Optimizing Neuron Positions

• Optimizing neuron positions is a diverse set of methods that treat neuron locations as explicit parameters to enhance network alignment, interpretability, and efficiency.
• It incorporates both index-based schemes like position-aware neurons and spatial embedding approaches, where connection properties depend on inter-neuron distances.
• These techniques improve federated learning, promote biologically plausible clustering, and enable visual interpretability in both artificial and computational neuroscience.

Optimizing neuron positions refers to a diverse class of methodologies in which the spatial or functional location of each neuron—either in a geometric, topological, or index-based sense—is treated as an explicit object of optimization, rather than a static or anonymous part of the network. This approach spans topics from encoding permutation invariance in artificial neural networks, to biologically grounded wiring efficiencies, spatial regularization, function-space adaptation, and mathematical neuroscience. Across these domains, optimizing neuron positions is often motivated by alignment, interpretability, resource efficiency, or the drive to capture biological connectivity phenomena.

1. Position as Model Parameter: Geometric and Index-Based Encodings

Neuron position can be understood strictly as a geometric object (e.g., an embedding in $\mathbb{R}^d$) or as a unique label or index within a layer. This distinction underpins several lines of work.

Index-based approaches: In deep learning models, hidden neuron outputs are traditionally permutation invariant, which poses challenges in collaborative settings such as federated learning (FL). Position-Aware Neurons (PANs) (Li et al., 2022) inject fixed position encodings (additive or multiplicative) into each neuron's output to break this symmetry:

• Additive: $h_l = f_l(W_l h_{l-1} + b_l + e_l)$
• Multiplicative: $h_l = f_l((W_l h_{l-1} + b_l) \odot e_l)$ with $e_l$ a fixed, non-trainable sinusoidal encoding vector representing neuron indices within the layer.

Spatial embedding approaches: Architectural paradigms embedding each neuron in Euclidean space and tying synaptic weights to physical distances have been proposed for both parameter efficiency and biological inspiration (Erb et al., 16 Jun 2025, Wołczyk et al., 2019, Mészáros et al., 3 Nov 2025). Here, positions $p_i \in \mathbb{R}^d$ are explicit learnable parameters, and connection properties (e.g., weights, delays) are functions of inter-neuron distances $\|p_i - p_j\|$.

2. Mathematical Formulations: Losses, Regularization, and Constraints

Index-Driven Coupling (PANs)

By embedding unique position labels, PANs introduce an implicit penalty for neuron permutations due to their influence on the forward computation. The emergent regularization term is: $$\mathcal{R}_{\mathrm{pan}} = \sum_{l=1}^{L-1}\|\Pi_l e_l - e_l\|^2$$ where $\Pi_l$ is a permutation of neurons in layer $l$. This regularization is implicit; no explicit term is added, and the fixed $e_l$ suffice to couple neuron identity to position.

Spatial Embedding with Distance-Dependent Wiring

In models with geometric neuron positions, connection strength and/or delay is determined by distance: $w_{ij} = \frac{1}{\|p_i - p_j\|_2}$

$\tau_{ij} = \|p_i - p_j\|_2$

Additional cost terms penalize wiring length (or delay): $$C_{\mathrm{dist}}(p, w) = \Lambda_2 \sum_{i, j} |w_{ji}| \|p_i - p_j\|$$ These appear in a global loss, e.g.: $$L(p, w) = L_\mathrm{task} + \Lambda_1 \sum_{i, j}|w_{ji}| + \Lambda_2 \sum_{i, j} |w_{ji}| \|p_i - p_j\|$$ where $\Lambda_1, \Lambda_2$ are regularization weights (Mészáros et al., 3 Nov 2025, Wołczyk et al., 2019).

Probabilistic/Information-Theoretic Objectives

For clustering or topographic mapping, neuron "centres" $\{x_i\}$ are optimized to minimize the expected code length for joint data/activation encoding (Luttrell, 2015): $$F = \frac{1}{2\sigma^2} \int dx\,P(x) \sum_i p(i|x) \|x - x_i\|^2$$ Stationarity yields centroid conditions: $$x_i = \frac{\int dx\,P(x)\,p(i|x)\,x}{\int dx\,P(x)\,p(i|x)}$$ producing Voronoi (Kohonen map) receptive fields for small $\sigma$.

3. Optimization Algorithms and Implementation Schemes

Coordinate and Geometric Updates

• PANs: Neuron positions are fixed (indices), encodings $e_l$ are not learned. Training proceeds with standard SGD/Adam on weights, with PAN modulation in the forward pass.
• Geometric optimization: Neuron coordinates $p_i$ are treated as first-class parameters and updated via automatic differentiation and backpropagation. For distance-based wiring, gradients on $p_i$ follow from: $$\frac{\partial L}{\partial p_i} = \cdots + \sum_j w_{ij} \frac{p_i - p_j}{\|p_i - p_j\|} + \mathrm{task/regularizer\,terms}$$
• Spatial network clustering: Both weight parameters and 2D neuron positions $p(n)$ are optimized via backprop, with explicit spatial cost gradients (Wołczyk et al., 2019).
• Finite neuron/PDE methods: Breakpoints or "neuron positions" are optimized via alternating minimization and subspace correction, with superlinear convergence for neuron subproblems (Levenberg–Marquardt) and optimal preconditioning for linear layers (Park et al., 2022).

