On the geometry of solutions of the quasi-geostrophic and Euler equations

We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.

Behavior of several two-dimensional fluid equations in singular scenarios

We give conditions that rule out formation of sharp fronts for certain two-dimensional incompressible flows. We show that a necessary condition of having a sharp front is that the flow has to have uncontrolled velocity growth. In the case of the quasi-geostrophic equation and two-dimensional Euler equation, we obtain estimates on the formation of semi-uniform fronts.

Calcium regulation of a slow post-spike hyperpolarization in vagal afferent neurons

Activation of distinct classes of potassium channels can dramatically affect the frequency and the pattern of neuronal firing. In a subpopulation of vagal afferent neurons (nodose ganglion neurons), the pattern of impulse activity is effectively modulated by a Ca2+-dependent K+ current. This current produces a post-spike hyperpolarization (AHPslow) that plays a critical role in the regulation of membrane excitability and is responsible for spike-frequency accommodation in these neurons. Inhibition of the AHPslow by a number of endogenous autacoids (e.g., histamine, serotonin, prostanoids, and bradykinin) results in an increase in the firing frequency of vagal afferent neurons from <0.1 to >10 Hz. After a single action potential, the AHPslow in nodose neurons displays a slow rise time to peak (0.3–0.5 s) and a long duration (3–15 s). The slow kinetics of the AHPslow are due, in part, to Ca2+ discharge from an intracellular Ca2+-induced Ca2+ release (CICR) pool. Action potential-evoked Ca2+ influx via either L or N type Ca2+ channels triggers CICR. Surprisingly, although L type channels generate 60% of action potential-induced CICR, only Ca2+ influx through N type Ca2+ channels can trigger the CICR-dependent AHPslow. These observations suggest that a close physical proximity exists between endoplasmic reticulum ryanodine receptors and plasma membrane N type Ca2+ channels and AHPslow potassium channels. Such an anatomical relation might be particularly beneficial for modulation of spike-frequency adaptation in vagal afferent neurons.

Membrane healing and restoration of contractility after mechanical injury in isolated skeletal muscle fibers of the frog.

In single isolated skeletal muscle fibers of the frog, we studied (i) the recovery from large sarcolemmal mechanical injuries of the response to electric stimulation and (ii) the integrity of the sarcolemma under the light microscope. In Ringer's solution, the damaged cells stopped contracting and deteriorated completely within 1 hr. In the presence of phosphatidylcholine (0.025 g/ml in Ringer's solution), the injured cells initially responded with local twitches. Within 0.5 hr, contractility and membrane integrity started to recover and both were back to control levels within 3 hr. When these cells were placed back in normal Ringer's solution, they remained viable and active for several hours. Our results suggest that phosphatidylcholine can protect muscle fibers from the effects of sarcolemmal injury.