Edge of chaos

Dynamic Memory, Metastability, and Perceptual Coalitions in DPSH: A Series of Falsification Experiments at the Edge of Chaos

Manuscript status: final internal research report
Date: September 12, 2026
Software: Cognia / DPSH runtime 2

Abstract

We tested the hypothesis that the dynamics of a discrete probabilistic spiking hybrid system (DPSH) near the edge of chaos improve temporal memory, nonlinear computational capacity, metastability, and the stability of perceptual coalitions. We developed a reproducible experimental framework with snapshots, paired perturbations, randomness replay, temporally separated training/validation/test protocols, ridge readouts, surrogate controls, and multi-seed sweeps. Leaky neurons exhibited non-zero short-term linear memory, with a maximum of approximately 0.395 in the tested map; however, that maximum occurred at zero or minimal recurrence. Neither basic nor structured reservoirs solved XOR tasks above chance. One configuration reached a balanced accuracy of 0.580 (95% CI 0.554–0.606) on short-lag XOR(1,2), but ablation showed that the result required neither recurrence, the second temporal branch, nor explicit multiplicative coincidence. An evoked four-aspect percept was decodable with an accuracy of 0.944, whereas direct input voting achieved 0.953, and inter-module coupling did not improve performance. Silent-tail clusters had dwell times 2.5–2.9 times longer than temporally shuffled surrogates, yet showed none of the returns, rich transitions, or scaling expected of emergent metastability. A two-axis map of coalition dynamics and recurrence contained no point that both qualified as a percept and lay in a divergent regime. The results do not support the claim that tuning DPSH to chaos alone creates functional memory, integration, or a coalition-based percept. They support the narrower conclusion that individual-neuron time constants and redundant synchronous input create measurable but predominantly input-bound representations.

Keywords: DPSH, edge of chaos, reservoir computing, metastability, spiking network, perceptual coalitions, falsification

1. Introduction

The “edge of chaos” hypothesis proposes that a system operating between rigid and divergent regimes may simultaneously preserve the past and respond sensitively to new input. This idea is attractive for DPSH because the runtime combines continuous state decay, discrete spikes, refractoriness, adaptation, stochastic hazard, and delayed synaptic events. Demonstrating a complex trajectory or clustering, however, is not sufficient. A functional claim requires a measurable computational benefit over simpler controls.

The study therefore tested four questions in sequence:

  1. Does proximity to a divergent regime increase linear temporal memory?
  2. Does recurrent DPSH create nonlinear temporal features?
  3. Do the observed long-lived states represent metastable dynamics, or merely a decaying aftereffect?
  4. Does a critical regime help stabilize perceptual coalitions and allow them to switch flexibly?

The main methodological principle was to distinguish a parameter from the measured regime. A high recurrent gain was not automatically called chaos; finite-time perturbation divergence determined the regime. Likewise, a cluster was not automatically treated as a metastable state, nor a decodable input as an emergent percept.

2. Methods

2.1 DPSH substrate

Models were declared in the Cognia language and executed by runtime 2. Neurons had exponential membrane and adaptation dynamics, absolute or relative refractoriness, and stochastic hazard. The synaptic graph used positive delays and separate excitatory and inhibitory polarities. Internal weights were not trained during measured runs; only an offline readout was fitted.

2.2 Experimental protocol

A typical run included a washout period and chronologically separated training, validation, and test splits. Ridge regularization was selected exclusively on the validation split from the range 10^-12 to 10^6. Results are reported over multiple deterministic seeds, with intervals based on Student's distribution. The current input was excluded from memory capacity.

The perturbation branch was created from a common snapshot. Control and perturbed branches used the same stimulus and frozen replay of a named random number generator. Divergence was estimated from the temporal evolution of the normalized state distance. A run was rejected if the firing-rate control or active fraction diverged, or if the randomness replay was exhausted.

2.3 Tasks and metrics

  • Linear memory: sum of test R² values for reconstructing past bits.
  • Nonlinear memory: balanced accuracy on lagged XOR(1,2) and XOR(5,10), using a linear readout over network state.
  • Metastability: dwell time, separability, jump-chain transition entropy, and a mandatory temporally shuffled surrogate in the silent tail.
  • Percept: balanced accuracy of latent identity, turnover of the spiking population, occlusion, increased noise, and temporal desynchronization of aspects.
  • Competition: successful stable switching, latency, and readout jump; these metrics were interpretable only when identity decoding exceeded chance.

