Excitation–inhibition balance controls coupling stability and network reorganization in a plastic Kuramoto model
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eLife Assessment
This useful study investigates how plasticity and homeostatic adaptation can lead to the emergence of synchronization patterns associated with different sleep phases. The ideas are novel and interesting, but the evidence at present remains incomplete. Further work is needed to determine whether the reported states persist for larger system sizes, longer integration times, and how robust the results are to different initial conditions.
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Abstract
Sleep and rest, characterized by synchronized neuronal activity that emerges under large shifts in the excitation–inhibition balance, are crucial for synaptic reorganization in the brain. However, the network dynamics that permit rewiring without erasing stable connections remain unclear. To address this question, we extended a Kuramoto framework with excitation–inhibition balance by adding Hebbian and homeostatic plasticity. The resulting model showed that networks with robust inhibition consistently exhibited desynchronized dynamics and stable couplings. In contrast, networks with weaker inhibition exhibited a bistable regime, in which couplings of intermediate strengths fluctuated while strong couplings remained stable. These findings suggest that the dynamic interplay between network activity and plasticity selectively stabilizes stronger connections while permitting flexible reorganization of weaker ones, providing a potential mechanism for network reorganization during sleep and rest.
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eLife Assessment
This useful study investigates how plasticity and homeostatic adaptation can lead to the emergence of synchronization patterns associated with different sleep phases. The ideas are novel and interesting, but the evidence at present remains incomplete. Further work is needed to determine whether the reported states persist for larger system sizes, longer integration times, and how robust the results are to different initial conditions.
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Reviewer #1 (Public review):
Summary:
The authors aim to understand how changes in the balance between excitatory and inhibitory interactions influence the stability and reorganization of network connections. To address this question, they extend a coupled-phase-oscillator model by adding plasticity rules. The central finding is that stronger inhibitory interactions lead to relatively stable and desynchronized network dynamics, whereas weaker inhibitory interactions produce a bistable regime in which intermediate-strength connections fluctuate while stronger connections are preserved.
Strengths:
This study offers a simple theoretical framework for linking network state, coupling stability, and reorganization. The model produces clear qualitative results, showing that different dynamical regimes are associated with different balances of …
Reviewer #1 (Public review):
Summary:
The authors aim to understand how changes in the balance between excitatory and inhibitory interactions influence the stability and reorganization of network connections. To address this question, they extend a coupled-phase-oscillator model by adding plasticity rules. The central finding is that stronger inhibitory interactions lead to relatively stable and desynchronized network dynamics, whereas weaker inhibitory interactions produce a bistable regime in which intermediate-strength connections fluctuate while stronger connections are preserved.
Strengths:
This study offers a simple theoretical framework for linking network state, coupling stability, and reorganization. The model produces clear qualitative results, showing that different dynamical regimes are associated with different balances of excitatory and inhibitory interactions. This could be useful as a conceptual starting point for considering how network states may regulate the stability and flexibility of connections. The manuscript also explores several model parameters.
Weaknesses:
The evidence is incomplete in supporting the biological interpretations. The model is a highly simplified coupled-phase-oscillator system and does not directly represent spiking activity, membrane potentials, synaptic currents, conduction delays, cellular excitability, or detailed biological plasticity mechanisms. Although the authors clarify that the model units are not actual neurons or synapses, the discussion often interprets the results in terms of neuronal inhibition, synaptic stability, sleep-related reorganization, and preservation of strong biological connections. This creates a gap between the abstract model and the biological conclusions. In particular, the manuscript does not sufficiently discuss what biological oscillatory activity the modeled phases are intended to represent, such as population-level activity reflected in electroencephalography or local field potentials. In several places, the manuscript appears to assume that neurons can generally be treated as oscillators, but this is not always a valid assumption. The authors should more clearly distinguish between rhythmic or phase-like activity at the population level and the dynamics of individual neurons, and should frame the model more cautiously as a phenomenological description of collective synchronization rather than a mechanistic model of spiking neuronal circuits.
