Signal transduction and nonlinearities revealed by white noise inputs in the fast adapting crayfish stretch receptor

Principal dynamic mode analysis of the Hodgkin-Huxley equations

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.

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.

On the noise-enhancing ability of stochastic Hodgkin-Huxley neuron systems

Recently noise has been shown to be useful in enhancing neuron sensitivity by stochastic resonance. In this study, in order to measure the noise-enhancing factor (NEF), a nonlinear stochastic model is introduced for the Hodgkin-Huxley (HH) neuron system with synaptic noise input stimulation and channel noises in the sodium and potassium channels. The enhancing factor of the HH neuron system is measured from the point of view of the noise-exploiting level of nonlinear stochastic H(infinity) signal processing. Since the nonlinear stochastic-enhancing measure problem of HH neuron systems requires a solution for the difficulty presented by the Hamilton Jacobi inequality (HJI), a fuzzy interpolation of locally linearized systems is employed to simplify the nonlinear noise-enhancing problems by solving only a set of linear matrix inequalities. The NEF of the HH neuron system is found to be related to the locations of eigenvalues of linearized HH neuron systems and can be estimated through the H(infinity) signal processing method. Based on a stochastic fuzzy linearized HH neuron system, we found that channel noises are enhanced by the active eigenvalues of ionic channels while synaptic noises are attenuated by the passive eigenvalues of synaptic process.

Random dynamics of the Hodgkin-Huxley neuron model

Noise can alter the response of neurons, enhancing their ability to detect weak inputs. We analyze how the Hodgkin-Huxley equations, a canonical neuron model, respond to white noise stimulation. We show that this model possesses a stochastic attractor, reduced to a unique stochastic equilibrium point that attracts all trajectories.

Coherence resonance and discharge time reliability in neurons and neuronal models

Neurons are subject to internal and external noise that have been known to modify the way they process incoming signals. Recent studies have suggested that such alterations have functional roles and can also be used in biomedical applications. The present work goes over experimental and theoretical descriptions of the response of neurons to white noise stimulation. It examines various forms of noise related behavior in a standard neuronal model, namely the leaky integrate and fire. This clarifies the conditions under which specific noise induced changes occur in neurons, and consequently can help in determining whether nervous systems operate under similar circumstances.