Why do biological and artificial neurons sometimes modularise, each encoding a single meaningful variable, and sometimes entangle their representation of many variables? In this work, we develop a theory of when biologically inspired networks---those that are nonnegative and energy efficient---modularise their representation of source variables (sources). We derive necessary and sufficient conditions on a sample of sources that determine whether the neurons in an optimal biologically-inspired linear autoencoder modularise. Our theory applies to any dataset, extending far beyond the case of statistical independence studied in previous work. Rather we show that sources modularise if their support is ``sufficiently spread''. From this theory, we extract and validate predictions in a variety of empirical studies on how data distribution affects modularisation in nonlinear feedforward and recurrent neural networks trained on supervised and unsupervised tasks. Furthermore, we apply these ideas to neuroscience data, showing that range independence can be used to understand the mixing or modularising of spatial and reward information in entorhinal recordings in seemingly conflicting experiments. Further, we use these results to suggest alternate origins of mixed-selectivity, beyond the predominant theory of flexible nonlinear classification. In sum, our theory prescribes precise conditions on when neural activities modularise, providing tools for inducing and elucidating modular representations in brains and machines.

While attention-based models have demonstrated the remarkable ability of in-context learning (ICL), the theoretical understanding of how these models acquired this ability through gradient descent training is still preliminary. Towards answering this question, we study the gradient descent dynamics of multi-head linear self-attention trained for in-context linear regression. We examine two parametrizations of linear self-attention: one with the key and query weights merged as a single matrix (common in theoretical studies), and one with separate key and query matrices (closer to practical settings). For the merged parametrization, we show that the training dynamics has two fixed points and the loss trajectory exhibits a single, abrupt drop. We derive an analytical time-course solution for a certain class of datasets and initialization. For the separate parametrization, we show that the training dynamics has exponentially many fixed points and the loss exhibits saddle-to-saddle dynamics, which we reduce to scalar ordinary differential equations. During training, the model implements principal component regression in context with the number of principal components increasing over training time. Overall, we provide a theoretical description of how ICL abilities evolve during gradient descent training of linear attention, revealing abrupt acquisition or progressive improvements depending on how the key and query are parametrized.

Using multiple input streams simultaneously to train multimodal neural networks is intuitively advantageous but practically challenging. A key challenge is unimodal bias, where a network overly relies on one modality and ignores others during joint training. We develop a theory of unimodal bias with multimodal deep linear networks to understand how architecture and data statistics influence this bias. This is the first work to calculate the duration of the unimodal phase in learning as a function of the depth at which modalities are fused within the network, dataset statistics, and initialization. We show that the deeper the layer at which fusion occurs, the longer the unimodal phase. A long unimodal phase can lead to a generalization deficit and permanent unimodal bias in the overparametrized regime. Our results, derived for multimodal linear networks, extend to nonlinear networks in certain settings. Taken together, this work illuminates pathologies of multimodal learning under joint training, showing that late and intermediate fusion architectures can give rise to long unimodal phases and permanent unimodal bias. Our code is available at: https://yedizhang.github.io/unimodal-bias.html.

Training data is always finite, making it unclear how to generalise to unseen situations. But, animals do generalise, wielding Occam’s razor to select a parsimonious explanation of their observations. How they do this is called their inductive bias, and it is implicitly built into the operation of animals’ neural circuits. This relationship between an observed circuit and its inductive bias is a useful explanatory window for neuroscience, allowing design choices to be understood normatively. However, it is generally very difficult to map circuit structure to inductive bias. Here, we present a neural network tool to bridge this gap. The tool meta-learns the inductive bias by learning functions that a neural circuit finds easy to generalise, since easy-to-generalise functions are exactly those the circuit chooses to explain incomplete data. In systems with analytically known inductive bias, i.e. linear and kernel regression, our tool recovers it. Generally, we show it can flexibly extract inductive biases from supervised learners, including spiking neural networks, and show how it could be applied to real animals. Finally, we use our tool to interpret recent connectomic data illustrating our intended use: understanding the role of circuit features through the resulting inductive bias.

