Computation in a single neuron: Hodgkin and Huxley revisited

A spiking neuron ``computes'' by transforming a complex dynamical input into a train of action potentials, or spikes. The computation performed by the neuron can be formulated as dimensional reduction, or feature detection, followed by a nonlinear decision function over the low dimensional space. Generalizations of the reverse correlation technique with white noise input provide a numerical strategy for extracting the relevant low dimensional features from experimental data, and information theory can be used to evaluate the quality of the low--dimensional approximation. We apply these methods to analyze the simplest biophysically realistic model neuron, the Hodgkin--Huxley model, using this system to illustrate the general methodological issues. We focus on the features in the stimulus that trigger a spike, explicitly eliminating the effects of interactions between spikes. One can approximate this triggering ``feature space'' as a two dimensional linear subspace in the high--dimensional space of input histories, capturing in this way a substantial fraction of the mutual information between inputs and spike time. We find that an even better approximation, however, is to describe the relevant subspace as two dimensional, but curved; in this way we can capture 90% of the mutual information even at high time resolution. Our analysis provides a new understanding of the computational properties of the Hodgkin--Huxley model. While it is common to approximate neural behavior as ``integrate and fire,'' the HH model is not an integrator nor is it well described by a single threshold.