Principal Dynamic Mode Analysis of the Hodgkin–Huxley Equations

Abstract

We develop an autoregressive model framework based on the concept of Principal Dynamic Modes (PDMs) for the process of action potential (AP) generation in the excitable neuronal membrane described by the Hodgkin–Huxley (H–H) equations. The model's exogenous input is injected current, and whenever the membrane potential output exceeds a specified threshold, it is fed back as a second input. The PDMs are estimated from the previously developed Nonlinear Autoregressive Volterra (NARV) model, and represent an efficient functional basis for Volterra kernel expansion. The PDM-based model admits a modular representation, consisting of the forward and feedback PDM bases as linear filterbanks for the exogenous and autoregressive inputs, respectively, whose outputs are then fed to a static nonlinearity composed of polynomials operating on the PDM outputs and cross-terms of pair-products of PDM outputs. A two-step procedure for model reduction is performed: first, influential subsets of the forward and feedback PDM bases are identified and selected as the reduced PDM bases. Second, the terms of the static nonlinearity are pruned. The first step reduces model complexity from a total of 65 coefficients to 27, while the second further reduces the model coefficients to only eight. It is demonstrated that the performance cost of model reduction in terms of out-of-sample prediction accuracy is minimal. Unlike the full model, the eight coefficient pruned model can be easily visualized to reveal the essential system components, and thus the data-derived PDM model can yield insight into the underlying system structure and function.

The essence of neuronal activity from the consistency of two different neuron models

“…Its scientific significance is that the evolution of neural energy at subthreshold state is directly related to the changes in the neural dark energy in the default mode network, so whether neural energy can modulate the state of neural activity as well as the study of the nature of consciousness has an important scientific significance [23,24]. As far as our current literature is concerned, this problem has not attracted enough attention from neuroscientists, which is because the widely used H-H model is based on a basic assumption that the operation of the ion pump can ensure that the Nernst potential is constant [25]. In addition, modern experimental techniques are also difficult to directly detect the consumed energy in firing procedure of an action potential for a single neuron.…”

Methodology of Recurrent Laguerre–Volterra Network for Modeling Nonlinear Dynamic Systems

“…The data-based nonparametric methods using the Volterra-Wiener formulation of nonlinear input-output models have been used over the last 40 years to study the dynamics of neural systems [5][6][21][35][36]. One key challenge for these methods has been the parsimonious model representation that allows computational efficiency, estimation accuracy and physiological interpretation of the obtained models.…”

“…A significant advance in this regard has been the practicable use of Laguerre expansions of the Volterra kernels [14]. Subsequent methodologies such as Principal Dynamic Modes [23] and Laguerre-Volterra Networks [24] are developed to achieve model parsimony, estimation efficiency and model interpretation, as demonstrated in various physiological systems [15] and, more recently, in the study of cerebral hemodynamics and neuronal dynamics [6][18–22]. …”

“…Using the obtained PDMs to represent/ expand the kernels, the Associated Nonlinear Functions (ANFs) are subsequently estimated via regression to obtain the Volterra-equivalent PDM-based model of the system [15]. This approach has been successfully used in various applications [6][14–17][51–53]. …”

“…The recent development of a stochastic H-H model is trying to capture some of these variabilities by introducing noise terms in the deterministic H-H equations [41–46]. On the other hand, the data-based non-parametric methods, such as nonlinear autoregressive Volterra (NARV) modeling [5], conventional PDM analysis [6], and recurrent ANN [47] have been utilized to approximate the AP generation process. These non-parametric methods, however, cannot provide biological insights through interpretation of the obtained models.…”