Abstract
Information is encoded in the brain by populations or clusters of cells, rather than by single cells. This encoding strategy is known as population coding. Here we review the standard use of population codes for encoding and decoding information, and consider how population codes can be used to support neural computations such as noise removal and nonlinear mapping. More radical ideas about how population codes may directly represent information about stimulus uncertainty are also discussed.
Key Points

Population codes:
Information about quantities in the world is represented by neural activity patterns in a characteristic general fashion. Single cells respond to a specific variety of values of the quantities; so each particular value leads to coordinated firing in a whole population of cells.

Encoding in the standard model:
Under the standard model, a single value of a quantity is encoded by the population. Each cell has a tuning curve for the quantity, which shows how its average response (in spikes per second) varies with the quantity. The actual population activity on any trial is noisy about these means and the noise has a variance that can be characterized.

Decoding by maximum likelihood and Bayes rule:
Under the standard model, a simple statistical technique can be used to find out what the activity of the population on any trial implies about the value of the quantity encoded. Under some further assumptions, the most likely value of the quantity can be extracted, essentially by a process of curve fitting the average responses predicted by the tuning curves of the cells to the actual responses recorded.

Decoding by recurrent interactions:
Decoding in the standard model seems to require complex mathematical operations. However, nonlinear recurrent networks of neurons can be constructed that have stable points corresponding to each value of the encoded variable, and can be shown to perform nearly optimal decoding using simple interactions.

Basis function mappings:
The tuning curves of neurons show that they collectively form a particular representation, called a basis function representation, of the quantity they encode. This means that any, even nonlinear, function of this quantity can be extracted as a linear sum over the activities of the population of neurons. This underlies a successful and predictive model of parietal cortex.

