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

Classification of fixed points

Izhikevich, Figures 4.15

T

Limit cycle in FN model

Gerstner & Kistler, Figure 3.4

Unstable fixed point

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

Separated time scales

Gerstner & Kistler, Figure 3.12

Stereotyped action potential