1. Periodic representation 2. Modular organization with different periods: – relatively narrow range of periods – no grid with very low or high spatial frequencies; e.g. 0.3 ~ 1.3 m [Hafting 2005, Stensola 2012] 3. Uniform phase coverage within a module. An Optimization Principle for the Emergence of Grid Cell Codes Tzuhsuan Ma1 and Ila Fiete1 1Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts Definition of grid cells Inferred properties of grid representation Background / past work on emergence of grid cells Results Advantage of using constant frequency ratio? Exploring contribution of the loss terms Conclusions 1. We find a minimal set of sufficient conditions for the emergence of grid cell codes. 2. These conditions demand a high-capacity, robust, and translationally-invariant code, and we find that grid cell code is the only solution satisfying them simultaneously. 3. We find no structure in period ratio in these conditions. The observed constant period ratio must come from the other constraints. 1. Remarkable advantages in coding: – exponential-capacity code [Fiete 2008] – robust, error-correcting code [Sreenivasan 2011] – accurate path-integration code [Burak 2009] – parallel operations [Fiete 2008], and many others. 1. Unsupervised, single module – Hebbian learning of recurrent weights with place-cell inputs → grid cells [Widloski 2014] – Hebbian learning of feedforward weights with place-cell inputs → grid cells [Dordek 2016] 2. Supervised – Learning of RNNs with a spatial integration task → periodic responses, no modules [Cueva 2018] – Learning of RNNs with a spatial integration task and a 50%-drop out layer → periodic responses with modules [Banino 2018] Cons: 1) Results largely depend on many constraints and fine-tuning of many hyperparameters. 2) No single explicit objective → architecture, dynamics, input/output encoding, and various other constraints all influence the solution that emegences. References 1. Banino, A. et al. Vector-based navigation using grid-like representations in artificial agents. Nature 26, (2018). 2. Burak, Y. & Fiete, I. R. Accurate path integration in continuous attractor network models of grid cells. PLoS Comput. Biol. 5, (2009). 3. Cueva, C. J. & Wei, X.-X. Emergence of grid-like representations by training recurrent neural networks to perform spatial localization. ICLR 2018 1–19 (2018). 4. Dordek, Y., Soudry, D., Meir, R. & Derdikman, D. Extracting grid cell characteristics from place cell inputs using non-negative principal compo- nent analysis. Elife 5, 1–36 (2016). 5. Fiete, I. R., Burak, Y. & Brookings, T. What Grid Cells Convey about Rat Location. J. Neurosci. 28, 6858–6871 (2008). 6. Hafting, T., Fyhn, M., Molden, S., Moser, M. B. & Moser, E. I. Microstructure of a spatial map in the entorhinal cortex. Nature 436, 801–806 (2005). 7. Sreenivasan, S. & Fiete, I. R. Grid cells generate an analog error-correcting code for singularly precise neural computation. Nat. Neurosci. 14, 1330–1337 (2011). 8. Stensola, H. et al. The entorhinal grid map is discretized. Nature 492, 72–78 (2012). 9. Widloski, J. & Fiete, I. R. A Model of Grid Cell Development through Spatial Exploration and Spike Time-Dependent Plasticity. Neuron 83, 481–495 (2014). 10. Wei, X. X., Prentice, J. & Balasubramanian, V. A principle of economy predicts the functional architecture of grid cells. Elife 4, 1–29 (2015). What do we want to optimize? The loss functions Loss 1 maximizes coding separability. → high capacity robust code. Loss 2 minimizes variance along the diagonals of distance matrix. → translationally invariant (TI) code. Loss 3 minimizes sparsity of the representation. → representation with narrow range of firing rate. Our appoach 1. We define an optimization problem purely from coding perspective. 2. Our setup is transparent without implicit dynamical constraints of RNNs position (m) cell i initial tuning curves initial spectrum final unsorted tuning