Hodgkin-Huxley Model Research Articles

Image Edge Detector with Gabor Type Filters Using a Spiking Neural Network of Biologically Inspired Neurons

We report the design of a Spiking Neural Network (SNN) edge detector with biologically inspired neurons that has a conceptual similarity with both Hodgkin-Huxley (HH) model neurons and Leaky Integrate-and-Fire (LIF) neurons. The computation of the membrane potential, which is used to determine the occurrence or absence of spike events, at each time step, is carried out by using the analytical solution to a simplified version of the HH neuron model. We find that the SNN based edge detector detects more edge pixels in images than those obtained by a Sobel edge detector. We designed a pipeline for image classification with a low-exposure frame simulation layer, SNN edge detection layers as pre-processing layers and a Convolutional Neural Network (CNN) as a classification module. We tested this pipeline for the task of classification with the Digits dataset, which is available in MATLAB. We find that the SNN based edge detection layer increases the image classification accuracy at lower exposure times, that is, for 1 < t < T /4, where t is the number of milliseconds in a simulated exposure frame and T is the total exposure time, with reference to a Sobel edge or Canny edge detection layer in the pipeline. These results pave the way for developing novel cognitive neuromorphic computing architectures for millisecond timescale detection and object classification applications using event or spike cameras.

Hearts are Poised Near the Edge of Chaos

The Cardiac Purkinje Fiber (CPF) is the last branch of the heart conduction system, which is meshed with the normal ventricular myocyte. Purkinje fiber plays a key role in the occurrence of ventricular arrhythmia and maintenance. Does the heart Purkinje fiber cells have the same memory function as the cerebral nerve? In this paper, the cardiac Hodgkin–Huxley equation is taken as the object of study. In particular, we find that the potassium ion-channel [Formula: see text] and the sodium ion-channel [Formula: see text] are memristors. We also derive the small-signal equivalent circuits about the equilibrium points of the CPF Hodgkin–Huxley model. According to the principle of local activity, the regions of Locally-Active domain, Edge of Chaos domain and Locally-Passive domain are partitioned under parameters [Formula: see text], and the domain exhibiting the normal human heartbeat frequency range (Goldilocks Zone) is identified. Meanwhile, the Super-Critical Hopf bifurcation of the CPF Hodgkin–Huxley model is identified. Finally, the migration changes between different state domains under external current [Formula: see text] excitation are analyzed in detail. All of the above complex nonlinear dynamics are distilled and mapped geometrically into a surreal union of intersecting two-dimensional manifolds, dubbed the Hodgkin–Huxley’s magic roof.

The New Travelling Wave Solutions of Time Fractional Fitzhugh-Nagumo Equation with Sine-Gordon Expansion Method

Authors aimed to employ the sine-Gordon expansion method to acquire the new exact solutions of fractional Fitzhugh-Nagumo equation which is a stripped type of the Hodgkin-Huxley model that expresses in extensive way activation and deactivation dynamics of neuron spiking. By using the wave transformations, by the practicality of chain rule and applicability of the conformable fractional derivative, the fractional nonlinear partial differential equation (FNPDE) changes to a nonlinear ordinary differential equation. So the exact solution of the considered equation can be obtained correctly with the aid of efficient and reliable analytical techniques.

A Combined Offline–Online Algorithm for Hodgkin–Huxley Neural Networks

Spiking neural networks are widely applied to simulate cortical dynamics in the brain and are regarded as the next generation of machine learning. The classical Hodgkin–Huxley (HH) neuron is the foundation of all spiking neural models. In numerical simulation, however, the stiffness of the nonlinear HH equations during an action potential (a spike) period prohibits the use of large time steps for numerical integration. Outside of this stiff period, the HH equations can be efficiently simulated with a relatively large time step. In this work, we present an efficient and accurate offline–online combined method that stops evolving the HH equations during an action potential period, uses a pre-computed (offline) high-resolution data set to determine the voltage value during the spike, and restarts the time evolution of the HH equations after the stiff period using reset values interpolated from the offline data set. Our method allows for time steps an order of magnitude larger than those used in the standard Runge–Kutta (RK) method, while accurately capturing dynamical properties of HH neurons. In addition, this offline–online method robustly achieves a maximum of a tenfold decrease in computation time as compared to RK methods, a result that is independent of network size.

