The What and Where of Adding Channel Noise to the Hodgkin-Huxley Equations

Abstract

Conductance-based equations for electrically active cells form one of the most widely studied mathematical frameworks in computational biology. This framework, as expressed through a set of differential equations by Hodgkin and Huxley, 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. Later 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 or their counterparts. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the equations of Hodgkin-Huxley type. 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 equations of Hodgkin-Huxley type as well as to more modern models of ion channel dynamics. 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.

In-phase and anti-phase synchronization in noisy Hodgkin–Huxley neurons

“…Within our work the intrinsic noise is due to the stochastic gating of the ion channels, i.e. the so-called channel noise which is inherently coupled to the electrical properties of the axonal cell membrane [19][20][21]. Interestingly, it has been shown that intrinsic channel noise does not only lead to the generation of spontaneous action potentials [22], but as well affects the neuronal dynamics at different levels, namely: (i) it can boost the signal quality [14,15], (ii) enhance the signal transmission reliability [23], (iii) cause frequency-and phase-synchronization features [24][25][26][27][28] and (iv) may result in a frequency stabilization [29], to name but a few.…”

Noise, transient dynamics, and the generation of realistic interspike interval variation in square-wave burster neurons

“…Finally, for the pre-Bötzinger neuron model, we used conductance noise [32,33], where the stochastic terms are added to the conductance terms in the voltage dynamics:…”

Feynman-Kac formula for stochastic hybrid systems

“…This could be the position of a molecular motor on a filament track [4], the length of a microtubule undergoing catastrophes [5] or a bacterium displaying run and tumble [6]. Another example at the single-cell level concerns membrane voltage fluctuations in a neuron due to the stochastic opening and closing of ion channels [7][8][9][10][11][12][13][14][15][16]. The discrete states of the ion channels evolve according to a continuous-time Markov process with voltage-dependent transition rates, whereas the membrane voltage evolves according to a piecewise deterministic equation that depends on the current state of the ion channels.…”