Brain networks face the challenge of optimizing information processing while remaining within resource limitations. The metabolic costs of growth and maintenance in physical space directly impact network structure and function. This trade-off likely explains the convergence of many brain organizational solutions across species. The paper investigates this hypothesis by introducing spatially embedded recurrent neural networks (seRNNs). seRNNs learn to perform a task while simultaneously facing the biophysical constraints of a 3D Euclidean space, where connection costs are proportional to distance. The model aims to demonstrate that the observed structural and functional motifs of the brain, including sparse connectivity, small-world structure, modularity, and characteristic neuronal tuning curves, might arise from this fundamental optimization problem. Existing computational models have shown that network modularity can emerge from spatial costs, orthogonal population dynamics from task optimization, and predictive coding from energy limitations. However, a unified model incorporating both brain anatomy and function to dynamically trade-off structural, functional, and behavioral objectives is lacking. seRNNs attempt to fill this gap.

The paper reviews existing literature on brain network development and optimization under resource constraints. It highlights previous computational models demonstrating the emergence of modularity from spatial costs, orthogonal population dynamics from task optimization, and predictive coding from energy limitations. However, it notes the absence of a unified model that integrates anatomical and functional aspects to dynamically balance these objectives during learning. The authors cite work on the optimization of functionality within resource constraints, emphasizing the role of metabolic costs in shaping brain network organization across species and scales. The literature on network modularity, small-world architecture, and functional clustering of neurons is also reviewed, providing a basis for comparing seRNN findings with empirical observations.

The study uses spatially embedded recurrent neural networks (seRNNs) trained on a one-step inference task. seRNNs are regularized RNNs with an additional term in the loss function that incorporates spatial and communication constraints. The regularization term penalizes long-distance connections in 3D Euclidean space and connections that do not strongly contribute to network communication, as measured by weighted communicability. This approach differs from standard L1 regularization which only minimizes the sum of absolute weights. A comparison group of 1000 standard L1 regularized RNNs serves as a baseline. Both seRNNs and L1 RNNs are trained for 10 epochs, and the regularization strength is systematically varied. The one-step inference task requires networks to remember a goal location and integrate it with new choice options, simulating basic working memory and decision-making processes. Network performance, structural properties (modularity, small-worldness, sparsity), and functional properties (functional clustering, mixed selectivity, energy efficiency) are analyzed. Generative network models are used to assess the topological resemblance of seRNNs to empirical neural networks. A spatial permutation test is employed to evaluate the spatial distribution of functionally related neurons. The study also analyzes the correlation between the selectivity for different task variables to assess the degree of mixed selectivity.

Both seRNNs and L1 RNNs successfully learned the task, but seRNNs exhibited brain-like structural and functional properties. seRNNs showed significantly higher modularity and small-worldness compared to L1 RNNs. Generative network modeling revealed that homophilic wiring rules best explain seRNN topology, consistent with empirical data. Functionally related units showed spatial clustering in seRNNs, particularly for goal information, while choice information remained distributed. This suggests an energy-efficient strategy of localized information processing. seRNNs demonstrated a more mixed-selective code compared to L1 RNNs, showing less anticorrelation between the explained variance of different task variables. Furthermore, seRNNs exhibited significantly lower energy consumption (mean unit activations) than L1 networks. Analysis of the parameter space revealed a critical window where optimal trade-offs between accuracy, sparsity, modularity, small-worldness, mixed selectivity, and energy efficiency emerged simultaneously. These features co-occur in a specific range of regularization strengths and training epochs, suggesting a strong interdependence driven by optimization under biophysical constraints.

The findings demonstrate that incorporating biophysical constraints (spatial distance and communication efficiency) into an artificial neural network model can lead to the emergence of several features commonly observed in biological brains. These features, previously studied independently, appear strongly co-dependent and arise from the simultaneous optimization of task performance, structural costs, and efficient network communication. The results suggest that the observed organizational principles in brains are not merely coincidental but may arise from fundamental biological constraints. The model provides a valuable tool for studying the intricate relationship between structure and function in the brain, offering a potential mechanism to explain how these features are linked. The findings also have implications for the development of neuro-inspired artificial intelligence.

The study's main contribution is the development of seRNNs, a novel model that links structural and functional features commonly observed in brains. The results demonstrate a strong interdependence between these features, arising from the simultaneous optimization of task performance, structural costs, and network communication under biophysical constraints. Future research could investigate the effects of different tasks and the incorporation of further biological details into the model to enhance its realism and predictive power.

The model, while incorporating biophysical constraints, omits many biological details such as molecular mechanisms of circuit development and heterogeneous neuronal spiking patterns. The task used is a simplified version of a more complex navigation task. These simplifications might limit the generalizability of the findings. The study focuses on a limited set of parameters, and a more extensive exploration could further refine the understanding of the critical window where brain-like features emerge.