Ion Channel Noise Research Articles

Intrinsic mechanism of influence of channel blocking and noise on response state of multicompartmental neurons

The fine structure of multi-compartment neurons can simultaneously capture both temporal and spatial characteristics, offering rich responses and intrinsic mechanisms. However, current studies of the effects of channel blockage and noise on neuronal response states are mainly limited to single-compartment neurons. This study introduces an analytical method to explore theintrinsic mechanism of channel blockage and noise effects on the response states of multi-compartment neurons, by using the smooth Pinsky-Rinzel two-compartment neuron model as a case study. Potassium, sodium, and calcium ion channel blockage coefficient are separately introduced to develop a smooth Pinsky-Rinzel neuron model with ion channel blockage. Methods such as single-parameter bifurcation analysis, double-parameter bifurcation analysis, coefficient of variation, and frequency characteristics analysis are utilized to examine the effects of various ion channel blockages on neuronal response states. Additionally, smooth Pinsky-Rinzel neuron Subunit noise model and conductance noise model are constructed to investigate their response characteristics by using interspike interval analysis and coefficient of variation indicators. Subthreshold stimulation is used to explore the presence of stochastic resonance phenomena. Single-parameter bifurcation analysis of the ion channel blockage model elucidates the dynamic processes of two torus bifurcations and limit point bifurcations in Pinsky-Rinzel neuron firing under potassium ion blocking. Double-parameter bifurcation analysis reveals a nearly linear increase in the Hopf bifurcation node of potassium ions with input current, whereas sodium ions exhibit a two-stage pattern of linear decline followed by exponential rise. The analysis of average firing frequency and coefficient of variation indicates that the moderate potassium channel blockage promotes firing, sodium channel blockage inhibits firing, and calcium channel blockage shows the complex characteristics but mainly promotes firing. Subthreshold stimulation of the channel noise model demonstrates the stochastic resonance phenomena in both models, accompanied by more intense chaotic firing, highlighting the positive role of noise in neural signal transmission. The interspike interval and coefficient of variation indicators show consistent variation levels for both noise models, with the conductance model displaying greater sensitivity to membrane area and stronger encoding capabilities. This study analyzes the general frequency characteristics of potassium and sodium ions in a multi-compartment neuron model through ion channel blocking model, providing special insights into the unique role of calcium ions. Further, the study explores stochastic resonance by using ion channel noise model, supporting the theory of noise-enhanced signal processing and offering new perspectives and tools for future studying complex information encoding in neural systems. By constructing an ion channel blockage model, the effects of potassium and sodium ions on the frequency characteristics of multi-compartment neurons are analyzed and the special influences of calcium ions are revealed. Using the ion channel noise model, the stochastic resonance is investigated, supporting the theory that the noise enhances signal processing. This research offers a new perspective and tool for studying the complex information encoding in neural systems.

A fast Markovian method for modeling channel noise in neurons

Channel noise results from rapid transitions of protein channels from closed to open state and is generally considered as the most dominant source of electrical noise causing membrane-potential fluctuations even in the absence of synaptic inputs. The simulation of a realistic channel noise remains a source of possible error. Although the Markovian method is considered as the golden standard for appropriate description of channel noise, its computation time increasing exponentially with the number of channels, it is poorly suitable to simulate realistic features. We describe here a novel algorithm at discrete time unit for simulating ion channel noise based on Markov chains (MC). Although this new algorithm refers to a Monte-Carlo process, it only needs few random numbers whatever the number of channels involved. Our fast MC (FMC) model does not exhibit the drawbacks due to approximations based on stochastic differential equations and the values of spike jitter are comparable to those obtained with the true Markovian method. In fact, we show here, that these drawbacks can be highlighted in the approximation based on stochastic differential equation methods even for a high number of channels (standard deviation of the 5th spike is about two-fold larger than that of MCF or true Markovian method for 5000 sodium channels). The FMC model appears therefore as the most accurate method to simulate channel noise with a fast execution time that does not depend on the channel number.

Optimal noise level for coding with tightly balanced networks of spiking neurons in the presence of transmission delays.

