Stochastic Hodgkin-Huxley Neuron Research Articles

On the long time behaviour of single stochastic Hodgkin-Huxley neurons with constant signal, and a construction of circuits of interacting neurons showing self-organized rhythmic oscillations

The stochastic Hodgkin-Huxley neurons considered in this paper replace time-constant deterministic input $a dt$ of the classical deterministic model by increments $\vartheta dt + dX_t$ of a stochastic process: $X$ is Ornstein-Uhlenbeck with volatility $\sigma>0$ and backdriving force $\tau>0$, and we call $\vartheta>0$ the signal. We have ergodicity and strong laws of large numbers for various functionals of the process, and characterize 'quiet behaviour' and 'regular spiking' as events whose probability depends on the parameters $(\tau,\sigma)$ and on the signal $\vartheta$. The notions of quiet behaviour and regular spiking allow for a construction of circuits of interacting stochastic Hodgkin-Huxley neurons, combining excitation with inhibition according to a bloc structure along the circuit, on which self-organized rhythmic oscillations can be observed.

Logical stochastic resonance and energy consumption in stochastic Hodgkin–Huxley neuron system

Logical stochastic resonance and energy consumption in stochastic Hodgkin–Huxley neuron system

Synchronization and metabolic energy consumption in stochastic Hodgkin-Huxley neurons: Patch size and drug blockers

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.

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

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

Anesthesia influences the subthreshold voltage characteristics on approach to spiking in a stochastic Hodgkin-Huxley neuron model with phasic GABA-A inhibition

Anesthesia influences the subthreshold voltage characteristics on approach to spiking in a stochastic Hodgkin-Huxley neuron model with phasic GABA-A inhibition

Effects of autapse and ion channel block on the collective firing activity of Newman–Watts small-world neuronal networks

An autapse is a special kind of synapse established between the axon and dendrites of the same neuron. In the present study, we have investigated the cooperative effects of autapse and ion channel block on the collective firing regularity of Newman–Watts small-world networks of stochastic Hodgkin–Huxley neurons. We obtain autaptic time delay induced multi-coherence resonance (MCR) phenomenon in the absence of ion channel block. When the ion channel block is considered, we find that this autaptic delay induced MCR phenomenon enhances with the increasing of potassium channel block, whereas it weakens with the increasing of sodium channel block at weak and intermediate autaptic conductance regimes. However, at strong autaptic conductance regime neither sodium nor potassium channel block have a significant effect on the collective firing regularity of the network. Besides, we investigate the effects of the coupling strength, the network randomness and the cell size on the regularity. We obtain an optimal coupling strength value and an optimal cell size leading to a more prominent MCR effect. We also show that the MCR phenomenon increases with the increasing of network randomness in potassium channel block, but it needs to a minimum network randomness for its appearing in case of sodium channel block.

Channel noise effects on first spike latency of a stochastic Hodgkin-Huxley neuron.

While it is widely accepted that information is encoded in neurons via action potentials or spikes, it is far less understood what specific features of spiking contain encoded information. Experimental evidence has suggested that the timing of the first spike may be an energy-efficient coding mechanism that contains more neural information than subsequent spikes. Therefore, the biophysical features of neurons that underlie response latency are of considerable interest. Here we examine the effects of channel noise on the first spike latency of a Hodgkin-Huxley neuron receiving random input from many other neurons. Because the principal feature of a Hodgkin-Huxley neuron is the stochastic opening and closing of channels, the fluctuations in the number of open channels lead to fluctuations in the membrane voltage and modify the timing of the first spike. Our results show that when a neuron has a larger number of channels, (i) the occurrence of the first spike is delayed and (ii) the variation in the first spike timing is greater. We also show that the mean, median, and interquartile range of first spike latency can be accurately predicted from a simple linear regression by knowing only the number of channels in the neuron and the rate at which presynaptic neurons fire, but the standard deviation (i.e., neuronal jitter) cannot be predicted using only this information. We then compare our results to another commonly used stochastic Hodgkin-Huxley model and show that the more commonly used model overstates the first spike latency but can predict the standard deviation of first spike latencies accurately. We end by suggesting a more suitable definition for the neuronal jitter based upon our simulations and comparison of the two models.

