Pattern recognition of Hodgkin-Huxley equations by auto-regressive Laguerre Volterra network

A nonparametric, data-driven nonlinear auto-regressive Volterra (NARV) [1] 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 [2].

Figure 1

A Structure of ASLVN for modeling H-H equations, where the input x(n) is the randomly injected current and the output y*(n) is the membrane potential. B The predictions results, z⁽¹⁾ represents the exogenous output, z⁽²⁾ represents the autoregressive output and z^(x) represents the cross term output.

We adopt stochastic global optimization algorithm Simulated Annealing [3] to train the ASLVN instead of Back-propagation method [2] to avoid local minima and convergence problems. We also use lasso regularization [4] 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.