The Hamiltonian Brain: Efficient Probabilistic Inference with Excitatory-Inhibitory Neural Circuit Dynamics

Signatures of Bayesian inference emerge from energy-efficient synapses

Biological synaptic transmission is unreliable, and this unreliability likely degrades neural circuit performance. While there are biophysical mechanisms that can increase reliability, for instance by increasing vesicle release probability, these mechanisms cost energy. We examined four such mechanisms along with the associated scaling of the energetic costs. We then embedded these energetic costs for reliability in artificial neural networks (ANNs) with trainable stochastic synapses, and trained these networks on standard image classification tasks. The resulting networks revealed a tradeoff between circuit performance and the energetic cost of synaptic reliability. Additionally, the optimised networks exhibited two testable predictions consistent with pre-existing experimental data. Specifically, synapses with lower variability tended to have (1) higher input firing rates and (2) lower learning rates. Surprisingly, these predictions also arise when synapse statistics are inferred through Bayesian inference. Indeed, we were able to find a formal, theoretical link between the performance-reliability cost tradeoff and Bayesian inference. This connection suggests two incompatible possibilities: evolution may have chanced upon a scheme for implementing Bayesian inference by optimising energy efficiency, or alternatively, energy-efficient synapses may display signatures of Bayesian inference without actually using Bayes to reason about uncertainty.

Joint modeling of choices and reaction times based on Bayesian contextual behavioral control

In cognitive neuroscience and psychology, reaction times are an important behavioral measure. However, in instrumental learning and goal-directed decision making experiments, findings often rely only on choice probabilities from a value-based model, instead of reaction times. Recent advancements have shown that it is possible to connect value-based decision models with reaction time models. However, typically these models do not provide an integrated account of both value-based choices and reaction times, but simply link two types of models. Here, we propose a novel integrative joint model of both choices and reaction times by combining a computational account of Bayesian sequential decision making with a sampling procedure. This allows us to describe how internal uncertainty in the planning process shapes reaction time distributions. Specifically, we use a recent context-specific Bayesian forward planning model which we extend by a Markov chain Monte Carlo (MCMC) sampler to obtain both choices and reaction times. As we will show this makes the sampler an integral part of the decision making process and enables us to reproduce, using simulations, well-known experimental findings in value based-decision making as well as classical inhibition and switching tasks. Specifically, we use the proposed model to explain both choice behavior and reaction times in instrumental learning and automatized behavior, in the Eriksen flanker task and in task switching. These findings show that the proposed joint behavioral model may describe common underlying processes in these different decision making paradigms.

Chaotic neural dynamics facilitate probabilistic computations through sampling

Cortical neurons exhibit highly variable responses over trials and time. Theoretical works posit that this variability arises potentially from chaotic network dynamics of recurrently connected neurons. Here, we demonstrate that chaotic neural dynamics, formed through synaptic learning, allow networks to perform sensory cue integration in a sampling-based implementation. We show that the emergent chaotic dynamics provide neural substrates for generating samples not only of a static variable but also of a dynamical trajectory, where generic recurrent networks acquire these abilities with a biologically plausible learning rule through trial and error. Furthermore, the networks generalize their experience in the stimulus-evoked samples to the inference without partial or all sensory information, which suggests a computational role of spontaneous activity as a representation of the priors as well as a tractable biological computation for marginal distributions. These findings suggest that chaotic neural dynamics may serve for the brain function as a Bayesian generative model.

Decentralized Neural Circuits of Multisensory Information Integration in the Brain

The brain combines multisensory inputs together to obtain a complete and reliable description of the world. Recent experiments suggest that several interconnected multisensory brain areas are simultaneously involved to integrate multisensory information. It was unknown how these mutually connected multisensory areas achieve multisensory integration. To answer this question, using biologically plausible neural circuit models we developed a decentralized system for information integration that comprises multiple interconnected multisensory brain areas. Through studying an example of integrating visual and vestibular cues to infer heading direction, we show that such a decentralized system is well consistent with experimental observations. In particular, we demonstrate that this decentralized system can optimally integrate information by implementing sampling-based Bayesian inference. The Poisson variability of spike generation provides appropriate variability to drive sampling, and the interconnections between multisensory areas store the correlation prior between multisensory stimuli. The decentralized system predicts that optimally integrated information emerges locally from the dynamics of the communication between brain areas and sheds new light on the interpretation of the connectivity between multisensory brain areas.

