1,126 research outputs found
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
Macroscopic equations governing noisy spiking neuronal populations
At functional scales, cortical behavior results from the complex interplay of
a large number of excitable cells operating in noisy environments. Such systems
resist to mathematical analysis, and computational neurosciences have largely
relied on heuristic partial (and partially justified) macroscopic models, which
successfully reproduced a number of relevant phenomena. The relationship
between these macroscopic models and the spiking noisy dynamics of the
underlying cells has since then been a great endeavor. Based on recent
mean-field reductions for such spiking neurons, we present here {a principled
reduction of large biologically plausible neuronal networks to firing-rate
models, providing a rigorous} relationship between the macroscopic activity of
populations of spiking neurons and popular macroscopic models, under a few
assumptions (mainly linearity of the synapses). {The reduced model we derive
consists of simple, low-dimensional ordinary differential equations with}
parameters and {nonlinearities derived from} the underlying properties of the
cells, and in particular the noise level. {These simple reduced models are
shown to reproduce accurately the dynamics of large networks in numerical
simulations}. Appropriate parameters and functions are made available {online}
for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley
models
Capacitance fluctuations causing channel noise reduction in stochastic Hodgkin-Huxley systems
Voltage-dependent ion channels determine the electric properties of axonal
cell membranes. They not only allow the passage of ions through the cell
membrane but also contribute to an additional charging of the cell membrane
resulting in the so-called capacitance loading. The switching of the channel
gates between an open and a closed configuration is intrinsically related to
the movement of gating charge within the cell membrane. At the beginning of an
action potential the transient gating current is opposite to the direction of
the current of sodium ions through the membrane. Therefore, the excitability is
expected to become reduced due to the influence of a gating current. Our
stochastic Hodgkin-Huxley like modeling takes into account both the channel
noise -- i.e. the fluctuations of the number of open ion channels -- and the
capacitance fluctuations that result from the dynamics of the gating charge. We
investigate the spiking dynamics of membrane patches of variable size and
analyze the statistics of the spontaneous spiking. As a main result, we find
that the gating currents yield a drastic reduction of the spontaneous spiking
rate for sufficiently large ion channel clusters. Consequently, this
demonstrates a prominent mechanism for channel noise reduction.Comment: 18 page
Spontaneous spiking in an autaptic Hodgkin-Huxley set up
The effect of intrinsic channel noise is investigated for the dynamic
response of a neuronal cell with a delayed feedback loop. The loop is based on
the so-called autapse phenomenon in which dendrites establish not only
connections to neighboring cells but as well to its own axon. The biophysical
modeling is achieved in terms of a stochastic Hodgkin-Huxley model containing
such a built in delayed feedback. The fluctuations stem from intrinsic channel
noise, being caused by the stochastic nature of the gating dynamics of ion
channels. The influence of the delayed stimulus is systematically analyzed with
respect to the coupling parameter and the delay time in terms of the interspike
interval histograms and the average interspike interval. The delayed feedback
manifests itself in the occurrence of bursting and a rich multimodal interspike
interval distribution, exhibiting a delay-induced reduction of the spontaneous
spiking activity at characteristic frequencies. Moreover, a specific
frequency-locking mechanism is detected for the mean interspike interval.Comment: 8 pages, 10 figure
Stochastic modeling of excitable dynamics: improved Langevin model for mesoscopic channel noise
Influence of mesoscopic channel noise on excitable dynamics of living cells
became a hot subject within the last decade, and the traditional biophysical
models of neuronal dynamics such as Hodgkin-Huxley model have been generalized
to incorporate such effects. There still exists but a controversy on how to do
it in a proper and computationally efficient way. Here we introduce an improved
Langevin description of stochastic Hodgkin-Huxley dynamics with natural
boundary conditions for gating variables. It consistently describes the channel
noise variance in a good agreement with discrete state model. Moreover, we show
by comparison with our improved Langevin model that two earlier Langevin models
by Fox and Lu also work excellently starting from several hundreds of ion
channels upon imposing numerically reflecting boundary conditions for gating
variables.Comment: V.M. Mladenov and P.C. Ivanov (Eds.): NDES 2014, Communications in
Computer and Information Science, vol. 438 (Springer, Switzerland, 2014), pp.