Indentical synchronization in complete networks of reaction-diffusion equations of FitzHugh-Nagumo

Synchronization is a ubiquitous feature in many natural systems and nonlinear science. This paper studies the synchronization in complete network consisting of n nodes. Each node is connected to all other nodes by linear coupling and represented by a reaction-diffusion system of FitzHugh-Nagumo type which can be obtained by simplifying the famous Hodgkin-Huxley model. From this complete network, the author seeks a sufficient condition on the coupling strength to achieve synchronization. The result shows that the more easily the nodes synchronize, the bigger the degrees of the networks. Based on this consequence, the author will test the theoretical result numerically to see if there is a compromise.

Optimal Control of Infinite-Dimensional Piecewise Deterministic Markov Processes: A BSDE Approach. Application to the Control of an Excitable Cell Membrane

In this paper we consider the optimal control of Hilbert space-valued infinite-dimensional Piecewise Deterministic Markov Processes (PDMP) and we prove that the corresponding value function can be represented via a Feynman–Kac type formula through the solution of a constrained Backward Stochastic Differential Equation. A fundamental step consists in showing that the corresponding integro-differential Hamilton–Jacobi–Bellman equation has a unique viscosity solution, by proving a suitable comparison theorem. We apply our results to the control of a PDMP Hodgkin-Huxley model with spatial component.

A biomorphic neuroprocessor based on a composite memristor-diode crossbar

A concept of biomorphic neuroprocessor that implements hardware spiking neural network for traditional tasks of information processing and can simulate operation of brain cortical column or its fragment is proposed. The key units of hardware neural network are memory and logic matrices, previously developed on the basis of composite memristor-diode crossbar. These matrices provide high element integration and energy efficiency compared to known neuroprocessors and individual matrices. Such efficiency is achieved by application of mixed analog-digital computations, including those that use memristors integrated in composite memristor-diode crossbars. Neuron electrical model was constructed on the basis of these matrices and the Hodgkin-Huxley biomorphic model for neuron membrane. Unlike existing neural networks with synapses based on discrete memristors, the generation of new association was demonstrated in memristor-diode crossbar by SPICE modeling of associative self-learning processes. The adaptation procedure for biomorphic neural network program to neuroprocessor hardware is defined. In essence, presented neuroprocessor is a prototype of new generation computers with artificial intelligence.

Exponential Time Differencing Algorithm for Pulse-Coupled Hodgkin-Huxley Neural Networks.

The exponential time differencing (ETD) method allows using a large time step to efficiently evolve stiff systems such as Hodgkin-Huxley (HH) neural networks. For pulse-coupled HH networks, the synaptic spike times cannot be predetermined and are convoluted with neuron's trajectory itself. This presents a challenging issue for the design of an efficient numerical simulation algorithm. The stiffness in the HH equations are quite different, for example, between the spike and non-spike regions. Here, we design a second-order adaptive exponential time differencing algorithm (AETD2) for the numerical evolution of HH neural networks. Compared with the regular second-order Runge-Kutta method (RK2), our AETD2 method can use time steps one order of magnitude larger and improve computational efficiency more than ten times while excellently capturing accurate traces of membrane potentials of HH neurons. This high accuracy and efficiency can be robustly obtained and do not depend on the dynamical regimes, connectivity structure or the network size.

On biological signaling

Abstract Presently, nerve pulse propagation is understood to take place by electric action pulses. The theoretical description is given by the Hodgkin-Huxley model. Recently, an alternative model was proclaimed, where signaling is carried out by acoustic solitons. The solitons are built by a local phase transition in the lyotropic liquid crystal (LLC) of a biologic membrane. We argue that the crystal structure arranging hydrogen bonds at the membrane surface do not allow such an acoustic soliton model. The bound water is a component of the LLC and the assumed phase transition represents a negative entropy step.

