Double-system-size resonance for spiking activity of coupled Hodgkin-Huxley neurons

Chemical Langevin Equation for Complex Reactions

The stochastic dynamical behaviors of an elementary reaction system can be investigated by the chemical Langevin equation (CLE). However, most of the reactions in engineering belong to complex reactions. It is not appropriate to describe the random evolution process of a complex chemical reaction system by directly using CLE because the foundation of the deviation of CLE is the equilibrium equation of the number of molecules in elementary reaction systems. In the study, the chemical Langevin equation for complex reactions (CLE-CR) is proposed based on the random process theory by introducing the extent of reactions to express the reaction rates of complex reactions. The reaction rates of complex reactions are regarded as some random variables following Poisson distribution. To illustrate the essential consistency of CLE-CR and CLE, the physical meaning of the propensity function in CLE is comprehensively discussed. A numerical example from chemical engineering is employed to demonstrate the effectiveness of CLE-CR and the solving procedure. The results show that CLE-CR can be conveniently applied into engineering to investigate the stochastic dynamical behaviors of complex reaction systems, giving the probabilistic information of the concentration evolution of chemical constituents.

Implementation of dynamic dual input multiple output logic gate via resonance in globally coupled Duffing oscillators

We have used a system of globally coupled double-well Duffing oscillators under an enhanced resonance condition to design and implement Dual Input Multiple Output (DIMO) logic gates. In order to enhance the resonance, the first oscillator in the globally coupled system alone is excited by two forces out of which one acts as a driving force and the other will be either sub-harmonic or super-harmonic in nature. We report that for an appropriate coupling strength, the second force coherently drives and enhances not only the amplitude of the weak first force to all the coupled systems but also drives and propagates the digital signals if any given to the first system. We then numerically confirm the propagation of any digital signal or square wave without any attenuation under an enhanced resonance condition for an amplitude greater than a threshold value. Further, we extend this idea for computing various logical operations and succeed in designing theoretically DIMO logic gates such as AND/NAND, OR/NOR gates with globally coupled systems.

Propagation of Collective Temporal Regularity in Noisy Hierarchical Networks

Neuronal communication between different brain areas is achieved in terms of spikes. Consequently, spike-time regularity is closely related to many cognitive tasks and timing precision of neural information processing. A recent experiment on primate parietal cortex reports that spike-time regularity increases consistently from primary sensory to higher cortical regions. This observation conflicts with the influential view that spikes in the neocortex are fundamentally irregular. To uncover the underlying network mechanism, we construct a multilayered feedforward neural information transmission pathway and investigate how spike-time regularity evolves across subsequent layers. Numerical results reveal that despite the obviously irregular spiking patterns in previous several layers, neurons in downstream layers can generate rather regular spikes, which depends on the network topology. In particular, we find that collective temporal regularity in deeper layers exhibits resonance-like behavior with respect to both synaptic connection probability and synaptic weight, i.e., the optimal topology parameter maximizes the spike-timing regularity. Furthermore, it is demonstrated that synaptic properties, including inhibition, synaptic transient dynamics, and plasticity, have significant impacts on spike-timing regularity propagation. The emergence of the increasingly regular spiking (RS) patterns in higher parietal regions can, thus, be viewed as a natural consequence of spiking activity propagation between different brain areas. Finally, we validate an important function served by increased RS: promoting reliable propagation of spike-rate signals across downstream layers.

Accurate Langevin approaches to simulate Markovian channel dynamics

The stochasticity of ion-channels dynamic is significant for physiological processes on neuronal cell membranes. Microscopic simulations of the ion-channel gating with Markov chains can be considered to be an accurate standard. However, such Markovian simulations are computationally demanding for membrane areas of physiologically relevant sizes, which makes the noise-approximating or Langevin equation methods advantageous in many cases. In this review, we discuss the Langevin-like approaches, including the channel-based and simplified subunit-based stochastic differential equations proposed by Fox and Lu, and the effective Langevin approaches in which colored noise is added to deterministic differential equations. In the framework of Fox and Lu's classical models, several variants of numerical algorithms, which have been recently developed to improve accuracy as well as efficiency, are also discussed. Through the comparison of different simulation algorithms of ion-channel noise with the standard Markovian simulation, we aim to reveal the extent to which the existing Langevin-like methods approximate results using Markovian methods. Open questions for future studies are also discussed.

The what and where of adding channel noise to the Hodgkin-Huxley equations

Conductance-based equations for electrically active cells form one of the most widely studied mathematical frameworks in computational biology. This framework, as expressed through a set of differential equations by Hodgkin and Huxley, synthesizes the impact of ionic currents on a cell's voltage--and the highly nonlinear impact of that voltage back on the currents themselves--into the rapid push and pull of the action potential. Later studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations or their counterparts. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the equations of Hodgkin-Huxley type. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic equations of Hodgkin-Huxley type as well as to more modern models of ion channel dynamics. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly MATLAB simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.

