A nonlinear autoregressive Volterra model of the Hodgkin-Huxley equations

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.