Heuristics for the Hodgkin–Huxley system

Hodgkin and Huxley (HH) discovered that voltages control ionic currents in nerve membranes. This led them to describe electrical activity in a neuronal membrane patch in terms of an electronic circuit whose characteristics were determined using empirical data. Due to the complexity of this model, a variety of heuristics, including relaxation oscillator circuits and integrate-and-fire models, have been used to investigate activity in neurons, and these simpler models have been successful in suggesting experiments and explaining observations. Connections between most of the simpler models had not been made clear until recently. Shown here are connections between these heuristics and the full HH model. In particular, we study a new model (Type III circuit): It includes the van der Pol-based models; it can be approximated by a simple integrate-and-fire model; and it creates voltages and currents that correspond, respectively, to the h and V components of the HH system.     

White-noise stimulation of the Hodgkin–Huxley model

 We studied the combined influence of noise and constant current stimulations on the Hodgkin–Huxley neuron model through time and frequency analysis of the membrane-potential dynamics. We observed that, in agreement with experimental data (Guttman et al. 1974), at low noise and low constant current stimulation the behavior of the model is well approximated by that of the linearized Hodgkin–Huxley system. Conversely, nonlinearities due to firing dominate at large noise or current stimulations. The transition between the two regimes is abrupt, and takes place in the same range of noise and current intensities as the noise-induced transition characterized by the qualitative change in the stationary distribution of the membrane potential (Tanabe and Pakdaman 2001a). The implications of these results are discussed. Received: 27 July 2001 / Accepted in revised form: 18 December 2001     

Noise-induced spatiotemporal patterns in Hodgkin–Huxley neuronal network

The effect of noise on the pattern selection in a regular network of Hodgkin–Huxley neurons is investigated, and the transition of pattern in the network is measured from subexcitable to excitable media. Extensive numerical results confirm that kinds of travelling wave such as spiral wave, circle wave and target wave could be developed and kept alive in the subexcitable network due to the noise. In the case of excitable media under noise, the developed spiral wave and target wave could coexist and new target-like wave is induced near to the border of media. The averaged membrane potentials over all neurons in the network are calculated to detect the periodicity of the time series and the generated traveling wave. Furthermore, the firing probabilities of neurons in networks are also calculated to analyze the collective behavior of networks.     

The influence of ion concentrations on the dynamic behavior of the Hodgkin–Huxley model-based cortical network

Action potentials (APs) in the form of very short pulses arise when the cell is excited by any internal or external stimulus exceeding the critical threshold of the membrane. During AP generation, the membrane potential completes its natural cycle through typical phases that can be formatted by ion channels, gates and ion concentrations, as well as the synaptic excitation rate. On the basis of the Hodgkin–Huxley cell model, a cortical network consistent with the real anatomic structure is realized with randomly interrelated small population of neurons to simulate a cerebral cortex segment. Using this model, we investigated the effects of Na⁺ and K⁺ ion concentrations on the outcome of this network in terms of regularity, phase locking, and synchronization. The results suggested that Na⁺ concentration does slightly affect the amplitude but not considerably affects the other parameters specified by depolarization and repolarization. K⁺ concentration significantly influences the form, regularity, and synchrony of the network-generated APs. No previous study dealing directly with the effects of both Na⁺ and K⁺ ion concentrations on regularity and synchronization of the simulated cortical network-generated APs, allowing for the comparison of results obtained using our methods, was encountered in the literature. The results, however, were consistent with those obtained through studies concerning resonance and synchronization from another perspective and with the information revealed through physiological and pharmacological experiments concerning changing ion concentrations or blocking ion channels. Our results demonstrated that the regularity and reliability of brain functions have a strong relationship with cellular ion concentrations, and suggested the management of the dynamic behavior of the cellular network with ion concentrations.     

Neural network model of selective visual attention using Hodgkin–Huxley equation

We propose a mathematical model of selective visual attention using a two-layered neural network with neurons described by the Hodgkin–Huxley equation in order to investigate part of the assumption proposed by Desimone and Duncan. The neural network consists of a layer of hippocampal formation and of visual cortex. A frequency of firing and a firing time for each neuron and also a correlation of the firing times between neurons are calculated numerically to clarify an attention state, a nonattention state, and an attention shift. We find that synchronous phenomena occur not only for the frequency but also for the firing time between the neurons in the hippocampal formation and those in a part of the visual cortex in our model. It also turns out that the attention shift is performed quickly in our model.Acknowledgements We are grateful to T. Omori for his valuable discussions and comments. K. K. was partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. This work was partially supported by Grant-In-Aid for Scientific Research No. 13680383 from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.     

