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Two mathematically distinct physiological concepts, the Goldman-Hodgkin-Katz eq. (GHK eq.) and the Hodgkin-Huxley model (HH model) were successfully associated with each other in a prior work. The previous work was performed on the following premises (i) The membrane potential is generated by ion adsorption, as opposed to the classical ion transport mechanisms, (ii) The living cell is a thermodynamically real system rather than an ideal system, and (iii) The conductance employed in the HH model is replaced by the ion activity coefficient, which is weighted with the role of conductance. Consequently, the GHK eq. was mathematically associated with the HH model through the intermediary of Boltzmann ion distribution and mass action law. To verify if our theoretical formularization could afford a physiologically, physically and chemically viable model, we performed computational analysis using the formulae (quantitative correlations between various variables) we derived in the previous work. The computational results obtained through associating the GHK eq. with the HH model validated our model and its predictions. This outcome suggests that the current prevailing physiological concepts could be expanded further, to incorporate the newly proposed mechanisms. That is, GHK eq. and HH model could be interpreted via another set of founding principles that incorporate the ubiquitous phenomena of ion-adsorption.

The partial phase synchronization (sometimes called cooperation) of neurons is fundamental for the understanding of the complex behavior of the brain. The lack or the excess of synchronization can generate brain disorders like Parkinson's disease and epilepsy. The phase synchronization phenomenon is strongly related to the regular or chaotic dynamics of individual neurons. The individual dynamics themselves are a function of the ion channel conductances, turning the conductances into important players in the process of neuron synchronized health depolarization/repolarization processes. It is well known that many diseases are related to alterations of the ion-channel conductance properties. To normalize their functioning, drugs are used to block or activate specific channels, changing their conductances. We investigate the synchronization process of a Hodgkin-Huxley-type neural network as a function of the values of the individual neuron conductances, showing the dynamics of the neurons must be taken into account in the synchronization process. Particular sets of conductances lead to non-chaotic individual neuron dynamics allowing synchronization states for very weak coupling and resulting in a non-monotonic transition to synchronized states, as the coupling strength among neurons is varied. On the other hand, a monotonic transition to synchronized states is observed for individual chaotic dynamics of the neurons. We conclude the analysis of the individual dynamics of isolated neurons allows the prediction of the synchronization process of the network. We provide alternative ways to achieve the desired network state (phase synchronized or desynchronized) without any changes in the synaptic current of neurons but making just small changes in the neuron ion-channel conductances. The mechanism behind the control is the close relation between ion-channel conductance and the regular or chaotic dynamics of neurons. Finally, we show that by changing at least two conductances simultaneously the control may be much more efficient since the second conductance makes the synchronization possible just by performing a small change in the first. The study presented here may have an impact on new drug development research.

In this work, a stochastic Langevin approach is adopted to coupled Hodgkin-Huxley neuron system to elucidate the role of channel noise in the kinetics and energetics of spiking of action potential and the synchronization between neurons. The size of patch plays a pivotal role in synchronization and metabolic energy consumption. The behavior of postsynaptic neuron is explored by taking three different patch size ranges from noise enhanced phase to dead range state before reaching a deterministic limit. We also discuss the effect of sodium and potassium channel blockers on kinetic and energetic characteristics of synchronization and metabolic energy consumption rate.

Despite the long and broad acceptance of the Goldman - Hodgkin - Katz equation (GHK eq.) and the Hodgkin - Huxley equation (HH eq.) as strong tools for the quantitative analysis of the membrane potential behavior, for a long time they have been utilized as separate concepts. That is the GHK eq. and the HH eq. have not been associated with each other mathematically. In this paper, an attempt to associate these equations to each other mathematically was demonstrated and was successful by viewing the system in question as a thermodynamically real system rather than an ideal system. For achieving that, two fundamental physical chemistry concepts, the mass action law, and the Boltzmann distribution were employed. Hence, this paper’s achievement is completely within the framework of common thermodynamics. Through this work, the origin of the membrane potential generation attributed to the ion adsorption-desorption process and governed by the mass action law and the Boltzmann distribution is expressed to be plausible, whereas the existing membrane potential generation mechanism states that membrane potential is generated by transmembrane ion transport. As at this moment, this work does not intend to deny the transmembrane ion transport as a membrane potential generation mechanism but urges the readers to reconsider its validity, since this work suggests that the ion adsorption-desorption mechanism is as plausible as the transmembrane ion transport mechanism as a cause of membrane potential generation.

