Stimulus-dependent Maximum Entropy Models of Neural Population Codes

Neural populations encode information about their stimulus in a collective fashion, by joint activity patterns of spiking and silence. A full account of this mapping from stimulus to neural activity is given by the conditional probability distribution over neural codewords given the sensory input. For large populations, direct sampling of these distributions is impossible, and so we must rely on constructing appropriate models. We show here that in a population of 100 retinal ganglion cells in the salamander retina responding to temporal white-noise stimuli, dependencies between cells play an important encoding role. We introduce the stimulus-dependent maximum entropy (SDME) model—a minimal extension of the canonical linear-nonlinear model of a single neuron, to a pairwise-coupled neural population. We find that the SDME model gives a more accurate account of single cell responses and in particular significantly outperforms uncoupled models in reproducing the distributions of population codewords emitted in response to a stimulus. We show how the SDME model, in conjunction with static maximum entropy models of population vocabulary, can be used to estimate information-theoretic quantities like average surprise and information transmission in a neural population. Author Summary: In the sensory periphery, stimuli are represented by patterns of spikes and silences across a population of sensory neurons. Because the neurons form an interconnected network, the code cannot be understood by looking at single cells alone. Recent recordings in the retina have enabled us to study populations of a hundred or more neurons that carry the visual information into the brain, and thus build probabilistic models of the neural code. Here we present a minimal (maximum entropy) yet powerful extension of well-known linear/nonlinear models for independent neurons, to an interacting population. This model reproduces the behavior of single cells as well as the structure of correlations in neural spiking. Our model predicts much better the complete set of patterns of spiking and silence across a population of cells, allowing us to explore the properties of the stimulus-response mapping, and estimate the information transmission, in bits per second, that the population carries about the stimulus. Our results show that to understand the code, we need to shift our focus from reproducing single-cell properties (such as firing rates) towards understanding the total “vocabulary” of patterns emitted by the population, and that network correlations play a central role in shaping the code of large neural populations.