Explosive, continuous and frustrated synchronization transition in spiking Hodgkin-Huxley neuronal networks: the role of topology and synaptic interaction
Synchronization is an important collective phenomenon in interacting oscillatory agents. Many functional features of the brain are related to synchronization of neurons. The type of synchronization transition that may occur (explosive vs. continuous) has been the focus of intense attention in recent years, mostly in the context of phase oscillator models for which collective behavior is independent of the mean-value of natural frequency. However, synchronization properties of biologically-motivated neural models depend on the firing frequencies. In this study we report a systematic study of gamma-band synchronization in spiking Hodgkin-Huxley neurons which interact via electrical or chemical synapses. We use various network models in order to define the connectivity matrix. We find that the underlying mechanisms and types of synchronization transitions in gamma-band differs from beta-band. In gamma-band, network regularity suppresses transition while randomness promotes a continuous transition. Heterogeneity in the underlying topology does not lead to any change in the order of transition, however, correlation between number of synapses and frequency of a neuron will lead to explosive synchronization in heterogenous networks with electrical synapses. Furthermore, small-world networks modeling a fine balance between clustering and randomness (as in the cortex), lead to explosive synchronization with electrical synapses, but a smooth transition in the case of chemical synapses. We also find that hierarchical modular networks, such as the connectome, lead to frustrated transitions. We explain our results based on various properties of the network, paying particular attention to the competition between clustering and long-range synapses.
Transient spike trains consisting of $M$ (= 1 - 5) pulses generated by single Hodgkin-Huxley (HH) neurons, have been analyzed by using both the continuous and discrete wavelet transformations (WT). We have studied effects of variations in the interspike intervals (ISI) of the spikes and effects of noises on the energy distribution and the wavelet entropy, which are expressed in terms of the WT expansion coefficients. The results obtained by the WT are discussed in connection with those obtained by the Fourier transformation.
We analyze the response of the Hodgkin-Huxley neuron to a large number of uncorrelated stochastic inhibitory and excitatory post-synaptic spike trains. In order to clarify the various mechanisms responsible for noise-induced spike triggering we examine the model in its silent regime. We report the coexistence of two distinct coherence resonances: the first one at low noise is due to the stimulation of "correlated" subthreshold oscillations; the second one at intermediate noise variances is instead related to the regularization of the emitted spike trains.
In a recent paper [A. Moujahid, A. d'Anjou, F. J. Torrealdea and F. Torrealdea, Phys. Rev. E {\bf 83}, 031912 (2011)], the authors have calculated the energy consumed in firing neurons by using the Hodgkin-Huxley (HH) model. The energy consumption rate adopted for the HH model yields a {\it negative} energy consumption meaning an energy transfer from an HH neuron to a source which is physically strange, although they have interpreted it as a biochemical energy cost. I propose an alternative expression for the power consumption which leads to a {\it positive} energy consumed in an HH neuron, pr…
Model calculations have been performed on the spike-train response of a pair of Hodgkin-Huxley (HH) neurons coupled by recurrent excitatory-excitatory couplings with time delay. The coupled, excitable HH neurons are assumed to receive the two kinds of spike-train inputs: the transient input consisting of $M$ impulses for the finite duration ($M$: integer) and the sequential input with the constant interspike interval (ISI). The distribution of the output ISI $T_{\rm o}$ shows a rich of variety depending on the coupling strength and the time delay. The comparison is made between the dependence …
We investigate the regularity of spontaneous spiking activity on Newman-Watts small-world networks consisting of biophysically realistic Hodgkin-Huxley neurons with a tunable intensity of intrinsic noise and fraction of blocked voltage-gated sodium and potassium ion channels embedded in neuronal membranes. We show that there exists an optimal fraction of shortcut links between physically distant neurons, as well as an optimal intensity of intrinsic noise, which warrant an optimally ordered spontaneous spiking activity. This doubly coherence resonance-like phenomenon depends significantly, and …
A dynamical mean-field approximation (DMA) previously proposed by the present author [H. Hasegawa, Phys. Rev E {\bf 67}, 041903 (2003)] has been extended to ensembles described by a general noisy spiking neuron model. Ensembles of $N$-unit neurons, each of which is expressed by coupled $K$-dimensional differential equations (DEs), are assumed to be subject to spatially correlated white noises. The original $KN$-dimensional {\it stochastic} DEs have been replaced by $K(K+2)$-dimensional {\it deterministic} DEs expressed in terms of means and the second-order moments of {\it local} and {\it glob…
Numerical investigations have been made of responses of a Hodgkin-Huxley (HH) neuron to spike-train inputs whose interspike interval (ISI) is modulated by deterministic, semi-deterministic (chaotic) and stochastic signals. As deterministic one, we adopt inputs with the time-independent ISI and with time-dependent ISI modulated by sinusoidal signal. The Rössler and Lorentz models are adopted for chaotic modulations of ISI. Stochastic ISI inputs with the Gamma distribution are employed. It is shown that distribution of output ISI data depends not only on the mean of ISIs of spike-train inputs bu…
An associative memory has been discussed of neural networks consisting of spiking N (=100) Hodgkin-Huxley (HH) neurons with time-delayed couplings, which memorize P patterns in their synaptic weights. In addition to excitatory synapses whose strengths are modified after the Willshaw-type learning rule with the 0/1 code for quiescent/active states, the network includes uniform inhibitory synapses which are introduced to reduce cross-talk noises. Our simulations of the HH neuron network for the noise-free state have shown to yield a fairly good performance with the storage capacity of $α_c = P_{…