Since more than a decade ago, three statements about spiking neuron (SN) implementations have been widely accepted: 1) Hodgkin and Huxley (HH) model is computationally prohibitive, 2) Izhikevich (IZH) artificial neuron is as efficient as Leaky Integrate-and-Fire (LIF) model, and 3) IZH model is more efficient than HH model ( Izhikevich, 2004 ). As suggested by Hodgkin and Huxley (1952) , their model operates in two modes: by using the α ’s and β ’s rate functions directly (HH model) and by storing them into tables (HHT model) for computational cost reduction. Recently, it has been stated that: 1) HHT model (HH using tables) is not prohibitive, 2) IZH model is not efficient, and 3) both HHT and IZH models are comparable in computational cost ( Skocik & Long, 2014 ). That controversy shows that there is no consensus concerning SN simulation capacities. Hence, in this work, we introduce a refined approach, based on the multiobjective optimization theory, describing the SN simulation capacities and ultimately choosing optimal simulation parameters. We have used normalized metrics to define the capacity levels of accuracy, computational cost, and efficiency. Normalized metrics allowed comparisons between SNs at the same level or scale. We conducted tests for balanced, lower, and upper boundary conditions under a regular spiking mode with constant and random current stimuli. We found optimal simulation parameters leading to a balance between computational cost and accuracy. Importantly, and, in general, we found that 1) HH model (without using tables) is the most accurate, computationally inexpensive, and efficient, 2) IZH model is the most expensive and inefficient, 3) both LIF and HHT models are the most inaccurate, 4) HHT model is more expensive and inaccurate than HH model due to α ’s and β ’s table discretization, and 5) HHT model is not comparable in computational cost to IZH model. These results refute the theory formulated over a decade ago ( Izhikevich, 2004 ) and go more in-depth in the statements formulated by Skocik and Long (2014) . Our statements imply that the number of dimensions or FLOPS in the SNs are theoretical but not practical indicators of the true computational cost. The metric we propose for the computational cost is more precise than FLOPS and was found to be invariant to computer architecture. Moreover, we found that the firing frequency used in previous works is a necessary but an insufficient metric to evaluate the simulation accuracy. We also show that our results are consistent with the theory of numerical methods and the theory of SN discontinuity. Discontinuous SNs, such LIF and IZH models, introduce a considerable error every time a spike is generated. In addition, compared to the constant input current, the random input current increases the computational cost and inaccuracy. Besides, we found that the search for optimal simulation parameters is problem-specific. That is important because most of the previous works have intended to find a general and unique optimal simulation. Here, we show that this solution could not exist because it is a multiobjective optimization problem that depends on several factors. This work sets up a renewed thesis concerning the SN simulation that is useful to several related research areas, including the emergent Deep Spiking Neural Networks.

The effects of channel blocks on the spontaneous spiking activity and neuron networks pattern selection are investigated by using an improved Hodgkin–Huxley (HH) model in which the electromagnetic induction is considered and the magnetic flux is used to describe the influence of electromagnetic field. The discharge behavior of neurons induced by potassium ion and sodium ion channel blocks was analyzed by numerical simulation in the improved HH model. The results suggest that changes in the maximum conductance of potassium channels can cause spontaneous discharge behavior of neurons. The poisoning of potassium ion can be weakened by the electromagnetic radiation in the neural network, and the neural network presents a state of spatial order in the case of spiral waves. It is interesting that the ordered waveform is generated by no-flux boundary condition when the initial states are selected as wedge-shaped type in the network. In addition, potassium channel blocks can promote the discharge of neurons and facilitate the formation of spiral waves in the neural network. By contrast, the electrical activity of neurons is inhibited by sodium channel blocks. The influence of membrane patch size on the electrical activity of single neuron is greater than that on the collective behavior of neural network. This research will enhance understanding of the role of toxins in neuronal firing and collective behavior of real neural systems.

To understand how single neurons process sensory information, it is necessary to develop suitable stochastic models to describe the response variability of the recorded spike trains. Spikes in a given neuron are produced by the synergistic action of sodium and potassium of the voltage-dependent channels that open or close the gates. Hodgkin and Huxley (HH) equations describe the ionic mechanisms underlying the initiation and propagation of action potentials, through a set of nonlinear ordinary differential equations that approximate the electrical characteristics of the excitable cell. Path integral provides an adequate approach to compute quantities such as transition probabilities, and any stochastic system can be expressed in terms of this methodology. We use the technique of path integrals to determine the analytical solution driven by a non-Gaussian colored noise when considering the HH equations as a stochastic system. The different neuronal dynamics are investigated by estimating the path integral solutions driven by a non-Gaussian colored noise q. More specifically we take into account the correlational structures of the complex neuronal signals not just by estimating the transition probability associated to the Gaussian approach of the stochastic HH equations, but instead considering much more subtle processes accounting for the non-Gaussian noise that could be induced by the surrounding neural network and by feedforward correlations. This allows us to investigate the underlying dynamics of the neural system when different scenarios of noise correlations are considered., Instituto de Física de Líquidos y Sistemas Biológicos, Facultad de Ciencias Exactas, Consejo Nacional de Investigaciones Científicas y Técnicas

A stabilization control method combined with the unscented Kalman filter (UKF) is proposed to control bifurcations in Hodgkin–Huxley neuronal system which are highly related to the occurrence of many dynamical diseases. In neuronal system, usually only the membrane potential can be measured with noise, thus the existing bifurcation controllers, which require exact information of all system states, are impractical. In our method, the system states used to construct the bifurcation controller are estimated by the UKF from partial noisy measurements. The stability of the controlled closed loop system is guaranteed by Lyapunov stability theory. Simulation results demonstrate the effectiveness of the proposed method. The designed controller may have potential applications in the therapy of dynamical diseases.

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.

Channel noise, which is generated by the random transitions of ion channels between open and closed states, is distinguished from external sources of physiological variability such as spontaneous synaptic release and stimulus fluctuations. This inherent stochasticity in ion-channel current can lead to variability of the timing of spikes occurring both spontaneously and in response to stimuli. In this paper, we investigate how intrinsic channel noise affects the response of stochastic Hodgkin–Huxley (HH) neuron to external fluctuating inputs with different amplitudes and correlation time. It is found that there is an optimal correlation time of input fluctuations for the maximal spiking coherence, where the input current has a fluctuating rate approximately matching the inherent oscillation of stochastic HH model and plays a dominating role in the timing of spike firing. We also show that the reliability of spike timing in the model is very sensitive to the properties of the current input. An optimal time scale of input fluctuations exists to induce the most reliable firing. The channel-noise-induced unreliability can be mostly overridden by injecting a fluctuating current with an appropriate correlation time. The spiking coherence and reliability can also be regulated by the size of channel stochasticity. As the membrane area (or total channel number) of the neuron increases, the spiking coherence decreases but the spiking reliability increases.

