Linear reinforcement learning in planning, grid fields, and cognitive control

Critical intelligence: Computing defensive behaviour

Characterising the mechanisms underlying naturalistic defensive behavior remains a significant challenge. While substantial progress has been made in unravelling the neural basis of tightly constrained behaviors, a critical gap persists in our comprehension of the circuits that implement algorithms capable of generating the diverse defensive responses observed outside experimental restrictions. Recent advancements in neuroscience technology now allow for an unprecedented examination of naturalistic behaviour. To help provide a theoretical grounding for this nascent experimental programme, we summarise the main computational and statistical challenges of defensive decision making, encapsulated in the concept of critical intelligence. Next, drawing from an extensive literature in biology, machine learning, and decision theory, we explore a range of candidate solutions to these challenges. While the proposed solutions offer insights into potential adaptive strategies, they also present inherent trade-offs and limitations in their applicability across different biological contexts. Ultimately, we propose series of experiments designed to differentiate between these candidate solutions, providing a roadmap for future investigations into the fundamental defensive algorithms utilized by biological agents and their neural implementation. Thus, our work aims to provide a roadmap towards broader understanding of how complex defensive behaviors are orchestrated in the brain, with implications for both neuroscience research and the development of more sophisticated artificial intelligence systems.

Constructing future behavior in the hippocampal formation through composition and replay

The hippocampus is critical for memory, imagination and constructive reasoning. Recent models have suggested that its neuronal responses can be well explained by state spaces that model the transitions between experiences. Here we use simulations and hippocampal recordings to reconcile these views. We show that if state spaces are constructed compositionally from existing building blocks, or primitives, hippocampal responses can be interpreted as compositional memories, binding these primitives together. Critically, this enables agents to behave optimally in new environments with no new learning, inferring behavior directly from the composition. We predict a role for hippocampal replay in building and consolidating these compositional memories. We test these predictions in two datasets by showing that replay events from newly discovered landmarks induce and strengthen new remote firing fields. When the landmark is moved, replay builds a new firing field at the same vector to the new location. Together, these findings provide a framework for reasoning about compositional memories and demonstrate that such memories are formed in hippocampal replay.

A unified neural representation model for spatial and conceptual computations

The hippocampus and entorhinal cortex encode spaces by spatially local and hexagonal grid activity patterns (place cells and grid cells), respectively. In addition, the same brain regions also implicate neural representations for nonspatial, semantic concepts (concept cells). These observations suggest that neurocomputational mechanisms for spatial knowledge and semantic concepts are related in the brain. However, the exact relationship remains to be understood. Here, we show a mathematical correspondence between a value function for goal-directed spatial navigation and an information measure for word embedding models in natural language processing. Based on this relationship, we integrate spatial and semantic computations into a neural representation model called "disentangled successor information" (DSI). DSI generates biologically plausible neural representations: spatial representations like place cells and grid cells, and concept-specific word representations which resemble concept cells. Furthermore, with DSI representations, we can perform inferences of spatial contexts and words by a common computational framework based on simple arithmetic operations. This computation can be biologically interpreted by partial modulations of cell assemblies of nongrid cells and concept cells. Our model offers a theoretical connection of spatial and semantic computations and suggests possible computational roles of hippocampal and entorhinal neural representations.

Humans rationally balance detailed and temporally abstract world models

How do people model the world's dynamics to guide mental simulation and evaluate choices? One prominent approach, the Successor Representation (SR), takes advantage of temporal abstraction of future states: by aggregating trajectory predictions over multiple timesteps, the brain can avoid the costs of iterative, multi-step mental simulation. Human behavior broadly shows signatures of such temporal abstraction, but finer-grained characterization of individuals' strategies and their dynamic adjustment remains an open question. We developed a task to measure SR usage during dynamic, trial-by-trial learning. Using this approach, we find that participants exhibit a mix of SR and model-based learning strategies that varies across individuals. Further, by dynamically manipulating the task contingencies within-subject to favor or disfavor temporal abstraction, we observe evidence of resource-rational reliance on the SR, which decreases when future states are less predictable. Our work adds to a growing body of research showing that the brain arbitrates between approximate decision strategies. The current study extends these ideas from simple habits into usage of more sophisticated approximate predictive models, and demonstrates that individuals dynamically adapt these in response to the predictability of their environment.