Neural circuits consist of many noisy, slow components, with individual neurons subject to ion channel noise, axonal propagation delays, and unreliable and slow synaptic transmission. This raises a fundamental question: how can reliable computation emerge from such unreliable components? A classic strategy is to simply average over a population of N weakly-coupled neurons to achieve errors that scale as . But more interestingly, recent work has introduced networks of leaky integrate-and-fire (LIF) neurons that achieve coding errors that scale superclassically as 1/N by combining the principles of predictive coding and fast and tight inhibitory-excitatory balance. However, spike transmission delays preclude such fast inhibition, and computational studies have observed that such delays can cause pathological synchronization that in turn destroys superclassical coding performance. Intriguingly, it has also been observed in simulations that noise can actually improve coding performance, and that there exists some optimal level of noise that minimizes coding error. However, we lack a quantitative theory that describes this fascinating interplay between delays, noise and neural coding performance in spiking networks. In this work, we elucidate the mechanisms underpinning this beneficial role of noise by deriving analytical expressions for coding error as a function of spike propagation delay and noise levels in predictive coding tight-balance networks of LIF neurons. Furthermore, we compute the minimal coding error and the associated optimal noise level, finding that they grow as power-laws with the delay. Our analysis reveals quantitatively how optimal levels of noise can rescue neural coding performance in spiking neural networks with delays by preventing the build up of pathological synchrony without overwhelming the overall spiking dynamics. This analysis can serve as a foundation for the further study of precise computation in the presence of noise and delays in efficient spiking neural circuits.

A New Neuron Ion Channel Model Under Time Varying Input Currents

In recent years, it has been argued and shown experimentally that ion channel noise in neurons can cause fundamental effects on the neuron’s dynamical behavior. Most profoundly, ion channel noise was seen to be able to cause spontaneous firing and stochastic resonance. However, Hodgkin-Huxley model affected when inserting some colored noise terms inside the conductance’s, where those effects captured by colored noise due to the gate multiplicity. This paper presents a new ion channel model under time varying periodic input currents. It firstly introduced the effect of without, with (colored noise), on the proposed model and the comparison of ion channel based on HH, Fox- Lu, and Linaro models. Additionally, in order to overcome the limitations of other parameter estimation methods, the proposed method fully constraints their models and obtains all models capabilities of reproducing the data. Finally, the relationship between the sequence of colored noise and the spike frequency are simulated efficiently each gate compared with microscopic simulations of the stochastic Markov process method. The simulation results revealed that above a critical value of the input frequency and also below a certain amplitude value, the colored terms play a very prominent role on the firing statistics. In addition, the spiking rate generated from the proposed model very close to microscopic simulations and doesn’t effect by the membrane size.

Resolving molecular contributions of ion channel noise to interspike interval variability through stochastic shielding.

Molecular fluctuations can lead to macroscopically observable effects. The random gating of ion channels in the membrane of a nerve cell provides an important example. The contributions of independent noise sources to the variability of action potential timing have not previously been studied at the level of molecular transitions within a conductance-based model ion-state graph. Here we study a stochastic Langevin model for the Hodgkin-Huxley (HH) system based on a detailed representation of the underlying channel state Markov process, the "[Formula: see text]D model" introduced in (Pu and Thomas in Neural Computation 32(10):1775-1835, 2020). We show how to resolve the individual contributions that each transition in the ion channel graph makes to the variance of the interspike interval (ISI). We extend the mean return time (MRT) phase reduction developed in (Cao et al. in SIAM JApplMath 80(1):422-447, 2020) to the second moment of the return time from an MRT isochron to itself. Because fixed-voltage spike detection triggers do not correspond to MRT isochrons, the inter-phase interval (IPI) variance only approximates the ISI variance. We find the IPI variance and ISI variance agree to within a few percent when both can be computed. Moreover, we prove rigorously, and show numerically, that our expression for the IPI variance is accurate in the small noise (large system size) regime; our theory is exact in the limit of small noise. By selectively including the noise associated with only those few transitions responsible for most of the ISI variance, our analysis extends the stochastic shielding (SS) paradigm (Schmandt and Galán in PhysRevLett 109(11):118101, 2012) from the stationary voltage clamp case to the current clamp case. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels. Finally, we demonstrate that the ISI variance is not an unambiguously defined quantity, but depends on the choice of voltage level set as the spike detection threshold. We find a small but significant increase in ISI variance, the higher the spike detection voltage, both for simulated stochastic HH data and for voltage traces recorded in in vitro experiments. In contrast, the IPI variance is invariant with respect to the choice of isochron used as a trigger for counting "spikes."

