Quantifying Information Stored in Synaptic Connections Rather Than in Firing Activities of Neural Networks

A cornerstone of our understanding of both biological and artificial neural networks is that they store information in the strengths of synaptic connections among the neurons. However, in contrast to the well-established theory for quantifying information encoded by the firing activity of neural networks, there does not exist a framework for quantifying information stored in the distribution of synaptic connection weights. Here, we develop a theoretical framework for synaptic information, by using densely connected Hebbian networks performing autoassociative memory tasks as the substrate, and by modeling data patterns that need to be stored as log-normal distributions. Specifically, we derive analytical approximations for Shannon mutual information between the data patterns and singletons, pairs, and arbitrary n-tuples of synaptic connections within the network. Our framework corroborates well-established insights regarding pattern storage capacity and the principle of distributed coding in neural firing activities. Notably, it formalizes the heterogeneity inherent in information encoding across synapses in a network, and 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 a powerful, interpretable framework for quantitatively understanding information storage in the synapses of neural networks, one that illustrates the duality of synaptic connectivity and neural population activity in learning and memory.