Autism spectrum disorders have been proposed to arise from impairments in the probabilistic integration of prior knowledge with sensory inputs. Circular inference is one such possible impairment, in which excitation-to-inhibition imbalances in the cerebral cortex cause the reverberation and amplification of prior beliefs and sensory information. Recent empirical work has associated circular inference with the clinical dimensions of schizophrenia. Inhibition impairments have also been observed in autism, suggesting that signal reverberation might be present in that condition as well. In this study, we collected data from 21 participants with self-reported diagnoses of autism spectrum disorders and 155 participants with a broad range of autistic traits in an online probabilistic decision-making task (the fisher task). We used previously established Bayesian models to investigate possible associations between autistic traits or autism and circular inference. There was no correlation between prior or likelihood reverberation and autistic traits across the whole sample. Similarly, no differences in any of the circular inference model parameters were found between autistic participants and those with no diagnosis. Furthermore, participants incorporated information from both priors and likelihoods in their decisions, with no relationship between their weights and psychiatric traits, contrary to what common theories for both autism and schizophrenia would suggest. These findings suggest that there is no increased signal reverberation in autism, despite the known presence of excitation-to-inhibition imbalances. They can be used to further contrast and refine the Bayesian theories of schizophrenia and autism, revealing a divergence in the computational mechanisms underlying the two conditions.

Cortical networks show a large heterogeneity of neuronal properties. However, traditional coding models have focused on homogeneous populations of excitatory and inhibitory neurons. Here, we analytically derive a class of recurrent networks of spiking neurons that close to optimally track a continuously varying input online, based on two assumptions: 1) every spike is decoded linearly and 2) the network aims to reduce the mean-squared error between the input and the estimate. From this we derive a class of predictive coding networks, that unifies encoding and decoding and in which we can investigate the difference between homogeneous networks and heterogeneous networks, in which each neurons represents different features and has different spike-generating properties. We find that in this framework, 'type 1' and 'type 2' neurons arise naturally and networks consisting of a heterogeneous population of different neuron types are both more efficient and more robust against correlated noise. We make two experimental predictions: 1) we predict that integrators show strong correlations with other integrators and resonators are correlated with resonators, whereas the correlations are much weaker between neurons with different coding properties and 2) that 'type 2' neurons are more coherent with the overall network activity than 'type 1' neurons.

Networks based on coordinated spike coding can encode information with high efficiency in the spike trains of individual neurons. These networks exhibit single-neuron variability and tuning curves as typically observed in cortex, but paradoxically coincide with a precise, non-redundant spike-based population code. However, it has remained unclear whether the specific synaptic connectivities required in these networks can be learnt with local learning rules. Here, we show how to learn the required architecture. Using coding efficiency as an objective, we derive spike-timing-dependent learning rules for a recurrent neural network, and we provide exact solutions for the networks' convergence to an optimal state. As a result, we deduce an entire network from its input distribution and a firing cost. After learning, basic biophysical quantities such as voltages, firing thresholds, excitation, inhibition, or spikes acquire precise functional interpretations.

When we face ambiguous images, the brain cannot commit to a single percept; instead, it switches between mutually exclusive interpretations every few seconds, a phenomenon known as bistable perception. While neuromechanistic models, e.g., adapting neural populations with lateral inhibition, may account for the dynamics of bistability, a larger question remains unresolved: how this phenomenon informs us on generic perceptual processes in less artificial contexts. Here, we propose that bistable perception is due to our prior beliefs being reverberated in the cortical hierarchy and corrupting the sensory evidence, a phenomenon known as "circular inference". Such circularity could occur in a hierarchical brain where sensory responses trigger activity in higher-level areas but are also modulated by feedback projections from these same areas. We show that in the face of ambiguous sensory stimuli, circular inference can change the dynamics of the perceptual system and turn what should be an integrator of inputs into a bistable attractor switching between two highly trusted interpretations. The model captures various aspects of bistability, including Levelt's laws and the stabilizing effects of intermittent presentation of the stimulus. Since it is related to the generic perceptual inference and belief updating mechanisms, this approach can be used to predict the tendency of individuals to form aberrant beliefs from their bistable perception behavior. Overall, we suggest that feedforward/feedback information loops in hierarchical neural networks, a phenomenon that could lead to psychotic symptoms when overly strong, could also underlie perception in nonclinical populations.

