Transcript Synchronization of Pulse-Coupled Biological Oscillators

Synchronization of PulseCoupled Biological Oscillators
By Renato E. Mirollo and Steven
H. Strogatz
Presented by Ramil Berner
Math 723 Spring Quarter 08
Overview
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Point of the Paper
Model for 2 Oscillators
Model for N Oscillators
Main Theorem
Conclusion
Synchronization of What coupled biological Who?
•Synchronization of Pulse-couple Biological
Oscillators
•Examples of Biological Oscillators
•Fireflies
•Crickets
•Menstruating Women
•Pacemaker
•Pancreas
•The Idea
•Oscillators will evolve to
synchronous firing.
•Shortest path to
synchronization will be
taken
What are we talking about?
• Oscillators assume to interact by a simple
form of pulse coupling
– When a given oscillator fires, its pulls all the
other oscillators up by an amount , or pulls
them up to firing.
– Whichever is less
What are we talking about? cont…
• Peskin Conjectured
– For arbitrary initial conditions, system
approaches a state in which all oscillators are
firing synchronously.
– This remains true even when the oscillators
are not quite identical.
• Proved the first for N=2 oscillators
– Further assumptions of small coupling
strength and small dissipation
What are we talking about? cont…
• Analysis of general version of Peskin’s
Model for all N
– Assume only the oscillators rise toward
threshold with a time-course which is
monotonic and concave down.
– Retain two of Peskin’s assumptions
• Oscillators have identical dynamics
• Each is coupled to all the others.
Two Oscillators – The Model
• Generalize the integrate and fire dynamics
– Assume only that x evolves according to
x
f
– ƒ:[0,1][0,1] is smooth, monotonically increasing and
concave down
0, 1
– Here
is a phase variable such that
•
d
1
, T is the cycle period
dt
T
• Ф=0 when the oscillator is at its lowest state x=0
• Ф=1 at the end of the cycle when the oscillator reaches the
threshold x=1.
•ƒ satisfies ƒ(0)=0, ƒ(1)=1
Two Oscillators – The Model cont…
• g denotes the inverse function of ƒ. g maps
states to their corresponding phases: g(x) = Ф.
Because of the hypothesis on ƒ, the function g
is increasing and concave up. The endpoint
conditions are g(0)=0 and g(1)=1
Two Oscillators – The Model cont…
• Example
– Here
–
– Consider Two oscillators governed by ƒ
• Interact by pulse coupling rule
Two Oscillators – The Model cont…
• (a) Two Points on a fixed curve
• (b) Just before firing
• (c) Immediately After Firing
Two Oscillators – The Model cont…
• Strategy
– To prove that the two oscillators always become
synchronized, we first calculate the return map and
then show the oscillators are driven closer together
each time the map is iterated.
– Perfect Synchrony when the oscillators have gotten
so close together that the firing of one brings the other
to threshold.
– They remain synchronized thereafter because there
dynamics are identical.
Calculating the Return Map
• Return Map
– Two Oscillators A and B
– Consider the system just after A has fired
– Ф denotes the phase of B
– The return map R(Ф) is defined to be the phase
of B immediately after the next firing of A.
Calculating the Return Map cont…
• Observe that after a time 1-Ф oscillator B
reaches threshold
• A moves from zero to an x-value given by
xA= ƒ(1-Ф).
Calculating the Return Map cont…
• An instant later B fires and xA jumps to
ε+ƒ(1-Ф) or 1. whichever is less.
• If xA=1, we are done, the oscillators have
synchronized. Hence assume that
xA=ε+ƒ(1-Ф)<1.
Calculating the Return Map cont…
• The corresponding phase of A is
g(ε+f(1-Ф), where g = ƒ-1
• h
(2.1)
• Two iterations of h will give us R(Ф)
– R(Ф)=h(h(Ф)) (2.2)
Calculating the Return Map cont…
• Domain of h and R: we assumed that
ε+ƒ (1-Ф)<1. this assumption is satisfied for ε in
[0,1) and Ф in (δ,1) where δ=1-g(1-ε).
• Thus the domain of h is the subinterval (δ,1)
• Similarly the domain of R is the subinterval
(δ,h-1(δ))
• This interval is non empty since δ<h-1 (δ) for ε<1
Lemma 2.1
Proposition 2.2
By this we see R has simple dynamics. The system is always driven to synchrony.
Example
• We must construct an f function
• From the lemma we know h’(Ф)<-1
– Let h’(Ф)=-λ, λ>1 is independent of Ф
– h(Ф)=-λ(Ф-Ф*)+Ф*
– R(Ф)=λ2(Ф-Ф*)+Ф*
– ƒ-1 must satisfy g’(ε+u)/g’(u)=λ for all u
– Solution is a ebu
– ƒ(Ф)=(1/b) (ln(1+eb-1) Ф)
Example cont…
• As b goes to zero ƒ approaches the dashed line. For
large b ƒ rises very rapidly and then levels off.
– b measures how concave down the graph will be.
– b is analogous to the conductance γ in the “leaky
capacitor model
Example cont…
• Implications
– Synchrony emerges more rapidly when the
dissipation b or the pulse strength ε is large.
– The time to synchronize is inversely
proportional to the product εb
Example cont…
• Where is the fixed point?
