Emergent Composite Clustering and Gauge-Inspired Adaptive Connectivity in Agent-Based Neural Networks

Emergent Composite Clustering and Gauge-Inspired Adaptive Connectivity in Agent-Based Neural Networks Andrei T. Patrascu1 1FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We introduce a novel neural modeling framework inspired by composite fermion theories and gauge symmetries from particle physics, explicitly constructed within agent-based neuron models. In this framework, neurons dynamically form strongly interconnected clusters—termed ”composite neurons”—whose collective states profoundly influence individual neuron parameters through a mechanism analogous to partial compositeness. Adaptive connections among neurons are governed by emergent gauge fields, ensuring local invariance and robust functional dynamics. We demonstrate numerically that this composite-gauge approach naturally leads to coherent neuron clustering, adaptive connectivity, robust signal propagation, and enhanced resilience to perturbations. Our results reveal clear emergent correlations and synchronized behaviors within neuronal clusters, validating the practical significance of this novel theoretical approach for computational neuroscience and robust neural network design. INTRODUCTION Gauge theories have profoundly shaped our understanding of particle physics, providing a unifying mathematical framework to describe fundamental interactions. A particularly intriguing development within this context is the concept of composite fermions and partial compositeness [9]. In composite models, elementary fermions such as quarks and leptons obtain their masses through mixing with strongly bound composite states at higher energy scales. Mathematically, this mixing can be elegantly described by coupling elementary fields fwith composite operators OΨ through interactions of the form: Lmix =λf¯ fOΨ+ h.c.,(1) where λfdetermines the strength of this coupling. Diagonalizing the resulting mass matrices naturally leads to mass hierarchies among fermions, thereby elegantly solving long-standing puzzles related to particle mass distributions and hierarchies. Inspired by these profound physical insights, similar principles can be powerfully applied to neuroscience and machine learning. Neurons in biological or artificial neural networks are traditionally modeled as individual processing units with fixed intrinsic parameters, analogous to elementary fermions. However, evidence increasingly suggests that neuron properties are dynamically influenced by their interactions within local neuronal groups or clusters, mirroring the partial compositeness concept. In this work, we propose a neural network paradigm directly inspired by gauge symmetry and composite fermion theories. Here, clusters of neurons dynamically form strong interconnections, acting as composite neurons whose collective states significantly modulate individual neuron parameters and behaviors. Mathematically, this can be represented by effective neuron parameters such as excitability thresholds (θeff i) and adaptive connectivity patterns (Aij), governed by local gauge invariance principles: DtVi=∂tVi−iqiAijVj,(2) where the gauge fields Aij ensure robustness and adaptability of neural computations, and the gauge charges qi represent individual neuron sensitivities to local network modulations. This gauge-composite neural model exhibits compelling advantages in neuroscience and machine learning contexts. For instance, it naturally fosters robust, flexible, and adaptive neuronal computations, reducing vulnerability to perturbations and adversarial effects in artificial neural networks [10]. Moreover, the formation of composite neuronal clusters significantly enhances interpretability, modularity, and generalization, aligning neural computations closer to observed biological neural dynamics [11]. Through detailed numerical simulations and agent-based modeling, we demonstrate explicitly how gauge-inspired adaptive connections and composite neuron clusters emerge, revealing stable collective neural states and rich dynamic behaviors that substantially outperform traditional neural network models. The integration of gauge theory concepts and composite fermion analogies into neural networks thus offers a powerful, unifying theoretical and practical framework, promising significant advancements in our understanding of biological neural systems and the design of robust, adaptive machine learning architectures. 