## A42 Stability of a Nonlinear Steady State

What we have learned about stability of steady states for linear systems can be transferred partially to nonlinear ODEs. To be specific, let us consider a biological membrane with a gated ion channel. To do this we combine a model of ion gating with an expression that governs the membrane potential (see Chapter 1 and Chapter 2). For simplicity, we will consider only one conductance. If n represents the gating variable and V the voltage, then the two are coupled by the differential equations...

## Why Do Oscillations Occur

If the following three conditions on the Morris Lecar equations hold, then oscillations will occur the V nullcline has the inverted N shape like that in Figure 2.9B a single intersection of the V- and w-nullclines occurs between the maximum and Figure 2.10 (A) The K+, Ca2+, and total current (1Ca + 1K + ieak Iapp) when w 0.35. States 1 and 3 are stable steady states, and state 2 is unstable as indicated by the velocity vectors. (B) The phase plane for the Morris-Lecar model for Iapp 150 pA,...

## B22 Nullclines and Fixed Points

As discussed in earlier chapters, a powerful technique for the analysis of planar differential equations and related to the direction fields is the use of nullclines. Nullclines are curves in the plane along which the rate of change of one or the other variable is zero. The x-nullcline is the curve where dx dt 0, that is, f (x, y) 0. Similarly, the Figure B.9 Direction fields and some trajectories for the FitzHugh-Nagumo equations. y-nullcline is the curve where g(x, y) 0. The usefulness of...

## A4 Phase Plane Analysis

Obtaining a solution to first-order ODEs means that you have expressed all of the dependent variables as functions of time. In the case of the 2 X 2 linear equations in Section A.3, this means that we have the time series for x, and x2. A great deal can be learned about these solutions by plotting the dependent variables as a function of time as done in Figure A.1. However, there are other ways of plotting solutions that give additional insight. For example, one can plot x, versus time, or some...

## The Voltage Clamp

In order to measure the voltage across a cell membrane or the current flowing through a membrane, microelectrodes are inserted into cells. These electrodes can be used both to measure current and voltage and to apply external current. In order to measure the Figure 2.7 Simulation of voltage clamp experiment using (2.28) and (2.29). (A) Current records resulting from 40 ms depolarizations from the holding potential of -60 mV to the indicated test potentials. (B) the maximum (steady state)...

## Traveling Wave Solutions

An interesting and important problem is to determine the behavior of the bistable equation when a portion of the region is initially above the threshold a and the remainder is initially at zero. To get some idea of what to expect it is useful to perform a numerical simulation. For this numerical simulation we use the method of lines to solve the differential equations ddt0 ax (ci(t) - c (t))+f (c ), (7.77) dCn D (c +i (t) - 2c (t) + c _i (t)) + f (c ), n 1,2, ,N - 1, (7.78) (cn-i(t) - Cn(t))+ f...

## Glucose Dependent Insulin Secretion

Insulin is secreted from -cells in the pancreas in an oscillatory fashion. Glucose must be metabolized by the -cell to stimulate insulin secretion, and the insulin, which is Figure 4.4 Flow system for experimental study of insulin secretion. Vbed is the volume of the reaction bed, f is the volume flow rate, and Go is the inflow concentration of glucose. Redrawn from Maki and Keizer, 1995 . prepackaged in secretory vesicles, is secreted from the -cell into the capillary system by exocytosis....

## John J Tyson

Biochemical and biophysical rhythms are ubiquitous characteristics of living organisms, from rapid oscillations of membrane potential in nerve cells to slow cycles of ovulation in mammals. One of the first biochemical oscillations to be discovered was the periodic conversion of sugar to alcohol glycolysis in anaerobic yeast cultures Chance et al., 1973 . The oscillation can be observed as periodic changes in fluorescence from an essential intermediate, NADH see Figure 9.1. In the laboratories...

## Bullfrog Sympathetic Ganglion Neuron Closed Cell Model

We proceed to write down the equations for a closed cell using 5.15 with jpnM jOM 0 together with appropriate expressions for the fluxes due to the ryanodine receptors, a leak out of the ER, and a SERCA pump i jryr Jleak - Jserca - 5.32 jRyR VRyRPo Ca2 er - Ca2 i , 5.33 where vRyR also in M s is proportional to the number of RyRs and PO is the probability that a receptor is open. Note that here we are generalizing 5.26 for a symmetric leak channel, but symmetry is only an approximation for the...

## Model of the Fertilization Calcium Wave

When mature Xenopus laevis oocytes eggs are loaded with an indicator dye e.g., Ca2 -green dextran and stimulated by the fusion of sperm, a propagating wave of intracellular Ca2 release can be observed by backcalculating the free Ca2 concentration Ca2 i x, t from a time-dependent fluorescence signal F x,t according to 8.3 or 8.5 . This fertilization Ca2 wave is an important step in early development. It triggers Figure 8.2 Schematic diagram of the fertilization Ca2 wave model. Ca2 enters the...