Artificial intelligence has a strong need for faster and more energy-efficient solutions, especially for computation performed at the sensor edge. On-chip photonic neural networks (PNNs) offer a promising solution for high speeds and energy efficiency. A less explored side of PNNs is their application to time-series data, which is often the case for real-world sensor applications. While PNNs promise high speeds and energy-efficient solutions, no good use cases have been proposed. This report will first review the state of the art of PNNs. It will be seen that to solve time-series tasks, photonic continuous-time recurrent neural networks (CTRNNs) are required. The dynamics of CTRNNs are thoroughly explored to leverage obtained insights to recommend novel practical applications. This is done through simple examples, and applied to a real machine learning task. It was seen that on a real classification task the network learned two distinct fixed points corresponding to the classes. A link between the time constant of the continuous-time neurons and the temporal dynamics of the task is also found. Two general directions for novel applications are then proposed. Firstly, photonic PNNs can be slowed down to match the task. Opto-electronic PNNs allow for more control of the time constant, and on-chip photonic filter neurons are suggested. Secondly, the high speeds of photonic neural networks can also be directly leveraged. Extremely fast convergence of photonic CTRNNs can be utilized for Modern Hopfield networks, pathfinding algorithms, and time-dependent optimization problems such as for example Model Predictive Control.