introducing the MatSumSig model.

I haven't really given this beasty much thought, but it was the original motivator for a lot of my later ideas, so I should mention it! And at this point it doesn't even matter if my MatSumSig model is wrong! The BKO scheme it motivated, by itself is interesting and useful.

It makes use of my function matrix notation, and I consider it a simplified model of a single neuron (in the physics tradition of if something is too hard, simplify it until you have something you can actually work with, at least as a starting point).

[ y1 ]   [ s1[x1,t1] ] [ sum[x1,t] ] [ a1 a2 a3 a4 a5 ] [ x1 ]
[ y2 ]   [ s2[x1,t2] ]                                  [ x2 ]
[ y3 ]   [ s3[x1,t3] ]                                  [ x3 ]
[ y4 ] = [ s4[x1,t4] ]                                  [ x4 ]
[ y5 ]   [ s5[x1,t5] ]                                  [ x5 ]
[ y6 ]   [ s6[x1,t6] ]                                  
[ y7 ]   [ s7[x1,t7] ]                                  
[ y8 ]   [ s8[x1,t8] ] 

where:

{a1,a2,a3,a4,a5} are reals/floats and can be positive or negative.
sum[x,t] sums the input x for a time-slice of length t (with output 0 during this time slice), then spits out the result at the end of that time slice. 
If we don't include the sum[] term, assume t = 0.
Indeed, we only need t > 0 if we want time-dependence. 
s_k[x,t_k] are sigmoids, with passed in parameter t_k.

Note that there are a lot of free parameters here, and I have no idea how the brain tweaks them! We have {a1,a2,a3,a4,a5}, {t,t1,t2,...,t8}, and then we have the sigmoids {s1,s2,..,s8} Indeed, until we have some idea how to fill in all these parameters we can't actually make use of this model/representation.