Skip Connections Increase the Capacity of Associative Memories in Variable Binding Mechanisms

The flexibility of intelligent behavior is fundamentally attributed to the ability to separate and assign structural information from content in sensory inputs. Variable binding is the atomic computation that underlies this ability. In this work, we investigate the implementation of variable binding via pointers of assemblies of neurons, which are sets of excitatory neurons that fire together. The Assembly Calculus is a framework that describes a set of operations to create and modify assemblies of neurons. We focus on the $\texttt{project}$ (which creates assemblies) and $\texttt{reciprocal-project}$ (which performs variable binding) operations and study the capacity of networks in terms of the number of assemblies that can be reliably created and retrieved. We find that assembly calculus networks implemented through Hebbian plasticity resemble associative memories in their structure and behavior. However, for networks with $N$ neurons per brain area, the capacity of variable binding networks ($0.01N$) is an order of magnitude lower than the capacity of assembly creation networks ($0.22N$). To alleviate this drop in capacity, we propose a $\textit{skip connection}$ between the input and variable assembly, which boosts the capacity to a similar order of magnitude ($0.1N$) as the $\texttt{Project}$ operation, while maintaining its biological plausibility.