2.4 Nature of the study

Development proceeded sequentially: a negative result motivated the next architecture or ablation. The study is therefore primarily exploratory and includes adaptive model selection. Confidence intervals are not corrected for all comparisons performed and must not be interpreted as confirmatory p-values. We regard results that survived explicit causal controls as the strongest; one-off maxima are candidates only.

3. Results

3.1 Linear memory arose in leaky neurons, not from recurrence

The original explicit_memory mode lost nearly all information during spontaneous resets, and changing tau_m did not have the expected effect. After switching to a leaky membrane, memory capacity became non-zero:

effective tau_m memory capacity 95% CI
12.5 ms 0.163 0.086–0.241
18.75 ms 0.250 0.231–0.269
25 ms 0.221 0.173–0.269
50 ms 0.117 0.031–0.204

The paired map of time constant and recurrence produced the highest measured value, 0.395, at tau_m=22.5 ms and recurrent_gain=0.35. The mandatory zero-gain control nevertheless reached 0.389 in a shortened run, and gains from 0.1 to 0.35 did not improve upon it. Recurrence increased the number of active observed neurons, but not predictive information. Linear memory is therefore best explained by a decaying local trace rather than by a collective recurrent reservoir.

3.2 Basic nonlinear tasks did not support a role for recurrence

A homogeneous reservoir, time-shifted input copies, heterogeneous input populations, and structured fast/slow microcircuits all remained near a balanced accuracy of 0.5 on XOR(1,2) and XOR(5,10). Projections that included adaptation and spike bins did not change the result. Recurrent gain produced no consistent improvement over zero recurrence.

An explicit coincidence operator did not help either. The additive control achieved 0.580 (95% CI 0.554–0.606) on XOR(1,2), whereas the multiplicative variant reached only 0.557 (0.540–0.574) at the same gain of 0.65. A subsequent ablation showed:

variant at gain 0.65 XOR(1,2) 95% CI
reference, two inputs, state(32) 0.580 0.554–0.606
state(16) 0.548 0.527–0.568
single direct input, state(32) 0.591 0.580–0.602
spiking relay, state(32) 0.500 0.500–0.500

Thus, neither the second temporal branch nor the explicit product caused the positive result. The short-lag XOR signal was carried mainly by intrinsic neuronal nonlinearity and richer observation. Long-lag XOR(5,10) remained inconclusive.

3.3 Silent-tail structure was not collective metastability

After the input was disabled, real dwell times were 79–86 ms, compared with 30–32 ms after temporal shuffling—a ratio of approximately 2.5–2.9. Temporal order therefore contained structure that could not be explained by the marginal cluster distribution alone. Transition entropy was nevertheless zero under every measured condition, separability was approximately 0.459–0.471, and neither the number of modules nor inter-module coupling had a consistent effect. The data are more consistent with a regular relaxation tail than with repeated visits to metastable basins.

3.4 The stable macrorepresentation was locked to its source

Four heterogeneous modules received independently corrupted aspects of the same 100 ms latent percept. Their joint state reached an accuracy of 0.944 without inter-module coupling and 0.932 with coupling. The best individual module achieved 0.930, while direct voting over the input aspects achieved 0.953. Turnover of the spiking population was approximately 0.999, meaning that a stable macro-identity coexisted with almost complete replacement of active neurons.

Causal controls changed the interpretation:

  • temporal desynchronization of the aspects reduced accuracy from 0.944 to 0.500 without coupling;
  • with coupling, desynchronized inputs reached 0.557, but the 95% CI of 0.460–0.654 included chance;
  • occluding the strongest aspect left accuracy at 0.885–0.890;
  • increasing noise by 20 percentage points left accuracy at 0.707–0.732;
  • inter-module coupling provided no significant protective or integrative benefit.

This is therefore a robust redundant, but source-locked, representation. The stability of a macrostate is not by itself evidence of an autonomous percept.

3.5 Competition and its combination with a critical regime failed

The original two-channel competition produced accuracies of 0.471–0.523. Apparently high switching success was therefore uninterpretable: a randomly fluctuating readout would eventually produce a short correct sequence.

The follow-up model used two independent aspects for each of two coalitions. The two-axis map contained 12 combinations of inhibitory_gain 0.25/0.5/1 and recurrent_gain 4/16/32/64, each evaluated over three seeds. No point met the predefined percept qualification threshold of 0.65; the maximum was 0.638 at inhibition 0.5 and recurrence 4. All 12 mean divergences were negative (−1.29e−8 to −2.01e−8 / ns). At recurrence 64, accuracy fell to 0.503–0.541 and memory capacity to 0–0.019. Higher inhibition of 1.5 exhausted the frozen RNG replay in the perturbed branch, so that condition was correctly rejected and excluded from the result.

