Binocular rivalry is investigated in a continuum model of the primary visual cortex that includes neural excitation and inhibition, stimulus orientation preference, and spike-rate adaptation. Visual stimuli consisting of bars or edges result in localized states of neural activity described by solitons. Stability analysis shows binocular fusion gives way to binocular rivalry when the orientation difference between left-eye and right-eye stimuli destabilizes one or more solitons. The model yields conditions for binocular rivalry, and two types of competitive dynamics are found: either one soliton oscillates between two stimulus regions or two solitons fixed in position at the stimulus regions oscillate out of phase with each other.

The rate of magnetization reversal due to the nucleation of soliton-antisoliton pairs at point-like defects is found for a uniaxial ferromagnet in an applied magnetic field. Point-like defects are considered as local variations in the magnetic anisotropy over a length scale smaller than the domain-wall width. A weak magnetic field applied along the easy axis causes the magnetization to become metastable, and the lowest activation barrier for reversal involves the nucleation of a soliton-antisoliton pair pinned to a point-like defect. Formulas are derived for the activation energy and field of reversal, and the reversal-rate prefactor is calculated using Langer's theory for the decay of a metastable state. As the applied field tends to zero, the lowest activation energy is found to be exactly half that of an unpinned soliton-antisoliton pair, and results from the formation of a spatially nonuniform metastable state when the defect strength become large. The smallest field of reversal is exactly half of the anisotropy field. The reversal-rate prefactor is found to increase with the number of point-like defects but decreases with increase in the defect strength due to a decrease in the activation entropy when translational symmetry is broken by the point-like defects, and soliton-antisoliton pairs become more strongly localized to the pinning sites.

Spike-rate adaptation is investigated within a mean-field model of brain activity. Two different mechanisms of negative feedback are considered; one involving modulation of the mean firing threshold, and the other, modulation of the mean synaptic strength. Adaptation to a constant stimulus is shown to take place for both mechanisms, and limit-cycle oscillations in the firing rate corresponding to bursts of neuronal activity are investigated. These oscillations are found to result from a Hopf bifurcation when the equilibrium lies between the local maximum and local minimum of a given nullcline. Oscillations with amplitudes significantly below the maximum firing rate are found over a narrow range of possible equilibriums.

A graphical technique for finding equilibrium magnetic configurations in exchange coupled multilayer magnetic nanowires is presented. For the case of a two layer wire this technique is used to find two domain wall configurations localized near the interface between the layers. Both configurations are demonstrated to satisfy some important requirements for use in a calculation of the rate of magnetization reversal due to thermal activation. It is described how the graphical technique can be used for other types of multilayer nanowires.

The effects of cortical boundary conditions and resulting modal aspects of continuum corticothalamic electrodynamics are explored, including feedbacks. Dispersion relations, electroencephalographic spectra, and stimulus response functions are calculated from the underlying physiology, and the effects of discrete mode structure are determined. Conditions under which modal effects are important are obtained, along with estimates of the point at which modal series can be truncated, and the limit in which only a single globally uniform mode need be retained. It is found that for physiologically plausible parameters only the lowest cortical spatial eigenmode together with the set of next-lowest modes can produce distinct modal structure in spectra and response functions, and then only at frequencies where corticothalamic resonances reduce dissipation to the point where the spatial eigenmodes are weakly damped. The continuum limit is found to be a good approximation, except at very low frequencies and, under some circumstances, near the alpha resonance. It is argued that the major electroencephalographic rhythms result from corticothalamic feedback resonances, but that cortical modal effects can contribute to weak substructure in the alpha resonance. This mechanism is compared and contrasted with purely cortical and pacemaker-based alternatives and testable predictions are formulated to enable experimental discrimination between these possibilities.

The energy barrier for thermally driven magnetization reversal in a two section nanowire was calculated, based on a mechanism for domain wall nucleation at the interface between sections. The wire was assumed to be cylindrical, uniform in diameter, and consisting of two different types of ferromagnetic material. It was found that the wall either overcomes the barrier and continues through the wire or falls back to its equilibrium position centered in the local minimum.

The existence of visual hallucinations with prominent temporal oscillations is well documented in conditions such as Charles Bonnett Syndrome. To explore these phenomena, a continuum model of cortical activity that includes additional physiological features of axonal propagation and synapto-dendritic time constants, is used to study the generation of hallucinations featuring both temporal and spatial oscillations. A detailed comparison of the physiological features of this model with those of two others used previously in the modeling of hallucinations is made, and differences, particularly regarding temporal dynamics, relevant to pattern formation are analyzed. Linear analysis and numerical calculation are then employed to examine the pattern forming behavior of this new model for two different forms of spatiotemporal coupling between neurons. Numerical calculations reveal an oscillating mode whose frequency depends on synaptic, dendritic, and axonal time constants not previously simultaneously included in such analyses. Its properties are qualitatively consistent with descriptions of a number of physiological disorders and conditions with temporal dynamics, but the analysis implies that corticothalamic effects will need to be incorporated to treat the consequences quantitatively.

Hopfield's Lyapunov function is used to view the stability and topology of equilibria in neuronal networks for visual rivalry and pattern formation. For two neural populations with reciprocal inhibition and slow adaptation, the dynamics of neural activity is found to include a pair of limit cycles: one for oscillations between states where one population has high activity and the other has low activity, as in rivalry, and one for oscillations between states where both populations have the same activity. Hopfield's Lyapunov function is used to find the dynamical mechanism for oscillations and the basin of attraction of each limit cycle. For a spatially continuous population with lateral inhibition, stable equilibria are found for local regions of high activity (solitons) and for bound states of two or more solitons. Bound states become stable when moving two solitons together minimizes the Lyapunov function, a result of decreasing activity in regions between peaks of high activity when the firing rate is described by a sigmoid function. Lowering the barrier to soliton formation leads to a pattern-forming instability, and a nonlinear solution to the dynamical equations is found to be given by a soliton lattice, which is completely characterized by the soliton width and the spacing between neighboring solitons. Fluctuations due to noise create lattice vacancies analogous to point defects in crystals, leading to activity which is spatially inhomogeneous.

Nucleation at the interface between two adjoining regions with dissimilar physical properties is investigated using a model for magnetization reversal of a two-layer ferromagnetic nanowire. Each layer of the nanowire is considered to have a different degree of magnetic anisotropy, representing a hard magnetic layer exchange-coupled to a softer layer. A magnetic field applied along the easy axis causes the softer layer to reverse, forming a domain wall close to the interface. For small applied fields this state is metastable and complete reversal of the nanowire takes place via activation over a barrier. A reversal mechanism involving nucleation at an interface is proposed, whereby a domain wall changes in width as it passes from the soft layer to the hard layer during activation. Langer's statistical theory for the decay of a metastable state is used to derive rates of magnetization reversal, and simple formulas are found in limiting cases for the activation energy, rate of reversal, and critical field at which the metastable state becomes unstable. These formulas depend on the anisotropy difference between each layer, and the behavior of the reversal rate prefactor is interpreted in terms of activation entropy and domain-wall dynamics.