Table 1.

Effect of the Hebbian-like error learning rule ${\mathcal{F}}^{\mathrm{eHebb}}$ on the alignment of the modulating signals α_ℓ for different layers

${\mathcal{F}}^0$	${\mathcal{F}}^{\mathrm{eHebb}}$	α0	α1	α2
W0,1, W1,2, W2,3	–	89.89	76.69	82.04
W0,1, W2,3	W1,2	89.95	59.95	72.14
W0,1, W1,2	W2,3	90.03	75.18	29.02
W2,3	W0,1, W1,2	75.29	61.23	72.56
W0,1	W1,2, W2,3	90.2	49.4	27.9
W1,2	W0,1, W2,3	84.86	74.25	30.33
–	W0,1, W1,2, W2,3	77.93	49.93	28.4

The leftmost column includes the parameters updated using ${\mathcal{F}}^0$ with feedback alignment, and the next column indicates layers trained with ${\mathcal{F}}^{\mathrm{eHebb}}$ (Eq. (8)). Angles α_ℓ represent the alignment between the modulatory signal e_ℓ and the backpropagated counterpart ${\mathbf{e}}_{\ell }^{\mathrm{BP}}$ at each layer (in degrees). Since e₀ is a synthetic error, the effect of the ${\mathcal{F}}^{\mathrm{eHebb}}$ on W_0,1 alone has been excluded. The model is trained for 500 episodes, and the computed angles are averaged after a burn-in period of 100 episodes.