Nonlinear system cell membrane

Autonomous phenomena in multicellular systems are considered in the section on bioelectric patterning in cell aggregates. A mathematical model is considered which describes the cell aggregate via macroscopic variables--concentrations and voltage. Cells are "smeared out" to formulate the theory in terms of continuum variables. A nonlinear integral operator is introduced to model the intercellular transport that corrects the cruder diffusion-like terms usually assumed. This transport term is explicitly related to the properties of cell membranes. 

Besides serving to demonstrate the physical principles involved in applying the PB equation as well as checks for the numerical solution of more complicated geometries, these three systems are also excellent models for systems of extreme biophysical importance, modeling flat-cell membranes, extended polyelectrolytes such as DNA, and spherical micelles. For each geometry, we will solve the PB equation and discuss the resulting potential profile. For those cases in which both nonlinear PB as well as linear DH solutions exist, the similarities and differences (particularly near the interface) are emphasized. 

Finally, Section 2.4 analyses a simplified model of a bursting pancreatic /3-cell [12]. The purpose of this section is to underline the importance of complex nonlinear dynamic phenomena in biomedical systems. Living systems operate under far-from-equilibrium conditions. This implies that, contrary to the conventional assumption of homeostasis, many regulatory mechanisms are actually unstable and produce self-sustained oscillatory dynamics. The electrophysiological processes of the pancreatic /3-cell display (at least) two interacting oscillatory processes A fast process associated with the K+ dynamics and a much slower process associated with the Ca2+ dynamics. Together these two processes can explain the characteristic bursting dynamics in the membrane potential. 

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