Bayesian encoding and decoding as distinct perspectives on neural coding

The Bayesian brain hypothesis is one of the most influential ideas in neuroscience. However, unstated differences in how Bayesian ideas are operationalized make it difficult to draw general conclusions about how Bayesian computations map onto neural circuits. Here, we identify one such unstated difference: some theories ask how neural circuits could recover information about the world from sensory neural activity (Bayesian decoding), whereas others ask how neural circuits could implement inference in an internal model (Bayesian encoding). These two approaches require profoundly different assumptions and lead to different interpretations of empirical data. We contrast them in terms of motivations, empirical support and relationship to neural data. We also use a simple model to argue that encoding and decoding models are complementary rather than competing. Appreciating the distinction between Bayesian encoding and Bayesian decoding will help to organize future work and enable stronger empirical tests about the nature of inference in the brain.

Sampling-based Bayesian inference in recurrent circuits of stochastic spiking neurons

Two facts about cortex are widely accepted: neuronal responses show large spiking variability with near Poisson statistics and cortical circuits feature abundant recurrent connections between neurons. How these spiking and circuit properties combine to support sensory representation and information processing is not well understood. We build a theoretical framework showing that these two ubiquitous features of cortex combine to produce optimal sampling-based Bayesian inference. Recurrent connections store an internal model of the external world, and Poissonian variability of spike responses drives flexible sampling from the posterior stimulus distributions obtained by combining feedforward and recurrent neuronal inputs. We illustrate how this framework for sampling-based inference can be used by cortex to represent latent multivariate stimuli organized either hierarchically or in parallel. A neural signature of such network sampling are internally generated differential correlations whose amplitude is determined by the prior stored in the circuit, which provides an experimentally testable prediction for our framework.

Where is the error? Hierarchical predictive coding through dendritic error computation

Top-down feedback in cortex is critical for guiding sensory processing, which has prominently been formalized in the theory of hierarchical predictive coding (hPC). However, experimental evidence for error units, which are central to the theory, is inconclusive and it remains unclear how hPC can be implemented with spiking neurons. To address this, we connect hPC to existing work on efficient coding in balanced networks with lateral inhibition and predictive computation at apical dendrites. Together, this work points to an efficient implementation of hPC with spiking neurons, where prediction errors are computed not in separate units, but locally in dendritic compartments. We then discuss the correspondence of this model to experimentally observed connectivity patterns, plasticity, and dynamics in cortex.

Depersonalization disorder as a systematic downregulation of interoceptive signals

Depersonalisation disorder (DPD) is a psychopathological condition characterised by a feeling of detachment from one's own body and surrounding, and it is understood as emerging from the downregulation of interoceptive afferents. However, the precise mechanisms that drive this 'interoceptive silencing' are yet to be clarified. Here we present a computational and neurobiologically plausible model of DPD within the active inference framework. Specifically, we describe DPD as arising from disrupted interoceptive processing at higher levels of the cortical hierarchy where the interoceptive and exteroceptive streams are integrated. We simulated the behaviour of an agent subjected to a situation of high interoceptive activation despite the absence of a perceivable threat in the external environment. The simulation showed how a similar condition, if perceived as inescapable, would result in a downregulation of interoceptive signals, whilst leaving the exteroceptive ones unaffected. Such interoceptive silencing would force the agent to over-rely on exteroceptive information and would ultimately lead to the DPD phenomenology. Finally, our simulation shows that repeated exposure to similar situations over time will lead the agent to increasingly disengage from bodily responses even in the face of a less triggering situation, explaining how a single episode of depersonalization can lead to chronic DPD.

Fractional neural sampling as a theory of spatiotemporal probabilistic computations in neural circuits

A range of perceptual and cognitive processes have been characterized from the perspective of probabilistic representations and inference. To understand the neural circuit mechanism underlying these probabilistic computations, we develop a theory based on complex spatiotemporal dynamics of neural population activity. We first implement and explore this theory in a biophysically realistic, spiking neural circuit. Population activity patterns emerging from the circuit capture realistic variability or fluctuations of neural dynamics both in time and in space. These activity patterns implement a type of probabilistic computations that we name fractional neural sampling (FNS). We further develop a mathematical model to reveal the algorithmic nature of FNS and its computational advantages for representing multimodal distributions, a major challenge faced by existing theories. We demonstrate that FNS provides a unified account of a diversity of experimental observations of neural spatiotemporal dynamics and perceptual processes such as visual perception inference, and that FNS makes experimentally testable predictions.

