Dissipative stochastic mechanics for capturing neuronal dynamics under the influence of ion channel noise: formalism using a special membrane

Strong diffusion formulation of Markov chain ensembles and its optimal weaker reductions

Two self-contained diffusion formulations, in the form of coupled stochastic differential equations, are developed for the temporal evolution of state densities over an ensemble of Markov chains evolving independently under a common transition rate matrix. Our first formulation derives from Kurtz's strong approximation theorem of density-dependent Markov jump processes [Stoch.

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Their Appl. 6, 223 (1978)STOPB70304-414910.1016/0304-4149(78)90020-0] and, therefore, strongly converges with an error bound of the order of lnN/N for ensemble size N. The second formulation eliminates some fluctuation variables, and correspondingly some noise terms, within the governing equations of the strong formulation, with the objective of achieving a simpler analytic formulation and a faster computation algorithm when the transition rates are constant or slowly varying. There, the reduction of the structural complexity is optimal in the sense that the elimination of any given set of variables takes place with the lowest attainable increase in the error bound. The resultant formulations are supported by numerical simulations.

Efficient variants of the minimal diffusion formulation of Markov chain ensembles

This study is concerned with ensembles of continuous-time Markov chains evolving independently under a common transition rate matrix in some finite state space. In this context, our prior work [Phys. Rev. E 91, 062116 (2015)] has formulated an approximation scheme, called the minimal diffusion formulation, to deduce how the number of chains in a prescribed relevant state evolves in time. The formulation consists of two specifically coupled Ornstein-Uhlenbeck processes in a stochastic differential equation representation; it is minimal in the sense that its structure does not change with the state space size or the transition matrix density, and it requires no matrix square-root operations. In the present study, we first calculate the autocorrelation function of the relevant state density in the minimal diffusion formulation, which is fundamental to the identification of the ensemble dynamics. The obtained autocorrelation function is then employed to develop two diffusion formulations that reduce the structural complexity of the minimal diffusion formulation without significant loss of accuracy in the dynamics. One of these variant formulations includes one less noise term than the minimal diffusion formulation and still satisfies the above-mentioned autocorrelation function in its dynamics. The second variant is in the form of a one-dimensional Langevin equation, therefore it is the simplest possible diffusion formulation one can obtain for the problem, yet its autocorrelation function is first-order accurate in time gap. Numerical simulations supporting the theoretical analysis are delivered.

Minimal diffusion formulation of Markov chain ensembles and its application to ion channel clusters

We study ensembles of continuous-time Markov chains evolving independently under a common transition rate matrix in some finite state space. A diffusion approximation, composed of two specifically coupled Ornstein-Uhlenbeck processes in stochastic differential equation representation, is formulated to deduce how the number of chains in a given particular state evolves in time. This particular form of the formulation builds upon a theoretical argument adduced here. The formulation is minimal in the sense that it is always a two-dimensional stochastic process regardless of the state space size or the transition matrix density, and that it requires no matrix square root operations. A set of criteria, put forward here as to be necessarily captured by any consistent approximation scheme, is used together with the master equation to determine uniquely the parameter values and noise variances in the formulation. The model is applied to the gating dynamics in ion channel clusters.

An Investigation of the Stochastic Hodgkin-Huxley Models Under Noisy Rate Functions

The effects of ion channel fluctuations on the transmembrane voltage activity are potentially profound in small-size excitable membrane patches. Different groups have extended Hodgkin-Huxley equations into stochastic differential equations to capture the effects of ion channel noise analytically (Fox & Lu, 1994 ; Linaro, Storace, & Giugliano, 2011 ; Güler, 2013 ). Studies have shown that the accuracy of spiking statistics by Fox and Lu's model does not match well with the corresponding statistics from the exact microscopic simulations. The models of both Linaro et al. and Güler, however, were found to produce highly accurate statistics. Here we extend the examination of these models to the case in which the rate functions for the opening and closing of gates are under the influence of noise. For that purpose, the usual rate functions are accompanied additively by Ornstein-Uhlenbeck–type stochastic angular variables. Moreover, we argue that the existence of such noise in the rate functions is a plausible physiological phenomenon for finite-size membranes. It is observed that the presence of noise in the rates is not effective on the degree of inaccuracies within the Fox and Lu model. Güler model's accuracy is found to remain high as in the case of noise free rates. But the performance of Linaro et al.’s model is seen to degrade seriously with the increasing strength of the introduced rate function noise. We attribute this failure of Linaro et al.’s model to the use of the covariance function of open channels at the steady state, in its derivation.

Stochastic Hodgkin-Huxley Equations with Colored Noise Terms in the Conductances

The excitability of cells is facilitated by voltage-gated ion channels. These channels accommodate a multiple number of gates individually. The possible impact of that gate multiplicity on the cell's function, specifically when the membrane area is of limited size, was investigated in the author's prior work (Güler, 2011 ). There, it was found that a nontrivially persistent correlation takes place between the transmembrane voltage fluctuations (also between the fluctuations in the gating variables) and the component of open channel fluctuations attributed to the gate multiplicity. This nontrivial phenomenon was found to be playing a major augmentative role for the elevation of excitability and spontaneous firing in small cells. In addition, the same phenomenon was found to be enhancing spike coherence significantly. Here we extend Fox and Lu's ( 1994 ) stochastic Hodgkin-Huxley equations by incorporating colored noise terms into the conductances there to obtain a formalism capable of capturing the addressed cross-correlations. Statistics of spike generation, spike coherence, firing efficiency, latency, and jitter from the articulated set of equations are found to be highly accurate in comparison with the corresponding statistics from the exact microscopic Markov simulations. This way, it is demonstrated vividly that our formulation overcomes the inherent inadequacy of the Fox and Lu equations. Finally, a recently proposed diffusion approximation method (Linaro, Storace, & Giugliano, 2011 ) is taken into consideration, and a discussion on its character is pursued.