Computational processes of simultaneous learning of stochasticity and volatility in humans

Making adaptive decisions requires predicting outcomes, and this in turn requires adapting to uncertain environments. This study explores computational challenges in distinguishing two types of noise influencing predictions: volatility and stochasticity. Volatility refers to diffusion noise in latent causes, requiring a higher learning rate, while stochasticity introduces moment-to-moment observation noise and reduces learning rate. Dissociating these effects is challenging as both increase the variance of observations. Previous research examined these factors mostly separately, but it remains unclear whether and how humans dissociate them when they are played off against one another. In two large-scale experiments, through a behavioral prediction task and computational modeling, we report evidence of humans dissociating volatility and stochasticity solely based on their observations. We observed contrasting effects of volatility and stochasticity on learning rates, consistent with statistical principles. These results are consistent with a computational model that estimates volatility and stochasticity by balancing their dueling effects.

Understanding dual process cognition via the minimum description length principle

Dual-process theories play a central role in both psychology and neuroscience, figuring prominently in domains ranging from executive control to reward-based learning to judgment and decision making. In each of these domains, two mechanisms appear to operate concurrently, one relatively high in computational complexity, the other relatively simple. Why is neural information processing organized in this way? We propose an answer to this question based on the notion of compression. The key insight is that dual-process structure can enhance adaptive behavior by allowing an agent to minimize the description length of its own behavior. We apply a single model based on this observation to findings from research on executive control, reward-based learning, and judgment and decision making, showing that seemingly diverse dual-process phenomena can be understood as domain-specific consequences of a single underlying set of computational principles.

Linking cognitive strategy, neural mechanism, and movement statistics in group foraging behaviors

Foraging for food is a rich and ubiquitous animal behavior that involves complex cognitive decisions, and interactions between different individuals and species. There has been exciting recent progress in understanding multi-agent foraging behavior from cognitive, neuroscience, and statistical perspectives, but integrating these perspectives can be elusive. This paper seeks to unify these perspectives, allowing statistical analysis of observational animal movement data to shed light on the viability of cognitive models of foraging strategies. We start with cognitive agents with internal preferences expressed as value functions, and implement this in a biologically plausible neural network, and an equivalent statistical model, where statistical predictors of agents' movements correspond to the components of the value functions. We test this framework by simulating foraging agents and using Bayesian statistical modeling to correctly identify the factors that best predict the agents' behavior. As further validation, we use this framework to analyze an open-source locust foraging dataset. Finally, we collect new multi-agent real-world bird foraging data, and apply this method to analyze the preferences of different species. Together, this work provides an initial roadmap to integrate cognitive, neuroscience, and statistical approaches for reasoning about animal foraging in complex multi-agent environments.

Humans adaptively deploy forward and backward prediction

The formation of predictions is essential to our ability to build models of the world and use them for intelligent decision-making. Here we challenge the dominant assumption that humans form only forward predictions, which specify what future events are likely to follow a given present event. We demonstrate that in some environments, it is more efficient to use backward prediction, which specifies what present events are likely to precede a given future event. This is particularly the case in diverging environments, where possible future events outnumber possible present events. Correspondingly, in six preregistered experiments (n = 1,299) involving both simple decision-making and more challenging planning tasks, we find that humans engage in backward prediction in divergent environments and use forward prediction in convergent environments. We thus establish that humans adaptively deploy forward and backward prediction in the service of efficient decision-making.

Spatial evolution of the proton-coupled Mott transition in correlated oxides for neuromorphic computing

The proton-electron coupling effect induces rich spectrums of electronic states in correlated oxides, opening tempting opportunities for exploring novel devices with multifunctions. Here, via modest Pt-aided hydrogen spillover at room temperature, amounts of protons are introduced into SmNiO3-based devices. In situ structural characterizations together with first-principles calculation reveal that the local Mott transition is reversibly driven by migration and redistribution of the predoped protons. The accompanying giant resistance change results in excellent memristive behaviors under ultralow electric fields. Hierarchical tree-like memory states, an instinct displayed in bio-synapses, are further realized in the devices by spatially varying the proton concentration with electric pulses, showing great promise in artificial neural networks for solving intricate problems. Our research demonstrates the direct and effective control of proton evolution using extremely low electric field, offering an alternative pathway for modifying the functionalities of correlated oxides and constructing low-power consumption intelligent devices and neural network circuits.