2 EMERGENCE OF GAUGE FIELDS, CHARGES, AND PARAMETERS IN NEURAL NETWORK DYNAMICS Gauge theories, originally formulated to describe fundamental particle interactions, introduce fields, charges, and symmetry parameters that ensure invariance under local transformations [12, 13]. Extending this powerful theoretical structure into neural networks offers novel insights into neural dynamics and connectivity. In neural network systems, the state of a neuron iat time tis typically described by membrane potential Vi(t). Consider a local gauge transformation of neuronal states defined by: Vi(t)→V0 i(t) = eiαi(t)Vi(t),(3) where αi(t) is the neuron-specific gauge parameter representing local recalibration of neuronal reference states [14, 15]. Invariance of the neural dynamics under these transformations necessitates the introduction of gauge fields Aij (t), which adaptively modulate connections between neurons. A gauge-invariant derivative, or covariant derivative, is introduced to preserve local invariance: DtVi(t) = ∂tVi(t)−iX j Aij(t)Vj(t).(4) This covariant derivative ensures the neuron dynamics remain consistent under local gauge parameter adjustments. The gauge fields Aij(t) thus emerge naturally as adaptive connectivity parameters, dynamically ensuring robustness and consistency of network computations [16, 17]. Neurons can further be assigned neuron-specific gauge charges qi, characterizing their sensitivity or coupling strength to the gauge fields. With explicit charges, the covariant derivative takes the more general form: DtVi(t) = ∂tVi(t)−iqiX j Aij(t)Vj(t).(5) These charges reflect intrinsic neuronal properties such as excitability or synaptic sensitivity, which determine how individual neurons respond dynamically to adaptive connectivity adjustments. The gauge fields themselves evolve according to neural adaptation principles, for instance, through gradient-based learning rules: dAij dt =−η∂ ∂Aij |DtVi−(∂t−iqiAij)Vj|2,(6) where ηis a learning rate controlling adaptive connectivity strength [18, 19]. The gauge field strength Fij, defined as: Fij(t) = ∂iAj−∂jAi−i[Ai, Aj],(7) describes nontrivial network connectivity structures, such as loops and synchronized oscillations, crucial for functional brain dynamics and computational power [20–22]. Introducing these gauge principles into neural networks significantly enhances robustness, flexibility, and functional modularity, naturally aligning network dynamics closer to biological neural systems [23, 24]. The adaptive connectivity described by gauge fields allows neural networks to dynamically adjust their computational structures, enhancing resilience against perturbations and noise, a crucial feature for both biological brains and artificial neural networks [25, 26]. In summary, the emergence of gauge fields, charges, and local gauge parameters within neural network dynamics provides a powerful mathematical framework for understanding and designing robust, adaptive, and biologically plausible neural systems. I present an intuitive explanation as well as the neural equivalent of various gauge structures in Table 1 below COMPOSITE NEURON DYNAMICS: EMERGENCE OF NEURONAL PARAMETERS FROM CLUSTERED INTERACTIONS The concept of composite fermions in particle physics offers a compelling approach to understanding fermion mass hierarchies and interaction strengths through strong coupling and partial compositeness. Inspired by this fundamental principle, we introduce an analogous model in neural network dynamics, where individual neurons are viewed as elementary units whose intrinsic properties emerge from their interactions with strongly coupled neuronal clusters, termed composite neurons. 3 Gauge Theory Concept Neural Network Interpretation Gauge Symmetry Local invariance under neuronal parameter redefinitions, indicating redundancy in neural coding. Gauge Field (Aij ) Connection adjustments (synaptic modulations) preserving neural invariances under local transformations. Gauge Charge (qi) Neuron’s sensitivity to local modulation: some neurons strongly affected by local network state; others robust and insensitive. Gauge Parameter (αi) Local baseline or reference setting neurons freely adjust without affecting global computation. Gauge Field Strength (Fij ) Presence of local rhythmic or geometric structures (oscillations, loops, traveling waves) arising due to nontrivial interactions among neurons. TABLE I: Detailed neural interpretations of gauge theory concepts. Composite Neuron Dynamics: Core Intuition Consider a neural network consisting of neurons whose activation dynamics are described by membrane potentials Vi(t). Individual neurons are conventionally modeled as elementary computational units with fixed intrinsic parameters. By contrast, in the composite neuron model, single neurons dynamically form clusters characterized by strong internal interactions. These composite clusters exhibit coherent collective dynamics, significantly altering the parameters of constituent neurons through interactions analogous to partial compositeness in gauge theories. Mathematical Formulation of Neuronal Clusters Let us formally define a neuronal cluster Cas a strongly interacting subset of neurons. The strength of internal interactions within cluster Cis quantified by a coupling parameter Λ, satisfying: Jij ≈Λ for i, j ∈C, (8) where Jij