Spatial Partial Functionalization of Neural Networks based on Noise Fields

We investigate how internal representations emerge across hierarchical processing systems by introducing a neuroscience-inspired framework for analyzing deep spiking neural networks (SNN) through the lens of functional connectivity. Drawing on concepts from systems neuroscience and information theory, we form the first-order functionally-connected (1FC) group of a neuron based on its statistically significant pairwise correlations with neurons from the previous layer of a trained SNN architecture. We then track its response properties during inference under various conditions. Our analysis shows that several principles of functional connectivity previously observed in biological cortex are preserved in spiking ResNet architectures. These 1FC ensembles display interesting properties: their aggregate cofiring reliably predicts downstream neuronal responses through a robust, ReLU-like input-output relationship, whose gain scales systematically with ensemble size. Reliable encoding of the presented class emerges only during high 1FC cofiring events, which themselves occur infrequently, indicating that informative representations are concentrated in rare but highly coordinated activity patterns. Under uniform random noise or adversarial perturbations, these response profiles are disrupted, particularly in early and intermediate layers. This enables a targeted high-resolution interrogation at specific nodes and pathways. We showed that the functional connectivity structure is shaped by learning and this structure breaks under weight permutation. These establish 1FC ensembles as a functionally meaningful substrate for input encoding and information transfer, with potential implications in designing targeted fine-grained diagnostics on the information flow.

The pruning of network connections is key to brain function but, despite its importance, there exist few biologically-plausible pruning rules with demonstrated good performance. In this work we evaluate noise-prune, a recently introduced unsupervised local pruning rule for recurrent networks that uses noisy fluctuations to determine the importance of connections. Noise-prune has previously only been empirically tested on random networks without a specific computational function. We show that noise-prune preserves task-performance in task-trained recurrent neural networks, greatly outperforming a strategy that only uses the magnitude of connections and performing on par with or exceeding a non-local strategy that uses second-order information. Rather than deterministically removing connections that fall below a certain threshold importance, noise-prune samples connections to preserve based on their importance and strengthens retained connections to preserve average synaptic strength. We show that this sampling and rescaling is essential to good performance, but that the optimal empirical degree of rescaling is lower than that predicted by the original theoretical argument. Our work thus validates noise-prune as a biologically-plausible pruning rule for functional recurrent network architectures and characterizes its optimal parameter settings.

Understanding how the architecture of neural networks shapes the computations they carry is a central challenge in neuroscience and machine learning. While specific circuit architectures have been linked to particular network computations and theoretical bounds on expressivity of broad classes of networks have been found, we are still missing general principles connecting the structure of finite networks to their computational capabilities. Here, we characterize the computational abilities of recurrent neural networks as a function of their connectivity by training a large collection of different networks to compute a large set of Boolean functions. For small networks, we constructed the complete ``catalogs'' of network-function performance, which revealed that computational capacity varies widely across architectures and that most networks show poor performance, and most functions are hard to compute. However, we show that having local 2- and 3-cycles in a network strongly enhances its computational ability, and networks with such cycles are often the minimal architectures that can solve particular functions. We further show that a small set of structural statistics accurately predict networks' performance. Extending our analysis to large networks showed that typical networks fail even to approximate a randomly selected function. Surprisingly, adding a small number of sparsely connected biologically-inspired interneurons to the network dramatically increases computational capacity. As in small networks, adding short cycles improved networks' capacity, outperforming acyclic or reachability-matched controls. Thus, our results identify local cycles as design principles linking neural connectivity to computational power, and offer a general framework to explore structure-function relations in computing networks.