Local prediction-learning in high-dimensional spaces enables neural networks to plan

Planning and problem solving are cornerstones of higher brain function. But we do not know how the brain does that. We show that learning of a suitable cognitive map of the problem space suffices. Furthermore, this can be reduced to learning to predict the next observation through local synaptic plasticity. Importantly, the resulting cognitive map encodes relations between actions and observations, and its emergent high-dimensional geometry provides a sense of direction for reaching distant goals. This quasi-Euclidean sense of direction provides a simple heuristic for online planning that works almost as well as the best offline planning algorithms from AI. If the problem space is a physical space, this method automatically extracts structural regularities from the sequence of observations that it receives so that it can generalize to unseen parts. This speeds up learning of navigation in 2D mazes and the locomotion with complex actuator systems, such as legged bodies. The cognitive map learner that we propose does not require a teacher, similar to self-attention networks (Transformers). But in contrast to Transformers, it does not require backpropagation of errors or very large datasets for learning. Hence it provides a blue-print for future energy-efficient neuromorphic hardware that acquires advanced cognitive capabilities through autonomous on-chip learning.

Prioritizing replay when future goals are unknown

Although hippocampal place cells replay nonlocal trajectories, the computational function of these events remains controversial. One hypothesis, formalized in a prominent reinforcement learning account, holds that replay plans routes to current goals. However, recent puzzling data appear to contradict this perspective by showing that replayed destinations lag current goals. These results may support an alternative hypothesis that replay updates route information to build a "cognitive map." Yet no similar theory exists to formalize this view, and it is unclear how such a map is represented or what role replay plays in computing it. We address these gaps by introducing a theory of replay that learns a map of routes to candidate goals, before reward is available or when its location may change. Our work extends the planning account to capture a general map-building function for replay, reconciling it with data, and revealing an unexpected relationship between the seemingly distinct hypotheses.

The successor representation subserves hierarchical abstraction for goal-directed behavior

Humans have the ability to craft abstract, temporally extended and hierarchically organized plans. For instance, when considering how to make spaghetti for dinner, we typically concern ourselves with useful "subgoals" in the task, such as cutting onions, boiling pasta, and cooking a sauce, rather than particulars such as how many cuts to make to the onion, or exactly which muscles to contract. A core question is how such decomposition of a more abstract task into logical subtasks happens in the first place. Previous research has shown that humans are sensitive to a form of higher-order statistical learning named "community structure". Community structure is a common feature of abstract tasks characterized by a logical ordering of subtasks. This structure can be captured by a model where humans learn predictions of upcoming events multiple steps into the future, discounting predictions of events further away in time. One such model is the "successor representation", which has been argued to be useful for hierarchical abstraction. As of yet, no study has convincingly shown that this hierarchical abstraction can be put to use for goal-directed behavior. Here, we investigate whether participants utilize learned community structure to craft hierarchically informed action plans for goal-directed behavior. Participants were asked to search for paintings in a virtual museum, where the paintings were grouped together in "wings" representing community structure in the museum. We find that participants' choices accord with the hierarchical structure of the museum and that their response times are best predicted by a successor representation. The degree to which the response times reflect the community structure of the museum correlates with several measures of performance, including the ability to craft temporally abstract action plans. These results suggest that successor representation learning subserves hierarchical abstractions relevant for goal-directed behavior.

Grid codes underlie multiple cognitive maps in the human brain

Grid cells fire at multiple positions that organize the vertices of equilateral triangles tiling a 2D space and are well studied in rodents. The last decade witnessed rapid progress in two other research lines on grid codes-empirical studies on distributed human grid-like representations in physical and multiple non-physical spaces, and cognitive computational models addressing the function of grid cells based on principles of efficient and predictive coding. Here, we review the progress in these fields and integrate these lines into a systematic organization. We also discuss the coordinate mechanisms of grid codes in the human entorhinal cortex and medial prefrontal cortex and their role in neurological and psychiatric diseases.

Curiosity-driven exploration: foundations in neuroscience and computational modeling

Curiosity refers to the intrinsic desire of humans and animals to explore the unknown, even when there is no apparent reason to do so. Thus far, no single, widely accepted definition or framework for curiosity has emerged, but there is growing consensus that curious behavior is not goal-directed but related to seeking or reacting to information. In this review, we take a phenomenological approach and group behavioral and neurophysiological studies which meet these criteria into three categories according to the type of information seeking observed. We then review recent computational models of curiosity from the field of machine learning and discuss how they enable integrating different types of information seeking into one theoretical framework. Combinations of behavioral and neurophysiological studies along with computational modeling will be instrumental in demystifying the notion of curiosity.

