Neural Information Processing Systems http://nips.cc/

We propose a topological framework for the detection of Hopf bifurcations directly from time series, based on persistent homology applied to phase space reconstructions via Takens embedding within the framework of Topological Data Analysis. The central idea is that changes in the dynamical regime are reflected in the emergence or disappearance of a dominant one-dimensional homological features in the reconstructed attractor. To quantify this behavior, we introduce a simple and interpretable scalar topological functional defined as the maximum persistence of homology classes in dimension one. This functional is used to construct a computable criterion for identifying critical parameters in families of dynamical systems without requiring knowledge of the underlying equations. The proposed approach is validated on representative systems of increasing complexity, showing consistent detection of the bifurcation point. The results support the interpretation of dynamical transitions as topological phase transitions and demonstrate the potential of topological data analysis as a model-free tool for the quantitative analysis of nonlinear time series.

Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals.Unfortunately, continuous attractors suffer from severe structural instability in general---they are destroyed by most infinitesimal changes of the dynamical law that defines them.This fragility limits their utility especially in biological systems as their recurrent dynamics are subject to constant perturbations.We observe that the bifurcations from continuous attractors in theoretical neuroscience models display various structurally stable forms.Although their asymptotic behaviors to maintain memory are categorically distinct, their finite-time behaviors are similar.We build on the persistent manifold theory to explain the commonalities between bifurcations from and approximations of continuous attractors.Fast-slow decomposition analysis uncovers the existence of a persistent slow manifold that survives the seemingly destructive bifurcation, relating the flow within the manifold to the size of the perturbation. Moreover, this allows the bounding of the memory error of these approximations of continuous attractors.Finally, we train recurrent neural networks on analog memory tasks to support the appearance of these systems as solutions and their generalization capabilities.Therefore, we conclude that continuous attractors are functionally robust and remain useful as a universal analogy for understanding analog memory.

The dynamical stability of the iterates during training plays a key role in determining the minima obtained by optimization algorithms. For example, stable solutions of gradient descent (GD) correspond to flat minima, which have been associated with favorable features. While prior work often relies on linearization to determine stability, it remains unclear whether linearized dynamics faithfully capture the full nonlinear behavior. Recent work has shown that GD may stably oscillate near a linearly unstable minimum and still converge once the step size decays, indicating that linear analysis can be misleading. In this work, we explicitly study the effect of nonlinear terms. Specifically, we derive an exact criterion for stable oscillations of GD near minima in the multivariate setting. Our condition depends on high-order derivatives, generalizing existing results. Extending the analysis to stochastic gradient descent (SGD), we show that nonlinear dynamics can diverge in expectation even if a single batch is unstable. This implies that stability can be dictated by a single batch that oscillates unstably, rather than an average effect, as linear analysis suggests. Finally, we prove that if all batches are linearly stable, the nonlinear dynamics of SGD are stable in expectation.

Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals.

Data from naturalistic systems are often modeled asgenerated from an underlying low dimensional manifold embedded inahighdimensional measurement space.