Dynamics of neural circuits mapping brain functions such as sensory processing and decision making, can be characterized by probabilistic representations and inference. The authors elaborate the role of spatiotemporal neural dynamics for more efficient performance of probabilistic computations.

Towards a cross-level understanding of Bayesian inference in the brain

Perception emerges from unconscious probabilistic inference, which guides behaviour in our ubiquitously uncertain environment. Bayesian decision theory is a prominent computational model that describes how people make rational decisions using noisy and ambiguous sensory observations. However, critical questions have been raised about the validity of the Bayesian framework in explaining the mental process of inference. Firstly, some natural behaviours deviate from Bayesian optimum. Secondly, the neural mechanisms that support Bayesian computations in the brain are yet to be understood. Taking Marr's cross level approach, we review the recent progress made in addressing these challenges. We first review studies that combined behavioural paradigms and modelling approaches to explain both optimal and suboptimal behaviours. Next, we evaluate the theoretical advances and the current evidence for ecologically feasible algorithms and neural implementations in the brain, which may enable probabilistic inference. We argue that this cross-level approach is necessary for the worthwhile pursuit to uncover mechanistic accounts of human behaviour.

Cortical oscillations support sampling-based computations in spiking neural networks

Being permanently confronted with an uncertain world, brains have faced evolutionary pressure to represent this uncertainty in order to respond appropriately. Often, this requires visiting multiple interpretations of the available information or multiple solutions to an encountered problem. This gives rise to the so-called mixing problem: since all of these "valid" states represent powerful attractors, but between themselves can be very dissimilar, switching between such states can be difficult. We propose that cortical oscillations can be effectively used to overcome this challenge. By acting as an effective temperature, background spiking activity modulates exploration. Rhythmic changes induced by cortical oscillations can then be interpreted as a form of simulated tempering. We provide a rigorous mathematical discussion of this link and study some of its phenomenological implications in computer simulations. This identifies a new computational role of cortical oscillations and connects them to various phenomena in the brain, such as sampling-based probabilistic inference, memory replay, multisensory cue combination, and place cell flickering.

Bridging physiological and perceptual views of autism by means of sampling-based Bayesian inference

Theories for autism spectrum disorder (ASD) have been formulated at different levels, ranging from physiological observations to perceptual and behavioral descriptions. Understanding the physiological underpinnings of perceptual traits in ASD remains a significant challenge in the field. Here we show how a recurrent neural circuit model that was optimized to perform sampling-based inference and displays characteristic features of cortical dynamics can help bridge this gap. The model was able to establish a mechanistic link between two descriptive levels for ASD: a physiological level, in terms of inhibitory dysfunction, neural variability, and oscillations, and a perceptual level, in terms of hypopriors in Bayesian computations. We took two parallel paths-inducing hypopriors in the probabilistic model, and an inhibitory dysfunction in the network model-which lead to consistent results in terms of the represented posteriors, providing support for the view that both descriptions might constitute two sides of the same coin.

Dynamical Mechanism of Sampling-Based Probabilistic Inference under Probabilistic Population Codes

Animals make efficient probabilistic inferences based on uncertain and noisy information from the outside environment. It is known that probabilistic population codes, which have been proposed as a neural basis for encoding probability distributions, allow general neural networks (NNs) to perform near-optimal point estimation. However, the mechanism of sampling-based probabilistic inference has not been clarified. In this study, we trained two types of artificial NNs, feedforward NN (FFNN) and recurrent NN (RNN), to perform sampling-based probabilistic inference. Then we analyzed and compared their mechanisms of sampling. We found that sampling in RNN was performed by a mechanism that efficiently uses the properties of dynamical systems, unlike FFNN. In addition, we found that sampling in RNNs acted as an inductive bias, enabling a more accurate estimation than in maximum a posteriori estimation. These results provide important arguments for discussing the relationship between dynamical systems and information processing in NNs.