Influence of a nanorod molecular layer on the biological activity of neuronal cells. A semiclassical model for complex solid/liquid interfaces with carbon nanotubes

A general account of electric effects is given for a biological phase interacting with a nanorod molecular layer by means of the formed hard-soft and solid-liquid interfaces. In particular, the frequency enhancement previously detected for the spontaneous activity of neuronal cultures interfaced with carbon nanotubes is quantitatively explained upon a quantum/semiclassical description, where the duration of a biological signal is viewed as the (average) lifetime of a decaying state (or population of states), and the effect of the carbon phase as a linewidth broadening. Four contributions were principally accounted for, one biological, for the synaptic strength, one electrochemical, for the overall capacitance increase implied by the nanotube double layers, one geometric, for the typical scales ruling the electron and ion conduction mechanisms, and one electromagnetic-like, translating the membrane polarization changes. These calculations predict an enhancement factor equal on average to ≃6.39, against a former experimental value ≃6.08.

Accurate and fast simulation of channel noise in conductance-based model neurons by diffusion approximation

Stochastic channel gating is the major source of intrinsic neuronal noise whose functional consequences at the microcircuit- and network-levels have been only partly explored. A systematic study of this channel noise in large ensembles of biophysically detailed model neurons calls for the availability of fast numerical methods. In fact, exact techniques employ the microscopic simulation of the random opening and closing of individual ion channels, usually based on Markov models, whose computational loads are prohibitive for next generation massive computer models of the brain. In this work, we operatively define a procedure for translating any Markov model describing voltage- or ligand-gated membrane ion-conductances into an effective stochastic version, whose computer simulation is efficient, without compromising accuracy. Our approximation is based on an improved Langevin-like approach, which employs stochastic differential equations and no Montecarlo methods. As opposed to an earlier proposal recently debated in the literature, our approximation reproduces accurately the statistical properties of the exact microscopic simulations, under a variety of conditions, from spontaneous to evoked response features. In addition, our method is not restricted to the Hodgkin-Huxley sodium and potassium currents and is general for a variety of voltage- and ligand-gated ion currents. As a by-product, the analysis of the properties emerging in exact Markov schemes by standard probability calculus enables us for the first time to analytically identify the sources of inaccuracy of the previous proposal, while providing solid ground for its modification and improvement we present here.

A possible approach to understanding the neuronal bases of the computational properties of the nervous system consists of modelling its basic building blocks, neurons and synapses, and then simulating their collective activity emerging in large networks. In developing such models, a satisfactory description level must be chosen as a compromise between simplicity and faithfulness in reproducing experimental data. Deterministic neuron models – i.e., models that upon repeated simulation with fixed parameter values provide the same results – are usually made up of ordinary differential equations and allow for relatively fast simulation times. By contrast, they do not describe accurately the underlying stochastic response properties arising from the microscopical correlate of neuronal excitability. Stochastic models are usually based on mathematical descriptions of individual ion channels, or on an effective macroscopic account of their random opening and closing. In this contribution we describe a general method to transform any deterministic neuron model into its effective stochastic version that accurately replicates the statistical properties of ion channels random kinetics.

Detailed numerical investigation of the dissipative stochastic mechanics based neuron model

Recently, a physical approach for the description of neuronal dynamics under the influence of ion channel noise was proposed in the realm of dissipative stochastic mechanics (Güler, Phys Rev E 76:041918, 2007). Led by the presence of a multiple number of gates in an ion channel, the approach establishes a viewpoint that ion channels are exposed to two kinds of noise: the intrinsic noise, associated with the stochasticity in the movement of gating particles between the inner and the outer faces of the membrane, and the topological noise, associated with the uncertainty in accessing the permissible topological states of open gates. Renormalizations of the membrane capacitance and of a membrane voltage dependent potential function were found to arise from the mutual interaction of the two noisy systems. The formalism therein was scrutinized using a special membrane with some tailored properties giving the Rose-Hindmarsh dynamics in the deterministic limit. In this paper, the resultant computational neuron model of the above approach is investigated in detail numerically for its dynamics using time-independent input currents. The following are the major findings obtained. The intrinsic noise gives rise to two significant coexisting effects: it initiates spiking activity even in some range of input currents for which the corresponding deterministic model is quiet and causes bursting in some other range of input currents for which the deterministic model fires tonically. The renormalization corrections are found to augment the above behavioral transitions from quiescence to spiking and from tonic firing to bursting, and, therefore, the bursting activity is found to take place in a wider range of input currents for larger values of the correction coefficients. Some findings concerning the diffusive behavior in the voltage space are also reported.