Algorithms for inverting Hodgkin-Huxley type neuron models
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The study of neurons is of fundamental importance in biology and medicine. Neurons are the most basic unit of information processing in the nervous system of humans and all other vertebrates and in complex invertebrates. In addition, networks of neurons (the human brain) are the most sophisticated computational devices known, and the study of neurons individually and working in concert is seen as a step toward understanding consciousness and cognition. In the 1950's Hodgkin and Huxley developed a system of nonlinear ordinary differential equations to describe the behavior of a neuron found in the squid. Equations of this form have since been used to model the behavior of a multitude of neurons across a broad spectrum of species. Hodgkin-Huxley type neuron models helped lay the foundation for computational neuroscience, and they remain widely used in the study of neuron behavior almost sixty years after their development. Hodgkin-Huxley type models accept a set of parameters as input and generate data describing the electrical activity of the neuron as a function of time. We develop inversion algorithms to predict a set of input parameter values from the voltage trace data generated by the model. We test our algorithm on data from the Hodgkin-Huxley equations, and we extend the algorithm to solve the inverse problem associated with a more complex Hodgkin-Huxley type model for a lobster stomatogastric neuron. We find strong empirical evidence that the algorithms produce parameter values that generate a good fit to the target voltage trace, and we prove that under certain conditions the inversion algorithm for the Hodgkin-Huxley equations converges to a perfect match. To our knowledge this is the first parameter optimization procedure for which convergence has been shown theoretically. Understanding the relationship between the parameters of a neuron model and its output has implications for designing effective neuron models and for explaining the mechanisms by which neurons regulate their behavior. Inversion algorithms for Hodgkin-Huxley type neuron models are an important theoretical and practical step toward understanding the relationship between model parameters and model behavior, and toward the larger problem of inferring neuronal parameters from behavior observed experimentally.