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Spatially–Localized Synchronous Oscillations in the E-I Map:
 1:2 Resonance Bifurcation

1:2 Resonance Bifurcation


The following movies are sequences of phase portraits for the E-I map in the vicinity of the codimension 2 point for a strong resonance bifurcation. In this case, the bifurcation occurs on a period 2 orbit (typically a fixed point).

E-I map   — a 2–D implicit discrete map describing spatially–localized synchronous oscillations exhibited in a network of neuronal conductance-based models extended along one spatial dimension. The network is composed of an excitatory–inhibitory pair of neuronal populations that are mutually coupled through distance–dependent synaptic coupling where:
  — width of the band of spikes on the nth cycle in the excitatory population
  — width of the band of spikes on the nth cycle in the inhibitory population

For details see the relevant publication:

S. E. FOLIAS  &  G. B. ERMENTROUT, Spatially–localized synchronous oscillations in
synaptically–coupled neuronal networks: conductance–based models & discrete maps.
SIAM J. Applied Dynamical Systems  9: 1019-1060 (2010). pdf

Movies   — each movie animates two orbits in the   – phase plane for decreasing θi and fixed σii. Each orbit contains up to 100,000 iterates and is colored with two colors distinguishing transient from long–term behavior:

The light blue transient leads to dark purple long–term behavior
The yellow-orange transient leads to red long–term behavior.
If both purple and red overlap, then only purple is shown.


·  Most movies are mpeg–4 part 2 (mpeg–4 improved) unless indicated otherwise



Instructions for viewing
·  Mouse–over an image (i.e., place cursor over image) to animate a sequence of snapshots for the movie.

·  Click a link next to FORMATS to download the movie.




Special Edition Movie  (new!)



σ=1.5775
SIZE: 609 MB
PHASE PORTRAITS: 8200
FORMATS:   .mp4 (mpeg4)
CONTENT: An equilibrium undergoes a flip bifurcation to a stable period 2 orbit which subsequently undergoes a 1:2 strong resonance bifurcation exhibiting a flip bifurcation, Neimark–Sacker bifurcation, weak resonances and chaos, a homoclinic bifurcation with a pair of double homoclinic tangles, and the analogue of a saddle–node bifurcation of limit cycles that causes orbits to diverge. The movie ends with an evolution of the seahorse attractor.



Short Movies




σ=1.58
SIZE: 24 MB
PHASE PORTRAITS: 447
FORMATS:   .mp4  (mpeg4)
CONTENT: A close–up near the homoclinic bifurcation corresponding to the longer movie for σii=1.58 above.

σ=1.575
SIZE: 8 MB
PHASE PORTRAITS: 1
FORMATS:   .mp4  (mpeg4)
CONTENT: Animation of a single trajectory when the double homoclinic tangle is present to demonstrate how the phase portrait is generated. A set of consecutive iterates is colored blue with darker blues indicated more advanced iterates. The set of iterates is then advanced leaving behind gray dots that denote previous iterates.



High Resolution Movies




σ=1.585
SIZE: 20 MB
PHASE PORTRAITS: 1000
FORMATS:   .mp4  (mpeg4)
CONTENT: Beginning well after the Neimark–Sacker bifurcation, the movie exhibits a partial sequence of the bifurcations, including weak resonances and chaos and the homoclinic bifurcation that leads to a large stable closed curve exhibiting resonances.

σ=1.58
SIZE: 57 MB
PHASE PORTRAITS: 2000
FORMATS:   .mp4  (mpeg4)
CONTENT: Beginning well after the Neimark–Sacker bifurcation, the movie exhibits a partial sequence of the bifurcations, including weak resonances and chaos and the homoclinic bifurcation that leads to a large stable closed curve exhibiting resonances.

σ=1.5775
SIZE: 117 MB
PHASE PORTRAITS: 2850
FORMATS:     .mov   |   .mp4 (H.264)
CONTENT: An equilibrium undergoes a flip bifurcation to a stable period 2 orbit which subsequently undergoes a 1:2 strong resonance bifurcation exhibiting a flip bifurcation, Neimark–Sacker bifurcation, weak resonances and chaos, a homoclinic bifurcation with a pair of double homoclinic tangles, and the analogue of a saddle–node bifurcation of limit cycles that causes orbits to diverge.

σ=1.575
SIZE: 292 MB  (154 MB  H.264)
PHASE PORTRAITS: 3977
FORMATS:    .mp4 (mpeg4)   |   .mp4 (H.264)
CONTENT: Beginning well after the Neimark–Sacker bifurcation, the movie exhibits a partial sequence of the bifurcations, including weak resonances and chaos and the homoclinic bifurcation that leads to intricate dynamics before orbits start to diverge.



Medium Resolution Movies




σ=1.5775
SIZE: 71 MB
PHASE PORTRAITS: 2850
FORMATS:    .mov   |   .mp4 (H.264)
CONTENT: An equilibrium undergoes a flip bifurcation to a stable period 2 orbit which subsequently undergoes a 1:2 strong resonance bifurcation exhibiting a flip bifurcation, Neimark–Sacker bifurcation, weak resonances and chaos, a homoclinic bifurcation with a pair of double homoclinic tangles, and the analogue of a saddle–node bifurcation of limit cycles that causes orbits to diverge.



  Stefanos E. Folias
    10.10.2010