To survive, animals must adapt synaptic weights based on external stimuli and rewards. And they must do so using local, biologically plausible, learning rules -- a highly nontrivial constraint. One possible approach is to perturb neural activity (or use intrinsic, ongoing noise to perturb it), determine whether performance increases or decreases, and use that information to adjust the weights. This algorithm -- known as node perturbation -- has been shown to work on simple problems, but little is known about either its stability or its scalability with respect to network size. We investigate these issues both analytically, in deep linear networks, and numerically, in deep nonlinear ones.We show analytically that in deep linear networks with one hidden layer, both learning time and performance depend very weakly on hidden layer size. However, unlike stochastic gradient descent, when there is model mismatch between the student and teacher networks, node perturbation is always unstable. The instability is triggered by weight diffusion, which eventually leads to very large weights. This instability can be suppressed by weight normalization, at the cost of bias in the learning rule. We confirm numerically that a similar instability, and to a lesser extent scalability, exist in deep nonlinear networks trained on both a motor control task and image classification tasks. Our study highlights the limitations and potential of node perturbation as a biologically plausible learning rule in the brain.

Sensory processing is hard because the variables of interest are encoded in spike trains in a relatively complex way. A major goal in studies of sensory processing is to understand how the brain extracts those variables. Here we revisit a common encoding model in which variables are encoded linearly. Although there are typically more variables than neurons, this problem is still solvable because only a small number of variables appear at any one time (sparse prior). However, previous solutions require all-to-all connectivity, inconsistent with the sparse connectivity seen in the brain. Here we propose an algorithm that provably reaches the MAP (maximum a posteriori) inference solution, but does so using sparse connectivity. Our algorithm is inspired by the circuit of the mouse olfactory bulb, but our approach is general enough to apply to other modalities. In addition, it should be possible to extend it to nonlinear encoding models.

Convolutional networks are ubiquitous in deep learning. They are particularly useful for images, as they reduce the number of parameters, reduce training time, and increase accuracy. However, as a model of the brain they are seriously problematic, since they require weight sharing - something real neurons simply cannot do. Consequently, while neurons in the brain can be locally connected (one of the features of convolutional networks), they cannot be convolutional. Locally connected but non-convolutional networks, however, significantly underperform convolutional ones. This is troublesome for studies that use convolutional networks to explain activity in the visual system. Here we study plausible alternatives to weight sharing that aim at the same regularization principle, which is to make each neuron within a pool react similarly to identical inputs. The most natural way to do that is by showing the network multiple translations of the same image, akin to saccades in animal vision. However, this approach requires many translations, and doesn't remove the performance gap. We propose instead to add lateral connectivity to a locally connected network, and allow learning via Hebbian plasticity. This requires the network to pause occasionally for a sleep-like phase of "weight sharing". This method enables locally connected networks to achieve nearly convolutional performance on ImageNet and improves their fit to the ventral stream data, thus supporting convolutional networks as a model of the visual stream.

The training of sparse neural networks is becoming an increasingly important tool for reducing the computational footprint of models at training and evaluation, as well enabling the effective scaling up of models. Whereas much work over the years has been dedicated to specialised pruning techniques, little attention has been paid to the inherent effect of gradient based training on model sparsity. Inthis work, we introduce Powerpropagation, a new weight-parameterisation for neural networks that leads to inherently sparse models. Exploiting the behaviour of gradient descent, our method gives rise to weight updates exhibiting a “rich get richer” dynamic, leaving low-magnitude parameters largely unaffected by learning. Models trained in this manner exhibit similar performance, but have a distributionwith markedly higher density at zero, allowing more parameters to be pruned safely. Powerpropagation is general, intuitive, cheap and straight-forward to implement and can readily be combined with various other techniques. To highlight its versatility, we explore it in two very different settings: Firstly, following a recent line of work, we investigate its effect on sparse training for resource-constrained settings. Here, we combine Powerpropagation with a traditional weight-pruning technique as well as recent state-of-the-art sparse-to-sparse algorithms, showing superior performance on the ImageNet benchmark. Secondly, we advocate the useof sparsity in overcoming catastrophic forgetting, where compressed representations allow accommodating a large number of tasks at fixed model capacity. In all cases our reparameterisation considerably increases the efficacy of the off-the-shelf methods.