All-Optical Spiking Neuron Based on Passive Microresonator

Neuromorphic photonics that aims to process and store information simultaneously like human brains has emerged as a promising alternative for next generation intelligent computing systems. The implementation of hardware emulating the basic functionality of neurons and synapses is the fundamental work in this field. However, previously proposed optical neurons implemented with SOA-MZIs, modulators, lasers or phase change materials are all dependent on active devices and pose challenges for integration. Meanwhile, although the nonlinearity in nanocavities has long been of interest, the previous theories are intended for specific situations, e.g., self-pulsation in microrings, and there is still a lack of systematic studies in the excitability behavior of the nanocavities including the silicon photonic crystal cavities. Here, we report for the first time a universal coupled mode theory model for all side-coupled passive microresonators. The excitability of microresonators resulting from the subcritical Andronov-Hopf bifurcation is categorized as Class 2 neural excitability and can be used to mimic the Hodgkin-Huxley neuron model. We demonstrate that the microresonator-based neuron can exhibit the typical characteristics of spiking neurons: excitability threshold, leaky integrating dynamics, refractory period, cascadability and inhibitory spiking behavior, paving the way to realize all-optical spiking neural networks.

ANALYZING THE IMPACT OF TEMPERATURE ON AXOPLASMIC FLUID PROPERTIES DEFINING NEURONAL EXCITATION

Axoplasmic fluid properties for neuronal excitation have been investigated with respect to temperature. Density, the mass fraction of ions and rate of addition of ions are the parameters considered for characterizing axoplasmic fluid properties. The behavior of these parameters has been analyzed with respect to the changes in temperature ranging from -5 degree Celsius to 35 degree Celsius. The temperature has been defined using Q10of3 coefficient as done in the Hodgkin-Huxley model. The trend of these parameters at different temperatures has been depicted along the axonal length represented through x-axis of the graphs. The conduction velocities of the above said parameters have also been recorded at different temperatures. The range [-5,35] degree Celsius has been increased by 20 degrees, 10 degree on the lower side and 10 degree on the upper side of the range [-5,25] degree Celsius and it is found that temperature dependency using Q10of3 coefficient for said parameters is valid only in the temperature ranging from 5 degree Celsius to 25 degree Celsius as it is for membrane voltage in the Hodgkin-Huxley model. These findings strongly support the obtained results and also suggest obtaining the temperature coefficient value which is applicable for a wider range of temperatures impacting neuronal excitation.

Action Potential: A Vortex Phenomena; Driving Membrane Oscillations.

Hodgkin-Huxley (HH) model has been one of the most successful electrical interpretation of nerve membrane which led to revolutions in the field of computational neuroscience. On the contrary, experimental observations indicate that, an Action Potential (AP) is accompanied with certain physiological changes in the nerve membrane such as, production and absorption of heat; variation of axon diameter, pressure and length. Although, in the early 1900's a Pressure Wave Theory was proposed by E. Wilke, but, due to lack of sophisticated experimental techniques it was left uncharted. Until recently, when Heimburg-Jackson, Hady-Machta and Rvachev, independently proposed Soliton Theory (thermodynamic interpretation of nerve membrane), Mechanical Surface Waves theory (electro-mechanical interpretation) and Rvachev Model (mechano-electrical activation of voltage gated sodium ion channels) respectively; encouraging a deviation from the traditional HH interpretation with justification for the physical changes in the nerve membrane observed experimentally. But, these theories lead to a “hit and miss” scenario because, they do explain certain features (increase/decrease in axon diameter) but miss to explain, correlation between the strength of stimuli and spike rate of AP. Bio-physical models of nerve membrane are thus important for enhancing our understanding regarding the governing dynamics of neural activities encompassing the experimental observations. A novel theory is proposed here which, unravels vortex ring formation due to ion currents in the intracellular and extracellular region leading to variation of pressure causing the increment/decrement in axon diameter. These formations manifest as membrane oscillations which are used to establish a correlation between the strength of stimuli and spike rate of AP. The theory proposed in this paper, brings a paradigm shift in our understanding of neural dynamics from a thorough bio-physical and physiological perspective with promising applications.