No regime combining a qualified coalition with critical or divergent dynamics was found in the searched region.

4. Discussion

4.1 What the data support

  1. A leaky membrane trace provides short-term linear memory.
  2. The original DPSH hazard and reset can create a weak short-term nonlinear feature that is decodable with sufficiently wide observation.
  3. Multiple synchronous redundant channels create a robust macrorepresentation despite rapid turnover of the spiking microstate.
  4. The experimental framework can distinguish a valid negative result from a failure of snapshots, readouts, or RNG pairing.

4.2 What the data do not support

  1. That recurrence increased linear memory relative to a non-recurrent control.
  2. That explicit multiplicative coincidence caused the short-lag XOR effect.
  3. That silent-tail clusters represent rich emergent metastability.
  4. That inter-module coupling integrated the percept better than direct inputs.
  5. That competing coalitions created bistability, hysteresis, or autonomous decision-making.
  6. That chaos or its edge improved memory, percepts, or flexibility in the tested DPSH systems.

4.3 Practical implications for DPSH

According to these data, searching parametrically for “more chaos” is not a productive development strategy. DPSH remains useful as a deterministically reproducible event-driven substrate with heterogeneous time constants, local stochasticity, and an exact experimental API. Functional coalitions or intuition, however, require a separately designed mechanism and their own qualification tests; they cannot be expected to emerge merely by increasing recurrent gain.

For future orchestration, this leads to a conservative recommendation: measure dynamic stability, memory, integration, and decision-making as separate properties. “Criticality” may later become one regulated variable, but it should be neither an architectural assumption nor a substitute for a functional module.

5. Limitations

  • The models are small and the simulations relatively short; most conditions used 3–5 seeds.
  • A ridge readout captures only linearly accessible information in the selected projection.
  • Finite-time divergence in a hybrid stochastic system is not a classical Lyapunov exponent.
  • Some conditions were designed adaptively in response to earlier results; the study is not a single preregistered confirmatory experiment.
  • Channel occlusion is not a complete ablation of a neuronal module.
  • Competition used step changes in the source; a true ascending/descending hysteresis test was not performed because the coalition failed the basic qualification criterion.
  • The negative result applies to the implemented topologies, parameters, and timescales, not to every possible DPSH network.

6. Conclusion

The series of experiments found no scientific support for the strong hypothesis that tuning DPSH to the edge of chaos by itself creates useful memory, emergent metastability, or perceptual coalitions. The best-supported effects have simpler explanations: a local leaky trace, intrinsic neuronal nonlinearity, and redundancy of synchronous input. Recurrence and inter-module coupling were mostly neutral or harmful, and the explored coalition map remained contractive.

This is not a failure of the experiment. A strong and attractive hypothesis was decomposed into measurable components, confronted with zero-recurrence controls, surrogate data, ablations, and out-of-training-distribution tests, and falsified in its present form. The edge-of-chaos topic is therefore closed for the current generation of DPSH. It should be reopened only for a new coalition architecture that first demonstrates functional representation, retention, and switching independently.

7. Code availability and reproduction

The implementation is part of the Cognia repository:

  • production runner: cognia_edge_of_chaos_dpsh;
  • models and launch scripts: Cognia/examples/19-EdgeOfChaos/;
  • two-axis runner: run-coalition-chaos.ps1;
  • specifications, partial results, and decision rules: Docs/EdgeOfChaos/;
  • validation targets: cognia_validate_edge_of_chaos, cognia_edge_of_chaos_runner_test, and cognia_window_engine_test.

Raw experimental artifacts are created locally in the specified output directory and are intentionally excluded from Git history. The aggregate numerical results used in this article are recorded in EOC-09 through EOC-15.

Internal references

  1. EOC-01 through EOC-08: protocol, readout, perturbations, sweeps, and the validation gate.
  2. EOC-09: production DPSH runner, linear and nonlinear memory, microcircuits, coincidence, and ablation.
  3. EOC-10: silent-tail metastability and module scaling.
  4. EOC-11 through EOC-13: evoked percept, causal synchrony, and robustness.
  5. EOC-14 through EOC-15: percept competition and the coalition × dynamic regime map.