On Laws of Thought-A Quantum-like Machine Learning Approach.

Incorporating insights from quantum theory, we propose a machine learning-based decision-making model, including a logic tree and a value tree; a genetic programming algorithm is applied to optimize both the logic tree and value tree. The logic tree and value tree together depict the entire decision-making process of a decision-maker. We applied this framework to the financial market, and a "machine economist" is developed to study a time series of the Dow Jones index. The "machine economist" will obtain a set of optimized strategies to maximize profits, and discover the efficient market hypothesis (random walk).

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.

Revealing degree distribution of bursting neuron networks.

We present a method to infer the degree distribution of a bursting neuron network from its dynamics. Burst synchronization (BS) of coupled Morris-Lecar neurons has been studied under the weak coupling condition. In the BS state, all the neurons start and end bursting almost simultaneously, while the spikes inside the burst are incoherent among the neurons. Interestingly, we find that the spike amplitude of a given neuron shows an excellent linear relationship with its degree, which makes it possible to estimate the degree distribution of the network by simple statistics of the spike amplitudes. We demonstrate the validity of this scheme on scale-free as well as small-world networks. The underlying mechanism of such a method is also briefly discussed.

Stochastic thermodynamics in mesoscopic chemical oscillation systems.

Stochastic thermodynamics in mesoscopic chemical oscillation systems is discussed on the basis of chemical Langevin equation for the state variables with particular attention paid to a parameter region close to the deterministic Hopf bifurcation. The Langevin dynamics defines stochastic trajectories in the state space and therefore trajectory dependent entropy and entropy production according to the schemes proposed by Udo Seifert (Phys. Rev. Lett. 2005, 95, 040602). The total entropy change along a stochastic trajectory obeys the fluctuation theorems. By using the stochastic normal form theory, we derive explicit theoretical expressions for the mean entropy production in the stationary state. The resulting entropy production in the large system volume V limit can scale linearly or independent with V when the control parameter is above or below the Hopf bifurcation while it is of V(1/2) at the bifurcation. We verify the above relations by direct simulation with a stochastic circadian clock model.

Resonant response of forced complex networks: the role of topological disorder.

We investigate the effect of topological disorder on a system of forced threshold elements, where each element is arranged on top of complex heterogeneous networks. Numerical results indicate that the response of the system to a weak signal can be amplified at an intermediate level of topological disorder, thus indicating the occurrence of topological-disorder-induced resonance. Using mean field method, we obtain an analytical understanding of the resonant phenomenon by deriving the effective potential of the system. Our findings might provide further insight into the role of network topology in signal amplification in biological networks.

Entropy production and fluctuation theorem along a stochastic limit cycle.

Entropy production along a trajectory in the stochastic irreversible Brusselator model of chemical oscillating reactions is discussed. Particular attention is paid to a parameter region near the deterministic supercritical Hopf bifurcation. In the stationary state, detailed fluctuation theorem holds due to the reversibility in the state space, which is verified by direct simulations via Gillespie's algorithm [J. Comput. Phys. 22, 403 (1976); J. Phys. Chem. 81, 2340 (1977)]. In addition, we have considered how the entropy production along a noisy limit cycle depends on the system size. Interestingly, in the large system size limit, the entropy production approaches a constant value when the control parameter stays at the deterministic steady state region, while it increases linearly in the deterministic oscillatory region. Such simulation results can be well understood by a stochastic normal form analysis.

Transition to burst synchronization in coupled neuron networks.

Transition to burst synchronization (BS), where all the neurons start and end bursting simultaneously, has been studied on a diffusively coupled network of Hindmash-Rose bursting neurons. When the coupling strength epsilon is increased from zero, spatiotemporal chaos can be first tamed into one type of BS states with fold-homoclinic bursting, which then undergoes spike-adding and transits into another type of BS states with fold-Hopf bursting. The latter transition takes place via dynamic cluster separation, during which all the neurons with degree k>k(c) change to show fold-Hopf bursting, with epsilonk(c) nearly a constant. A reasonable mechanism behind such phenomena is given by using local mean field approximation, and the role of network topology is also discussed. The case when the neurons are coupled via chemical synapses is also briefly discussed.

