Associative memory and recurrent neural networks: a dynamical systems approach

This thesis explores the theory of associative memory in recurrent neural networks through the lens of dynamical systems. Starting from the classical literature on voltage-based and firing-rate models, we analyze their fundamental dynamical properties, including the existence of fixed points and the conditions for local and global stability. These results provide a unified framework to understand how recurrent networks store and retrieve information. Building on this foundation, we introduce a novel framework for input-driven memory retrieval, addressing a long-standing gap in the study of associative networks. We investigate this mechanism in both deterministic and stochastic settings, showing how external inputs and intrinsic noise can guide transitions between memories, smooth retrieval dynamics, and shape the underlying energy landscape. Finally, we extend classical estimates of memory storage capacity to a broader class of activation functions, thereby widening the applicability of associative memory models. Together, these contributions advance the understanding of recurrent neural networks as dynamical systems, bridging theoretical neuroscience with modern machine learning perspectives.