Effects of temperature and ion channel blocks on propagation of action potential in myelinated axons.

Potassium ion and sodium ion channels play important roles in the propagation of action potentials along a myelinated axon. The random opening and closing of ion channels can cause the fluctuation of action potentials. In this paper, an improved Hodgkin-Huxley chain network model is proposed to study the effects of ion channel blocks, temperature, and ion channel noise on the propagation of action potentials along the myelinated axon. It is found that the chain network has minimum coupling intensity threshold and maximum tolerance temperature threshold that allow the action potentials to pass along the whole axon, and the blockage of ion channels can change these two thresholds. A striking result is that the simulated value of the optimum membrane size (inversely proportional to noise intensity) coincides with the area range of feline thalamocortical relay cells in biological experiments.

Ion channel noise shapes the electrical activity of endocrine cells.

Endocrine cells in the pituitary gland typically display either spiking or bursting electrical activity, which is related to the level of hormone secretion. Recent work, which combines mathematical modelling with dynamic clamp experiments, suggests the difference is due to the presence or absence of a few large-conductance potassium channels. Since endocrine cells only contain a handful of these channels, it is likely that stochastic effects play an important role in the pattern of electrical activity. Here, for the first time, we explicitly determine the effect of such noise by studying a mathematical model that includes the realistic noisy opening and closing of ion channels. This allows us to investigate how noise affects the electrical activity, examine the origin of spiking and bursting, and determine which channel types are responsible for the greatest noise. Further, for the first time, we address the role of cell size in endocrine cell electrical activity, finding that larger cells typically display more bursting, while the smallest cells almost always only exhibit spiking behaviour.

Interplay between channel and shot noise at the onset of spiking activity in neural membranes

The role played by ion channel noise and ion shot noise around the threshold conditions for spiking activity in biological membranes is studied by means of a stochastic model based on the Hodgkin–Huxley equations, considering membrane voltage-dependent gating channels for sodium and potassium cations, and leakage channels. Ion channel noise, that is, the noise linked to the random opening and closing of the ion channels, is included by means of Langevin sources. Ion shot noise, associated with the random passage of ions through the cell membrane, is considered by using the Gillespie’s method, in terms of the probabilities for different ions to cross the membrane. The threshold for spiking activity is reached by applying increasing values of an external excitation Iapp in a large membrane patch S, for which the strength of channel noise is insufficient for the onset of spikes in the absence of Iapp. On the other hand, since by decreasing S the strength of both noise sources increases, spiking activity is also achieved for small enough values of S when Iapp = 0. The noise behavior of this biological system is analyzed in terms of the autocorrelation function and the spectral density of the membrane voltage fluctuations. Even if ion shot noise is typically considered as negligible when other electrical sources of neural noise are taken into account, the results indicate that, particularly around the onset of instabilities, the signature of shot noise, by the interplay with channel noise, is evident in the spectral density of the membrane voltage fluctuations.

Effect of potassium channel noise on nerve discharge based on the Chay model.

BACKGROUND: The nervous system senses and transmits information through the firing behavior of neurons, and this process is affected by various noises. However, in the previous study of the influence of noise on nerve discharge, the channel of some noise effects is not clear, and the difference from other noises was not examined.OBJECTIVE: To construct ion channel noise which is more biologically significant, and to clarify the basic characteristics of the random firing rhythm of neurons generated by different types of noise acting on ion channels.Method: Based on the dynamics of the ion channel, we constructed ion channel noise. We simulated the nerve discharge based on the Chay model of potassium ion channel noise, and used the nonlinear time series analysis method to measure the certainty and randomness of nerve discharge.RESULTS: In the Chay model with potassium ion noise, the chaotic rhythm defined by the original model could be effectively unified with the random rhythm simulated by the previous random Chay model into a periodic bifurcation process.CONCLUSION: This method clarified the influence of ion channel noise on nerve discharge, better understood the randomness of nerve discharge and provided a more reasonable explanation for the mechanism of nerve discharge.