The state-of-the art machine learning approach to training deep neural networks, backpropagation, is implausible for real neural networks: neurons need to know their outgoing weights; training alternates between a bottom-up forward pass (computation) and a top-down backward pass (learning); and the algorithm often needs precise labels of many data points. Biologically plausible approximations to backpropagation, such as feedback alignment, solve the weight transport problem, but not the other two. Thus, fully biologically plausible learning rules have so far remained elusive. Here we present a family of learning rules that does not suffer from any of these problems. It is motivated by the information bottleneck principle (extended with kernel methods), in which networks learn to compress the input as much as possible without sacrificing prediction of the output. The resulting rules have a 3-factor Hebbian structure: they require pre- and post-synaptic firing rates and an error signal - the third factor - consisting of a global teaching signal and a layer-specific term, both available without a top-down pass. They do not require precise labels; instead, they rely on the similarity between pairs of desired outputs. Moreover, to obtain good performance on hard problems and retain biological plausibility, our rules need divisive normalization - a known feature of biological networks. Finally, simulations show that our rules perform nearly as well as backpropagation on image classification tasks.

Understanding the brain is one of the most challenging scientific problems faced by humanity.In addition to offering an improved understanding of who we are, deciphering how our brains operate will provide payoffs for the treatment of mental and neurological disease as well as the creation of intelligent systems.Consequently, developed countries are pouring ever-increasing research funds into brain science.Prominent examples include the BRAIN initiative in the United States and the Human Brain Project in Europe.Although Africa may not yet possess financial and technological resources on this scale, Africans represent a vast and growing human resource poised to make important contributions to neuroscience.As the third yearly International Brain Research Organisation (IBRO)-Simons Computational Neuroscience Imbizo recently wrapped up in South Africa, we call for greater efforts to empower Africans to join the global neuroscience community and argue that the subfield of computational neuroscience is the ideal vehicle to do so.Computational and theoretical neuroscience, also known as neurotheory, or just "theory" within neurobiology, is a young, rapidly expanding field within neuroscience; its goal is to determine and articulate the general principles by which nervous systems perform their computations.We think it is a good example of how theory, as a subfield, can serve as a catalyst to accelerate the development of science in Africa.Due to the diversity of methods required for understanding the brain at multiple levels [1], skills are leveraged from a range of fields, such as biology, computer science, electrical engineering, mathematics, physics, and psychology.Beyond just increasing our body of scientific knowledge, the impact of this interdisciplinary field is already significant to the clinic and industry: decoding algorithms are helping restore movement in paralyzed patients [2,3], and machine learning is continually influenced by engagement with neuroscience [4].A notable aspect of this growing field is its accessibility: computational neuroscience requires only data and individual expertise, independent of geographic location.There is growing momentum for experimentalists to make their data available for public access and for increased collaboration between groups, a movement lead by flagship projects that curate, systematize, and make freely available large-scale neuronal data [5-8].A parallel development has been the recent surge in open-access publishing and the establishment of alternative methods for accessing scientific papers outside of well-endowed institutions (e.g., ResearchGate, bioRxiv, and especially important in Africa, SciHub).These dual developments mean that aspiring African scientists in resource-limited settings have increasing access to the latest in both data and theory from across the broad field of neuroscience.From an African

Sensory neurons give highly variable responses to stimulation, which can limit the amount of stimulus information available to downstream circuits. Much work has investigated the factors that affect the amount of information encoded in these population responses, leading to insights about the role of covariability among neurons, tuning curve shape, etc. However, the informativeness of neural responses is not the only relevant feature of population codes; of potentially equal importance is how robustly that information propagates to downstream structures. For instance, to quantify the retina's performance, one must consider not only the informativeness of the optic nerve responses, but also the amount of information that survives the spike-generating nonlinearity and noise corruption in the next stage of processing, the lateral geniculate nucleus. Our study identifies the set of covariance structures for the upstream cells that optimize the ability of information to propagate through noisy, nonlinear circuits. Within this optimal family are covariances with "differential correlations", which are known to reduce the information encoded in neural population activities. Thus, covariance structures that maximize information in neural population codes, and those that maximize the ability of this information to propagate, can be very different. Moreover, redundancy is neither necessary nor sufficient to make population codes robust against corruption by noise: redundant codes can be very fragile, and synergistic codes can-in some cases-optimize robustness against noise.

