Real-time Instantaneous Phase Estimation Using a Deep Dual-Branch Complex Neural Network

Estimating the instantaneous phase of neural oscillations is crucial for technology that interfaces with the brain, such as brain-computer interfaces (BCIs) and neuromodulation systems. In these systems, phase information from the oscillating neural signal can be used to guide subsequent decisions to apply experimental perturbation. Traditional methods for phase estimation rely on the Hilbert transform computed using the Discrete Fourier Transform (DFT), which introduces a phase lag due to dependency on past and present signal values. This paper proposes a deep learning algorithm utilizing a dual-branch structure similar to the complex wavelet transform to generate a pseudo-complex valued signal for instantaneous phase estimation. The network has Discrete Cosine Transform (DCT) layers, which help to extract latent space representations for the real and imaginary signal components, respectively. An additional design goal was to make this Deep Learning (DL)-based algorithm suitable for deployment on portable edge devices with limited computing resources such as field-programmable gate arrays (FPGAs). This work demonstrates a proof-of-principle for real-time instantaneous phase estimation in neuromodulation applications. Our generalized model achieves an improvement of 40.3% in phase estimation accuracy over the endpoint-corrected Hilbert Transform (ecHT) method and an improvement of 9.2% over conventional deep learning model architectures.