Artificial neuron model (Izhikevich 2003, 2004, 2007)

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Accession:39948
A set of models is presented based on 2 related parameterizations to reproduce spiking and bursting behavior of multiple types of cortical neurons and thalamic neurons. These models combine the biologically plausibility of Hodgkin Huxley-type dynamics and the computational efficiency of integrate-and-fire neurons. Using these model, one can simulate tens of thousands of spiking cortical neurons in real time (1 ms resolution) using a desktop PC.
References:
1 . Izhikevich EM (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting
2 . Izhikevich EM (2004) Which model to use for cortical spiking neurons? IEEE Trans Neural Netw 15:1063-70 [PubMed]
3 . Izhikevich EM (2003) Simple model of spiking neurons. IEEE Trans Neural Netw 14:1569-72 [PubMed]
Model Information (Click on a link to find other models with that property)
Model Type: Neuron or other electrically excitable cell;
Brain Region(s)/Organism:
Cell Type(s): Abstract Izhikevich neuron;
Channel(s):
Gap Junctions:
Receptor(s):
Gene(s):
Transmitter(s):
Simulation Environment: NEURON; MATLAB;
Model Concept(s): Action Potential Initiation; Activity Patterns; Parameter Fitting; Simplified Models; Tutorial/Teaching; Action Potentials;
Implementer(s): Lytton, William [billl at neurosim.downstate.edu]; Dura-Bernal, Salvador [salvadordura at gmail.com];
%   This MATLAB file generates figure 2 in the paper by 
%                   Izhikevich E.M. (2004) 
%           "Simple model of spiking neurons"
%
%          and figure 1 in the paper by
%               Izhikevich E.M. (2004) 
%   "Which Model to Use For Cortical Spiking Neurons?" 
%
%   use MATLAB R13 or later. November 2003. San Diego, CA 

%%
figure;
%%%%%%%%%%%%%%% regular spiking (RS) %%%%%%%%%%%%%%%%%%%%%%
subplot(2,4,1) 
a=0.02; b=0.2;  c=-65;  d=8;
V=-63;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:150;
T1=tspan(end)/10;
for t=tspan
    if (t>T1) 
        I=14;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('regular spiking (RS)');


%%%%%%%%%%%%%%% intrinsically bursting (IB) %%%%%%%%%%%%%%%%%%%%%%
subplot(2,4,2) 
a=0.02; b=0.2;  c=-55;  d=4;
V=-70;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:150;
T1=tspan(end)/10;
for t=tspan
    if (t>T1) 
        I=11;%14;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('intrinsically bursting (IB)');

%%%%%%%%%%%%%%% chattering (CH) %%%%%%%%%%%%%%%%%%%%%%
subplot(2,4,3) 
a=0.02; b=0.2;  c=-50;  d=2;
V=-70;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:150;
T1=tspan(end)/10;
for t=tspan
    if (t>T1) 
        I=10;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('chattering (CH)');


%%%%%%%%%%%%%%% fast spiking (FS) %%%%%%%%%%%%%%%%%%%%%%
subplot(2,4,4) 
a=0.1; b=0.2;  c=-65;  d=2;
V=-70;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:150;
T1=tspan(end)/10;
for t=tspan
    if (t>T1) 
        I=10;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title(' fast spiking (FS)');



%%%%%%%%%%%%%%% thalamo-cortical (TC) %%%%%%%%%%%%%%%%%%%%%%
subplot(2,4,5) 
a=0.02; b=0.25;  c=-65;  d=0.05;
V=-63;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:150;
T1=2*tspan(end)/10;
for t=tspan
    if (t>T1) 
        I=1.5;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('thalamo-cortical (TC)');


%%%%%%%%%%%%%%% thalamo-cortical burst (TC) %%%%%%%%%%%%%%%%%%%%%%
subplot(2,4,6) 
a=0.02; b=0.25;  c=-65;  d=0.05;
V=-87;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:150;
T1=3*tspan(end)/10;
for t=tspan
    if (t>T1) 
        I=0.0;
    else
        I=-25;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('thalamo-cortical burst (TC)');

