Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter.Comment: 40 pages, 13 figure

Intradomain phase transitions in flexible block copolymers with self-aligning segments

We study a model of flexible block copolymers (BCPs) in which there is an enlthalpic preference for local alignment, among like-block segments. We describe a generalization of the self-consistent field theory (SCFT) of flexible BCPs to include inter-segment orientational interactions via a Landau-DeGennes free energy associated with a polar or nematic order parameter for segments of one component of a diblock copolymer. We study the equilibrium states of this model numerically, using a pseudo-spectral approach to solve for chain conformation statistics in the presence of a self-consistent torque generated by inter-segment alignment forces. Applying this theory to the structure of lamellar domains composed of symmetric diblocks possessing a single block of "self-aligning", polar segments, we show the emergence of spatially complex segment order parameters (segment director fields) within a given lamellar domain. Because BCP phase separation gives rise to spatially inhomogeneous orientation order of segments even in the absence of explicit intra-segment aligning forces, the director fields of BCPs, as well as thermodynamics of lamellar domain formation, exhibit a highly non-linear dependence on both the inter-block segregation ($\chi N$) and the enthalpy of alignment ($\varepsilon$). Specifically, we predict the stability of new phases of lamellar order in which distinct regions of alignment coexist within a single mesodomain, and which spontaneously break the symmetry of the lamella pattern of composition in the melt via in-plane tilt of the director in the centers of the like-composition domains. We show further that, in analogy to a Freedericksz transition in confined nematics, that the elastic costs to reorient segments within the domain, as described by Frank elasticity, increase the threshold value $\varepsilon$ needed to induce this intra-domain phase transition

The role of ongoing dendritic oscillations in single-neuron dynamics

The dendritic tree contributes significantly to the elementary computations a neuron performs while converting its synaptic inputs into action potential output. Traditionally, these computations have been characterized as temporally local, near-instantaneous mappings from the current input of the cell to its current output, brought about by somatic summation of dendritic contributions that are generated in spatially localized functional compartments. However, recent evidence about the presence of oscillations in dendrites suggests a qualitatively different mode of operation: the instantaneous phase of such oscillations can depend on a long history of inputs, and under appropriate conditions, even dendritic oscillators that are remote may interact through synchronization. Here, we develop a mathematical framework to analyze the interactions of local dendritic oscillations, and the way these interactions influence single cell computations. Combining weakly coupled oscillator methods with cable theoretic arguments, we derive phase-locking states for multiple oscillating dendritic compartments. We characterize how the phase-locking properties depend on key parameters of the oscillating dendrite: the electrotonic properties of the (active) dendritic segment, and the intrinsic properties of the dendritic oscillators. As a direct consequence, we show how input to the dendrites can modulate phase-locking behavior and hence global dendritic coherence. In turn, dendritic coherence is able to gate the integration and propagation of synaptic signals to the soma, ultimately leading to an effective control of somatic spike generation. Our results suggest that dendritic oscillations enable the dendritic tree to operate on more global temporal and spatial scales than previously thought

An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patterns

This paper presents an introduction to phase transitions and critical phenomena on the one hand, and nonequilibrium patterns on the other, using the Ginzburg-Landau theory as a unified language. In the first part, mean-field theory is presented, for both statics and dynamics, and its validity tested self-consistently. As is well known, the mean-field approximation breaks down below four spatial dimensions, where it can be replaced by a scaling phenomenology. The Ginzburg-Landau formalism can then be used to justify the phenomenological theory using the renormalization group, which elucidates the physical and mathematical mechanism for universality. In the second part of the paper it is shown how near pattern forming linear instabilities of dynamical systems, a formally similar Ginzburg-Landau theory can be derived for nonequilibrium macroscopic phenomena. The real and complex Ginzburg-Landau equations thus obtained yield nontrivial solutions of the original dynamical system, valid near the linear instability. Examples of such solutions are plane waves, defects such as dislocations or spirals, and states of temporal or spatiotemporal (extensive) chaos

Periodic solutions to a mean-field model for electrocortical activity

We consider a continuum model of electrical signals in the human cortex, which takes the form of a system of semilinear, hyperbolic partial differential equations for the inhibitory and excitatory membrane potentials and the synaptic inputs. The coupling of these components is represented by sigmoidal and quadratic nonlinearities. We consider these equations on a square domain with periodic boundary conditions, in the vicinity of the primary transition from a stable equilibrium to time-periodic motion through an equivariant Hopf bifurcation. We compute part of a family of standing wave solutions, emanating from this point.Comment: 9 pages, 5 figure

Discrete Breathers in Two-Dimensional Anisotropic Nonlinear Schrodinger lattices

We study the structure and stability of discrete breathers (both pinned and mobile) in two-dimensional nonlinear anisotropic Schrodinger lattices. Starting from a set of identical one-dimensional systems we develop the continuation of the localized pulses from the weakly coupled regime (strongly anisotropic) to the homogeneous one (isotropic). Mobile discrete breathers are seen to be a superposition of a localized mobile core and an extended background of two-dimensional nonlinear plane waves. This structure is in agreement with previous results on onedimensional breather mobility. The study of the stability of both pinned and mobile solutions is performed using standard Floquet analysis. Regimes of quasi-collapse are found for both types of solutions, while another kind of instability (responsible for the discrete breather fission) is found for mobile solutions. The development of such instabilities is studied, examining typical trajectories on the unstable nonlinear manifold.Comment: 13 pages, 9 figure