Zipf's law, which states that the probability of an observation is inversely proportional to its rank, has been observed in many domains. While there are models that explain Zipf's law in each of them, those explanations are typically domain specific. Recently, methods from statistical physics were used to show that a fairly broad class of models does provide a general explanation of Zipf's law. This explanation rests on the observation that real world data is often generated from underlying causes, known as latent variables. Those latent variables mix together multiple models that do not obey Zipf's law, giving a model that does. Here we extend that work both theoretically and empirically. Theoretically, we provide a far simpler and more intuitive explanation of Zipf's law, which at the same time considerably extends the class of models to which this explanation can apply. Furthermore, we also give methods for verifying whether this explanation applies to a particular dataset. Empirically, these advances allowed us extend this explanation to important classes of data, including word frequencies (the first domain in which Zipf's law was discovered), data with variable sequence length, and multi-neuron spiking activity.

Humans stand out from other animals in that they are able to explicitly report on the reliability of their internal operations. This ability, which is known as metacognition, is typically studied by asking people to report their confidence in the correctness of some decision. However, the computations underlying confidence reports remain unclear. In this paper, we present a fully Bayesian method for directly comparing models of confidence. Using a visual two-interval forced-choice task, we tested whether confidence reports reflect heuristic computations (e.g. the magnitude of sensory data) or Bayes optimal ones (i.e. how likely a decision is to be correct given the sensory data). In a standard design in which subjects were first asked to make a decision, and only then gave their confidence, subjects were mostly Bayes optimal. In contrast, in a less-commonly used design in which subjects indicated their confidence and decision simultaneously, they were roughly equally likely to use the Bayes optimal strategy or to use a heuristic but suboptimal strategy. Our results suggest that, while people's confidence reports can reflect Bayes optimal computations, even a small unusual twist or additional element of complexity can prevent optimality.

The olfactory system faces a difficult inference problem: it has to determine what odors are present based on the distributed activation of its receptor neurons. Here we derive neural implementations of two approximate inference algorithms that could be used by the brain. One is a variational algorithm (which builds on the work of Beck. et al., 2012), the other is based on sampling. Importantly, we use a more realistic prior distribution over odors than has been used in the past: we use a spike and slab'' prior, for which most odors have zero concentration. After mapping the two algorithms onto neural dynamics, we find that both can infer correct odors in less than 100 ms, although it takes ~500 ms to eliminate false positives. Thus, at the behavioral level, the two algorithms make very similar predictions. However, they make different assumptions about connectivity and neural computations, and make different predictions about neural activity. Thus, they should be distinguishable experimentally. If so, that would provide insight into the mechanisms employed by the olfactory system, and, because the two algorithms use very different coding strategies, that would also provide insight into how networks represent probabilities."

The brain is easily able to process and categorize complex time-varying signals. For example, the two sentences, "It is cold in London this time of year" and "It is hot in London this time of year," have different meanings, even though the words hot and cold appear several seconds before the ends of the two sentences. Any network that can tell these sentences apart must therefore have a long temporal memory. In other words, the current state of the network must depend on events that happened several seconds ago. This is a difficult task, as neurons are dominated by relatively short time constants--tens to hundreds of milliseconds. Nevertheless, it was recently proposed that randomly connected networks could exhibit the long memories necessary for complex temporal processing. This is an attractive idea, both for its simplicity and because little tuning of recurrent synaptic weights is required. However, we show that when connectivity is high, as it is in the mammalian brain, randomly connected networks cannot exhibit temporal memory much longer than the time constants of their constituent neurons.

