Task-specific programming of chaos in neural circuits

Jungyoon Kim; Kunwoo Park; Kyuho Kim; Namkyoo Park; Sunkyu Yu

arxiv: 2605.19465 · v1 · pith:QIYLPTVKnew · submitted 2026-05-19 · 🌊 nlin.CD · physics.comp-ph

Task-specific programming of chaos in neural circuits

Jungyoon Kim , Kyuho Kim , Kunwoo Park , Namkyoo Park , Sunkyu Yu This is my paper

Pith reviewed 2026-05-20 02:12 UTC · model grok-4.3

classification 🌊 nlin.CD physics.comp-ph

keywords neural circuitschaotic dynamicsreservoir computingnetwork topologysmall-world networksneuromorphic computingchaos controltask-specific computation

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The pith

Small-world connectivity lets neural circuits switch chaos on and off by rewiring edges.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that in a continuous-time neural-circuit model, changing the network's connection pattern shifts the system between ordered and chaotic regimes while also altering correlation times, stability, and how signals travel. Small-world topologies stand out because they allow this switch to happen quickly and with low latency simply by adding or removing a few edges. This control pairs with adjustments to individual circuit elements to create a single phase diagram that maps both chaos level and response speed. Reservoir-computing tests across many topologies then show that each regime performs differently on practical tasks, turning network shape into a practical knob for matching the circuit to the job at hand.

Core claim

In a continuous-time neural-circuit model, tuning network topology drives an ordered-to-chaotic transition accompanied by changes in correlation timescales, stability characteristics, and signal propagation. Small-world connectivity enables low-latency on-off switching of chaos via edge rewiring. Joint control of element-level properties and network topology yields a unified chaos-latency phase diagram, and reservoir-computing benchmarks across topological regimes confirm that network topology functions as a reconfigurable parameter for task-specific computation and tunable randomness.

What carries the argument

Small-world connectivity as the reconfigurable parameter that permits low-latency on-off switching of chaotic dynamics through edge rewiring in the neural-circuit model.

If this is right

Tuning network topology produces distinct correlation timescales and signal-propagation behaviors suited to different computational needs.
Small-world networks support low-latency on-off switching of chaos without retuning individual elements.
A single phase diagram unifies element-level and topological controls over both chaos intensity and response speed.
Reservoir-computing performance varies systematically with topology, allowing task-specific selection of network structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Hardware versions of these circuits could use physical rewiring mechanisms to adapt dynamically to changing tasks.
The same topological principle might help explain how biological networks reconfigure computation without altering neuron properties.
Testing the approach on larger networks or with different neuron models would reveal how far the small-world advantage extends.

Load-bearing premise

The continuous-time neural-circuit model and its observed dynamical transitions are representative enough to guide real physical implementations or apply beyond the specific equations and parameters used.

What would settle it

A physical realization of the model in which edge rewiring in small-world topologies fails to produce rapid, low-latency toggling between ordered and chaotic states would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.19465 by Jungyoon Kim, Kunwoo Park, Kyuho Kim, Namkyoo Park, Sunkyu Yu.

**Figure 1.** Figure 1: Network topology on neuronal dynamics. a, A directed neural circuit network with node-wise positive (orange) or negative (black) edges. The initial values are uniformly sampled from the interval [–5, 5]. The node states evolve dynamically over t. b, Illustration of Dale-type sign constraints, where individual neurons provide strictly excitatory or inhibitory connections. ce, Dynamics in the regular (c) sm… view at source ↗

**Figure 2.** Figure 2: Topological classification of circuit dynamics. a, Network parameters: clustering coefficient C and average path length L, as functions of β. b, Temporal metrics: memory timescale τc and first-passage time τp, as functions of β. ε = 0.01 in estimating τp. c,d, Time autocorrelation Q(ς) (c) and perturbation spreading (d) for regular (top; β = 10–4 ), small-world (middle; β = 0.1), [PITH_FULL_IMAGE:figures/… view at source ↗

**Figure 3.** Figure 3: Chaos-latency phase diagrams. Phase diagrams are shown over the (β, E) parameter space for the memory metric τc (a), latency metric τp (b), and the LLE (c). The parameters β and E [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Topologically programmable chaos for task-specific computation. a, Schematic for reservoir-computing framework based on our neural circuit, applied to three representative tasks: MNIST image classification (left), remote signal tracking (centre), and temporal signal prediction (right). b-d, Task performances as a function of β at E = 0.2, spanning regular, small-world, and random network regimes: classific… view at source ↗

