# Chaotic Balanced State in a Model of Cortical Circuits

**Date**: 03/09/2026
**Domain**: natural_sciences/computational_neuroscience
**Taxonomy**: academic/research_paper
**Filter**: Active comments

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## Overall Feedback

Here are some overall reactions to the document.

**Chaotic Characterization Rests on a Non-Standard, Divergent Lyapunov Exponent**

Sections 8 and equation 8.9 show that the maximum Lyapunov exponent λ_L diverges as D_k → 0 because dD_k/dt ∝ √D_k for binary neurons. All three reviewers flag that an infinite λ_L is a consequence of the discontinuous gain function of binary threshold units and would arise in any such network regardless of whether the macroscopic dynamics are genuinely chaotic or merely irregular. The standard criterion for deterministic chaos is a finite positive Lyapunov exponent, and the paper does not compute any finite-K or finite-perturbation divergence rate, nor does it measure how quickly D_k saturates relative to its equilibrium value 2(m_k − q_k). It would be helpful to supplement the theoretical argument with at least one direct simulation measuring the decorrelation timescale of two nearby trajectories across parameter regimes, and to provide a finite-time or finite-perturbation divergence measure (e.g., the rate at which the overlap Q_k(t) decays from its initial value) that can distinguish the chaotic balanced state from a merely noisy one. Clarifying that the 'infinite Lyapunov exponent' reflects binary discreteness rather than a generic property of the balanced state would also sharpen the paper's central dynamical claim.

**Independence Assumption in the Mean-Field Derivation May Be Circular in the Chaotic Regime**

The mean-field derivation in Appendix A (equations A.4–A.14) and the self-consistent autocorrelation equation 5.17 both rely on the statistical independence of inputs to different neurons, justified by K ≪ N_k following Derrida et al. (1987). However, the paper simultaneously argues in Section 8 that the balanced state is strongly chaotic, a regime in which microscopic trajectories diverge rapidly and cross-neuron correlations could be amplified. The factorization used in equation A.13 treats joint neural activities as products of independent distributions, yet the chaotic dynamics is precisely what could sustain the equal-time cross-correlations that would invalidate this factorization. The O(1/N) bound on cross-correlations cited in Section 10.1 addresses the connectivity structure but does not explicitly bound the dynamically generated temporal correlations in the chaotic regime. It would be helpful to show that the O(1/√K) cross-correlations acknowledged in Section 10.1 contribute only O(1/K) corrections to α_k and β_k(τ) in equations 3.12 and A.13, confirming that the self-consistent solution remains internally consistent when the network is chaotic.

**Global Stability Analysis Uses an Approximation Whose Validity Range Is Not Established**

Section 6.2 derives the global stability condition by replacing H(−u_k/√α_k) with the Heaviside function Θ(u_k), motivated by the argument that large perturbations drive u_k to O(√K). While this is qualitatively reasonable, the paper does not specify the minimum perturbation amplitude for which the approximation is valid, nor does it analyze the intermediate regime where perturbations are neither small relative to 1/√K (the local regime) nor large enough to fully saturate the sigmoid. The closed-form expression for τ_G (equation 6.6) is therefore presented as an exact threshold despite being derived from an approximate dynamical equation, and the possibility that intermediate-amplitude perturbations could support additional attractors or limit cycles — which would invalidate the global stability conclusion — is not ruled out. This matters especially because Figure 9 shows τ_L and τ_G crossing as a function of I/E ratio, and Section 6.3 identifies qualitatively different dynamical regimes depending on their ordering. It would be helpful to either provide a Lyapunov function argument covering all perturbation sizes, or to validate equation 6.6 against direct numerical integration of the full mean-field equations 3.3 across the parameter space of Figure 9.

