type i type ii, prcs and coupling - ima.umn.edu · with such a mechanism • oscillations abruptly...
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Type I Type II, PRCs and Coupling
Daniel Forger
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Through what mechanism do oscillations appear?
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Type 1 Integrator
200 400 600 800 1000
0.2
0.4
0.6
0.8
1
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With such a mechanism
• Oscillations abruptly start or stop• The period changes drastically when
parameters are changed• The system does not resonate, instead
it needs to get kicked
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Type 2 Resonator
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With such a mechanism
• Oscillations switch from decaying very slowly to growing very slowly
• The period of the oscillations barely changes when going through the bifurcation
• The system “resonates”
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Chaotic Entrainment
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Type I vs. Type IIPloS Computational Biology 2:e30
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Design I
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Design II
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Coupling by sharing a protein
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Coupling by sharing a protein
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Coupling by extracellular signaling
Garcia-Ojalvo et al. PNAS 101:10955
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Order Parameter
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R =M 2 − M
2
bi2 − bi
2
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Coupling can reduce noise
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Phase ShiftingGlass and Winfree AJP 246:R251
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PRC Atlas
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Formula for the PRC• Point D is on a limit Cycle• We perturb in the positive horizontal
direction
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EGGF
= tanφ ∆z sinθ(r − ∆z cosθ )
= tanφ
φ = arctan∆z
2 − ∆z
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0-D phaseless region
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1-D phaseless region
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2-D Phaseless Region
11 12
1 2
( ) m m m m
m mf C
K C K C= +
+ +
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Winfree, The Geometry of Biological Time
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Ukai, Nature Cell Biology 9:1327
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