Event-based vision sensors encode local pixel-wise brightness changes in streams of events rather than image frames and yield sparse, energy-efficient encodings of scenes, in addition to low latency, high dynamic range, and lack of motion blur. Recent progress in object recognition from event-based sensors has come from conversions of deep neural networks, trained with backpropagation. However, using these approaches for event streams requires a transformation to a synchronous paradigm, which not only loses computational efficiency, but also misses opportunities to extract spatio-temporal features. In this article we propose a hybrid architecture for end-to-end training of deep neural networks for event-based pattern recognition and object detection, combining a spiking neural network (SNN) backbone for efficient event-based feature extraction, and a subsequent analog neural network (ANN) head to solve synchronous classification and detection tasks. This is achieved by combining standard backpropagation with surrogate gradient training to propagate gradients through the SNN. Hybrid SNN-ANNs can be trained without conversion, and result in highly accurate networks that are substantially more computationally efficient than their ANN counterparts. We demonstrate results on event-based classification and object detection datasets, in which only the architecture of the ANN heads need to be adapted to the tasks, and no conversion of the event-based input is necessary. Since ANNs and SNNs require different hardware paradigms to maximize their efficiency, we envision that SNN backbone and ANN head can be executed on different processing units, and thus analyze the necessary bandwidth to communicate between the two parts. Hybrid networks are promising architectures to further advance machine learning approaches for event-based vision, without having to compromise on efficiency.