The brain uses spikes in neural circuits to perform many dynamical computations. The computations are performed with properties such as spiking efficiency, i.e. minimal number of spikes, and robustness to noise. A major obstacle for learning computations in artificial spiking neural networks with such desired biological properties is due to lack of our understanding of how biological spiking neural networks learn computations. Here, we consider the credit assignment problem, i.e. determining the local contribution of each synapse to the network's global output error, for learning nonlinear dynamical computations in a spiking network with the desired properties of biological networks. We approach this problem by fusing the theory of efficient, balanced neural networks (EBN) with nonlinear adaptive control theory to propose a local learning rule. Locality of learning rules are ensured by feeding back into the network its own error, resulting in a learning rule depending solely on presynaptic inputs and error feedbacks. The spiking efficiency and robustness of the network are guaranteed by maintaining a tight excitatory/inhibitory balance, ensuring that each spike represents a local projection of the global output error and minimizes a loss function. The resulting networks can learn to implement complex dynamics with very small numbers of neurons and spikes, exhibit the same spike train variability as observed experimentally, and are extremely robust to noise and neuronal loss.

In order to respond reliably to specific features of their environment, sensory neurons need to integrate multiple incoming noisy signals. Crucially, they also need to compete for the interpretation of those signals with other neurons representing similar features. The form that this competition should take depends critically on the noise corrupting these signals. In this study we show that for the type of noise commonly observed in sensory systems, whose variance scales with the mean signal, sensory neurons should selectively divide their input signals by their predictions, suppressing ambiguous cues while amplifying others. Any change in the stimulus context alters which inputs are suppressed, leading to a deep dynamic reshaping of neural receptive fields going far beyond simple surround suppression. Paradoxically, these highly variable receptive fields go alongside and are in fact required for an invariant representation of external sensory features. In addition to offering a normative account of context-dependent changes in sensory responses, perceptual inference in the presence of signal-dependent noise accounts for ubiquitous features of sensory neurons such as divisive normalization, gain control and contrast dependent temporal dynamics.

Spontaneous activity is commonly observed in a variety of cortical states. Experimental evidence suggested that neural assemblies undergo slow oscillations with Up ad Down states even when the network is isolated from the rest of the brain. Here we show that these spontaneous events can be generated by the recurrent connections within the network and understood as signatures of neural circuits that are correcting their internal representation. A noiseless spiking neural network can represent its input signals most accurately when excitatory and inhibitory currents are as strong and as tightly balanced as possible. However, in the presence of realistic neural noise and synaptic delays, this may result in prohibitively large spike counts. An optimal working regime can be found by considering terms that control firing rates in the objective function from which the network is derived and then minimizing simultaneously the coding error and the cost of neural activity. In biological terms, this is equivalent to tuning neural thresholds and after-spike hyperpolarization. In suboptimal working regimes, we observe spontaneous activity even in the absence of feed-forward inputs. In an all-to-all randomly connected network, the entire population is involved in Up states. In spatially organized networks with local connectivity, Up states spread through local connections between neurons of similar selectivity and take the form of a traveling wave. Up states are observed for a wide range of parameters and have similar statistical properties in both active and quiescent state. In the optimal working regime, Up states are vanishing, leaving place to asynchronous activity, suggesting that this working regime is a signature of maximally efficient coding. Although they result in a massive increase in the firing activity, the read-out of spontaneous Up states is in fact orthogonal to the stimulus representation, therefore interfering minimally with the network function.

To predict sensory inputs or control motor trajectories, the brain must constantly learn temporal dynamics based on error feedback. However, it remains unclear how such supervised learning is implemented in biological neural networks. Learning in recurrent spiking networks is notoriously difficult because local changes in connectivity may have an unpredictable effect on the global dynamics. The most commonly used learning rules, such as temporal back-propagation, are not local and thus not biologically plausible. Furthermore, reproducing the Poisson-like statistics of neural responses requires the use of networks with balanced excitation and inhibition. Such balance is easily destroyed during learning. Using a top-down approach, we show how networks of integrate-and-fire neurons can learn arbitrary linear dynamical systems by feeding back their error as a feed-forward input. The network uses two types of recurrent connections: fast and slow. The fast connections learn to balance excitation and inhibition using a voltage-based plasticity rule. The slow connections are trained to minimize the error feedback using a current-based Hebbian learning rule. Importantly, the balance maintained by fast connections is crucial to ensure that global error signals are available locally in each neuron, in turn resulting in a local learning rule for the slow connections. This demonstrates that spiking networks can learn complex dynamics using purely local learning rules, using E/I balance as the key rather than an additional constraint. The resulting network implements a given function within the predictive coding scheme, with minimal dimensions and activity.

How are firing rates in a spiking network related to neural input, connectivity and network function? This is an important problem because firing rates are one of the most important measures of network activity, in both the study of neural computation and neural network dynamics. However, it is a difficult problem, because the spiking mechanism of individual neurons is highly non-linear, and these individual neurons interact strongly through connectivity. We develop a new technique for calculating firing rates in optimal balanced networks. These are particularly interesting networks because they provide an optimal spike-based signal representation while producing cortex-like spiking activity through a dynamic balance of excitation and inhibition. We can calculate firing rates by treating balanced network dynamics as an algorithm for optimizing signal representation. We identify this algorithm and then calculate firing rates by finding the solution to the algorithm. Our firing rate calculation relates network firing rates directly to network input, connectivity and function. This allows us to explain the function and underlying mechanism of tuning curves in a variety of systems.

Two observations about the cortex have puzzled neuroscientists for a long time. First, neural responses are highly variable. Second, the level of excitation and inhibition received by each neuron is tightly balanced at all times. Here, we demonstrate that both properties are necessary consequences of neural networks that represent information efficiently in their spikes. We illustrate this insight with spiking networks that represent dynamical variables. Our approach is based on two assumptions: We assume that information about dynamical variables can be read out linearly from neural spike trains, and we assume that neurons only fire a spike if that improves the representation of the dynamical variables. Based on these assumptions, we derive a network of leaky integrate-and-fire neurons that is able to implement arbitrary linear dynamical systems. We show that the membrane voltage of the neurons is equivalent to a prediction error about a common population-level signal. Among other things, our approach allows us to construct an integrator network of spiking neurons that is robust against many perturbations. Most importantly, neural variability in our networks cannot be equated to noise. Despite exhibiting the same single unit properties as widely used population code models (e.g. tuning curves, Poisson distributed spike trains), balanced networks are orders of magnitudes more reliable. Our approach suggests that spikes do matter when considering how the brain computes, and that the reliability of cortical representations could have been strongly underestimated.

How do neural networks learn to represent information? Here, we address this question by assuming that neural networks seek to generate an optimal population representation for a fixed linear decoder. We define a loss function for the quality of the population read-out and derive the dynamical equations for both neurons and synapses from the requirement to minimize this loss. The dynamical equations yield a network of integrate-and-fire neurons undergoing Hebbian plasticity. We show that, through learning, initially regular and highly correlated spike trains evolve towards Poisson-distributed and independent spike trains with much lower firing rates. The learning rule drives the network into an asynchronous, balanced regime where all inputs to the network are represented optimally for the given decoder. We show that the network dynamics and synaptic plasticity jointly balance the excitation and inhibition received by each unit as tightly as possible and, in doing so, minimize the prediction error between the inputs and the decoded outputs. In turn, spikes are only signalled whenever this prediction error exceeds a certain value, thereby implementing a predictive coding scheme. Our work suggests that several of the features reported in cortical networks, such as the high trial-to-trial variability, the balance between excitation and inhibition, and spike-timing dependent plasticity, are simply signatures of an efficient, spike-based code.

Compelling behavioral evidence suggests that humans can make optimal decisions despite the uncertainty inherent in perceptual or motor tasks. A key question in neuroscience is how populations of spiking neurons can implement such probabilistic computations. In this article, we develop a comprehensive framework for optimal, spike-based sensory integration and working memory in a dynamic environment. We propose that probability distributions are inferred spike-per-spike in recurrently connected networks of integrate-and-fire neurons. As a result, these networks can combine sensory cues optimally, track the state of a time-varying stimulus and memorize accumulated evidence over periods much longer than the time constant of single neurons. Importantly, we propose that population responses and persistent working memory states represent entire probability distributions and not only single stimulus values. These memories are reflected by sustained, asynchronous patterns of activity which make relevant information available to downstream neurons within their short time window of integration. Model neurons act as predictive encoders, only firing spikes which account for new information that has not yet been signaled. Thus, spike times signal deterministically a prediction error, contrary to rate codes in which spike times are considered to be random samples of an underlying firing rate. As a consequence of this coding scheme, a multitude of spike patterns can reliably encode the same information. This results in weakly correlated, Poisson-like spike trains that are sensitive to initial conditions but robust to even high levels of external neural noise. This spike train variability reproduces the one observed in cortical sensory spike trains, but cannot be equated to noise. On the contrary, it is a consequence of optimal spike-based inference. In contrast, we show that rate-based models perform poorly when implemented with stochastically spiking neurons.

