Stochastic analysis of the electromagnetic induction effect on a neuron’s action potential dynamics

Non-intrusive polynomial chaos expansion (PCE) and stochastic collocation (SC) methods are attractive techniques for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic variability. PCE estimates coefficients for known orthogonal polynomial basis functions based on a set of response function evaluations, using sampling, linear regression, tensor-product quadrature, or Smolyak sparse grid approaches. SC, on the other hand, forms interpolation functions for known coefficients, and requires the use of structured collocation point sets derived from tensor product or sparse grids. When tailoring the basis functions or interpolation grids to match the forms of the input uncertainties, exponential convergence rates can be achieved with both techniques for a range of probabilistic analysis problems. In addition, analytic features of the expansions can be exploited for moment estimation and stochastic sensitivity analysis. In this paper, the latest ideas for tailoring these expansion methods to numerical integration approaches will be explored, in which expansion formulations are modified to best synchronize with tensor-product quadrature and Smolyak sparse grids using linear and nonlinear growth rules. The most promising stochastic expansion approaches are then carried forward for use in new approaches for mixed aleatory-epistemic UQ, employing second-order probability approaches, and design under uncertainty, employing bilevel, sequential, and multifidelity approaches.

1. Na currents and fluctuations of Na currents were studied under voltage clamp in the same myelinated nerve fibres of Rana esculenta at 13 degrees C. The results were used to test several kinetic models for the gating process of Na channels.2. Long voltage pulses, depolarizing the membrane by 16-48 mV from a hyperpolarizing holding level of - 28 mV, were applied in 4 sec intervals. The d.c. and a.c. components of the membrane current were recorded during the last 328 msec of the 473 msec pulses. For each depolarization, ninety-six trials were made with the node in Ringer solution and, again, after adding 300 nm-tetrodotoxin (TTX) in that solution.3. The TTX-sensitive d.c. component declined during the 328 msec records by 14-51% of its time average. The a.c. component was corrected for this trend by subtracting the first from the second of each pair of subsequent records. The TTX-sensitive part of its variance declined, on the average, in parallel to the current, as if the open probability rather than the conductance of the individual Na channels was reduced by a slow process.4. Single-channel conductances, gamma, were calculated on the assumption that Na channels have only one non-zero conductance and were corrected for the limited band width (5 kHz) of the a.c. records. Values of gamma increased slightly (< 30% from 16 to 40 mV), and averaged 8.85 +/- 0.7 pS (s.e. of mean, seventeen measurements on ten fibres). This small degree of change in gamma suggests that deviations from the all-or-none gating are very small.5. Power spectral densities of the fluctuations between 3 Hz and 5 kHz were calculated from the trend-free a.c. records and corrected for the TTX-insensitive noise component. Control calculations showed that the only effect of the nonstationarity in the Na current was to enhance the low-frequency points of such spectra by less than 10%. The spectra revealed at least two Lorentzian components with cut-off frequencies in the range expected from the activation and inactivation kinetics. The low-frequency component became dominant as depolarization was increased.6. Na currents recorded during brief (< 40 msec) depolarizations were analysed in terms of various all-or-none gating models, in which inactivation either was independent of activation (Hodgkin-Huxley (HH) model) or could occur only from the partly or fully activated states (coupled models). The transient Na currents were reproduced by all models.7. With the parameters from such fits, the fluctuation spectra expected for each model were calculated. The predictions differed in the fraction, r(h), of the variance contributed by the slow (inactivation) fluctuations; r(h) was larger in the coupled models than in the HH model.8. The experimental spectra were divided into two spectral components to yield empirical values for r(h). We used as templates the spectral curves derived for the fast and for the slow fluctuations of the HH model. The empirical r(h) values were one (48 mV) to four (16 mV) times larger than those expected for the HH model. They were also larger than the theoretical r(h) of the coupled models at the small depolarizations, but became equal or smaller than those at the largest depolarization. Direct comparison of the measured and theoretical spectra revealed the same discrepancies.9. We conclude that all of the simple gating models considered in this paper are inconsistent with the fluctuation measurements, the coupled models giving slightly smaller deviations than the model with independent activation and inactivation.