325-33
The space-clamped Hodgkin-Huxley system with random synaptic input: inhibition of spiking by weak noise and analysis with moment equations
We consider a classical space-clamped Hodgkin-Huxley model neuron stimulated
by synaptic excitation and inhibition with conductances represented by
Ornstein-Uhlenbeck processes. Using numerical solutions of the stochastic model
system obtained by an Euler method, it is found that with excitation only there
is a critical value of the steady state excitatory conductance for repetitive
spiking without noise and for values of the conductance near the critical value
small noise has a powerfully inhibitory effect. For a given level of inhibition
there is also a critical value of the steady state excitatory conductance for
repetitive firing and it is demonstrated that noise either in the excitatory or
inhibitory processes or both can powerfully inhibit spiking. Furthermore, near
the critical value, inverse stochastic resonance was observed when noise was
present only in the inhibitory input process.
The system of 27 coupled deterministic differential equations for the
approximate first and second order moments of the 6-dimensional model is
derived. The moment differential equations are solved using Runge-Kutta methods
and the solutions are compared with the results obtained by simulation for
various sets of parameters including some with conductances obtained by
experiment on pyramidal cells of rat prefrontal cortex. The mean and variance
obtained from simulation are in good agreement when there is spiking induced by
strong stimulation and relatively small noise or when the voltage is
fluctuating at subthreshold levels. In the occasional spike mode sometimes
exhibited by spinal motoneurons and cortical pyramidal cells the assunptions
underlying the moment equation approach are not satisfied
Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons
We derive the mean-field equations arising as the limit of a network of
interacting spiking neurons, as the number of neurons goes to infinity. The
neurons belong to a fixed number of populations and are represented either by
the Hodgkin-Huxley model or by one of its simplified version, the
Fitzhugh-Nagumo model. The synapses between neurons are either electrical or
chemical. The network is assumed to be fully connected. The maximum
conductances vary randomly. Under the condition that all neurons initial
conditions are drawn independently from the same law that depends only on the
population they belong to, we prove that a propagation of chaos phenomenon
takes places, namely that in the mean-field limit, any finite number of neurons
become independent and, within each population, have the same probability
distribution. This probability distribution is solution of a set of implicit
equations, either nonlinear stochastic differential equations resembling the
McKean-Vlasov equations, or non-local partial differential equations resembling
the McKean-Vlasov-Fokker- Planck equations. We prove the well-posedness of
these equations, i.e. the existence and uniqueness of a solution. We also show
the results of some preliminary numerical experiments that indicate that the
mean-field equations are a good representation of the mean activity of a finite
size network, even for modest sizes. These experiment also indicate that the
McKean-Vlasov-Fokker- Planck equations may be a good way to understand the
mean-field dynamics through, e.g., a bifurcation analysis.Comment: 55 pages, 9 figure
Dynamics of a FitzHugh-Nagumo system subjected to autocorrelated noise
We analyze the dynamics of the FitzHugh-Nagumo (FHN) model in the presence of
colored noise and a periodic signal. Two cases are considered: (i) the dynamics
of the membrane potential is affected by the noise, (ii) the slow dynamics of
the recovery variable is subject to noise. We investigate the role of the
colored noise on the neuron dynamics by the mean response time (MRT) of the
neuron. We find meaningful modifications of the resonant activation (RA) and
noise enhanced stability (NES) phenomena due to the correlation time of the
noise. For strongly correlated noise we observe suppression of NES effect and
persistence of RA phenomenon, with an efficiency enhancement of the neuronal
response. Finally we show that the self-correlation of the colored noise causes
a reduction of the effective noise intensity, which appears as a rescaling of
the fluctuations affecting the FHN system.Comment: 13 pages, 10 figure
Effect of channel block on the spiking activity of excitable membranes in a stochastic Hodgkin-Huxley model
The influence of intrinsic channel noise on the spontaneous spiking activity
of poisoned excitable membrane patches is studied by use of a stochastic
generalization of the Hodgkin-Huxley model. Internal noise stemming from the
stochastic dynamics of individual ion channels is known to affect the
collective properties of the whole ion channel cluster. For example, there
exists an optimal size of the membrane patch for which the internal noise alone
causes a regular spontaneous generation of action potentials. In addition to
varying the size of ion channel clusters, living organisms may adapt the
densities of ion channels in order to optimally regulate the spontaneous
spiking activity. The influence of channel block on the excitability of a
membrane patch of certain size is twofold: First, a variation of ion channel
densities primarily yields a change of the conductance level. Second, a
down-regulation of working ion channels always increases the channel noise.
While the former effect dominates in the case of sodium channel block resulting
in a reduced spiking activity, the latter enhances the generation of
spontaneous action potentials in the case of a tailored potassium channel
blocking. Moreover, by blocking some portion of either potassium or sodium ion
channels, it is possible to either increase or to decrease the regularity of
the spike train.Comment: 10 pages, 3 figures, published 200
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