FPGA Realization of Hodgkin-Huxley Neuronal Model.

One of the appealing cases of the neuromorphic research area is the implementation of biological neural networks. The current study offers Multiplierless Hodgkin-Huxley Model (MHHM). This modified model may reproduce various spiking behaviors, like the biological HH neurons, with high accuracy. The presented modified model, in comparison to the original HH model, due to its exact similarity to the original model, has more top performances in the case of FPGA saving and more achievable frequency (speed-up). In this approach, the proposed model has a 69 % saving in FPGA resources and also the maximum frequency of 85 MHz that is more than other similar works. In this modification, all spiking behaviors of the original model have been generated with low error calculations. To validate the MHHM neuron, this proposed model has been implemented on digital hardware FPGA. This approach demonstrates that the original HH model and the proposed model have high similarity in terms of higher performance and digital hardware cost reduction.

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

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

Synthesis and Analysis of the Fixed-Point Hodgkin–Huxley Neuron Model

In many tasks related to realistic neurons and neural network simulation, the performance of desktop computers is nowhere near enough. To overcome this obstacle, researchers are developing FPGA-based simulators that naturally use fixed-point arithmetic. In these implementations, little attention is usually paid to the choice of numerical method for the discretization of the continuous neuron model. In our study, the implementation accuracy of a neuron described by simplified Hodgkin–Huxley equations in fixed-point arithmetic is under investigation. The principle of constructing a fixed-point neuron model with various numerical methods is described. Interspike diagrams and refractory period analysis are used for the experimental study of the synthesized discrete maps of the simplified Hodgkin–Huxley neuron model. We show that the explicit midpoint method is much better suited to simulate the neuron dynamics on an FPGA than the explicit Euler method which is in common use.

Skin pain sensation of epidermal electronic device/skin system considering non-Fourier heat conduction

Epidermal electronic devices (EEDs) have drawn much attention recently due to their wide applications in biological monitoring and biomedicine. Due to the thermo-mechanical coupling between EEDs and skin, skin pain sensation is susceptive to noxious stimuli from EEDs in practical applications. An analytical thermo-mechanical model is developed to predict the temperature and stress distributions in the EED/skin system accounting for the non-Fourier effect in skin tissues via Dual-Phase-Lag (DPL) model. The phase lag parameters in the DPL model are obtained from the two-temperature model and then validated by experiments. The holistic analytical frame, consisting of the thermo-mechanical model for temperature and stress, Arrhenius burn model for thermal damage, modified Hodgkin–Huxley model for nociceptor transduction and gate control theory for pain modulation and perception, is adopted to quantify the skin pain sensation in terms of the noxious stimuli from EEDs. Skin pain sensations of EED/skin system are fully investigated in both constant and periodic operation modes. The model in this paper paves the theoretical foundation to study wearing comfort of EEDs and can be easily extended to study other bio-integrated systems for different types of applications.

Excitatory autapse induces different cases of reduced neuronal firing activities near Hopf bifurcation

Excitatory autapse with short time delay can lower the firing frequency of spiking behaviors near subcritical Hopf bifurcation, which is interpreted with type II phase response curve wherein an excitatory impulse current applied at the phase following a spike can induce the next spike delayed. Such a nonlinear phenomenon is different from the common viewpoint that excitatory effect typically induces enhancement of neuronal firing activities. In this paper, more cases of reduction of firing activities induced by excitatory autapse than the one mentioned above are simulated in the Hodgkin-Huxley model with relatively short time delay, and the dynamical mechanism is provided with phase trajectories modulated by excitatory autaptic current and bifurcations. For the monostable spiking near the subcritical Hopf bifurcation, one novel case is induced. The delayed autaptic current pulse evoked by a spike induces the trajectory moved to the neighborhood of the unstable focus, and after the pulse, the subthreshold oscillations with ascending amplitude appear until they evolve to the next spike, which results in mixed subthreshold oscillations and a single spike. For the spiking coexisting with the resting state, which locates between the subcritical Hopf bifurcation and saddle-node bifurcation of limit cycles, two novel cases are induced. One is the mixed-mode oscillations, and the trajectory of subthreshold oscillations begins from the neighborhood outside of the unstable limit cycle, which is induced by the delayed autaptic current pulse. The other is induced by the autaptic current pulse that can cause the trajectory to run into the unstable limit cycle, which leads to the appearance of subthreshold oscillations with descending amplitude after the pulse, and finally evolves to the resting state corresponding to the coexisting stable focus. The results present novel cases that excitatory effects can suppress rather than enhance firing activities and the corresponding dynamical mechanism related to bifurcations, which enrich the study of nonlinear dynamics and neurodynamics, and provide the potential functions of excitatory autapse.

