Bain and Nishiyama-Wassermann transition path separation in the martensitic transitions of Fe.

The importance of martensitic transformations has led to tremendous efforts to explore the microscopic martensitic transition paths. There are five possible transformation paths (for γ → α transition) known for Fe at present, and at an arbitrary activation energy, any of the five paths might be followed. It then becomes considerably difficult to monitor the microscopic phase transition mechanism in experiments. Therefore, it is helpful to realize only one of the paths in a physical process. Based on first-principles calculations, we show that at suitable activation energies the Nishiyama-Wassermann (N-W) transformation path can be realized without the involvement of the Bain path, since the condition E NW(θ) < E < E Bain can be satisfied by pure Fe. E is the activation energy of the system, and E NW(θ) and E Bain are the energy barriers for the N-W and Bain transformations, respectively. In particular, the potential energy surface (PES) for the N-W transformation has been calculated as being four-dimensional, i.e., E = E(a,b,c,θ), where (a, b, c) are the lattice constants and θ is the shear angle involved in the shear distortion of the N-W path.

Multistability in a neuron model with extracellular potassium dynamics.

Experiments show a primary role of extracellular potassium concentrations in neuronal hyperexcitability and in the generation of epileptiform bursting and depolarization blocks without synaptic mechanisms. We adopt a physiologically relevant hippocampal CA1 neuron model in a zero-calcium condition to better understand the function of extracellular potassium in neuronal seizurelike activities. The model neuron is surrounded by interstitial space in which potassium ions are able to accumulate. Potassium currents, Na{+}-K{+} pumps, glial buffering, and ion diffusion are regulatory mechanisms of extracellular potassium. We also consider a reduced model with a fixed potassium concentration. The bifurcation structure and spiking frequency of the two models are studied. We show that, besides hyperexcitability and bursting pattern modulation, the potassium dynamics can induce not only bistability but also tristability of different firing patterns. Our results reveal the emergence of the complex behavior of multistability due to the dynamical [K{+}]{o} modulation on neuronal activities.

Law of mass action, detailed balance, and the modeling of calcium puffs.

Using deterministic-stochastic simulations we show that for intracellular calcium puffs the mixing assumption for reactants does not hold within clusters of receptor channels. Consequently, the law of mass action does not apply and useful definitions of averaged calcium concentrations in the cluster are not obvious. Effective reaction kinetics can be derived, however, by separating concentrations for self-coupling of channels and coupling to different channels, thus eliminating detailed balance in the reaction scheme. A minimal Markovian model can be inferred, describing well calcium puffs in neuronal cells and allowing insight into the functioning of calcium puffs.

Calcium domains around single and clustered IP3 receptors and their modulation by buffers.

We study Ca(2+) release through single and clustered IP(3) receptor channels on the ER membrane under presence of buffer proteins. Our computational scheme couples reaction-diffusion equations and a Markovian channel model and allows our investigating the effects of buffer proteins on local calcium concentrations and channel gating. We find transient and stationary elevations of calcium concentrations around active channels and show how they determine release amplitude. Transient calcium domains occur after closing of isolated channels and constitute an important part of the channel's feedback. They cause repeated openings (bursts) and mediate increased release due to Ca(2+) buffering by immobile proteins. Stationary domains occur during prolonged activity of clustered channels, where the spatial proximity of IP(3)Rs produces a distinct [Ca(2+)] scale (0.5-10 microM), which is smaller than channel pore concentrations (>100 microM) but larger than transient levels. While immobile buffer affects transient levels only, mobile buffers in general reduce both transient and stationary domains, giving rise to Ca(2+) evacuation and biphasic modulation of release amplitude. Our findings explain recent experiments in oocytes and provide a general framework for the understanding of calcium signals.

Puff-wave transition in an inhomogeneous model for calcium signals.

In many cell types, calcium ion channels on the endoplasmic reticulum membrane occur in a clustered distribution. The channels generate either localized puffs, each comprising channels of only one cluster, or global calcium waves. In this work we model the calcium system as a two-dimensional lattice of active elements distributed regularly in an otherwise passive space. We address an important feature of the puff-wave transition, which is the difference in lifetime of puffs at a few hundred milliseconds and long-lived global waves with periods of several seconds. We show that such a lifetime difference between puffs and waves can be understood with strongly reduced ordinary differential equations modified by a time-scale factor that takes into account the coupling strength of active and passive regions determined by the Ca2+ diffusion coefficient. Furthermore, we show that the point model can also describe very well the dependence of Ca2+ oscillation characteristics on the cluster-cluster distance in the case of large diffusivity.

