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1. Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells

   Xu, D. ; Gao, R. ; Zhang, W. H. ; Wei, X. X. ; Wu, Y. N. ( December 2022 , Symmetry and Geometry in Neural Representations (NeurReps Workshop), Neural Information Processing Systems (NeurIPS 2022))

   The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the self-position of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of the recurrent transformation as a representation of self-motion. Geometrically, we study the conformal isometry of the Lie group representation where the local displacement of the activity vector in the neural space is proportional to the local displacement of the agent in the 2D physical space. Topologically, the compact abelian Lie group representation automatically leads to the torus topology commonly assumed and observed in neuroscience. We then focus on a simple non-linear recurrent model that underlies the continuous attractor neural networks of grid cells. Our numerical experiments show that conformal isometry leads to hexagon periodic patterns in the grid cell responses and our model is capable of accurate path integration. 

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2. Explaining heterogeneity in medial entorhinal cortex with task-driven neural networks

   https://doi.org/10.1101/2021.10.30.466617  

   Nayebi, Aran ; Attinger, Alexander ; Campbell, Malcolm G ; Hardcastle, Kiah ; Low, Isabel IC ; Mallory, Caitlin S ; Mel, Gabriel C ; Sorscher, Ben ; Williams, Alex ; Ganguli, Surya ; et al ( December 2021 , NeurIPS)

   Medial entorhinal cortex (MEC) supports a wide range of navigational and memory related behaviors. Well-known experimental results have revealed specialized cell types in MEC — e.g. grid, border, and head-direction cells — whose highly stereotypical response profiles are suggestive of the role they might play in sup- porting MEC functionality. However, the majority of MEC neurons do not exhibit stereotypical firing patterns. How should the response profiles of these more “het- erogeneous” cells be described, and how do they contribute to behavior? In this work, we took a computational approach to addressing these questions. We first performed a statistical analysis that shows that heterogeneous MEC cells are just as reliable in their response patterns as the more stereotypical cell types, suggest- ing that they have a coherent functional role. Next, we evaluated a spectrum of candidate models in terms of their ability to describe the response profiles of both stereotypical and heterogeneous MEC cells. We found that recently developed task-optimized neural network models are substantially better than traditional grid cell-centric models at matching most MEC neuronal response profiles — including those of grid cells themselves — despite not being explicitly trained for this pur- pose. Specific choices of network architecture (such as gated nonlinearities and an explicit intermediate place cell representation) have an important effect on the ability of the model to generalize to novel scenarios, with the best of these models closely approaching the noise ceiling of the data itself. We then performed in silico experiments on this model to address questions involving the relative functional relevance of various cell types, finding that heterogeneous cells are likely to be just as involved in downstream functional outcomes (such as path integration) as grid and border cells. Finally, inspired by recent data showing that, going beyond their spatial response selectivity, MEC cells are also responsive to non-spatial rewards, we introduce a new MEC model that performs reward-modulated path integration. We find that this unified model matches neural recordings across all variable-reward conditions. Taken together, our results point toward a conceptually principled goal-driven modeling approach for moving future experimental and computational efforts beyond overly-simplistic single-cell stereotypes. 

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3. Multi-scale representation learning for spatial feature distributions using grid cells

   Mai, Gengchen ( January 2020 , ICLR 2020)

   Unsupervised text encoding models have recently fueled substantial progress in NLP. The key idea is to use neural networks to convert words in texts to vector space representations based on word positions in a sentence and their contexts, which are suitable for end-to-end training of downstream tasks. We see a strikingly similar situation in spatial analysis, which focuses on incorporating both absolute positions and spatial contexts of geographic objects such as POIs into models. A general-purpose representation model for space is valuable for a multitude of tasks. However, no such general model exists to date beyond simply applying discretization or feed-forward nets to coordinates, and little effort has been put into jointly modeling distributions with vastly different characteristics, which commonly emerges from GIS data. Meanwhile, Nobel Prize-winning Neuroscience research shows that grid cells in mammals provide a multi-scale periodic representation that functions as a metric for location encoding and is critical for recognizing places and for path-integration. Therefore, we propose a representation learning model called Space2Vec to encode the absolute positions and spatial relationships of places. We conduct experiments on two real-world geographic data for two different tasks: 1) predicting types of POIs given their positions and context, 2) image classification leveraging their geo-locations. Results show that because of its multi-scale representations, Space2Vec outperforms well-established ML approaches such as RBF kernels, multi-layer feed-forward nets, and tile embedding approaches for location modeling and image classification tasks. Detailed analysis shows that all baselines can at most well handle distribution at one scale but show poor performances in other scales. In contrast, Space2Vec's multi-scale representation can handle distributions at different scales. 

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4. Acoustic metamaterials for realizing a scalable multiple phi-bit unitary transformation

   https://doi.org/10.1063/5.0188462  

   Runge, K ; Deymier, P A ; Hasan, M A ; Lata, T D ; Levine, J A ( February 2024 , AIP Advances)

   The analogy between acoustic modes in nonlinear metamaterials and quantum computing platforms constituted of correlated two-level systems opens new frontiers in information science. We use an inductive procedure to demonstrate scalable initialization of and scalable unitary transformations on superpositions of states of multiple correlated logical phi-bits, classical nonlinear acoustic analog of qubits. A multiple phi-bit state representation as a complex vector in a high-dimensional, exponentially scaling Hilbert space is shown to correspond with the state of logical phi-bits represented in a low-dimensional linearly scaling physical space of an externally driven acoustic metamaterial. Manipulation of the phi-bits in the physical space enables the implementation of a non-trivial multiple phi-bit unitary transformation that scales exponentially. This scalable transformation operates in parallel on the components of the multiple phi-bit complex state vector, requiring only a single physical action on the metamaterial. This work demonstrates that acoustic metamaterials offer a viable path toward achieving massively parallel information processing capabilities that can challenge current quantum computing paradigms. 

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5. Deformation Robust Roto-Scale-Translation Equivariant CNNs

   Mars Gao, L ; Lin, Guang ; Zhu, Wei ( July 2022 , Transactions on machine learning research)

   Incorporating group symmetry directly into the learning process has proved to be an effective guideline for model design. By producing features that are guaranteed to transform covariantly to the group actions on the inputs, group-equivariant convolutional neural networks (G-CNNs) achieve significantly improved generalization performance in learning tasks with intrinsic symmetry. General theory and practical implementation of G-CNNs have been studied for planar images under either rotation or scaling transformation, but only individually. We present, in this paper, a roto-scale-translation equivariant CNN (RST -CNN), that is guaranteed to achieve equivariance jointly over these three groups via coupled group convolutions. Moreover, as symmetry transformations in reality are rarely perfect and typically subject to input deformation, we provide a stability analysis of the equivariance of representation to input distortion, which motivates the truncated expansion of the convolutional filters under (pre-fixed) low-frequency spatial modes. The resulting model provably achieves deformation-robust RST equivariance, i.e., the RST symmetry is still “approximately” preserved when the transformation is “contaminated” by a nuisance data deformation, a property that is especially important for out-of-distribution generalization. Numerical experiments on MNIST, Fashion-MNIST, and STL-10 demonstrate that the proposed model yields remarkable gains over prior arts, especially in the small data regime where both rotation and scaling variations are present within the data. 

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