In this study, we propose a model of the dendritic structure of the neuron (referred to as a neural network – NN), which can be viewed as an extension of the models that are currently used in the description of the potential on the neuron's membrane. The proposed extensions augment the generic model and offer a fuller description of the neuron's nature. The common assumption being used in most of the previous models stating a single channel (forming component of the neuron's membrane) can be positioned in only one of the two states (permissive – open and non-permissive – closed), is now relaxed by allowing the channel to be positioned in more states (five or eight states). The relationship between these states is expressed in terms of Markov kinetic schemes. In the paper, we demonstrate that the new approach is more suitable for a larger number of applications than the conventional Hodgkin–Huxley model. The study, by providing the mathematical background of the new extended model, forms a significant step towards a hardware implementation of the biologically realistic neural network (NN) of this type. To reduce the number of components required in such implementation, we propose a new optimization technique that significantly reduces the computational complexity of a single neuron.

The development of mathematical models to describe neuronal interaction processes in the brain is a challenging task of nonlinear dynamics. Recent advances in biochemistry and neuroscience allow better understanding of biological mechanisms underlying the neuron functioning and synaptic connections between neurons. Moreover, significant progress in brain imaging sheds light on the structure of the brain network and certain aspects of neuronal dynamics. However, dynamical mechanisms leading to synchronization between different brain areas still remain unknown and require further investigation. To shed light on this issue, we consider two small-world networks of Hodgkin-Huxley neurons interacting via inhibitory coupling. We found that synchronization indices (SI) in both networks oscillate periodically in time, so that time intervals of high SI alternate with time intervals of low SI. Depending on the coupling strength, the two coupled networks can be in the regime of either in-phase or anti-phase synchronization. We suppose that the inherent mechanism behind such a behavior lies in the cognitive resource redistribution between neuronal ensembles of the brain.

In this paper, we introduce a general method for the rigorous verification of saddle–node bifurcations in ordinary differential equations. The approach is constructive in the sense that we obtain precise and explicit bounds within which the saddle–node bifurcation occurs. After introducing a set of sufficient generic conditions, an algorithm to verify rigorously the conditions is introduced. The approach is applied to prove existence of saddle–node bifurcations in the Hodgkin–Huxley model.

Degeneracy is a fundamental source of biological robustness, complexity and evolvability in many biological systems. However, degeneracy is often confused with redundancy. Furthermore, the quantification of degeneracy has not been addressed for realistic neuronal networks. The objective of this paper is to characterize degeneracy in neuronal network models via quantitative mathematic measures. Firstly, we establish Hodgkin–Huxley neuronal networks with Newman–Watts small world network architectures. Secondly, in order to calculate the degeneracy, redundancy and complexity in the ensuing networks, we use information entropy to quantify the information a neuronal response carries about the stimulus – and mutual information to measure the contribution of each subset of the neuronal network. Finally, we analyze the interdependency of degeneracy, redundancy and complexity – and how these three measures depend upon network architectures. Our results suggest that degeneracy can be applied to any neuronal network as a formal measure, and degeneracy is distinct from redundancy. Qualitatively degeneracy and complexity are more highly correlated over different network architectures, in comparison to redundancy. Quantitatively, the relationship between both degeneracy and redundancy depends on network coupling strength: both degeneracy and redundancy increase with complexity for small coupling strengths; however, as coupling strength increases, redundancy decreases with complexity (in contrast to degeneracy, which is relatively invariant). These results suggest that the degeneracy is a general topologic characteristic of neuronal networks, which could be applied quantitatively in neuroscience and connectomics.

The use of Hodgkin-Huxley (HH) equations abounds in the literature, but the identifiability of the HH model parameters has not been broadly considered. Identifiability analysis addresses the question of whether it is possible to estimate the model parameters for a given choice of measurement data and experimental inputs. Here we explore the structural identifiability properties of a generalized form of HH from voltage clamp data. Through a scaling argument, we conclude that the steady-state gating variables are not identifiable from voltage clamp data, and then further show that their product together with the conductance term forms an identifiable combination. We additionally show that these parameters become identifiable when the initial conditions for each of the gating variables are known. The time constants for each gating variable are shown to be identifiable, and a novel method for estimating them is presented. Finally, the exponents of the gating variables are shown to be identifiable in the two-gate case, and we conjecture these to be identifiable in the general case. These results are broadly applicable to models using HH-like formalisms, and show in general which parameters and combinations of parameters are possible to estimate from voltage clamp data. HighlightsWe consider the structural identifiability of the Hodgkin-Huxley equations.The product of all steady-state parameters with conductance is identifiable.When initial conditions are known, the steady-state parameters can be identified.The time constants of the generalized model are identifiable.The input-output equation can be used to compute the time constants.

We study an array of activity rhythms generated by a half-center oscillator (HCO), represented by a pair of reciprocally coupled neurons with post-inhibitory rebounds (PIR). Such coupling induced bursting possesses two time scales, one for fast spiking and another for slow quiescent periods, is shown to exhibit an array of synchronization properties. We discuss several HCO configurations constituted by two endogenous bursters, by tonic-spiking and quiescent neurons, as well as mixed-mode configurations composed of neurons of different type. We demonstrate that burst synchronization can be accompanied by complex, often chaotic, interactions of fast spikes within synchronized bursts. (C) 2015 Elsevier B.V. All rights reserved.

In this paper, we study the effects of spike timing-dependent plasticity on synchronisation in a network of Hodgkin–Huxley neurons. Neuron plasticity is a flexible property of a neuron and its network to change temporarily or permanently their biochemical, physiological, and morphological characteristics, in order to adapt to the environment. Regarding the plasticity, we consider Hebbian rules, specifically for spike timing-dependent plasticity (STDP), and with regard to network, we consider that the connections are randomly distributed. We analyse the synchronisation and desynchronisation according to an input level and probability of connections. Moreover, we verify that the transition for synchronisation depends on the neuronal network architecture, and the external perturbation level.