An investigation of models of the IP3R channel in Xenopus oocyte.

We consider different models of inositol 1,4,5-trisphosphate (IP(3)) receptor (IP(3)R) channels in order to fit nuclear membrane patch clamp data of the stationary open probability, mean open time, and mean close time of channels in the Xenopus oocyte. Our results indicate that rather than to treat the tetrameric IP(3)R as four independent and identical subunits, one should assume sequential binding-unbinding processes of Ca(2+) ions and IP(3) messengers. Our simulations also favor the assumption that a channel opens through a conformational transition from a close state to an active state.

Global noise and oscillations in clustered excitable media.

We study the effects of global noise on waves in heterogeneous, spatially clustered, reaction-diffusion systems with possible applications to calcium signaling. We first discuss how clustering of the excitability determines the dynamics by shifting bifurcation points and creating new oscillatory solutions. We then consider the specific situation, where intrinsic noise, due to the smallness of the excitable patches, destroys the global oscillatory state. We show that additional small global fluctuations, however, can partially restore temporal and spatial coherence of the oscillatory signal.

Hybrid stochastic and deterministic simulations of calcium blips.

Intracellular calcium release is a prime example for the role of stochastic effects in cellular systems. Recent models consist of deterministic reaction-diffusion equations coupled to stochastic transitions of calcium channels. The resulting dynamics is of multiple time and spatial scales, which complicates far-reaching computer simulations. In this article, we introduce a novel hybrid scheme that is especially tailored to accurately trace events with essential stochastic variations, while deterministic concentration variables are efficiently and accurately traced at the same time. We use finite elements to efficiently resolve the extreme spatial gradients of concentration variables close to a channel. We describe the algorithmic approach and we demonstrate its efficiency compared to conventional methods. Our single-channel model matches experimental data and results in intriguing dynamics if calcium is used as charge carrier. Random openings of the channel accumulate in bursts of calcium blips that may be central for the understanding of cellular calcium dynamics.

The dynamics of small excitable ion channel clusters.

Through computational modeling we predict that small sodium ion channel clusters on small patches of membrane can encode electric signals most efficiently at certain magic cluster sizes. We show that this effect can be traced back to algebraic features of small integers and are universal for channels with a simple gating dynamics. We further explore physiologic conditions under which such effects can occur.

Entropically enhanced excitability in small systems.

We consider the dynamics of small excitable systems, ubiquitous in physics, chemistry, and biology. Spontaneous excitation rates induced by system-size fluctuations exhibit sharp maxima at multiple, small system sizes at which also the system's response to external perturbations is strongly enhanced. This novel effect is traced back to algebraic features of small integers and thus generic.

A secure communication scheme based on the phase synchronization of chaotic systems.

Phase synchronization of chaotic systems with both weak and strong couplings has recently been investigated extensively. Similar to complete synchronization, this type of synchronization can also be applied in secure communications. We develop a digital secure communication scheme that utilizes the instantaneous phase as the signal transmitted from the drive to the response subsystems. Simulation results show that the scheme is difficult to be broken by some traditional attacks. Moreover, it operates with a weak positive conditional Lyapunov exponent in the response subsystem.

Selection of intracellular calcium patterns in a model with clustered Ca2+ release channels.

A two-dimensional model is proposed for intracellular Ca2+ waves, which incorporates both the discrete nature of Ca2+ release sites in the endoplasmic reticulum membrane and the stochastic dynamics of the clustered inositol 1,4,5-triphosphate (IP3) receptors. Depending on the Ca2+ diffusion coefficient and concentration of IP3, various spontaneous Ca2+ patterns, such as calcium puffs, local waves, abortive waves, global oscillation, and tide waves, can be observed. We further investigate the speed of the global waves as a function of the IP3 concentration and the Ca2+ diffusion coefficient and under what conditions the spatially averaged Ca2+ response can be described by a simple set of ordinary differential equations.

Optimal ion channel clustering for intracellular calcium signaling.

Ion channels and receptors in the cell membranes and internal membranes are often distributed in discrete clusters. One particularly well-studied example is the distribution of inositol 1,4,5-triphosphate receptors in the plasma membrane that controls the flux of Ca2+ from the endoplasmic reticulum into the cytosol. By using mathematical modeling, we show that channel clustering can enhance the cell's Ca2+ signaling capability. Furthermore, we predict optimal signaling cellular capability at cluster sizes and distances that agree with experimentally found values in Xenopus oocyte.

