Performance-Evolved Reservoirs Encode Wilson-Cowan Dynamics as Material Memory in Their Connectivity Sign Structure

Authors Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Yadav’s performance-evolved reservoir networks show that repeated exposure to Wilson-Cowan population dynamics can turn an initially generic recurrent graph into a compact structure whose internal connectivity carries the excitatory-inhibitory organization of the target system. Pudding Theory reads this result through Material Memory. The reservoir does not merely approximate a trajectory. It stores the history of repeated dynamical signals as a persistent topological trace in its node roles, gains, edge signs, and population-level connectivity. What the source frames as performance-driven structural adaptation is, in this reading, the physical inscription of a signal into a receptive substrate. The anomalous feature is not prediction accuracy alone, but sign recovery without sign supervision. The Wilson-Cowan interaction pattern becomes a memory constraint on the evolved graph. If the evolved population-level E-to-E, E-to-I, and I-to-E connectivity signs were measured to decorrelate from the Wilson-Cowan sign matrix across repeated PDNE runs at matched prediction error, this Postulate would be falsified.

Source Synopsis

Yadav studies whether a compact reservoir computing system can learn not only the time series generated by a Wilson-Cowan neuronal population model, but also a structurally interpretable internal organization. The Wilson-Cowan system is a two-population model of excitatory activity \(E(t)\) and inhibitory activity \(I(t)\), driven by an external pulse stimulus \(s(t)\). Its interaction matrix contains recurrent excitation, excitatory drive to inhibition, inhibitory feedback to excitation, and no explicit inhibitory self-coupling in the parameterization used.

The source applies performance-dependent network evolution, or PDNE, to reservoir computing. A reservoir begins as a minimal random directed network. Its topology is then modified through node addition and deletion. Additions are accepted when test prediction error strictly decreases across output channels. Deletions are accepted when error does not increase. The readout is trained by ridge regression, while the recurrent reservoir topology evolves according to performance.

The evolved reservoirs predict both \(E(t)\) and \(I(t)\) across withheld stimulus amplitudes. They also generalize without retraining to new pulse configurations, including different pulse counts, positions, and amplitudes. This matters because the reservoirs were trained only on two-pulse sequences. Their zero-shot performance indicates that they learned the reactive dynamics of the Wilson-Cowan system rather than memorizing a fixed input-output map.

The structural result is the core finding. Nodes can be classified as E-specific, I-specific, shared, or peripheral by readout contribution. Across stochastic repetitions, the evolved networks show stable proportions of these categories. At the population-connectivity level, the evolved reservoirs recover the Wilson-Cowan sign pattern for three of four interaction types: E-to-E positive, E-to-I negative, and I-to-E positive. The main deviation is a negative I-to-I interaction, absent from the source Wilson-Cowan equations but consistent with known lateral inhibitory motifs in cortical interneuron circuitry. Yadav concludes that performance-driven evolution can produce accurate, compact, and structurally interpretable digital twins of neuronal population dynamics.

Postulate Lens

The applicable Pudding Theory framing is Material Memory. The source phenomenon is a direct case of repeated signal exposure becoming persistent structure. A reservoir begins without imposed Wilson-Cowan anatomy. After many performance-filtered contacts with the same dynamical family, it carries a trace of that family in its edge signs, node roles, gain distribution, and population-level organization.

Material Memory states that matter retains the trace of repeated signals, and that the trace biases future probability. In this paper, the “matter” is the computational substrate of the reservoir graph. The “repeated signal” is the family of Wilson-Cowan trajectories generated under varying pulse amplitudes. The retained trace is not a scalar error minimum. It is a structured memory: a sign-constrained organization that continues to bias the reservoir’s responses to new stimuli.

The source already contains the form required by the Postulate. The network is repeatedly driven by a physiological rhythm model. Its topology is changed only when the change improves prediction. After this history, unseen stimuli no longer meet an unmarked random network. They meet a substrate that has been marked by the past. Prediction becomes possible because the prior signal has been deposited into the graph.

Pudding Theory Reading

Pudding Theory reads the evolved reservoir as a material record of Wilson-Cowan dynamics. The reservoir is not an abstract approximator whose topology happens to be interpretable after training. It is a receptive substrate whose recurrent graph has accumulated the dynamical trace of the source system. PDNE supplies the inscription mechanism. Each accepted node addition or deletion is a small act of retention. Each rejected change is a failure to carry the signal. Over many cycles, the substrate preserves only those structural variations compatible with the repeated E-I waveform family.

This changes the interpretation of the learned sign matrix. In the source framing, the sign recovery is evidence that performance pressure can discover structure. In the Pudding Theory reading, sign recovery is the diagnostic of memory. The Wilson-Cowan signal has left a durable trace in the reservoir’s connectivity. The trace is not stored as a copy of the two differential equations. It is distributed across node classes, recurrent weights, gains, and output-relevant subspaces. This is why the graph does not reproduce the two-node Wilson-Cowan architecture. Memory need not preserve literal form. It preserves future bias. The evolved graph is organized so that new pulse histories are preferentially mapped into Wilson-Cowan-like transients.

