i Endscripts normal w Subscript italic i j Superscript l Baseline normal a Subscript i Superscript l minus 1 Baseline left-parenthesis k right-parenthesis plus w Subscript b Superscript l Baseline equals sigma-summation Underscript i Endscripts s Subscript italic i j Superscript l Baseline left-parenthesis k right-parenthesis plus w Subscript b Superscript l Baseline comma"/>
where intermediate variable are defined for convenience. The partial derivative in Eq. (3.21) is easily evaluated as
(3.23)
This holds for all layers in the network. Defining allows us to rewrite Eq. (3.21) as
(3.24)
We now show that a simple recursive formula exists for finding . Starting with the output layer, we observe that influences only the instantaneous output node error ej(k). Thus, we have
(3.25)
For a hidden layer, has an impact on the error indirectly through all node values in the subsequent layer. Due to the tap delay lines, also has an impact on the error across time. Therefore, the chain rule now becomes
where by definition . Continuing with the remaining term
(3.27)
Now
(3.27a)
since the only influence has on is via the synapse connecting unit j in layer l to unit m in layer l + 1. The definition of the synapse is explicitly given as