**Finite-K Corrections Are Assessed Only for Population Rates, Leaving Autocorrelation and Chaos Results Unvalidated at Biologically Realistic Connectivity**

Section 4.2 and Figure 3 validate finite-K corrections to population rates m_k, and the paper asserts that these corrections are only quantitative. However, the autocorrelation equations (5.17), the rate distribution (equations 5.12–5.13 and 7.12–7.14), and the chaos analysis (equations 8.5–8.9) are all presented only in the K → ∞ limit with no analogous finite-K treatment. For biologically realistic values (K ~ 10^3, N ~ 10^4–10^5), the ratio K/N is not negligibly small, and the O(1/√K) residuals in the balance equations may not be negligible relative to the quantities appearing on the right-hand sides of equations 4.17–4.18, particularly when m_0 is small. The paper does not provide a perturbative expansion in 1/√K for the stability boundaries τ_L and τ_G, nor does it verify that the stability phase diagram of Figure 9 is robust at finite K. It would be helpful to include a systematic comparison of analytical predictions against direct network simulations spanning at least one order of magnitude in K, and to derive leading finite-K corrections to equation 5.17 or verify the autocorrelation and rate distribution predictions numerically at finite K.

**Comparison with Experimental Rate Distributions Is Qualitative and Lacks Statistical Rigor**

Section 7.3 compares the theoretically predicted power-law rate distribution (equations 7.12–7.14, Figure 11B) with a single experimental histogram from prefrontal cortex (Figure 11C, Abeles et al., 1988, unpublished data) and concludes the results are 'consistent,' but no quantitative goodness-of-fit measure or parameter estimation is provided. The theoretical tail ρ_k(m) ∝ 1/(m√|log m|) is derived using the approximation h(m) ≈ −√(2|log m|), which is valid only for m ≪ 1, yet the upper bound m_+ in equation 7.14 is stated to satisfy m_+ ≫ m_k, potentially violating this approximation over part of the support. The normalization of ρ_k(m) over [m_−, m_+] is not verified explicitly, and the paper does not demonstrate that the bounded uniform threshold distribution is the only model consistent with the data, nor does it rule out alternative explanations such as heterogeneous external inputs. The use of unpublished data also prevents independent verification. It would be helpful to either perform a quantitative fit with explicit parameter values (Δ, α_k, m_0) matched to the observed mean rate of 15.8 Hz, or to clearly frame this comparison as illustrative and acknowledge that a single dataset from one brain region under one behavioral condition cannot distinguish the bounded-threshold prediction from alternative models.

**Stationarity and Ergodicity of the Balanced State Are Assumed but Not Demonstrated**

Throughout Sections 5 and 8, time averages (equations 5.3–5.4, 5.16) are used interchangeably with ensemble averages, implicitly assuming that the chaotic balanced state is ergodic and stationary. The self-consistent equations for q_k (equation 5.12) and q_k(τ) (equation 5.17) are derived under this stationarity assumption. However, the paper also shows in Section 8 that trajectories desynchronize on a timescale comparable to τ_k, and the existence of the frozen-state solution (q_k = m_k, noted as unstable) and the balanced limit-cycle regime (Section 6.3, case 2) suggests that non-ergodic attractors coexist with the ergodic balanced fixed point in parts of parameter space. Desynchronization of pairs of trajectories is consistent with ergodicity but does not establish it, and in a system with an infinite Lyapunov exponent and binary degrees of freedom the ergodic properties of the invariant measure are non-trivial. It would be helpful to explicitly verify stationarity of q_k and q_k(τ) in direct simulations — for example, by checking that time-averaged statistics computed over early versus late portions of a trajectory agree — for at least the parameter values used in the figures, and to state explicitly the conditions under which ergodicity is expected to hold.

**The thermodynamic-limit order of limits (N → ∞ first, then K → ∞) may not reflect the finite-size regime where the theory is applied**

Assumption involved: The mean-field theory is stated to be exact in the limit N → ∞ with K fixed, followed by K → ∞ (Section 3, Eq. 3.1: 1 ≪ K ≪ N_k). The balance conditions (Eqs. 4.1–4.4) and the Gaussian input statistics (Eq. 3.3) are both derived under this sequential limit. The paper then applies the theory to a network with K = 1000 (Figures 3, 6, 7, 13–16) without specifying N, and uses the finite-K correction equations (4.17–4.18) as the operative model for all quantitative figures. The condition K ≪ N_k is never numerically verified for the illustrated cases. For K = 1000, the condition requires N_k ≫ 1000, meaning each sub-population must contain many thousands of neurons. Cortical columns of realistic size (roughly 10^4–10^5 neurons total, split between excitatory and inhibitory populations) may satisfy this only marginally. More concretely, the Gaussian approximation for input fluctuations (central limit theorem applied to K inputs) requires K to be large, while the independence of inputs to different neurons requires K/N_k to be small. At K = 1000 and, say, N_k = 5000, the fraction K/N_k = 0.2 is not negligible, and pairwise input correlations of order K/N_k could meaningfully inflate the variance α_k beyond Eq. 3.6. It would be helpful to include a sensitivity analysis showing how the predicted autocorrelation decay and rate distribution change as K/N_k varies from 0.01 to 0.2, so readers can assess the robustness of the theory at biologically plausible network sizes.