Optimal time scales of input fluctuations for spiking coherence and reliability in stochastic Hodgkin–Huxley neurons

Abstract 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.

Effect of Spike-Timing-Dependent Plasticity on Intrinsic Coherence Resonance in Newman–Watts Stochastic Hodgkin–Huxley Neuronal Networks

In this paper, we numerically study the effect of spike-timing-dependent plasticity (STDP) on coherence resonance (CR) induced by channel noise in adaptive Newman–Watts stochastic Hodgkin–Huxley neuron networks. It is found that STDP can either enhance or suppress the intrinsic CR when the adjusting rate of STDP decreases or increases. STDP can alter the effects of network randomness and network size on the intrinsic CR. Under STDP, for electrical coupling there are optimal network randomness and network size by which the intrinsic CR becomes strongest, however, for chemical coupling the intrinsic CR is always enhanced as network randomness or network size increases, which are different from the results for fixed coupling. These results show that the intrinsic CR of the neuronal networks can be either enhanced or suppressed by STDP, and there are optimal network randomness and network size by which the intrinsic CR becomes strongest. These findings could provide a new insight into the role of STDP for the information processing and transmission in neural systems.

Autapse-induced multiple coherence resonance in single neurons and neuronal networks.

We study the effects of electrical and chemical autapse on the temporal coherence or firing regularity of single stochastic Hodgkin-Huxley neurons and scale-free neuronal networks. Also, we study the effects of chemical autapse on the occurrence of spatial synchronization in scale-free neuronal networks. Irrespective of the type of autapse, we observe autaptic time delay induced multiple coherence resonance for appropriately tuned autaptic conductance levels in single neurons. More precisely, we show that in the presence of an electrical autapse, there is an optimal intensity of channel noise inducing the multiple coherence resonance, whereas in the presence of chemical autapse the occurrence of multiple coherence resonance is less sensitive to the channel noise intensity. At the network level, we find autaptic time delay induced multiple coherence resonance and synchronization transitions, occurring at approximately the same delay lengths. We show that these two phenomena can arise only at a specific range of the coupling strength, and that they can be observed independently of the average degree of the network.

Response of autaptic Hodgkin–Huxley neuron with noise to subthreshold sinusoidal signals

Abstract In this work, we investigated the response of a stochastic Hodgkin–Huxley (HH) neuron with an autapse to subthreshold sinusoidal signals. It is found that the autapse not only adjusts the stochastic responses, but also improves the detection of subthreshold signals. In the case of weak noise, the autapse facilitates the response of neuron to the subthreshold sinusoidal signals with a small parameter region in t d e l a y - ω space. The increased noise intensity enlarges this parameter region and increases the corresponding response frequency in such range. As the autaptic intensity increases, however, this parameter region shrunks. We also observed that there is an optimal range of the delay time of autapse, within which the stochastic HH neuron fires action potentials with high frequency. The corresponding response spike train for the optimal delay time is nearly a regular sequence with the interspike intervals approximated to the delay time. The current results reveal a novel resonance phenomenon facilitated by autapse, named autaptic delay-induced coherence resonance.

Effect of Autaptic Activity on Intrinsic Coherence Resonance in Newman–Watts Networks of Stochastic Hodgkin–Huxley Neurons

In this paper, we study the effect of autaptic activity on intrinsic coherence resonance (CR) induced by channel noise in Newman–Watts (NW) networks of stochastic Hodgkin–Huxley (HH) neurons. It is found that autaptic strength and autaptic delay have a big effect on the intrinsic CR. As autaptic strength increases, there is optimal autaptic strength by which the intrinsic CR is most highly enhanced. Autaptic delay can enhance, reduce, or destroy the intrinsic CR, depending on the delay length. Moreover, there are optimal coupling strength and network randomness by which autaptic activity can most highly enhance the intrinsic CR. These results show that autaptic activity has different effects on the intrinsic CR in the neuronal networks, and it can most highly enhance the intrinsic CR at optimal coupling strength and network randomness. These findings could find potential implications of channel noise and autaptic activity for the information processing and transmission in neural systems.

