Fluid limit theorems for stochastic hybrid systems with application to neuron models

Feynman-Kac formula for stochastic hybrid systems

We derive a Feynman-Kac formula for functionals of a stochastic hybrid system evolving according to a piecewise deterministic Markov process. We first derive a stochastic Liouville equation for the moment generator of the stochastic functional, given a particular realization of the underlying discrete Markov process; the latter generates transitions between different dynamical equations for the continuous process. We then analyze the stochastic Liouville equation using methods recently developed for diffusion processes in randomly switching environments. In particular, we obtain dynamical equations for the moment generating function, averaged with respect to realizations of the discrete Markov process. The resulting Feynman-Kac formula takes the form of a differential Chapman-Kolmogorov equation. We illustrate the theory by calculating the occupation time for a one-dimensional velocity jump process on the infinite or semi-infinite real line. Finally, we present an alternative derivation of the Feynman-Kac formula based on a recent path-integral formulation of stochastic hybrid systems.

Numerical simulations of piecewise deterministic Markov processes with an application to the stochastic Hodgkin-Huxley model

The stochastic Hodgkin-Huxley model is one of the best-known examples of piecewise deterministic Markov processes (PDMPs), in which the electrical potential across a cell membrane, V(t), is coupled with a mesoscopic Markov jump process representing the stochastic opening and closing of ion channels embedded in the membrane. The rates of the channel kinetics, in turn, are voltage-dependent. Due to this interdependence, an accurate and efficient sampling of the time evolution of the hybrid stochastic systems has been challenging. The current exact simulation methods require solving a voltage-dependent hitting time problem for multiple path-dependent intensity functions with random thresholds. This paper proposes a simulation algorithm that approximates an alternative representation of the exact solution by fitting the log-survival function of the inter-jump dwell time, H(t), with a piecewise linear one. The latter uses interpolation points that are chosen according to the time evolution of the H(t), as the numerical solution to the coupled ordinary differential equations of V(t) and H(t). This computational method can be applied to all PDMPs. Pathwise convergence of the approximated sample trajectories to the exact solution is proven, and error estimates are provided. Comparison with a previous algorithm that is based on piecewise constant approximation is also presented.

Simple, fast and accurate implementation of the diffusion approximation algorithm for stochastic ion channels with multiple states

BACKGROUND

The phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled gating particles, while the DA was modeled using uncoupled gating particles. Implementations of DA with coupled particles, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies.

MAIN CONTRIBUTIONS

We derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable--allowing an easy, transparent and efficient DA implementation, avoiding unnecessary approximations. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods, except when short time steps or low channel numbers were used.

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

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.

Intrinsic variability of latency to first-spike

The assessment of the variability of neuronal spike timing is fundamental to gain understanding of latency coding. Based on recent mathematical results, we investigate theoretically the impact of channel noise on latency variability. For large numbers of ion channels, we derive the asymptotic distribution of latency, together with an explicit expression for its variance. Consequences in terms of information processing are studied with Fisher information in the Morris-Lecar model. A competition between sensitivity to input and precision is responsible for favoring two distinct regimes of latencies.