**Status**: [Pending]

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## Detailed Comments (25)

### 1. Infinite Lyapunov Exponent Reflects Binary Discreteness, Not Generic Chaos

**Status**: [Pending]

**Quote**:
> Since $\alpha_{k}-\gamma_{k} \propto D_{k}$, equation 8.9 has a growing solution even if $D_{k}(0)=0$. This implies that the Lyapunov exponent $\lambda_{L}$ is infinitely large in the balanced state.

**Feedback**:
The divergence of λ_L follows directly from dD_k/dt ∝ √D_k (equation 8.9), which gives λ_L = lim_{D_k→0} D_k^{-1} dD_k/dt ∝ D_k^{-1/2} → ∞. This is a consequence of the discontinuous Heaviside gain function of binary threshold units: any network of binary neurons would exhibit the same divergence regardless of whether its macroscopic dynamics are genuinely chaotic. The standard criterion for deterministic chaos requires a finite positive Lyapunov exponent. It would be helpful to supplement the argument with a finite-time or finite-perturbation divergence measure — for example, the rate at which D_k(t) rises from a small initial value to its equilibrium 2(m_k − q_k) — and to clarify explicitly that the infinite λ_L is an artifact of binary discreteness rather than a property distinguishing the balanced state from other regimes.

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### 2. Instability of Locked Solution Q_k = m_k Promised but Never Demonstrated

**Status**: [Pending]

**Quote**:
> This equation has two stationary solutions. One is
> 
> $Q_{k}=m_{k},$ (8.7)
> 
> which corresponds to a fully locked trajectories. This solution is unstable, as will be shown below.

**Feedback**:
The paper promises to show below that Q_k = m_k is unstable, but no subsequent derivation appears in Section 8. The instability is the central claim supporting the chaotic characterization. It follows directly from equation 8.9: since dD_k/dt ∝ √D_k > 0 for any D_k > 0, the locked state D_k = 0 is repelling for all positive perturbations. A sentence after equation 8.9 should make this explicit — for instance, 'since dD_k/dt > 0 for all D_k > 0, the locked fixed point Q_k = m_k is unstable, fulfilling the promise made after equation 8.7' — so the proof is traceable rather than deferred to a location that never appears.

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### 3. Equation 8.9 Coefficient 2/π Requires Explicit Intermediate Derivation

**Status**: [Pending]

**Quote**:
> To find the initial rate of divergence, we expand equation 8.5 for small $D_{k}$ and find that to leading order, the distances satisfy
> 
> $\tau_{k}\frac{dD_{k}}{dt}=\frac{2}{\pi}\frac{e^{-u_{k}^{2}/2\alpha_{k}}}{\sqrt{\alpha_{k}}}\sqrt{\alpha_{k}-\gamma_{k}}.$

**Feedback**:
The coefficient 2/π arises from two separate factors: one from dD_k = −2 dQ_k (equation 8.4) and one from differentiating [H(...)]² with respect to γ_k, which gives 2H·H'·∂f/∂γ_k. The standard Gaussian integral identity then yields a factor of 1/√(2π) from H', and the combination produces 2/π. Without showing these intermediate steps it is not possible to verify whether the coefficient is 2/π, 1/π, or 4/π. It would be helpful to add at least one intermediate line in Appendix A showing how the squared-H derivative and the dD_k = −2dQ_k relation combine to give the stated prefactor.

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### 4. Diagonal Variance Term Uses Total N Rather Than Population Size N_l

**Status**: [Pending]

**Quote**:
> Observing that $[(J_{kl}^{ij}\sigma_{l}^{j}(t))^{2}]=J_{kl}^{2}m_{l}K/N$, whereas $[(J_{kl}^{ij}\sigma_{l}^{j}(t))]^{2}=J_{kl}^{2}m_{l}^{2}K/N^{2}$, which is negligible, one obtains equation 3.6.

