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     

Library-based numerical reduction of the Hodgkin–Huxley neuron for network simulation

We present an efficient library-based numerical method for simulating the Hodgkin–Huxley (HH) neuronal networks. The key components in our numerical method involve (i) a pre-computed high resolution data library which contains typical neuronal trajectories (i.e., the time-courses of membrane potential and gating variables) during the interval of an action potential (spike), thus allowing us to avoid resolving the spikes in detail and to use large numerical time steps for evolving the HH neuron equations; (ii) an algorithm of spike-spike corrections within the groups of strongly coupled neurons to account for spike-spike interactions in a single large time step. By using the library method, we can evolve the HH networks using time steps one order of magnitude larger than the typical time steps used for resolving the trajectories without the library, while achieving comparable resolution in statistical quantifications of the network activity, such as average firing rate, interspike interval distribution, power spectra of voltage traces. Moreover, our large time steps using the library method can break the stability requirement of standard methods (such as Runge–Kutta (RK) methods) for the original dynamics. We compare our library-based method with RK methods, and find that our method can capture very well phase-locked, synchronous, and chaotic dynamics of HH neuronal networks. It is important to point out that, in essence, our library-based HH neuron solver can be viewed as a numerical reduction of the HH neuron to an integrate-and-fire (I&F) neuronal representation that does not sacrifice the gating dynamics (as normally done in the analytical reduction to an I&F neuron).     

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     

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.     

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.     

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.     

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.     

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.     

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 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.     

Day–night contrast as source of health for the human circadian system

Modern societies are characterized by a 24/7 lifestyle (LS) with no environmental differences between day and night, resulting in weak zeitgebers (weak day light, absence of darkness during night, constant environmental temperature, sedentary LS and frequent snacking), and as a consequence, in an impaired circadian system (CS) through a process known as chronodisruption. Both weak zeitgebers and CS impairment are related to human pathologies (certain cancers, metabolic syndrome and affective and cognitive disorders), but little is known about how to chronoenhance the CS. The aim of this work is to propose practical strategies for chronoenhancement, based on accentuating the day/night contrast. For this, 131 young subjects were recruited, and their wrist temperature (WT), activity, body position, light exposure, environmental temperature and sleep were recorded under free-living conditions for 1 week. Subjects with high contrast (HC) and low contrast (LC) for each variable were selected to analyze the HC effect in activity, body position, environmental temperature, light exposure and sleep would have on WT. We found that HC showed better rhythms than LC for every variable except sleep. Subjects with HC and LC for WT also demonstrated differences in LS, where HC subjects had a slightly advanced night phase onset and a general increase in day/night contrast. In addition, theoretical high day/night contrast calculated using mathematical models suggests an improvement by means of LS contrast. Finally, some individuals classified as belonging to the HC group in terms of WT when they are exposed to the LS characteristic of the LC group, while others exhibit WT arrhythmicity despite their good LS habits, revealing two different WT components: an exogenous component modified by LS and another endogenous component that is refractory to it. Therefore, intensifying day/night contrast in subject’s LS has proven to be a feasible measure to chronoenhance the CS.     

Evidence for the Predator Attraction Hypothesis in an amphibian predator–prey system

Many species possess damage-released chemical alarm cues that function in alerting nearby individuals to a predator attack. One hypothesis for the evolution and/or maintenance of such cues is the Predator Attraction Hypothesis, where predators, rather than prey, are the “intended” recipients of these cues. If a predator attack attracts additional predators, these secondary predators might interfere with the predation event, providing the prey with a better chance to escape. In this study, we conducted two experiments to explore this hypothesis in an amphibian predator/prey system. In Experiment 1, we found that tiger salamanders (Ambystoma mavortium) showed a foraging attraction to chemical cues from wood frog (Lithobates sylvaticus) tadpoles. Salamanders that were experienced with tadpole prey, in particular, were strongly attracted to tadpole alarm cues. In Experiment 2, we observed experimental encounters between a tadpole and either one or two salamanders. The presence of the second predator caused salamanders to increase attack speed at the cost of decreased attack accuracy (i.e., increasing the probability that the tadpole would escape attacks). We also found that the mere presence of visual and chemical cues from a second predator did not affect this speed/accuracy trade-off but did cause enough of a distraction to increase tadpole survival. Thus, our findings are consistent with the Predator Attraction Hypothesis for the evolution and/or maintenance of alarm cues.     

Interatomic potential for Fe–Cr–Ni–N system based on the second nearest-neighbor modified embedded-atom method

The interatomic potential for Fe–Cr–Ni–N system based on the second nearest-neighbour modified embedded-atom method has been developed in this work. The potential is based on those for the corresponding lower order systems. The potential parameters for the binary systems, Cr–N, Ni–N, Ni–Fe and Ni–Cr, were determined by fitting the lattice constants, elastic properties, heat of solution and defect binding energies. The potential parameters for the ternary systems were calculated based on the corresponding binary systems. Then, all of them were applied to the quaternary system Fe–Cr–Ni–N to confirm their validity by a simulation of the lattice constants of AISI 316 austenitic stainless steel with a range of nitrogen content. The results were in good agreement with the previous observations and calculations.     

Unmarked gene deletion and host–vector system for the hyperthermophilic crenarchaeon Sulfolobus islandicus

Sulfolobus islandicus is being used as a model for studying archaeal biology, geo-biology and evolution. However, no genetic system is available for this organism. To produce an S. islandicus mutant suitable for genetic analyses, we screened for colonies with a spontaneous pyrEF mutation. One mutant was obtained containing only 233 bp of the original pyrE sequence in the mutant allele and it was used as a host to delete the β-glycosidase (lacS) gene. Two unmarked gene deletion methods were employed, namely plasmid integration and segregation, and marker replacement and looping out, and unmarked lacS mutants were obtained by each method. A new alternative recombination mechanism, i.e., marker circularization and integration, was shown to operate in the latter method, which did not yield the designed deletion mutation. Subsequently, Sulfolobus–E. coli plasmid shuttle vectors were constructed, which genetically complemented ΔpyrEFΔlacS mutation after transformation. Thus, a complete set of genetic tools was established for S. islandicus with pyrEF and lacS as genetic markers.