Dr. Shusen Pu

In this thesis, we present a natural 14-dimensional Langevin model for the HodgkinHuxley (HH) conductance-based neuron model in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14×28 noise coefficient matrix. We show that (i) the corresponding 14D mean-field ordinary differential equation system is consistent with the classical 4D representation of the HH system; (ii) the 14D representation leads to a noise coefficient matrix that can be obtained cheaply on each timestep, without requiring a matrix decomposition; (iii) sample trajectories of the 14D representation are pathwise equivalent to trajectories of several existing Langevin models,
including one proposed by Fox and Lu in 1994; (iv) our 14D representation give the most accurate interspike-interval distribution, not only with respect to moments but under both the L1 and L∞ metric-space norms; and (v) the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models. We combine the stochastic shielding (SS) approximation, introduced by Schmandt and Galan in 2012, with Langevin versions of the HH model to derive an analytic decomposition of the variance of the interspike intervals (ISI), based on the mean–return-time oscillator phase. We prove in theory, and demonstrate numerically, that in the limit of small noise, the variance of the ISI decomposes linearly into a sum of contributions from each directed edge. Unlike prior analyses, our results apply to current clamp rather than voltage clamp conditions. Under current clamp, a stochastic conductance-based model is an example of a piecewise-deterministic Markov process. Our theory is exact in the limit of small channel noise. Through numerical simulations we demonstrate its applicability over a range from small to moderate noise levels. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels.