4. Output: prediction vs target
Black = trained network output. Dashed = desired target. Light gray = input pulse.
ht+1 =
ht +
(dt/τ)[−ht +
tanh(Wrecht + Winxt + b)]
yt = Woutht+1 + bout
We numerically discretize the continuous-time equation with Euler's method. BPTT then differentiates through this entire sequence of Euler steps.
1. The task: delayed response
Input channel x₁ receives a pulse from 1 to 2 s. The desired response appears later on output channel y₁, from 3 to 5 s. Any additional input and output channels are present but set to zero for this task.
x₁(t): pulse
→ recurrent memory →
target y₁(t): delayed pulse
3. What is training doing?
Epoch
0 / 1500
MSE loss
—
Gradient norm
—
Status
Ready
① Forward
simulate h(t) and y(t)
→
② Loss
compare y(t) with target
→
③ BPTT
send error backward through time
→
④ Update
W ← W − η∇W
What does “backward through time” mean?
h(t) influences h(t+dt), which influences later states. Therefore an output error at a late time can be caused by recurrent activity much earlier in the trial. BPTT follows these causal links backward through the unrolled Euler steps.
5. Training loss
MSE should generally decrease as the weights learn the task.
6. Hidden-unit activity
These four curves are the learned internal recurrent state h₁(t), …, h₄(t).
7. Population trajectory
PCA projects the 4D hidden trajectory into PC1 and PC2. Green = trial start, black = trial end.
8. Learned network
Blue = positive weight, orange = negative weight, line thickness = |weight|. These lines change during training because SGD changes the matrices.
9. Weight matrices
These are the current learned parameters. The values update during training, so after training you can inspect exactly what the network learned.
Win
Input → hidden weights
Wrec
Hidden → hidden recurrent weights
Wout
Hidden → output weights
Biases
b and bout
Hidden and output bias terms
negative
near zero
positive
10. BPTT mathematics implemented in JavaScript
9. The BPTT mathematics implemented in JavaScript
Forward Euler step
α = dt/τ
at = Wrecht + Winxt + b
ht+1 = (1−α)ht + α tanh(at)
Loss
L = (1/T) Σt(yt − yttarget)²
Backward recurrent signal
qt =
α [∂L/∂ht+1] ⊙ tanh′(at)
∂L/∂Wrec += qthtT
SGD update
W ← W − η ∂L/∂W
The code computes these derivatives explicitly. There is no autograd engine.
Important note
The current setup keeps the task fixed while allowing the architecture to change. The delayed-response task remains fixed so architecture changes can be compared cleanly. Later versions can expose task timing, activation, optimizer, multiple trials and task families.