The Bayesian sampler: Generic Bayesian inference causes incoherence in human probability judgments

Human probability judgments are systematically biased, in apparent tension with Bayesian models of cognition. But perhaps the brain does not represent probabilities explicitly, but approximates probabilistic calculations through a process of sampling, as used in computational probabilistic models in statistics. Naïve probability estimates can be obtained by calculating the relative frequency of an event within a sample, but these estimates tend to be extreme when the sample size is small. We propose instead that people use a generic prior to improve the accuracy of their probability estimates based on samples, and we call this model the Bayesian sampler. The Bayesian sampler trades off the coherence of probabilistic judgments for improved accuracy, and provides a single framework for explaining phenomena associated with diverse biases and heuristics such as conservatism and the conjunction fallacy. The approach turns out to provide a rational reinterpretation of "noise" in an important recent model of probability judgment, the probability theory plus noise model (Costello & Watts, 2014, 2016a, 2017; Costello & Watts, 2019; Costello, Watts, & Fisher, 2018), making equivalent average predictions for simple events, conjunctions, and disjunctions. The Bayesian sampler does, however, make distinct predictions for conditional probabilities and distributions of probability estimates. We show in 2 new experiments that this model better captures these mean judgments both qualitatively and quantitatively; which model best fits individual distributions of responses depends on the assumed size of the cognitive sample. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Learning probabilistic neural representations with randomly connected circuits

We present a theory of neural circuits’ design and function, inspired by the random connectivity of real neural circuits and the mathematical power of random projections. Specifically, we introduce a family of statistical models for large neural population codes, a straightforward neural circuit architecture that would implement these models, and a biologically plausible learning rule for such circuits. The resulting neural architecture suggests a design principle for neural circuit—namely, that they learn to compute the mathematical surprise of their inputs, given past inputs, without an explicit teaching signal. We applied these models to recordings from large neural populations in monkeys’ visual and prefrontal cortices and show them to be highly accurate, efficient, and scalable.

The brain represents and reasons probabilistically about complex stimuli and motor actions using a noisy, spike-based neural code. A key building block for such neural computations, as well as the basis for supervised and unsupervised learning, is the ability to estimate the surprise or likelihood of incoming high-dimensional neural activity patterns. Despite progress in statistical modeling of neural responses and deep learning, current approaches either do not scale to large neural populations or cannot be implemented using biologically realistic mechanisms. Inspired by the sparse and random connectivity of real neuronal circuits, we present a model for neural codes that accurately estimates the likelihood of individual spiking patterns and has a straightforward, scalable, efficient, learnable, and realistic neural implementation. This model’s performance on simultaneously recorded spiking activity of >100 neurons in the monkey visual and prefrontal cortices is comparable with or better than that of state-of-the-art models. Importantly, the model can be learned using a small number of samples and using a local learning rule that utilizes noise intrinsic to neural circuits. Slower, structural changes in random connectivity, consistent with rewiring and pruning processes, further improve the efficiency and sparseness of the resulting neural representations. Our results merge insights from neuroanatomy, machine learning, and theoretical neuroscience to suggest random sparse connectivity as a key design principle for neuronal computation.

Cortical-like dynamics in recurrent circuits optimized for sampling-based probabilistic inference

Sensory cortices display a suite of ubiquitous dynamical features, such as ongoing noise variability, transient overshoots and oscillations, that have so far escaped a common, principled theoretical account. We developed a unifying model for these phenomena by training a recurrent excitatory-inhibitory neural circuit model of a visual cortical hypercolumn to perform sampling-based probabilistic inference. The optimized network displayed several key biological properties, including divisive normalization and stimulus-modulated noise variability, inhibition-dominated transients at stimulus onset and strong gamma oscillations. These dynamical features had distinct functional roles in speeding up inferences and made predictions that we confirmed in novel analyses of recordings from awake monkeys. Our results suggest that the basic motifs of cortical dynamics emerge as a consequence of the efficient implementation of the same computational function-fast sampling-based inference-and predict further properties of these motifs that can be tested in future experiments.

Representation of visual uncertainty through neural gain variability

Uncertainty is intrinsic to perception. Neural circuits which process sensory information must therefore also represent the reliability of this information. How they do so is a topic of debate. We propose a model of visual cortex in which average neural response strength encodes stimulus features, while cross-neuron variability in response gain encodes the uncertainty of these features. To test this model, we studied spiking activity of neurons in macaque V1 and V2 elicited by repeated presentations of stimuli whose uncertainty was manipulated in distinct ways. We show that gain variability of individual neurons is tuned to stimulus uncertainty, that this tuning is specific to the features encoded by these neurons and largely invariant to the source of uncertainty. We demonstrate that this behavior naturally arises from known gain-control mechanisms, and illustrate how downstream circuits can jointly decode stimulus features and their uncertainty from sensory population activity.

