Neural population codes
Introduction
In vertebrates, information is often encoded as patterns of activity within populations of neurons subserving a similar function. This may partly explain the brain’s resilience to injury, precision of action, and ability to learn. Interest in understanding these patterns of activity is spurred by two experimental goals [1]. The first goal is decoding: understanding how the ‘neural code’ is read in order to control external electrical interfaces and prosthetic devices. The second goal is computation: understanding how the brain processes information to accomplish behavior. Approaching these two goals requires different strategies. In the first case, we attempt to extract the best possible estimate of an underlying variable. In the second case, we need to know not only what information is encoded but also what mechanisms exist for computation, as a limitation in processing within the brain could mean that not all of the information in a population is actually used.
Here, we discuss the analysis of population coding in several stages. In the first stage, we look at the firing patterns for the individual neurons that make up the population. Information coding in individual neurons is often described as being either a ‘rate code’ or a ‘temporal code’. In a rate code, all the information is contained in the average firing rate of the cell. In a temporal code, information may also be contained in the precise time at which a spike occurs, or in the precise interval between different spikes 2., 3., 4., 5., 6., 7., 8.. We will confine this discussion to neurons for which a rate code contains all the behaviorally relevant information 9., 10.•, 11., 12.. For a neuron that uses a rate code, the probability of firing is often modeled using Poisson statistics. In this case, the behavior of each cell in the population is described by tuning curves that relate the probability of firing to external variables that depend on sensation or motor performance. Therefore, this review begins with a description of cells with Poisson spike statistics, followed by an analysis of the formation of tuning curves and the types of tuning curves that are observed.
Given a neural population with a known set of tuning curves, the problem of reading the ‘neural code’ during a physiology experiment becomes one of estimation; we must attempt to estimate the true value of an external variable on the basis of recordings of spike data from the population. We first discuss linear estimation methods, which were the techniques initially applied to this problem. We then discuss optimal Bayesian methods, which are based on maximum likelihood estimators.
In order to perform a calculation on values that are represented by population codes, the brain must be able to transform one or more ‘input’ population codes into a new population code that represents the result of the desired computation. Therefore, computation in the brain can be described as a mapping between different population codes, in which values in one code are translated into values in a new code with different properties. As the population codes in our formulation will be described completely by the tuning curves for each individual cell, this problem is one of approximating the desired tuning curve for the cell in the output population as a function of the activities of the cells in the input population. We describe the algorithms that have been developed to accomplish this task.
We often would like to be able to answer the question, ‘what is coded in a particular neural population?’ This question is difficult to answer, as there may be more than one variable that is correlated with neural activity in the population. In the final section of this review, we examine techniques for determining whether there is a particular variable that is best represented by a population. As with algorithms for reading the neural code, there are both linear methods (which are based on correlation) and probability theory methods (which are based on calculating information measures).
The mathematical basis for understanding population codes is still in its infancy, and many of the techniques described here will no doubt be significantly modified in the future. At this time, however, we believe that it is possible to describe and analyze population codes using techniques from probability and estimation theory, as we will show below.
Section snippets
Single-cell representations
To model the behavior of individual cells in a population, we consider an external variable x that determines the average firing of a cell. As the firing rate depends on x, we can think of this as a tuning curve s(x). The cell generates a sequence of spikes n(t), in which the average firing rate is given by s(x). At each time t, n(t) is either 1 (meaning a spike occurred) or 0, (meaning no spike occurred). Figure 1 shows the basic structure of this model, with an example of a tuning curve and
Population representations and tuning curves
As a population is composed of individual cells, all of the information in a population of Poisson firing cells is determined by the set of firing rates si(x), where i indexes each of the different cells in the population. Figure 2 shows how a population might encode a single variable x. In order to model a cell or population that responds to more than one variable, we can consider the variable x to be a vector (x1,x2) encoding more than one feature of movement or the environment [13•]. If the
Interpreting neural activity
To interpret neural activity, we must estimate the value of the variable x on the basis of measurements of the spike activity of a population of cells (see Figure 6). For example, we might attempt to measure the direction of arm movement on the basis of measurements of firing in primary motor cortex 34., 35.. There are several experimental paradigms for estimating the value of a variable x(t) given observations of neural firing rates ni(t) and knowledge of receptive fields si(x). Most of these
Computation in population codes
Computation in the brain involves translations from one internal representation into another. For example, a sensory representation that indicates a target for movement must be translated into the motor representation of the pattern of muscle activity that will achieve that target 17., 57., 58.•. There are two basic goals of computation: first, the transformation of one function of the input into another, and second, the combination of information from multiple sources [59]. In both cases, we
Optimal representations
If we assume that a population code is optimal, then we can ask ‘optimal for what?’ 29., 71.•. Understanding optimality criteria within populations may be helpful for determining both their purpose and limitations to their function. For example, some authors have attempted to determine whether motor cortical cells are more closely related to intrinsic coordinates (such as joint angles) or extrinsic coordinates (such as Cartesian components of hand position) 23., 24., 72.•, and whether they are
Conclusions
Although population representations appear to be ubiquitous in neural systems, different brain regions have evolved to perform different specific tasks, and each region of the brain may perform its computations in very different ways. Nevertheless, an understanding of the fundamental properties of populations of spiking neurons may allow us to interpret population activity and hence to control prosthetic devices [34], and it may allow us to gain insight into the limitations and abilities of
References and recommended reading
Papers of particular interest, published within the annual period of review, have been highlighted as:
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of special interest
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of outstanding interest
Acknowledgements
The author was supported during this research by the Stanford University Department of Neurology, and by grant K23-NS41243 from the National Institute of Neurological Disorders and Stroke.
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