Dynamical Systems in Neuroscience:

Solutions to Exercises, Chap. 10 51512. [MATLAB] An example is the file adjoint.mfunction Q=adjoint(F,t,x0)% finds solution to the Malkin’s adjoint equation Q’ = -DF^t Q% at time-points t with t(end) being the period% ’x0’ is a point on the limit cycle with zero phasetran=3; % the number of skipped cyclesdx = 0.000001; dy = 0.000001; % for evaluation of JacobianQ(1,:)=feval(F,0,x0)’;[t,x] = ode23s(F,t,x0);% initial point;% find limit cyclefor k=1:tranQ(length(t),:)=Q(1,:); % initial point;for i=length(t):-1:2% backward integrationL = [(feval(F,t(i),x(i,:)+[dx 0])-feval(F,t(i),x(i,:)))/dx,...(feval(F,t(i),x(i,:)+[0 dy])-feval(F,t(i),x(i,:)))/dy];Q(i-1,:) = Q(i,:) + (t(i)-t(i-1))*(Q(i,:)*L);end;end;Q = Q/(Q(1,:)*feval(F,0,x0)); % normalizationAn example of a call of the function is Q=adjoint(’F’,0:0.01:2*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)];13. [MATLAB] We assume that PRC (ϑ) is given by its truncated Fourier series with unknownFourier coefficients. Then, we find the coefficients that minimize the difference between predictedand actual interspike intervals. MATLAB file findprc.m takes the row vector of spikemoments, not counting the spike at time zero, and the input function p(t), determines thesampling frequency, the averaged period of oscillation and then calls file prcerror.m to findPRC.function PRC=findprc(sp,pp)global spikes p tau n% finds PRC of an oscillator theta’= 1 + PRC(theta)pp(t)% using the row-vector of spikes ’sp’ (when theta(t)=0)spikes = [0 sp];p=pp;tau = spikes(end)/length(p) % time step (sampling period)n=8; % The number of Fourier terms approximating PRCcoeff=zeros(1,2*n+1); % initial approximationcoeff(2*n+2) = spikes(end)/length(spikes); % initial periodcoeff=fminsearch(’prcerror’,coeff);a = coeff(1:n) % Fourier coefficients for sinb = coeff(n+1:2*n) % Fourier coefficients for cosb0= coeff(2*n+1) % dc termT = coeff(2*n+2) % period of oscillationPRC=b0+sum((ones(floor(T/tau),1)*a).*sin((tau:tau:T)’*(1:n)*2*pi/T),2)+...sum((ones(floor(T/tau),1)*b).*cos((tau:tau:T)’*(1:n)*2*pi/T),2);