Firing dynamics in hybrid coupled populations of bistable neurons

We study the firing behavior of bistable neurons that are coupled by both electrical and chemical synapses, constituting a hybrid coupled population. We use stochastic Hodgkin–Huxley model neurons for the dynamics of individual bistable units within the population and the connectivity is implemented with a scale-free network topology which is a widely used network paradigm in theoretical studies of neural circuits. By analyzing the electrical activity of both whole population and individual neurons, it is shown that population mean firing rate exhibits a non-monotonic behavior in the space of electrical and chemical coupling strengths. We identify dynamical mechanisms that shape such non-monotonic spiking behavior and show that different types of population spiking patterns can emerge by fine-tuning the coupling strengths of electrical and chemical synapses. Furthermore, we map the transitions between observed dynamical states in the parameter space of interest depending on the level of individual neuron bistability, existence probability of a synapse type and intrinsic ion channel noise.

Lévy noise induced escape in the Morris–Lecar model

The phenomenon of an excitable system producing a pulse under external or internal stimulation may be interpreted as a stochastic escape problem. This work addresses this issue by examining the Morris–Lecar neural model driven by symmetric α-stable Lévy motion (non-Gaussian noise) as well as Brownian motion (Gaussian noise). Two deterministic quantities: the first escape probability and the mean first exit time, are adopted to analyze the state transition from the resting state to the excited state and the stability of this stochastic model. Additionally, a recent geometric concept, the stochastic basin of attraction is used to explore the basin stability of the escape region. Our main results include: (i) the larger Lévy motion index with smaller jump magnitude and the relatively small noise intensity are conducive for the Morris–Lecar model to produce pulses; (ii) a smaller noise intensity and a larger Lévy motion index make the mean first exit time longer, which means the stability of the resting state can be enhanced in this case; (iii) the effect of ion channel noise is more pronounced on the stochastic Morris–Lecar model than the current noise. This work provides some numerical simulations about the impact of non-Gaussian, heavy-tailed, burst-like fluctuations on excitable systems such as the Morris–Lecar system.

Güler and Linaro et al Model in an Investigation of the Neuronal Dynamics using noise Comparative Study

Recently, theoretical arguments, numerical simulation and experiments shown that ion channel noise in neurons can have deep impact on the behavior of the neuron's dynamical when there is a limited size for the membrane space. It can be create different models of Linaro al equations by using stochastic differential equations to find the impacts of ion channel noise, and it has been analytically put forward the Güler model. More recently, Güler has discussed that in small neurons the rate functions for the closing and opening of gates are under the effect of the noise. In this research, the investigation of dynamics neurons are determined with noise rate functions. The exact Markov simulations will be employ during the investigation with above analytical models. Comparatively, the results will be presented from these models. The research aims to show more details on the phenomenon recently outlined by Güler.

Anesthesia modifies subthreshold critical slowing down in a stochastic Hodgkin-Huxley-like model with inhibitory synaptic input.

The dynamics of a stochastic type-I Hodgkin-Huxley-like point neuron model exposed to inhibitory synaptic noise are investigated as a function of distance from spiking threshold and the inhibitory influence of the general anesthetic agent propofol. The model is biologically motivated and includes the effects of intrinsic ion-channel noise via a stochastic differential equation description as well as inhibitory synaptic noise modeled as multiple Poisson-distributed impulse trains with saturating response functions. The effect of propofol on these synapses is incorporated through this drug's principal influence on fast inhibitory neurotransmission mediated by γ-aminobutyric acid (GABA) type-A receptors via reduction of the synaptic response decay rate. As the neuron model approaches spiking threshold from below, we track membrane voltage fluctuation statistics of numerically simulated stochastic trajectories. We find that for a given distance from spiking threshold, increasing the magnitude of anesthetic-induced inhibition is associated with augmented signatures of critical slowing: fluctuation amplitudes and correlation times grow as spectral power is increasingly focused at 0 Hz. Furthermore, as a function of distance from threshold, anesthesia significantly modifies the power-law exponents for variance and correlation time divergences observable in stochastic trajectories. Compared to the inverse square root power-law scaling of these quantities anticipated for the saddle-node bifurcation of type-I neurons in the absence of anesthesia, increasing anesthetic-induced inhibition results in an observable exponent <-0.5 for variance and >-0.5 for correlation time divergences. However, these behaviors eventually break down as distance from threshold goes to zero with both the variance and correlation time converging to common values independent of anesthesia. Compared to the case of no synaptic input, linearization of an approximating multivariate Ornstein-Uhlenbeck model reveals these effects to be the consequence of an additional slow eigenvalue associated with synaptic activity that competes with those of the underlying point neuron in a manner that depends on distance from spiking threshold.