Effects of ion channel blocks on electrical activity of stochastic Hodgkin–Huxley neural network under electromagnetic induction

This study focuses on fatigue reliability in steel bridges, a critical concern due to increasing traffic loads. Employing the S-N curve method, a Stochastic Sensitivity Analysis (SSA) is conducted to assess the impact of input parameters. Miner’s rule and fracture mechanics are compared for fatigue assessment, considering uncertainties through stochastic sensitivity analysis. Material-related factors and the equivalent stress range are identified as pivotal contributors to fatigue reliability. The study emphasizes the importance of understanding influential parameters for improved design and predictive insights in structural behavior.

Synchronous behavior in networks of coupled systems : with applications to neuronal dynamics

In this thesis we investigate the phase space structure of biophysical neuron models with dynamic ion concentrations. We start with the classical Hodgkin–Huxley model of an electrically excitable neural membrane with constant intra– and extracellular ion concentrations. The extension of this model to include changes in ion concentrations that result from transmembrane currents is carefully reviewed. This extension describes a closed system, in which no particle exchange with the surroundings is considered, however the neuron contains ion pumps that dissipate energy to keep it far from the thermodynamic equilibrium. Exploiting all symmetries of the model and applying one commonly used and one novel approximation of the gating dynamics, we obtain a reduced ion–based neuron model with only four dynamical variables. The dynamics of the closed neuron system is investigated in the next part. A new stable fixed point that coexists with the physiological resting state is found. Numerical simulations show that the new fixed point is extremely close to the Donnan equilibrium, i.e., the thermodynamic equilibrium of the system. The neuron cannot fire action potentials in this state, because the electric energy that is usually stored in the ion gradients is almost fully dissipated. We refer to this condition as free energy–starvations (FES). This is the first bistability of neuron states with completely different intra– and extracellular ion concentrations ever reported. Perturbations that cause the transition from the physiological resting state to FES are, for example, long stimulations with applied currents or a temporary interruption of the pump activity. We perform the first bifurcation analysis of a unified model for action potentials and ion dynamics. We vary the pump rate as a bifurcation parameter, and thereby prove the coexistence of a physiological resting state and FES in a large number of reduced ion–based model variants. The result is also replicated for a very biophysically detailed model developed by Kager et al.. Most importantly, the bifurcation analysis shows that a closed neuron system can only recover from FES if the rate of the ion pumps is extremely enhanced. This leads to our first major conclusion: ion homeostasis cannot rely on the pumps alone. Spreading depression (SD) is an important example of neural

We study the effects of noise and diffusion in an excitable slow-fast population system of the Leslie-Gower type. The phenomenon of noise-induced excitement is investigated in the zone of stable equilibria near the Andronov-Hopf bifurcation with the Canard explosion. The stochastic generation of mixed-mode oscillations is studied by numerical simulation and stochastic sensitivity analysis. Effects of the diffusion are considered for the spatially distributed variant of this slow-fast population model. The phenomenon of the diffusion-induced generation of spatial patterns-attractors in the Turing instability zone is demonstrated. The multistability and variety of transient processes of the pattern formation are discussed. This article is part of the theme issue 'Patterns in soft and biological matters'.

Nonequilibrium dynamics in a noise-induced predator–prey model

A conductance-based model of Na+ and K+ currents underlying action potential generation is introduced by simplifying the quantitative model of Hodgkin and Huxley (HH). If the time course of rate constants can be approximated by a pulse, HH equations can be solved analytically. Pulse-based (PB) models generate action potentials very similar to the HH model but are computationally faster. Unlike the classical integrate-and-fire (IAF) approach, they take into account the changes of conductances during and after the spike, which have a determinant influence in shaping neuronal responses. Similarities and differences among PB, IAF, and HH models are illustrated for three cases: high-frequency repetitive firing, spike timing following random synaptic inputs, and network behavior in the presence of intrinsic currents.

