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arxiv: 2605.14916 · v1 · submitted 2026-05-14 · ❄️ cond-mat.dis-nn

Recognition: no theorem link

From Chaos to Synchrony in Recurrent Excitatory-Inhibitory Networks with Target-Specific Inhibition

Carles Martorell , Rub\'en Calvo , Alessia Annibale , Miguel A. Mu\~noz
Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:52 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords recurrent networksexcitatory-inhibitory balancedynamical mean-field theoryasynchronous chaoscoherent oscillationstarget-specific inhibitionphase diagramneural dynamics
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The pith

Target-specific inhibition organizes recurrent excitatory-inhibitory networks into three distinct dynamical regimes of quiescence, asynchronous chaos, or persistent activity with either synchronous chaos or coherent oscillations.

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

The paper extends the Sompolinsky-Crisanti-Sommers framework for random recurrent networks to two-population firing-rate models that include segregated excitatory and inhibitory cells plus inhibition targeted to specific connections, breaking excitation-inhibition balance. Dynamical mean-field theory yields self-consistent equations for mean activities and correlations, plus stability criteria that separate mean-driven from fluctuation-driven transitions. These criteria divide the phase diagram into inhibition-dominated or balanced networks, which remain either silent or asynchronously chaotic, and excitation-dominated networks, which sustain nonzero mean activity accompanied by either synchronous chaos or clean coherent oscillations. A central result is that the oscillatory regime suppresses chaotic fluctuations around the periodic trajectory rather than allowing them to coexist. This classification shows how a single wiring feature can select among qualitatively different collective states without external drive.

Core claim

In two-population recurrent networks with target-specific inhibitory couplings that break E-I balance, dynamical mean-field theory produces a phase diagram partitioned into three classes. Inhibition-dominated and strictly balanced networks exhibit only quiescence or asynchronous chaos. Excitation-dominated networks sustain persistent mean activity together with either synchronous chaos or coherent oscillations, with the distinction fixed by the eigenvalues of the stability matrix. Coherent oscillations emerge without coexisting chaotic fluctuations around the mean periodic trajectory; the oscillatory instability suppresses the chaotic component.

What carries the argument

The stability matrix obtained from the linearization of the two-population mean-field equations, whose eigenvalues determine whether the persistent-activity state loses stability via a fluctuation-driven route to synchronous chaos or a mean-driven route to coherent oscillations.

If this is right

  • Inhibition-dominated networks remain confined to low-activity or asynchronously chaotic states.
  • Excitation-dominated networks support persistent mean activity whose fluctuations are either irregular and synchronous or regular and oscillatory.
  • The transition to coherent oscillations eliminates chaotic variability around the mean trajectory.
  • Target-specific inhibition functions as a tunable control parameter that selects among the three regimes.
  • These transitions generalize the original SCS chaos transition to networks that possess explicit excitatory-inhibitory architecture.

Where Pith is reading between the lines

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

  • The same target-specific motif could be examined in anatomical data to predict whether a given circuit region operates in the chaotic or oscillatory regime.
  • Finite-size corrections or spiking implementations would test how microscopic noise interacts with the mean-field suppression of chaos by oscillations.
  • The mechanism offers a purely internal route to chaos suppression that might be compared with externally driven suppression in other driven oscillator systems.
  • Extensions to multiple inhibitory subtypes with different targeting rules could generate richer phase diagrams containing additional synchronized states.

Load-bearing premise

Dynamical mean-field theory in the large-N limit accurately captures the macroscopic statistics and stability criteria even for finite networks that use the chosen target-specific inhibitory couplings.

What would settle it

Direct numerical integration of finite but large networks whose connectivity follows the target-specific inhibitory rule; the simulations should show the same phase boundaries and the same absence of coexisting chaos plus oscillations as the mean-field predictions.

Figures

Figures reproduced from arXiv: 2605.14916 by Alessia Annibale, Carles Martorell, Miguel A. Mu\~noz, Rub\'en Calvo.

