A Wasserstein Geometric Framework for Hebbian Plasticity

Neural population geometry shapes downstream computation. Recent empirical findings in neurobiology suggest that a hyperbolic structure underlies population activity in the hippocampus. Here we provide a theoretical framework for this phenomenon. First, we propose a plausible construction of hippocampal tuning curves that statistically induces hyperbolic geometry. Next, we establish a connection between neural decoding and associative memory by demonstrating that the Modern Hopfield Network update rule computes the minimum mean-squared-error (MMSE) estimator. Finally, we introduce a novel associative memory model defined in hyperbolic space that yields significantly larger capacity than leading models. Our results suggest that animals encode spatial information as a latent hyperbolic cognitive map, improving both memory capacity and decoding accuracy.

Introduction: Biological systems face anatomical and metabolic constraints, including costly synaptic maintenance and limited connectivity. These constraints favor neural codes that compress behaviorally relevant information into low-redundancy patterns. We test whether an excitatory competitive Hebbian rule can support synaptic resource allocation under such constraints and whether the resulting representations occupy a more favorable cost-performance regime than reference learning rules. Methods: Representational cost is quantified using mutual-information-based measures derived from the Variational Information Bottleneck. Experiments use fixed audiovisual embeddings from three audiovisual benchmarks (AVE, Kinetics-Sounds, VGGSound100) to isolate downstream associative plasticity. Hebbian learning is compared with Dense Difference Target Propagation (DDTP) and backpropagation (BP) under matched sparsity and architectural constraints. Results: Hebbian learning achieves lower task-information cost (CTI) than sparse BP and DDTP in the main compressed comparisons, while reaching CTI values comparable to shallow BP with nonnegative weights. Rather than uniformly improving classification performance, Hebbian learning shifts the trade-off between task-relevant information and representational cost, yielding lower CTI at comparable functional performance in several settings. Discussion: The results indicate a cost-performance trade-off rather than uniform accuracy gains. For a given level of task-relevant information, Hebbian representations retain less input information while preserving functional performance, although accuracy is slightly reduced on some datasets. These findings support interpreting Hebbian learning as a mechanism for synaptic resource allocation rather than as a general strategy for maximizing audiovisual classification accuracy.

We construct a minimal but biologically plausible spiking neuron model operating in discrete time, combining multiplicative spike-timing-dependent plasticity (WSTDP), divisive normalization of synaptic integration, homeostatic threshold adaptation, and a one-step refractory period. We show that this normalization admits a biologically plausible dendritic implementation in which each binary junction operates using only locally available information. Assembling excitatory-inhibitory pairs of such neurons into a two-dimensional recurrent network and applying periodic localized stimulation, we find that the network spontaneously gives rise to stable, self-propagating wave packets with the properties of dissipative solitons: they maintain a stable spatial profile, propagate at constant speed, and annihilate upon frontal collision. Their emergence requires a geometric asymmetry between excitatory and inhibitory connection radii, and initial inhibitory synapses stronger than excitatory ones. WSTDP engraves the direction of propagation into the synaptic weight profile, so that the network learns by itself to sustain propagation in one direction while suppressing the reverse. When two sources are active simultaneously, the resulting waves annihilate upon collision, defining a semi-persistent boundary whose position encodes the relative phase and frequency of the two sources. These results provide a minimal computational framework for studying the emergence of cortical traveling waves, activity zone delimitation, and spatial memory from local plasticity rules alone.