Memory Uncertainty Relation and Harmonic Memory in Random Recurrent Networks

Reliable attractor recall conventionally requires broad basins of attraction. However, in reservoir-computing based associative memory, temporal cues reliably recover dynamical memories despite basins dominated by unpredictable, riddled-like regions. We reveal that memory basins exhibit an octopus-like'' structure: a robust head'' near the attractor and thin, intertwined ``tentacles'' spanning state space. Initial states in tentacular regions yield near-zero uncertainty exponents, making the recalled memory effectively unpredictable at finite precision. Yet, cue-driven generalized synchronization bypasses this unpredictability, driving the system into the robust basin head. This mechanism yields a quantitative relation linking minimum cue duration, synchronization rate, and basin-head radius. Trained recurrent neural networks exhibit similar geometry, suggesting this phenomenon extends beyond reservoir computing.

The linear recurrent neural network (LRNN) is a simple model for studying how much memory a network builds up as it trains. For uncorrelated inputs, earlier work found that training itself settles the network between keeping the past and reacting only to the present. Real sequences are correlated, and we solve the learning dynamics exactly for correlated inputs. In the solution, keeping the past carries a cost. The whole effect of correlation lands on that cost. This cost reduces to the earlier one when inputs are uncorrelated and grows once they are positively correlated. Three findings follow. (1) Correlation reshapes the course of learning, not only its end. Memory builds, overshoots, and is partly removed, and the settled network keeps less of the past. (2) Memory switches off at a threshold set by one number, how much each input resembles the one just before it. Neither sequence length nor longer-range correlation moves this threshold. Memory is worth keeping only when the task needs the previous input more than the current input already supplies it through correlation with the past. (3) The best network changes too. Zero error demands a feedthrough, a path that passes the current input straight to the network's output and remembers nothing, and training builds it unprompted when given one spare hidden dimension. Our work turns one property of the input into a prediction of whether a network learns memory and explains why correlated data turns recurrent networks into change detectors.

Recurrent neural networks are canonical models of biological memory. In these models, memories are represented by distributed patterns of neural activity that are stored in the recurrent connections between neurons, such that they become attractors of the network's dynamics. During memory recall, network dynamics thus converge toward one of these memory patterns when started from a noisy or partial cue. Therefore, memory performance critically hinges on the dynamical stability of the stored patterns. However, previous theoretical approaches only studied dynamical stability under highly restrictive conditions that do not readily apply to biological neural circuits. Here, we develop a theory of the local stability of discrete fixed points in a broad class of networks with graded neural activities and in the presence of noise. Using methods from random matrix theory, we analyze the bulk and outliers of the eigenvalue spectra of the Jacobians that characterize network dynamics around fixed points. We show that either all fixed points are stable or all of them are unstable, depending on whether their number is below a ``critical load for stability'', which is distinct from the classical critical capacity that measures the maximal number of achievable fixed points regardless of their stability. We further analyze the dependence of this critical load for stability on experimentally measurable quantities characterizing the statistics of memory patterns and the activation functions of neurons. Our analysis highlights the computational benefits of sparse-like patterns and threshold-linear activation functions and offers testable predictions for neural circuits supporting memory.