Neurons generate diverse patterns of activity for various functions. Revealing factors which determine neuronal firing patterns is fundamental to a better understanding of brain activity and coding. Traditionally, the space clamp model has been used to investigate neuronal electrical activity. In this paper, we study the Hodgkin–Huxley cable model, taking into consideration axonal geometry. We examine the influence of morphology on neuronal activity, exploring neuronal response to constant current stimuli injected into one end of fiber-like axons of different lengths and radii. We demonstrate novel patterns of firing, including a finite number of spikes and series followed by failures, and under some specific current stimulus regimes, we also detect irregular behaviors. Our results illustrate various means in which the pattern of activity may be regulated by axonal structure, suggesting this mechanism is instrumental in information coding of physiological, as well as deforming pathological conditions.

A complex network of bistable Hodgkin-Huxley (HH) neurons with excitatory coupling can exhibit a partially spiking chimera behavior. We propose to use this chimera-like state for classification of the entering stimulus amplitude in the neural network with coexisting resting and spiking states. Due to different additive noise applied to each neuron in the network, the neurons are nonidentical. Therefore, depending on the amplitude of the external current, a part of the neurons stays in the resting state, while another part oscillates. Keeping fixed the coupling strength between neurons inside the network, we train the neural network on external pulses with two different amplitudes to adjust the coupling strength between the network neurons and two output neurons. We consider two variants of the classifier, in the presence and in the absence of inhibitory coupling between output neurons, and study how the output neurons respond to the external pulses of different amplitudes. The accuracy of the proposed classifier reaches 100% when the output neurons are inhibitory coupled, so that only one of these neurons is activated.

Synchronization in neural networks is strongly tied to the implementation of cognitive processes, but abnormal neuronal synchronization has been linked to a number of brain disorders such as epilepsy and schizophrenia. Here we examine the effects of channel noise on the synchronization of small Hodgkin-Huxley neuronal networks. The principal feature of a Hodgkin-Huxley neuron is the existence of protein channels that transition between open and closed states with voltage dependent rate constants. The Hodgkin-Huxley model assumes infinitely many channels, so fluctuations in the number of open channels do not affect the voltage. However, real neurons have finitely many channels which lead to fluctuations in the membrane voltage and modify the timing of the spikes, which may in turn lead to large changes in the degree of synchronization. We demonstrate that under mild conditions, neurons in the network reach a steady state synchronization level that depends only on the number of neurons in the network. The channel noise only affects the time it takes to reach the steady state synchronization level., 7 Figures

It is well known that the classical Hodgkin–Huxley (HH) equations exhibit complex nonlinear dynamic properties. This paper examines the ion conductance equations of 28 modified HH models, calculating the ion channel conductance and simulating neuronal firing patterns. It is interesting to discover and confirm that the ion conductance is exactly the coefficient of the fourth-order term at the origin of Taylor’s formula. It is demonstrated that the equations for these models are closely related to the two-variable Taylor’s formula of the conductance around the origin in terms of the channel parameters. We consider one specific model in great detail as an example. We find that the conductance curves are generally of the same form, but that the conductance peaks differ. Several firing patterns are observed in the modified HH models, including bursting and mixed-mode oscillations. The emergence of mixed-mode oscillations in particular may be of interest for future work.

Cellular membrane potential plays a key role in the formation and retrieval of memories in the metazoan brain, but it remains unclear whether such memory can also be encoded in simpler organisms like bacteria. Here, we show that single-cell-level memory patterns can be imprinted in bacterial biofilms by light-induced changes in the membrane potential. We demonstrate that transient optical perturbations generate a persistent and robust potassium-channel-mediated change in the membrane potential of bacteria within the biofilm. The light-exposed cells respond in an anti-phase manner, relative to unexposed cells, to both natural and induced oscillations in extracellular ion concentrations. This anti-phase response, which persists for hours following the transient optical stimulus, enables a direct single-cell resolution visualization of spatial memory patterns within the biofilm. The ability to encode robust and persistent membrane-potential-based memory patterns could enable computations within prokaryotic communities and suggests a parallel between neurons and bacteria. We thank Leticia Galera-Laporta, Dong-yeon D. Lee, and Kaito Kikuchi for useful discussions; G.M.S. acknowledges support for this research from the National Institute of General Medical Sciences (grant R01 GM121888 to G.M.S.) and the Howard Hughes Medical Institute-Simons Foundation Faculty Scholars program. J.G.-O. acknowledges support from the Spanish Ministry of Science, Innovation and Universities and FEDER (project PGC2018-101251-B-I00 and “' Maria de Maeztu ” Programme for Units of Excellence in R\&D, grant CEX2018-000792-M) , and from the Generalitat de Catalunya (ICREA Academia programme).

In this work, we revisit two traditional models of action potential generation in excitable cells, the Hodgkin-Huxley (HH) and the FitzHugh-Nagumo (FHN) models. The main goal is to evaluate the possibility of modeling the generation of the action potential via a single delay-differential equation (DDE). Delay-differential equations are important mathematical tools and can reproduce a great diversity of biological phenomena. However, their use in modeling of action potentials is still incipient. In this paper, we present new models based on single delay-differential equation. The solutions of the new models are similar to those of the original HH and FHN models. Based on these results, we claim that delay-differential equations can also be used as building blocks in the development of models of cell electrophysiology.