Slowing of the Time Course of Acidification Decreases the Acid-Sensing Ion Channel 1a Current Amplitude and Modulates Action Potential Firing in Neurons.

Acid-sensing ion channels (ASICs) are H+-activated neuronal Na+ channels. They are involved in fear behavior, learning, neurodegeneration after ischemic stroke and in pain sensation. ASIC activation has so far been studied only with fast pH changes, although the pH changes associated with many roles of ASICs are slow. It is currently not known whether slow pH changes can open ASICs at all. Here, we investigated to which extent slow pH changes can activate ASIC1a channels and induce action potential signaling. To this end, ASIC1a current amplitudes and charge transport in transfected Chinese hamster ovary cells, and ASIC-mediated action potential signaling in cultured cortical neurons were measured in response to defined pH ramps of 1–40 s duration from pH 7.4 to pH 6.6 or 6.0. A kinetic model of the ASIC1a current was developed and integrated into the Hodgkin-Huxley action potential model. Interestingly, whereas the ASIC1a current amplitude decreased with slower pH ramps, action potential firing was higher upon intermediate than fast acidification in cortical neurons. Indeed, fast pH changes (<4 s) induced short action potential bursts, while pH changes of intermediate speed (4–10 s) induced longer bursts. Slower pH changes (>10 s) did in many experiments not generate action potentials. Computer simulations corroborated these observations. We provide here the first description of ASIC function in response to defined slow pH changes. Our study shows that ASIC1a currents, and neuronal activity induced by ASIC1a currents, strongly depend on the speed of pH changes. Importantly, with pH changes that take >10 s to complete, ASIC1a activation is inefficient. Therefore, it is likely that currently unknown modulatory mechanisms allow ASIC activity in situations such as ischemia and inflammation.

Firing patterns of the modified Hodgkin–Huxley models subject to Taylor ’s formula

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.

Dynamic clamp constructed phase diagram for the Hodgkin and Huxley model of excitability

Excitability-a threshold-governed transient in transmembrane voltage-is a fundamental physiological process that controls the function of the heart, endocrine, muscles, and neuronal tissues. The 1950s Hodgkin and Huxley explicit formulation provides a mathematical framework for understanding excitability, as the consequence of the properties of voltage-gated sodium and potassium channels. The Hodgkin-Huxley model is more sensitive to parametric variations of protein densities and kinetics than biological systems whose excitability is apparently more robust. It is generally assumed that the model's sensitivity reflects missing functional relations between its parameters or other components present in biological systems. Here we experimentally assembled excitable membranes using the dynamic clamp and voltage-gated potassium ionic channels (Kv1.3) expressed in Xenopus oocytes. We take advantage of a theoretically derived phase diagram, where the phenomenon of excitability is reduced to two dimensions defined as combinations of the Hodgkin-Huxley model parameters, to examine functional relations in the parameter space. Moreover, we demonstrate activity dependence and hysteretic dynamics over the phase diagram due to the impacts of complex slow inactivation kinetics. The results suggest that maintenance of excitability amid parametric variation is a low-dimensional, physiologically tenable control process. In the context of model construction, the results point to a potentially significant gap between high-dimensional models that capture the full measure of complexity displayed by ion channel function and the lower dimensionality that captures physiological function.