Pattern recognition of Hodgkin-Huxley equations by auto-regressive Laguerre Volterra network
BMC Neuroscience volume 16, Article number: P156 (2015)
A nonparametric, data-driven nonlinear auto-regressive Volterra (NARV)  model has been successfully applied for capturing the dynamics in the generation of action potentials, which is classically modeled by Hodgkin-Huxley (H-H) equations. However, the compactness still need to be improved for further interpretations. Therefore, we propose a novel Auto-regressive Sparse Laguerre Volterra Network (ASLVN) model (shown in Figure 1A), which is developed from traditional Laguerre Volterra Network (LVN) and principal dynamic mode (PDM) framework .
We adopt stochastic global optimization algorithm Simulated Annealing  to train the ASLVN instead of Back-propagation method  to avoid local minima and convergence problems. We also use lasso regularization  to enhance the spasity of the network and prune redundant branches for parsimony. The prediction results are shown in Fig.1B, it can be seen that the exogenous output z(1) represents the subthreshold dynamics in phase III, and the autoregressive output z(2) dominates in the spike shape in phase I, and the cross term output z(x) helps to maintain the refractory period by cancelling the effect of z(1) in phase II and we also observe that refractory inhibition effect decays after initiation of AP, which explains the absolute refractory period and relative refractory period in physiology.
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Geng, K., Marmarelis, V.Z. Pattern recognition of Hodgkin-Huxley equations by auto-regressive Laguerre Volterra network. BMC Neurosci 16 (Suppl 1), P156 (2015). https://doi.org/10.1186/1471-2202-16-S1-P156