We show that the dynamics of spiking neurons can be interpreted as a form of Bayesian inference in time. Neurons that optimally integrate evidence about events in the external world exhibit properties similar to leaky integrate-and-fire neurons with spike-dependent adaptation and maximally respond to fluctuations of their input. Spikes signal the occurrence of new information-what cannot be predicted from the past activity. As a result, firing statistics are close to Poisson, albeit providing a deterministic representation of probabilities.

In the companion letter in this issue ("Bayesian Spiking Neurons I: Inference"), we showed that the dynamics of spiking neurons can be interpreted as a form of Bayesian integration, accumulating evidence over time about events in the external world or the body. We proceed to develop a theory of Bayesian learning in spiking neural networks, where the neurons learn to recognize temporal dynamics of their synaptic inputs. Meanwhile, successive layers of neurons learn hierarchical causal models for the sensory input. The corresponding learning rule is local, spike-time dependent, and highly nonlinear. This approach provides a principled description of spiking and plasticity rules maximizing information transfer, while limiting the number of costly spikes, between successive layers of neurons.

We present an online version of the expectation-maximization (EM) algorithm for hidden Markov models (HMMs). The sufficient statistics required for parameters estimation is computed recursively with time, that is, in an online way instead of using the batch forward-backward procedure. This computational scheme is generalized to the case where the model parameters can change with time by introducing a discount factor into the recurrence relations. The resulting algorithm is equivalent to the batch EM algorithm, for appropriate discount factor and scheduling of parameters update. On the other hand, the online algorithm is able to deal with dynamic environments, i.e., when the statistics of the observed data is changing with time. The implications of the online algorithm for probabilistic modeling in neuroscience are briefly discussed.