Influences of autapse and channel blockage on multiple coherence resonance in a single neuron

We study how the spiking regularity of a single stochastic Hodgkin–Huxley neuron is effected in the presence of ion channel blocking and autaptic connection. In this study, we consider a chemical autapse expressed by its coupling strength and delay time. It is found that the neuron exhibits multiple coherence resonance (MCR) behavior induced by autaptic time delay at an appropriate level of ion channel blocking and autaptic coupling strength. This MCR behavior increases with the decrement of working potassium ion channels, whereas it decreases or completely disappears with the increment of a fraction of sodium ion channels blocking, regardless of autaptic coupling strength. Furthermore, this behavior is more explicit at intermediate autaptic coupling strength regardless of the ion channel blocking type. We briefly discuss the obtained results with the underlying reasons in terms of ion channel blocking type and autapse parameters. We also showed that ion channel noise, thus membrane patch size, should be at an optimal level to obtain MCR behavior otherwise, this behavior would be destroyed. The obtained results also showed that autaptic time delay is more operative on regularity than its coupling strength regardless of ion channel blocking. Considering the importance of spiking regularity on neuronal information processing, our results may help to understand the intersection of ion channel blocking and autaptic connections of a single neuron.

Stochastic resonance, coherence resonance, and spike timing reliability of Hodgkin–Huxley neurons with ion-channel noise

The random transitions of ion channels between open and closed states are a major source of noise in neurons. In this study, we investigate the stochastic dynamics of a single Hodgkin–Huxley (HH) neuron with realistic, physiological channel noise, which depends on the channel number and the voltage potential of the membrane. Without external input, the stochastic HH model can generate spontaneous spikes induced by ion-channel noise, and the variability of inter-spike intervals attains a minimum for an optimal membrane area, a phenomenon known as coherence resonance. When a subthreshold periodic input current is added, the neuron can optimally detect the input frequency for an intermediate membrane area, corresponding to the phenomenon of stochastic resonance. We also investigate spike timing reliability of neuronal responses to repeated presentations of the same stimulus with different realizations of channel noise. We show that, with increasing membrane area, the reliability of neuronal response decreases for subthreshold periodic inputs, and attains a minimum for suprathreshold inputs. Furthermore, Arnold tongues of high reliability arise in a two-dimensional plot of frequency and amplitude of the sinusoidal input current, resulting from the resonance effect of spike timing reliability.

Effects of ion channel noise on neural circuits: an application to the respiratory pattern generator to investigate breathing variability.

A major source of noise in neuronal circuits is the "flickering" of ion currents passing through the neurons' membranes (channel noise), which cannot be suppressed experimentally. Computational simulations are therefore the best way to investigate the effects of this physiological noise by manipulating its level at will. We investigate the role of noise in the respiratory pattern generator and show that endogenous, breath-to-breath variability is tightly linked to the respiratory pattern.

Ion channel noise can explain firing correlation in auditory nerves

Neural spike trains are commonly characterized as a Poisson point process. However, the Poisson assumption is a poor model for spiking in auditory nerve fibres because it is known that interspike intervals display positive correlation over long time scales and negative correlation over shorter time scales. We have therefore developed a biophysical model based on the well-known Meddis model of the peripheral auditory system, to produce simulated auditory nerve fibre spiking statistics that more closely match the firing correlations observed in empirical data. We achieve this by introducing biophysically realistic ion channel noise to an inner hair cell membrane potential model that includes fractal fast potassium channels and deterministic slow potassium channels. We succeed in producing simulated spike train statistics that match empirically observed firing correlations. Our model thus replicates macro-scale stochastic spiking statistics in the auditory nerve fibres due to modeling stochasticity at the micro-scale of potassium channels.

Editorial: Neuronal Stochastic Variability: Influences on Spiking Dynamics and Network Activity.

