% Hasselmo 2008's persistent firing oscillatory interference model
% eric zilli - 20110912 - v1.0
%
% In this variation of the classic oscillatory interference model,
% the sinusoidal oscillators are thresholded into a square wave that
% is intended to represent a dense train of spikes. The activities (*)
% of three(**) square wave trains encoding position along 120-degree-spaced
% directions are multiplied together to produce the output, which also
% has a rectangular-pulse shape.
%
% (*) Actually the activities had a nonzero threshold but this was not
% mentioned in the paper.
%
% (**) The paper says 3 but he used 6 oscillators with 60 degree spacings,
% but oscillators only change frequency when the animal is going within
% 90 degrees of their preferred directions, so only 3 of the oscillators
% may change in frequency at any time. Instead of using these oscillators
% directly (see BurgessEtAl2007_precession.m), the differences between the
% phases of oppositely directed pairs of oscillators are used as the
% oscillator phases to produce the main output. This is briefly mentioned
% without much detail on page 1217, bottom left.
%
% The manuscript examines grid cell theta phase precession in the model.
% Phase precession occurs when the spike times of the grid cell occur at
% earlier and earlier phases of a reference oscillation (in vivo it is the
% theta LFP) as an animal passes through a grid field. This model does not
% use a baseline oscillation, though, so we must be careful about how we
% determine the phase of what the reference oscillation would be. Figure
% 3's caption in the manuscript appears to suggest that a fixed frequency
% baseline oscillation is used to provide the phase reference, but in
% reality that plot was generated using one of the oscillators itself
% (whose phase is the sum of two oppositely directed VCOs) as the
% reference. As simulations show, this mechanism only produces precession
% for one direction through the field, and shows procession when the field
% is crossed in the other direction (e.g. set useRealTrajectory=0 and try
% running both positive and negatives directions along the x axis by
% changing the sign of the constant velocity input).
%
% As shown by our simulations below, when the proper baseline reference is
% used, the model does not show proper precession. Instead, when the grid
% cell is active, its steady input train of boxcars causes it to spike
% constantly over the entire range of baseline oscillator phases and often
% for more than one cycle.
%
% NB. Figure 3 is easily misinterpretted because the phase plots lack
% y axis labels. As a result, the plot appears to show 360 degrees of
% precession, when in reality it is much less than 90 degrees as careful
% comparison of lines 3 and 4 in that figure demonstrate.
%
% Note: This code is not necessarily optimized for speed, but is meant as
% a transparent implementation of the model as described in the manuscript.
%
% This code is released into the public domain. Not for use in skynet.
% if >0, plots the sheet of activity during the simulation on every livePlot'th step
livePlot = 20;
% plot spike phases vs. time at end
plotSpikePhases = 1;
% The model doesn't use a baseline, but we can make one up for the purpose
% of defining a baseline phase. If =0 reference the summed phases of
% the oscillations pointed both ways along one direction (if that makes sense.)
% If =2, reference the phase of individual VCO 3 (not summing its phase
% with the opposite VCO's).
referenceBaselinePhase = 0;
% if =0, just give constant velocity. if =1, load trajectory from disk
useRealTrajectory = 1;
constantVelocity = 1*[.5; 0*0.5]; % m/s
%% Simulation parameters
dt = .02; % time step, s
simdur = 200; % total simulation time, s
tind = 1; % time step number for indexing
t = 0; % simulation time variable, s
x = 0; % position, m
y = 0; % position, m
%% Model parameters
ncells = 1;
% Basline maintains a fixed frequency
baseFreq = 5; % Hz
% Grid orientation
orientation = 0; % rad
% Directional preference of each VCO (this also sets the number of VCOs)
dirPreferences = [0:2:5 1:2:5]*pi/3;
% dirPreferences = [0 pi];
% Basis vectors for each head direction
H = [cos(dirPreferences+orientation)' sin(dirPreferences+orientation)'];
% This will let us add/subtract dendritic values with opposite direction preferences:
oppositeVCOs = [1 0 0 -1 0 0; 0 1 0 0 -1 0; 0 0 1 0 0 -1];
% oppositeVCOs = [1 -1];
% Scaling factor relating speed to oscillator frequencies
% Paper (e.g. Figure 3) uses P(z) = 0.0193 or P(z) = 0.0048 with units found
% elsewhere in the paper to be cycles/cm = Hz/(cm/s).
% Pz = 0.48; % ventral; Hz/(m/s)
Pz = 1.93; % dorsal; Hz/(m/s)
% Threshold that determines whether the VCO output is 0 or 1
VCOThreshold = 0.5;
% Threshold that determines whether the grid cell output is a spike or not
spikeThreshold = 0;
%% History variables
speed = zeros(1,ceil(simdur/dt));
curDir = zeros(1,ceil(simdur/dt));
vhist = zeros(1,ceil(simdur/dt));
fhist = zeros(1,ceil(simdur/dt));
x = zeros(1,ceil(simdur/dt));
y = zeros(1,ceil(simdur/dt));
%% Firing field plot variables
nSpatialBins = 60;
minx = -0.90; maxx = 0.90; % m
miny = -0.90; maxy = 0.90; % m
occupancy = zeros(nSpatialBins);
spikes = zeros(nSpatialBins);
spikeTimes = [];
spikeCoords = [];
spikePhases = [];
%% Initial conditions
% Oscillators will start at phase 0:
VCOPhases = zeros(1,length(dirPreferences)); % rad
basePhase = 0; % rad
VCOActivity = zeros(length(dirPreferences)/2,1);
%% Make optional figure of sheet of activity
if livePlot
figh = figure('color','w');
if useRealTrajectory
set(figh,'position',[520 378 1044 420])
end
drawnow
end
%% Possibly load trajectory from disk
if useRealTrajectory
load data/HaftingTraj_centimeters_seconds.mat;
% interpolate down to simulation time step
pos = [interp1(pos(3,:),pos(1,:),0:dt:pos(3,end));
interp1(pos(3,:),pos(2,:),0:dt:pos(3,end));
interp1(pos(3,:),pos(3,:),0:dt:pos(3,end))];
pos(1:2,:) = pos(1:2,:)/100; % cm to m
vels = [diff(pos(1,:)); diff(pos(2,:))]/dt; % m/s
x = pos(1,1); % m
y = pos(2,1); % m
end
%% !! Main simulation loop
fprintf('Simulation starting. Press ctrl+c to end...\n')
while t0);
% VCO frequencies are pushed up or down from the baseline frequency
% depending on the speed and head direction, with a scaling factor Pz
% that sets the spacing between the spatial grid fields.
