Predicting bifurcation-induced transitions in oscillatory systems with deep learning

Critical transitions in oscillatory systems are often driven by bifurcations of periodic orbits, such as period-doubling and period-adding bifurcations. However, accurately predicting these transitions and classifying the specific oncoming bifurcation types remains a significant challenge. Inspired by recent advances showing that deep learning classifiers, trained on synthetic data, can effectively anticipate and classify bifurcations of equilibria, this study extends the approach to oscillatory dynamics. A hybrid deep learning classifier, integrating convolutional neural networks (CNNs) and long short-term memory (LSTM) networks, is developed and trained using noisy synthetic inter-spike interval (ISI) sequences derived from a theoretical neuronal model exhibiting period-2 oscillations. This classifier provides accurate early warnings of oncoming bifurcation-induced transitions under the investigated conditions and distinguishes between period-doubling and period-adding bifurcations in the tested datasets. Notably, it substantially outperforms conventional statistical early-warning signals methods, such as variance, autocorrelation and local return-map features, which in oscillatory systems exhibit poor predictive capabilities under the influence of non-monotonic trends and complex real noise. Moreover, without any retraining, the classifier shows transferability in predicting and classifying oncoming bifurcations in experimental recordings from injured rat sciatic nerves, as well as in diverse theoretical models spanning discrete maps and differential equations. By capturing transferable pre-bifurcation signatures across the tested oscillatory systems, this approach shows the potential of deep learning in predicting oscillatory critical transitions.