**Feedback**:
Re-deriving: J_{kl}^{ij} is nonzero with probability K/N_l and equals J_{kl}/√K when nonzero, so [(J_{kl}^{ij})²] = (J_{kl}²/K)·(K/N_l) = J_{kl}²/N_l. With [(σ_l^j)²] = m_l for binary units, the per-term expectation is J_{kl}² m_l/N_l. Summing over j = 1,…,N_l gives J_{kl}² m_l, matching equation 3.6. The text instead writes J_{kl}² m_l K/N (using total network size N rather than population size N_l), which is off by a factor of N/N_l unless N_l = N. The sum over N_l terms of J_{kl}² m_l K/N yields J_{kl}² m_l K N_l/N ≠ J_{kl}² m_l. Readers might note that rewriting '[(J_{kl}^{ij}σ_l^j(t))²] = J_{kl}² m_l K/N' as '= J_{kl}² m_l/N_l' would make the derivation consistent with the connection probability K/N_l.

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### 5. Mean Input Derivation Uses [J_{kl}^{ij}] = J_{kl}√K/N Rather Than J_{kl}√K/N_l

**Status**: [Pending]

**Quote**:
> The population average $[J_{kl}^{ij}]$ is equivalent to a quenched average over the random connectivity and is therefore equal to $J_{kl}\sqrt{K}/N$, yielding equation 3.5.

**Feedback**:
Each neuron in population k receives K inputs from population l chosen from N_l cells, so [J_{kl}^{ij}] = (J_{kl}/√K)·(K/N_l) = J_{kl}√K/N_l, not J_{kl}√K/N. Summing over j = 1,…,N_l then gives N_l·(J_{kl}√K/N_l) = J_{kl}√K, which correctly yields equation 3.5. Using N instead of N_l introduces a factor of N_l/N that cancels only if N_l = N (single population). The text should either write [J_{kl}^{ij}] = J_{kl}√K/N_l explicitly, or note that J_{kl} is defined to absorb the factor N_l/N, so that the reduction to equation 3.5 is transparent.

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### 6. Equation 5.2 Implicitly Sets J_{kE} = 1 Without Stating This Convention

**Status**: [Pending]

**Quote**:
> $$
> \beta_{k} = q_{E} + J_{k}^{2} q_{I}. \tag{5.2}
> $$

**Feedback**:
From equation 5.7, β_k = Σ_l J_{kl}² q_l = J_{kE}² q_E + J_{kI}² q_I. The formula β_k = q_E + J_k² q_I holds only if J_{kE} = 1. Yet Figure 3 uses J_E = 2, J_I = 1.8, so J_{kE} ≠ 1 in general. The same implicit normalization appears in equation 5.9 and equation 5.15. If J_E = 2, the excitatory contribution to β_k should carry a factor J_E² = 4, propagating into equations 5.11, 5.12, 5.17, and all downstream results. It would be helpful to state the convention J_{kE} = 1 explicitly when equation 5.2 is introduced, or to write the formula as β_k = J_{kE}² q_E + J_{kI}² q_I with J_{kI} ≡ J_k.

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### 7. Autocorrelation Equation 5.17 Integrates Over Future Lags, Making It Non-Causal

**Status**: [Pending]

**Quote**:
> $$
> \begin{array}{l}
> \tau_{k} \frac{d q_{k}(\tau)}{d \tau} = - q_{k}(\tau) + \int_{0}^{\infty} \frac{d t}{\tau_{k}} \\
> \quad \times \exp(-t / \tau_{k}) \int D x \left[ H \left(\frac{- u_{k} - \sqrt{\beta_{k}(t + \tau)} x}{\sqrt{\alpha_{k} - \beta_{k}(t + \tau)}}\right) \right]^{2}. \tag{5.17}
> \end{array}
> $$

**Feedback**:
The right-hand side involves β_k(t + τ) for t ≥ 0, requiring knowledge of q_k at lags strictly greater than τ. This makes the equation non-causal: the derivative at lag τ depends on q_k at all larger lags. The physical argument is that t is the time elapsed since the last update before t+τ, which should correspond to a backward lag of t, giving β_k(t) not β_k(t+τ). The same issue appears in Appendix A equation A.10. If the argument should be β_k(t) (or β_k(|τ−t|)), the equation becomes a standard Volterra equation solvable by forward integration. It would be helpful to clarify the intended argument and the solution procedure, since the current form requires a global fixed-point iteration rather than forward integration.

