• The Pure and Applied Mathematics
• /
• v.9 no.1
• /
• pp.19-30
• /
• 2002

We consider one class of bursting oscillation models, that is square-wave burster. One of the interesting features of these models is that periodic bursting solution need not to be unique or stable for arbitrarily small values of a singular perturbation parameter $\epsilon$. Recent results show that the bursting solution is uniquely determined and stable for most of the ranges of the small parameter $\epsilon$. In this paper, we present a condition of uniqueness and stability of periodic bursting solutions for all sufficiently small values of $\epsilon$ > 0.

• The Korean Journal of Physiology and Pharmacology
• /
• v.15 no.6
• /
• pp.371-382
• /
• 2011

We performed experiments using Aplysia neurons to identify the mechanism underlying the changes in the firing patterns in response to temperature changes. When the temperature was gradually increased from $11^{\circ}C$ to $31^{\circ}C$ the firing patterns changed sequentially from the silent state to beating, doublets, beating-chaos, bursting-chaos, square-wave bursting, and bursting-oscillation patterns. When the temperature was decreased over the same temperature range, these sequential changes in the firing patterns reappeared in reverse order. To simulate this entire range of spiking patterns we modified nonlinear differential equations that Chay and Lee made using temperature-dependent scaling factors. To refine the equations, we also analyzed the spike pattern changes in the presence of potassium channel blockers. Based on the solutions of these equations and potassium channel blocker experiments, we found that, as temperature increases, the maximum value of the potassium channel relaxation time constant, ${\tau}_n(t)$ increases, but the maximum value of the probabilities of openings for activation of the potassium channels, n(t) decreases. Accordingly, the voltage-dependent potassium current is likely to play a leading role in the temperature-dependent changes in the firing patterns in Aplysia neurons.

• Progress in Medical Physics
• /
• v.18 no.2
• /
• pp.73-80
• /
• 2007

Fruitful findings have been produced from five out of sixty cells which were obtained from each 63 individual Aplisia caught at the Jeju coast. Spiking patterns of three out of five cells, such as relaxation oscillator, bursting within a short time of the inter-burst interval, chaotic bursting, period doubling sequences, bursting with long trains of action potentials separated by short silent periods, regular repeated beating or elliptic bursting, and silent states had been changed in order as the temperature was lowered to $10^{\circ}C\;from\;32^{\circ}C$. In the intervals of every about 40 minutes repeated ups and downs of temperature produced similar firing patterns at the allowable temperature ranges. The other two cells showed difference from these. The amplitudes of the action potentials of the two cells will not be highly decreased in 24 hours. Average spike frequencies, the inter-burst interval, peak to peak spike amplitude of action potentials, minimum potential values are compared and analyzed by using the computer programme. The spike frequencies according to temperature show the distribution of bell type, with maximal spike frequencies at intermediate temperatures and minimal ones at either end. The most common pattern consist of high spike frequency during failing and low one during rising temperatures.