1. Biointelligence Laboratory, Seoul National University
Ch 9. Rhythms and Synchrony 9.5 ~ 9.6 Adaptive Cooperative Systems, Martin Beckerman, 1997.
Summarized by
Kwonill, Kim
Biointelligence Laboratory, Seoul National University

2. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Contents
9.5 Phase Space
9.5.1 Linear Stability Analysis
9.5.2 Nullclines
9.5.3 Poincare-Bendixson Theorem
9.5.4 Hopf Bifurcation
9.6 Population Dynamics in the Wilson-Cowan Model
(C) 2009, SNU Biointelligence Lab, 

3. Linear Stability Analysis
Stability test near a fixed point with the linear approximation
where , by Taylor’s series,
Linearized equations of motion
(9.50)
(9.51)
(9.52)
(C) 2009, SNU Biointelligence Lab, 

4. Linear Stability Analysis
Characteristic equation for eigenvalues of Jacobian
(9.53)
(9.54)
(C) 2009, SNU Biointelligence Lab, 

5. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Nullclines
x nulcline :
y nulcline :
Providing useful information on the dynamics
Ex. The van der Pol oscillator
(9.55)
, where
(9.56)
(9.57)
(C) 2009, SNU Biointelligence Lab, 

6. Poincare-Bendixson Theorem
If for some time t> t0 a trajectory is bounded in some region of phase space, and that resion does not enclose any stable steady states, then that trajectory is either a closed periodic obit (or limit cycle) or approaches a closed periodic obit
Two cases in which a limit cycle will form
(1) The flow is trapped in a particular region containing a repelling unstable node
(2) flow is trapped in an annular region
(C) 2009, SNU Biointelligence Lab, 

7. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Hopf Bifurcation
Bifurcation
a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior
Hopf bifurcation
Fixed point  Limit cycle
Hopf bifurcation theorem
(C) 2009, SNU Biointelligence Lab, 

8. Population Dynamics in the Wilson-Cowan Model
Assumptions of the Wilson-Cowan Model
A population of excitatory & inhibitory cells interacts randomly
The population is sufficiently dense
, : the proportion of excitatory(inhibitory) cells firing per unit time at time t
At resting state,
Adding the refractory property & the thresholded excitation
Fraction of cells that are not refractory (i.e., sensitive)
(C) 2009, SNU Biointelligence Lab, 

9. Population Dynamics in the Wilson-Cowan Model
, : Refractory periods for excitatory and inhibitory neurons
The fraction of cells that are not refractory (i.e., sensitive) :
The average excitation generated in an excitatory cell at time t:
The average excitation generated in an inhibitory cell at time t:
Decay term
External input
(C) 2009, SNU Biointelligence Lab, 

10. Population Dynamics in the Wilson-Cowan Model
Assumption: the sensitivity of the cells is not correlated with their level of excitation.
Simplifying equations with moving temporal averages
sigmoid
(9.58)
(9.59)
(9.63)
(C) 2009, SNU Biointelligence Lab, 

11. Population Dynamics in the Wilson-Cowan Model
With Taylor’s series about and
Modify the sigmoid and refractory terms so that to ensure that is a steady state solution when
The final form of the Wilson-Cowan dynamic equations
(9.64)
(9.65)
(9.66)
(9.67)
(C) 2009, SNU Biointelligence Lab, 

12. Population Dynamics in the Wilson-Cowan Model
Nullclines
More stable states are possible in the other a’s and c’s.
(9.66)
(9.67)
(9.68)
(C) 2009, SNU Biointelligence Lab, 

13. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Summary
Phase Space
Linear Stability Analysis
Nullclines
Poincare-Bendixson Theorem
Hopf Bifurcation
Population Dynamics in the Wilson-Cowan Model
(C) 2009, SNU Biointelligence Lab,