Inhibitor-Induced Wavetrains and Spiral Waves in an Extended FitzHugh-Nagumo Model of Nerve Cell Dynamics.

It is well known that the FitzHugh-Nagumo model is one of the simplified forms of the four-variable Hodgkin-Huxley model that can reflect most of the significant phenomena of nerve cell action potential. However, this model cannot capture the irregular action potentials of sufficiently large periods in a one-parameter family of solutions. Motivated by this, we propose a modified FitzHugh-Nagumo reaction-diffusion system by changing its recovery kinetics. First, we investigate the parameter regime to know the existence of the wavetrains. Second, we conceive the occurrence of Eckhaus bifurcations of solutions that divide the solution region into two parts. The essential spectra at different grid points explore the occurrence of bifurcations of the waves. We find that the wavetrains of sufficiently large periods cross the stability boundary. This characteristic phenomenon is absent in the standard FitzHugh-Nagumo model. Finally, we observe a reasonable agreement between the direct PDE simulations and the solutions in the traveling wave ODEs. Furthermore, the model exhibits spiral wave for monotone and non-monotone cases that agrees with the waves observed in cellular activity.

Mathematical Modeling of Dynamic Cellular Association Patterns in Seminiferous Tubules.

In vertebrates, sperm is generated in testicular tube-like structures called seminiferous tubules. The differentiation stages of spermatogenesis exhibit a dynamic spatiotemporal wavetrain pattern. There are two types of pattern-the vertical type, which is observed in mice, and the helical type, which is observed in humans. The mechanisms of this pattern difference remain little understood. In the present study, we used a three-species reaction-diffusion model to reproduce the wavetrain pattern observed in vivo. We hypothesized that the wavelength of the pattern in mice was larger than that in humans and undertook numerical simulations. We found complex patterns of helical and vertical pattern frequency, which can be understood by pattern selection using boundary conditions. From these theoretical results, we predicted that a small number of vertical patterns should be present in human seminiferous tubules. We then found vertical patterns in histological sections of human tubules, consistent with the theoretical prediction. Finally, we showed that the previously reported irregularity of the human pattern could be reproduced using two factors: a wider unstable wavenumber range and the irregular geometry of human compared with mouse seminiferous tubules. These results show that mathematical modeling is useful for understanding the pattern dynamics of seminiferous tubules in vivo.

Competition between global feedback and diffusion in coupled Belousov-Zhabotinsky oscillators.

The Belousov-Zhabotinsky (BZ) reaction is a famous experimental model for chemical oscillatory reaction and pattern formation. We herein study a diffusive coupled system of two oscillators with global feedback using the photosensitive BZ reaction both experimentally and theoretically. The coupled oscillator showed in-phase and antiphase oscillations depending on the strength of diffusive coupling and light feedback. Moreover, we analyzed our model to locate the bifurcational origin and found the reconnection of the bifurcation branches for antiphase oscillation, which was induced by the competition between global feedback and the diffusion effect.

Alternans and Spiral Breakup in an Excitable Reaction-Diffusion System: A Simulation Study.

The determination of the mechanisms of spiral breakup in excitable media is still an open problem for researchers. In the context of cardiac electrophysiological activities, spiral breakup exhibits complex spatiotemporal pattern known as ventricular fibrillation. The latter is the major cause of sudden cardiac deaths all over the world. In this paper, we numerically study the instability of periodic planar traveling wave solution in two dimensions. The emergence of stable spiral pattern is observed in the considered model. This pattern occurs when the heart is malfunctioning (i.e., ventricular tachycardia). We show that the spiral wave breakup is a consequence of the transverse instability of the planar traveling wave solutions. The alternans, that is, the oscillation of pulse widths, is observed in our simulation results. Moreover, we calculate the widths of spiral pulses numerically and observe that the stable spiral pattern bifurcates to an oscillatory wave pattern in a one-parameter family of solutions. The spiral breakup occurs far below the bifurcation when the maximum and the minimum excited states become more distinct, and hence the alternans becomes more pronounced.

A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment.

In this paper, a mathematical model of a prey-predator system is proposed to resolve the paradox of enrichment in ecosystems. The model is based on the natural strategy that a predator takes, i.e, it produces resting eggs in harsh environment. Our result gives a criterion for a functional response, which ensures that entering dormancy stabilizes the population dynamics. It is also shown that the hatching of resting eggs can stabilize the population dynamics when the switching between non-resting and resting eggs is sharp. Furthermore, the bifurcation structure of our model suggests the simultaneous existence of a stable equilibrium and a large amplitude cycle in natural enriched environments.

Selection principle for various modes of spatially nonuniform electrochemical oscillations.

The pattern selection principle for various modes of spatially nonuniform oscillation was investigated by taking a current oscillation of negative differential resistance type, appearing in H2O2 reduction on platinum (Pt) ring electrodes, as a model system. In experiments, various modes of spatiotemporal oscillation, such as a spatially uniform oscillation, standing wave oscillation, and rotating wave oscillation, appeared depending on the applied potential and the distance between the Pt-ring electrode and the reference electrode. A simple mathematical model for the spatiotemporal patterns at the electrode surface was proposed. Numerical calculations and nonlinear bifurcation analysis based on the proposed model reproduced all the essential features of the experimental results and clarified the pattern selection principle.

Isolation and expression of five-amino-acid-deleted variant of feline hepatocyte growth factor (HGF) cDNA.

Hepatocyte growth factor (HGF) is a pleiotropic cytokine that stimulates a wide array of cellular targets, including hepatocytes and other epithelial cells, melanocytes, endothelial and hematopoietic cells. We have cloned a different form of cDNA, with a deletion of 15 base pairs predicted to result in the loss of 5 amino acids from the first kringle domain. To investigate the biological activity, original and deleted variant of feline HGF cDNAs were transiently expressed in COS-7 cells. Both recombinant feline HGFs showed almost the same dose-response curves in the stimulation of the growth of BNL CL.2 cells (a mouse hepatocyte cell line) and scatter activity of Madin-Darby canine kidney (MDCK) cells. The findings reported here suggest that the deleted variant of feline HGF has almost the same biological activity as the original in terms of the proliferation and scatter activity.