Maximum likelihood estimation of a stochastic integrateandfire neural encoding modelLiam Paninski, Jonathan W. Pillow and Eero P. Simoncelli 
Neural Computation, 16:25332561. (2004)
One of the central problems in systems neuroscience is that of characterizing the functional relationship between sensory stimuli and neural spike responses. This problem is referred to as the neural coding problem, because the spike trains of neurons can be considered a code by which the brain represents information about the state of the external world. One approach to understanding this code is to build mathematical models of the mapping between stimuli and spike responses; the code can then be interpreted by using the model to predict the neural response to a stimulus, or to decode the stimulus that gave rise to a particular response. In this chapter, we will examine likelihoodbased approaches, which use the explicit probability of response to a given stimulus for both fitting the model and assessing its validity. We will show how the likelihood can be derived for several types of neural models, and discuss theoretical considerations underlying the formulation and estimation of such models. Finally, we will discuss several ideas for evaluating model performance, including timerescaling of spike trains and optimal decoding using Bayesian inversion of the likelihood function. 
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