Thanks to Will at Stack Overflow , who suggested using histc(). This is still not using the Mapping Toolbox though, which would be nice.
my_colormap = [204 204 204
153 153 153
255 255 178
254 204 92
253 141 60
240 59 32
189 0 38
0 0 0]/255 ;
binEdges = [0 0.001 0.005 0.01 0.05 0.1 0.25 0.5 1] ;
labels = textscan(num2str(binEdges*100),'%s') ;
labels = labels{1} ;
labels{length(labels)} = [labels{length(labels)} '%'] ;
[~,indices] = histc(map_data,binEdges);
indices(isnan(map_data)) = NaN ;
indices(isinf(map_data)) = NaN ;
figure ;
pcolor(indices-1) % Instead of image(), to display NaN values as white
shading flat
axis equal tight
colormap(gca,my_colormap); % gca as first argument prevents
% colormap from changing for all
% subsequent plots
h = colorbar;
caxis([0 length(binEdges)-1])
h.YTickLabel = labels ;