The International Commission on Radiological Protection (ICRP) has developed the reference Human Respiratory Tract Model (HRTM), detailed in ICRP Publications 66 and 130, to estimate the deposition and clearance of inhaled radionuclides. These models utilize reference anatomical and physiological parameters for particle deposition (PD). Biokinetic models further estimate retention and excretion of internalized particulates, aiding the derivation of inhalation dose coefficients (DC). This study aimed to assess variability in deterministic131I biokinetic and dosimetry models through stochastic analysis using the updated HRTM from ICRP Publication 130. The complexities of the ICRP PD model were reconstructed into a new, independent computational model. Comparison with reference data for total PD fractions for reference worker, solely a nose breather, covering activity median aerodynamic diameters from 0.3μm to 20μm, showed a 1.04% relative and 0.7% absolute difference, demonstrating good agreement with ICRP deposition fractions. The deterministic DC module was reconstructed in Python and expanded for stochastic analysis, systematically expanding deposition components from HRTM and assigning probability distribution functions to uncertain parameters. These were integrated into an in-house stochastic radiological exposure dose calculator, utilizing latin hypercube sampling. A case of an occupational radionuclide intake was explored, in which biodistribution and committed effective DC (CEDC) were computed for131I type F, considering a lognormal particle size distribution with a median of 5μm. Results showed the published ICRP reference CEDC marginally exceeds the 75th percentile of observed samples, with log-gamma distribution as the best-fit probability distribution. A Random Forest regression model with SHapley Additive exPlanations was employed for sensitivity analysis to predict feature importance. The analysis identified the HRTM particle transport rates scaling factor, followed by the aerodynamic deposition efficiency in the alveolar interstitial region as the most impactful parameters. This study offers a unique stochastic approach on inhaled particulate metabolism, enhancing radiation consequence management, medical countermeasures, and dose reconstruction for epidemiological studies.

We introduce a method for systematically reducing the dimension of biophysically realistic neuron models with stochastic ion channels exploiting time-scales separation. Based on a combination of singular perturbation methods for kinetic Markov schemes with some recent mathematical developments of the averaging method, the techniques are general and applicable to a large class of models. As an example, we derive and analyze reductions of different stochastic versions of the Hodgkin Huxley (HH) model, leading to distinct reduced models. The bifurcation analysis of one of the reduced models with the number of channels as a parameter provides new insights into some features of noisy discharge patterns, such as the bimodality of interspike intervals distribution. Our analysis of the stochastic HH model shows that, besides being a method to reduce the number of variables of neuronal models, our reduction scheme is a powerful method for gaining understanding on the impact of fluctuations due to finite size effects on the dynamics of slow fast systems. Our analysis of the reduced model reveals that decreasing the number of sodium channels in the HH model leads to a transition in the dynamics reminiscent of the Hopf bifurcation and that this transition accounts for changes in characteristics of the spike train generated by the model. Finally, we also examine the impact of these results on neuronal coding, notably, reliability of discharge times and spike latency, showing that reducing the number of channels can enhance discharge time reliability in response to weak inputs and that this phenomenon can be accounted for through the analysis of the reduced model.

Due to the importance of the brain and neurons, a vast area of research has been conducted in this field. However, due to the complexity of the neural behavior, each study investigated the functionality of neurons from one perspective such as electrophysiological, chemical, or mechanical perspective. In spite of the large number of research conducted on the brain injury topic, there is no study investigating the interaction of the mechanical and electrical characteristics of the neurons and its effect on the cell functionality. Understating the interaction between the mechanical and electrical properties of a neuron will have a substantial effect on treating neurological diseases such as traumatic brain injury and improving treatment methods such as ultrasound. As a result, there is a vital need to simulate the effect of mechanical forces on the electrophysiological behavior of a neuron. This study is one of the few attempts to achieve this goal by taking into account the mechanosensitivity of ion channels which affects the action potentials. Our proposed comprehensive model is based on power law equation (fractional dashpot) for mechanical modeling, Hodgkin Huxley (HH) equation for electrophysiological model and recent experiments for combination of these two equations. Based on the model, the calculated strain from the power law equation affects the activation and inactivation of ion channels. By changing the activation and inactivation variable in the HH equation, we can evaluate the effect of strain and mechanical stimulation on neural function. The results reveal neuron functions’ deficiency during neuron mechanical damage. As a result, action potential signal’s amplitude reduces. This reduction in amplitude of the action potential may be reversible or irreversible based on the amount of damage (plastic deformation).

