Interactive Playground
=======================

The panel runs a small first-to-fire LIF **toy** network live (2 inputs, 2 output
neurons). The input set is a **generalized star** (Definition on
:doc:`spike_star`), :math:`\mathcal{I} = \{\mathbf{x} = \mathbf{c} + V\boldsymbol{\alpha}
: \boldsymbol{\alpha}\in[-1,1]^m\}` — a **box** when :math:`V` is diagonal, a
**zonotope** otherwise. Drag :math:`\mathbf{c}`, resize with :math:`\epsilon`, and
**shear** or **add a generator** to deform the box into a zonotope and watch the
certificate respond.

Each input fires once at its F2F latency
:math:`\mathcal{L}(x) = \mathrm{round}((T-1)(1-x)) + 1`; the output neurons integrate
the (binary) input spikes, and the class score is :math:`y_c = T+1-t_c^{\mathrm{out}}`,
so the class whose output **spikes first** wins. The star is partitioned into
**latency cells**; on each cell the spike pattern is fixed, so the output is exact,
and the reachable scores are the union of the per-cell outcomes.

Four linked views update as you drag:

- **Input star & latency cells** — the star (box or zonotope) and the latency-cell pieces it
  is partitioned into, each tinted by its winning class.
- **Output decision: class-score enclosure** — the reachable :math:`(y_0, y_1)` score points
  (one per cell), the certified **box enclosure** :math:`[\underline y_c, \overline y_c]`, and
  the tie diagonal. Robust when the box sits strictly on one side.
- **Membrane potentials + envelope** — each output's membrane as a band (min/max over the
  reachable cells), the center trace, the threshold :math:`\theta`, and the first-spike marker.
- **Spike raster + envelope** — for each neuron, the *could-fire* range of timesteps over the
  star (the spike-train envelope) and the center spike.

The verdict certifies a **strict score margin** :math:`\underline y_{c^*} > \overline y_c`
over all reachable cells (a sound check, validated against sampling). Deforming the box into a
zonotope changes which latency cells are reachable — so a skewed star can certify where an
axis-aligned box cannot.

.. raw:: html

   <link rel="stylesheet" href="_static/spikestar_playground/style.css" />
   <div id="snnv-playground"></div>
   <script type="module" src="_static/spikestar_playground/app.js"></script>