Logarithmic spiral solutions of the Kopell-Howard lambda-omega reaction-diffusion equations.

Our investigation of logarithmic spirals is motivated by disparate experimental results: (i) the discovery of logarithmic spiral shaped precipitate formation in chemical garden experiments. Understanding precipitate formation in chemical gardens is important since analogous precipitates form in deep ocean hydrothermal vents, where conditions may be compatible with the emergence of life. (ii) The discovery that logarithmic spiral shaped waves of spreading depression can spontaneously form and cause macular degeneration in hypoglycemic chick retina. The role of reaction-diffusion mechanisms in spiral formation in these diverse experimental settings is poorly understood. To gain insight, we use the topological shooting to prove the existence of 0-bump stationary logarithmic spiral solutions, and rotating logarithmic spiral wave solutions, of the Kopell-Howard lambda-omega reaction-diffusion model.

Nighttime instabilities of neurophysiological, cardiovascular, and respiratory activity: integrative modeling and preliminary results.

Unstable (cyclical alternating pattern, or CAP) sleep is associated with surges of sympathetic nervous system activity, increased blood pressure and vasoconstriction, heightened baroreflex sensitivity, and unstable heart rhythm and breathing. In susceptible persons, CAP sleep provokes clinically significant events, including hypertensive crises, sleep-disordered breathing, and cardiac arrhythmias. Here we explore the neurophysiology of CAP sleep and its impact on cardiovascular and respiratory functions. We show that: (i) an increase in neurophysiological recovery rate can explain the emergence of slow, self-sustained, hypersynchronized A1 CAP-sleep pattern and its transition to the faster A2-A3 CAP-sleep patterns; (ii) in a two-dimensional, continuous model of cardiac tissue with heterogeneous action potential duration (APD) distribution, heart rate accelerations during CAP sleep may encounter incompletely recovered electrical excitability in cell clusters with longer APD. If the interaction between short cycle length and incomplete, spatially heterogeneous repolarization persists over multiple cycles, irregularities and asymmetry of depolarization front may accumulate and ultimately lead to a conduction block, retrograde conduction, breakup of activation waves, reentrant activity, and arrhythmias; and (iii) these modeling results are consistent with the nighttime data obtained from patients with structural heart disease (N=13) that show clusters of atrial and ventricular premature beats occurring during the periods of unstable heart rhythm and respiration that accompany CAP sleep. In these patients, CAP sleep is also accompanied by delayed adaptation of QT intervals and T-wave alternans.

From baseline to epileptiform activity: a path to synchronized rhythmicity in large-scale neural networks.

In large-scale neural networks in the brain the emergence of global behavioral patterns, manifested by electroencephalographic activity, is driven by the self-organization of local neuronal groups into synchronously functioning ensembles. However, the laws governing such macrobehavior and its disturbances, in particular epileptic seizures, are poorly understood. Here we use a mean-field population network model to describe a state of baseline physiological activity and the transition from the baseline state to rhythmic epileptiform activity. We describe principles which explain how this rhythmic activity arises in the form of spatially uniform self-sustained synchronous oscillations. In addition, we show how the rate of migration of the leading edge of the synchronous oscillations can be theoretically predicted, and compare the accuracy of this prediction with that measured experimentally using multichannel electrocorticographic recordings obtained from a human subject experiencing epileptic seizures. The comparison shows that the experimentally measured rate of migration of the leading edge of synchronous oscillations is within the theoretically predicted range of values. Computer simulations have been performed to investigate the interactions between different regions of the brain and to show how organization in one spatial region can promote or inhibit organization in another. Our theoretical predictions are also consistent with the results of functional magnetic resonance imaging (fMRI), in particular with observations that lower-frequency electroencephalographic (EEG) rhythms entrain larger areas of the brain than higher-frequency rhythms. These findings advance the understanding of functional behavior of interconnected populations and might have implications for the analysis of diverse classes of networks.

Spiral waves in disinhibited mammalian neocortex.

Spiral waves are a basic feature of excitable systems. Although such waves have been observed in a variety of biological systems, they have not been observed in the mammalian cortex during neuronal activity. Here, we report stable rotating spiral waves in rat neocortical slices visualized by voltage-sensitive dye imaging. Tissue from the occipital cortex (visual) was sectioned parallel to cortical lamina to preserve horizontal connections in layers III-V (500-mum-thick, approximately 4 x 6 mm(2)). In such tangential slices, excitation waves propagated in two dimensions during cholinergic oscillations. Spiral waves occurred spontaneously and alternated with plane, ring, and irregular waves. The rotation rate of the spirals was approximately 10 turns per second, and the rotation was linked to the oscillations in a one-cycle- one-rotation manner. A small (<128 mum) phase singularity occurred at the center of the spirals, about which were observed oscillations of widely distributed phases. The phase singularity drifted slowly across the tissue ( approximately 1 mm/10 turns). We introduced a computational model of a cortical layer that predicted and replicated many of the features of our experimental findings. We speculate that rotating spiral waves may provide a spatial framework to organize cortical oscillations.