Biological networks have so many possible states that exhaustive sampling is impossible. Successful analysis thus depends on simplifying hypotheses, but experiments on many systems hint that complicated, higher-order interactions among large groups of elements have an important role. Here we show, in the vertebrate retina, that weak correlations between pairs of neurons coexist with strongly collective behaviour in the responses of ten or more neurons. We find that this collective behaviour is described quantitatively by models that capture the observed pairwise correlations but assume no higher-order interactions. These maximum entropy models are equivalent to Ising models, and predict that larger networks are completely dominated by correlation effects. This suggests that the neural code has associative or error-correcting properties, and we provide preliminary evidence for such behaviour. As a first test for the generality of these ideas, we show that similar results are obtained from networks of cultured cortical neurons.

The reproducibility and precision of biological patterning is limited by the accuracy with which concentration profiles of morphogen molecules can be established and read out by their targets. We consider four measures of precision for the Bicoid morphogen in the Drosophila embryo: the concentration differences that distinguish neighboring cells, the limits set by the random arrival of Bicoid molecules at their targets (which depends on absolute concentration), the noise in readout of Bicoid by the activation of Hunchback, and the reproducibility of Bicoid concentration at corresponding positions in multiple embryos. We show, through a combination of different experiments, that all of these quantities are approximately 10%. This agreement among different measures of accuracy indicates that the embryo is not faced with noisy input signals and readout mechanisms; rather, the system exerts precise control over absolute concentrations and responds reliably to small concentration differences, approaching the limits set by basic physical principles.

The nervous system represents time dependent signals in sequences of discrete, identical action potentials or spikes; information is carried only in the spike arrival times. We show how to quantify this information, in bits, free from any assumptions about which features of the spike train or input signal are most important, and we apply this approach to the analysis of experiments on a motion sensitive neuron in the fly visual system. This neuron transmits information about the visual stimulus at rates of up to 90 bits͞s, within a factor of 2 of the physical limit set by the entropy of the spike train itself.

Patterning in multicellular organisms results from spatial gradients in morphogen concentration, but the dynamics of these gradients remain largely unexplored. We characterize, through in vivo optical imaging, the development and stability of the Bicoid morphogen gradient in Drosophila embryos that express a Bicoid-eGFP fusion protein. The gradient is established rapidly (approximately 1 hr after fertilization), with nuclear Bicoid concentration rising and falling during mitosis. Interphase levels result from a rapid equilibrium between Bicoid uptake and removal. Initial interphase concentration in nuclei in successive cycles is constant (+/-10%), demonstrating a form of gradient stability, but it subsequently decays by approximately 30%. Both direct photobleaching measurements and indirect estimates of Bicoid-eGFP diffusion constants (D < or = 1 microm(2)/s) provide a consistent picture of Bicoid transport on short ( approximately min) time scales but challenge traditional models of long-range gradient formation. A new model is presented emphasizing the possible role of nuclear dynamics in shaping and scaling the gradient.

Traditional approaches to neural coding characterize the encoding of known stimuli in average neural responses. Organisms face nearly the opposite task--extracting information about an unknown time-dependent stimulus from short segments of a spike train. Here the neural code was characterized from the point of view of the organism, culminating in algorithms for real-time stimulus estimation based on a single example of the spike train. These methods were applied to an identified movement-sensitive neuron in the fly visual system. Such decoding experiments determined the effective noise level and fault tolerance of neural computation, and the structure of the decoding algorithms suggested a simple model for real-time analog signal processing with spiking neurons.