Spiking neural networks are a promising approach towards next-generation models of the brain in computational neuroscience. Moreover, compared to classic artificial neural networks, they could serve as an energy-efficient deployment of AI by enabling fast computation in specialized neuromorphic hardware. However, training deep spiking neural networks, especially in an unsupervised manner, is challenging and the performance of a spiking model is significantly hindered by dead or bursting neurons. Here, we apply end-to-end learning with membrane potential-based backpropagation to a spiking convolutional auto-encoder with multiple trainable layers of leaky integrate-and-fire neurons. We propose bio-inspired regularization methods to control the spike density in latent representations. In the experiments, we show that applying regularization on membrane potential and spiking output successfully avoids both dead and bursting neurons and significantly decreases the reconstruction error of the spiking auto-encoder. Training regularized networks on the MNIST dataset yields image reconstruction quality comparable to non-spiking baseline models (deterministic and variational auto-encoder) and indicates improvement upon earlier approaches. Importantly, we show that, unlike the variational auto-encoder, the spiking latent representations display structure associated with the image class.