Coupling large numbers of relatively simple elements often results in networks with complex computational abilities. Examples abound in biological systems - from genetic to neural networks, from metabolic networks to immune systems, from networks of proteins to networks of economic and social agents. Recent and continuing increases in the experimental ability to simultaneously track the dynamics of many constituent elements within these networks present a challenge to theorists: to provide conceptual frameworks and develop mathematical and numerical tools for the analysis of such vast data. The subject poses great challenges, as the systems of interest are noisy and the available information is incomplete.
For the specific case of neural activity, Generalized Linear Models provide a useful framework for a systematic description. The formulation of these models is based on the exponential family of probability distributions; the Bernoulli and Poisson distributions are relevant to the case of stochastic spiking. In this approach, the time-dependent activity of each individual neuron is modeled in terms of experimentally accessible correlates: preceding patterns of activity of this neuron and other monitored neurons in the network, inputs provided through various sensory modalities or by other brain areas, and outputs such as muscle activity or motor responses. Model parameters are fit to maximize the likelihood of the observed firing statistics; smoothness and sparseness constraints can be incorporated via regularization techniques. When applied to neural data, this modeling approach provides a powerful tool for mapping the spatiotemporal receptive fields of individual neurons, characterizing network connectivity through pairwise interactions, and monitoring synaptic plasticity.