%%%%%%%%%%%%%%%%%%  resonator (RZ) %%%%%%%%%%%%%%%%%%%%%%%%
subplot(2,4,7) 
a=0.1;  b=0.26; c=-65;  d=2;
V=-70;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:100;
T1=tspan(end)/10;
T2=T1+50;
for t=tspan
    if ((t>T2) & (t<T2+5))
        I=10;
    elseif (t>T1)
        I=-0.5;
    else
        I=-2;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 T2 T2 T2+5 T2+5 T2+5 max(tspan)],-90+[0 0 10 10 20 20 10 10 10]); 
axis([0 max(tspan) -90 30])
axis off;
title('resonator (RZ)');


%%%%%%%%%%%%%%% low-threshold spiking (LTS) %%%%%%%%%%%%%%%%%%%%%%
subplot(2,4,8) 
a=0.02; b=0.25;  c=-65;  d=2;
V=-63;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:250;
T1=1*tspan(end)/10;
for t=tspan
    if (t>T1) 
        I=10.0;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('low-threshold spiking (LTS)');



%%
figure;

%%%%%%%%%%%%%%% (A) tonic spiking %%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,1) 
a=0.02; b=0.2;  c=-65;  d=6;
V=-70;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:100;
T1=tspan(end)/10;
for t=tspan
    if (t>T1) 
        I=14;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('(A) tonic spiking');

%%%%%%%%%%%%%%%%%% (B) phasic spiking %%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,2)%  
a=0.02; b=0.25; c=-65;  d=6;
V=-64; u=b*V;
VV=[];  uu=[];
tau = 0.25;tspan = 0:tau:200;
T1=20;
for t=tspan
    if (t>T1) 
        I=0.5;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('(B) phasic spiking');

%%%%%%%%%%%%%% (C) tonic bursting %%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,3)  
a=0.02; b=0.2;  c=-50;  d=2;
V=-70;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:220;
T1=22;
for t=tspan
    if (t>T1) 
        I=15;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('(C) tonic bursting');

%%%%%%%%%%%%%%% (D) phasic bursting %%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,4)   
a=0.02; b=0.25; c=-55;  d=0.05;
V=-64;  u=b*V;
VV=[];  uu=[];
tau = 0.2;  tspan = 0:tau:200;
T1=20;
for t=tspan
    if (t>T1) 
        I=0.6;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('(D) phasic bursting');


%%%%%%%%%%%%%%% (E) mixed mode %%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,5) 
a=0.02; b=0.2;  c=-55;  d=4;
V=-70;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:160;
T1=tspan(end)/10;
for t=tspan
    if (t>T1) 
        I=10;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('(E) mixed mode');


%%%%%%%%%%%%%%%% (F) spike freq. adapt %%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,6)  
a=0.01; b=0.2;  c=-65;  d=8;
V=-70;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:85;
T1=tspan(end)/10;
for t=tspan
    if (t>T1) 
        I=30;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 max(tspan)],-90+[0 0 10 10]);
axis([0 max(tspan) -90 30])
axis off;
title('(F) spike freq. adapt');

%%%%%%%%%%%%%%%%% (G) Class 1 exc. %%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,7)  
a=0.02; b=-0.1; c=-55; d=6;
V=-60; u=b*V;
VV=[]; uu=[];
tau = 0.25; tspan = 0:tau:300;
T1=30;
for t=tspan
    if (t>T1) 
        I=(0.075*(t-T1)); 
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+4.1*V+108-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 max(tspan) max(tspan)],-90+[0 0 20 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(G) Class 1 excitable');