The response of a classical Hodgkin–Huxley neuron to an inhibitory input pulse

A population of uncoupled neurons can often be brought close to synchrony by a single strong inhibitory input pulse affecting all neurons equally. This mechanism is thought to underlie some brain rhythms, in particular gamma frequency (30–80 Hz) oscillations in the hippocampus and neocortex. Here we show that synchronization by an inhibitory input pulse often fails for populations of classical Hodgkin–Huxley neurons. Our reasoning suggests that in general, synchronization by inhibitory input pulses can fail when the transition of the target neurons from rest to spiking involves a Hopf bifurcation, especially when inhibition is shunting, not hyperpolarizing. Surprisingly, synchronization is more likely to fail when the inhibitory pulse is stronger or longer-lasting. These findings have potential implications for the question which neurons participate in brain rhythms, in particular in gamma oscillations.     

A nonlinear autoregressive Volterra model of the Hodgkin–Huxley equations

We propose a new variant of Volterra-type model with a nonlinear auto-regressive (NAR) component that is a suitable framework for describing the process of AP generation by the neuron membrane potential, and we apply it to input-output data generated by the Hodgkin–Huxley (H–H) equations. Volterra models use a functional series expansion to describe the input-output relation for most nonlinear dynamic systems, and are applicable to a wide range of physiologic systems. It is difficult, however, to apply the Volterra methodology to the H–H model because is characterized by distinct subthreshold and suprathreshold dynamics. When threshold is crossed, an autonomous action potential (AP) is generated, the output becomes temporarily decoupled from the input, and the standard Volterra model fails. Therefore, in our framework, whenever membrane potential exceeds some threshold, it is taken as a second input to a dual-input Volterra model. This model correctly predicts membrane voltage deflection both within the subthreshold region and during APs. Moreover, the model naturally generates a post-AP afterpotential and refractory period. It is known that the H–H model converges to a limit cycle in response to a constant current injection. This behavior is correctly predicted by the proposed model, while the standard Volterra model is incapable of generating such limit cycle behavior. The inclusion of cross-kernels, which describe the nonlinear interactions between the exogenous and autoregressive inputs, is found to be absolutely necessary. The proposed model is general, non-parametric, and data-derived.     

Extending the Conditions of Application of an Inversion of the Hodgkin–Huxley Gating Model

We present an inversion of the Hodgkin–Huxley formalism to estimate initial conditions and model parameters, including functions of voltage, from the solutions of the underlying ordinary differential equation (ODE) subjected to multiple voltage step stimulations. As such, the procedure constitutes a means to estimate the parameters including functions of voltage of an Hodgkin–Huxley formalism from experimental data. The basic idea was developed in a previous communication (SIAM J. Appl. Math. 64:1264–1274, 2009). The inversion in question applies to currents exhibiting activation and inactivation, but the version, as published previously, cannot estimate the unknowns for channels that rapidly inactivate just after a brief opening. In such cases, the amplitude of the current, in a given voltage range, is too small to be detectable by the instrumentation using previously applied experimental protocols. This is, for example, the case for the sodium channels in a number of excitable tissue for potential in the vicinity of the cell resting potential. The current communication extends the inversion procedure in a manner to overcome this limitation. Furthermore, within the inversion framework, we can determine whether the data at the basis of the estimation sufficiently constrains the estimation problem, i.e., whether it is complete. We exploit this element of our method to document a set of stimulation protocols that constitute a complete data set for the purpose of inverting the Hodgkin–Huxley formalism.     

Giant squid-hidden canard: the 3D geometry of the Hodgkin–Huxley model

This work is motivated by the observation of remarkably slow firing in the uncoupled Hodgkin-Huxley model, depending on parameters tau( h ), tau( n ) that scale the rates of change of the gating variables. After reducing the model to an appropriate nondimensionalized form featuring one fast and two slow variables, we use geometric singular perturbation theory to analyze the model's dynamics under systematic variation of the parameters tau( h ), tau( n ), and applied current I. As expected, we find that for fixed (tau( h ), tau( n )), the model undergoes a transition from excitable, with a stable resting equilibrium state, to oscillatory, featuring classical relaxation oscillations, as I increases. Interestingly, mixed-mode oscillations (MMO's), featuring slow action potential generation, arise for an intermediate range of I values, if tau( h ) or tau( n ) is sufficiently large. Our analysis explains in detail the geometric mechanisms underlying these results, which depend crucially on the presence of two slow variables, and allows for the quantitative estimation of transitional parameter values, in the singular limit. In particular, we show that the subthreshold oscillations in the observed MMO patterns arise through a generalized canard phenomenon. Finally, we discuss the relation of results obtained in the singular limit to the behavior observed away from, but near, this limit.     