represents the synaptic coupling strength between neurons iand j. The average cluster activation VC(t) is thus given by: VC(t) = 1 |C|X i∈C Vi(t),(9) with |C|denoting the cardinality of the cluster. Partial Compositeness in Neuronal Dynamics The partial compositeness concept posits that single neuron states are linear combinations of intrinsic neuron dynamics and collective cluster dynamics. The effective neuronal state Veff i(t) is thus defined as: Veff i(t) = ZiVi(t) + X C iC VC(t),(10) where Zirepresents the remaining elementary neuronal dynamics, and iC are mixing parameters quantifying the degree of partial compositeness, analogous to mixing angles in particle physics. Emergence of Effective Neuronal Parameters The interaction between individual neurons and composite clusters results in modified neuronal parameters. In analogy to particle physics, the effective ”mass” of neurons corresponds to the inverse excitability threshold (θ−1 i). Due to partial compositeness, the effective threshold is: θeff i=θi 1 + PC|iC |2Λ θC ,(11) where θCdenotes the cluster threshold. Neurons strongly interacting with composite clusters thus become either more or less excitable depending on cluster properties. 4 Effective Yukawa Coupling: Synaptic Strength Analogous to Yukawa couplings in particle physics, the effective synaptic strength between neurons iand jemerges from their interaction with clusters: Jeff ij ≈Jij +gCiC jC ,(12) where gCcharacterizes the composite cluster’s internal coupling strength. Stronger mixing with composite clusters enhances synaptic strength, dynamically adjusting connectivity. Effective Gauge Charge: Network Connectivity The gauge charge in this neural analogy corresponds to the neuron’s coupling sensitivity to global network modulation. The effective charge for neuron iis: qeff i=qi+X C QC|iC |2,(13) with QCrepresenting the composite cluster’s global connectivity strength. Neurons with larger effective charges play significant roles in network-wide synchronization. Adaptive Gauge Fields in Neural Networks Adaptive neuronal connectivity is governed by emergent gauge fields Aij(t), ensuring robustness and local invariance under neuron-specific recalibrations: DtVi(t) = ∂tVi(t)−iqiX j Aij(t)Vj(t).(14) Gauge fields dynamically evolve according to adaptive learning rules derived from neural network optimization criteria: dAij dt =−η∂ ∂Aij |DtVi−(∂t−iqiAij)Vj|2.(15) Cluster Formation and Stability Composite neuronal clusters naturally form through adaptive connectivity adjustments governed by gauge fields. Stable cluster dynamics arise due to the balancing of internal strong coupling and external network interactions, demonstrated by high correlation in neuronal activity within clusters: ρij =cov(Vi, Vj) σViσVj ≈1, i, j ∈C, (16) where ρij is the correlation coefficient, indicating robust cluster formation. The introduction of composite neuron dynamics and gauge-inspired adaptive connectivity offers a rigorous mathematical framework to explain emergent neuronal properties. Individual neuron parameters such as excitability thresholds, synaptic strengths, and connectivity charges naturally arise through partial compositeness with neuronal clusters, enriching our theoretical understanding and practical capabilities in modeling complex neural systems. EMERGENCE OF EFFECTIVE NEURON PARAMETERS FROM COMPOSITE CLUSTERING Understanding the emergence of effective neuronal parameters through composite clustering provides profound insight into neural network dynamics. Drawing inspiration from gauge theories in particle physics, we analogously 5 define effective parameters within neural systems that arise due to interactions of individual neurons with neuronal clusters. In gauge theories, fundamental particle parameters such as mass, Yukawa coupling, and gauge charge emerge through interactions with composite states. Similarly, in neural networks, neuron parameters like excitability thresholds, synaptic strengths, and connectivity strengths emerge naturally from their interaction with strongly coupled neuronal clusters. We call this mechanism partial compositeness in neural networks. Correspondence of Parameters Table II summarizes the correspondence between neural network parameters and gauge theory parameters: Neural Network Parameter Symbol Gauge Theory Equivalent Excitability Threshold θeff iMass (m) Synaptic Strength Jeff ij Yukawa Coupling (y) Connectivity Strength qeff iGauge Charge (q) TABLE II: Neuron parameters and their gauge theory analogues. MATHEMATICAL DERIVATION OF EFFECTIVE NEURON PARAMETERS Consider a neuronal cluster C, composed of neurons that interact strongly. Define the cluster’s activation state as: VC(t) = 1 |C|X k∈C Vk(t),(17) where |C|is the number of neurons in cluster C. The effective state of an individual neuron idue to mixing with clusters is given by: Veff i(t) = ZiVi(t) + X C iC VC(t),(18) where Ziis the intrinsic state