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 http://bi.snu.ac.kr/

(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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

Linear Stability Analysis Characteristic equation for eigenvalues of Jacobian (9.53) (9.54) (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

(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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

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, http://bi.snu.ac.kr/

(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, http://bi.snu.ac.kr/