We propose a new interpretation of spiking neurons as Bayesian integra- tors accumulating evidence over time about events in the external world or the body, and communicating to other neurons their certainties about these events. In this model, spikes signal the occurrence of new infor- mation, i.e. what cannot be predicted from the past activity. As a result, firing statistics are close to Poisson, albeit providing a deterministic rep- resentation of probabilities. We proceed to develop a theory of Bayesian inference in spiking neural networks, recurrent interactions implement- ing a variant of belief propagation. Many perceptual and motor tasks performed by the central nervous system are probabilis- tic, and can be described in a Bayesian framework [4, 3]. A few important but hidden properties, such as direction of motion, or appropriate motor commands, are inferred from many noisy, local and ambiguous sensory cues. These evidences are combined with priors about the sensory world and body. Importantly, because most of these inferences should lead to quick and irreversible decisions in a perpetually changing world, noisy cues have to be integrated on-line, but in a way that takes into account unpredictable events, such as a sudden change in motion direction or the appearance of a new stimulus. This raises the question of how this temporal integration can be performed at the neural level. It has been proposed that single neurons in sensory cortices represent and compute the log probability that a sensory variable takes on a certain value (eg Is visual motion in the neuron's preferred direction?) [9, 7]. Alternatively, to avoid normalization issues and provide an appropriate signal for decision making, neurons could represent the log proba- bility ratio of a particular hypothesis (eg is motion more likely to be towards the right than towards the left) [7, 6]. Log probabilities are convenient here, since under some assump- tions, independent noisy cues simply combine linearly. Moreover, there are physiological evidence for the neural representation of log probabilities and log probability ratios [9, 6, 7]. However, these models assume that neurons represent probabilities in their firing rates. We argue that it is important to study how probabilistic information are encoded in spikes. Indeed, it seems spurious to marry the idea of an exquisite on-line integration of noisy cues with an underlying rate code that requires averaging on large populations of noisy neurons and long periods of time. In particular, most natural tasks require this integration to take place on the time scale of inter-spike intervals. Spikes are more efficiently signaling events Institute of Cognitive Science, 69645 Bron, France than analog quantities. In addition, a neural theory of inference with spikes will bring us closer to the physiological level and generate more easily testable predictions. Thus, we propose a new theory of neural processing in which spike trains provide a de- terministic, online representation of a log-probability ratio. Spikes signals events, eg that the log-probability ratio has exceeded what could be predicted from previous spikes. This form of coding was loosely inspired by the idea of "energy landscape" coding proposed by Hinton and Brown [2]. However, contrary to [2] and other theories using rate-based representation of probabilities, this model is self-consistent and does not require different models for encoding and decoding: As output spikes provide new, unpredictable, tempo- rally independent evidence, they can be used directly as an input to other Bayesian neurons. Finally, we show that these neurons can be used as building blocks in a theory of approx- imate Bayesian inference in recurrent spiking networks. Connections between neurons implement an underlying Bayesian network, consisting of coupled hidden Markov models. Propagation of spikes is a form of belief propagation in this underlying graphical model. Our theory provides computational explanations of some general physiological properties of cortical neurons, such as spike frequency adaptation, Poisson statistics of spike trains, the existence of strong local inhibition in cortical columns, and the maintenance of a tight balance between excitation and inhibition. Finally, we discuss the implications of this model for the debate about temporal versus rate-based neural coding. 1 Spikes and log posterior odds 1.1 Synaptic integration seen as inference in a hidden Markov chain We propose that each neuron codes for an underlying "hidden" binary variable, xt, whose state evolves over time. We assume that xt depends only on the state at the previous time step, xt-dt, and is conditionally independent of other past states. The state xt can switch from 0 to 1 with a constant rate ron = 1 lim dt dt0 P (xt = 1|xt-dt = 0), and from 1 to 0 with a constant rate roff . For example, these transition rates could represent how often motion in a preferred direction appears the receptive field and how long it is likely to stay there. The neuron infers the state of its hidden variable from N noisy synaptic inputs, considered to be observations of the hidden state. In this initial version of the model, we assume that these inputs are conditionally independent homogeneous Poisson processes, synapse i emitting a spike between time t and t + dt (sit = 1) with constant probability qiondt if xt = 1, and another constant probability qi dt off if xt = 0. The synaptic spikes are assumed to be otherwise independent of previous synaptic spikes, previous states and spikes at other synapses. The resulting generative model is a hidden Markov chain (figure 1-A). However, rather than estimating the state of its hidden variable and communicating this estimate to other neurons (for example by emitting a spike when sensory evidence for xt = 1 goes above a threshold) the neuron reports and communicates its certainty that the current state is 1. This certainty takes the form of the log of the ratio of the probability that the hidden state is 1, and the probability that the state is 0, given all the synaptic inputs received so far: Lt = log P (xt=1|s0t) P (xt=0|s0t) . We use s0t as a short hand notation for the N synaptic inputs received at present and in the past. We will refer to it as the log odds ratio. Thanks to the conditional independencies assumed in the generative model, we can com- pute this Log odds ratio iteratively. Taking the limit as dt goes to zero, we get the following differential equation: L = ron 1 + e-L - roff 1 + eL + w i i(sit - 1) - A. B. r .r r .r on off on off x x x t dt i t t dt st Ot q , q q , q q , q on off j on off on off st I O G t Lt t t t t s s s t dt t t dt C. E. 4 sd 2 do g 0 oL -2 -4 500 1000 1500 2000 2500 3000 D. 2 20 0 L Count t Ot -2 0 500 1000 1500 2000 2500 3000 0 200 400 600 Time ISI Figure 1: A. Generative model for the synaptic input. B. Schematic representation of log odds ratio encoding and decoding. The dashed circle represents both eventual downstream elements and the self-prediction taking place inside the model neuron. A spike is fired only when Lt exceeds Gt. C. One example trial, where the state switches from 0 to 1 (shaded area) and back to 0. plain: Lt, dotted: Gt. Black stripes at the top: corresponding spikes train. D. Mean Log odds ratio (dark line) and mean output firing rate (clear line). E. Output spike raster plot (1 line per trial) and ISI distribution for the neuron shown is C. and D. Clear line: ISI distribution for a poisson neuron with the same rate. wi, the synaptic weight, describe how informative synapse i is about the state of the hidden variable, e.g. wi = log qion . Each synaptic spike (si qi t = 1) gives an impulse to the log off odds ratio, which is positive if this synapse is more active when the hidden state if 1 (i.e it increases the neuron's confidence that the state is 1), and negative if this synapse is more active when xt = 0 (i.e it decreases the neuron's confidence that the state is 1). The bias, , is determined by how informative it is not to receive any spike, e.g. = qi i on - qioff . By convention, we will consider that the "bias" is positive or zero (if not, we need simply to invert the status of the state x). 1.2 Generation of output spikes The spike train should convey a sparse representation of Lt, so that each spike reports new information about the state xt that is not redundant with that reported by other, preceding, spikes. This proposition is based on three arguments: First, spikes, being metabolically expensive, should be kept to a minimum. Second, spikes conveying redundant information would require a decoding of the entire spike train, whereas independent spike can be taken into account individually. And finally, we seek a self consistent model, with the spiking output having a similar semantics to its spiking input. To maximize the independence of the spikes (conditioned on xt), we propose that the neu- ron fires only when the difference between its log odds ratio Lt and a prediction Gt of this log odds ratio based on the output spikes emitted so far reaches a certain threshold. Indeed, supposing that downstream elements predicts Lt as best as they can, the neuron only needs to fire when it expects that prediction to be too inaccurate (figure 1-B). In practice, this will happen when the neuron receives new evidence for xt = 1. Gt should thereby follow the same dynamics as Lt when spikes are not received. The equation for Gt and the output Ot (Ot = 1 when an output spike is fired) are given by: G = ron 1 + e-L - roff 1 + eL + go(Ot - 1) (1) g O o t = 1. when Lt > Gt + , 0 otherwise, (2) 2 Here go, a positive constant, is the only free parameter, the other parameters being con- strained by the statistics of the synaptic input.

