Slow coherent spontaneous fluctuations (<0.1 Hz) in functional magnetic resonance imaging blood-oxygen-level-dependent signals have been observed for a resting state of the human brain. In this paper, considering feed-forward inhibition in addition to excitation between brain areas, which we assume to be in up (active) or down (quiescent) states, we propose a model for the generation and organization of the slow fluctuations. Connectivity with feed-forward excitation and inhibition between the areas makes the system have multiple stable states and organized slow fluctuations manifest as noise-induced slow transitions between the states. With various connectivities, we observe slow fluctuations and various organizations, including anticorrelated clusters, through numerical simulations.

The notion of a toric origami manifold, which weakens the notion of a symplectic toric manifold, was introduced by A. Cannas da Silva, V. Guillemin and A.R. Pires. They showed that toric origami manifolds bijectively correspond to origami templates via moment maps, where an origami template is a collection of Delzant polytopes with some folding data. Like a fan is associated to a Delzant polytope, a multi-fan introduced by A. Hattori and M. Masuda can be associated to an oriented origami template. In this paper, we discuss their relationship and show that any simply connected compact smooth 4-manifold with a smooth action of $T^2$ can be a toric origami manifold. We also characterize products of even dimensional spheres which can be toric origami manifolds.

When the cohomology ring of a generalized Bott manifold with $\mathbb{Q}$-coefficient is isomorphic to that of a product of complex projective spaces $\mathbb{C}P^{n_i}$ , the generalized Bott manifold is said to be $\mathbb{Q}$-trivial. We find a necessary and sufficient condition for a generalized Bott manifold to be $\mathbb{Q}$-trivial. In particular, every $\mathbb{Q}$-trivial generalized Bott manifold is diffeomorphic to a $\prod_{n_i>1} \mathbb{C}P^{n_i}$ -bundle over a $\mathbb{Q}$-trivial Bott manifold.

Many motile bacteria display wiggling trajectories, which correspond to helical swimming paths. Wiggling trajectories result from flagella pushing off-axis relative to the cell body and making the cell wobble. The spatial extent of wiggling trajectories is controlled by the swimming velocity and flagellar torque, which leads to rotation of the cell body. We employ the method of regularized Stokeslets to investigate the wiggling trajectories produced by flagellar bundles, which can form at many locations and orientations relative to the cell body for peritrichously flagellated bacteria.Modeling the bundle as a rigid helix with fixed position and orientation relative to the cell body, we show that the wiggling trajectory depends on the position and orientation of the flagellar bundle relative to the cell body. We observe and quantify the helical wiggling trajectories of Bacillus subtilis, which show a wide range of trajectory pitches and radii, many with pitch larger than 4 μm. For this bacterium, we show that flagellar bundles with fixed orientation relative to the cell body are unlikely to produce wiggling trajectories with pitch larger than 4 μm. An estimate based on torque-balance shows that this constraint on pitch is a result of the large torque exerted by the flagellar bundle. On the other hand, multiple rigid bundles with fixed orientation, similar to those recently observed experimentally, are able to produce wiggling trajectories with large pitches.

We discuss hysteretic behaviors of dilute viscoelastic polymeric fuids with moment-closure approximation approach in extensional/enlongational fows. Polymeric fuids are modeled by the nite extensible nonlinear elastic (FENE) spring dumbbell model. Hysteresis is one of key features to describe FENE model. We here investigate the hysteretic behavior of FENE-D model introduced in [Y. Hyon et al., Multiscale Model. Simul., 7(2008), pp.978-1002]. The FENE-D model is established from a special equilibrium solution of the Fokker-Planck equation to catch extreme behavior of FENE model in large extensional fow rates. Since the hysteresis of FENE model can be seen during a relaxation in simple extensional fow employing the normal stress/the elongational viscosity versus the mean-square extension, we simulate FENED in simple extensional fows to investigate its hysteretic behavior comparing to FENE-P, FENE-L [G. Lielens et al., J. Non-Newtonian Fluid Mech., 76(1999), pp.249{279]. The FENE-P is a well-known pre-averaged approximated model, and it shows a good agreement to macroscopic induced stresses. However, FENE-P does not catch any hysteretic phenomenon. In contrast, the FENE-L shows a better hysteretic behavior than the other models to FENE, but it has a limitation for macroscopic induced stresses in large shear rates. On the other hand, FENE-D presents a good agreement to macroscopic induced stresses even in large shear rates, and moreover, it shows a hysteretic phenomenon in certain large fow rates.

We introduce a mathematical model to study the transport of ions through ion channels. The system is derived in the frame of the energetic variational approaches, taking into account the coupling between electrostatics, diffusion, and protein (ion channel) structure. The geometric constraints of the ion channel are introduced through a potential energy controlling the localmaximum volume inside the ion channel. A diffusive interface (labeling) deion is also employed to describe the geometric configuration of the channels. The surrounding bath and channel are smoothly connected with the antechamber region by this label function. A corresponding modified Poisson–Nernst–Planck channel system for ion channels is derived using the variational derivatives of the total energy functional. The functional consists of the entropic free energy for diffusion of the ions, the electrostatic potential energy, the repulsive potential energy for the excluded volume effect of the ion particles, and the potential energy for the geometric constraints of the ion channel. For the biological application of such a system,we consider channel recordings of voltage clamp tomeasure the current flowing through the ion channel. The results of one-dimensional numerical simulations are presented to demonstrate some signature effects of the channel, such as the current output produced by single-step and double-step voltage inputs.