There are also important methodological limitations. Although the manuscript presents the model equations, parameter values, time step, simulation duration, and coupling update rules, several other essential details are not clearly specified, including the number of simulation runs, the procedure for setting initial conditions, and the numerical method used to solve the ordinary differential equations. Critically, technical details such as the integration scheme, solver settings, initialization procedure, and random seed handling are essential for reproducibility. Because the main findings depend on the interaction between phase dynamics and adaptive coupling, even small implementation differences could affect the reported dynamical regimes and coupling fluctuations.
A further concern is the presentation of the mathematical formulation. Several equations appear to contain notation errors and inconsistencies, making it difficult to follow the exact model definition. The authors should carefully revise the mathematical notation throughout the manuscript to ensure that the model can be understood and reproduced unambiguously.
Overall, the study provides a useful but limited theoretical account of how network dynamics may regulate coupling stability and reorganization. The results support the internal behavior of the proposed model, but the broader biological claims are not yet fully convincing. The likely impact of the work is therefore mainly conceptual: it may stimulate further modeling studies, but additional methodological detail, stronger justification of the modeling assumptions, and comparison with more biologically grounded models would be needed before the conclusions can be applied confidently to neuronal circuit dynamics or sleep-related synaptic reorganization.
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Reviewer #2 (Public review):
Summary:
This manuscript investigates the impact of plasticity mechanisms in an excitation-inhibition (EI) network model on the emergence of synchronization patterns that the authors associate with different sleep phases. The model consists of an EI Kuramoto network in which recurrent excitatory couplings evolve according to Hebbian and homeostatic adaptation rules. Through numerical simulations, the authors analyze how these plasticity mechanisms modify both the collective dynamics and the structure of the coupling matrix.
Strengths:
The topic addressed in the manuscript is timely and potentially relevant, as understanding the interplay between synaptic adaptation and collective neural dynamics remains an important challenge in theoretical neuroscience.
Weaknesses:
In its current form, the work suffers from …
Reviewer #2 (Public review):
Summary:
This manuscript investigates the impact of plasticity mechanisms in an excitation-inhibition (EI) network model on the emergence of synchronization patterns that the authors associate with different sleep phases. The model consists of an EI Kuramoto network in which recurrent excitatory couplings evolve according to Hebbian and homeostatic adaptation rules. Through numerical simulations, the authors analyze how these plasticity mechanisms modify both the collective dynamics and the structure of the coupling matrix.
Strengths:
The topic addressed in the manuscript is timely and potentially relevant, as understanding the interplay between synaptic adaptation and collective neural dynamics remains an important challenge in theoretical neuroscience.
Weaknesses:
In its current form, the work suffers from substantial conceptual, methodological, and technical limitations that significantly weaken the conclusions.
From a biological perspective, the model is highly abstract and qualitative. The connection between the model variables and the physiological processes that the authors aim to describe remains unclear. Consequently, the manuscript does not provide sufficient evidence to support biologically meaningful conclusions regarding sleep dynamics. In my opinion, the work is more naturally positioned within the framework of theoretical or computational dynamical systems than within the scope of a biology-oriented journal.
From a mathematical and dynamical-systems perspective, the analysis is incomplete, and several important technical aspects are either missing or inadequately addressed. In particular, the characterization of the dynamical regimes is often imprecise, the numerical evidence is not sufficiently robust, and little effort is made to interpret the results within the broader context of synchronization theory, adaptive networks, or collective dynamics.
More specifically:
(1) The biological interpretation of the model variables is ambiguous throughout the manuscript. At several points, the authors suggest that individual oscillators should not be interpreted as neurons but rather as abstract biological units (lines 78-82, 86-89, 418-420). However, other parts of the manuscript refer to the coupling matrix entries, particularly $J_{ee}$, as synaptic weights (e.g., line 118). These two interpretations are not obviously compatible. If the oscillators represent coarse-grained or abstract units, the biological meaning of the adaptive couplings should be carefully justified. More generally, the manuscript lacks a clear discussion of what aspects of neural circuits are captured by the model and which aspects are intentionally neglected.
