1Physiology 472/572 - 2009 - Quantitative modeling of biological systems Lecture 25: Limit cycles Oscillations in linear systems • in linear systems, solutions must be combinations of eλ1t, eλ2t • if λ1 and λ2 are complex, they must be complex conjugates, i.e., λ = μ ± iω • consider a solution x = ½(a - ib)e(μ + iω)t + ½(a + ib)e(μ - iω)t • i.e. x = eμt [a cos(ωt) + b sin(ωt)] • μ > 0: oscillations grow; μ = 0: fixed amplitude oscillations; μ < 0: oscillations decay Nonlinear oscillations • consider the Van der Pol oscillator udtdvu3/u G(u)whereG(u) - vdtdu3−=−== • nullclines are v = G(u), u = 0 • equilibrium point is at (0,0) • Jacobian is ⎥⎦⎤⎢⎣⎡− 0111, λ2 - λ + 1 = 0, λ = (1 ± i√3)/2 • equilibrium point is unstable to growing oscillations • oscillations reach a finite amplitude and stabilize • this is a limit cycle • biological examples of limit cycles include predator-prey systems, neural oscillators and calcium oscillations • other examples include chemical systems and nonlinear electronic circuits2Calcium oscillations • many types of cells show spontaneous oscillations in calcium levels • examples: hepatocytes, fibroblasts, endothelial cells, variety of hormone-secreting cells • oscillations are believed to be important to cell function and communication • represented in the two-pool model by increasing the calcium influx rate μ • limit cycle oscillations are obtained • rapid spikes of cytosolic calcium deplete Ca2+-sensitive stores, which recover slowly • point at which varying a parameter leads to a transition from steady state to oscillation is known as a Hopf bifurcation point, a much studied phenomenon • here the bifurcation occurs at about μ = 0.31 0 0.5 1 1.5 2 2.5 300.511.5uvμ = 0.40 10 20 30 40
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