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

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

Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks

In this paper, we study the effect of delayed autaptic self-feedback activity on the spiking temporal coherence and spatial synchronization of Newman–Watts networks of stochastic Hodgkin–Huxley neurons. As autaptic delay is varied, the spiking behaviors of the neurons intermittently become synchronous and non-synchronous, and meanwhile they are ordered and disordered, exhibiting both synchronization transitions (ST) and multiple coherence resonance (MCR). Moreover, the autaptic delays for CR and synchronization are close. When coupling strength or network randomness increases, or when channel noise intensity decreases, MCR is enhanced, but ST becomes strongest at optimal coupling strength, network randomness, or channel noise intensity. These results show that autaptic activity can induce both MCR and ST in the neuronal networks. This implies that autaptic activity can simultaneously enhance or reduce the temporal coherence and synchronization of the neuronal network. These findings could find potential implications for the information processing and transmission in neural systems.

Autaptic pacemaker mediated propagation of weak rhythmic activity across small-world neuronal networks

We study the effects of an autapse, which is mathematically described as a self-feedback loop, on the propagation of weak, localized pacemaker activity across a Newman–Watts small-world network consisting of stochastic Hodgkin–Huxley neurons. We consider that only the pacemaker neuron, which is stimulated by a subthreshold periodic signal, has an electrical autapse that is characterized by a coupling strength and a delay time. We focus on the impact of the coupling strength, the network structure, the properties of the weak periodic stimulus, and the properties of the autapse on the transmission of localized pacemaker activity. Obtained results indicate the existence of optimal channel noise intensity for the propagation of the localized rhythm. Under optimal conditions, the autapse can significantly improve the propagation of pacemaker activity, but only for a specific range of the autaptic coupling strength. Moreover, the autaptic delay time has to be equal to the intrinsic oscillation period of the Hodgkin–Huxley neuron or its integer multiples. We analyze the inter-spike interval histogram and show that the autapse enhances or suppresses the propagation of the localized rhythm by increasing or decreasing the phase locking between the spiking of the pacemaker neuron and the weak periodic signal. In particular, when the autaptic delay time is equal to the intrinsic period of oscillations an optimal phase locking takes place, resulting in a dominant time scale of the spiking activity. We also investigate the effects of the network structure and the coupling strength on the propagation of pacemaker activity. We find that there exist an optimal coupling strength and an optimal network structure that together warrant an optimal propagation of the localized rhythm.

Effects of channel noise on synchronization transitions in delayed scale-free network of stochastic Hodgkin–Huxley neurons**Project supported by the Natural Science Foundation of Shandong Province of China (Grant No. ZR2012AM013).

We numerically study the effect of the channel noise on the spiking synchronization of a scale-free Hodgkin–Huxley neuron network with time delays. It is found that the time delay can induce synchronization transitions at an intermediate and proper channel noise intensity, and the synchronization transitions become strongest when the channel noise intensity is optimal. The neurons can also exhibit synchronization transitions as the channel noise intensity is varied, and this phenomenon is enhanced at around the time delays that can induce the synchronization transitions. It is also found that the synchronization transitions induced by the channel noise are dependent on the coupling strength and the network average degree, and there is an optimal coupling strength or network average degree with which the synchronization transitions become strongest. These results show that by inducing synchronization transitions, the channel noise has a big regulation effect on the synchronization of the neuronal network. These findings could find potential implications for the information transmission in neural systems.

