In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Decades of research on the neural code underlying spatial navigation have revealed a diverse set of neural response properties. The Entorhinal Cortex (EC) of the mammalian brain contains a rich set of spatial correlates, including grid cells which encode space using tessellating patterns. However, the mechanisms and functional significance of these spatial representations remain largely mysterious. As a new way to understand these neural representations, we trained recurrent neural networks (RNNs) to perform navigation tasks in 2D arenas based on velocity inputs. Surprisingly, we find that grid-like spatial response patterns emerge in trained networks, along with units that exhibit other spatial correlates, including border cells and band-like cells. All these different functional types of neurons have been observed experimentally. The order of the emergence of grid-like and border cells is also consistent with observations from developmental studies. Together, our results suggest that grid cells, border cells and others as observed in EC may be a natural solution for representing space efficiently given the predominant recurrent connections in the neural circuits.

To our knowledge, this is the first study to show how neural representations of space, including grid-like cells and border cells as observed in the brain, could emerge from training a recurrent neural network to perform navigation tasks.

Whereas it is believed that techniques such as Adam, batch normalization and, more recently, SeLU nonlinearities ``solve'' the exploding gradient problem, we show that this is not the case and that in a range of popular MLP architectures, exploding gradients exist and that they limit the depth to which networks can be effectively trained, both in theory and in practice. We explain why exploding gradients occur and highlight the {\it collapsing domain problem}, which can arise in architectures that avoid exploding gradients. 

 ResNets have significantly lower gradients and thus can circumvent the exploding gradient problem, enabling the effective training of much deeper networks, which we show is a consequence of a surprising mathematical property. By noticing that {\it any neural network is a residual network}, we devise the {\it residual trick}, which reveals that introducing skip connections simplifies the network mathematically, and that this simplicity may be the major cause for their success.
We show that in contras to popular wisdom, the exploding gradient problem has not been solved and that it limits the depth to which MLPs can be effectively trained. We show why gradients explode and how ResNet handles them.

Partially observable Markov decision processes (POMDPs) are a widely-used framework to model decision-making with uncertainty about the environment and under stochastic outcome. In conventional POMDP models, the observations that the agent receives originate from fixed known distribution. However, in a variety of real-world scenarios the agent has an active role in its perception by selecting which observations to receive. Due to combinatorial nature of such selection process, it is computationally intractable to integrate the perception decision with the planning decision. To prevent such expansion of the action space, we propose a greedy strategy for observation selection that aims to minimize the uncertainty in state. 
 We develop a novel point-based value iteration algorithm that incorporates the greedy strategy to achieve near-optimal uncertainty reduction for sampled belief points. This in turn enables the solver to efficiently approximate the reachable subspace of belief simplex by essentially separating computations related to perception from planning.
 Lastly, we implement the proposed solver and demonstrate its performance and computational advantage in a range of robotic scenarios where the robot simultaneously performs active perception and planning.

We develop a point-based value iteration solver for POMDPs with active perception and planning tasks.