VCOFreqs = baseFreq + Pz*h'.*VCOMask; % Hz
% Advance oscillator phases
% Radial frequency (2pi times frequency in Hz) is the time derivative of phase.
VCOPhases = VCOPhases + dt*2*pi*VCOFreqs; % rad
% Note: baseline oscillation is not used in model, but we use it to
% determine spike phase (though, in vivo, it is not clear why theta phase
% would be equal to this phase).
basePhase = basePhase + dt*2*pi*baseFreq; % rad
% Sum phases of oscillators with opposite direction preferences
% NB. not clear how this could be carried out biologically
summedPhases = oppositeVCOs*VCOPhases';
% Take cosine of phases to get current oscillation activity
summedOpposites = cos(summedPhases);
% Sum each VCO activation
% "~VCOActivity" is 1 for those VCOs that were not over threshold
% on the previous step and is 0 for those VCOs that were. We can implement
% the refractory period that was not stated in the manuscript by
% multiplying the thresholded oscillator values on the current step by
% that vector.
VCOActivity = heaviside(summedOpposites-VCOThreshold);
% Final activity is the product of the oscillations.
f = prod(VCOActivity);
% f = sum(VCOActivity);
% Save for later
fhist(tind) = f;
% Save firing field information
if f>spikeThreshold
spikeTimes = [spikeTimes; t];
spikeCoords = [spikeCoords; x(tind) y(tind)];
if referenceBaselinePhase==1
spikePhases = [spikePhases; basePhase];
elseif referenceBaselinePhase==2
spikePhases = [spikePhases; VCOPhases(3)];
else
spikePhases = [spikePhases; summedPhases(3)];
end
end
if useRealTrajectory
xindex = round((x(tind)-minx)/(maxx-minx)*nSpatialBins)+1;
yindex = round((y(tind)-miny)/(maxy-miny)*nSpatialBins)+1;
occupancy(yindex,xindex) = occupancy(yindex,xindex) + dt;
spikes(yindex,xindex) = spikes(yindex,xindex) + double(f>spikeThreshold);
end
if livePlot>0 && (livePlot==1 || mod(tind,livePlot)==1)
if ~useRealTrajectory
figure(figh);
subplot(121);
plot((0:tind-1)*dt,fhist(1:tind));
set(gca,'ylim',[-0.1 1.1]);
title('Activity');
xlabel('Time (s)')
axis square
set(gca,'ydir','normal')
title(sprintf('t = %.1f s',t))
subplot(122);
plot(x(1:tind),y(1:tind))
hold on;
if ~isempty(spikeCoords)
cmap = jet;
cmap = [cmap((end/2+1):end,:); cmap(1:end/2,:)];
phaseInds = mod(spikePhases,2*pi)*(length(cmap)-1)/2/pi;
pointColors = cmap(ceil(phaseInds)+1,:);
scatter3(spikeCoords(:,1), ...
spikeCoords(:,2), ...
zeros(size(spikeCoords(:,1))), ...
30*ones(size(spikeCoords(:,1))), ...
pointColors, ...
'o','filled');
end
axis square
title({'Trajectory (blue) and',...
'spikes (colored by theta phase',...
'blues before baseline peak, reds after)'})
drawnow
else
figure(figh);
subplot(131);
plot((0:tind-1)*dt,fhist(1:tind));
set(gca,'ylim',[-0.1 1.1]);
hold on;
title('Activity (blue)');
xlabel('Time (s)')
axis square
set(gca,'ydir','normal')
subplot(132);
imagesc(spikes./occupancy);
axis square
set(gca,'ydir','normal')
title({'Rate map',sprintf('t = %.1f s',t)})
subplot(133);
plot(x(1:tind),y(1:tind))
hold on;
if ~isempty(spikeCoords)
cmap = jet;
cmap = [cmap((end/2+1):end,:); cmap(1:end/2,:)];
phaseInds = mod(spikePhases,2*pi)*(length(cmap)-1)/2/pi;
pointColors = cmap(ceil(phaseInds)+1,:);
scatter3(spikeCoords(:,1), ...
spikeCoords(:,2), ...
zeros(size(spikeCoords(:,1))), ...
30*ones(size(spikeCoords(:,1))), ...
pointColors, ...
'o','filled');
end
axis square
title({'Trajectory (blue) and',...
'spikes (colored by theta phase',...
'blues before baseline peak, reds after)'})
drawnow
end
end
end
if plotSpikePhases
figure;
% shift spikePhases by pi to center pattern in plot
plot(spikeTimes,mod(pi+spikePhases,2*pi),'.')
title('Spike phase vs. time')
xlabel('Time (s)')
ylabel('Phase (rad)')
end