Effects of channel noise on synchronization transitions in Newman–Watts neuronal network with time delays

In this paper, we numerically study the effect of ion channel noise on the synchronization of delayed Newman–Watts network of stochastic Hodgkin–Huxley neurons. It is found that time delay can induce synchronization transitions only when channel noise intensity is intermediate, and the synchronization transitions become most obvious when channel noise intensity is optimal. As channel noise intensity is varied, the neurons can also exhibit synchronization transitions, and the phenomenon is enhanced when time delay is tuned at around time delays that can induce synchronization transitions. Moreover, this phenomenon becomes most obvious when the value of coupling strength or the value of network randomness is optimal. These results show that channel noise has a subtle regulation effect on the synchronization of the neuronal network. These findings may find potential implications for the normal and pathological functions in the brain and the information processing and transmission in neural systems.

Enhancement of temporal coherence via time-periodic coupling strength in a scale-free network of stochastic Hodgkin–Huxley neurons

We investigate the effects of time-periodic coupling strength on the temporal coherence or firing regularity of a scale-free network consisting of stochastic Hodgkin–Huxley (H–H) neurons. The temporal coherence exhibits a resonance-like behavior depending on the cell size or the channel noise intensity. The best temporal coherence requires an optimal channel noise intensity, and this coherence can be significantly increased by time-periodic coupling strength when its frequency matches the integer multiples of the intrinsic subthreshold oscillation frequency of H–H neuron. Particularly, we find the multiple-coherence resonance depending on frequency of time-periodic coupling strength at the optimal noise intensity. We also obtain a resonance-like dependence of temporal coherence on the amplitude of time-periodic coupling strength. Additionally, we investigate the effects of average degree on the temporal coherence and find that the temporal coherence exhibits a resonance-like behavior with respect to the network average degree, indicating that the best regularity requires an optimal average degree.

Delayed feedback and detection of weak periodic signals in a stochastic Hodgkin–Huxley neuron

We study the effect of the delayed feedback loop on the weak periodic signal detection performance of a stochastic Hodgkin–Huxley neuron. We consider an electrical autapse characterized by its coupling strength and delay time. The stochastic Hodgkin–Huxley neuron exhibits subthreshold oscillations, and thus has an intrinsic time scale with the subthreshold oscillations. Therefore, we investigate the interplay of the subthreshold oscillations, coupling strength and delay time on the weak periodic signal detection. Results indicate that the delayed feedback either enhances or suppresses the weak signal detection depending on its parameters, when compared to that without the feedback. The delayed feedback augments the weak periodic signal detection for the optimal values of the intrinsic noise and the coupling strength when the delay time is close to the integer multiples of the period of the intrinsic oscillations, due to the multiple resonance among the weak signal, the intrinsic oscillations, and the delayed feedback.We analyze the interspike interval histograms and show that the delayed feedback enhances or suppresses the weak periodic signal detection by increasing or decreasing the phase locking (synchronization) between the spiking and the weak periodic signal. We also show that an optimal phase locking is obtained when the delay time is close to the period of the intrinsic oscillations, leading a single dominant time scale in the spike trains.

The ISI distribution of the stochastic Hodgkin-Huxley neuron.

The simulation of ion-channel noise has an important role in computational neuroscience. In recent years several approximate methods of carrying out this simulation have been published, based on stochastic differential equations, and all giving slightly different results. The obvious, and essential, question is: which method is the most accurate and which is most computationally efficient? Here we make a contribution to the answer. We compare interspike interval histograms from simulated data using four different approximate stochastic differential equation (SDE) models of the stochastic Hodgkin-Huxley neuron, as well as the exact Markov chain model simulated by the Gillespie algorithm. One of the recent SDE models is the same as the Kurtz approximation first published in 1978. All the models considered give similar ISI histograms over a wide range of deterministic and stochastic input. Three features of these histograms are an initial peak, followed by one or more bumps, and then an exponential tail. We explore how these features depend on deterministic input and on level of channel noise, and explain the results using the stochastic dynamics of the model. We conclude with a rough ranking of the four SDE models with respect to the similarity of their ISI histograms to the histogram of the exact Markov chain model.