Quantifying information stored in synaptic connections rather than in firing patterns of neural networks

A cornerstone of our understanding of both biological and artificial neural networks is that they store information in the strengths of connections among the constituent neurons. However, in contrast to the well-established theory for quantifying information encoded by the firing patterns of neural networks, little is known about quantifying information encoded by its synaptic connections. Here, we develop a theoretical framework using continuous Hopfield networks as an exemplar for associative neural networks, and data that follow mixtures of broadly applicable multivariate log-normal distributions. Specifically, we analytically derive the Shannon mutual information between the data and singletons, pairs, triplets, quadruplets, and arbitrary n-tuples of synaptic connections within the network. Our framework corroborates well-established insights about storage capacity of, and distributed coding by, neural firing patterns. Strikingly, it discovers synergistic interactions among synapses, revealing that the information encoded jointly by all the synapses exceeds the 'sum of its parts'. Taken together, this study introduces an interpretable framework for quantitatively understanding information storage in neural networks, one that illustrates the duality of synaptic connectivity and neural population activity in learning and memory.