Differentiation between fetal and postnatal iron deficiency in altering brain substrates of cognitive control in pre-adolescence

BACKGROUND

Early iron deficiency (ID) is a common risk factor for poorer neurodevelopment, limiting children's potential and contributing to global burden. However, it is unclear how early ID alters the substrate of brain functions supporting high-order cognitive abilities and whether the timing of early ID matters in terms of long-term brain development. This study aimed to examine the effects of ID during fetal or early postnatal periods on brain activities supporting proactive and reactive cognitive control in pre-adolescent children.

METHODS

Participants were part of a longitudinal cohort enrolled at birth in southeastern China between December 2008 and November 2011. Between July 2019 and October 2021, 115 children aged 8-11 years were invited to participate in this neuroimaging study. Final analyses included 71 children: 20 with fetal ID, 24 with ID at 9 months (postnatal ID), and 27 iron-sufficient at birth and 9 months. Participants performed a computer-based behavioral task in a Magnetic Resonance Imaging scanner to measure proactive and reactive cognitive control. Outcome measures included accuracy, reaction times, and brain activity. Linear mixed modeling and the 3dlme command in Analysis of Functional NeuroImages (AFNI) were separately used to analyze behavioral performance and neuroimaging data.

RESULTS

Faster responses in proactive vs. reactive conditions indicated that all groups could use proactive or reactive cognitive control according to contextual demands. However, the fetal ID group was lower in general accuracy than the other 2 groups. Per the demands of cues and targets, the iron-sufficient group showed greater activation of wide brain regions in proactive vs. reactive conditions. In contrast, such condition differences were reversed in the postnatal ID group. Condition differences in brain activation, shown in postnatal ID and iron-sufficient groups, were not found in the fetal ID group. This group specifically showed greater activation of brain regions in the reward pathway in proactive vs. reactive conditions.

CONCLUSIONS

Early ID was associated with altered brain functions supporting proactive and reactive cognitive control in childhood. Alterations differed between fetal and postnatal ID groups. The findings imply that iron supplement alone is insufficient to prevent persisting brain alterations associated with early ID. Intervention strategies in addition to the iron supplement should consider ID timing.

Rapid learning of predictive maps with STDP and theta phase precession

The predictive map hypothesis is a promising candidate principle for hippocampal function. A favoured formalisation of this hypothesis, called the successor representation, proposes that each place cell encodes the expected state occupancy of its target location in the near future. This predictive framework is supported by behavioural as well as electrophysiological evidence and has desirable consequences for both the generalisability and efficiency of reinforcement learning algorithms. However, it is unclear how the successor representation might be learnt in the brain. Error-driven temporal difference learning, commonly used to learn successor representations in artificial agents, is not known to be implemented in hippocampal networks. Instead, we demonstrate that spike-timing dependent plasticity (STDP), a form of Hebbian learning, acting on temporally compressed trajectories known as 'theta sweeps', is sufficient to rapidly learn a close approximation to the successor representation. The model is biologically plausible - it uses spiking neurons modulated by theta-band oscillations, diffuse and overlapping place cell-like state representations, and experimentally matched parameters. We show how this model maps onto known aspects of hippocampal circuitry and explains substantial variance in the temporal difference successor matrix, consequently giving rise to place cells that demonstrate experimentally observed successor representation-related phenomena including backwards expansion on a 1D track and elongation near walls in 2D. Finally, our model provides insight into the observed topographical ordering of place field sizes along the dorsal-ventral axis by showing this is necessary to prevent the detrimental mixing of larger place fields, which encode longer timescale successor representations, with more fine-grained predictions of spatial location.

Opponent Learning with Different Representations in the Cortico-Basal Ganglia Circuits

The direct and indirect pathways of the basal ganglia (BG) have been suggested to learn mainly from positive and negative feedbacks, respectively. Since these pathways unevenly receive inputs from different cortical neuron types and/or regions, they may preferentially use different state/action representations. We explored whether such a combined use of different representations, coupled with different learning rates from positive and negative reward prediction errors (RPEs), has computational benefits. We modeled animal as an agent equipped with two learning systems, each of which adopted individual representation (IR) or successor representation (SR) of states. With varying the combination of IR or SR and also the learning rates from positive and negative RPEs in each system, we examined how the agent performed in a dynamic reward navigation task. We found that combination of SR-based system learning mainly from positive RPEs and IR-based system learning mainly from negative RPEs could achieve a good performance in the task, as compared with other combinations. In such a combination of appetitive SR-based and aversive IR-based systems, both systems show activities of comparable magnitudes with opposite signs, consistent with the suggested profiles of the two BG pathways. Moreover, the architecture of such a combination provides a novel coherent explanation for the functional significance and underlying mechanism of diverse findings about the cortico-BG circuits. These results suggest that particularly combining different representations with appetitive and aversive learning could be an effective learning strategy in certain dynamic environments, and it might actually be implemented in the cortico-BG circuits.

