A nonlinear autoregressive Volterra model of the Hodgkin–Huxley equations

Abstract

We propose a new variant of Volterra-type model with a nonlinear auto-regressive (NAR) component that is a suitable framework for describing the process of AP generation by the neuron membrane potential, and we apply it to input-output data generated by the Hodgkin-Huxley (H-H) equations. Volterra models use a functional series expansion to describe the input-output relation for most nonlinear dynamic systems, and are applicable to a wide range of physiologic systems. It is difficult, however, to apply the Volterra methodology to the H-H model because is characterized by distinct subthreshold and suprathreshold dynamics. When threshold is crossed, an autonomous action potential (AP) is generated, the output becomes temporarily decoupled from the input, and the standard Volterra model fails. Therefore, in our framework, whenever membrane potential exceeds some threshold, it is taken as a second input to a dual-input Volterra model. This model correctly predicts membrane voltage deflection both within the subthreshold region and during APs. Moreover, the model naturally generates a post-AP afterpotential and refractory period. It is known that the H-H model converges to a limit cycle in response to a constant current injection. This behavior is correctly predicted by the proposed model, while the standard Volterra model is incapable of generating such limit cycle behavior. The inclusion of cross-kernels, which describe the nonlinear interactions between the exogenous and autoregressive inputs, is found to be absolutely necessary. The proposed model is general, non-parametric, and data-derived.

Principal Dynamic Mode Analysis of the Hodgkin–Huxley Equations

“…We have also found that, for pruned models, the optimal threshold for detection, θ D , changes with input power; namely, lower input power leads to lower detection thresholds for pruned PDM-based models (results not shown). While not emphasized in this work, we have previously found the NARV model trained with high-power input is applicable to lower power out-of-sample inputs, 2 and we have similarly found a single high-power input adequate to train PDM-based models applicable to lower power inputs (results not shown).…”

“…The H–H model is well known to exhibit rich dynamics, the most famous and basic being threshold phenomena, refractoriness, and limit cycle behavior. The NARV model was previously shown to produce these behaviors, 2 and we have found that the 27-P, 8-P, and 4-P PDM models all yield these basic dynamics as well (results not shown). The H–H model also exhibits more complex dynamics 32 such as hysteresis, bistability, and the phenomenon of anode-break excitation.…”

“…2 We find that a two-step model reduction procedure is advisable for this purpose, whereby a set of PDMs is first identified that efficiently represents the equivalent Volterra model, and subsequent pruning of the PDM-based model terms yields models of dramatically reduced complexity with comparable predictive capability. This procedure identifies an eight-parameter (“8-P”) model, representing an eightfold reduction relative to the full NARV model, and a threefold reduction relative to the differential equations representation of the H–H system.…”

“…This paper follows our previous work 2 in which we developed the Nonlinear Auto-Regressive Volterra-type (NARV) model for neuronal membrane dynamics, which we applied to the H–H equations. The NARV model is a nonlinear “black-box” description of the input–output relation between injected current and membrane potential for a spiking neuron.…”

Volterra dendritic stimulus processors and biophysical spike generators with intrinsic noise sources

“…DSPs are modeled using Volterra series. Volterra series have been used for analyzing nonlinear neuronal responses in many contexts (Lu et al, 2011; Eikenberry and Marmarelis, 2012), and have been applied to the identification of single neurons in many of sensory areas (Benardete and Kaplan, 1997; Theunissen et al, 2000; Clark et al, 2011). Volterra dendritic processors can model a wide range of nonlinear effects commonly seen in sensory systems (Lazar and Slutskiy, in press).…”