Manuscript Psy03-C@1 · Psy03-C-v1.tex · section · d3380b90

Exact cell Jacobians

Gate-aware recurrent stability › Exact cell Jacobians · lines 592–623

Psy03-C@1#eq:jacobians

Let

D_{r} = diag (r_{i} (1 - r_{i})), D_{ζ} = diag (ζ_{i} (1 - ζ_{i})), D_{n} = diag (1 - n_{i}^{2}), w = B_{n} h + b_{hn} .

Direct differentiation gives

N_{h} N_{x} J_{h} J_{x} = D_{n} [diag (r) B_{n} + diag (w) D_{r} B_{r}], = D_{n} [A_{n} + diag (w) D_{r} A_{r}], = diag (ζ) + diag (1 - ζ) N_{h} + diag (h - n) D_{ζ} B_{ζ}, = diag (1 - ζ) N_{x} + diag (h - n) D_{ζ} A_{ζ} .

The hidden bias in $w$ is essential.

For the ordinary post-cell map

Ψ_{t} (h) = φ (a F_{θ} (u_{t}, h) - o_{p} 1),

its Jacobian at a differentiability point is

D Ψ_{t} (h) = a D_{φ} J_{h},

where $D_{φ}$ is evaluated at the post-cell preactivation.

Displayed formulas in this passage

Psy03-C@1#eq:jacobians · line 602
$N_{h} N_{x} J_{h} J_{x} = D_{n} [diag (r) B_{n} + diag (w) D_{r} B_{r}], = D_{n} [A_{n} + diag (w) D_{r} A_{r}], = diag (ζ) + diag (1 - ζ) N_{h} + diag (h - n) D_{ζ} B_{ζ}, = diag (1 - ζ) N_{x} + diag (h - n) D_{ζ} A_{ζ} .$
unlabeled · line 595
$D_{r} = diag (r_{i} (1 - r_{i})), D_{ζ} = diag (ζ_{i} (1 - ζ_{i})), D_{n} = diag (1 - n_{i}^{2}), w = B_{n} h + b_{hn} .$
unlabeled · line 614
$Ψ_{t} (h) = φ (a F_{θ} (u_{t}, h) - o_{p} 1),$
unlabeled · line 618
$D Ψ_{t} (h) = a D_{φ} J_{h},$