Alternative Models to Hodgkin–Huxley Equations

The Hodgkin and Huxley equations have served as the benchmark model in electrophysiology since 1950s. But it suffers from four major drawbacks. Firstly, it is only phenomenological not mechanistic. Secondly, it fails to exhibit the all-or-nothing firing mechanism for action potential generation. Thirdly, it lacks a theory for ion channel opening and closing activation across the cell membrane. Fourthly, it does not count for the phenomenon of voltage-gating which is vitally important for action potential generation. In this paper, a mathematical model for excitable membranes is constructed by introducing circuit characteristics for ion pump exchange, ion channel activation, and voltage-gating. It is demonstrated that the model is capable of re-establishing the Nernst resting potentials, explicitly exhibiting the all-or-nothing firing mechanism, and most important of all, filling the long-lasting theoretical gap by a unified theory on ion channel activation and voltage-gating. It is also demonstrated that the new model has one half fewer parameters but fits significantly better to experiment than the HH model does. The new model can be considered as an alternative template for neurons and excitable membranes when one looks for simpler models for mathematical studies and for forming large networks with fewer parameters.     

Frequency sensitivity in Hodgkin–Huxley systems

 The frequency sensitivity of weak periodic signal detection has been studied via numerical simulations for both a single neuron and a neuronal network. The dependence of the critical amplitude of the signal upon its frequency and a resonance between the intrinsic oscillations of a neuron and the signal could account for the frequency sensitivity. In the presence of both a subthreshold periodic signal and noise, the signal-to-noise ratio (SNR) of the output of either a single neuron or a neuronal network present the typical characteristics of stochastic resonance. In particular, there exists a frequency-sensitive range of 30–100 Hz, and for signals with frequencies within this range the SNRs have large values. This implies that the system under consideration (a single neuron or a neuronal network) is more sensitive to the detection of periodic signals, and the frequency sensitivity may be of a functional significance to signal processing. Received: 26 October 1999 / Accepted in revised form: 25 July 2000     

Minimal Hodgkin–Huxley type models for different classes of cortical and thalamic neurons

We review here the development of Hodgkin–Huxley (HH) type models of cerebral cortex and thalamic neurons for network simulations. The intrinsic electrophysiological properties of cortical neurons were analyzed from several preparations, and we selected the four most prominent electrophysiological classes of neurons. These four classes are “fast spiking” “regular spiking” “intrinsically bursting” and “low-threshold spike” cells. For each class, we fit “minimal” HH type models to experimental data. The models contain the minimal set of voltage-dependent currents to account for the data. To obtain models as generic as possible, we used data from different preparations in vivo and in vitro, such as rat somatosensory cortex and thalamus, guinea-pig visual and frontal cortex, ferret visual cortex, cat visual cortex and cat association cortex. For two cell classes, we used automatic fitting procedures applied to several cells, which revealed substantial cell-to-cell variability within each class. The selection of such cellular models constitutes a necessary step towards building network simulations of the thalamocortical system with realistic cellular dynamical properties.     

Hodgkin–Huxley type modelling and parameter estimation of GnRH neurons

In this paper a simple one compartment Hodgkin–Huxley type electrophysiological model of GnRH neurons is presented, that is able to reasonably reproduce the most important qualitative features of the firing pattern, such as baseline potential, depolarization amplitudes, sub-baseline hyperpolarization phenomenon and average firing frequency in response to excitatory current. In addition, the same model provides an acceptable numerical fit of voltage clamp (VC) measurement results. The parameters of the model have been estimated using averaged VC traces, and characteristic values of measured current clamp traces originating from GnRH neurons in hypothalamic slices. The resulting parameter values show a good agreement with literature data in most of the cases. Applying parametric changes, which lead to the increase of baseline potential and enhance cell excitability, the model becomes capable of bursting. The effects of various parameters to burst length have been analyzed by simulation.     

In-phase and anti-phase synchronization in noisy Hodgkin–Huxley neurons

We numerically investigate the influence of intrinsic channel noise on the dynamical response of delay-coupling in neuronal systems. The stochastic dynamics of the spiking is modeled within a stochastic modification of the standard Hodgkin–Huxley model wherein the delay-coupling accounts for the finite propagation time of an action potential along the neuronal axon. We quantify this delay-coupling of the Pyragas-type in terms of the difference between corresponding presynaptic and postsynaptic membrane potentials. For an elementary neuronal network consisting of two coupled neurons we detect characteristic stochastic synchronization patterns which exhibit multiple phase-flip bifurcations: The phase-flip bifurcations occur in form of alternate transitions from an in-phase spiking activity towards an anti-phase spiking activity. Interestingly, these phase-flips remain robust for strong channel noise and in turn cause a striking stabilization of the spiking frequency.     