Specialized Scheduling, Regularization, and Preconditioning

• Hyperparameters for spatial penalties are fixed per paper (e.g., $\alpha=1, \beta=3$ for clustering (Wołczyk et al., 2019)).
• Preconditioned or whitened gradient updates (matrix-free estimation of local feature covariances) have been proposed for general architectures, offering accelerated convergence by moving each neuron in a local function space sensitive to its input distribution (Munoz, 3 Feb 2025): $\Delta \theta = -G^{-1} g$ where $G = E[x x^T]$ is the feature covariance matrix per neuron.

4. Empirical Phenomena: Clustering, Alignment, and Efficiency

Federated and Collaborative Settings

PANs enforce alignment across federated learning clients, enabling simple coordinate-wise averaging with marked improvements in model accuracy under severe non-i.i.d. distributions:

• Matching ratio VGG9 Conv5: $\approx 0.62 \to 0.85$
• Weight divergence drops 30–50%
• FL accuracy (Dirichlet $\alpha=0.1$): e.g., FedAvg improves from $82.8\%$ to $84.5\%$
• Convergence speed increases by 10–20% (Li et al., 2022)

Emergent Spatial and Functional Structure

Spatially regularized or distance-penalized networks self-organize:

• Clustering: Task-specific neuron assemblies localize in position space, recapitulating biological modularity (Wołczyk et al., 2019, Mészáros et al., 3 Nov 2025).
• Small-world and modular topologies: Distance-penalized SNNs develop high small-worldness and modularity, as measured by appropriate clustering metrics (Mészáros et al., 3 Nov 2025).
• Pruning and robustness: Geometric models remain accurate at extreme sparsity levels (≥80% pruned), outperforming dense conventional networks with similar parameter counts (Erb et al., 16 Jun 2025).

Interpretability and Visualization

Geometric position optimization enables spatial plots of neuron activity and clustering by function, providing immediate interpretability not afforded by dense parameterization (Erb et al., 16 Jun 2025, Mészáros et al., 3 Nov 2025).

5. Methodological Extensions and Generalization

• PANs can be deployed in MLPs (per-unit), CNNs (per-channel), with proposals for per-token or per-attention-head in Transformer-like models; position encoding hyperparameters ($A$, $T$) are tuned carefully to balance coupling strength and signal fidelity (Li et al., 2022).
• Spatial regularization naturally extends to layers beyond 2D (e.g., higher-dimensional embeddings), with ablation studies up to $d=32$ (Erb et al., 16 Jun 2025).
• In geometric mapping to curved surfaces, columns of neurons are shifted/rotated to fit prescribed layer curvature under density and smoothness constraints, with iterative refinement for uniformity and biological plausibility (Long et al., 2014).
• Position learning applies to both rate-based and spiking neural networks; for the latter, delay dynamics and event-based gradients (EventProp) support biologically motivated time-space coupling (Mészáros et al., 3 Nov 2025).
• In function approximation (e.g., finite-neuron methods for PDEs), neuron breakpoint repositioning enables high accuracy in capturing oscillatory or singular solutions, outperforming standard first-order optimization (Park et al., 2022).

6. Trade-Offs, Limitations, and Open Directions

Approach	Strength	Limitation
PANs	Simple alignment, no extra tuning	Coupling sensitive to $A$
Spatially embedded networks	Parameter efficiency, interpretability	Sometimes lower accuracy for identical param counts
Information-theoretic centers	Guarantees topographic mapping	Computationally costly for large $M$ or high $D$
Subspace correction (PDE)	Fast convergence, high fidelity	Currently 1D, adaptation to deep networks unclear

• Coupling parameters (amplitude $A$ in PANs, spatial cost weights) require task-specific tuning to avoid under- or over-regularization.
• Non-learnability of position encodings in PANs is crucial—if $e_l$ is learned or altered, alignment breaks down.
• For geometric models, unconstrained positional drift can hinder cross-instance interpretability; practical embeddings often constrain or regularize drift.
• Curve-fitting for cortical mapping becomes unstable with extreme curvature, suggesting the need for more robust surface-fitting techniques (Long et al., 2014).
• Extension to more complex architectures, or to continuous time and space dynamics, remains an open direction.

7. Context and Future Prospects

Optimizing neuron positions represents a convergence of computational neuroscience, machine learning, information theory, and numerical methods. The outlined methodologies enable:

• Alignment and parameter averaging in distributed learning.
• Emergence of interpretable, modular, and biologically plausible structures.
• Parameter-efficient architectures with competitive accuracy and strong pruning resilience.
• New avenues for visualization and interpretability by direct mapping between functional and geometric neuron locality.

Further research is underway on integrating position learning into more complex dynamical systems, developing scalable algorithms for multi-dimensional spatial regularization, and exploring applications to continual learning and hardware-efficient deployment. Questions of global optimality, generalization, and transferability of learned positional configurations are of ongoing interest.