Delay-enhanced spatiotemporal order in coupled neuronal systems

In a network of noisy neuron oscillators with time-delayed coupling, we uncover a phenomenon of delay-enhanced spatiotemporal order. We find that time delay in the coupling can dramatically enhance the temporal coherence and spatial synchrony of the noise-induced spike trains. In addition, if the delay time is tuned to nearly match the intrinsic spiking period of the neuronal network, both the coherence and the synchrony reach maximum levels, demonstrating an interesting type of resonance phenomenon with delay. Such findings are shown to be robust to the change of the noise intensity and the rewiring probability of small-world network.

Timing cellular decision making under noise via cell-cell communication

Many cellular processes require decision making mechanisms, which must act reliably even in the unavoidable presence of substantial amounts of noise. However, the multistable genetic switches that underlie most decision-making processes are dominated by fluctuations that can induce random jumps between alternative cellular states. Here we show, via theoretical modeling of a population of noise-driven bistable genetic switches, that reliable timing of decision-making processes can be accomplished for large enough population sizes, as long as cells are globally coupled by chemical means. In the light of these results, we conjecture that cell proliferation, in the presence of cell-cell communication, could provide a mechanism for reliable decision making in the presence of noise, by triggering cellular transitions only when the whole cell population reaches a certain size. In other words, the summation performed by the cell population would average out the noise and reduce its detrimental impact.

Evaluation of stochastic differential equation approximation of ion channel gating models

Fox and Lu derived an algorithm based on stochastic differential equations for approximating the kinetics of ion channel gating that is simpler and faster than "exact" algorithms for simulating Markov process models of channel gating. However, the approximation may not be sufficiently accurate to predict statistics of action potential generation in some cases. The objective of this study was to develop a framework for analyzing the inaccuracies and determining their origin. Simulations of a patch of membrane with voltage-gated sodium and potassium channels were performed using an exact algorithm for the kinetics of channel gating and the approximate algorithm of Fox & Lu. The Fox & Lu algorithm assumes that channel gating particle dynamics have a stochastic term that is uncorrelated, zero-mean Gaussian noise, whereas the results of this study demonstrate that in many cases the stochastic term in the Fox & Lu algorithm should be correlated and non-Gaussian noise with a non-zero mean. The results indicate that: (i) the source of the inaccuracy is that the Fox & Lu algorithm does not adequately describe the combined behavior of the multiple activation particles in each sodium and potassium channel, and (ii) the accuracy does not improve with increasing numbers of channels.

Detection of subthreshold pulses in neurons with channel noise

Neurons are subject to various kinds of noise. In addition to synaptic noise, the stochastic opening and closing of ion channels represents an intrinsic source of noise that affects the signal-processing properties of the neuron. We study the response of a stochastic Hodgkin-Huxley neuron to transient input subthreshold pulses. It is found that the average response time decreases but variance increases as the amplitude of channel noise increases. In the case of single-pulse detection, we show that channel noise enables one neuron to detect the subthreshold signals and an optimal membrane area (or channel noise intensity) exists for a single neuron to achieve optimal performance. However, the detection ability of a single neuron is limited by large errors. Here, we test a simple neuronal network that can enhance the pulse-detecting abilities of neurons and find that dozens of neurons can perfectly detect subthreshold pulses. The phenomenon of intrinsic stochastic resonance is also found at both the level of single neurons and the level of networks. At the network level, the detection ability of networks can be optimized for the number of neurons comprising the network.

Enhancement of stochastic oscillations by colored noise or internal noise in NO reduction by CO on small platinum surfaces

In this paper, based on the stochastic model of NO reduction by CO on Pt crystal surfaces and taking Gaussian colored noise as external fluctuations of the NO partial pressure, we study the effect of the colored noise on the internal noise-induced stochastic oscillations (INSOs) and the effect of internal noise on the colored noise-induced stochastic oscillations (CNSOs). It is found that the INSO can be enhanced by the colored noise with appropriate correlation time or noise strength and, interestingly, the CNSO can be enhanced by the internal noise as well and, moreover, the enhanced CNSO can reach the best oscillatory states repetitively via proper internal noises. This effect of the internal noise is different from its effect on the stochastic oscillations induced by the external Gaussian white noise, which probably results from the interaction of the correlated colored noise and the internal noise.

Proximity to periodic windows in bifurcation diagrams as a gateway to coherence resonance in chaotic systems

We show that chaotic states situated in the proximity of periodic windows in bifurcation diagrams are eligible for the observation of coherence resonance. In particular, additive Gaussian noise of appropriate intensity can enhance the temporal order in such chaotic states in a resonant manner. Results obtained for the logistic map and the Lorenz equations suggest that the presented mechanism of coherence resonance is valid beyond particularities of individual systems. We attribute the findings to the increasing attraction of imminent periodic orbits and the ability of noise to anticipate their existence and use a modified wavelet analysis to support our arguments.

