%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This page contains MatCont input and MatLab commands to accompany tutorials on MatCont. Please look at the instructions file on my website for more information. There are multiple models, so scroll down to the one you need. Feel free to use, and adjust. This file page is plain ASCII so copy and paste is possible. Last edit: H.G.E. Meijer, 25-2-2021 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% MorrisLecar %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Name MorrisLecar Coordinates V,n Parameters phi,gCa,V3,V4,Iapp Equations ECa=120 EK=-84 EL=-60 gK=8 gL=2 V1=-1.2 V2=18 CM=20 minf=(1+tanh((V-V1)/V2))/2 taun=1/cosh((V-V3)/(2*V4)) ninf=(1+tanh((V-V3)/V4))/2 V'=(-gCa*minf*(V-ECa)-gK*n*(V-EK)-gL*(V-EL)+Iapp)/CM n'=phi*(ninf-n)/taun %Parameter values for Hopf case: phi=0.04, gCa=4.4, V3=2.0, V4=30 %Parameter values for SNIC case: phi=0.067, gCa=4.0, V3=12.0, V4=17.4 %Parameter values for Homoclinic case: phi=0.23, gCa=4.0, V3=12.0, V4=17.4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% JansenRitslow %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Name JR_slow Coordinates y0,y4,y1,y5,y2,y6,y3,y7 Parameters A,a,Bs,bs,Bf,bf,e0,r,v0,theta,CC,II Equations C1=CC; C2=0.8*CC; C3=0.25*CC; C4=0.25*CC; S4=2*e0/(1+exp(r*(v0-(y1-theta*y2-(1-theta)*y3)))); S5=2*e0/(1+exp(r*(v0-(C1*y0)))); S6=2*e0/(1+exp(r*(v0-(C3*y0)))); S7=S6; y0'=y4; y4'=A*a*S4-2*a*y4-a*a*y0; y1'=y5; y5'=A*a*(II+C2*S5)-2*a*y5-a*a*y1; y2'=y6; y6'=Bf*bf*C4*S6-2*bf*y6-bf*bf*y2; y3'=y7; y7'=Bs*bs*C4*S7-2*bs*y7-bs*bs*y3; %Plot profile ii=10;jj=1;ndim=8;ntst=40;ncol=4; mesh=sort([reshape(repmat(f(1:ntst,ii),1,ncol)+... repmat((0:ncol-1)/ncol,ntst,1).*repmat(diff(f(1:ntst+1,ii)),1,ncol),ntst*ncol,1); 1]); plot(mesh*x(end-1,ii),x(jj:ndim:end-2,ii)) %SWD commands load Systems\JR_slow\diagram\lc1.mat out=x(3:8:end-2,:)-(x(5:8:end-2,:)+x(7:8:end-2,:))/2; extrema=nan(6,size(out,2)); for i=1:size(out,2) diff_tr=diff(out(1:end-1,i)); diff_tr_shift=diff(out(2:end,i)); temp=[out(find([ (diff_tr>0).*(diff_tr_shift<0) ]),i);... out(find([ (diff_tr<0).*(diff_tr_shift>0) ]),i)]; extrema(1:size(temp,1),i)=temp; end figure(1);hold on;plot(x(end,:),sort(extrema)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% LORENZ-84 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Name Lorenz84 Coordinates x,y,z Parameters a,b,F,G Equations x'=-y^2-z^2-a*x+a*F y'=x*y-b*x*z-y+G z'=b*x*y+x*z-z Plotting commands: %%% load Systems/diagram/H_LC(1).mat ntst=20;ii=4;jj=40;ndim=3; figure;plot(f(1:ntst+1,ii)*x(end-1,ii),x(jj:ndim:end-2,ii)) %%% ntst=20;ii=4;jj=40;ndim=3;ncol=4 plot(f(1:ntst+1,ii)*x(end-1,ii),x(jj:ndim*ncol:end-2,ii)) %%% ii=240;jj=1;ndim=3;ntst=20;ncol=4; mesh=sort(reshape(repmat(f(1:ntst,ii),1,ncol)+... repmat((0:ncol-1)/ncol,ntst,1).*repmat(diff(f(1:ntst+1,ii)),1,ncol),ntst*ncol,1); 1]); plot(mesh*x(end-1,ii),x(jj:ndim:end-2,ii)) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Duffing Periodically Forced; Frequency Response %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Name DuffingForced Coordinates x,y,u,v Parameters alpha,beta,gamma,delta,omega Equations x'=y y'=gamma*u-alpha*x-delta*y-beta*x^3 u'=-omega*v+u*(1-u^2-v^2) v'=omega*u+v*(1-u^2-v^2) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% SEIR Periodically Forced %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Name SEIR Coordinates ss,ee,ii,x,y Parameters alpha,beta0,gamma,delta,mu,omega Equations beta=beta0*(1+delta*x) ss'=mu*exp(-ss)-mu-beta*exp(ii) ee'=beta*exp(ss+ii-ee)-(mu+alpha) ii'=alpha*exp(ee-ii)-(mu+gamma) x'=-omega*y+x*(1-x^2-y^2) y'=omega*x+y*(1-x^2-y^2) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Hodgkin-Huxley %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Name HodgkinHuxley Coordinates V,m,h,n Parameters Istim Equations gna=120 gk=36 gleak=.003 Ena=50 Ek=-90 Eleak=-70 am=0.1*(V+35)/(1-exp(-(V+35)/10)) bm=4*exp(-(V+60)/18) ah=0.07*exp(-(V+60)/20) bh=1/(1+exp(-(V+30)/10)) an=0.01*(V+50)/(1-exp(-(V+50)/10)) bn=0.125*exp(-(V+60)/80) minf=am/(am+bm) hinf=ah/(ah+bh) ninf=an/(an+bn) taum=1/(am+bm) tauh=1/(ah+bh) taun=1/(an+bn) V'=-(gna*m^3*h*(V-Ena)+gk*n^4*(V-Ek)+gleak*(V-Eleak))+Istim m'=(minf-m)/taum h'=(hinf-h)/tauh n'=(ninf-n)/taun %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% FHN in travelling coordinates %%% for waves and pulses %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Name FitzhughNagumo_wave Coordinates U,W,V Parameters aa,bb,cc Equations U'=W W'=cc*W+U*(U-aa)*(U-1.0)+V V'=bb*U/cc