**Figure 1.** Figure 1: Network topology on neuronal dynamics. a, A directed neural circuit network with node-wise positive (black) or negative (orange) edges. The initial values are uniformly sampled from the interval [–5, 5]. The node states evolve dynamically over t. b, Illustration of Dale-type sign constraints, where individual neurons provide strictly excitatory or inhibitory connections. ce, Dynamics in the regular (c) sm… view at source ↗

**Figure 4.** Figure 4: Topologically programmable chaos for task [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗

read the original abstract

Chaotic dynamics have emerged as a versatile resource for neuromorphic and probabilistic computing, enabling high-dimensional nonlinear processing and classical analogues of quantum randomness. Exploiting chaos for computation requires task-dependent control over complexity, as demonstrated in reservoir computing, random-number generation, and probabilistic inference. Existing approaches have focused on tuning element-level parameters, leaving the collective, many-body origin of chaos largely unexplored as a design freedom. Here, we demonstrate programmable chaotic dynamics for task-specific reservoir computing. Using a continuous-time neural-circuit model, we show that tuning network topology drives an ordered-to-chaotic transition, accompanied by transitions in correlation timescales, stability characteristics, and signal propagation. By jointly controlling element-level properties and network topology, we establish a unified chaos-latency phase diagram, revealing that small-world connectivity enables low-latency on-off switching of chaos via edge rewiring. Supported by distinct reservoir-computing benchmarks across various topological regimes, our results demonstrate that network topology serves as a reconfigurable parameter for task-specific computation and tunable randomness.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Small-world rewiring gives low-latency chaos switching in this neural model, but the advantage may stay tied to the specific equations.

read the letter

Small-world connectivity lets you flip chaos on and off quickly by rewiring edges in this continuous-time neural circuit model, and the benchmarks suggest it helps with task-specific reservoir computing. That's the core new angle. The paper moves past the usual focus on local parameters by treating network topology as a control knob. They show an ordered-to-chaotic transition driven by topology, along with shifts in correlation times, stability, and signal propagation. The unified phase diagram that combines chaos measures with latency is a clean way to present the joint control of element properties and topology. The reservoir computing tests across different regimes provide some evidence that these changes translate to performance differences. What works here is the clear link between topology changes and dynamical features, plus the practical demonstration via benchmarks. It builds on known network dynamics but applies it to programmable chaos for computation. The soft spots are around generalizability. All the transitions and the low-latency switching claim come from one specific model. If the vector field or the balance of local and global coupling changes, it's not obvious the small-world advantage holds up. The abstract and summary don't indicate tests for invariance to those details, so the reconfigurability might be tied to the chosen ODEs rather than a broad many-body effect. Also, without error bars or precise exclusion criteria in the reported results, it's hard to gauge how solid the benchmark distinctions are. This work is for people designing neuromorphic systems or exploring chaos in networks for computing tasks. Someone looking for topology-based design rules in reservoir computing would get something out of it. The ideas are coherent and engage with the literature on dynamical networks, so it deserves a serious referee who can push on the model dependence and ask for more robustness checks. I'd send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that in a continuous-time neural-circuit model, tuning network topology (especially small-world rewiring) induces an ordered-to-chaotic transition with accompanying changes in correlation timescales, stability, and signal propagation. Joint control of element-level parameters and topology yields a unified chaos-latency phase diagram in which small-world connectivity permits low-latency on-off switching of chaos via edge rewiring; this is supported by reservoir-computing benchmarks that differ across topological regimes, positioning topology as a reconfigurable design parameter for task-specific computation.

Significance. If the reported transitions and benchmark distinctions prove robust, the work supplies a collective, topology-based mechanism for programming chaotic resources that complements element-level tuning. The unified phase diagram and explicit demonstration of rewiring-based latency control constitute concrete strengths that could inform neuromorphic and probabilistic computing architectures.

major comments (2)

[§4] §4 (dynamical transitions and phase diagram): the central attribution of low-latency chaos switching to small-world topology requires explicit checks that the ordered-to-chaotic boundary, correlation timescales, and stability metrics remain qualitatively unchanged under variations in the vector field, nonlinearity, or relative scale of local versus global coupling; without such tests the reconfigurability advantage may be specific to the chosen ODEs rather than a general topological effect.
[Benchmark results] Benchmark results (reservoir-computing section): quantitative performance differences across topologies are reported, yet the text provides no error bars, number of independent trials, or statistical tests; this weakens the claim that small-world regimes produce distinctly superior task-specific behavior.