(2) More fundamentally, the manuscript inherits the well-known limitations associated with interpreting Kuramoto oscillators as neural elements. Kuramoto phase oscillators provide a minimal description of synchronization phenomena, but they do not explicitly represent membrane dynamics, firing rates, spiking activity, synaptic currents, or realistic neuronal timescales.
Under certain assumptions, Kuramoto-like models can be rigorously derived from more detailed neuronal models through phase-reduction techniques (see, for instance, Chapter 10 of Izhikevich's \textit{Dynamical Systems in Neuroscience}). However, the authors do not employ such a reduction procedure, nor do they establish a formal connection between their model variables and the dynamics of neuronal populations. As a consequence, the biological interpretation of the model remains unclear.
The authors should therefore explicitly discuss these limitations and carefully justify why the synchronization patterns observed in such a highly reduced model can be related to neural sleep states. At present, the biological interpretation appears considerably stronger than what the model itself can support, for instance, the claims in lines 321-322 or 330-337. In particular, it remains unclear whether the reported dynamical regimes should be interpreted as genuine mechanisms underlying sleep rhythms or merely as generic synchronization phenomena arising in adaptive oscillator networks.
(3) The numerical methodology raises serious concerns regarding the robustness of the reported results. According to the Methods section, simulations are performed using only $N=100$ oscillators and integration times of approximately 50 time units. Such choices may be sufficient for illustrative purposes but are generally inadequate for drawing conclusions about asymptotic collective behavior in adaptive dynamical systems. Finite-size fluctuations can strongly affect synchronization measures, and adaptive networks are well known to exhibit extremely long transients, metastability, and slow convergence processes. No systematic finite-size analysis is provided, nor is there any demonstration that the reported states persist for larger system sizes or longer integration times.
(4) The use of the term "bistable regime" to describe the dynamics shown in Figure 1C is incorrect. A bistable regime refers to a parameter region in which multiple attractors coexist and the asymptotic state depends on the initial condition. The figure instead presents a single trajectory displaying oscillatory dynamics. No evidence is provided for the coexistence of attractors, nor are multiple initial conditions explored. Furthermore, the displayed time series are too short to determine whether the observed dynamics correspond to a stable limit cycle, quasiperiodic motion, intermittent behavior, or a long transient approaching another attractor. The authors should perform a proper dynamical characterization of this regime. Similar collective oscillatory states have been extensively studied in synchronization and adaptive-network models and should be discussed in relation to the existing literature.
In particular, several time series shown in the manuscript (e.g., Figures 1C and 2F) exhibit trends that suggest the possibility of unresolved transient dynamics. The authors should demonstrate convergence of the reported regimes by substantially extending simulation times and by performing finite-size analyses. Simulations with at least one order of magnitude more units ($N\gtrsim 1000$) and integration times sufficient to establish asymptotic behavior would be expected in a study whose main claims rely on collective dynamical phenomena.
(5) The manuscript lacks several standard tools routinely employed in the analysis of nonlinear dynamical systems. The conclusions are largely based on visual inspection of time series and order parameters. However, no bifurcation analysis, stability analysis, phase-space characterization, attractor reconstruction, or systematic exploration of parameter dependence is provided. As a consequence, many of the identified "phases" or "regimes" remain only qualitatively described. A more rigorous dynamical-systems treatment would substantially strengthen the work and would help distinguish genuine asymptotic states from finite-size or transient phenomena.
(6) A substantial fraction of the results appears to extend the authors' previous work by incorporating plastic adaptation mechanisms. While incremental advances are acceptable, the manuscript would benefit from a broader theoretical context. The discussion is heavily centered on previous studies by the same authors, whereas there exists an extensive literature on synchronization, adaptive networks, neural mass models, balanced EI systems, and sleep-related oscillations that is largely absent from the discussion. The novelty and significance of the present contribution would be easier to assess if the results were more carefully compared with alternative theoretical approaches.
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