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.

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.

The thalamus is introduced to a recent model of the visual cortex to examine its effect on pattern formation in general and the generation of temporally oscillating patterns in particular. By successively adding more physiological details to a basic corticothalamic model, it is determined which features are responsible for which effects. In particular, with the addition of a thalamic population, several changes occur in the spatiotemporal power spectrum: power increases at resonances of the corticothalamic loop, while the loop acts as a spatiotemporal low-pass filter, and synaptic and dendritic dynamics temporally low-pass filter the activity more generally. Investigation of the effect of altering parameters and gains reveals new parameter regimes where activity that corresponds to hallucinations is induced by both spatially homogeneous and inhomogeneous temporally oscillating modes. This suggests that the thalamus and corticothalamic loops are essential components of a model of oscillating visual hallucinations.

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.