Probabilistic population codes:
Although the standard model is a powerful way of characterizing population codes, it has some shortcomings. In particular, it cannot correctly represent the uncertainty the animal might have about the quantity encoded. An extension to the standard model can be defined, in which the population of neurons is treated as encoding uncertainty (and also multiplicity, in the case that multiple values of the quantity are present).
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References
Bair, W. Spike timing in the mammalian visual system. Curr. Opin. Neurobiol. 9, 447–453 (1999).
Borst, A. & Theunissen, F. E. Information theory and neural coding. Nature Neurosci. 2, 947– 957 (1999).
Usrey, W. & Reid, R. Synchronous activity in the visual system. Annu. Rev. Physiol. 61, 435– 456 (1999).
Zemel, R., Dayan, P. & Pouget, A. Probabilistic interpretation of population codes. Neural Comput. 10, 403–430 (1998).
Tolhurst, D., Movshon, J. & Dean, A. The statistical reliability of signals in single neurons in cat and monkey visual cortex. Vision Res. 23, 775–785 (1982).
Földiak, P. in Computation and Neural Systems (eds Eeckman, F. & Bower, J.) 55–60 (Kluwer Academic Publishers, Norwell, Massachusetts, 1993).
Salinas, E. & Abbot, L. Vector reconstruction from firing rate. J. Comput. Neurosci. 1, 89– 108 (1994).
Sanger, T. Probability density estimation for the interpretation of neural population codes. J. Neurophysiol. 76, 2790– 2793 (1996).
Zhang, K., Ginzburg, I., McNaughton, B. & Sejnowski, T. Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells. J. Neurophysiol. 79, 1017–1044 (1998).
Cox, D. & Hinckley, D. Theoretical statistics (Chapman and Hall, London, 1974).
Ferguson, T. Mathematical statistics: a decision theoretic approach (Academic, New York, 1967).
Paradiso, M. A theory of the use of visual orientation information which exploits the columnar structure of striate cortex. Biol. Cybern. 58, 35–49 (1988).A pioneering study of the statistical properties of population codes, including the first use of Bayesian techniques to read out and analyse population codes.
Seung, H. & Sompolinsky, H. Simple model for reading neuronal population codes. Proc. Natl Acad. Sci. USA 90, 10749–10753 (1993).
Deneve, S., Latham, P. & Pouget, A. Reading population codes: A neural implementation of ideal observers. Nature Neurosci. 2, 740 –745 (1999).Shows how a recurrent network of units with bellshaped tuning curves can be wired to implement a close approximation to a maximum likelihood estimator. Maximum likelihood estimation is widely used in psychophysics to analyse human performance in simple perceptual tasks in a class of model known as `ideal observer analysis'.
Georgopoulos, A., Kalaska, J. & Caminiti, R. On the relations between the direction of twodimensional arm movements and cell discharge in primate motor cortex. J. Neurosci. 2, 1527–1537 ( 1982).
Pouget, A., Deneve, S., Ducom, J. & Latham, P. Narrow vs wide tuning curves: what's best for a population code? Neural Comput. 11, 85–90 ( 1999).
Pouget, A. & Thorpe, S. Connectionist model of orientation identification. Connect. Sci. 3, 127– 142 (1991).
Regan, D. & Beverley, K. Postadaptation orientation discrimination . J. Opt. Soc. Am. A 2, 147– 155 (1985).
Zhang, K. Representation of spatial orientation by the intrinsic dynamics of the headdirection cell ensemble: a theory. J. Neurosci. 16, 2112–2126 (1996). Head direction cells in rats encode the heading direction of the rat in worldcentred coordinates. This internal compass is calibrated with sensory cues, maintained in the absence of these cues and updated after each movement of the head. This model shows how attractor networks of neurons with bellshaped tuning curves to head direction can be wired to account for these properties.
Seung, H. How the brain keeps the eyes still. Proc. Natl Acad. Sci. USA 93, 13339–13344 (1996).
Poggio, T. A theory of how the brain might work. Cold Spring Harbor Symp. Quant. Biol. 55, 899–910 (1990).Introduces the idea that many computations in the brain can be formalized in terms of nonlinear mappings, and as such can be solved with population codes computing basis functions. Although completely living up to the title would be a tall order, this remains a most interesting proposal. A wide range of available neurophysiological data can be easily understood within this framework.
Pouget, A. & Sejnowski, T. Spatial transformations in the parietal cortex using basis functions. J. Cogn. Neurosci. 9, 222–237 (1997).
Andersen, R., Essick, G. & Siegel, R. Encoding of spatial location by posterior parietal neurons . Science 230, 456–458 (1985).
Squatrito, S. & Maioli, M. Gaze field properties of eye position neurons in areas MST and 7a of macaque monkey. Visual Neurosci. 13, 385–398 ( 1996).
Rumelhart, D., Hinton, G. & Williams, R. in Parallel Distributed Processing (eds Rumelhart, D., McClelland, J. & Group, P. R.) 318–362 (MIT Press, Cambridge, Massachusetts, 1986).
Zipser, D. & Andersen, R. A backpropagation programmed network that stimulates reponse properties of a subset of posterior parietal neurons . Nature 331, 679–684 (1988).
Burnod, Y. et al. Visuomotor transformations underlying arm movements toward visual targets: a neural network model of cerebral cortical operations. J. Neurosci. 12, 1435–1453 (1992).A model of the coordinate transformation required for arm movements using a representation very similar to basis functions. This model was one of the first to relate the tuning properties of cells in the primary motor cortex to their computational role. In particular, it explains why cells in M1 change their preferred direction to hand movement with starting hand position.
Salinas, E. & Abbot, L. Transfer of coded information from sensory to motor networks. J. Neurosci. 15, 6461–6474 (1995). A model showing how a basis function representation can be used to learn visuomotor transformations with a simple hebbian learning rule.
Olshausen, B. & Field, D. Emergence of simplecell receptive field properties by learning a sparse code for natural images. Nature 381, 607–609 ( 1996).
Bishop, C., Svenson, M. & Williams, C. GTM: The generative topographic mapping. Neural Comput. 10, 215–234 (1998).
Lewicki, M. & Sejnowski, T. Learning overcomplete representations . Neural Comput. 12, 337– 365 (2000).
Hinton, G. E. in Proceedings of the Ninth International Conference on Artificial Neural Networks 1–6 (IEEE, London, England, 1999).
Poggio, T. & Edelman, S. A network that learns to recognize threedimensional objects. Nature 343, 263 –266 (1990).
Salinas, E. & Abbott, L. Invariant visual responses from attentional gain fields. J. Neurophysiol. 77, 3267– 3272 (1997).
Deneve, S. & Pouget, A. in Advances in Neural Information Processing Systems (eds Jordan, M., Kearns, M. & Solla, S.) (MIT Press, Cambridge, Massachusetts, 1998).
Groh, J. & Sparks, D. Two models for transforming auditory signals from headcentered to eyecentered coordinates. Biol. Cybern. 67, 291–302 ( 1992).
Pouget, A. & Sejnowski, T. A neural model of the cortical representation of egocentric distance. Cereb. Cortex 4, 314–329 (1994).
Olshausen, B., Anderson, C. & Essen, D. V. A multiscale dynamic routing circuit for forming size and positioninvariant object representations. J. Comput. Neurosci. 2, 45–62 ( 1995).
Treue, S., Hol, K. & Rauber, H. Seeing multiple directions of motionphysiology and psychophysics. Nature Neurosci. 3, 270–276 (2000).
Shadlen, M., Britten, K., Newsome, W. & Movshon, T. A computational analysis of the relationship between neuronal and behavioral responses to visual motion. J. Neurosci. 16, 1486– 1510 (1996).
Zemel, R. & Dayan, P. in Advances in Neural Information Processing Systems 11 (eds Kearns, M., Solla, S. & Cohn, D.) 174–180 (MIT Press, Cambridge, Massachusetts, 1999).
Simoncelli, E., Adelson, E. & Heeger, D. in Proceedings 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition 310– 315 (Los Alamitos, Los Angeles, 1991).
Watamaniuk, S., Sekuler, R. & Williams, D. Direction perception in complex dynamic displays: the integration of direction information. Vision Res. 29 , 47–59 (1989).
Anderson, C. in Computational Intelligence Imitating Life 213– 222 (IEEE Press, New York, 1994).Neurons are frequently suggested to encode single values (or at most 2–3 values for cases such as transparency). Anderson challenges this idea and argues that population codes might encode probability distributions instead. Being able to encode a probability distribution is important because it would allow the brain to perform Bayesian inference, an efficient way to compute in the face of uncertainty.
Recanzone, G., Wurtz, R. & Schwarz, U. Responses of MT and MST neurons to one and two moving objects in the receptive field. J. Neurophysiol. 78 , 2904–2915 (1997).
Wezel, R. V., Lankheet, M., Verstraten, F., Maree, A. & Grind, W. V. D. Responses of complex cells in area 17 of the cat to bivectorial transparent motion. Vision Res. 36, 2805–2813 ( 1996).
Riesenhuber, M. & Poggio, T. Hierarchical models of object recognition in cortex. Nature Neurosci. 2 , 1019–1025 (1999).
Perrett, D., Mistlin, A. & Chitty, A. Visual neurons responsive to faces. Trends Neurosci. 10, 358–364 ( 1987).
Bruce, V., Cowey, A., Ellis, A. & Perret, D. Processing the Facial Image (Clarendon, Oxford, 1992).
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Glossary
 NEURONAL NOISE