Coherence resonance induced by colored noise near Hopf bifurcation.

Effects of colored noise near supercritical Hopf bifurcation, especially noise induced oscillation (NIO) and coherence resonance (CR), have been studied analytically in the Brusselator model, using the stochastic normal form method. Two types of colored noise are considered: one is the standard Gaussian colored noise generated by the Ornstein-Uhlenbeck (OU) process and the other is the so-called power-limited (PL) process. Depending on the noise intensity and noise type, it is found that the autocorrelation time, most probable radius and signal to noise ratio of the NIO may show nontrivial dependencies on the noise correlation time tau(c). Interestingly, for OU-type noise with intensity above a threshold, SNR is a bell-shaped function of tau(c), indicating enhancement of CR by noise correlation; and for PL-type noise, SNR may show double maxima when tau(c) is changed, demonstrating a new kind of multiresonance phenomenon. These theoretical predictions are well reproduced by numerical simulations.

Engineered internal noise stochastic resonator in gene network: a model study.

Based on a genetic bistable switch model coupled with a gene oscillator model, we have constructed a mesoscopic stochastic model for the coupled synthetic gene network, and studied how internal noise would influence the oscillation of such a system. We found that the state-to-state transitions can occur if the internal noise is taken into account, and the performance of resulting oscillation can reach a maximum in a certain internal noise level, which indicates the occurrence of internal noise stochastic resonance (SR) and makes the coupled gene network work as a stochastic resonator. The potential role of such an effect on gene expression systems is also discussed.

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.

Ordering spatiotemporal chaos in complex thermosensitive neuron networks.

We have studied the effect of random long-range connections in chaotic thermosensitive neuron networks with each neuron being capable of exhibiting diverse bursting behaviors, and found stochastic synchronization and optimal spatiotemporal patterns. For a given coupling strength, the chaotic burst-firings of the neurons become more and more synchronized as the number of random connections (or randomness) is increased and, rather, the most pronounced spatiotemporal pattern appears for an optimal randomness. As the coupling strength is increased, the optimal randomness shifts towards a smaller strength. This result shows that random long-range connections can tame the chaos in the neural networks and make the neurons more effectively reach synchronization. Since the model studied can be used to account for hypothalamic neurons of dogfish, catfish, etc., this result may reflect the significant role of random connections in transferring biological information.

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.

Oscillator death on small-world networks.

We have investigated the oscillator death behavior on small-world networks. On one hand, we find that small-world connectivity can eliminate the oscillator death present in the regular lattice. On the other hand, the small-world connectivity can also lead to global oscillator death which is absent in the regular lattice or the completely random network.

Nucleation in scale-free networks.

We have studied nucleation dynamics of the Ising model in scale-free networks whose degree distribution follows a power law with the exponent γ by using the forward flux sampling method and focusing on how the network topology would influence the nucleation rate and pathway. For homogeneous nucleation, the new phase clusters grow from those nodes with smaller degree, while the cluster sizes follow a power-law distribution. Interestingly, we find that the nucleation rate R{Hom} decays exponentially with network size and, accordingly, the critical nucleus size increases linearly with network size, implying that homogeneous nucleation is not relevant in the thermodynamic limit. These observations are robust to the change of γ and are also present in random networks. In addition, we have also studied the dynamics of heterogeneous nucleation, wherein w impurities are initially added either to randomly selected nodes or to targeted ones with the largest degrees. We find that targeted impurities can enhance the nucleation rate R{Het} much more sharply than random ones. Moreover, ln(R{Het}/R{Hom}) scales as w{(γ-2)/(γ-1)} and w for targeted and random impurities, respectively. A simple mean-field analysis is also present to qualitatively illustrate the above simulation results.

