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Register a new userOn stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons
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The random transitions of ion channels between conducting and non-conducting states generate a source of internal fluctuations in a neuron, known as channel noise. The standard method for modeling fluctuations in the states of ion channels uses continuous-time Markov chains nonlinearly coupled to a differential equation for voltage. Beginning with the work of Fox and Lu, there have been attempts to generate simpler models that use stochastic differential equation (SDEs) to approximate the stochastic spiking activity produced by Markov chain models. Recent numerical investigations, however, have raised doubts that SDE models can preserve the stochastic dynamics of Markov chain models. We analyze three SDE models that have been proposed as approximations to the Markov chain model: one that describes the states of the ion channels and two that describe the states of the ion channel subunits. We show that the former channel-based approach can capture the distribution of channel noise and its effect on spiking in a Hodgkin-Huxley neuron model to a degree not previously demonstrated, but the latter two subunit-based approaches cannot. Our analysis provides intuitive and mathematical explanations for why this is the case: the temporal correlation in the channel noise is determined by the combinatorics of bundling subunits into channels, and the subunit-based approaches do not correctly account for this structure. Our study therefore confirms and elucidates the findings of previous numerical investigations of subunit-based SDE models. Moreover, it presents the first evidence that Markov chain models of the nonlinear, stochastic dynamics of neural membranes can be accurately approximated by SDEs. This finding opens a door to future modeling work using SDE techniques to further illuminate the effects of ion channel fluctuations on electrically active cells.
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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 cells 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 doesnt, 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 stochast
We consider a pair of stochastic integrate and fire neurons receiving correlated stochastic inputs. The evolution of this system can be described by the corresponding Fokker-Planck equation with non-trivial boundary conditions resulting from the refractory period and firing threshold. We propose a finite volume method that is orders of magnitude faster than the Monte Carlo methods traditionally used to model such systems. The resulting numerical approximations are proved to be accurate, nonnegative and integrate to 1. We also approximate the transient evolution of the system using an Ornstein--Uhlenbeck process, and use the result to examine the properties of the joint output of cell pairs. The results suggests that the joint output of a cell pair is most sensitive to changes in input variance, and less sensitive to changes in input mean and correlation.
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.
Gamma frequency oscillations (25-140 Hz), observed in the neural activities within many brain regions, have long been regarded as a physiological basis underlying many brain functions, such as memory and attention. Among numerous theoretical and computational modeling studies, gamma oscillations have been found in biologically realistic spiking network models of the primary visual cortex. However, due to its high dimensionality and strong nonlinearity, it is generally difficult to perform detailed theoretical analysis of the emergent gamma dynamics. Here we propose a suite of Markovian model reduction methods with varying levels of complexity and applied it to spiking network models exhibiting heterogeneous dynamical regimes, ranging from homogeneous firing to strong synchrony in the gamma band. The reduced models not only successfully reproduce gamma band oscillations in the full model, but also exhibit the same dynamical features as we vary parameters. Most remarkably, the invariant measure of the coarse-grained Markov process reveals a two-dimensional surface in state space upon which the gamma dynamics mainly resides. Our results suggest that the statistical features of gamma oscillations strongly depend on the subthreshold neuronal distributions. Because of the generality of the Markovian assumptions, our dimensional reduction methods offer a powerful toolbox for theoretical examinations of many other complex cortical spatio-temporal behaviors observed in both neurophysiological experiments and numerical simulations.
The Hodgkin-Huxley (HH) model is a powerful model to explain different aspects of spike generation in excitable cells. However, the HH model was proposed in 1952 when the real structure of the ion channel was unknown. It is now common knowledge that in many ion-channel proteins the flow of ions through the pore is governed by a gate, comprising a so-called selectivity filter inside the ion channel, which can be controlled by electrical interactions. The selectivity filter is believed to be responsible for the selection and fast conduction of particular ions across the membrane of an excitable cell. Other (generally larger) parts of the molecule such as the pore-domain gate control the access of ions to the channel protein. In fact, two types of gates are considered here for ion channels: the external gate, which is the voltage sensitive gate, and the internal gate which is the selectivity filter gate (SFG). Some quantum effects are to expected in the SFG due to its small dimensions, which may play an important role in the operation of an ion channel. Here, we examine parameters in a generalized model of HH to see whether any parameter affects the spike generation. Our results indicate that the previously suggested semi-quantum-classical equation proposed by Bernroider and Summhammer (BS) agrees strongly with the HH equation under different conditions and may even provide a better explanation in some cases. We conclude that the BS model can refine the classical HH model substantially.
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