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Phase portrait Fitzugh-Nagumo model
Gerstner & Kistler, Figure 3.2
0w
0w
0w 0 nullclinew
0 nullclinev
, flowv w
Vertical
Horizontal
Phase portraits new fixed points
Khalil, Nonlinear Systems, Figures 2.3-2.7
Real eigenvalues and eigenvectors
Stable f.p. Unstable f.p.
Complex eigenvalues and eigenvectors
Saddle
Real eigenvalues and eigenvectors
Linearization around a fixed point
0a
0a
Gerstner & Kistler, Figure 3.3
0b
0
0
T
D
T a
D b a
0
0
T
D
0or 0
0
T T
D
Different system
Stable fixed point and oscillation in the FN model
Gerstner & Kistler, Figure 3.5
Stable fixed point –
Zero input
Limit cycle –
Nonzero input (I=2)
Upward shift of v-nullcline
Single f.p. 1 1b 3 f.p. 1 1b
Unstable fixed point
Nullclines of type I model
Gerstner & Kistler, Figure 3.6
Zero input – 3 fixed points
Nonzero input – 1 unstable fixed points
Morris-Lecar model
Stable
Saddle
Unstable
Gain functions for models of type I and II
Gerstner & Kistler, Figure 3.7
Type I – continuous transition to oscillation
Type II – discontinuous transition to oscillation
Threshold in type I model
Gerstner & Kistler, Figure 3.9
Stable manifold of saddle-point No spike – below threshold
Spike generated – above threshold
Stable
Saddle Unstable
Morris-Lecar model
Threshold-like effect in FN Model (Type II)
Gerstner & Kistler, Figure 3.7
• For v0 < -0.25 trajectory returns rapidly to rest
• For v0 > -0.1 a voltage pulse develops
• Amplitude of v(t) varies smoothly (was stereotyped for type I)
Continuously varying behavior
FN model with separated time scales
Vertical arrows: length O(ε)
Plays the role of stable manifold (separating)
Gerstner & Kistler, Figure 3.11
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