Synchronization of PulseCoupled Biological Oscillators

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Synchronization of Pulse-Coupled Biological
Oscillators

• By Renato E. Mirollo and Steven H. Strogatz

Presented by Ramil Berner Math 723 Spring Quarter
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Overview

• Point of the Paper
• Model for 2 Oscillators
• Model for N Oscillators
• Main Theorem
• Conclusion

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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

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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

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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 N2 oscillators
• Further assumptions of small coupling strength
  and small dissipation

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What are we talking about? cont

• Analysis of general version of Peskins Model for
  all N
• Assume only the oscillators rise toward threshold
  with a time-course which is monotonic and concave
  down.
• Retain two of Peskins assumptions
• Oscillators have identical dynamics
• Each is coupled to all the others.

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Two Oscillators The Model

• Generalize the integrate and fire dynamics
• Assume only that x evolves according to
• 0,1?0,1 is smooth, monotonically increasing
  and concave down
• Here is a phase variable such
  that
• ?0 when the oscillator is at its lowest state
  x0
• ?1 at the end of the cycle when the oscillator
  reaches the threshold x1.

• satisfies (0)0, (1)1

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Two Oscillators The Model cont

• g denotes the inverse function of . g maps
  states to their corresponding phases g(x) ?.
  Because of the hypothesis on , the function g
  is increasing and concave up. The endpoint
  conditions are g(0)0 and g(1)1

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Two Oscillators The Model cont

• Example
• Here
• Consider Two oscillators governed by
• Interact by pulse coupling rule

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Two Oscillators The Model cont

• (a) Two Points on a fixed curve
• (b) Just before firing
• (c) Immediately After Firing

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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.

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Calculating the Return Map

• Return Map
• Two Oscillators A and B
• Consider the system just after A has fired
• ? denotes the phase of B
• The return map R(?) is defined to be the phase of
  B immediately after the next firing of A.

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Calculating the Return Map cont

• Observe that after a time 1-? oscillator B
  reaches threshold
• A moves from zero to an x-value given by xA
  (1-?).

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Calculating the Return Map cont

• An instant later B fires and xA jumps to e(1-?)
  or 1. whichever is less.
• If xA1, we are done, the oscillators have
  synchronized. Hence assume that xAe(1-?)lt1.

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Calculating the Return Map cont

• The corresponding phase of A is
• g(ef(1-?), where g -1
• h ? (2.1)?
• Two iterations of h will give us R(?)?
• R(?)h(h(?)) (2.2)?

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Calculating the Return Map cont

• Domain of h and R we assumed that
  e (1-?)lt1. this assumption is satisfied for e
  in 0,1) and ? in (d,1) where d1-g(1-e).
• Thus the domain of h is the subinterval (d,1)?
• Similarly the domain of R is the subinterval
  (d,h-1(d))?
• This interval is non empty since dlth-1 (d) for elt1

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Lemma 2.1
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Proposition 2.2
By this we see R has simple dynamics. The system
is always driven to synchrony.
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Example

• We must construct an f function
• From the lemma we know h(?)lt-1
• Let h(?)-?, ?gt1 is independent of ?
• h(?)-?(?-?)?
• R(?)?2(?-?)?
• -1 must satisfy g(eu)/g(u)? for all u
• Solution is a ebu
• (?)(1/b) (ln(1eb-1) ?)?

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Example cont

• As b goes to zero approaches the dashed line.
  For large b rises very rapidly and then levels
  off.
• b measures how concave down the graph will be.
• b is analogous to the conductance ? in the leaky
  capacitor model

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Example cont

• Implications
• Synchrony emerges more rapidly when the
  dissipation b or the pulse strength e is large.
• The time to synchronize is inversely proportional
  to the product eb

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Example cont

• Where is the fixed point?
• Proposition 2.2
• F(?) ?-h(?)0
• Rewrite as e (?)- (1- ?)?
• (?)(1/b) (ln(1eb-1) ?)?
• ?(eb(1 e)-1)/((eb-1)(ebe1))?
• Graph of ?

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Example cont

• As e ? 0 the fixed point ? ½
• Holds for general
• In the limit of small coupling repelling fixed
  point always occurs with the oscillators in
  antiphase.

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Example cont

• What determines the Stability Type
• The Eigenvalues ?ebe
• Determines the stability
• egt0, bgt0
• In this case ?gt1 and ? is a repeller.
• Note, if either elt0 or blt0 ? is automatically
  stabilized
• If both elt0 blt0 then the fixed point is again a
  repeller

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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

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Computer Simulation and the Two Main Proofs

• State of the system is characterized by the
  phases ?1, ?n of the remaining nN-1 Oscillators
• The possible states are
• S(?1, ?n) e Rn s.t. 0lt ?1lt ?2ltlt ?nlt1
• ?0 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
• ?n is the next to fire. Then ?n-1 and so on.
• After ?n fires we re-label it to ?0 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

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The firing map

• Let ?(?1, ?n) be the vector of phases
  immediately after firing.
• Want to find the firing map h that transforms ?
  to the vector phases right after the next firing.
• Note next firing occurs after a time 1-?n
• Oscillator I has drifted to the phase ?i1-?n,
  where I 1,2,3,,n-1
• Let s(?1, ?n) )(1- ?n),?11-?n,..?n-11-?n) )
  (s1, sn)
• After firing, new phases are given by the map
• t(s1, sn)(g(f(s1) e),, g(f(sn) e))?
• h(?) t(s(?))?
• Describes the new phases of the oscillators after
  one firing.

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Absorption

• Set S is invariant under the affine map s
• Not invariant under the map t
• Since f(sn) e?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

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Absorption cont

• Two complications of absorption
• Problem 1
• Domain of h is not all of S, instead we have
• Se (?1, ?n) e S s.t f(?n-11-?n ) e lt1
• Or Se (?1, ?n) e S s.t ?n-?n-1 gt1-g(1-e)
• 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.

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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.

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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.

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Theorem 3.1 and Lebesgue measure

• The set A has Lebesgue Measure zero.
• For measure zero, set is empty

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Theorem 3.1 and Lebesgue measure cont
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Theorem 3.2

• Assume another set B exists which is in Sn but
  never achieves synchrony
• The set B has Lebesgue Measure zero.

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Theorem 3.2 cont
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Numerical Results

• Computer Simulation
• Let N100
• S0 2
• ? 1
• e 0.3

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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 t9T Perfect synchronization

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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.

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Numerical Results cont

• Evolution of the system in state space.
• System strobed after each firing of oscillator
  i1
• In the beginning some shallow parts
• By seventh firing parts have become completely
  flat.

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Numerical Results cont

• Meaning the corresponding oscillators are firing
  in unison
• Dominant group has emerged by the tenth firing.

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Conclusion

• Point of the Paper
• Model for 2 Oscillators
• Model for N Oscillators
• Main Theorem
• Conclusion

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References

• 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

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• Questions?