a–f, Schematics of the distributions of responses by two cells to two distinct stimuli in six different cases. Cyan dots indicate joint responses of the cell pair to stimulus 1; orange dots indicate responses to stimulus 2. Ellipses convey the shapes of the statistical distributions of the responses to each stimulus. Three types of noise correlation are depicted. In a and d, the two cells have statistically independent noise correlations. In b and e, the cells share positively correlated noise fluctuations. In c and f, the cells share negatively correlated noise fluctuations. In all six cases, dashed lines indicate optimal linear boundaries for stimulus discrimination. The information in a–f is based on similar plots published previously5,11,30. a–c, When both neurons have similar stimulus-response properties (for example, as schematized, when both cells have a smaller mean response to stimulus 1 than stimulus 2), positively correlated noise fluctuations (b) increase the overlap between the two response distributions and thereby impair stimulus discrimination, whereas negatively correlated noise fluctuations (c) improve stimulus discrimination as compared to the case with independent noise fluctuations (a). d–f, When both neurons have opposite stimulus tuning (for example, as schematized, when neuron 1 responds more vigorously to stimulus 1 and neuron 2 responds more vigorously to stimulus 2), positively correlated noise fluctuations (e) decrease the overlap between the two response distributions as compared to the case with independent noise fluctuations (d) and thereby improve stimulus discrimination, whereas negatively correlated noise fluctuations (f) impair stimulus discrimination by increasing the overlap of the two response distributions. g, Cells in visual cortical areas, denoted by red circles, integrate signals from earlier stages of the visual pathway, as schematized by the input connections to two example cortical neurons. Thus, as visual information propagates through neural circuitry, noise fluctuations become correlated between cells with similar receptive fields, leading to an upper bound on the amount of information that a neural ensemble can encode. h, Example receptive fields for cells in g. Cells in early stages of the visual processing pathway have relatively simple receptive fields. Integration of their activity patterns leads to more complex visual receptive fields in downstream visual areas. Dashed boxes enclose receptive fields (right) for the two example cells marked in g, as well as the receptive fields of cells providing visual inputs (left). i, A network’s pattern of synaptic connectivity constrains the dimensionality of the activity in downstream visual circuits12. Left, in the early layers of the visual pathway, the dimensionality of ensemble activity is about the same order of magnitude as the number of photoreceptors. In downstream visual areas, due to the extraction of visual features, neural activity is constrained to a manifold of lower dimensionality (indicated by the red-shaded manifold in the space of all possible photoreceptor inputs). This manifold is determined by the set of receptive fields and hence the visual features that the downstream visual area detects. Grey ellipses (left) depict the distributions of photoreceptor responses to two distinct visual stimuli; after propagating through the visual circuitry these distributions are confined to the lower-dimensional manifold (red ellipses). Right, for a family of visual stimuli parameterized by a single variable, the mean neural ensemble responses lie along a corresponding tuning curve. Noise in the input circuitry propagates to downstream areas and leads to noise fluctuations in downstream neurons that are statistically correlated for cells with similar receptive fields. This, in turn, implies that the magnitude of noise fluctuations along the neural tuning curve becomes proportional to the number of cells in a neural ensemble and indistinguishable from the encoded visual signals, which also increase in proportion to the number of cells. This proportional growth of noise and signal ultimately limits the ability to discriminate two visual stimuli. Thus, for neural ensembles with more than a certain number of cells, the encoded information reaches an upper bound. j, We simulated a two-layer, linear feedforward neural network, to illustrate that information-limiting correlations are intrinsic to feed-forward neural networks with overlapping receptive fields12. Top, for three example output cells, the plot shows the synaptic weights of the inputs from cells in the first layer of the network. Bottom, diagram of connections between the two layers of the network. Symbols are defined as follows: x is the mean activity of cells in the first layer in response to a given stimulus; n is the noise in the activity of the input cells; r is the activity of the output cells. k, Digitized plots of spike counts for simulated activity in the network of j, for the two example input cells (yellow and black) and three example output cells (red, green, blue). The noise traces for the input cells came from independent Poisson random processes. External inputs to the network selectively drove either the yellow or the black cell, but owing to the presence of noise the two cells are occasionally active concurrently. l, Frequency plots of pairwise activity levels (rounded to the nearest integer) for pairs of output cells in the network of j. Yellow and black circles denote which of the two corresponding input cells received external input. The diameter of each circle denotes the number of time bins with a given pair of activity levels in the two cells. Σ values are noise correlation coefficients and are larger for pairs of output cells with greater overlap in their receptive fields. m, Plot of the distribution of activity responses in the output cell layer, for the three example cells coloured green, red and blue in j. Data points are coloured either yellow or black, to indicate whether the output activity is a response to stimulation of the yellow- or black-coloured cell in the input layer. The red plane denotes the optimal linear classification boundary between the two stimulation conditions.