Synchronized firing of bursting neurons has been found in many real neural systems. In this paper, a modified bursting Hodgkin–Huxley neuronal model is proposed and used to construct neuronal systems with heterogeneous neurons coupled by delayed chemical synapses, then synchronization of the systems is studied in various conditions. First, synchronization for a two-neuron system is explored. It is found that if the synapse is excitatory, the two neurons could achieve burst synchronization easily. If the synapse is inhibitory, they can achieve anti-phase burst synchronization, which is robust to variation of systematic parameters. Second, robust synchronization for a globally coupled neuronal network is investigated. It is found that the robustness of the synchronization is sensitive to the parameter values of synaptic conductance and synaptic delay, but is less sensitive to the parameter values of synaptic decay time. It is also found that the oscillation frequency of the whole network is sensitive to the parameter values of synaptic delay and synaptic decay time, but is less sensitive to the parameter value of synaptic conductance.

According to Stevens’ law the relationship between stimulus and response is a power-law within an interval called the dynamic range. The dynamic range of sensory organs is found to be larger than that of a single neuron, suggesting that the network structure plays a key role in the behavior of both the scaling exponent and the dynamic range of neuron assemblies. In order to verify computationally the relationships between stimulus and response for spiking neurons, we investigate small-world networks of neurons described by the Hodgkin–Huxley equations connected by chemical synapses. We found that the dynamic range increases with the network size, suggesting that the enhancement of the dynamic range observed in sensory organs, with respect to single neurons, is an emergent property of complex network dynamics.

The main focus in most of the continuum based muscle models is the mechanics of muscle contraction while other physiological processes governing muscle contraction, e.g., cell membrane excitation and activation, are ignored. These latter processes are essential to initiate contraction and to determine the amount of generated force, and by excluding them, the developed model cannot replicate the true behavior of the muscle in question. The aim of this study is to establish a thermodynamically and physiologically consistent framework which allows us to model smooth muscle contraction by including cell membrane excitability and kinetics of myosin phosphorylation, along with dynamics of smooth muscle contraction. The model accounts for these processes through a set of coupled dissipative constitutive equations derived by applying first principles. To show the performance of the derived model, it is evaluated for two different cases: a chemo-mechanical study of pig taenia coli cells where the excitation process is excluded, and an electro-chemo-mechanical study of rat myometrium. The results show that the model is able to replicate important aspects of the smooth muscle excitation-contraction process.

In this paper, we present an accurate numerical method for solving fractional Hodgkin–Huxley model. A non-standard finite difference method (NSFDM) is implemented to study the dynamic behaviors of the proposed model. The Grunwald–Letinkov definition is used to approximate the fractional derivatives. Numerical results are presented graphically reveal that NSFDM is easy to implement, effective and convenient for solving the proposed model.

We examine how the synchronization of the series of action potentials (APs) of realistic neurons interconnected in a lattice is influenced by variations of both the direction and magnitude of neuron–neuron connectivity in a noisy environment. We first demonstrate the existence of an optimal noise level that brings about the highest average number of APs per unit time, for a single Hodgkin–Huxley neuron. We then show that synchronization, as a collective response of interconnected neurons forming an N × N lattice, is optimal at different noise strengths σ c = σ c ( p ) , depending on the degree of random-link malfunction parameterized by flipping direction probability p . Thus, even without the scale-free structure of neuronal networks, proper combinations of both randomness in reconnection (flipping) and noisy environment can be beneficial to the collective functioning of neurons.

Information processing in the brain results from the spread and interaction of electrical and chemical signals among neurons. The Hodgkin-Huxley model, describes the spiking and refractory properties of real neurons and serves as a paradigm based on nonlinear conductance of ion channels. Activation of various ion channels establish the spiking behavior of a neuron which determines the communication and coding in nervous system. Firing rates, timing of spikes, and spiking threshold properties are major firing properties of neuronal sparking activities. The temperature influence on neuron spiking threshold regulates the neuronal activities as electrophysiological experiments on neural spiking actions are rarely conducted outside of room temperature in vitro or body temperature in vivo. Temperature changes affect the spiking dynamics of excitable neurons via maximum ion channel conductances and ion channel gating kinetics. The latter has a major impact on changing the shape and amplitude of action potential as well as the generation and propagation of spikes. The purpose of this research is to study, simulate and analyze how the temperature change influence spiking properties of a neuron, channel kinetics as well as activation and inactivation variables of the potassium and sodium channels which determine the dynamics of Hodgkin-Huxley model.

Background Parameterization of neuronal membrane conductance models relies on data acquired from current clamp (CC) or voltage clamp (VC) recordings. Although the CC approach provides key information on a neuron’s firing properties, it is often difficult to disentangle the influence of multiple conductances that contribute to the excitation properties of a real neuron. Isolation of a single conductance using pharmacological agents or heterologous expression simplifies analysis but requires extensive VC evaluation to explore the complete state behavior of the channel of interest. New Method We present an improved parameterization approach that uses data derived from dynamic action potential clamp (DAPC) recordings to extract conductance equation parameters. We demonstrate the utility of the approach by applying it to the standard Hodgkin-Huxley conductance model although other conductance models could be easily incorporated as well. Results Using a fully simulated setup we show that, with as few as five action potentials previously recorded in DAPC mode, sodium conductance equation parameters can be determined with average parameter errors of less than 4% while action potential firing accuracy approaches 100%. In real DAPC experiments, we show that by “training” our model with five or fewer action potentials, subsequent firing lasting for several seconds could be predicted with ˜96% mean firing rate accuracy and 94% temporal overlap accuracy. Comparison with existing methods Our DAPC-based approach surpasses the accuracy of VC-based approaches for extracting conductance equation parameters with a significantly reduced temporal overhead. Conclusion DAPC-based approach will facilitate the rapid and systematic characterization of neuronal channelopathies.

