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Title: Emergence of grid-like representations by training recurrent neural networks to perform spatial localization

Paper content:
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.
Understanding the neural code in the brain has long been driven by studying feed-forward architectures, starting from Hubel and Wiesel's famous proposal on the origin of orientation selectivity in primary visual cortex BID19 .
Inspired by the recent development in deep learning BID25 BID30 BID18 BID39 , there has been a burst of interest in applying deep feedforward models, in particular convolutional neural networks (CNN) BID29 , to study the sensory systems, which hierarchically extract useful features from sensory inputs (see e.g., BID61 ; BID24 ; BID22 ; BID60 ).For
more cognitive tasks, neural systems often need to maintain certain internal representations of relevant variables in the absence of external stimuli-a process that requires more than feature extraction. We
will focus on spatial navigation, which typically requires the brain to maintain a representation of self-location and update it according to the animal's movements and landmarks of the environment. Physiological
studies done in rodents and other mammals (including humans, non-human primates and bats) have revealed a variety of neural correlates of space in Hippocampus and Entorhinal Cortex (EC), including place cells BID41 , grid cells BID10 BID15 BID11 BID62 BID23 BID20 , along with border cells BID49 , band-like cells BID27 and others (see FIG0 ). In particular
, each grid cell only fires when the animal occupies a distinct set of physical locations, and strikingly these locations lie on a lattice. The study of
the neural underpinning of spatial cognition has provided an important window into how high-level cognitive functions are supported in the brain BID0 .How might the
spatial navigation task be solved using a network of neurons? Recurrent neural
networks (RNNs) BID18 BID12 BID43 BID54 BID13 BID53 seem particularly useful for these tasks. Indeed, recurrent-based
continuous attractor networks have been one popular type of models proposed for the formation of grid cells BID4 BID5 and place cells BID45 . Such models have provided
valuable insights into one set of possible mechanisms that could support the formation of the grids. However, these models typically
rely on fine-tuned connectivity patterns, in particular the models need a subtle yet systematic asymmetry in the connectivity pattern to move the attractor state according to the animal's own movement. The existence of such a specific
2D connectivity in rodent EC remains unclear. Additionally, previous models have
mainly focused on grid cells, while other types of responses that co-exist in the Entorhinal Cortex have been largely ignored. It would be useful to have a unified
model that can simultaneously explain different types of neural responses in EC.Motivated by these considerations, here we present an alternative modeling approach for understanding the representation of space in the neural system. Specifically, we trained a RNN to perform
some spatial navigation tasks. By leveraging the recent development in RNN
training and knowledge of the navigation system in the brain, we show that training a RNN with biologically relevant constraints naturally gives rise to a variety of spatial response profiles as observed in EC, including grid-like responses. To our knowledge, this is the first study to
show that grid-like responses could emerge from training a RNN to perform navigation. Our result implies that the neural representation
in EC may be seen as a natural way for the brain to solve the navigation task efficiently BID55 . More generally, it suggests that RNNs can be a powerful
tool for understanding the neural mechanisms of certain high-level cognitive functions. recorded when an animal navigates in a square environment
, replotted from BID27 , with the heat map representing the firing rate of this neuron as a function of the animal's location (red corresponds to high firing rate); a "band-like" cell from BID27 ; a border cell from BID49 ; an irregular spatially tuned cell from BID7 ; a "speed cell" from BID26 , which exhibits roughly linear dependence on the rodent's running speed; a "heading direction cell" from BID46 , which shows systematic change of firing rate depending on animal's heading direction. b) The network consists of N = 100 recurrently connected
units (or neurons) which receive two external inputs, representing the animal's speed and heading direction. The two outputs linearly weight the neurons in the RNN.
The goal of training is to make the responses of the two
output neurons accurately represent the animal's physical location. c) Typical trajectory after training. As shown, the output
of the RNN can accurately, though not
perfectly, track the animal's location during navigation.
In this paper, we trained RNNs to perform path integration (dead-reckoning) in 2D arenas.
We found that after training RNNs with appropriate regularization, the model neurons exhibit a variety of spatial and velocity tuning profiles that match neurophysiology in EC.
What's more, there is also similarity in terms of when these distinct neuron types emerge during training/development.
The EC has long been thought to be involved in path integration and localization of the animal's location .