weight and iC are mixing parameters representing neuron-cluster coupling strengths. The intrinsic excitability threshold of neuron iis θi. Interaction with clusters modifies this to an effective threshold, derived as follows: θeff i=θi 1 + PC|iC |2Λ θC ≈θiθC |iC |2Λ,(for strong coupling) (19) where θCis the composite cluster threshold, and Λ quantifies strong internal interactions within the cluster. Thus, a neuron’s excitability is substantially influenced by cluster interactions. Synaptic strength between neurons iand jcan be derived by considering their interaction through composite clusters: Jeff ij =Jij +gCiC jC ,(20) where gCcharacterizes internal composite cluster coupling strength. This relation highlights how neuronal interactions are enhanced through mutual interactions with neuronal clusters. The neuron’s connectivity to the entire network can be analogously described by a gauge charge. The effective connectivity (charge) for neuron iemerges naturally from mixing: qeff i=qi+X C QC|iC |2,(21) with QCrepresenting the composite cluster’s global connectivity strength. Thus, neuron connectivity strength increases significantly when strongly interacting with influential clusters. 6 Influence of Clustering on Individual Neuron Dynamics To further illustrate the cluster influence, consider the activation dynamics of neuron i: τdVi dt =−Vi+X j Jeff ij Veff j+Ii(t),(22) where τis the neuronal membrane time constant and Ii(t) is external input. Substituting the effective neuron parameters explicitly demonstrates cluster influence: τdVi dt =−Vi+X j (Jij +gCiC jC ) ZjVj+X C jC0VC0!+Ii(t).(23) This equation shows how neuronal clustering modifies both excitability and synaptic strengths, profoundly shaping individual neuronal dynamics. Intuitive Understanding of Parameter Emergence Intuitively, neuronal parameters are not static intrinsic properties but dynamically evolve through interactions with clusters. Clusters act as powerful modulators, enhancing synaptic coupling (Yukawa), adjusting excitability thresholds (mass), and modulating global network connectivity (charge). Neurons strongly interacting with clusters adopt collective behaviors, while those weakly coupled remain more independent and adaptable. The composite neuron approach illustrates that essential neuronal parameters naturally emerge through strong neuron-cluster interactions. This understanding bridges particle physics concepts and neuroscience, providing robust theoretical foundations for analyzing neural dynamics and designing adaptable, efficient neural networks. ENHANCEMENT OF NEURAL NETWORKS THROUGH COMPOSITE AND GAUGE-INSPIRED METHODS Recent advances inspired by gauge theory and composite fermion concepts from particle physics offer promising pathways to substantially enhance neural network architectures. This chapter rigorously details the mathematical foundations and practical improvements resulting from integrating composite and gauge-inspired methodologies into neural networks. Composite neuron dynamics and gauge-inspired adaptivity introduce powerful innovations to neural networks. We summarize the key innovations as follows: •Composite Neurons: Dynamically formed clusters characterized by strongly interconnected neurons. •Partial Compositeness: Single neuron properties emerge from interactions with composite neuron clusters. •Gauge Symmetry: Local invariance under neuron-specific parameter adjustments. •Gauge Fields: Adaptive synaptic connectivity modulating interactions dynamically. •Gauge Charges: Neuron sensitivity measures to network modulation. Improvement of Current Neural Networks We now provide rigorous mathematical explanations of how composite and gauge concepts concretely improve neural network functionalities. Neural networks enhanced by gauge symmetry possess inherent robustness through local invariance: Vi(t)→eiαi(t)Vi(t).(24) 7 Adaptive gauge fields Aij (t) ensure robustness via covariant derivatives: DtVi(t) = ∂tVi(t)−iX j Aij(t)Vj(t).(25) This invariance significantly reduces sensitivity to adversarial perturbations. Composite clusters form via strong neuron-neuron couplings (Λ), creating functional modules: Jij ≈Λ (for neurons i, j within a cluster).(26) Neurons thus naturally form independent modules, enhancing interpretability and compositionality. Partial compositeness induces natural sparsity. Neurons interact sparsely with clusters characterized by mixing parameters iC , reducing computational overhead: Veff i(t) = ZiVi(t) + X C iC VC(t),|iC |  1.(27) Gauge fields adapt connectivity dynamically, optimizing routing and network efficiency: dAij(t) dt =−η∂ ∂Aij |DtVi−(∂t−iAij)Vj|2,(28) where ηcontrols adaptation speed. Composite regularization and gauge constraints improve network training convergence and stability, guided by loss functions: Ltotal =Ltask +λcLcomposite +λgLgauge,(29) where Lcomposite =X CX i,j∈C (Jij −Λ)2,(30) and Lgauge =|DtVi−(∂t−iAij)Vj|2.