Retrospective Inference as a Form of Bounded Rationality, and Its Beneficial Influence on Learning

Probabilistic models of cognition typically assume that agents make inferences about current states by combining new sensory information with fixed beliefs about the past, an approach known as Bayesian filtering. This is computationally parsimonious, but, in general, leads to suboptimal beliefs about past states, since it ignores the fact that new observations typically contain information about the past as well as the present. This is disadvantageous both because knowledge of past states may be intrinsically valuable, and because it impairs learning about fixed or slowly changing parameters of the environment. For these reasons, in offline data analysis it is usual to infer on every set of states using the entire time series of observations, an approach known as (fixed-interval) Bayesian smoothing. Unfortunately, however, this is impractical for real agents, since it requires the maintenance and updating of beliefs about an ever-growing set of states. We propose an intermediate approach, finite retrospective inference (FRI), in which agents perform update beliefs about a limited number of past states (Formally, this represents online fixed-lag smoothing with a sliding window). This can be seen as a form of bounded rationality in which agents seek to optimize the accuracy of their beliefs subject to computational and other resource costs. We show through simulation that this approach has the capacity to significantly increase the accuracy of both inference and learning, using a simple variational scheme applied to both randomly generated Hidden Markov models (HMMs), and a specific application of the HMM, in the form of the widely used probabilistic reversal task. Our proposal thus constitutes a theoretical contribution to normative accounts of bounded rationality, which makes testable empirical predictions that can be explored in future work.

Accelerated Physical Emulation of Bayesian Inference in Spiking Neural Networks

The massively parallel nature of biological information processing plays an important role due to its superiority in comparison to human-engineered computing devices. In particular, it may hold the key to overcoming the von Neumann bottleneck that limits contemporary computer architectures. Physical-model neuromorphic devices seek to replicate not only this inherent parallelism, but also aspects of its microscopic dynamics in analog circuits emulating neurons and synapses. However, these machines require network models that are not only adept at solving particular tasks, but that can also cope with the inherent imperfections of analog substrates. We present a spiking network model that performs Bayesian inference through sampling on the BrainScaleS neuromorphic platform, where we use it for generative and discriminative computations on visual data. By illustrating its functionality on this platform, we implicitly demonstrate its robustness to various substrate-specific distortive effects, as well as its accelerated capability for computation. These results showcase the advantages of brain-inspired physical computation and provide important building blocks for large-scale neuromorphic applications.

Stochasticity from function - Why the Bayesian brain may need no noise

An increasing body of evidence suggests that the trial-to-trial variability of spiking activity in the brain is not mere noise, but rather the reflection of a sampling-based encoding scheme for probabilistic computing. Since the precise statistical properties of neural activity are important in this context, many models assume an ad-hoc source of well-behaved, explicit noise, either on the input or on the output side of single neuron dynamics, most often assuming an independent Poisson process in either case. However, these assumptions are somewhat problematic: neighboring neurons tend to share receptive fields, rendering both their input and their output correlated; at the same time, neurons are known to behave largely deterministically, as a function of their membrane potential and conductance. We suggest that spiking neural networks may have no need for noise to perform sampling-based Bayesian inference. We study analytically the effect of auto- and cross-correlations in functional Bayesian spiking networks and demonstrate how their effect translates to synaptic interaction strengths, rendering them controllable through synaptic plasticity. This allows even small ensembles of interconnected deterministic spiking networks to simultaneously and co-dependently shape their output activity through learning, enabling them to perform complex Bayesian computation without any need for noise, which we demonstrate in silico, both in classical simulation and in neuromorphic emulation. These results close a gap between the abstract models and the biology of functionally Bayesian spiking networks, effectively reducing the architectural constraints imposed on physical neural substrates required to perform probabilistic computing, be they biological or artificial.

Modeling subjective belief states in computational psychiatry: interoceptive inference as a candidate framework

The nascent field computational psychiatry has undergone exponential growth since its inception. To date, much of the published work has focused on choice behaviors, which are primarily modeled within a reinforcement learning framework. While this initial normative effort represents a milestone in psychiatry research, the reality is that many psychiatric disorders are defined by disturbances in subjective states (e.g., depression, anxiety) and associated beliefs (e.g., dysmorphophobia, paranoid ideation), which are not considered in normative models. In this paper, we present interoceptive inference as a candidate framework for modeling subjective-and associated belief-states in computational psychiatry. We first introduce the notion and significance of modeling subjective states in computational psychiatry. Next, we present the interoceptive inference framework, and in particular focus on the relationship between interoceptive inference (i.e., belief updating) and emotions. Lastly, we will use drug craving as an example of subjective states to demonstrate the feasibility of using interoceptive inference to model the psychopathology of subjective states.