Personalized visual encoding model construction with small data

Quantifying population heterogeneity in brain stimuli-response mapping may allow insight into variability in bottom-up neural systems that can in turn be related to individual's behavior or pathological state. Encoding models that predict brain responses to stimuli are one way to capture this relationship. However, they generally need a large amount of fMRI data to achieve optimal accuracy. Here, we propose an ensemble approach to create encoding models for novel individuals with relatively little data by modeling each subject's predicted response vector as a linear combination of the other subjects' predicted response vectors. We show that these ensemble encoding models trained with hundreds of image-response pairs, achieve accuracy not different from models trained on 20,000 image-response pairs. Importantly, the ensemble encoding models preserve patterns of inter-individual differences in the image-response relationship. We also show the proposed approach is robust against domain shift by validating on data with a different scanner and experimental setup. Additionally, we show that the ensemble encoding models are able to discover the inter-individual differences in various face areas' responses to images of animal vs human faces using a recently developed NeuroGen framework. Our approach shows the potential to use existing densely-sampled data, i.e. large amounts of data collected from a single individual, to efficiently create accurate, personalized encoding models and, subsequently, personalized optimal synthetic images for new individuals scanned under different experimental conditions.

Compositional Sequence Generation in the Entorhinal-Hippocampal System

Neurons in the medial entorhinal cortex exhibit multiple, periodically organized, firing fields which collectively appear to form an internal representation of space. Neuroimaging data suggest that this grid coding is also present in other cortical areas such as the prefrontal cortex, indicating that it may be a general principle of neural functionality in the brain. In a recent analysis through the lens of dynamical systems theory, we showed how grid coding can lead to the generation of a diversity of empirically observed sequential reactivations of hippocampal place cells corresponding to traversals of cognitive maps. Here, we extend this sequence generation model by describing how the synthesis of multiple dynamical systems can support compositional cognitive computations. To empirically validate the model, we simulate two experiments demonstrating compositionality in space or in time during sequence generation. Finally, we describe several neural network architectures supporting various types of compositionality based on grid coding and highlight connections to recent work in machine learning leveraging analogous techniques.

Making Sense of the Multiplicity and Dynamics of Navigational Codes in the Brain

Since the discovery of conspicuously spatially tuned neurons in the hippocampal formation over 50 years ago, characterizing which, where, and how neurons encode navigationally relevant variables has been a major thrust of navigational neuroscience. While much of this effort has centered on the hippocampal formation and functionally-adjacent structures, recent work suggests that spatial codes, in some form or another, can be found throughout the brain, even in areas traditionally associated with sensation, movement, and executive function. In this review, we highlight these unexpected results, draw insights from comparison of these codes across contexts, regions, and species, and finally suggest an avenue for future work to make sense of these diverse and dynamic navigational codes.

Predictive maps in rats and humans for spatial navigation

Much of our understanding of navigation comes from the study of individual species, often with specific tasks tailored to those species. Here, we provide a novel experimental and analytic framework integrating across humans, rats, and simulated reinforcement learning (RL) agents to interrogate the dynamics of behavior during spatial navigation. We developed a novel open-field navigation task ("Tartarus maze") requiring dynamic adaptation (shortcuts and detours) to frequently changing obstructions on the path to a hidden goal. Humans and rats were remarkably similar in their trajectories. Both species showed the greatest similarity to RL agents utilizing a "successor representation," which creates a predictive map. Humans also displayed trajectory features similar to model-based RL agents, which implemented an optimal tree-search planning procedure. Our results help refine models seeking to explain mammalian navigation in dynamic environments and highlight the utility of modeling the behavior of different species to uncover the shared mechanisms that support behavior.