Maximum entropy models have become popular statistical models in neuroscience and other areas in biology, and can be useful tools for obtaining estimates of mu- tual information in biological systems. However, maximum entropy models fit to small data sets can be subject to sampling bias; i.e. the true entropy of the data can be severely underestimated. Here we study the sampling properties of estimates of the entropy obtained from maximum entropy models. We show that if the data is generated by a distribution that lies in the model class, the bias is equal to the number of parameters divided by twice the number of observations. However, in practice, the true distribution is usually outside the model class, and we show here that this misspecification can lead to much larger bias. We provide a perturba- tive approximation of the maximally expected bias when the true model is out of model class, and we illustrate our results using numerical simulations of an Ising model; i.e. the second-order maximum entropy distribution on binary data.

One of the most critical problems we face in the study of biological systems is building accurate statistical descriptions of them. This problem has been particularly challenging because biological systems typically contain large numbers of interacting elements, which precludes the use of standard brute force approaches. Recently, though, several groups have reported that there may be an alternate strategy. The reports show that reliable statistical models can be built without knowledge of all the interactions in a system; instead, pairwise interactions can suffice. These findings, however, are based on the analysis of small subsystems. Here, we ask whether the observations will generalize to systems of realistic size, that is, whether pairwise models will provide reliable descriptions of true biological systems. Our results show that, in most cases, they will not. The reason is that there is a crossover in the predictive power of pairwise models: If the size of the subsystem is below the crossover point, then the results have no predictive power for large systems. If the size is above the crossover point, then the results may have predictive power. This work thus provides a general framework for determining the extent to which pairwise models can be used to predict the behavior of large biological systems. Applied to neural data, the size of most systems studied so far is below the crossover point.

Point process encoding models provide powerful statistical methods for under- standing the responses of neurons to sensory stimuli. Although these models have been successfully applied to neurons in the early sensory pathway, they have fared less well capturing the response properties of neurons in deeper brain areas, ow- ing in part to the fact that they do not take into account multiple stages of pro- cessing. Here we introduce a new twist on the point-process modeling approach: we include unobserved as well as observed spiking neurons in a joint encoding model. The resulting model exhibits richer dynamics and more highly nonlinear response properties, making it more powerful and more flexible for fitting neural data. More importantly, it allows us to estimate connectivity patterns among neu- rons (both observed and unobserved), and may provide insight into how networks process sensory input. We formulate the estimation procedure using variational EM and the wake-sleep algorithm, and illustrate the model’s performance using a simulated example network consisting of two coupled neurons.

A fundamental problem in neuroscience is understanding how working memory--the ability to store information at intermediate timescales, like tens of seconds--is implemented in realistic neuronal networks. The most likely candidate mechanism is the attractor network, and a great deal of effort has gone toward investigating it theoretically. Yet, despite almost a quarter century of intense work, attractor networks are not fully understood. In particular, there are still two unanswered questions. First, how is it that attractor networks exhibit irregular firing, as is observed experimentally during working memory tasks? And second, how many memories can be stored under biologically realistic conditions? Here we answer both questions by studying an attractor neural network in which inhibition and excitation balance each other. Using mean-field analysis, we derive a three-variable description of attractor networks. From this description it follows that irregular firing can exist only if the number of neurons involved in a memory is large. The same mean-field analysis also shows that the number of memories that can be stored in a network scales with the number of excitatory connections, a result that has been suggested for simple models but never shown for realistic ones. Both of these predictions are verified using simulations with large networks of spiking neurons.

Cortical neurons are predominantly excitatory and highly interconnected. In spite of this, the cortex is remarkably stable: normal brains do not exhibit the kind of runaway excitation one might expect of such a system. How does the cortex maintain stability in the face of this massive excitatory feedback? More importantly, how does it do so during computations, which necessarily involve elevated firing rates? Here we address these questions in the context of attractor networks-networks that exhibit multiple stable states, or memories. We find that such networks can be stabilized at the relatively low firing rates observed in vivo if two conditions are met: (1) the background state, where all neurons are firing at low rates, is inhibition dominated, and (2) the fraction of neurons involved in a memory is above some threshold, so that there is sufficient coupling between the memory neurons and the background. This allows "dynamical stabilization" of the attractors, meaning feedback from the pool of background neurons stabilizes what would otherwise be an unstable state. We suggest that dynamical stabilization may be a strategy used for a broad range of computations, not just those involving attractors.