minor comments (2)

[Methods] Clarify the precise definition and numerical implementation of the rewiring probability and the edge-rewiring protocol used for on-off switching.
[Figures] Figure captions and axis labels for the phase diagram should explicitly state the fixed parameter values and the range over which element-level properties are varied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important aspects of robustness and statistical rigor that strengthen the manuscript. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: §4 (dynamical transitions and phase diagram): the central attribution of low-latency chaos switching to small-world topology requires explicit checks that the ordered-to-chaotic boundary, correlation timescales, and stability metrics remain qualitatively unchanged under variations in the vector field, nonlinearity, or relative scale of local versus global coupling; without such tests the reconfigurability advantage may be specific to the chosen ODEs rather than a general topological effect.

Authors: We agree that demonstrating robustness to changes in the underlying dynamics is essential to support the generality of the topological mechanism. In the revised manuscript we have performed additional simulations in which we vary the nonlinearity strength (by scaling the sigmoid gain) and the relative weight of local versus global coupling while keeping the small-world rewiring probability fixed. The ordered-to-chaotic transition, correlation-time scaling, and Lyapunov-spectrum signatures remain qualitatively unchanged across these variations; the low-latency rewiring control of chaos persists. These results are now reported in a new subsection of §4 together with a supplementary figure that overlays the phase boundaries for the different vector-field realizations. revision: yes
Referee: Benchmark results (reservoir-computing section): quantitative performance differences across topologies are reported, yet the text provides no error bars, number of independent trials, or statistical tests; this weakens the claim that small-world regimes produce distinctly superior task-specific behavior.

Authors: We acknowledge that the original presentation lacked quantitative uncertainty measures. In the revision we have re-run all reservoir-computing benchmarks over 20 independent random initializations and network realizations for each topology. Error bars now indicate one standard deviation, and we have added two-tailed t-tests (with Bonferroni correction) comparing small-world performance against regular and random topologies on each task. The statistical results confirm that the small-world advantage is significant (p < 0.01) for the majority of tasks examined. These updates appear in the revised reservoir-computing section and in the caption of the corresponding figure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; topology treated as independent input

full rationale

The derivation begins with a standard continuous-time neural-circuit ODE model whose vector field and parameters are specified independently of the target chaotic behavior. Network topology (small-world rewiring, etc.) is introduced as an exogenous control parameter, and the ordered-to-chaotic transition, correlation timescales, stability metrics, and reservoir-computing performance differences are obtained by direct numerical integration and benchmarking across topological regimes. No equation is defined in terms of its own output, no fitted constant is relabeled as a prediction, and no load-bearing step reduces to a self-citation or prior ansatz by the same authors. The phase diagram and benchmark distinctions therefore remain falsifiable against changes in the underlying ODE or parameter values and do not collapse by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on a standard continuous-time neural circuit model whose parameters and topology are adjusted to produce the reported transitions; no new physical entities are postulated.

free parameters (1)

rewiring probability and related topology controls
Chosen to achieve small-world regimes and drive the ordered-to-chaotic transition in the model.

axioms (1)

domain assumption Neural dynamics follow continuous-time differential equations with standard activation and coupling terms
Invoked to define the base model in which topology changes produce the observed dynamical transitions.

pith-pipeline@v0.9.0 · 5713 in / 1260 out tokens · 58392 ms · 2026-05-20T02:12:19.180404+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We extend this random-network model to a range of network topologies using the following procedure... β ∈ [0,1] denotes the disorder parameter... from a regular graph (β → 0), through a small-world graph, to an Erdős-Rényi graph (β → 1)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the memory timescale τc and the propagation latency τp... Q(ς) = ⟨φi(t)φi(t+ς)⟩i,t

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

& Ganguli, S

1 Poole, B., Lahiri, S., Raghu, M., Sohl -Dickstein, J. & Ganguli, S. Exponential expressivity in deep neural networks through transient chaos. Advances in neural information processing systems 29 (2016)

work page 2016

[1] [1]

& Ganguli, S

1 Poole, B., Lahiri, S., Raghu, M., Sohl -Dickstein, J. & Ganguli, S. Exponential expressivity in deep neural networks through transient chaos. Advances in neural information processing systems 29 (2016)

work page 2016