Register Neurons: State Storage & Registration

• Register Neurons are computational units with internal state registers that enable storage, pattern matching, communication, and symbolic reasoning.
• Methodologies like volumetric and neural field registration allow robust mapping and tracking of neuronal identities even under large deformations and incomplete data.
• Applications span neuroscience, neuro-inspired computing, robotics, and symbolic AI, driving advancements in neuron classification and connectomic analysis.

Register Neurons are computational units, physical neurons, or algorithmic modules equipped with internal state "registers" that facilitate storage, pattern matching, communication, or symbolic reasoning. The concept spans biological neuroscience, neuro-inspired computing, and neuro-symbolic architectures, with core methodologies encompassing volumetric registration, neural field pose estimation, machine learning-driven identification, and symbolic memory encoding. These neurons leverage state representation and registration—whether of spatial position, identity, symbolic attractor, or functional role—to enable the tracking, classification, manipulation, and retrieval of neuronal information at scale.

1. Methodologies for Neuronal Registration

Register Neurons are central to automated workflows for tracking and identifying neurons within living tissue and virtualized architectures.

• Volumetric Registration: In whole-brain calcium imaging of freely moving specimens, such as C. elegans, neuron registration is achieved by deformable alignment of 3D fluorescent image volumes (Nguyen et al., 2016). The imaging pipeline includes:
  • Extraction and straightening of animal centerline using active contour algorithms on dark-field images.
  • Reinterpolation of image intensities along centerline normals, suppressing posture-induced deformation.
  • Rigid alignment via cross-correlation with a reference volume.
  • Neuron detection by Hessian-based curvature filtering and watershed segmentation.
• Neuron Registration Vector Encoding (NRVE): NRVE reframes registration as a time-independent matching problem. Each neuron’s matches to a set of $K$ reference volumes (spanning the full data acquisition time) are encoded as a registration vector. Hierarchical clustering with a correlation distance metric assigns neuron identities, avoiding the error compounding inherent in consecutive (temporal) tracking.
• Neural Field Registration: In object-centric neural field modeling, the 6-DoF pose of an object’s neural field within a scene is estimated via multi-view spatial sampling, robust point-wise registration (FPFH/RANSAC/ICP), and bidirectional @@@@2@@@@ alignment (Hall et al., 2024). Loss functions over neural field surfaces drive pose optimization:

$$L_s(S_a, S_b^q; T) = \mathbb{E}_{x \in A}\,\kappa(r(x; S_a, S_b^q, T); p, \alpha) + \mathbb{E}_{x \in B^q}\,\kappa(r(x; S_b^q, S_a, T^{-1}); p, \alpha) + w \cdot L_r,$$

with $r(x; S_a, S_b^q, T) = \| S_a(x) - S_b^q(T(x)) \|$.

These procedures collectively facilitate robust, scalable mapping of neuron identities or object geometries in settings with large deformation, viewpoint ambiguity, or incomplete data.

2. Architectures and Internal State Representation

Register Neurons encode state through explicit modular mechanisms, bridging direct biological functions and engineering abstractions.

• Neural Status Register (NSR) (Faber et al., 2020): Inspired by CPU status registers, NSRs are fully differentiable modules that operate on continuous quantities. Operand selection is performed by learnable weighted selectors. The difference is processed to obtain relaxed sign and zero bits:
  • $B_+(d) \approx \tanh(\lambda d)$ (sign)
  • $B_0(d) \approx 1 - 2 (\tanh(\lambda d))^2$ (zero)
  • These outputs are linearly combined and squashed via a sigmoid to yield the register's output. Multiple NSR units can operate in redundant parallel to mitigate learning difficulties in tasks such as XOR.
• Prime Attractors as Symbolic Registers (Lizée, 2022): Spiking neural networks are trained to sustain prime attractors—noise-driven, self-sustaining states representing atomic symbols. Register neurons hold these attractors, with winner-take-all mechanisms ensuring robust retrieval from noisy signals. Hebbian mechanisms enable one-shot binding and unbinding of attractor states, operationalizing symbolic variable storage and register switching.
• Single-use Register Automata (Bojańczyk et al., 2019): Register automata process streams ("data words") over infinite alphabets, where registers store and compare atomic values. Imposing the single-use restriction—resetting registers upon use—ensures robust equivalence among distinct automata models, and supports modular algebraic decompositions (Krohn-Rhodes), facilitating analysis and composition.

This unified state encoding underpins quantitative reasoning, symbolic computation, and robust registration across biological and artificial domains.

3. Machine Learning in Neuron Registration and Classification

The application of register neurons to large-scale data relies on sophisticated machine learning techniques for detection, mapping, and classification.

• Convolutional Neural Network Detection and Affine Atlas Registration (Iqbal et al., 2018): DeNeRD splits brain sections into manageable patches, detects neurons with modified Faster R-CNNs, and affinely registers detected patches to a reference atlas. The result is region-attributed detection compatible with automated density quantification.
• Permutation-Invariant Encoders for Neuron Classification (Liao et al., 2023):
  • Skeleton Encoder leverages farthest point sampling and ball query grouping, followed by Conv1D (kernel size 1) ensuring point-order invariance, producing global morphological representations.
  • Connectome Encoder applies GNN propagation over synaptic graphs:

$$X^{(l+1)} = \text{ReLU}\left( [ (1-\alpha)\hat{H}X^{(l)} + \alpha X^{(0)} ][(1-\beta)I + \beta \Theta] \right)$$

Fused representations yield high classification accuracy on whole-brain datasets.