Optimal intracellular calcium signaling.

In many cell types, calcium is released from internal stores through calcium release channels upon external stimulation (e.g., pressure or receptor binding). These channels are clustered with a typical cluster size of about 20 channels, generating stochastic calcium puffs. We find that the clustering of the release channels in small clusters increases the sensitivity of the calcium response, allowing for coherent calcium responses at signals to which homogeneously distributed channels would not respond.

Phase synchronization in coupled chaotic oscillators with time delay.

The phase synchronization (PS) of two Rössler oscillators with time-delayed signal coupling is studied. We find that time delay can always lead to PS even when the delay is very long. Moreover, with the increase of time delay, the coupling strength at the transition to PS undergoes a nearly periodic wave distribution. At some fixed time-delayed signal coupling, a PS region is followed by a non-PS region when the coupling strength increases. However, an increase of the coupling leads to the PS state again. This phenomenon occurs in systems with a relatively large PS transition point.

n:m phase synchronization with mutual coupling phase signals.

We generalize the n:m phase synchronization between two chaotic oscillators by mutual coupling phase signals. To characterize this phenomenon, we use two coupled oscillators to demonstrate their phase synchronization with amplitudes practically noncorrelated. We take the 1:1 phase synchronization as an example to show the properties of mean frequencies, mean phase difference, and Lyapunov exponents at various values of coupling strength. The phase difference increases with 2pi phase slips below the transition. The scaling rules of the slip near and away from the transition are studied. Furthermore, we demonstrate the transition to a variety of n:m phase synchronizations and analyze the corresponding coupling dynamics. (c) 2002 American Institute of Physics.

Positive Lyapunov exponents calculated from time series of strange nonchaotic attractors.

Time-series methods for estimating Lyapunov exponents may give a positive exponent when they are applied to the time series of strange nonchaotic systems. Strange nonchaotic systems are characterized by expanding and contracting regions in phase space that result in repeatedly expanding or contracting trajectories. Using time-series methods, the maximum time-series Lyapunov exponent is calculated as an average of the locally most expanding exponents that characterize the divergence of nearby trajectories following a reconstructed attractor over time. A positive exponent is reported by time-series methods for trajectories in an expanding region. While in a converging region, the most expanding dynamics are related to the quasiperiodic driving force. Statistically, a zero exponent related to the quasiperiodic force is obtained through time-series methods within converging regions. As a result, the calculated maximum Lyapunov exponent is positive.

Intermittent phase synchronization of coupled spatiotemporal chaotic systems.

Phase synchronization is studied with a discrete system formed by two coupled map lattices, in which phases are measured in two-dimensional vectors. Simulation results show that by imposing external coupling between the two lattices, phase synchronization can be found in all two-dimensional phase planes between them. When the system is approaching the phase synchronizing state, unstable phase synchronization is observed. This is referred to as intermittent phase synchronization that appears when the trajectories on two interacting phase planes have opposite directions of rotation but with only a small phase difference. The intermittent phase synchronization could also be observed in coupled autonomous systems with diffusive attractors although their phase concepts are inconsistent. Our results show that the intermittent phase synchronization of both discrete and autonomous systems relates to the diffusion or the complexity of the attractors.

Phase signal coupling induced n:m phase synchronization in drive-response oscillators.

We have studied phase synchronization between two identical Rössler oscillators connected in the drive-response configuration by a single phase signal. Before the transition to phase synchronization, the distribution of the time interval between consecutive 2pi jumps shows several sharp peaks. With a strong phase signal coupling, the n:m phase synchronization between the oscillators can be achieved. For the n (not equal) m phase synchronizing state, some values of coupling strength result in a phenomenon characterized by a reduction in the mean amplitude of the response termed amplitude reduction. In these regions, the mean rotation speed of the response remains approximately constant while the locking ratio n:m varies.

Simple approach to the creation of a strange nonchaotic attractor in any chaotic system.

A simple approach to the creation of a strange nonchaotic attractor in any chaotic system is described. The main idea is to control the parameter of the system in such a manner that the system dynamics is expanding at some times, but converging at others. With this approach, a strange nonchaotic attractor can be found in a large region in the parameter space near the boundaries between chaotic and regular phases or within the chaotic region far from the regular one. The maximum nontrivial Lyapunov exponent of the system can pass through zero nonsmoothly and cannot be fitted by a linear function.