– Proposition 2.2
• F(Ф)= Ф-h(Ф)=0
• Rewrite as ε= ƒ(Ф*)- ƒ(1- Ф*)
– ƒ(Ф)=(1/b) (ln(1+eb-1) Ф)
– Ф*=(eb(1+ ε)-1)/((eb-1)(ebε+1))
– Graph of Ф*
Example cont…
• As ε  0 the fixed point  ½
– Holds for general ƒ
– In the limit of small coupling; repelling fixed point
always occurs with the oscillators in antiphase.
Example cont…
• What determines the Stability Type
– The Eigenvalues λ=ebε
• Determines the stability
• ε>0, b>0
• In this case λ>1 and Ф* is a repeller.
– Note, if either ε<0 or b<0; Ф* is automatically stabilized
– If both ε<0 & b<0 then the fixed point is again a repeller
Population of Oscillators
• As System evolves,
– oscillators clump together
– Groups fire at the same time
• As Groups get bigger they create a larger
pulse when they fire
– Absorbs Other oscillators
• Ultimately 1 group remains
– The population is then synchronized
Computer Simulation and the Two Main Proofs
• State of the system is characterized by the phases Ф1,…
Фn of the remaining n=N-1 Oscillators
– The possible states are
• S={(Ф1,… Фn) ε Rn s.t. 0< Ф1< Ф2<…< Фn<1}
• Ф0 =0
– The flow preserves cyclic ordering of the oscillators
• Because Oscillators are assumed to have identical dynamics and
coupling is all to all
• Same frequencies so no order change between firings
• Monotonicity ensures the order is maintained
– Фn is the next to fire. Then Фn-1 and so on.
• After Фn fires we re-label it to Ф0 and increase the indices by one
when the next fires.
• If the oscillators had different frequencies this indexing
scheme would fail.
– Oscillators could pass another and the dynamics would be more
difficult to analyze
The firing map
• Let Ф=(Ф1,… Фn) be the vector of phases
immediately after firing.
– Want to find the firing map h that transforms Ф to the
vector phases right after the next firing.
– Note: next firing occurs after a time 1-Фn
• Oscillator I has drifted to the phase Фi+1-Фn, where I =
1,2,3,…,n-1
– Let σ=(Ф1,… Фn) )=(1- Фn),Ф1+1-Фn,..Фn-1+1-Фn) )
=(σ1,… σn)
• After firing, new phases are given by the map:
– τ(σ1,… σn)=(g(f(σ1) + ε),…, g(f(σn) + ε))
– h(Ф)= τ(σ(Ф))
• Describes the new phases of the oscillators after one firing.
Absorption
• Set S is invariant under the affine map σ
– Not invariant under the map τ
• Since f(σn)+ ε≧1
– When this happens
• Firing of oscillator n has also brought oscillator n-1
to threshold with it.
– Both oscillators now act as one.
– This is what is meant by absorption
Absorption cont…
• Two complications of absorption
– Problem 1
• Domain of h is not all of S, instead we have
– Sε ={(Ф1,… Фn) ε S s.t f(Фn-1+1-Фn ) + ε <1}
– Or Sε ={(Ф1,… Фn) ε S s.t Фn-Фn-1 >1-g(1-ε)}
– Problem 2
• Groups are created that will have enhanced
pulse strength.
– We must now allow for the possibility of
different pulse strengths in the population.
Main Theorem
• Two parts
– Part I
• Rules out the possibility that elements of a set will
not all be absorbed
– Part II
• Rules out the possibility that there might exist sets
which never get absorbed.
Main Theorem
•Consider two sets
•A which is the set of all oscillators within S that did not
absorb
•B which is another set in S that did not absorb
•If B or A are not empty we have a problem.
Theorem 3.1 and Lebesgue measure
• The set A has Lebesgue Measure zero.
• For measure zero, set is empty
Theorem 3.1 and Lebesgue measure cont…
Theorem 3.2
• Assume another set B exists which is in Sn but never achieves
synchrony
• The set B has Lebesgue Measure zero.
Theorem 3.2 cont…
Numerical Results
• Computer Simulation
– Let N=100
– S0 = 2
–γ = 1
– ε = 0.3
Numerical Results cont…
• Number of Oscillators over time
– Little coherence in the beginning
• Organization is rather slow.
– Down the road the synchrony builds up
• By t=9T – Perfect synchronization
Numerical Results cont…
• The slow beginning makes sense
– Real world example
• Asian Fireflies at dawn
– Buildup to synchrony is slow, due to stimulus of light from
many sources.
– Conflicting pulses in the incoherent initial stage.
Numerical Results cont…
• Evolution of the system in state space.
– System strobed after each firing of oscillator i=1
– In the beginning some shallow parts
– By seventh firing parts have become completely flat.
Numerical Results cont…
• Meaning the corresponding oscillators are
firing in unison
– Dominant group has emerged by the tenth
firing.
Conclusion
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Point of the Paper
Model for 2 Oscillators
Model for N Oscillators
Main Theorem
Conclusion
References
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“Synchronization of Pulse-Coupled Biological Oscillators”, Renato E. Mirollo and Steven H. Strogatz, Siam Journal of
Applied Mathematics, Vol. 50, No. 6 (Dec 1990), PP. 1645-1662
•Questions?