China National Funds for Distinguished Young Scientists [11125419, 10925525]; Leading Talents of Fujian Province; Deutsche Forschungsgemeinschaft [IRTG 1740, RU1660]

A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin-Huxley equations. This work is the first demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial differential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.

Sodium channel Na(v)1.7 is preferentially expressed in dorsal root ganglion (DRG) and sympathetic ganglion neurons. Gain-of-function Na(v)1.7 mutations/variants have been identified in the painful disorders inherited erythromelalgia and small-fiber neuropathy (SFN). DRG neurons transfected with these channel variants display depolarized resting potential, reduced current-threshold, increased firing-frequency and spontaneous firing. Whether the depolarizing shift in resting potential and enhanced spontaneous firing are due to persistent activity of variant channels, or to compensatory changes in other conductance(s) in response to expression of the variant channel, as shown in model systems, has not been studied. We examined the effect of wild-type Na(v)1.7 and a Na(v)1.7 mutant channel, D623N, associated with SFN, on resting potential and membrane potential during interspike intervals in DRG neurons. Resting potential in DRG neurons expressing D623N was depolarized compared to neurons expressing WT-Na(v)1.7. Exposure to TTX hyperpolarized resting potential by 7 mV, increased current-threshold, decreased firing-frequency, and reduced NMDG-induced-hyperpolarization in DRG neurons expressing D623N. To assess the contribution of depolarized resting potential to DRG neuron excitability, we mimicked the mutant channel's depolarizing effect by current injection to produce equivalent depolarization; the depolarization decreased current threshold and increased firing-frequency. Voltage-clamp using ramp or repetitive action potentials as commands showed that D623N channels enhance the TTX-sensitive inward current, persistent at subthreshold membrane voltages, as predicted by a Hodgkin-Huxley model. Our results demonstrate that a variant of Na(v)1.7 associated with painful neuropathy depolarizes resting membrane potential and produces an enhanced inward current during interspike intervals, thereby contributing to DRG neuron hyperexcitability.

Since the initial demonstration of nerve excitation and the subsequent action potential generation by Hodgkin and Huxley in 1952, most efforts in modulating or restoring neural activity to cure diseases or injury have concentrated on using neural interfaces for electrical stimulation with electrodes. However, it was soon appreciated that repeated chronic stimulations necessary for lasting rehabilitation could have its drawbacks. Namely, the eventual degradation of tissue and electrodes, issues of biocompatibility and immune responses to foreign objects. However, new innovative methods are emerging which can improve the quality and duration of neural stimulations. Here, we demonstrate that ectopic expression of channelrhodopsin-2 (ChR2) using solid lipid nanoparticles (SLNs) as a novel delivery system can prove to be a viable alternative to electrode based stimulations for exploring network dynamics or when used in therapeutic rehabilitation.

The dynamical behaviors of a two-dimensional simplified Hodgkin–Huxley model exposed to external electric fields are investigated based on the qualitative analysis and numerical simulation. A necessary and sufficient condition is given for the existence of the Hopf bifurcation. The stability of equilibrium points and limit cycles is also studied. Moreover, the canards and bifurcation are discussed in the simplified model and original model. The dynamical behaviors of the simplified model are consistent with the original model. It would be a great help to further investigations of the original model.

Neurons communicate via electrochemical currents, thus simulation is typically accomplished through modeling the dynamical nature of the neuron's electrical properties. In this paper we utilize Hodgkin-Huxley model and briefly compare it to Leaky integrate-and-fire model. The Hodgkin-Huxley model is a conductance-based model where current flows across the cell membrane due to charging of the membrane capacitance, and movement of ions across ion channels. The leaky integrate-and-fire model is widely used example of formal spiking neuron model. In it the action potentials are generated when the membrane potential crosses a fixed threshold value and the dynamics of the membrane potential is governed by a ‘leaky current’. Conductance-based models (HH models) for excitable cells are developed to help understand underlying mechanisms that contribute to action potential generation, repetitive firing and oscillatory patterns. These factors contribute in modeling the olfactory bulb's dynamic behaviors. Due to these characteristics, we have focused on the conductance-based neuronal models in this work. The model consists of input, mitral and granule layer, connected by synapses. A series of simulations accounting for various olfactory activities are run to explain certain effects of the dynamic behavior of the olfactory bulb (OB). These simulation results are verified against documented evidence in published Journal papers.

The electric activities of neurons are often affected by ion channel poisoning, in particularly, interrupting normal transduction of signals within the brain. This may be due to changes in conductance and the number of active channels. Tetraethylammonium, for example, is known to cause ion channel poisoning of potassium channels, while tetrodotoxin has similar detrimental effects on sodium channels. The occurrence of spiral waves in neuronal systems was observed frequently in the past, and it was argued that these waves of excitation may play an important role by the propagation of electric signals across the quiescent regions of the brain. In this work, the parameters x k and x Na determine the ratio, with regards to the total number of ion channels, of active potassium and sodium channels, respectively, and they are taken to be representative also for the degree of channel poisoning. In the numerical studies, a well developed stable rotating spiral wave is used as the initial state to be controlled by the ion channel poisoning. We show that, under noise-free conditions, spiral waves are terminated whenever x k and x Na are set lower than a given threshold. However, breakup of spiral wave occurs if the intensity of the channel noise increases. In order to quantify these observations, we use a simple but robust synchronization measure, which captures succinctly the transition from spiral waves to homogeneous neuronal activity and/or broken turbulent state. The critical thresholds can be inferred from the abrupt changes occurring in the corresponding dependencies of synchronization versus the x k and x Na ratios. Furthermore, the sampled membrane potentials of a single neuron are recorded to detect the periodical spiral wave in a feasible way and the results could be dependent of the position of node (or site) to be monitored. Notably, small synchronization factors can be tightly associated to states where the formation of spiral waves is robust to channel poisoning and weak channel noise.

