Bayesian Approach to Estimating Phase Response Curves

Abstract

Many research groups have sought to measure phase response curves (PRCs) from real neurons. In contrast to the numerical calculations of the PRCs for the mathematical neuron models, electrophysiological experiments on real neurons face serious problems whereby PRCs have to be retrieved from noisy data. However, methods for estimating PRCs from noisy spike response data have yet to be established. In this chapter, we explain our Bayesian approach to estimating the PRCs and its application to physiological data. In the first half of this chapter, we describe a Bayesian algorithm for estimating PRCs from noisy spike response data. This algorithm is based on a likelihood function derived from a detailed model of the spike response in PRC measurements that is formulated as a Langevin phase equation. We construct a maximum a posteriori (MAP) estimation algorithm based on the analytically obtained likelihood function. This algorithm gives estimates of not only the PRC but also the Langevin force intensity. In the last half of this chapter, we apply the MAP estimation algorithm to physiological data measured from a hippocampal CA1 pyramidal neuron. We explain the protocol of the PRC measurement in a dynamic clamp, which maintains the baseline firing frequencies as close to a target value for as long as the perturbation experiment lasts. Finally, we verify the reliability of the estimated PRC by testing whether the Fokker–Planck equation based on the estimated PRC and Langevin force intensity captures the stochastic oscillatory behavior of the same neuron disturbed by periodic perturbations.