The general agreement between the different response properties in our model and the neurophysiology provide strong evidence supporting the hypothesis that the neural population in EC may provide an efficient code for representation self-locations based on the velocity input.Recently, there has been increased interest in using complex neural network models to understand the neural code.
But the focus has been on using feedforward architectures, in particular CNNs BID29 .
Given the abundant recurrent connections in the brain, it seems a particularly fruitful avenue to take advantage of the recent development in RNNs to help with neuroscience questions BID34 BID50 BID37 BID53 .
Here, we only show one instance following this approach.
However, the insight from this work could be general, and potentially useful for other cognitive functions as well.The finding that metabolic constraints lead to the emergence of grid-like responses may be seen as conceptually related to the efficient coding hypothesis in visual processing BID1 , in particular the seminal work on the emergence of the V1-like Gabor filters in a sparse coding model by BID42 .
Indeed, our work is partly inspired by these results.
While there are conceptual similarities, however, we should also note there are differences between the sparse coding work and ours.
First, the sparsity constraint in sparse coding can be naturally viewed as a particular prior while in the context of the recurrent network, it is difficult to interpret that way.
Second, the grid-like responses are not the most sparse solution one could imagine.
In fact, they are still quite dense compared to a more spatially localized representation.
Third, the grid-like patterns that emerged in our network are not filters based on the raw input, rather the velocity inputs need to be integrated first in order to encode spatial locations.
Our work is also inspired by recent work using the efficient coding idea to explain the functional architecture of the grid cells BID55 .
It has been shown that efficient coding considerations could explain the particular set of grid scales observed in rodents BID52 .
However, in that work, the firing patterns of the neurons are assumed to have a lattice structure to start with.
Furthermore, our work is related to the study by Sussillo and others BID53 , in which they show that regularization of RNN models are important for generating solutions that are similar to the neural activity observed in motor cortex.
In Sussillo et al., a smoothness constraint together with others lead to simple oscillatory neural dynamics that well matches the neural data.
We have not incorporated a smoothness constraint into our network.Additionally, we note that there are a few recent studies which use place cells as the input to generate grid cells BID8 BID51 , which are fundamentally different from our work.
In these feedforward network models, the grid cells essentially perform dimensionality reduction based on the spatial input from place cells.
However, the main issue with these models is that, it is unclear how place cells acquire spatial tuning in the first place.
To the contrary, our model takes the animal's velocity as the input, and addresses the question of how the spatial tuning can be generated from such input, which are known to exist in EC BID46 BID26 .
In another related study BID21 , the authors train a RNN with LSTM units BID18 to perform different navigation tasks.
However, no grid-like spatial firing patterns are reported.Although our model shows a qualitative match to the neural responses observed in the EC, nonetheless it has several major limitations, with each offering interesting future research directions.
First, the learning rule we use seems to be biologically implausible.
We are interested in exploring how a more biologically plausible learning rule could give rise to similar results BID32 BID37 BID14 .
Second, the simulation results do not show a variety of spatial scales in grid-like cells.
Experimentally, it is known that grid cells have multiple spatial scales, that scale geometrically with a ratio 1.4 BID52 , and this particular scale ratio is predicted by efficient coding of space BID55 .
We are investigating how to modify the model to get a hierarchy of spatial scales, perhaps by incorporating more neurons or modifying the regularization.
Last but not least, we have focused on the representation produced by the trained RNN.
An equally important set of questions concern how the networks actually support the generation of such a representation.
As a preliminary effort, we have examined the connectivity patterns of the trained network, and they do not seem to resemble the connectivity patterns required by standard attractor network models.
Maybe this should not be seen as too surprising.
After all, the trained networks can produce a diverse set of neural responses, while the previous models only led to grid responses.
It would be interesting for future work to systematically examine the questions related to the underlying mechanisms.
To quantify the speed selectivity of each unit we first fit a line to the tuning curve of unit activity as a function of speed.
The speed selectivity is the absolute value of the slope.
If the unit activity is not modulated by speed then the speed selectivity is 0.
To quantify the direction selectivity of each unit we calculated the average unit activity as a function of direction input and then took the maximum minus minimum of this tuning curve.
If the unit activity is not modulated by direction then the direction selectivity is 0.
To quantify the spatial selectivity we used lifetime sparseness BID56 .
If the unit activity is not modulated by spatial location then the spatial selectivity is 0.
Each dot in the figures below show the selectivity for a single unit.

assistant: text: 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.