In previous articles we have investigated the firing properties of the standard Hodgkin-Huxley (HH) systems of ordinary and partial differential equations in response to input currents composed of a drift (mean) and additive Gaussian white noise. For certain values of the mean current, as the noise amplitude increased from zero, the firing rate exhibited a minimum and this phenomenon was called inverse stochastic resonance (ISR). Here we analyse the underlying transitions from a stable equilibrium point to the limit cycle and vice-versa. Focusing on the case of a mean input current density $\mu=6.8$ at which repetitive firing occurs and ISR had been found to be pronounced, some of the properties of the corresponding stable equilibrium point are found. A linearized approximation around this point has oscillatory solutions from whose maxima spikes tend to occur. A one dimensional diffusion is also constructed for small noise based on the correlations between the pairs of HH variables and the small magnitudes of the fluctuations in two of them. Properties of the basin of attraction of the limit cycle (spike) are investigated heuristically and also the nature of distribution of spikes at very small noise corresponding to trajectories which do not ever enter the basin of attraction of the equilibrium point. Long term trials of duration 500000 ms are carried out for values of the noise parameter $\sigma$ from 0 to 2.0, with results appearing in Section 3. The graph of mean spike count versus $\sigma$ is divided into 4 regions $R_1,...,R_4,$ where $R_3$ contains the minimum associated with ISR., Comment: 27 pages, 16 figures

Diverse behaviors of the original Hodgkin-Huxley (HH) model, depending on the parameter values, have been studied extensively. This paper proposes modified HH equations exposed to externally applied extremely low frequency (ELF) electric fields. We investigate the effect of the DC electric fields on the dynamics of the modified HH model using bifurcation analysis. The obtained bifurcation sets partition the two dimensional parameter space, representing intensity of externally applied DC current and trans-membrane voltage induced by external DC electric fields, in terms of the qualitatively different behaviors of the HH model. Thus the neuronal information encodes the stimulus information, and vice versa. We also illustrate that the multi-stability phenomena in the HH model are associated with Hopf and double cycle bifurcations.

Since ion channels move electrical charge during their activity, they have traditionally been studied using electrophysiological approaches. This was sometimes combined with mathematical models, for example with the description of the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon by Hodgkin and Huxley. The methods for studying ion channels also have strong roots in protein chemistry (limited proteolysis, the use of antibodies, etc.). The advent of the molecular cloning and the identification of genes coding for specific ion channel subunits in the late 1980s introduced a multitude of new techniques with which to study ion channels and the field has been rapidly expanding ever since (e.g. antibody development against specific peptide sequences, mutagenesis, the use of gene targeting in animal models, determination of their protein structures) and new methods are still in development. This review focuses on techniques commonly employed to examine ion channel function in an electrophysiological laboratory. The focus is on the K ATP channel, but many of the techniques described are also used to study other ion channels.

The model of mammalian cortical hypercolumn was simulated using liquid state machines built of simple Hodgkin-Huxley neurons. The influence of cell electrical parameters on the system dynamics was investigated. The systematic analysis of the hypercolumn separation ability in the function of time constants, cell membrane capacitance, resistance, Hodgkin-Huxley equilibrium potential and sodium and potassium channel conductances was performed. Optimal ranges of time constants for the most effective computational abilities of the model were estimated.

Identifiability analysis of a single Hodgkin-Huxley (HH) type voltage dependent ion channel model under voltage clamp circumstances is performed in order to decide if one can uniquely determine the model parameters from measured data in this simple case. It is shown that the two steady-state parameters (m"~, h"~) and the conductance (g) are not globally identifiable together using a single step voltage input. Moreover, no pair from these three parameters is identifiable. Based on the results of the identifiability analysis, a novel optimization-based identification method is proposed and demonstrated on in silico data. The proposed method is based on the decomposition of the parameter estimation problem into two parts using multiple voltage step traces. The results of the article are used to formulate explicit criteria for the design of voltage clamp protocols.

An additional gradient force is often used to simulate the polarization effect induced by the external field in the reaction–diffusion systems. The polarization effect of weak electric field on the regular networks of Hodgkin–Huxley neurons is measured by imposing an additive term VE on physiological membrane potential at the cellular level, and the dynamical evolution of spiral wave subjected to the external electric field is investigated. A statistical variable is defined to study the dynamical evolution of spiral wave due to polarization effect. In the numerical simulation, 40000 neurons placed in the 200 × 200 square array with nearest neighbor connection type. It is found that spiral wave encounters death and the networks become homogeneous when the intensity of electric field exceeds the critical value, otherwise, spiral wave keeps alive completely. On the other hand, breakup of spiral wave occurs as the intensity of electric field exceeds the critical value in the presence of weak channel noise, otherwise, spiral wave keeps robustness to the external field completely. The critical value can be detected from the abrupt changes in the curve for factors of synchronization vs. control parameter, a smaller factor of synchronization is detected when the spiral wave keeps alive.

We study the effects of the ionic reversal potentials for sodium (E"N"a) and potassium (E"K) on the dynamics and energetics of the action potential in the Hodgkin-Huxley model of the squid giant axon, finding that larger action potentials, whose size is controlled primarily by the difference between E"N"a and E"K, can be evoked more rapidly, travel at a faster speed, and consume more metabolic energy than smaller ones. We then systematically investigate whether the biological values of the reversal potentials are optimal for any of a large class of objective functions combining common functional properties of the axon. There appear to be no unconstrained local maxima or minima in these functions over either the two-dimensional {E"N"a,E"K} parameter space or, more significantly, over higher-dimensional parameter spaces which also include the ionic conductances and the axon diameter. This implies that any optimization of these functions in the Hodgkin-Huxley model must necessarily involve some kind of constraint among the parameters. We identify and discuss some possible constrained optimizations: if E"N"a is fixed, then the biological value of E"K is optimal for action potential velocity and for the metabolic energy divided by the maximum firing frequency.