Internal Models in Biological Control

Rationality principles such as optimal feedback control and Bayesian inference underpin a probabilistic framework that has accounted for a range of empirical phenomena in biological sensorimotor control. To facilitate the optimization of flexible and robust behaviors consistent with these theories, the ability to construct internal models of the motor system and environmental dynamics can be crucial. In the context of this theoretic formalism, we review the computational roles played by such internal models and the neural and behavioral evidence for their implementation in the brain.

Building machines that adapt and compute like brains

Building machines that learn and think like humans is essential not only for cognitive science, but also for computational neuroscience, whose ultimate goal is to understand how cognition is implemented in biological brains. A new cognitive computational neuroscience should build cognitive-level and neural-level models, understand their relationships, and test both types of models with both brain and behavioral data.

Cognitive computational neuroscience

To learn how cognition is implemented in the brain, we must build computational models that can perform cognitive tasks, and test such models with brain and behavioral experiments. Cognitive science has developed computational models that decompose cognition into functional components. Computational neuroscience has modeled how interacting neurons can implement elementary components of cognition. It is time to assemble the pieces of the puzzle of brain computation and to better integrate these separate disciplines. Modern technologies enable us to measure and manipulate brain activity in unprecedentedly rich ways in animals and humans. However, experiments will yield theoretical insight only when employed to test brain-computational models. Here we review recent work in the intersection of cognitive science, computational neuroscience and artificial intelligence. Computational models that mimic brain information processing during perceptual, cognitive and control tasks are beginning to be developed and tested with brain and behavioral data.

Recognition Dynamics in the Brain under the Free Energy Principle

We formulate the computational processes of perception in the framework of the principle of least action by postulating the theoretical action as a time integral of the variational free energy in the neurosciences. The free energy principle is accordingly rephrased, on autopoetic grounds, as follows: all viable organisms attempt to minimize their sensory uncertainty about an unpredictable environment over a temporal horizon. By taking the variation of informational action, we derive neural recognition dynamics (RD), which by construction reduces to the Bayesian filtering of external states from noisy sensory inputs. Consequently, we effectively cast the gradient-descent scheme of minimizing the free energy into Hamiltonian mechanics by addressing only the positions and momenta of the organisms' representations of the causal environment. To demonstrate the utility of our theory, we show how the RD may be implemented in a neuronally based biophysical model at a single-cell level and subsequently in a coarse-grained, hierarchical architecture of the brain. We also present numerical solutions to the RD for a model brain and analyze the perceptual trajectories around attractors in neural state space.

With or without you: predictive coding and Bayesian inference in the brain

Two theoretical ideas have emerged recently with the ambition to provide a unifying functional explanation of neural population coding and dynamics: predictive coding and Bayesian inference. Here, we describe the two theories and their combination into a single framework: Bayesian predictive coding. We clarify how the two theories can be distinguished, despite sharing core computational concepts and addressing an overlapping set of empirical phenomena. We argue that predictive coding is an algorithmic/representational motif that can serve several different computational goals of which Bayesian inference is but one. Conversely, while Bayesian inference can utilize predictive coding, it can also be realized by a variety of other representations. We critically evaluate the experimental evidence supporting Bayesian predictive coding and discuss how to test it more directly.

Periodic population codes: From a single circular variable to higher dimensions, multiple nested scales, and conceptual spaces

Across the nervous system, neurons often encode circular stimuli using tuning curves that are not sine or cosine functions, but that belong to the richer class of von Mises functions, which are periodic variants of Gaussians. For a population of neurons encoding a single circular variable with such canonical tuning curves, computing a simple population vector is the optimal read-out of the most likely stimulus. We argue that the advantages of population vector read-outs are so compelling that even the neural representation of the outside world's flat Euclidean geometry is curled up into a torus (a circle times a circle), creating the hexagonal activity patterns of mammalian grid cells. Here, the circular scale is not set a priori, so the nervous system can use multiple scales and gain fields to overcome the ambiguity inherent in periodic representations of linear variables. We review the experimental evidence for this framework and discuss its testable predictions and generalizations to more abstract grid-like neural representations.