Cognitive maps are mental representations of spatial and conceptual relationships in an environment, and are critical for flexible behavior. To form these abstract maps, the hippocampus has to learn to separate or merge aliased observations appropriately in different contexts in a manner that enables generalization and efficient planning. Here we propose a specific higher-order graph structure, clone-structured cognitive graph (CSCG), which forms clones of an observation for different contexts as a representation that addresses these problems. CSCGs can be learned efficiently using a probabilistic sequence model that is inherently robust to uncertainty. We show that CSCGs can explain a variety of cognitive map phenomena such as discovering spatial relations from aliased sensations, transitive inference between disjoint episodes, and formation of transferable schemas. Learning different clones for different contexts explains the emergence of splitter cells observed in maze navigation and event-specific responses in lap-running experiments. Moreover, learning and inference dynamics of CSCGs offer a coherent explanation for disparate place cell remapping phenomena. By lifting aliased observations into a hidden space, CSCGs reveal latent modularity useful for hierarchical abstraction and planning. Altogether, CSCG provides a simple unifying framework for understanding hippocampal function, and could be a pathway for forming relational abstractions in artificial intelligence.

We study the behavior of a fundamental tool in sparse statistical modeling -the bestsubset selection procedure (aka "best-subsets"). Assuming that the underlying linear model is sparse, it is well known, both in theory and in practice, that the best-subsets procedure works extremely well in terms of several statistical metrics (prediction, estimation and variable selection) when the signal to noise ratio (SNR) is high. However, its performance degrades substantially when the SNR is low -it is outperformed in predictive accuracy by continuous shrinkage methods, such as ridge regression and the Lasso. We explain why this behavior should not come as a surprise, and contend that the original version of the classical best-subsets procedure was, perhaps, not designed to be used in the low SNR regimes. We propose a close cousin of best-subsets, namely, its q -regularized version, for q ∈ {1, 2}, which (a) mitigates, to a large extent, the poor predictive performance of best-subsets in the low SNR regimes; (b) performs favorably and generally delivers a substantially sparser model when compared to the best predictive models available via ridge regression and the Lasso. Our estimator can be expressed as a solution to a mixed integer second order conic optimization problem and, hence, is amenable to modern computational tools from mathematical optimization. We explore the theoretical properties of the predictive capabilities of the proposed estimator and complement our findings via several numerical experiments.

Cognitive maps enable us to learn the layout of environments, encode and retrieve episodic memories, and navigate vicariously for mental evaluation of options. A unifying model of cognitive maps will need to 5 explain how the maps can be learned scalably with sensory observations that are non-unique over multiple spatial locations (aliased), retrieved efficiently in the face of uncertainty, and form the fabric of efficient hierarchical planning. We propose learning higher-order graphs -structured in a specific way that allows efficient learning, hierarchy formation, and inference -as the general principle that connects these different desiderata. We show that these graphs can be learned efficiently from experienced sequences using a 10 cloned Hidden Markov Model (CHMM), and uncertainty-aware planning can be achieved using messagepassing inference. Using diverse experimental settings, we show that CHMMs can be used to explain the emergence of context-specific representations, formation of transferable structural knowledge, transitive inference, shortcut finding in novel spaces, remapping of place cells, and hierarchical planning. Structured higher-order graph learning and probabilistic inference might provide a simple unifying framework for un-15 derstanding hippocampal function, and a pathway for relational abstractions in artificial intelligence.properties of place cells and grid cells [8]. Yet another recent model casts spatial and non-spatial problems as a connected graph with neural responses as efficient representations of this graph [9]. Unfortunately, 30 both these models fail to explain several experimental observations such as the discovery of place cells that encode routes [10,11], remapping in place cells [12], a recent discovery of place cells that do not encode goal value [13], and flexible planning after learning the environment.Here, we propose that learning higher-order graphs of sequential events might be an underlying principle of cognitive maps, and propose a specific representational structure that aids in learning, memory integration, 35 retrieval of episodes, and navigation. In particular, we demonstrate that this representational structure can be represented as a probabilistic sequence model -the cloned Hidden Markov Model (CHMM). We show that sequence learning in CHMMs can explain a variety of cognitive maps phenomena such as discovering spatial maps from random walks under aliased and disjoint sensory experiences, transferable structural knowledge, finding shortcuts, and hierarchical planning and physiological findings such as remapping of place cells, 40 and route-specific encoding. Notably, all these properties emerge from a simple model that is easy to train, scale, and perform inference on. Cloned Hidden Markov Model as a model of cognitive mapsCHMMs are based on Dynamic Markov Coding (DMC) [14], an idea for representing higher-order sequences by splitting, or cloning, observed states. For example, a first order Markov chain representing the 45 sequences A-C-E and B-C-D will also assi...

Understanding the information processing roles of cortical circuits is an outstanding problem in neuroscience and artificial intelligence. Theory-driven efforts will be required to tease apart the functional logic of cortical circuits from the vast amounts of experimental data on cortical connectivity and physiology. Although the theoretical setting of Bayesian inference has been suggested as a framework for understanding cortical computation, making precise and falsifiable biological mappings need models that tackle the challenge of real world tasks. Based on a recent generative model, Recursive Cortical Networks, that demonstrated excellent performance on visual task benchmarks, we derive a family of anatomically instantiated and functional cortical circuit models. Efficient inference and generalization guided the representational choices in the original computational model. The cortical circuit model is derived by systematically comparing the computational requirements of this model with known anatomical constraints. The derived model suggests precise functional roles for the feed-forward, feedback, and lateral connections observed in different laminae and columns, assigns a computational role for the path through the thalamus, predicts the interactions between blobs and inter-blobs, and offers an algorithmic explanation for the innate inter-laminar connectivity between clonal neurons within a cortical column. The model also explains several visual phenomena, including the subjective contour effect, and neon-color spreading effect, with circuit-level precision. Our work paves a new path forward in understanding the logic of cortical and thalamic circuits.

Learning Compact High-Dimensional Models in Noisy Environments Building compact, interpretable statistical models where the output depends upon a small number of input features is a well-known problem in modern analytics applications. A fundamental tool used in this context is the prominent best subset selection (BSS) procedure, which seeks to obtain the best linear fit to data subject to a constraint on the number of nonzero features. Whereas the BSS procedure works exceptionally well in some regimes, it performs pretty poorly in out-of-sample predictive performance when the underlying data are noisy, which is quite common in practice. In this paper, we explore this relatively less-understood overfitting behavior of BSS in low-signal noisy environments and propose alternatives that appear to mitigate such shortcomings. We study the theoretical statistical properties of our proposed regularized BSS procedure and show promising computational results on various data sets, using tools from integer programming and first-order methods.