Spiking neural networks (SNNs) based on the leaky integrate and fire (LIF) model have been applied to energy-efficient temporal and spatiotemporal processing tasks. Due to the bioplausible neuronal dynamics and simplicity, LIF-SNN benefits from event-driven processing, however, usually face the embarrassment of reduced performance. This may because, in LIF-SNN, the neurons transmit information via spikes. To address this issue, in this work, we propose a leaky integrate and analog fire (LIAF) neuron model so that analog values can be transmitted among neurons, and a deep network termed LIAF-Net is built on it for efficient spatiotemporal processing. In the temporal domain, LIAF follows the traditional LIF dynamics to maintain its temporal processing capability. In the spatial domain, LIAF is able to integrate spatial information through convolutional integration or fully connected integration. As a spatiotemporal layer, LIAF can also be used with traditional artificial neural network (ANN) layers jointly. In addition, the built network can be trained with backpropagation through time (BPTT) directly, which avoids the performance loss caused by ANN to SNN conversion. Experiment results indicate that LIAF-Net achieves comparable performance to the gated recurrent unit (GRU) and long short-term memory (LSTM) on bAbI question answering (QA) tasks and achieves state-of-the-art performance on spatiotemporal dynamic vision sensor (DVS) data sets, including MNIST-DVS, CIFAR10-DVS, and DVS128 Gesture, with much less number of synaptic weights and computational overhead compared with traditional networks built by LSTM, GRU, convolutional LSTM (ConvLSTM), or 3-D convolution (Conv3D). Compared with traditional LIF-SNN, LIAF-Net also shows dramatic accuracy gain on all these experiments. In conclusion, LIAF-Net provides a framework combining the advantages of both ANNs and SNNs for lightweight and efficient spatiotemporal information processing.