27,502 research outputs found
Simple, Fast and Accurate Implementation of the Diffusion Approximation Algorithm for Stochastic Ion Channels with Multiple States
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 activation subunits,
while the DA was modeled using uncoupled activation subunits. Implementations
of DA with coupled subunits, 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. We derived the SDE explicitly for any given ion channel kinetic
scheme. The resulting generic equations were surprisingly simple and
interpretable - allowing an easy and efficient DA implementation. 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.Comment: 32 text pages, 10 figures, 1 supplementary text + figur
Stochastic Differential Equation Model for Cerebellar Granule Cell Excitability
Neurons in the brain express intrinsic dynamic behavior which is known to be stochastic in nature. A crucial question in building models of neuronal excitability is how to be able to mimic the dynamic behavior of the biological counterpart accurately and how to perform simulations in the fastest possible way. The well-established Hodgkin-Huxley formalism has formed to a large extent the basis for building biophysically and anatomically detailed models of neurons. However, the deterministic Hodgkin-Huxley formalism does not take into account the stochastic behavior of voltage-dependent ion channels. Ion channel stochasticity is shown to be important in adjusting the transmembrane voltage dynamics at or close to the threshold of action potential firing, at the very least in small neurons. In order to achieve a better understanding of the dynamic behavior of a neuron, a new modeling and simulation approach based on stochastic differential equations and Brownian motion is developed. The basis of the work is a deterministic one-compartmental multi-conductance model of the cerebellar granule cell. This model includes six different types of voltage-dependent conductances described by Hodgkin-Huxley formalism and simple calcium dynamics. A new model for the granule cell is developed by incorporating stochasticity inherently present in the ion channel function into the gating variables of conductances. With the new stochastic model, the irregular electrophysiological activity of an in vitro granule cell is reproduced accurately, with the same parameter values for which the membrane potential of the original deterministic model exhibits regular behavior. The irregular electrophysiological activity includes experimentally observed random subthreshold oscillations, occasional spontaneous spikes, and clusters of action potentials. As a conclusion, the new stochastic differential equation model of the cerebellar granule cell excitability is found to expand the range of dynamics in comparison to the original deterministic model. Inclusion of stochastic elements in the operation of voltage-dependent conductances should thus be emphasized more in modeling the dynamic behavior of small neurons. Furthermore, the presented approach is valuable in providing faster computation times compared to the Markov chain type of modeling approaches and more sophisticated theoretical analysis tools compared to previously presented stochastic modeling approaches
Stochastic Modeling and Simulation of Ion Transport through Channels
Ion channels are of major interest and form an area of intensive research in
the fields of biophysics and medicine since they control many vital
physiological functions. The aim of this work is on one hand to propose a fully
stochastic and discrete model describing the main characteristics of a multiple
channel system. The movement of the ions is coupled, as usual, with a Poisson
equation for the electrical field; we have considered, in addition, the
influence of exclusion forces. On the other hand, we have discussed about the
nondimensionalization of the stochastic system by using real physical
parameters, all supported by numerical simulations. The specific features of
both cases of micro- and nanochannels have been taken in due consideration with
particular attention to the latter case in order to show that it is necessary
to consider a discrete and stochastic model for ions movement inside the
channels
Channel noise induced stochastic facilitation in an auditory brainstem neuron model
Neuronal membrane potentials fluctuate stochastically due to conductance
changes caused by random transitions between the open and close states of ion
channels. Although it has previously been shown that channel noise can
nontrivially affect neuronal dynamics, it is unknown whether ion-channel noise
is strong enough to act as a noise source for hypothesised noise-enhanced
information processing in real neuronal systems, i.e. 'stochastic
facilitation.' Here, we demonstrate that biophysical models of channel noise
can give rise to two kinds of recently discovered stochastic facilitation
effects in a Hodgkin-Huxley-like model of auditory brainstem neurons. The
first, known as slope-based stochastic resonance (SBSR), enables phasic neurons
to emit action potentials that can encode the slope of inputs that vary slowly
relative to key time-constants in the model. The second, known as inverse
stochastic resonance (ISR), occurs in tonically firing neurons when small
levels of noise inhibit tonic firing and replace it with burst-like dynamics.
Consistent with previous work, we conclude that channel noise can provide
significant variability in firing dynamics, even for large numbers of channels.
Moreover, our results show that possible associated computational benefits may
occur due to channel noise in neurons of the auditory brainstem. This holds
whether the firing dynamics in the model are phasic (SBSR can occur due to
channel noise) or tonic (ISR can occur due to channel noise).Comment: Published by Physical Review E, November 2013 (this version 17 pages
total - 10 text, 1 refs, 6 figures/tables); Associated matlab code is
available online in the ModelDB repository at
http://senselab.med.yale.edu/ModelDB/ShowModel.asp?model=15148
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
Stochastic Representations of Ion Channel Kinetics and Exact Stochastic Simulation of Neuronal Dynamics
In this paper we provide two representations for stochastic ion channel
kinetics, and compare the performance of exact simulation with a commonly used
numerical approximation strategy. The first representation we present is a
random time change representation, popularized by Thomas Kurtz, with the second
being analogous to a "Gillespie" representation. Exact stochastic algorithms
are provided for the different representations, which are preferable to either
(a) fixed time step or (b) piecewise constant propensity algorithms, which
still appear in the literature. As examples, we provide versions of the exact
algorithms for the Morris-Lecar conductance based model, and detail the error
induced, both in a weak and a strong sense, by the use of approximate
algorithms on this model. We include ready-to-use implementations of the random
time change algorithm in both XPP and Matlab. Finally, through the
consideration of parametric sensitivity analysis, we show how the
representations presented here are useful in the development of further
computational methods. The general representations and simulation strategies
provided here are known in other parts of the sciences, but less so in the
present setting.Comment: 39 pages, 6 figures, appendix with XPP and Matlab cod
Physical properties of voltage gated pores
Experiments on single ionic channels have contributed to a large extent to our current view on the function of cell membrane. In these experiments the main observables are the physical quantities: ionic concentration, membrane electrostatic potential and ionic fluxes, all of them presenting large fluctuations. The classical theory of Goldman–Hodking–Katz assumes that an open channel can be well described by a physical pore where ions follow statistical physics. Nevertheless real molecular channels are active pores with open and close dynamical states. By skipping the molecular complexity of real channels, here we present the internal structure and calibration of two active pore models. These models present a minimum set of degrees of freedom, specifically ion positions and gate states, which follow Langevin equations constructed from a unique potential energy functional and by using standard rules of statistical physics. Numerical simulations of both models are implemented and the results show that they have dynamical properties very close to those observed in experiments of Na and K molecular channels. In particular a significant effect of the external ion concentration on gating dynamics is predicted, which is consistent with previous experimental observations. This approach can be extended to other channel types with more specific phenomenology.Peer ReviewedPostprint (published version
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