(31) Composite modules and gauge parameters provide clear functional roles and adaptive connectivity patterns, significantly enhancing interpretability and explainability: ρij =cov(Vi, Vj) σViσVj ≈1,within clusters,(32) qeff i=qi+X C QC|iC |2,(33) where ρij indicates correlated dynamics, and qeff idescribes connectivity sensitivity. Gauge invariance acts as an implicit regularizer, preventing overfitting by enforcing: f(Vi)≈f(eiαiVi),(34) thus enhancing network generalization. The integration of composite neuron dynamics and gauge-inspired adaptive connectivity represents a significant advance for neural network architectures, addressing major current limitations. These mathematical enhancements improve robustness, modularity, sparsity, adaptability, convergence, interpretability, and generalization, providing a comprehensive theoretical and practical framework for the next generation of neural networks. 8 COMPOSITE AND GAUGE-INSPIRED CNN FOR ROBUST IMAGE CLASSIFICATION This chapter introduces a practical neural network construction utilizing composite and gauge-inspired concepts, explicitly applied to robust image classification under adversarial perturbations. We focus on a clear numerical scenario: enhancing convolutional neural networks (CNNs) robustness against adversarial perturbations. Performance metrics include standard accuracy and adversarial robustness. The proposed architecture integrates composite neurons and gauge-inspired adaptive connectivity: Input →Composite Conv Blocks (CCB) → Gauge-Invariant Routing Layers (GIRL) →Composite Dense Layers (CDL) →Output (Softmax). (35) Composite Convolutional Block (CCB) CCBs are convolutional layers forming neuron clusters characterized by strong coupling parameters Λ: J(conv) ij ≈Λ for i, j ∈cluster,(36) where J(conv) ij are convolutional layer weights. Gauge-Invariant Routing Layer (GIRL) GIRL ensures local gauge invariance of neuron feature maps x(l) i,j: x(l) i,j →eiαi,j x(l) i,j,(37) introducing adaptive gauge fields A(l) ij for dynamic routing: Dtx(l) i,j =∂tx(l) i,j −iA(l) i,jx(l) i,j.(38) Gauge-invariance regularization: Lgauge =λg|Dtx(l)−(∂t−iA(l))x(l)|2.(39) Composite Dense Layers (CDL) Dense layers similarly form composite neuron clusters, with cluster-specific coupling Λd: J(dense) ij ≈Λdfor i, j ∈dense cluster.(40) Potential New Applications •Natural Language Processing (NLP): Improved robustness and modularity against textual adversarial attacks. •Reinforcement Learning (RL): Adaptive policies robust to environment changes. •Neuroscience-Inspired Computational Models: Robust, biologically plausible cortical simulations utilizing adaptive composite modules. The composite and gauge-inspired CNN framework presented significantly enhances robustness, modularity, and generalization in neural networks. 9 ADAPTIVE SEMI-COMPOSITE REGULARIZATION In this chapter, we analytically describe the Adaptive Semi-Composite Regularization approach for neural networks. This technique is inspired by composite models in particle physics, where elementary particles (fermions) mix with composite states at higher energies, gaining their effective parameters, such as masses and couplings, from composite interactions. By analogy, we consider clusters of neurons (composite neurons) that collectively determine adaptive parameters influencing other neurons (elementary neurons), providing both interpretability and robustness to the network. Definitions and Notations Consider a neural network layer with weight matrix: W∈Rnin×nout ,(41) where nin and nout are input and output dimensions respectively. Output neurons are divided into two distinct groups: •Composite neurons: Partitioned explicitly into NCclusters, denoted by sets: Cj={i1 j, i2 j, . . . , i|Cj| j}, j = 1,2, . . . , NC(42) •Elementary neurons: Remaining neurons outside clusters, denoted by: E={e1, e2, . . . , eNE}.(43) Adaptive Composite Parameters Λj Each composite cluster Cjis characterized by an adaptive internal parameter Λj, derived from internal cluster dynamics. Λjis not externally imposed, but adaptively learned during optimization. We define the regularization cost for composite neurons within the cluster Cjas: Lcomposite = NC X j=1 X i∈Cj kWi−Λjk2,(44) where Wirepresents the weight vector corresponding to the composite neuron i. The optimal adaptive parameter Λjfor cluster Cjis obtained by minimizing Lcomposite with respect to Λj: ∂ ∂Λj Lcomposite = 0 ⇒Λj=1 |Cj|X i∈Cj Wi.(45) Thus, each Λjemerges as the centroid (mean) of the weight vectors of neurons within its corresponding cluster, naturally reflecting internal cluster dynamics. Interaction with Elementary Neurons Elementary neurons do not belong directly to clusters but gain their effective parameters by adaptively mixing with composite clusters. Each elementary neuron ekis thus described by a weighted combination of adaptive cluster parameters: Λeff ek= NC X j=1 αkjΛj,(46)