The diphtheria toxin channel is believed to be a homooligomer of its T domain in which each subunit consists of two alpha-helices, lying within the membrane, connected by a short interhelical loop of four amino acids (residues 349-352). To investigate the validity and implications of this model, we singly mutated each of these amino acids to cysteines, formed channels with the mutant T-domain proteins in planar lipid bilayers, and added to the trans compartment sulfhydryl-specific reagents [methanethiosulfonate derivatives (MTS-ER)] that introduce a positive or negative charge to reacted cysteines. The introduction of a positive charge at residue 351 or 352 (through the MTS-ER reactions) resulted in a step decrease in single-channel conductance, whereas the introduction of a negative charge resulted in a step increase. The opposite sign of these effects indicates the predominantly electrostatic nature of the phenomenon and implies that residues 351 and 352 lie close to the channel entrance. The same reactions at residue 350 resulted in very little change in channel conductance but instead changed the character of the natural rapid flickering of the channel between open and closed states to one in which the channel spent more time in the closed state; this may have resulted from the group introduced at position 350 acting as a tethered channel blocker. The MTS derivatives had no effect on channels containing a cysteine at position 349, suggesting that this residue faces away from the channel entrance. We propose that the step changes in conductance or flickering pattern result from the chemical reaction of one MTS-ER molecule with one cysteine, and thus a bimolecular chemical reaction is being witnessed at the single molecule level. From the distribution of waiting times between the appearance (i.e., the opening) of a channel and the step change in its conductance or flickering pattern, we can calculate a pseudo-first-order rate constant, which can then be converted to a second-order rate constant, for the chemical reaction.

In the stochastic sensitivity analysis, a large number of simulation models lead to low computational e‐ciency. With the dimension increasing in assigned problems, the accurate is di‐cult to achieve by popular regression methodologies. Without considering the correlations of parameters, inaccurate sensitivity coe‐cients would be calculated by analyzing the efiect of parametric variables on structures, moreover, the existing methods only calculate the local gradient as sensitivity. According to these problems, approximate model and global sensitivity method are employed for design sensitivity analysis of structures. The approximate model is constructed by the hybrid neural network which possesses signiflcant learning capacity and generalization capability with a small amount of information. The improved chaotic particle swarm is applied to optimize the neural network so as to improve the computational e‐ciency and accuracy. The proposed sensitivity analysis method values the global response of the outputs by varying all the input parameters at a time with the correlations of parameters. Uniform design and Latin hypercube sampling are used to sample points. Numerical analysis show that the proposed method can successfully measure the actual sensitivity.

SUMMARYThis paper presents three new computational methods for calculating design sensitivities of statistical moments and reliability of high‐dimensional complex systems subject to random input. The first method represents a novel integration of the polynomial dimensional decomposition (PDD) of a multivariate stochastic response function and score functions. Applied to the statistical moments, the method provides mean‐square convergent analytical expressions of design sensitivities of the first two moments of a stochastic response. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD‐saddlepoint approximation (SPA) or PDD‐SPA method, entailing SPA and score functions; and the PDD‐Monte Carlo simulation (MCS) or PDD‐MCS method, utilizing the embedded MCS of the PDD approximation and score functions. For all three methods developed, the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Numerical examples, including a 100‐dimensional mathematical problem, indicate that the new methods developed provide not only theoretically convergent or accurate design sensitivities, but also computationally efficient solutions. A practical example involving robust design optimization of a three‐hole bracket illustrates the usefulness of the proposed methods. Copyright © 2014 John Wiley &amp; Sons, Ltd.