The source treats the final network size, density, and edge distribution as outcomes of stochastic evolution. Pudding Theory treats them as constrained memory variables. The final network size \(66 \pm 17\) is not merely an implementation detail. It marks the amount of substrate required, under the chosen update rules, to retain the E-I trace with sufficient stability for zero-shot use. The broad gain distribution is likewise not incidental. It is the reservoir’s way of storing timescale separation. The inhibitory channel rises faster and demands sharper representation, so the trace allocates a slightly larger I-specific population and heterogeneous gain.

The I-to-I deviation is especially informative. Wilson-Cowan sets \(w_{II}=0\), yet the evolved reservoir develops strong negative I-to-I connectivity. The source identifies this as a biologically plausible motif. Pudding Theory goes further. It says the repeated signal contains more than the explicit parameter table. A dynamical trajectory carries implicit stabilization requirements. When those requirements are written into a recurrent substrate, the graph may store a stabilizing feature absent from the source equations but latent in the target behavior. The trace is therefore not a transcription of model parameters. It is a memory of the full response geometry.

The source’s “background” stochasticity becomes part of the signal. Ten independent runs do not erase the sign structure. They expose it. Variation in exact topology shows that many microscopic traces can carry the same macroscopic memory. The stable fact is the population-level sign organization. Material Memory predicts that this is where the durable trace should appear: not at the identity of single nodes, but at the level where repeated exposure has constrained future dynamics.

Falsifiable Observable

The distinguishing observable is the cross-run correlation between the evolved reservoir’s population-level connectivity sign matrix and the Wilson-Cowan E-I sign matrix, evaluated among reservoirs with matched prediction error and comparable size. Material Memory predicts that high predictive accuracy under repeated Wilson-Cowan exposure should carry a persistent sign trace at population level, not merely arbitrary topology with a trained readout. If the evolved population-level E-to-E, E-to-I, and I-to-E connectivity signs were measured to decorrelate from the Wilson-Cowan sign matrix across repeated PDNE runs at matched prediction error, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The result may not require this interpretation. PDNE selects networks by prediction error. If the target system has E-I dynamics, then some compatible topology will be favored. Calling that memory risks relabeling optimization.

Sterling: Optimization names the procedure. It does not name what persists. The procedure could have produced accurate reservoirs whose internal signs were arbitrary, with the readout absorbing the interpretation. The reported result is stronger. The recurrent substrate itself carries the E-I sign relation for three of four interaction types. That is the relevant physical fact.

Tanaka: But this is a simulated reservoir. No biological material is retaining a trace.

Sterling: Material Memory is not restricted to carbon tissue. It concerns substrates whose states are modified by repeated signals and whose later responses are biased by those modifications. A recurrent graph with retained edges, gains, and node roles satisfies that condition.

Tanaka: The I-to-I term argues against direct memory. It is absent in the Wilson-Cowan equations.

Sterling: It argues against literal copying. It supports memory of dynamics. The reservoir stores the response geometry, including stabilization pressure implicit in the trajectories. A memory trace can preserve the operational demand better than the symbolic parameter list.

Discussion

This reading buys a sharper account of interpretability. The source shows that a performance-evolved reservoir can become structurally meaningful. Pudding Theory explains why the meaning resides in the substrate rather than only in the readout. Repeated exposure to a coherent dynamical family leaves a trace that constrains future response. That trace appears at the population level as sign organization and at the node level as specialization.

The limitation is that the present source uses synthetic Wilson-Cowan data. The target system is known, and the sign matrix can be checked. Experimental neural data would be harder. There, the claim would require comparing learned traces against independently measured circuit constraints or perturbation responses.

The conclusion would change if equally accurate reservoirs consistently failed to retain population-level E-I signs. It would also change if the sign structure appeared before exposure or under unrelated training signals. In that case, the graph would not be carrying a Wilson-Cowan trace. It would be expressing an architectural bias. The present result points the other way: history becomes structure, and structure becomes future bias.

References

1. Manish Yadav, “Emergent E-I Structure in Performance-Evolved Reservoir Networks of Neuronal Population Dynamics,” arXiv:2603.13635, DOI: doi:10.48550/arxiv.2603.13635, 2026.

2. S. Ochs, “Pudding Theory: A Topological Theory of Information Fields,” QBist Lab Working Paper, 2026.

3. H. R. Wilson and J. D. Cowan, “Excitatory and inhibitory interactions in localized populations of model neurons,” Biophysical Journal 12, 1-24, 1972.

4. H. Jaeger, “The ‘echo state’ approach to analysing and training recurrent neural networks,” GMD Report 148, 2001.

5. W. Maass, T. Natschlager, and H. Markram, “Real-time computing without stable states: A new framework for neural computation based on perturbations,” Neural Computation 14, 2531-2560, 2002.

6. M. Yadav, S. Sinha, and M. Stender, “Evolution beats random chance: Performance-dependent network evolution for enhanced computational capacity,” Physical Review E 111, 014320, 2025.

7. R. Tremblay, S. Lee, and B. Rudy, “Gabaergic interneurons in the neocortex: From cellular properties to circuits,” Neuron 91, 260-292, 2016.