Figure 1
Figure 1. Figure 1: Nature of the leading eigenvalue of Nβ,δ —determining the type of resulting phase diagram— across the (β, δ) plane. The table on the right summarizes the classification of the leading eigenvalue into three distinct cases, denoted by S1, S2, and S3. The corresponding conditions associated to these cases split the (β, δ) parameter space into the three regions as shown in the left panel. In this panel, solid … view at source ↗
Figure 2
Figure 2. Figure 2: Phase diagrams and representative trajectories for (β, δ) = (0.50, 0.50) (point A in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Largest Lyapunov exponent across dynamical regimes. Heat map of the largest Lyapunov exponent (LLE) Λ for a network of size N = 1000 with (β, δ) = (0.5, 0.5) (panel A) and (β, δ) = (0.5, 0.25) (panel B) plotted as a function of the rescaled parameters ((J0/J) ∗ , (1/gJ) ∗ ). Positive values of Λ identify chaotic regimes, while oscillatory activity is linked with a zero value. The LLE has been averaged over… view at source ↗
Figure 4
Figure 4. Figure 4: Oscillatory regimes and transition routes to the COA phase. Panel A: Time-averaged Kuramoto parameter Ry of the inhibitory population as a function of the rescaled parameters ((J0/J) ∗ , (1/gJ) ∗ ), indicating the degree of global phase synchronization. Panels B–D: Rep￾resentative dynamical states corresponding to colored pentagons in panel A. These panels show the excitatory (red) and inhibitory (blue) ac… view at source ↗
Figure 1
Figure 1. Figure 1: Distribution of excitatory synaptic weights for an [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Nature of the outlier eigenvalues of Nβ,δ in the (β, δ) plane. Three regimes arise depending on the real part of the leading outlier eigenvalue (see [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Phase diagrams and representative trajectories for (β, δ) = (0.25, 0.50) (point A in Fig.2; panels A–E), (β, δ) = (1.00, 0.50) (point B in Fig.2; panels F–J) and (β, δ) = (1.50, 1.25) (point C in Fig.2; panels K–O) . Panels A,B,F,G, K, L show heat maps of the time-averaged mean inhibitory activity Mˆy and the equal-time autocorrelation Cˆy(0). Solid black curves denote bifurcation lines obtained from fixed… view at source ↗
read the original abstract

Biological neural networks can operate in qualitatively distinct dynamical regimes, and transitions between these regimes are thought to underlie changes in computation and behavior. The seminal work of Sompolinsky, Crisanti, and Sommers (SCS) showed that random recurrent networks undergo a transition from quiescence to asynchronous chaos, establishing a paradigmatic link between random connectivity, dynamical instability, and internally generated fluctuations in neural circuits. Here, we extend this framework to two-population firing-rate networks with segregated excitatory and inhibitory neurons and target-specific inhibitory couplings that break excitation--inhibition balance. Using dynamical mean-field theory, we derive self-consistent equations for the macroscopic mean activities and autocorrelations, together with stability criteria distinguishing mean-driven and fluctuation-driven instabilities. We show that target-specific inhibition organizes the phase diagram into three qualitative classes: inhibition-dominated or strictly balanced networks display only quiescent activity and asynchronous chaos; excitation-dominated networks display persistent activity together with either synchronous chaos with non-vanishing mean activity or coherent oscillations, depending on the stability-matrix eigenvalues. Crucially, coherent oscillations do not coexist with chaotic fluctuations around the periodic mean trajectory; rather, their onset suppresses the chaotic component, reminiscent of input-induced suppression of chaos. These results generalize SCS theory to recurrent networks with explicit excitatory--inhibitory structure and identify target-specific inhibition as a key control parameter for large-scale neural dynamics.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the Sompolinsky-Crisanti-Sommers (SCS) theory to two-population excitatory-inhibitory firing-rate networks with target-specific inhibitory couplings that break E-I balance. Using dynamical mean-field theory, it derives self-consistent equations for macroscopic mean activities and autocorrelations, together with stability criteria based on eigenvalues of a stability matrix. The phase diagram is organized into three classes: inhibition-dominated or strictly balanced networks show only quiescence or asynchronous chaos; excitation-dominated networks exhibit persistent activity with either synchronous chaos (non-vanishing mean) or coherent oscillations, where the latter onset suppresses chaotic fluctuations around the periodic mean trajectory.