The part of a neuronal response that cannot apparently be accounted for by the stimulus. Part of this factor may arise from truly random effects (such as stochastic fluctuations in neuronal channels), and part from uncontrolled, but nonrandom, effects.
 NONLINEAR FUNCTION

A linear function of a onedimensional variable (such as direction of motion) is any function that looks like a straight line, that is, any function that can be written as y = ax + b , where a and b are constant. Any other functions are nonlinear. In two dimensions and above, linear functions correspond to planes and hyperplanes. All other functions are nonlinear.
 GAUSSIAN FUNCTION

A bellshaped curve. Gaussian tuning curves are extensively used because their analytical expression can be easily manipulated in mathematical derivations.
 TUNING CURVE

A tuning curve to a feature is the curve describing the average response of a neuron as a function of the feature values.
 SYMMETRIC LATERAL CONNECTIONS

Lateral connections are formed between neurons at the same hierarchical level. For instance, the connections between cortical neurons in the same area and same layer are said to be lateral. Lateral connections are symmetric if any connection from neuron a to neuron b is matched by an identical connection from neuron b to neuron a.
 HYPERCOLUMN

In the visual cortex, an orientation hypercolumn refers to a patch of cortex containing neurons with similar spatial receptive fields but covering all possible preferred orientations. This concept can be generalized to other visual features and to other sensory and motor areas.
 OPTIMAL INFERENCE

This refers to the statistical computation of specifically extracting all the information implied about the stimulus from the (noisy) activities of the population. Ideal observers make optimal inferences.
 IDENTITY MAPPING

A mapping is a transformation from a variable x to a variable y, such as y = x^{2}. The identity mapping is the simplest form of such mapping in which y is simply equal to x.
 BASIS SET

In linear algebra, a set of vectors such that any other vector can be expressed in terms of a weighted sum of these vectors is known as a basis. By analogy, sine and cosine functions of all possible frequencies are said to form a basis set.
 FOURIER TRANSFORM

A transformation that expresses any function in terms of a weighted sum of sine and cosine functions of all possible frequencies. The weights assigned to each frequency are specific to the function being considered and are known as the Fourier coefficients for this function.
 BACKPROPAGATION

A learning algorithm based on the chain rule in calculus, in which error signals computed in the output layer are propagated back through any intervening layers to the input layer of the network.
 HEBBIAN LEARNING RULE

A learning rule in which the synaptic strength of a connection is changed according to the correlation in the activities of its presynaptic and postsynaptic sides.
 DELTA LEARNING RULE

A learning rule that adjusts synaptic weights according to the product of the presynaptic activity and a postsynaptic error signal obtained by computing the difference between the actual output activity and a desired or required output activity.
 UNSUPERVISED OR SELFORGANIZING METHODS

An adaptation in which a network is trained to uncover and represent the statistical structure within a set of inputs, without reference to a set of explicitly desired outputs. This contrasts with supervised learning, in which a network is trained to produce particular desired outputs in response to given inputs.
 MOTION TRANSPARENCY

A situation in which several directions of motion are perceived simultaneously at the same location. This occurs when looking through the windscreen of a car. At each location, the windscreen is perceived as being still while the background moves in a direction opposite to the motion of the car.
 FULLFIELD GRATING

A grating is a visual stimulus consisting of alternating light and dark bars, like the stripes on the United States flag. A fullfield grating is a very wide grating that occupies the entire visual field.
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Pouget, A., Dayan, P. & Zemel, R. Information processing with population codes. Nat Rev Neurosci 1, 125–132 (2000). https://doi.org/10.1038/35039062
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DOI: https://doi.org/10.1038/35039062
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