Most models of spatial representations in the cortex assume cells with limited receptive fields that are defined in a particular egocen(cid:173) tric frame of reference. However, cells outside of primary sensory cortex are either gain modulated by postural input or partially shifting. We show that solving classical spatial tasks, like sen(cid:173) sory prediction, multi-sensory integration, sensory-motor transfor(cid:173) mation and motor control requires more complicated intermediate representations that are not invariant in one frame of reference. We present an iterative basis function map that performs these spatial tasks optimally with gain modulated and partially shifting units, and tests it against neurophysiological and neuropsycholog(cid:173) ical data. In order to perform an action directed toward an object, it is necessary to have a representation of its spatial location. The brain must be able to use spatial cues coming from different modalities (e.g. vision, audition, touch, proprioception), combine them to infer the position of the object, and compute the appropriate movement. These cues are in different frames of reference corresponding to different sensory or motor modalities. Visual inputs are primarily encoded in retinotopic maps, auditory inputs are encoded in head centered maps and tactile cues are encoded in skin-centered maps. Going from one frame of reference to the other might seem easy. For example, the head-centered position of an object can be approximated by the sum of its retinotopic position and the eye position. However, positions are represented by population codes in the brain, and computing a head-centered map from a retinotopic map is a more complex computation than the underlying sum. Moreover, as we get closer to sensory-motor areas it seems reasonable to assume

Neurophysiologists are often faced with the problem of evaluating the quality of a code for a sensory or motor variable, either to relate it to the performance of the animal in a simple discrimination task or to compare the codes at various stages along the neuronal pathway. One common belief that has emerged from such studies is that sharpening of tuning curves improves the quality of the code, although only to a certain point; sharpening beyond that is believed to be harmful. We show that this belief relies on either problematic technical analysis or improper assumptions about the noise. We conclude that one cannot tell, in the general case, whether narrow tuning curves are better than wide ones; the answer depends critically on the covariance of the noise. The same conclusion applies to other manipulations of the tuning curve profiles such as gain increase.

Coarse codes are widely used throughout the brain to encode sensory and motor variables. Methods designed to interpret these codes, such as population vector analysis, are either inefficient (the variance of the estimate is much larger than the smallest possible variance) or biologically implausible, like maximum likelihood. Moreover, these methods attempt to compute a scalar or vector estimate of the encoded variable. Neurons are faced with a similar estimation problem. They must read out the responses of the presynaptic neurons, but, by contrast, they typically encode the variable with a further population code rather than as a scalar. We show how a nonlinear recurrent network can be used to perform estimation in a near-optimal way while keeping the estimate in a coarse code format. This work suggests that lateral connections in the cortex may be involved in cleaning up uncorrelated noise among neurons representing similar variables.