function[Starts Goals Decoded Vec_l FirstError LastError Steps] = PhaseModel(PNoise,GC_cpm,DrawFig)
%% Phase coded vector cell model of vector navigation with grid cells
% Daniel Bush, UCL Institute of Cognitive Neuroscience
% Reference: Using Grid Cells for Navigation (2015) Neuron (in press)
% Contact: drdanielbush@gmail.com
%
% Inputs:
% PNoise = Standard deviation of phase noise (rad)
% GC_cpm = Number of grid cells per unique phase, in each module
% DrawFig = Plot figure of errors (0 / 1)
%
% Outputs:
% Starts = Random 2D start locations (m)
% Goals = Random 2D goal locations (m)
% Decoded = 2D translation vector decoded from grid cell activity (m)
% Vec_l = Length of decoded 2D translation vector (m)
% FirstError = Error in first decoded translation vector (m)
% LastError = Error in final decoded translation vector (m)
% Steps = No. of iterative steps used to compute translation vector
% Provide some parameters for the simulation
iterations = 1000; % How many iterations to run?
Range = 500; % Range (m)
GC_mps = 20; % Unique grid cell phases on each axis, per module
GC_scales = 0.25.*1.4.^(0:9); % Grid cell scales (m)
dt = 0.1; % Length of a theta cycle (s)
Emax_k = 0.01; % E-max winner-take-all parameter (see de Almeida et al., 2009)
N_vec_f = 12500; % Number of fine-grained vector cells (dendrites?)
N_vec_c = 1250; % Number of coarse-grained vector cells
% Assign the vectors encoded by the vector cells
Vectors_f = linspace(0,Range,N_vec_f+1); % Assign the linear range of vectors encoded by the fine-grained vector cells (dendrites)
Vectors_c = 0; % Assign the psuedo-exponential range of vectors encoded by the coarse-grained vector cells
for scale = 1 : length(GC_scales)
Vectors_c = [Vectors_c Vectors_c(end)+linspace(GC_scales(scale).*(Range/(sum(GC_scales)*100)),GC_scales(scale).*Range/sum(GC_scales),round(N_vec_c/length(GC_scales)))];
end
clear scale
% Compute grid cell peak firing locations and generate the delay line connectivity
Locations = (meshgrid(1:GC_mps,1:length(GC_scales))-1) / GC_mps .* repmat(GC_scales',1,GC_mps);
DelayLines = nan(length(GC_scales),length(Vectors_f));
for scale = 1 : length(GC_scales)
DelayLines(scale,:) = mod(GC_scales(scale)/2 - Vectors_f,GC_scales(scale))/GC_scales(scale)*dt;
end
clear scale
% Generate the the fine to coarse grained vector cell synaptic connectivity
Vec_fc_w = zeros(N_vec_f,N_vec_c);
for c = 1 : length(Vectors_f)
[i ind] = min(abs(Vectors_f(c)-Vectors_c));
Vec_fc_w(c,ind) = 1;
clear i ind
end
clear c
% Assign some memory for the output
Starts = nan(iterations,2); % Log of start positions
Goals = nan(iterations,2); % Log of goal positions
Decoded = nan(iterations,2,5); % Log of active vector cells on each axis
Vec_l = nan(iterations,1); % Log of true vector lengths
FirstError = nan(iterations,1); % Log of first distance error for each computed vector
LastError = nan(iterations,1); % Log of last distance error for each computed vector
Steps = nan(iterations,1); % Number of steps taken
% Then, for each iteration...
for i = 1 : iterations
% Update the user
if mod(i,iterations/10)==0
disp([int2str(i/iterations*100) '% complete...']);
drawnow
end
% Choose a random start and goal location
Starts(i,:) = [Range*rand Range*rand];
Goals(i,:) = [Range*rand Range*rand];
% Start the iterative vector navigation process
start = Starts(i,:);
step = 1;
stop = false;
while ~stop
% Then, for each axis...
