Dynamical stability in biologically plausible attractor neural networks

Recurrent neural networks are canonical models of biological memory, but previous theoretical approaches to auto-associative memory have studied only highly restrictive conditions that do not readily apply to biological neural circuits. Those approaches ensured memory stability by engineering the model dynamics (e.g., to follow the gradient of a Lyapunov function), so they do not apply to more general classes of dynamical systems. We present an alternative approach that uses a general dynamical system where memory stability is not guaranteed but emerges for certain parameter choices. 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 find that either all fixed points are stable or all are unstable, depending on whether their number is below a “critical load for stability”, which differs from the classical “critical capacity”, which measures the maximal number of achievable fixed points regardless of stability. By analyzing how this critical load for stability depends on experimentally measurable quantities that characterize memory-pattern statistics and neuronal activation functions, our analysis highlights the computational benefits of sparse-like patterns and threshold-linear activation functions and offers testable predictions for neural circuits supporting memory. Focusing on hippocampal area CA3, a neural circuit previously suggested to implement auto-associative memory, we partner with experimental collaborators to test the prediction that CA3 pyramidal neurons should use threshold-linear activation. Curating a large dataset of in vivo patch-clamp results for CA3 pyramidal neurons from behaving mice, we provide initial support for this prediction.