Coarse-grained Monte Carlo simulations of the phase transition of the Potts model on weighted networks.

Developing an effective coarse-grained (CG) approach is a promising way for studying dynamics on large size networks. In the present work, we have proposed a strength-based CG (s-CG) method to study critical phenomena of the Potts model on weighted complex networks. By merging nodes with close strengths together, the original network is reduced to a CG network with much smaller size, on which the CG Hamiltonian can be well defined. In particular, we make an error analysis and show that our s-CG approach satisfies the condition of statistical consistency, which demands that the equilibrium probability distribution of the CG model matches that of the microscopic counterpart. Extensive numerical simulations are performed on scale-free networks and random networks, without or with strength correlation, showing that this s-CG approach works very well in reproducing the phase diagrams, fluctuations, and finite-size effects of the microscopic model, while the d-CG approach proposed in our recent work [Phys. Rev. E 82, 011107 (2010)] does not.

Statistically consistent coarse-grained simulations for critical phenomena in complex networks.

We propose a degree-based coarse-graining approach that not just accelerates the evaluation of dynamics on complex networks, but also satisfies the consistency conditions for both equilibrium statistical distributions and nonequilibrium dynamical flows. For the Ising model and susceptible-infected-susceptible epidemic model, we introduce these required conditions explicitly and further prove that they are satisfied by our coarse-grained network construction within the annealed network approximation. Finally, we numerically show that the phase transitions and fluctuations on the coarse-grained network are all in good agreements with those on the original one.

Theoretical study on the effects of internal noise for rate oscillations during CO oxidation on platinum(110) surfaces.

Very recently, the effects of internal molecular noise in mesoscopic chemical reaction systems have gained growing attention. Using a mesoscopic stochastic model, the effect of internal noise for rate oscillation during CO oxidation on Pt(110) surface is studied analytically. In a parameter region outside but close to the supercritical Hopf bifurcation, a stochastic normal form is obtained from the chemical Langevin equation. By stochastic averaging procedure, the system is simplified and solvable. Noise-induced oscillation and internal noise coherent resonance (which is related to an optimal system size), observed from simulations, are well reproduced by the theory. The theoretical analysis helps to clearly figure out when and how the internal noise affects the system's oscillating dynamics.

Internal noise coherent resonance for mesoscopic chemical oscillations: a fundamental study.

The effect of internal noise for a mesoscopic chemical oscillator is studied analytically in a parameter region outside, but close to, the supercritical Hopf bifurcation. By normal form calculation and a stochastic averaging procedure, we obtain stochastic differential equations for the oscillation amplitude r and phase theta that is solvable. Noise-induced oscillation and internal noise coherent resonance, which has been observed in many numerical experiments, are reproduced well by the theory.

Frequency-selective response of FitzHugh-Nagumo neuron networks via changing random edges.

We consider a network of FitzHugh-Nagumo neurons; each neuron is subjected to a subthreshold periodic signal and independent Gaussian white noise. The firing pattern of the mean field changes from an internal-scale dominant pattern to an external-scale dominant one when more and more edges are added into the network. We find numerically that (a) this transition is more sensitive to random edges than to regular edges, and (b) there is a saturation length for random edges beyond which the transition is no longer sharpened. The influence of network size is also investigated.

Effects of internal noise for calcium signaling in a coupled cell system.

We have studied the collective behavior of calcium signaling in a linear array of coupled cells, taking into account the influence of internal noise. For a single cell, the performance of the stochastic calcium oscillations shows two maxima as a function of the cell size, indicating the occurrence of system size bi-resonance. When the cells are coupled together, we find that the increase of the coupling strength or the number of cells can considerably enhance the first peak, but decrease the second one. The occurrence of the bi-resonance and the distinct dependence on the coupling are shown to be relevant with regard to the system's deterministic bifurcation features.