%%%%%%%%%%%%%%%%%% (H) Class 2 exc. %%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,8)  
a=0.2;  b=0.26; c=-65;  d=0;
V=-64;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:300;
T1=30;
for t=tspan
    if (t>T1) 
        I=-0.5+(0.015*(t-T1)); 
    else
        I=-0.5;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 max(tspan) max(tspan)],-90+[0 0 20 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(H) Class 2 excitable');

%%%%%%%%%%%%%%%%% (I) spike latency %%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,9) 
a=0.02; b=0.2;  c=-65;  d=6;
V=-70;  u=b*V;
VV=[];  uu=[];
tau = 0.2; tspan = 0:tau:100;
T1=tspan(end)/10;
for t=tspan
    if t>T1 & t < T1+3 
        I=7.04;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 T1+3 T1+3 max(tspan)],-90+[0 0 10 10 0 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(I) spike latency');


%%%%%%%%%%%%%%%%% (J) subthresh. osc. %%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,10) 
a=0.05; b=0.26; c=-60;  d=0;
V=-62;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:200;
T1=tspan(end)/10;
for t=tspan
    if (t>T1) & (t < T1+5) 
        I=2;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 (T1+5) (T1+5) max(tspan)],-90+[0 0 10 10 0 0],...
      tspan(220:end),-10+20*(VV(220:end)-mean(VV)));
axis([0 max(tspan) -90 30])
axis off;
title('(J) subthreshold osc.');


%%%%%%%%%%%%%%%%%% (K) resonator %%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,11) 
a=0.1;  b=0.26; c=-60;  d=-1;
V=-62;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:400;
T1=tspan(end)/10;
T2=T1+20;
T3 = 0.7*tspan(end);
T4 = T3+40;
for t=tspan
    if ((t>T1) & (t < T1+4)) | ((t>T2) & (t < T2+4)) | ((t>T3) & (t < T3+4)) | ((t>T4) & (t < T4+4)) 
        I=0.65;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 (T1+8) (T1+8) T2 T2 (T2+8) (T2+8) T3 T3 (T3+8) (T3+8) T4 T4 (T4+8) (T4+8) max(tspan)],-90+[0 0 10 10 0 0 10 10 0 0 10 10 0 0 10 10 0 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(K) resonator');

%%%%%%%%%%%%%%%% (L) integrator %%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,12) 
a=0.02; b=-0.1; c=-55; d=6;
V=-60; u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:100;
T1=tspan(end)/11;
T2=T1+5;
T3 = 0.7*tspan(end);
T4 = T3+10;
for t=tspan
    if ((t>T1) & (t < T1+2)) | ((t>T2) & (t < T2+2)) | ((t>T3) & (t < T3+2)) | ((t>T4) & (t < T4+2)) 
        I=9;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+4.1*V+108-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 (T1+2) (T1+2) T2 T2 (T2+2) (T2+2) T3 T3 (T3+2) (T3+2) T4 T4 (T4+2) (T4+2) max(tspan)],-90+[0 0 10 10 0 0 10 10 0 0 10 10 0 0 10 10 0 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(L) integrator');

%%%%%%%%%%%%%%%%% (M) rebound spike %%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,13)  
a=0.03; b=0.25; c=-60;  d=4;
V=-64;  u=b*V;
VV=[];  uu=[];
tau = 0.2;  tspan = 0:tau:200;
T1=20;
for t=tspan
    if (t>T1) & (t < T1+5) 
        I=-15;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 (T1+5) (T1+5) max(tspan)],-85+[0 0 -5 -5 0 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(M) rebound spike');

%%%%%%%%%%%%%%%%% (N) rebound burst %%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,14)  
a=0.03; b=0.25; c=-52;  d=0;
V=-64;  u=b*V;
VV=[];  uu=[];
tau = 0.2;  tspan = 0:tau:200;
T1=20;
for t=tspan
    if (t>T1) & (t < T1+5) 
        I=-15;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 (T1+5) (T1+5) max(tspan)],-85+[0 0 -5 -5 0 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(N) rebound burst');

