The what and where of adding channel noise to the Hodgkin-Huxley equations

One of the most celebrated successes in computational biology is the Hodgkin-Huxley framework for modeling electrically active cells. This framework, expressed through a set of differential equations, synthesizes the impact of ionic currents on a cell's voltage -- and the highly nonlinear impact of that voltage back on the currents themselves -- into the rapid push and pull of the action potential. Latter studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the Hodgkin-Huxley equations. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic Hodgkin-Huxley equations. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly Matlab simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.Comment: 14 pages, 3 figures, review articl

A Model of Electrically Stimulated Auditory Nerve Fiber Responses with Peripheral and Central Sites of Spike Generation

A computational model of cat auditory nerve fiber (ANF) responses to electrical stimulation is presented. The model assumes that (1) there exist at least two sites of spike generation along the ANF and (2) both an anodic (positive) and a cathodic (negative) charge in isolation can evoke a spike. A single ANF is modeled as a network of two exponential integrateand-fire point-neuron models, referred to as peripheral and central axons of the ANF. The peripheral axon is excited by the cathodic charge, inhibited by the anodic charge, and exhibits longer spike latencies than the central axon; the central axon is excited by the anodic charge, inhibited by the cathodic charge, and exhibits shorter spike latencies than the peripheral axon. The model also includes subthreshold and suprathreshold adaptive feedback loops which continuously modify the membrane potential and can account for effects of facilitation, accommodation, refractoriness, and spike-rate adaptation in ANF. Although the model is parameterized using data for either single or paired pulse stimulation with monophasic rectangular pulses, it correctly predicts effects of various stimulus pulse shapes, stimulation pulse rates, and level on the neural response statistics. The model may serve as a framework to explore the effects of different stimulus parameters on psychophysical performance measured in cochlear implant listeners

Neural Masking by Sub-threshold Electric Stimuli: Animal and Computer Model Results

Electric stimuli can prosthetically excite auditory nerve fibers to partially restore sensory function to individuals impaired by profound or severe hearing loss. While basic response properties of electrically stimulated auditory nerve fibers (ANF) are known, responses to complex, time-changing stimuli used clinically are inadequately understood. We report that forward-masker pulse trains can enhance and reduce ANF responsiveness to subsequent stimuli and the novel observation that sub-threshold (nonspike-evoking) electric trains can reduce responsiveness to subsequent pulse-train stimuli. The effect is observed in the responses of cat ANFs and shown by a computational biophysical ANF model that simulates rate adaptation through integration of external potassium cation (K) channels. Both low-threshold (i.e., Klt) and high-threshold (Kht) channels were simulated at each node of Ranvier. Model versions without Klt channels did not produce the sub-threshold effect. These results suggest that some such accumulation mechanism, along with Klt channels, may underlie sub-threshold masking observed in cat ANF responses. As multichannel auditory prostheses typically present sub-threshold stimuli to various ANF subsets, there is clear relevance of these findings to clinical situations

The Dependence of Auditory Nerve Rate Adaptation on Electric Stimulus Parameters, Electrode Position, and Fiber Diameter: A Computer Model Study

This paper describes results from a stochastic computational neuron model that simulates the effects of rate adaptation on the responses to electrical stimulation in the form of pulse trains. We recently reported results from a single-node computational model that included a novel element that tracks external potassium ion concentration so as to modify membrane voltage and cause adaptation-like responses. Here, we report on an improved version of the model that incorporates the anatomical components of a complete feline auditory nerve fiber (ANF) so that conduction velocity and effects of manipulating the site of excitation can be evaluated. Model results demonstrate rate adaptation and changes in spike amplitude similar to those reported for feline ANFs. Changing the site of excitation from a central to a peripheral axonal site resulted in plausible changes in latency and relative spread (i.e., dynamic range). Also, increasing the distance between a modeled ANF and a stimulus electrode tended to decrease the degree of rate adaptation observed in pulse-train responses. This effect was clearly observed for high-rate (5,000 pulse/s) trains but not low-rate (250 pulse/s) trains. Finally, for relatively short electrode-to-ANF distances, increases in modeled ANF diameter increased the degree of rate adaptation. These results are compared against available feline ANF data, and possible effects of individual parameters are discussed