In this paper, we study the effect of time delay on the spiking activity in Newman-Watts small-world networks of Hodgkin-Huxley neurons with non-Gaussian noise, and investigate how the non-Gaussian noise affects the delay-induced behaviors. It was found that, as the delay increases, the neuron spiking intermittently performs the most ordered and synchronized behavior when the delay lengths are integer multiples of the spiking periods, which shows multiple temporal resonances and spatial synchronizations, and reveals that the locking between the delay lengths and the spiking periods might be the mechanism behind the behaviors. It was also found that the delay-optimized spiking behaviors could be enhanced when non-Gaussian noise's deviation from the Gaussian noise is appropriate. These results show that time delay and non-Gaussian noise would cooperate to play more constructive and efficient roles in the information processing of neural networks.

Ever since the pioneering work of Hodgkin and Huxley, biological neuron models have consisted of ODEs representing the evolution of the transmembrane voltage and the dynamics of ionic conductances. It is only recently that discrete dynamical systems–also known as maps–have begun to receive attention as valid phenomenological neuron models. The present review tries to provide a coherent perspective of map-based biological neuron models, describing their dynamical properties; stressing the similarities and differences, both among them and in relation to continuous-time models; exploring their behavior in networks; and examining their wide-ranging possibilities of application in computational neuroscience.

Information processing in the brain results from the spread and interaction of electrical and chemical signals within and among neurons. The equations that describe brain dynamics generally do not have analytical solutions. The recent expansion in the use of simulation tools in the field of neuroscience has been encouraged by the rapid growth of quantitative observations that both stimulate and constrain the formulation of new hypotheses of neuronal function. The purpose of this research is to study, simulate and analyze the influence of Ca concentration on the Na channel. Ca deviation from its normal levels show major clinical problems resulting from the decreased (increased) excitability of neurons as fatigue, depression, confusion, cardiac arrhythmias etc. become evident. We simulate the sensitivity of the Na channel to the concentration of Ca and its “stabilizing” effect on nerve and muscle excitability. Our research is based on Hodgkin and Huxley research, Moore-Cox model of the Na channel as well as NEURON simulation environment. The latter is a powerful and flexible tool for implementing biologically realistic models of electrical and chemical signalling in neurons while adding the necessary expansion and modifications required by the stated goals.

Transition of spiral wave in the regular networks of Hodgkin–Huxley (H–H) neurons is simulated and discussed in detail when the effect of membrane temperature and forcing current is considered. Neurons are distributed in the sites of two-dimensional array, neurons are connected with complete nearest-neighbor connections, no-flux boundary conditions, appropriate initial values and physiological parameters are used to develop a stable rotating spiral wave. A statistic factor of synchronization is defined to discuss the transition and development of spiral wave in the two parameters space (membrane temperature T and forcing current I), and it is found that spiral wave keeps alive due to positive current forcing and the spiral wave can be removed completely when the temperature is increased to a threshold about T = 22.3 °C at a fixed current intensity. Periodical forcing current is imposed on the networks of neurons globally and locally, respectively. It is found that spiral wave could be suppressed by the new generated traveling wave or target wave when periodical forcing current is imposed on the border of networks of neurons, and the most effective frequency of the external forcing current is close to the intrinsic frequency of the spiral wave of the networks.

We sought to determine the feasibility of directly studying neural tissue activity by analysis of differential phase shifts in MRI signals that occurred when trickle currents were applied to a bath containing active or resting neural tissue. We developed a finite element bidomain model of an aplysia abdominal ganglion in order to estimate the sensitivity of this contrast mechanism to changes in cell membrane conductance occurring during a gill-withdrawal reflex. We used our model to determine both current density and magnetic potential distributions within a sample chamber containing an isolated ganglion when it was illuminated with current injected synchronously with the MR imaging sequence, and predicted the resulting changes in MRI phase images. This study provides the groundwork for attempts to image neural function using Magnetic Resonance Electrical Impedance Tomography (MREIT). We found that phase noise in a candidate 17.6 T MRI system should be sufficiently low to detect phase signal differences between active and resting membrane states at resolutions around 1 mm3. We further delineate the broad dependencies of signal to noise ratio on activity frequency, current application time and active tissue fractions, and outline strategies that can be used to lower phase noise below that presently observed in conventional MREIT techniques. We also propose the idea of using MREIT as an alternative means of studying neuromodulation.

The role of repetitive firing in neurophysiologic or neuropsychiatric disorders, such as Parkinson, epilepsy and bipolar type disorders, has always been a topic of medical research as therapies target either the cease of firing or a decrease in its frequency. In electrotherapy, one of the mechanisms to achieve the purpose in point is to apply a low density electric current to the nervous system. In this study, a computer simulation is provided of a treatment in which the stimulation current is computed by nerve fiber cell membrane potential feedback so that the level of the current is automatically instead of manually adjusted. The behavior of the nerve cell is represented by the Hodgkin-Huxley (HH) model, which is slightly modified into a linear model with state dependent coefficients. Due to this modification, the algebraic and differential Riccati equations can be applied, which allows an optimal controller minimizing a quadratic performance index given by the user. Using a controlled current injection can decrease unnecessarily long current injection times that may be harmful to the neuronal network. This study introduces a prototype for a possible future application to a network of neurons as it is more realistic than a single neuron.