# Event-triggered phase desynchronisation for recurrent hidden states

- ID: 2308
- Canonical URL: https://synthcore.org/idea/2308/event-triggered-phase-desynchronisation-for-recurrent-hidden
- API JSON: https://synthcore.org/api/idea/2308.json
- API Markdown: https://synthcore.org/api/idea/2308.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.14907](https://arxiv.org/abs/2608.14907)
- Category: dynamics
- Solves: stability, accuracy, scalability
- ML areas: rnn, ssm, training-dynamics, regularization
- Math tags: dynamical-systems, control-theory, geometry, stochastic-processes
- Ratings: usefulness 7/10; difficulty 6/10; novelty 8/10

## Idea description

Augment each recurrent or state-space hidden channel with a two-dimensional oscillatory state and periodically compute a pseudo-phase from its Cartesian coordinates. Use sparse event-triggered feedback to reduce the squared phase order parameter, preventing hidden channels from synchronising while avoiding the computation and communication cost of continuously recomputing the control signal. The controller acts as a tangent rotation of each two-dimensional hidden state, changing phase diversity without directly changing channel amplitude.

## Mathematical statement

For hidden oscillator state $h_j=(x_j,y_j)\in\mathbb{R}^2$, define the pseudo-phase $\hat\theta_j=\operatorname{atan2}(y_j,x_j)$ and complex phase $z_j=e^{i\hat\theta_j}$. The synchrony order parameter is $r=N^{-1}\sum_{j=1}^N z_j$ and the Lyapunov candidate is $V=|r|^2$. Its phase gradient is $\partial V/\partial\hat\theta_j=-(2/N)\operatorname{Im}(z_j\bar r)$. Continuous feedback $u_j=-k\partial V/\partial\hat\theta_j=(2k/N)\operatorname{Im}(z_j\bar r)$, with gain $k>0$, gives $\dot V=-k\|\nabla_\theta V\|^2$ when uncontrolled phase drift is absent or cancelled. In the neural implementation, apply $u_jJh_j$ with $J=[[0,-1],[1,0]]$, which rotates $h_j$ without changing its norm. At event time $t_\ell$, compute and store $u(t_\ell)$; between events use the held value $u(t)=u(t_\ell)$. Trigger the next update when $\|u(t)-u(t_\ell)\|_2\geq\delta$, where $\delta>0$. If $\|\dot u(t)\|_2\leq M$ during an interval, every inter-event time satisfies $t_{\ell+1}-t_\ell\geq\delta/M$, excluding Zeno behaviour. With held control, the ideal descent law becomes $\dot V=-k\|\nabla V\|^2+\nabla V^{\mathsf T}e$, where $e=u_{\mathrm{held}}-u_{\mathrm{exact}}$; thus a sufficiently small trigger tolerance gives practical rather than exact Lyapunov descent.

## Key formulas

- $$\hat\theta_j=\operatorname{atan2}(y_j,x_j),\qquad z_j=e^{i\hat\theta_j},\qquad r=\frac{1}{N}\sum_{j=1}^{N}z_j,\qquad V(\hat\theta)=|r|^2.$$
- $$\frac{\partial V}{\partial\hat\theta_j}=-\frac{2}{N}\operatorname{Im}(z_j\bar r),\qquad u_j=-k\frac{\partial V}{\partial\hat\theta_j}=\frac{2k}{N}\operatorname{Im}(z_j\bar r),\qquad \dot V=-k\|\nabla_{\hat\theta}V\|_2^2.$$
- $$\dot h_j=F_j(h,x;\theta)+u_jJh_j,\qquad J=\begin{bmatrix}0&-1\\1&0\end{bmatrix},\qquad u(t)=u(t_\ell)\quad\text{for }t\in[t_\ell,t_{\ell+1}),$$
- $$\|u(t)-u(t_\ell)\|_2\geq\delta\quad\text{triggers an event},\qquad t_{\ell+1}-t_\ell\geq\frac{\delta}{M}\quad\text{if }\|\dot u(t)\|_2\leq M.$$

## Implementation notes

First, integrate the mechanism into a continuous-time RNN or state-space model whose width is $2N$, interpreted as $N$ two-dimensional hidden channels $h_j=(x_j,y_j)$. After every solver step or discrete recurrent update, compute the pseudo-phases using `atan2`; for better numerical robustness, use normalized complex states `z_j=(x_j+i*y_j)/(sqrt(x_j^2+y_j^2)+epsilon)`. Compute `r`, then `u_exact[j]=(2*k/N)*imag(z_j*conj(r))`. Initialize `u_hold` to zero. At each step, if `norm(u_exact-u_hold) >= delta`, replace `u_hold` with `u_exact` and record an event; otherwise retain the old value. Evolve each pair with `h_j <- h_j + dt*(F_j(h,x;theta)+u_hold[j]*(J@h_j))`, where `J=[ [0,-1], [1,0] ]`. The paper supplies the order-parameter Lyapunov mechanism, feedback gradient, trigger condition, and dwell-time argument. Estimate the bound $M$ empirically from a high percentile of finite differences of `u_exact`; tune $k$ and $delta$ on a validation set. Run a cheap experiment with a 32-channel continuous-time RNN on sequential MNIST, comparing no control, continuous control, and event-triggered control. Measure accuracy, $V(t)$, $\|\nabla V\|$, event rate, and minimum dwell time. The predicted signature is $\dot V<0$ under continuous control when intrinsic phase drift is negligible; event-triggered control should preserve the decay slope for small $\delta$, achieve event rate roughly proportional to $1/\delta$, and have minimum dwell time near or above $0.8\delta/\hat M$.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: no_effect
- Verdict: Built a reproducible phase-controller verification and small recurrent classification experiment. The continuous Lyapunov law matched finite differences across gains k=0.25–2.0 with ratios 1.000002–1.000015, and event rates decreased approximately inversely with delta while dwell times tracked the empirical delta/M bound. The mechanism manifested, but it did not improve this classifier: baseline accuracy was 0.875 versus 0.852 event-triggered and 0.813 continuous.

### Mechanism check

- Verdict: Built a reproducible phase-controller verification and small recurrent classification experiment. The continuous Lyapunov law matched finite differences across gains k=0.25–2.0 with ratios 1.000002–1.000015, and event rates decreased approximately inversely with delta while dwell times tracked the empirical delta/M bound. The mechanism manifested, but it did not improve this classifier: baseline accuracy was 0.875 versus 0.852 event-triggered and 0.813 continuous.
- Confidence: 8/10
- Limitations: Only one seed and one tiny synthetic sequence-classification task were tested. Sequential MNIST, multi-seed statistics, tuning, FLOP or communication benchmarks, and larger recurrent/state-space models were not tested; the largest-delta run had only one event, so its minimum dwell time was unavailable.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 1
- Paired wins: 0/8
- Benchmark verdict: no measurable effect

## Artifacts

- [bench_event.py](https://synthcore.org/code/918/bench_event.py)
- [bench_report.json](https://synthcore.org/code/918/bench_report.json)
- [experiment.py](https://synthcore.org/code/918/experiment.py)
- [report.md](https://synthcore.org/code/918/report.md)
- [report_bench_2026-09-02T232842.md](https://synthcore.org/code/918/report_bench_2026-09-02T232842.md)
- [results.json](https://synthcore.org/code/918/results.json)
- [stage2_bench.py](https://synthcore.org/code/918/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/918)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.