Stochastic variability is present across all scales of brain activity. At the single-cell level, for instance, synaptic transmission is mediated by stochastic release of neurotransmitter and membrane potentials fluctuate due to random conformational changes of ion channels. When these cell-level sources of stochastic variability emerge at the network level, they generate fluctuating currents that drive complex network dynamics. Even if intrinsic cellular noise sources are neglected, the interaction of many nonlinear units in recurrent networks typically leads to an effective network noise which is often mathematically tractable in a stochastic framework. This Research Topic brings together works that address the pressing challenges of developing computational tools and mathematical theories that advance our understanding of stochastic neural dynamics. Six contributions cover stochastic variability at the single-cell level. Moezzi et al. study synaptic coupling between inner hair cells and auditory nerve fibers. Three works update our understanding of ion channel noise in stochastic versions of the Hodgkin-Huxley equations (O'Donnell and Van Rossum; Pezo et al.; Rowat and Greenwood). Puzerey and Galan quantify information transmission in a stochastic Hodgkin-Huxley neuron model that receives barrages of balanced excitatory and inhibitory inputs. Lazar and Zhou communicate a modeling framework that includes dendritic processing of noisy inputs and channel-noise influenced spike generation. The remaining four studies offer new perspectives on network dynamics. Dummer et al. works out the requirements for self-consistent input/output statistics for neurons embedded in recurrent networks. Lagzi and Rotter develop a Markov chain model that clarifies the stochastic dynamics of balanced networks. Mejias and Longtin explore effects of neural heterogeneity on network response properties. Lajoie et al. make elegant use of random dynamical systems theory to analyse stimulus encoding in in chaotic networks. Two commentary articles are also part of this research topic: the commentary of Thomas on Lajoie et al. and the commentary of Baroni and Mazzoni on Mejias and Longtin.

From individual spiking neurons to population behavior: Systematic elimination of short-wavelength spatial modes.

Mean-field models of the brain approximate spiking dynamics by assuming that each neuron responds to its neighbors via a naive spatial average that neglects local fluctuations and correlations in firing activity. In this paper we address this issue by introducing a rigorous formalism to enable spatial coarse-graining of spiking dynamics, scaling from the microscopic level of a single type 1 (integrator) neuron to a macroscopic assembly of spiking neurons that are interconnected by chemical synapses and nearest-neighbor gap junctions. Spiking behavior at the single-neuron scale ℓ≈10μm is described by Wilson's two-variable conductance-based equations [H. R. Wilson, J. Theor. Biol. 200, 375 (1999)], driven by fields of incoming neural activity from neighboring neurons. We map these equations to a coarser spatial resolution of grid length Bℓ, with B≫1 being the blocking ratio linking micro and macro scales. Our method systematically eliminates high-frequency (short-wavelength) spatial modes q(->) in favor of low-frequency spatial modes Q(->) using an adiabatic elimination procedure that has been shown to be equivalent to the path-integral coarse graining applied to renormalization group theory of critical phenomena. This bottom-up neural regridding allows us to track the percolation of synaptic and ion-channel noise from the single neuron up to the scale of macroscopic population-average variables. Anticipated applications of neural regridding include extraction of the current-to-firing-rate transfer function, investigation of fluctuation criticality near phase-transition tipping points, determination of spatial scaling laws for avalanche events, and prediction of the spatial extent of self-organized macrocolumnar structures. As a first-order exemplar of the method, we recover nonlinear corrections for a coarse-grained Wilson spiking neuron embedded in a network of identical diffusively coupled neurons whose chemical synapses have been disabled. Intriguingly, we find that reblocking transforms the original type 1 Wilson integrator into a type 2 resonator whose spike-rate transfer function exhibits abrupt spiking onset with near-vertical takeoff and chaotic dynamics just above threshold.

An investigation of the coefficient of variation using voltage clamps techniques

In recent years, it has been argued and experimentally shown that ion channel noise in neurons can have profound effects on the neuron’s dynamical behavior. Most profoundly, ion channel noise was seen to be able to cause spontaneous firing and stochastic resonance. It has been recently found that a non-trivially persistent cross correlation takes place between the transmembrane voltage fluctuations and the component of open channel fluctuations attributed to gate multiplicity. This non-trivial phenomenon was found to play a major augmentative role for the elevation of excitability and spontaneous firing in the small size cell. In addition, the same phenomenon was found to significantly enhance the spike coherence. In this paper, statistics of the coefficient of variation, to be obtained from the colored stochastic Hodgkin-Huxley equations using voltage clamps techniqueswill be studied. The simulation result shows the coefficient of variation; enhance the agreement with the microscopeinthe case of the noisy currents.