Presets
Playback
Training
Input features
Data
Decision model (drift-diffusion)
Probe (also drag it on the surface)
Camera
Layers
Quality
Model check
The science
The network
A multilayer perceptron maps the chosen input features (x, y, x², y², xy, sinπx, sinπy) through hidden layers a(l) = f(W(l)a(l−1) + b(l)) to one output logit z; the probability of class 1 is p = 1/(1 + e−z). Training minimises the cross-entropy L = −mean[y ln p + (1−y) ln(1−p)] + (λ/2)ΣW² by back-propagation (the chain rule, output error δ = p − y) with plain SGD, momentum or Adam on random mini-batches. 25 % of the points are held out as a test set; the gap between train and test loss shows over-fitting.
Reading the picture
The colour map is p over the whole plane: the decision boundary is where it turns from blue to orange. Each hidden neuron has its own little map of what it responds to; deeper neurons combine the simpler shapes of earlier ones. Line width is |weight|. Drag the probe to feed a new stimulus and watch every activation, number and pulse change. Why hidden layers matter: XOR cannot be separated by a single linear boundary, but a hidden layer (or the feature xy) solves it.
From network output to a decision (drift-diffusion)
Brains do not decide instantly: noisy evidence is accumulated until a threshold is hit. In the drift-diffusion model dx = v dt + σ dW the accumulator x starts at 0 and the choice is made when it reaches ±a. Here the drift is the network's confidence, v = k · z. For this process the probability of hitting the upper bound is P = 1/(1 + e−2va/σ²) and the mean decision time is E[T] = (a/v) tanh(av/σ²); the page adds a fixed non-decision time t₀ for sensory and motor delays. A higher threshold makes choices slower but more accurate (speed-accuracy trade-off); a probe near the boundary has v≈0 and gives slow, uncertain decisions, as in real reaction-time data.
Independent checks (shown in the page)
(1) The back-propagated gradient of every sampled weight and bias equals a central finite-difference derivative of the loss (relative error far below 10⁻⁴). (2) A Monte Carlo simulation of 5000 diffusion trials reproduces the closed-form choice probability within four binomial standard errors. (3) The simulated mean decision time matches (a/v) tanh(av/σ²) within 5 % (the simulation uses a continuity-corrected barrier to remove the Euler step bias).
Honest simplifications
This is a tiny two-dimensional toy: real neurons spike, have dendrites and learn with local, not global, rules, and back-propagation is not how cortex learns. Linking the artificial logit to a drift rate is a modelling choice (a hypothesis in cognitive neuroscience, not a measured law), the gain k, threshold a, noise σ and t₀ are free illustrative parameters, and the accumulator has no leak, bias or collapsing bounds. Datasets are synthetic; accuracy numbers say nothing about real tasks.
What the 3D scene shows
Everything is computed by the same code as the 2D page. Each glowing neuron is one unit of the network (orange = positive activation for the probe, blue = negative); cables are the weights (thickness |w|, orange positive, blue negative) and the pulses carry the probe signal. The coloured, hilly surface is the output probability p(class 1) over the whole input plane (height = p), with the training data as gems standing on it. At the front the ball on the track is the drift-diffusion accumulator: it drifts toward the gate of the network's favoured class at a speed proportional to the logit, and the first gate it touches is the decision. Sizes are artistic; real cortical neurons are about 20 µm across.