors at the level of the peripheral nervous system, and more so central sensory processing networks, are primarily designed to detect changes in the quantity and quality of the sensory profile of the external and internal environment over different time-windows. For example, the vestibular system in humans is activated by changes in linear and angular acceleration over an approximately 7 s time window [166]. Hence, when constructing rollercoasters to stimulate the vestibular system, a turn needs to be introduced for every 7 s of travel to satisfy the stimulus expectations of the customers. Let us now tackle the question of how knowledge is gained from these sensations.

Some neurons themselves function as sensors by having receptors on specialized segments of their cell membrane, such as olfactory receptors present on olfactory cilia that are specialized forms of dendrites. Similarly, intraspinal cord mechanosensory neurons within the spinal cord can detect axial bending of the body [167,168]. Neurons within the central nervous system can also be regarded as sensors that monitor the body's physiological, physical and emotive states by monitoring the activity of other neurons via receptors for neurotransmitters as well as for the hormonal and neuromodulator milieu within the extracellular space. Thus, changes in both externally driven and internally sensed neural activity eventually determine the behavior of an organism. Neuronal plasticity occurs at the synaptic level via both active and silent synapses that can control learning during critical periods [169,170]. Synaptic modification is less likely to occur during the elicitation of the reflexive type of responses, where a referential knowledge or prediction network is not invoked. During information gain, however, each new active synapse has the potential to significantly bias the output of a KN.

An ultimate goal of knowledge creation is to minimize perceived complexity and maximize predictability (minimizing entropy within an information-theoretic framework) without sacrificing accuracy. To achieve this, KNs may tolerate a certain amount of fuzziness, maximizing their applicability to real world scenarios that are rarely identical [171,172]. Neuro-fuzzy systems appear to be good at pattern recognition for solving real-world problems [173]. Feature extraction, cross-validation, synchronization and consensus finding are important components of such systems that have been implemented for knowledge discovery via artificial neural nets [174,175]. Consensus maps have been proposed to encode naturalistic smells via the odotopic mapping of odorants in the olfactory bulb and forebrain [176–179], and social calls via combination-sensitivity within the auditory cortex [50,61,180]. How cross-validation and synchronization is automatically accomplished within real neural networks is less clear. Oscillations in neural activity within different brain regions, such as the amygdala and hippocampus, may provide a mechanism for cross-validation, typically involving cross-modal processing, via temporal coherence, phase-amplitude coupling and other such mechanisms [137,181–186]. Oscillations in the form of traveling waves may also provide a mechanism for spreading information (memory encoding and recall) within the brain to strengthen multimodal consensus and the binding of common features within an information scene or landscape [187–190]. In this regard, the state of network dynamics and Hebbian plasticity appear to be essential for the optimization of network topology within real and artificial neural nets [191,192].

Action selection can occur at multiple timescales and involve different amounts of
knowledge for action decisions. For fast reflexive activity, information is locally processed
and triggers a quick (few milliseconds) reflexive action that can be monosynaptic. Other
actions can take hundreds of milliseconds to minutes and even days to be planned and