simm in the MatSumSig model

So, we have basics like simple logic, and set union and intersection in the MatSumSig model. Interestingly, we have a version of simm too! ie, simm becomes somewhat biologically plausible. Heh, but to tell the truth, I don't care if the brain doesn't actually use simm, since I have found it to be very useful.

Anyway, recall one definition of simm:

simm(w,f,g) = \Sum_k w[k] min(f[k],g[k]) / max(w*f,w*g)

If we ignore the max(w*f,w*g) denominator, here is a MatSumSig version of simm:

[ r ] = [ sigmoid[x1] ] [ w1 w2 w3 w4 ] [ pos[x1] ] [ 1 -1 -1  0  0  0  0  0  0  0  0  0 ] [ pos[x1]  ] [  1  1  0  0  0  0  0  0 ] [ f1 ]
                                        [ pos[x2] ] [ 0  0  0  1 -1 -1  0  0  0  0  0  0 ] [ pos[x2]  ] [  1 -1  0  0  0  0  0  0 ] [ g1 ]
                                        [ pos[x3] ] [ 0  0  0  0  0  0  1 -1 -1  0  0  0 ] [ pos[x3]  ] [ -1  1  0  0  0  0  0  0 ] [ f2 ]
                                        [ pos[x4] ] [ 0  0  0  0  0  0  0  0  0  1 -1 -1 ] [ pos[x4]  ] [  0  0  1  1  0  0  0  0 ] [ g2 ]
                                                                                           [ pos[x5]  ] [  0  0  1 -1  0  0  0  0 ] [ f3 ]
                                                                                           [ pos[x6]  ] [  0  0 -1  1  0  0  0  0 ] [ g3 ]
                                                                                           [ pos[x7]  ] [  0  0  0  0  1  1  0  0 ] [ f4 ]
                                                                                           [ pos[x8]  ] [  0  0  0  0  1 -1  0  0 ] [ g4 ]
                                                                                           [ pos[x9]  ] [  0  0  0  0 -1  1  0  0 ]
                                                                                           [ pos[x10] ] [  0  0  0  0  0  0  1  1 ]
                                                                                           [ pos[x11] ] [  0  0  0  0  0  0  1 -1 ]
                                                                                           [ pos[x12] ] [  0  0  0  0  0  0 -1  1 ]

where it is assumed w_k >= 0.

If we extract out the intersection component, see last post, we have:

[I1,I2,I3,I4] = 2* [min(f1,g1), min(f2,g2), min(f3,g3), min(f4,g4)]

[ r ] = [ sigmoid[x1] ] [ w1 w2 w3 w4 ] [ I1 ]
                                        [ I2 ]
                                        [ I3 ]
                                        [ I4 ]

Now, the above can be considered a space based simm. We can also do a time based one. I think it goes like this, though I haven't given this much thought in a long, long time!

[ r ] = [ sum[x1,t2] ] [ sigmoid[x1,t1] ] [ 1 -1 -1 ] [ pos[x1] ] [  1  1 ] [ f ]
                                                      [ pos[x2] ] [  1 -1 ] [ g ]
                                                      [ pos[x3] ] [ -1  1 ]

where [ sum[x1,t2] ] is the time based equivalent of [ w1 w2 w3 w4 ]