Synchrony_probability_Ising_model.m

clear all;
clf

CM = hsv(3);

% Choose between Matlab optimization = 1
% or minimize.m minimization function = 2.
alg_choice = 1;

% Number of neurons.
N = 100;

% Load data from a file. 
% The file.dat has the form:
% p(x_1) p(x_2) ... p(x_N) mu rho % trial number 1
% p(x_1) p(x_2) ... p(x_N) mu rho % trial number 2
% We keep mu and rho fixed over trials. We have multiple
% trials so that we can get an average and smooth out
% some of the inherent noise.
M{1} = importdata('figure_2a_0.325.dat',' ');
M{2} = importdata('figure_2a_0.49.dat',' ');
M{3} = importdata('figure_2a_0.73.dat',' ');

% List the possible states, using the symmetry argument.
% Symmetry argument can be found in Macke 2011.
% Essentially says that the parameters in the Ising model h_i, J_ij
% using symmetry conditions can be reduced to just 2 parameters. The 
% symmetry condition means that we talk about population spike 
% counts rather than particular probabilities. i.e P(101101) becomes 
% just P(4) as 4 spikes occurred. 
% Here we list the possible states which are just 1...N and at the
% same time we list their squares as needed in the Ising model.
states = [(0:N)',(0:N)'.^2];

% Generate a list of binomial coefficients.
warning off;
binom = zeros(N+1,1);
for i=0:N
    binom(i+1) = nchoosek(N,i);
end

% An initial guess at the parameters to be minimized:
param_list_init = [0;0];

for j=1:3
    % PP is a matrix where each row is a prob dist.
    PP  = M{j}(:,1:N+1);
    mu  = M{j}(:,N+2);
    rho = M{j}(:,N+3);

    % Take the mean of the input p(x).
    % i.e we average over the trials
    % p(x_1) p(x_2) ... p(x_N) mu rho % trial number 1
    % p(x_1) p(x_2) ... p(x_N) mu rho % trial number 2
    P = mean(PP,1)';

    % Output mean firing rate and corr coefficient.
    mean_mu = mean(mu);
    mean_rho = mean(rho);

    % Calculate the means and moments i.e. the quantities
    % sum(k*P_k) and sum(k^2*P_k).
    mean_feature = P'*states;

    % Choose the algorithm to use.
    if alg_choice == 1
        % MATLAB Optimization Toolbox minimization function.
        % Uses Optimization toolbox unconstrained minimization function.
        options = optimset('GradObj','on','LargeScale','on',...
            'Display','off',...
            'MaxFunEvals',1000,'MaxIter',1000,'TolFun',1e-10,'TolX',1e-10);
        % We set GradObj = on as we supply the gradient.
        % We set LargeScale = on as that is the algorithm that uses the
        % user supplied gradient.
        % Display just shows some accuracy output.
        % The final options are tolerances etc.

        % The minimization function
        [param_list] =...
              fminunc(@(x)neg_log_like_binom...
              (x,states,mean_feature,P,binom),param_list_init,options);
    elseif alg_choice == 2
        % Eric's minimization function, based on conjugate gradient.
        param_list = minimize(param_list_init,...
              'neg_log_like_binom',200,...
              states,mean_feature,P,binom);
    end  

    % Use the parameters found via minimization to calculate the probability
    % distribution from the Ising model.
    % The binomial prefactor is explained in Macke 2011. 
    Q_unnormalized = binom.*exp(states*param_list);

    % Normalize this Q.
    % This gives us a probability distribution Q(k), the probability of
    % finding a population with count k.
    Q = Q_unnormalized/sum(Q_unnormalized);
    
    % Plot output.
    semilogy(Q,'color',CM(j,:));
    hold on
end

title('Ising Model Prediction','fontsize',18)
fig2b_leg = legend('\rho = 0.05','\rho = 0.1','\rho = 0.25');
set(fig2b_leg,'FontSize',16);
xlabel('Population spike count i.e synchrony','fontsize',16)
ylabel('Probability','fontsize',16)
axis([0 N 1e-4 1]);