% SecondSite
%
% This program demonstrates by simulation that it is
% possible for background signals from one cone class to 
% influence thresholds for another cone class without
% violating cone independence for appearance experiments.
%
% Although this program simulates a very simple dichromatic
% visual system, we believe the point made holds
% generally.
%
% 11/9/99  dhb  Wrote it.

% Clean up at the start.
clear all; close all;

% The example is for two cone classes, A and B.
% There are two post-receptoral channels, 1 and
% 2.  Channel 1 receives signals from class A only.
% Channel 2 sums signals from classes A and B and
% applies a compressive (logarithmic) non-linearity
% applied to the sum.  Otherwise the system is
% static.  There are no gain changes, no subtractive
% adaptations, and the noise that limits threshold
% is independent of the stimulus.

% THRESHOLDS:  We consider thresholds for signals
% originating in class B.  Since channel 1 is
% unaffected by signals from class B, we need only consider
% channel 2.  We assume that the noise limiting
% detection is constant, and that this noise
% is injected after the channel 2 non-linearity.
% Thus threshold will be reached when change
% in channel 2 output caused by adding the test
% to the background produces a constant response
% difference in channel 2.  Let this constant
% difference be 1.

% Let the background produce 1 unit of quantal absorbtions
% in cone class B and varying amounts of absorptions in cone
% class A.  We compute thresholds for incremental lights
% that only stimulate cone class B as a function of the
% background level for class A.
bgB = 1;
bgsA = 0:0.1:2;
for i = 1:length(bgsA)
	bgA = bgsA(i);
	excite2Bg = bgA+bgB;
	resp2Bg = log10(excite2Bg);
	threshResp2 = resp2Bg+1;
	threshExcite2 = 10.^threshResp2;
	threshB = threshExcite2-excite2Bg;
	threshsB(i) = threshB;
end

% Plotting the results shows that, as expected,
% the threshold for a B cone isolating test depends
% on the level of A cone stimulation in the background.
figure(1); clf;
plot(bgsA,threshsB,'r:');
hold on
plot(bgsA,threshsB,'r*');
hold off
title('Thresholds');
xlabel('Background A level');
ylabel('Threshold for B cone stimuli');
drawnow;

% APPEARANCE:  We simulate asymmetric matches.
% Two lights seen on different backgrounds will
% match if and only if the outputs of both channels 
% for the test and match conditions are the same.
% For our purposes here, it is the response to the
% background and test together that matters, since
% there is no subtractive adaptation in our model.

% We work out the AB cone coordinates of a
% match stimulus seen on a match background
% required to produce the same appearance
% as a test stimulus seen on a test background.
% The simulation shows that the B cone component
% of the match stimulus is independent of the
% A cone level of the match background.
testBgA = 1;
testBgB = 1;
matchBgsA = 0:0.1:2;
matchBgB = 1;
testA = 1;
testB = 1;
for i = 1:length(matchBgsA)
	matchBgA = matchBgsA(i);
	testResp1 = testBgA+testA;
	testResp2 = log10(testBgA+testBgB+testA+testB);
	matchA = testResp1-matchBgA;
	matchExcite2 = 10.^testResp2;
	matchB = matchExcite2 - matchBgA - matchBgB - matchA;
	matchesB(i) = matchB;
end

% The plot shows the result claimed.
figure(2); clf
plot(matchBgsA,matchesB+matchBgB,'r:');
hold on
plot(matchBgsA,matchesB+matchBgB,'r*');
hold off
title('Appearance');
axisLength = axis;
axis([axisLength(1) axisLength(2) 0 4]);
xlabel('Match background A level');
ylabel('B component of asymmetric match');
drawnow;