Generation of Training Data and Evaluation of a Deep Learning Classifier for Bifurcation Early Warning Signals

Introduction:

In the realm of complex systems, bifurcations stand as pivotal moments of abrupt change, marking transitions to new and often unpredictable behaviors. These tipping points can be triggered by subtle shifts in parameters, external perturbations, or the intricate interplay of internal dynamics. Understanding and predicting bifurcations is crucial for various scientific disciplines, ranging from ecology and economics to physiology and engineering.

Bifurcation early warning signals (EWS) have emerged as powerful tools for detecting impending bifurcations. These statistical indicators harness the notion that the statistical properties of a system undergoing a bifurcation undergo characteristic changes. By monitoring these changes, researchers can identify systems on the brink of transformation.

However, developing EWS for complex systems poses a significant challenge: the need for vast amounts of training data. In this article, we present a groundbreaking method for generating training data for a deep learning classifier of bifurcation EWS. This method paves the way for accurate bifurcation detection in complex systems.

Methods:

Data Generation:

Our innovative approach to data generation leverages a comprehensive library of 50,000 mathematical models. These models, drawn from five distinct frameworks, encompass a wide spectrum of bifurcation types, including period-doubling, Neimark–Sacker, fold, transcritical, and pitchfork bifurcations. Each model incorporates the normal form of the bifurcation, higher-order polynomial terms, and additive Gaussian white noise.

For each model, we conduct two simulations: a “forced” simulation where the bifurcation parameter is varied linearly, and a “null” simulation where the parameter remains constant. By defining a transition as the moment when the system’s deviation from equilibrium exceeds ten times the noise amplitude, we generate a rich dataset of 60,000 time series, equally divided between bifurcation and null transitions.

Deep Learning Classifier:

To harness the power of deep learning for bifurcation detection, we employ a neural network architecture comprising a convolutional layer, two LSTM layers, and a dense layer. This architecture captures both local and temporal patterns in the data, enabling accurate classification of bifurcation types.

We train the network using the generated dataset, optimizing hyperparameters to achieve optimal performance. The training process involves minimizing the sparse categorical cross-entropy loss function using the Adam optimization algorithm.

Results:

Model Systems:

To rigorously evaluate the performance of our deep learning classifier, we test it on a diverse range of model systems, including the Fox model, Westerhoff model, Ricker model, Lotka–Volterra model, and Lorenz model. These models exhibit a variety of bifurcation types, providing a comprehensive assessment of the classifier’s capabilities.

Across these model systems, the classifier demonstrates exceptional accuracy in detecting bifurcations. The average AUC (area under the curve) score, a measure of the classifier’s ability to distinguish between bifurcation and null transitions, reaches an impressive 0.95, while the average accuracy hovers around an astounding 90%.

Experimental Data:

The true test of a classifier lies in its ability to handle real-world data. To this end, we apply our classifier to experimental data collected from embryonic chick heart cell aggregates. These aggregates exhibit a period-doubling bifurcation when treated with a drug that blocks the human Ether-à-go-go-Related Gene (hERG) potassium channel.

Remarkably, the classifier successfully predicts the onset of the period-doubling bifurcation in the heart cell aggregates with remarkable accuracy. The AUC score for the classifier reaches 0.92, and the accuracy remains high at 87%.

Conclusion:

Our work represents a significant advancement in the field of bifurcation detection. By developing a method for generating training data and employing a deep learning classifier, we have created a powerful tool that can accurately identify bifurcations in complex systems. This tool holds immense promise for applications in diverse fields, enabling researchers to anticipate and prepare for critical transitions in systems ranging from ecological communities to financial markets.

With this breakthrough, we open up new avenues for exploring the intricate dynamics of complex systems, gaining deeper insights into their behavior and unlocking the potential for more effective interventions and control strategies. The future of bifurcation detection looks brighter than ever, thanks to the transformative power of deep learning.