Two system-size-resonance behaviors for calcium signaling: for optimal cell size and for optimal network size

We have studied the collective calcium signaling behavior of an array of coupled N cells, taking into account the internal noises resulting from the small cell size V. The system's performance was characterized by the reciprocal coefficient of variance (RCV) of the calcium spike train. Two system-size resonances were observed, namely, the RCV value shows a clear peak when both N and V are optimal. Therefore, an optimal number of cells of optimal size work the best as a whole.

Enhancement of internal-noise coherence resonance by modulation of external noise in a circadian oscillator

A circadian oscillator driven by external noise and internal noise has been studied by use of the chemical Langevin equation method. When the system is near a Hopf bifurcation and driven by internal noise only, it is found that the coherence resonance phenomenon can be induced by the internal noise. When the system is simultaneously driven by internal and external noise, it is found that external-noise coherence resonance can be suppressed by internal noise, while internal-noise coherence resonance can be enhanced by modulation of the external noise intensity in a certain range of noise intensity. Another interesting result is that the external noise can regulate the optimal system size when the internal-noise coherence resonance occurs.

System-size resonance in a binary attractor neural network

System size resonance (SSR) is a phenomenon in which the response of a system is optimal for a certain finite size, but poorer as the size goes to zero or infinity. In order to show SSR effects in binary attractor neural networks, we study the response of a network, in the ferromagnetic phase, to an external, time-dependent stimulus. Under the presence of such a stimulus, the network shows SSR, as is demonstrated by the measure of the signal amplification both analytically and by simulation.

Role of chemical synapses in coupled neurons with noise

We examine the behavior in the presence of noise of an array of Morris-Lecar neurons coupled via chemical synapses. Special attention is devoted to comparing this behavior with the better known case of electrical coupling arising via gap junctions. In particular, our numerical simulations show that chemical synapses are more efficient than gap junctions in enhancing coherence at an optimal noise (what is known as array-enhanced coherence resonance): in the case of (nonlinear) chemical coupling, we observe a substantial increase in the stochastic coherence of the system, in comparison with (linear) electrical coupling. We interpret this qualitative difference between both types of coupling as arising from the fact that chemical synapses only act while the presynaptic neuron is spiking, whereas gap junctions connect the voltage of the two neurons at all times. This leads in the electrical coupling case to larger correlations during interspike time intervals, which are detrimental to the array-enhanced coherence effect. Finally, we report on the existence of a system-size coherence resonance in this locally coupled system, exhibited by the average membrane potential of the array.

Internal noise stochastic resonance for intracellular calcium oscillations in a cell system

By constructing a mesoscopic stochastic model for intracellular calcium oscillations in a cell system, we have investigated how the internal noise would influence the calcium oscillations of such a system using stochastic simulation methods and chemical Langevin method. It is found that stochastic calcium oscillations appear when the internal noise is considered, while the deterministic model only yields steady state. The performance of such oscillations undergoes a maximum with the variation of the internal noise level, indicating the occurrence of internal noise stochastic resonance. Interestingly, we find that the optimal system size matches well with the real cell size when the control parameter is tuned near the left Hopf bifurcation point, and such a match is robust to the variation of the control parameters.

Amphiphilic ABC triblock copolymer-assisted synthesis of core/shell structured CdTe nanowires

A new type of amphiphilic ABC triblock copolymer, poly(acrylic acid)(33)-poly(styrene)(47)-poly(ethylene oxide)(113) (PAA(33)-PS(47)-PEO(113)), was designed to assist the synthesis of core/shell structured CdTe nanowires via a one-step synthetic route. The PAA block was adopted to capture cadmium ions as the precursor of CdTe. Due to the bivalent coordination of Cd(2+), the copolymer in dioxane/H(2)O formed micelles with Cd(2+)-polychelate cores. Then CdTe nanocrystals were obtained within the micelles after introduction of NaHTe into the micelle solution. Transmission electron microscopy experiments revealed that the CdTe nanocrystals obtained simultaneously formed "pearl-necklace" aggregates in solution possibly driven by dipole interactions between neighboring particles, and then single crystalline CdTe nanowires upon reflux. Accompanying this morphology change, a phase transition from cubic zinc blende to wurtzite structure was observed by selected-area electron diffraction. The aggregation of the PS block in dioxane with a certain amount of H(2)O enabled the PS blocks to form a densely packed shell on the CdTe nanowires whose typical size is 700-800 nm in length and 15-20 nm in width. The third block of PEO was employed to render the finally formed CdTe nanowires dispersibility.