Spike-train responses of single Hodgkin-Huxley (HH) and integrate-and-fire (IF) neurons with and without the refractory period, are calculated and compared. The HH and IF neurons are assumed to receive spike-train inputs with the constant interspike intervals (ISIs) and stochastic ISIs given by the Gamma distribution, through excitatory and inhibitory synaptic couplings: for both the couplings the HH neuron can fire while the IF neuron can only for the excitatory one. It is shown that the response to the constant-ISI inputs of the IF neuron strongly depends on the refractory period and the syn…
By means of the concepts of factorial moment, return map and NM estimator, we analyze some responses of a HH neuron to various types of spike-train inputs. The corresponding fractal dimensions and values of NM estimators can describe the correlation characteristics quantitatively. It is found that all the intervals of successive output spikes may obey a power-law for some conditions. And in time scale the output series for a HH model for each type of inputs is correlated with typical correlation length and correlation strength. These power-law and correlation characteristics may be useful in t…
The response of the Hodgkin-Huxley neuronal model subjected to stochastic uncorrelated spike trains originating from a large number of inhibitory and excitatory post-synaptic potentials is analyzed in detail. The model is examined in its three fundamental dynamical regimes: silence, bistability and repetitive firing. Its response is characterized in terms of statistical indicators (interspike-interval distributions and their first moments) as well as of dynamical indicators (autocorrelation functions and conditional entropies). In the silent regime, the coexistence of two different coherence r…
Numerical calculations have been made on the spike-train response of a pair of Hodgkin-Huxley (HH) neurons coupled by synapses and axons with time delay. The recurrent excitatory-excitatory, inhibitory-inhibitory, excitatory-inhibitory, and inhibitory-excitatory couplings are adopted. The coupled, excitable HH neurons are assumed to receive the two kinds of spike-train inputs: the transient input consisting of M impulses for the finite duration (M: integer) and the sequential input with the constant interspike interval (ISI). The HH neurons in all the kinds of couplings are found to play a rol…
Spatial coherence resonance in a spatially extended system that is locally modeled by Hodgkin-Huxley (HH) neurons is studied in this paper. We focus on the ability of additive temporally and spatially uncorrelated Gaussian noise to extract a particular spatial frequency of excitatory waves in the medium, whereby examining also the impact of diffusive and small-world network topology determining the interactions amongst coupled HH neurons. We show that there exists an intermediate noise intensity that is able to extract a characteristic spatial frequency of the system in a resonant manner provi…
The Hodgkin-Huxley model describes the conduction of the nervous impulse through the axon, whose membrane's electric response can be described employing multiple connected electric circuits containing capacitors, voltage sources, and conductances. These conductances depend on previous depolarizing membrane voltages, which can be identified with a memory resistive element called memristor. Inspired by the recent quantization of the memristor, a simplified Hodgkin-Huxley model including a single ion channel has been studied in the quantum regime. Here, we study the quantization of the complete H…
This paper presents an overview of some techniques and concepts coming from dynamical system theory and used for the analysis of dynamical neural networks models. In a first section, we describe the dynamics of the neuron, starting from the Hodgkin-Huxley description, which is somehow the canonical description for the ``biological neuron''. We discuss some models reducing the Hodgkin-Huxley model to a two dimensional dynamical system, keeping one of the main feature of the neuron: its excitability. We present then examples of phase diagram and bifurcation analysis for the Hodgin-Huxley equatio…
Inferring directional couplings from the spike data of networks is desired in various scientific fields such as neuroscience. Here, we apply a recently proposed objective procedure to the spike data obtained from the Hodgkin--Huxley type models and in vitro neuronal networks cultured in a circular structure. As a result, we succeed in reconstructing synaptic connections accurately from the evoked activity as well as the spontaneous one. To obtain the results, we invent an analytic formula approximately implementing a method of screening relevant couplings. This significantly reduces the comput…
The activity of an isolated and auto-connected neuron is studied using Hodgkin--Huxley and Integrate-and-Fire frameworks. Main ingredients of the modeling are the auto-stimulating autaptic current observed in experiments, with a spontaneous synaptic liberation noise and a calcium--dependent negative feedback mechanism. The distributions of inter-spikes intervals and burst durations are analytically calculated, and show a good agreement with experimental data.
The dynamics of a multiplex heterogeneous network of oscillators is studied. Two types of similar models based on the Hodgkin-Huxley formalism are used as the basic elements of the networks. The first type model demonstrates bursting oscillations. The second model demonstrates bistability between bursting attractor and stable steady state. Basin of attraction of the stable equilibrium in the model is very small. Bistabilty is a result taking into account an additional ion channel, which has a non-monotonic characteristic and can be interpreted as a channel with a communication defect. Suggeste…
Noise-delayed decay (NDD) phenomenon emerges when the first-spike latency of a periodically forced stochastic neuron exhibits a maximum for a particular range of noise intensity. Here, we investigate the latency response dynamics of a single Hodgkin-Huxley neuron that is subject to both a suprathreshold periodic stimulus and a background activity arriving through dynamic synapses. We study the first spike latency response as a function of the presynaptic firing rate f. This constitutes a more realistic scenario than previous works, since f provides a suitable biophysically realistic parameter …
Synchronization is studied in a spatially-distributed network of weekly-coupled, excitatory neurons of Hodgkin-Huxley type. All neurons are coupled to each other synaptically with a fixed time delay and a coupling strength inversely proportional to the distance between two neurons. We found that a robust, noise-resistant phase clustering state occurred regardless of the initial phase distribution. This has not been shown in previous studies where similar clustering states were found only when the coupling was inhibitory. The spatial distribution of neurons in each synchronous cluster is determ…
