This paper introduces a new system for dynamic visual recognition that combines bio-inspired hardware with a brain-like spiking neural network. The system is designed to take data from a dynamic vision sensor (DVS) that simulates the functioning of the human retina by producing an address event output (spike trains) based on the movement of objects. The system then convolutes the spike trains and feeds them into a brain-like spiking neural network, called NeuCube, which is organized in a three-dimensional manner, representing the organization of the primary visual cortex. Spatio-temporal patterns of the data are learned during a deep unsupervised learning stage, using spike-timing-dependent plasticity. In a second stage, supervised learning is performed to train the network for classification tasks. The convolution algorithm and the mapping into the network mimic the function of retinal ganglion cells and the retinotopic organization of the visual cortex. The NeuCube architecture can be used to visualize the deep connectivity inside the network before, during, and after training and thereby allows for a better understanding of the learning processes. The method was tested on the benchmark MNIST-DVS dataset and achieved a classification accuracy of 92.90%. The paper discusses advantages and limitations of the new method and concludes that it is worth exploring further on different datasets, aiming for advances in dynamic computer vision and multimodal systems that integrate visual, aural, tactile, and other kinds of information in a biologically plausible way.