Dynamical Systems in Neuroscience:

Solutions to Exercises, Chap. 10 513hold off;The call of the function is isochrons(’F’,0:0.1:2*pi,[1;0]); withfunction dx = F(t,x);z=x(1)+1i*x(2);dz=(1+1i)*z-z*z*conj(z);dx=[real(dz); imag(dz)];4. (Hoppensteadt and Keener 1982) From calculus, B · C = |B| |C| cos(ψ). Since |B| = 1 andC = (A + cos ϑ, sin ϑ), see Fig. 10.46, we have B · C = A cos ϑ + cos 2 ϑ + sin 2 ϑ. Hence,cos ψ = (1+A cos ϑ)/ √ 1 + 2A cos ϑ + A 2 . When ϑ is in the upper (lower) half-plane, the phaseis delayed (advanced).5. An example is the file prc.mfunction PRC=prc(F,phases,x0,A)% plot phase-resetting curve (PRC) of system x’=F(t,x) + A delta(t)% at points given by the vector ’phases’.% ’x0’ is a point on the limit cycle with zero phase% A is the strength of stimulation (row-vector)% use peaks of spikes to find the phase differencesT= phases(end); % is the period of the cycletau = T/6000; % time step of integrationk=3;% the number of cycles needed to determine the new phasePRC=[];[tc,lc] = ode23s(F,0:tau:k*T,x0); % find limit cyclepeak=1+find(lc(2:end-1,1)>lc(1:end-2,1)&lc(2:end-1,1)>=lc(3:end,1));peak0 = tc(peak(end)); % the last peak is used for referencefor i=1:length(phases)[m,j]=min(abs(phases(i)-tc));[t,x] = ode23s(F,phases(i):tau:k*T,lc(j,:)+A); % stimulatepeaks=1+find(x(2:end-1,1)>x(1:end-2,1)&x(2:end-1,1)>=x(3:end,1));PRC=[PRC, mod(T/2+peak0-t(peaks(end)),T)-T/2];subplot(2,1,2);drawnow;plot(phases(1:length(PRC)),PRC);xlabel(’phase of stimulation’);ylabel(’induced phase difference’);subplot(2,1,1);plot(tc,lc(:,1),’r’,t,x(:,1),t(peaks(end)),x(peaks(end),1),’ro’);xlabel(’time’);ylabel(’membrane potential’);end;An example of a call of the function is PRC=prc(’F’,0:0.1:2*pi,[-1 0],[0.1 0]); withfunction dx = F(t,x);z=x(1)+1i*x(2);dz=(1+1i)*z-z*z*conj(z);dx=[real(dz); imag(dz)];6.grad ϑ(x) ==( ϑ(x + h1 ) − ϑ(x), . . . , ϑ(x + h )m) − ϑ(x)h 1h m( PRC1 (ϑ(x), h 1 ), . . . , PRC )m(ϑ(x), h m )h 1 h m