Research & AlgorithmsDANIEL SCHMITTER · RESEARCH & ENGINEERING

Neuromorphic ideas · Research & Algorithms

Learning between events instead of stepping through time

A quiet interval can contain a great deal of computation—and very little new information. Known dynamics let a learning system integrate through it. The surprise was how well a simple numerical rival could do the same.

Daniel Schmitter · Research story · Revised draft

Read the paper ↗Paper PDF
EXPLORE THE IDEA

Nothing arrives. The state still evolves.

An event is not the same thing as a simulation time step.

EVENT ARRIVALSThree impulses, continuous evolutionBETWEEN EVENTSEvaluate the state directlyz(t) = 0.683ż = −7zz(t + Δ) = exp(−7Δ) z(t)Until the next impulse changes the state
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Computed linear exponential responses to three specified impulses. The cursor reveals the exact state between arrivals; event positions are synthetic and do not represent a recorded spiking network.

Follow the information

From input to outcome

The clock is replaced by intervals whose dynamics are known. Each knot crossing changes the active polynomial, and each interval contributes to a continuous fitting objective. The target is explicitly held between events.

Scroll the diagram horizontally to follow the route. Keyboard: focus the diagram, then use the arrow keys.

Event arrivals → Known decay trace → Partition at knots → Integrate feature products → Fit spline readout. The clock is replaced by intervals whose dynamics are known. Each knot crossing changes the active polynomial, and each interval contributes to a continuous fitting objective. The target is explicitly held between events.
Information-flow map. Accurate event integration; no advantage over the strongest qualified quadrature control. Original vector schematic based on the method and evidence discussed in this article; signal shapes and icons are illustrative, not additional measurements. Open full-size diagram ↗

Read the main route from left to right; labelled side branches show additional inputs, checks or feedback. The sections below explain the operations and their experimental limits.

The clock is not always the right unit of work

Imagine an always-on sensor whose internal trace jumps when an event arrives and then decays predictably. A clock-driven implementation keeps revisiting the quiet interval. If the decay is already known, those steps need not discover anything. The interesting question is whether learning can use the entire interval without simulating it one small step at a time.

The learner accumulates time, not snapshots

Between impulses, the scalar state follows a known exponential trajectory. A spline readout changes polynomial pieces only when that trajectory crosses a knot. The event routine partitions there and integrates feature products over each interval, so the resulting Gram represents a continuous-time fitting objective. The target is held at its last event value by design.

The quiet tail creates the difficult numerical case. Ordinary time quadrature may spend its nodes poorly, but a comparator that removes the constant contribution analytically resolves the same issue at essentially the same cost. The strongest result is therefore an accurate event-integrated objective, not a speed advantage over every informed alternative.

Let events update the learning objective
Let events update the learning objective. Original scientific diagram; the stated component and information flow, not an additional experiment. Open full-size figure ↗

Integrate the learning signal, not just the state

Our readout is a learned spline of the trace amplitude. Within one cell its basis functions are cubic polynomials. As the trace decays through cells, their products can be integrated along the known trajectory. Each interval contributes directly to a Gram matrix and target vector. A later ridge solve uses those statistics rather than replaying a dense synthetic clock.

G+=∫tatbϕ(x(t))ϕ(x(t))Tdt\begin{gathered}G\mathrel{+}=\int_{t_a}^{t_b}\phi(x(t))\phi(x(t))^Tdt\end{gathered}
One interval contributes an integrated learning statistic. For a known exponential trace, local polynomial calculus computes it directly.

An assumption that makes the example possible

The experiment supplies event times, the decay constant, and a target held unchanged between events. Those are strong assumptions. We are not recovering an unobserved teacher signal between measurements or training a complete spiking network. The construction asks a cleaner computational question: given this objective, how accurately and cheaply can we accumulate it?

Accuracy was real. The speed headline was not.

Analytic accumulation passes all 48 tested conditions. Several fixed quadrature rules fail on long quiet tails, while midpoint stepping is both inaccurate and slower. But the declared twofold speed advantage over qualified quadrature does not appear. The lesson becomes clearer when we separate the nearly constant part of the final cell: three-point quadrature integrates the remaining polynomial exactly in real arithmetic.

The strong control changes the story

That improved control was constructed after the initial outcomes, so it is an explanatory audit rather than a fresh confirmation. It passes every condition and runs at essentially the same speed as analytic evaluation. The calculus remains useful. What disappears is the claim that one implementation has a distinctive speed advantage merely because its formula is analytic.

Archived numerical results. Left: all accumulation controls, including accuracy failures. Right: the separate equal-budget precision study discussed in the companion post. Neither panel measures neuromorphic hardware.
Archived numerical results. Left: all accumulation controls, including accuracy failures. Right: the separate equal-budget precision study discussed in the companion post. Neither panel measures neuromorphic hardware.

An event-driven objective, not yet an event-driven agent

A future event-driven learner could reuse this primitive when both between-event dynamics and supervision semantics are known. Multiple interacting traces, learned events, and delayed rewards need additional machinery. None is supplied by the closed-form scalar integral alone.

What to keep for event-driven ML

The reusable component is a continuous objective accumulated from events under known dynamics. Its value in a larger learner would depend on supervision, state interactions, and the total system cost. No energy or hardware advantage has been measured here. A good event-based algorithm should beat a comparator that understands the same dynamics—not just a clock that unnecessarily ignores them.

Evidence & further reading

The links below distinguish the project record from foundational literature. This revised story does not add a new application-validation experiment.

  1. Neuromorphic operator learning: mechanisms, evidence and limits. Spline research archive (2026). Local archive snapshot.
  2. EventProp: exact gradients for spiking neural networks. Timo C. Wunderlich and Christian Pehle (2020). Primary literature.

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