Video analysis is a computer vision task that is useful for many applications like surveillance, human-machine interaction, and autonomous vehicles. Deep Convolutional Neural Networks (CNNs) are currently the state-of-the-art methods for video analysis. However they have high computational costs, and need a large amount of labeled data for training. In this paper, we use Convolutional Spiking Neural Networks (CSNNs) trained with the unsupervised Spike Timing-Dependent Plasticity (STDP) learning rule for action classification. These networks represent the information using asynchronous low-energy spikes. This allows the network to be more energy efficient and neuromorphic hardware-friendly. However, the behaviour of CSNNs is not studied enough with spatio-temporal computer vision models. Therefore, we explore transposing two-stream neural networks into the spiking domain. Implementing this model with unsupervised STDP-based CSNNs allows us to further study the performance of these networks with video analysis. In this work, we show that two-stream CSNNs can successfully extract spatio-temporal information from videos despite using limited training data, and that the spiking spatial and temporal streams are complementary. We also show that using a spatio-temporal stream within a spiking STDP-based two-stream architecture leads to information redundancy and does not improve the performance.