Parasitic Lives in the Atlantic Ocean,

Mediterranean Sea, and Great Lakes Up to 35.5 inches

Pure entrainment: Phase differences all tend to a constant.

We add a forcer and look for all the oscillators to be entrained with the forcer.

Example of entrainment

Spinal

Cord

Sinusoidal Forcing arm

Electrodes

Dr. Kathleen Hoffman Dr. Eric Tytell Dr. Avis Cohen Dr. Tim Kiemel

Entrainment

No Entrainment

1 2 3

ωf

+ a sin(Θ2 – Θ1 +

Ψa)

+ d sin(Θ2 – Θ3 + Ψd) + f sin(Θf – Θ2 + Ψf)

Toy Model

+ d sin(Θ1 – Θ2 + Ψd) + a sin(Θ3 – Θ2 + Ψa)

ωf

Θ1’ =

Θ2’ =

Θ3’ =

ω

Θf’ =

ω

ω

Nearest neighbor coupling, any oscillator can be forced

With nearest neighbor coupling

1 2

ωf

+ a sin(Θ2 – Θ1) + d sin(Θ1 – Θ2 ) + f sin(Θf – Θ2)

ωf

Θ1’ =

Θ2’ =

ω

Θf’ =

ω

We determine the Jacobian

And evaluate the determinant,

And the trace

Fixed Points No Fixed Points

With nearest neighbor coupling

1 2 3

ωf

With nearest neighbor coupling

1 2 3

ωf

Three oscillators with one end forced.

Three oscillators with forcing in the middle.

2 3 4

ωf

1 5 n…

Five oscillators with the head forced

Five oscillators with forcing in the middle.

Calibrate our analysis with Eric’s data Look at different coupling systems

Symmetry in the two oscillator system. Phase shifting Entrainment range limits

N-Oscillators with the head forced.

N-Oscillators with the tail forced.

Dr. Joseph Previte Dr. Michael Rutter UMBC Team

› Dr. Kathleen Hoffman› Dr. Eric Tytell› Dr. Avis Cohen› Dr. Tim Kiemel

National Science Foundation