Explicit knowledge of task structure is a primary determinant of human model-based action

Explicit information obtained through instruction profoundly shapes human choice behaviour. However, this has been studied in computationally simple tasks, and it is unknown how model-based and model-free systems, respectively generating goal-directed and habitual actions, are affected by the absence or presence of instructions. We assessed behaviour in a variant of a computationally more complex decision-making task, before and after providing information about task structure, both in healthy volunteers and in individuals suffering from obsessive-compulsive or other disorders. Initial behaviour was model-free, with rewards directly reinforcing preceding actions. Model-based control, employing predictions of states resulting from each action, emerged with experience in a minority of participants, and less in those with obsessive-compulsive disorder. Providing task structure information strongly increased model-based control, similarly across all groups. Thus, in humans, explicit task structural knowledge is a primary determinant of model-based reinforcement learning and is most readily acquired from instruction rather than experience.

Cognitive Control as a Multivariate Optimization Problem

A hallmark of adaptation in humans and other animals is our ability to control how we think and behave across different settings. Research has characterized the various forms cognitive control can take-including enhancement of goal-relevant information, suppression of goal-irrelevant information, and overall inhibition of potential responses-and has identified computations and neural circuits that underpin this multitude of control types. Studies have also identified a wide range of situations that elicit adjustments in control allocation (e.g., those eliciting signals indicating an error or increased processing conflict), but the rules governing when a given situation will give rise to a given control adjustment remain poorly understood. Significant progress has recently been made on this front by casting the allocation of control as a decision-making problem. This approach has developed unifying and normative models that prescribe when and how a change in incentives and task demands will result in changes in a given form of control. Despite their successes, these models, and the experiments that have been developed to test them, have yet to face their greatest challenge: deciding how to select among the multiplicity of configurations that control can take at any given time. Here, we will lay out the complexities of the inverse problem inherent to cognitive control allocation, and their close parallels to inverse problems within motor control (e.g., choosing between redundant limb movements). We discuss existing solutions to motor control's inverse problems drawn from optimal control theory, which have proposed that effort costs act to regularize actions and transform motor planning into a well-posed problem. These same principles may help shed light on how our brains optimize over complex control configuration, while providing a new normative perspective on the origins of mental effort.

The whole prefrontal cortex is premotor cortex

We propose that the entirety of the prefrontal cortex (PFC) can be seen as fundamentally premotor in nature. By this, we mean that the PFC consists of an action abstraction hierarchy whose core function is the potentiation and depotentiation of possible action plans at different levels of granularity. We argue that the apex of the hierarchy should revolve around the process of goal-selection, which we posit is inherently a form of optimization over action abstraction. Anatomical and functional evidence supports the idea that this hierarchy originates on the orbital surface of the brain and extends dorsally to motor cortex. Accordingly, our viewpoint positions the orbitofrontal cortex in a key role in the optimization of goal-selection policies, and suggests that its other proposed roles are aspects of this more general function. Our proposed perspective will reframe outstanding questions, open up new areas of inquiry and align theories of prefrontal function with evolutionary principles. This article is part of the theme issue 'Systems neuroscience through the lens of evolutionary theory'.

Planning in the brain

Recent breakthroughs in artificial intelligence (AI) have enabled machines to plan in tasks previously thought to be uniquely human. Meanwhile, the planning algorithms implemented by the brain itself remain largely unknown. Here, we review neural and behavioral data in sequential decision-making tasks that elucidate the ways in which the brain does-and does not-plan. To systematically review available biological data, we create a taxonomy of planning algorithms by summarizing the relevant design choices for such algorithms in AI. Across species, recording techniques, and task paradigms, we find converging evidence that the brain represents future states consistent with a class of planning algorithms within our taxonomy-focused, depth-limited, and serial. However, we argue that current data are insufficient for addressing more detailed algorithmic questions. We propose a new approach leveraging AI advances to drive experiments that can adjudicate between competing candidate algorithms.

Dynamical self-organization and efficient representation of space by grid cells

To plan trajectories and navigate, animals must maintain a mental representation of the environment and their own position within it. This "cognitive map" is thought to be supported in part by the entorhinal cortex, where grid cells are active when an animal occupies the vertices of a scaling hierarchy of periodic lattices of locations in an enclosure. Here, we review computational developments which suggest that the grid cell network is: (a) efficient, providing required spatial resolution with a minimum number of neurons, (b) self-organizing, dynamically coordinating the structure and scale of the responses, and (c) adaptive, re-organizing in response to changes in landmarks and the structure of the boundaries of spaces. We consider these ideas in light of recent discoveries of similar structures in the mental representation of abstract spaces of shapes and smells, and in other brain areas, and highlight promising directions for future research.