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Information processing with population codes


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, non-linear 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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Figure 1: The standard population coding model.
Figure 2: A neural implementation of a maximum likelihood estimator.
Figure 3: Function approximation with basis functions.
Figure 4: Multiplicity and uncertainty in population codes.


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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 non-random, effects.


A linear function of a one-dimensional 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.


A bell-shaped curve. Gaussian tuning curves are extensively used because their analytical expression can be easily manipulated in mathematical derivations.


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


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.


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.


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.


A mapping is a transformation from a variable x to a variable y, such as y = x2. The identity mapping is the simplest form of such mapping in which y is simply equal to x.


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.


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.


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.


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.


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.


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.


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.


A grating is a visual stimulus consisting of alternating light and dark bars, like the stripes on the United States flag. A full-field 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).

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