By David S. Touretzky (Editor)
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3 Periodic (Toroidal) boundary conditions u i,0 = u i,N , i = 1, 2, . . , M Left virtual cells: yi,0 = yi,N , Right virtual cells: yi,N +1 = yi,1 , u i,N +1 = u i,1 , i = 1, 2, . . , M u 0, j = u M, j , j = 1, 2, . . , N Top virtual cells: y0, j = y M, j , Bottom virtual cells: y M+1, j = y1, j , u M+1, j = u 1, j , j = 1, 2, . . , N . A A M N D CNN D B B C C Fig. 13. The circuit interpretation of the Periodic (Toroidal) boundary condition. Identify each cell from the top row with the corresponding cell in the bottom row, identify each cell from the left column with the corresponding cell in the right column.
The same observation applies to the horizontal pairs, and all diagonal pairs of similarly-directed bold-light edges. Observe also that for each zero coefﬁcient akl = 0, or bkl = 0, two corresponding edges will disappear from the corresponding signal ﬂow graph. Hence, for templates with only a few non-zero entries, their associated synaptic signal ﬂow graphs are particularly simple. It is in such situations where useful insights can be obtained, specially when two or more such synaptic signal ﬂow graphs are interconnected to form a composite synaptic system graph.
The details will be discussed in Chapter 15. A very rough sketch of a typical living neuron with interacting neighbors is shown in Fig. 26. A more detailed discussion can be found in Chapter 16. 28, respectively. These two ﬂow graphs show explicitly the directions of the signal ﬂows from neighboring cells and their associated synaptic weights akl and bkl , respectively. 2 Mathematical foundations synaptic current sources controlled by the inputs of surround cells i– 1, j– 1 u , 0 b –1, ,1u i–1 b –1 b0,–1 bi, j–1 synaptic current sources controlled by the outputs of surround cells total fe curr edback ent total fe curr edforwa ent rd uij ,j a 1, –1 A Y ij y i–1 0 y i+ 1, j– 1 a –1, a0,1 yi, j+1 a0,–1yi, j–1 a1 i– 1, j– 1 j+1 b1,0ui+1, j u i+ 1, j– 1 b0,1ui, j+1 u i+1, b 1,1 1, –1 current summing node ij of cell C(ij) –1 , j+ 1 u 1 j+ yi ,j i–1 1,1 –1 ,– 1 a– b b ,1 + yi +1 ,j +1 a1,0 yi+1, j y –1 ,– 1 a 31 uij – xij bij uij zij input voltage of cell C(ij) internal core of cell C(ij) 1 xij + xij – aij y ij threshold current of cell C(ij) + f (xij ) 1 y ij – state voltage of cell C(ij) output voltage of cell C(ij) Fig.
Advances in Neural Information Processing Systems 2 by David S. Touretzky (Editor)