Dynamical Systems in Neuroscience:

Simple Models 287membrane potential, v (mV)40200-20-40(a) (b) v peak-600 200 400 600 800 1000time (ms)upstrokev resetAHP20 msdownstroke(reset)recovery variable, u500-50I=0 I=70AHPrestingu-nullclinev-nullclineafter-spike resetspiking limit cycle attractor-60-40-200 20 35v resetvpeakmembrane potential, v (mV)v r v t(c)Figure 8.6: (a) Output of the MATLAB code simulating the simple model (8.5, 8.6).(b) Comparison of the simulated (continuous curve) and experimental (dashed curve)voltage traces shows two major discrepancies marked by arrows. (c) Phase portrait ofthe model.v=vr*ones(1,n); u=0*v; % initial valuesI=[zeros(1,0.1*n),70*ones(1,0.9*n)];% pulse of input dc-currentfor i=1:n-1% forward Euler methodv(i+1)=v(i)+tau*(k*(v(i)-vr)*(v(i)-vt)-u(i)+I(i))/C;u(i+1)=u(i)+tau*a*(b*(v(i)-vr)-u(i));if v(i+1)>=vpeak% a spike is fired!v(i)=vpeak;% padding the spike amplitudev(i+1)=c;% membrane voltage resetu(i+1)=u(i+1)+d;% recovery variable updateend;end;plot(tau*(1:n), v);% plot the resultNotice that the spikes were padded to v peak to avoid amplitude jitter associatedwith the finite simulation time step tau=1 ms. In Fig. 8.6b we magnify the simulatedvoltage trace and compare it with recording of a neocortical pyramidal neuron (dashedcurve). There are two discrepancies, marked by arrows: The pyramidal neuron has(1) sharper spike upstroke and (2) smoother spike downstroke. The first discrepancycan be removed by assuming that the coefficient k of the square polynomial in (8.5)