%%%%%%%%%%%%%%%%% (O) thresh. variability %%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,15)  
a=0.03; b=0.25; c=-60;  d=4;
V=-64;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:100;
for t=tspan
   if ((t>10) & (t < 15)) | ((t>80) & (t < 85)) 
        I=1;
    elseif (t>70) & (t < 75)
        I=-6;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 10 10 15 15 70 70 75 75 80 80 85 85 max(tspan)],...
          -85+[0 0  5  5  0  0  -5 -5 0  0  5  5  0  0]);
axis([0 max(tspan) -90 30])
axis off;
title('(O) thresh. variability');


%%%%%%%%%%%%%% (P) bistability %%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,16) 
a=0.1;  b=0.26; c=-60;  d=0;
V=-61;  u=b*V;
VV=[];  uu=[];
tau = 0.25; tspan = 0:tau:300;
T1=tspan(end)/8;
T2 = 216;
for t=tspan
    if ((t>T1) & (t < T1+5)) | ((t>T2) & (t < T2+5)) 
        I=1.24;
    else
        I=0.24;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1 T1 (T1+5) (T1+5) T2 T2 (T2+5) (T2+5) max(tspan)],-90+[0 0 10 10 0 0 10 10 0 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(P) bistability');


%%%%%%%%%%%%%% (Q) DAP %%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,17) 
a=1;  b=0.2; c=-60;  d=-21;
V=-70;  u=b*V;
VV=[];  uu=[];
tau = 0.1; tspan = 0:tau:50;
T1 = 10;
for t=tspan
     if abs(t-T1)<1 
        I=20;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 T1-1 T1-1 T1+1 T1+1 max(tspan)],-90+[0 0 10 10 0 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(Q) DAP         ');



%%%%%%%%%%%%%% (R) accomodation %%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,18) 
a=0.02;  b=1; c=-55;  d=4;
V=-65;  u=-16;
VV=[];  uu=[];  II=[];
tau = 0.5; tspan = 0:tau:400;
for t=tspan
    if (t < 200)
        I=t/25;
    elseif t < 300
        I=0;
    elseif t < 312.5
        I=(t-300)/12.5*4;
    else
        I=0;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*(V+65));
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
    II(end+1)=I;
end;
plot(tspan,VV,tspan,II*1.5-90);
axis([0 max(tspan) -90 30])
axis off;
title('(R) accomodation');

%%%%%%%%%%%%%% (S) inhibition induced spiking %%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,19) 
a=-0.02;  b=-1; c=-60;  d=8;
V=-63.8;  u=b*V;
VV=[];  uu=[];
tau = 0.5; tspan = 0:tau:350;
for t=tspan
       if (t < 50) | (t>250)
        I=80;
    else
        I=75;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 50 50 250 250 max(tspan)],-80+[0 0 -10 -10 0 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(S) inh. induced sp.');

%%%%%%%%%%%%%% (T) inhibition induced bursting %%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(5,4,20) 
a=-0.026;  b=-1; c=-45;  d=-2;
V=-63.8;  u=b*V;
VV=[];  uu=[];
tau = 0.5; tspan = 0:tau:350;
for t=tspan
       if (t < 50) | (t>250)
        I=80;
    else
        I=75;
    end;
    V = V + tau*(0.04*V^2+5*V+140-u+I);
    u = u + tau*a*(b*V-u);
    if V > 30
        VV(end+1)=30;
        V = c;
        u = u + d;
    else
        VV(end+1)=V;
    end;
    uu(end+1)=u;
end;
plot(tspan,VV,[0 50 50 250 250 max(tspan)],-80+[0 0 -10 -10 0 0]);
axis([0 max(tspan) -90 30])
axis off;
title('(T) inh. induced brst.');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

set(gcf,'Units','normalized','Position',[0.3 0.1 0.6 0.8]);
 

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