for ax = 1 : 2
% Compute the phase of firing in each grid cell at the current
% location, in each direction along the axis
StartPhases1 = mod(mod((meshgrid(1:GC_mps,1:length(GC_scales))-1) / GC_mps .* repmat(GC_scales',1,GC_mps) - start(ax), repmat(GC_scales',1,GC_mps)) ./ repmat(GC_scales',1,GC_mps) * 2*pi + pi, 2*pi);
StartPhases2 = mod(mod(start(ax) - (meshgrid(1:GC_mps,1:length(GC_scales))-1) / GC_mps .* repmat(GC_scales',1,GC_mps), repmat(GC_scales',1,GC_mps)) ./ repmat(GC_scales',1,GC_mps) * 2*pi + pi, 2*pi);
% Compute the set of grid cells across modules that fire maximally at the goal
[Dist Cells] = min(abs(Locations - repmat(mod(Goals(i,ax),GC_scales)',1,GC_mps)),[],2);
Inds = sub2ind(size(Locations),(1:length(GC_scales))',Cells);
clear Dist Cells
% Compute the firing times of that set of grid cells across modules
FiringTimes1 = (repmat(StartPhases1(Inds),GC_cpm,1) + PNoise*randn(length(GC_scales)*GC_cpm,1))/(2*pi)*dt;
FiringTimes2 = (repmat(StartPhases2(Inds),GC_cpm,1) + PNoise*randn(length(GC_scales)*GC_cpm,1))/(2*pi)*dt;
clear Inds StartPhases1 StartPhases2
% Identify the coincidence with which these spikes arrive at each
% fine-grained vector cell (dendrite), in each direction
ArrivalTimes = [abs(sum(exp(1i.*(mod(repmat(DelayLines,GC_cpm,1) + repmat(FiringTimes1,1,N_vec_f+1),dt)/dt * 2 * pi))))./length(FiringTimes1) ; ...
abs(sum(exp(1i.*(mod(repmat(DelayLines,GC_cpm,1) + repmat(FiringTimes2,1,N_vec_f+1),dt)/dt * 2 * pi))))./length(FiringTimes2)];
clear FiringTimes1 FiringTimes2
% Implement the WTA algorithm and decode the vector as the weighted
% mean of all vector cells firing in each array
vector_out = double(ArrivalTimes>((1-Emax_k)*max(ArrivalTimes(:))));
Decoded(i,ax,step) = nanmean([Vectors_c((vector_out(1,:)*Vec_fc_w)>0) -Vectors_c((vector_out(2,:)*Vec_fc_w)>0)]);
clear ArrivalTimes vector_out
end
clear ax
% Then, unless the last position is within one metre of the
% goal, move to the new start position and take another step
if (sqrt(sum((start + 0.8 .*Decoded(i,:,step) - Goals(i,:)).^2)) > 1) && (step <= 5)
start = start + 0.8 .* Decoded(i,:,step);
step = step + 1;
else
stop = true;
end
end
% Compute the true vector length, first step error and total number of steps
FirstError(i,1) = sqrt(sum(((Goals(i,:) - Starts(i,:)) - Decoded(i,:,1)).^2,2));
LastError(i,1) = sqrt(sum((Goals(i,:) - start - Decoded(i,:,step)).^2,2));
Vec_l(i,1) = sqrt(sum((Goals(i,:) - Starts(i,:)).^2,2));
Steps(i,1) = step;
clear step start stop
end
% Plot vector length v error data, if required
if DrawFig
figure
subplot(2,2,1)
position = repmat(linspace(-0.5,0.5,100),100,1);
phase = -position*2*pi + PNoise*randn(100,100);
scatter(position(:),phase(:),'k.')
set(gca,'FontSize',14)
xlabel('Distance through the grid field (au)','FontSize',14)
ylabel('Theta firing phase (rad)','FontSize',14)
axis square
clear phase position
subplot(2,2,2)
scatter(mod(Vectors_f+GC_scales(1)/2,GC_scales(1))/GC_scales(1)-0.5,DelayLines(1,:)*1000,'k.','SizeData',800)
set(gca,'FontSize',14)
xlabel('Relative Position of Goal (s_i)','FontSize',14)
ylabel('Delay Line Length (ms)','FontSize',14)
axis square
subplot(2,2,3)
temp = histc(LastError,linspace(0,ceil(max(LastError)*100)/100,100)) ./ iterations;
bar(linspace(0,ceil(max(LastError)*100),100),temp,'FaceColor','k','EdgeColor','k')
set(gca,'FontSize',14)
xlabel('Error in Decoded Translation Vector (cm)','FontSize',14)
ylabel('Relative Frequency','FontSize',14)
axis square
subplot(2,2,4)
scatter(Vec_l,FirstError*100,'k.')
set(gca,'FontSize',14)
xlabel('True Translational Vector Length (m)','FontSize',14)
ylabel('First Decoded Translational Vector Error (cm)','FontSize',14)
b2 = regress(FirstError*100,[Vec_l ones(size(Vec_l,1),1)]);
hold on
plot(linspace(0,max(Vec_l),10),b2(2) + b2(1).*linspace(0,max(Vec_l),10),'r','LineWidth',3)
hold off
axis square
clear b2
end
clear i iterations DelayLines Locations dt error N_vec_f N_vec_c Vec_fc_w Vectors_f