A parameter-space search algorithm tested on a Hodgkin–Huxley model

We demonstrate a parameter-space search algorithm using a computational model of a single-compartment neuron with conductance-based Hodgkin-Huxley dynamics. To classify bursting (the desired behavior), we use a simple cost function whose inputs are derived from the frequency content of the neural output. Our method involves the repeated use of a stochastic gradient descent-type algorithm to locate parameter values that allow the neural model to produce bursting within a specified tolerance. We demonstrate good results, including those showing that the utility of our algorithm improves as the pre-defined allowable parameter ranges increase and that the initial approach to our method is computationally efficient.     

The enthalpy change for the reduction of nicotinamide–adenine dinucleotide

The heat of the reaction NAD(+)+propan-2-ol=NADH+acetone+H(+) was determined to be 42.5+/-0.6kJ/mol (10.17+/-0.15kcal/mol) from equilibrium measurements at 9-42 degrees C catalysed by yeast alcohol dehydrogenase. With the aid of thermochemical data for acetone and propan-2-ol the values of DeltaH=-29.2kJ/mol (-6.99kcal/mol) and DeltaG(0)=22.1kJ/mol (5.28kcal/mol) are derived for the reduction of NAD (NAD(+)+H(2)=NADH+H(+)). These values are consistent with analogous but less accurate data for the ethanol-acetaldehyde reaction. Thermodynamic data for the reduction of NAD and NADP are summarized.     

Centrifugal isolation of SARS-CoV-2: numerical simulation for purification of hospitals’ air

Biomechanics and Modeling in Mechanobiology - Coronavirus and its spread all over the world have been the most challenging crisis in 2020. Hospitals are categorized among the most vulnerable...     

On the role of synchrony for neuron–astrocyte interactions and perceptual conscious processing

Recent research on brain correlates of cognitive processes revealed the occurrence of global synchronization during conscious processing of sensory stimuli. In spite of technological progress in brain imaging, an explanation of the computational role of synchrony is still a highly controversial issue. In this study, we depart from an analysis of the usage of blood-oxygen-level-dependent functional magnetic resonance imaging for the study of cognitive processing, leading to the identification of evoked local field potentials as the vehicle for sensory patterns that compose conscious episodes. Assuming the “astrocentric hypothesis” formulated by James M. Robertson (astrocytes being the final stage of conscious processing), we propose that the role of global synchrony in perceptual conscious processing is to induce the transfer of information patterns embodied in local field potentials to astrocytic calcium waves, further suggesting that these waves are responsible for the “binding” of spatially distributed patterns into unitary conscious episodes.     

A phenomenological cohesive model for the macroscopic simulation of cell–matrix adhesions

Cell adhesion is crucial for cells to not only physically interact with each other but also sense their microenvironment and respond accordingly. In fact, adherent cells can generate physical forces that are transmitted to the surrounding matrix, regulating the formation of cell–matrix adhesions. The main purpose of this work is to develop a computational model to simulate the dynamics of cell–matrix adhesions through a cohesive formulation within the framework of the finite element method and based on the principles of continuum damage mechanics. This model enables the simulation of the mechanical adhesion between cell and extracellular matrix (ECM) as regulated by local multidirectional forces and thus predicts the onset and growth of the adhesion. In addition, this numerical approach allows the simulation of the cell as a whole, as it models the complete mechanical interaction between cell and ECM. As a result, we can investigate and quantify how different mechanical conditions in the cell (e.g., contractile forces, actin cytoskeletal properties) or in the ECM (e.g., stiffness, external forces) can regulate the dynamics of cell–matrix adhesions.     

Studies on the oxidation–reduction systems of the erythrocyte

1. Starvation for 3 days produces a decrease in methaemoglobin-reductase and glutathione-reductase activities, but it does not alter the glucose 6-phosphate-dehydrogenase activity of the rat erythrocyte. 2. The feeding of a protein-free diet for 11 days causes greater changes in the first two enzymes and also a diminution of the third. Under this experimental condition slight decreases in protein and haemoglobin contents were noted. 3. The experimental animals did not show methaemoglobinaemia, probably because the activity of methaemoglobin diaphorase is preserved. 4. The GSH content was not affected but the stability of the tripeptide in the presence of an oxidizing agent was diminished.