• Clustering and Identity Assignment: In NRVE, clustering across high-dimensional registration vectors yields stable neuron identities despite volumetric deformation.

These practices enable high-throughput, accurate registration and functional characterization of neuron populations, supporting connectomics, development studies, and disease diagnostics.

4. Symbolic, Structural, and Modular Considerations

Register Neurons serve not only for stateful data processing but also for symbolic, structural, and modular reasoning.

• Hierarchical Labeling and Mapping (Schaefer et al., 2021): Large-scale artificial neural networks paralleling human brain organization leverage region-specific labeling schemas reflecting anatomical hierarchy. Labels capture hemisphere, anatomical region, column, microcolumn, and layer (cortex), or hemisphere, functional region, lobule, microzone, and module (cerebellum), facilitating targeted analysis and comparative studies.
• Neuro-Symbolic Binding and Hash Tables (Lizée, 2022): Registers in spiking neural architectures act as loci for representing, binding, and unbinding symbols through attractor states. Combinatorial binding (via random second-order networks) enables the formation of hash tables and variable abstractions akin to symbolic computing, applicable to toy symbolic computers.
• Algebraic Theory of Automata (Bojańczyk et al., 2019): The single-use property in register automata reconstitutes a finite-alphabet-like algebraic foundation, supporting Krohn-Rhodes decomposition into prime modules.

These constructs ground register neurons in scalable, modular frameworks, directly supporting parallelism, symbolic binding, retrieval, and compositional design.

5. Practical Applications and Impact

Register Neurons and related registration frameworks power a broad suite of real-world applications.

• Neuroscience: Automated tracking of neurons in deforming tissue (e.g., C. elegans), region-specific quantification of inhibitory neuron development across brain-wide datasets, and classification of neurons in human and Drosophila cortex (Nguyen et al., 2016, Iqbal et al., 2018, Liao et al., 2023).
• Neuro-inspired Computing: NSR layers advance quantitative reasoning in deep networks, solving comparison, counting, minima-finding, and graph shortest-path problems with superior extrapolation (Faber et al., 2020).
• Symbolic AI and Neuro-symbolic Integration: Register neurons holding prime attractors enable sample-efficient, reusable symbolic manipulation; mechanisms for binding and unbinding prime attractors realize symbolic memory and computational modules (Lizée, 2022).
• Robotics and 3D Perception: Neural field registration yields accurate object pose and segmentation within scene neural fields, enabling object completion in partial scenes and scene synthesis via object substitution (Hall et al., 2024).
• Connectomics and Clinical Diagnostics: Large-scale neuron classification pipelines enable systematic mapping of neuron types and wiring patterns; hierarchical labeling supports region- and layer-specific retrieval and comparison (Liao et al., 2023, Schaefer et al., 2021).

Collectively, register neuron architectures and registration pipelines drive progress in large-scale data analysis, robust neural tracking, AI-augmented quantitative reasoning, and symbolic computing, with measurable increases in accuracy, throughput, and adaptability in both biological and engineered systems.

6. Theoretical Foundations and Limitations

Register Neurons and associated systems are anchored in well-established mathematical and computational theory, but notable limitations remain.

• Energy Minimization and Registration: TPS-based registration is governed by explicit objective functions penalizing mismatch and deformation:

$$E(u) = \int [ f(x; u[X]) - f(x; R) ]^2 dx + E_\text{deformation}(u)$$

Gaussian mixture models encode neuron centroids, amplitudes, and sizes.

• Single-use Restriction: The single-use register constraint in automata theory is essential for algebraic equivalence and eliminates infinite-memory pathologies, enabling modular decomposition and robust closure properties (Bojańczyk et al., 2019).
• Capacity Limits in Symbolic Bonding: The attractor-based register capacities, driven by winner-take-all filtering, are limited; excessive binding induces signal blurring, though unbinding methods restore register functionality (Lizée, 2022).
• Computational Resources: Full registration pipelines, as in NRVE, entail significant computational loads (e.g., 250 GB data processed over ~40 hours using ≥200 cores (Nguyen et al., 2016)). Efficient parallelization is critical.
• Hyperparameter Sensitivity and Activation Saturation: NSRs require careful lambda scaling for gradient stability across input orders; extreme scaling can induce vanishing gradients.

This foundational rigor ensures soundness and extensibility of register neuron models, especially when scaling to complete connectomes or integrating with symbolic reasoning modules.

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The register neuron paradigm encompasses a continuum from biological neurons and their computational proxies to engineered modules in AI architectures, united by the principle of local state representation and robust registration. Recent advances in volumetric neuroimaging, symbolic attractor encoding, neural field registration, and algebraic automata theory collectively define a scalable, modular, and formal basis for neuron registration, identification, and symbolic manipulation.