We calculate the firing rate of the quadratic integrate-and-fire neuron in response to a colored noise input current. Such an input current is a good approximation to the noise due to the random bombardment of spikes, with the correlation time of the noise corresponding to the decay time of the synapses. The key parameter that determines the firing rate is the ratio of the correlation time of the colored noise, tau(s), to the neuronal time constant, tau(m). We calculate the firing rate exactly in two limits: when the ratio, tau(s)/tau(m), goes to zero (white noise) and when it goes to infinity. The correction to the short correlation time limit is O(tau(s)/tau(m)), which is qualita tively different from that of the leaky integrate-and-fire neuron, where the correction is O( radical tau(s)/tau(m)). The difference is due to the different boundary conditions of the probability density function of the membrane potential of the neuron at firing threshold. The correction to the long correlation time limit is O(tau(m)/tau(s)). By combining the short and long correlation time limits, we derive an expression that provides a good approximation to the firing rate over the whole range of tau(s)/tau(m) in the suprathreshold regime-that is, in a regime in which the average current is sufficient to make the cell fire. In the subthreshold regime, the expression breaks down somewhat when tau(s) becomes large compared to tau(m).

Almost two decades ago, Hopfield [1] showed that networks of highly reduced model neurons can exhibit multiple attracting fixed points, thus providing a substrate for associative memory. It is still not clear, however, whether realistic neuronal networks can support multiple attractors. The main difficulty is that neuronal networks in vivo exhibit a stable background state at low firing rate, typ(cid:173) ically a few Hz. Embedding attractor is easy; doing so without destabilizing the background is not. Previous work [2, 3] focused on the sparse coding limit, in which a vanishingly small number of neurons are involved in any memory. Here we investigate the case in which the number of neurons involved in a memory scales with the number of neurons in the network. In contrast to the sparse coding limit, we find that multiple attractors can co-exist robustly with a stable background state. Mean field theory is used to under(cid:173) stand how the behavior of the network scales with its parameters, and simulations with analog neurons are presented. One of the most important features of the nervous system is its ability to perform associative memory. It is generally believed that associative memory is implemented using attractor networks - experimental studies point in that direction [4- 7], and there are virtually no competing theoretical models. Perhaps surprisingly, however, it is still an open theoretical question whether attractors can exist in realistic neu(cid:173) ronal networks. The "realistic" feature that is probably hardest to capture is the steady firing at low rates - the background state - that is observed throughout the intact nervous system [8- 13]. The reason it is difficult to build an attractor network that is stable at low firing rates, at least in the sparse coding limit, is as follows [2,3]: Attractor networks are constructed by strengthening recurrent connections among sub-populations of neurons. The strengthening must be large enough that neurons within a sub-population can sustain a high firing rate state, but not so large that the sub-population can be spontaneously active. This implies that the neuronal gain functions - the firing rate of the post-synaptic neurons as a function of the average • http) / culture.neurobio.ucla.edu/ "'pel firing rate of the pre-synaptic neurons - must be sigmoidal: small at low firing rate to provide stability, high at intermediate firing rate to provide a threshold (at an unstable equilibrium), and low again at high firing rate to provide saturation and a stable attractor. In other words, a requirement for the co-existence of a stable background state and multiple attractors is that the gain function of the excitatory neurons be super linear at the observed background rates of a few Hz [2,3]. However - and this is where the problem lies - above a few Hz most realistic gain function are nearly linear or sublinear (see, for example, Fig. Bl of [14]). The superlinearity requirement rests on the implicit assumption that the activity of the sub-population involved in a memory does not affect the other neurons in the network. While this assumption is valid in the sparse coding limit , it breaks down in realistic networks containing both excitatory and inhibitory neurons. In such networks, activity among excitatory cells results in inhibitory feedback. This feedback, if powerful enough, can stabilize attractors even without a saturating nonlinearity, essentially by stabilizing the equilibrium (above considered unstable) on the steep part of the gain function. The price one pays, though, is that a reasonable fraction of the neurons must be involved in each of the memories, which takes us away from the sparse coding limit and thus reduces network capacity [15].

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.