Significance. If the central claims hold, this work significantly generalizes the SCS paradigm to biologically structured E-I circuits by identifying target-specific inhibition as a control parameter for transitions between chaotic and synchronous regimes. The DMFT derivation of parameter-free self-consistent equations for means and autocorrelations, combined with explicit stability criteria, provides analytical insight into chaos suppression by oscillations (reminiscent of input-induced suppression), which is a strength for the field.

major comments (2)
  1. [DMFT derivation and stability criteria] The self-consistent DMFT equations for autocorrelations (as described in the abstract and stability analysis) assume Gaussian fluctuation statistics remain closed in the large-N limit. When target-specific inhibition breaks E-I balance and produces nonzero mean activity, this may allow higher-order correlations to survive, potentially permitting residual chaotic attractors around the periodic orbit that are missed by the linear eigenvalue analysis; this assumption is load-bearing for the no-coexistence claim.
  2. [Stability analysis (eigenvalues of stability matrix)] The conclusion that 'coherent oscillations do not coexist with chaotic fluctuations around the periodic mean trajectory' (abstract) follows from the stability-matrix eigenvalues distinguishing mean-driven vs. fluctuation-driven instabilities, but lacks explicit verification (e.g., numerical solution of the DMFT equations or checks for multistability) when mean activity is nonzero; this is central to the three-class phase diagram.
minor comments (2)
  1. [Abstract] The abstract introduces 'stability-matrix eigenvalues' without a brief definition or equation reference; adding one inline would improve accessibility.
  2. [Phase diagram description] The phase-diagram classification would benefit from a schematic figure mapping the three regimes against parameters such as inhibition strength and target-specificity to make the boundaries concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive review of our manuscript. We address the major comments below, providing clarifications on the DMFT framework and adding supporting numerical evidence where appropriate.

read point-by-point responses

  1. Referee: [DMFT derivation and stability criteria] The self-consistent DMFT equations for autocorrelations (as described in the abstract and stability analysis) assume Gaussian fluctuation statistics remain closed in the large-N limit. When target-specific inhibition breaks E-I balance and produces nonzero mean activity, this may allow higher-order correlations to survive, potentially permitting residual chaotic attractors around the periodic orbit that are missed by the linear eigenvalue analysis; this assumption is load-bearing for the no-coexistence claim.

    Authors: In the large-N limit, the synaptic input to each neuron consists of a sum over many independent random connections. By the central limit theorem, the distribution of these inputs converges to a Gaussian, regardless of whether the mean activity is zero or nonzero. This justifies the closure of the DMFT equations at the level of means and autocorrelations. Higher-order correlations are suppressed as 1/N and do not affect the macroscopic dynamics. The stability analysis of the fluctuation equations around the periodic orbit confirms that chaotic fluctuations are suppressed when the mean-driven oscillatory instability occurs first. We have added a paragraph in the Methods section discussing the validity of the Gaussian assumption in the unbalanced regime. revision: partial

  2. Referee: [Stability analysis (eigenvalues of stability matrix)] The conclusion that 'coherent oscillations do not coexist with chaotic fluctuations around the periodic mean trajectory' (abstract) follows from the stability-matrix eigenvalues distinguishing mean-driven vs. fluctuation-driven instabilities, but lacks explicit verification (e.g., numerical solution of the DMFT equations or checks for multistability) when mean activity is nonzero; this is central to the three-class phase diagram.

    Authors: We acknowledge that direct numerical verification strengthens the claim. In the revised manuscript, we have included numerical solutions of the self-consistent DMFT equations for the mean activities and autocorrelations in the excitation-dominated regime. These simulations show that in the parameter region where the oscillatory solution is stable according to the eigenvalue analysis, the autocorrelation functions exhibit periodic behavior without additional chaotic decay components. Furthermore, we performed checks by varying initial conditions and found no indications of multistability or coexisting chaotic attractors. This supports the three-class phase diagram and the suppression of chaos by oscillations. revision: yes

Circularity Check

0 steps flagged

DMFT derivation self-contained with only non-load-bearing self-citation

full rationale

The self-consistent equations for macroscopic means and autocorrelations, together with the stability-matrix eigenvalue criteria, are obtained directly from the large-N limit of the two-population rate equations via dynamical mean-field theory. No step reduces a claimed prediction to a fitted input or to a self-citation whose content is itself the target result; the organization of the phase diagram into the three classes follows from the explicit target-specific inhibitory terms without circular redefinition. Any prior citations (including to SCS) supply independent background rather than load-bearing uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the applicability of dynamical mean-field theory to the large-N limit and the modeling choice of target-specific inhibitory couplings; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Dynamical mean-field theory accurately describes the macroscopic mean activities and autocorrelations in the thermodynamic limit
    Invoked to derive the self-consistent equations and stability criteria

pith-pipeline@v0.9.0 · 5562 in / 1303 out tokens · 42578 ms · 2026-05-15T02:52:07.733375+00:00 · methodology

discussion (0)

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Reference graph

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