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### 8. Equation A.10 Autocorrelation Argument t'+τ Should Be t'

**Status**: [Pending]

**Quote**:
> \tau_{k} \frac{dq_{k}}{d\tau} = -q_{k}(\tau) + \int_{0}^{\infty} \frac{dt'}{\tau_{k}} \exp(-t'/\tau_{k}) F_{k}(m_{E}, m_{I}; q_{1}(t' + \tau), q_{2}(t' + \tau)), \tag{A.10}

**Feedback**:
The physical setup is that the current evaluation is at time t+τ and the last update before t+τ occurred at t+τ−t', so the lag between the two evaluations is t'. In a stationary process the autocorrelation at lag t' is q_k(t'), not q_k(t'+τ). The argument t'+τ in A.10 introduces an extra shift of τ inconsistent with the physical description. This is confirmed by the final result cited as equation 5.17, whose structure is consistent with an integrand involving q_l(t') rather than q_l(t'+τ). Rewrite the argument of F_k in equation A.10 as q_1(t'), q_2(t') to match the physical derivation and the cited result.

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### 9. Equation 6.4 Omits Self-Coupling Terms in Heaviside Argument

**Status**: [Pending]

**Quote**:
> $\tau_{k}\frac{d}{dt}\delta m_{k}(t)=-\delta m_{k}(t)+\Theta(\delta m_{E}-J_{k}\delta m_{I})-m_{k}.$ (6.4)

**Feedback**:
The large-perturbation approximation replaces H(−u_k/√α_k) with Θ(u_k). The perturbed local field is δu_k = J_{kE} δm_E − J_{kI} δm_I (from equations 3.8–3.9). For k = E this is J_{EE} δm_E − J_{EI} δm_I, and for k = I it is J_{IE} δm_E − J_{II} δm_I. The equation as written uses a single coefficient J_k multiplying only δm_I and drops the self-excitation/self-inhibition terms. Since J_{EE} = J_{IE} = 1 by the paper's normalization, the excitatory self-coupling term δm_E is missing from equation 6.4 for both populations. This affects the derivation of τ_G in equation 6.6. It would be helpful to rewrite equation 6.4 with the full argument Θ(J_{kE} δm_E − J_{kI} δm_I) and verify that equation 6.6 follows from the corrected system.

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### 10. Square-Root Factor in First Argument of τ_G Has No Apparent Derivation

**Status**: [Pending]

**Quote**:
> $\tau_{G}=J_{E}\,\min\left\{\sqrt{\frac{J_{I}m_{I}(1-m_{I})}{J_{E}m_{E}(1-m_{E})}},\frac{m_{I}}{m_{E}},\frac{1-m_{I}}{1-m_{E}}\right\}.$ (6.6)

**Feedback**:
The second and third arguments of the min, m_I/m_E and (1−m_I)/(1−m_E), are rational functions of rates that naturally arise from comparing slopes of the piecewise-linear branches of equation 6.4. The first argument, √(J_I m_I(1−m_I)/(J_E m_E(1−m_E))), contains a square root and mixes J_I and J_E in a way that does not follow straightforwardly from a piecewise-linear ODE analysis, which would produce rational (not square-root) combinations. No intermediate derivation steps are provided. If the square root is spurious — e.g., if the correct expression is J_I m_I(1−m_I)/(J_E m_E(1−m_E)) without the root — the stability boundary τ_G and the phase diagram of Figure 9 would change quantitatively. It would be helpful to add at least one intermediate step showing how each of the three arguments is derived from the piecewise-linear solution.

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### 11. Linearization Perturbation Bound Off by Factor √K

**Status**: [Pending]

**Quote**:
> We therefore consider a solution of equations 3.3 with an initial condition $m_{k}(0)=m_{k}+\delta m_{k}(0)$ with a small $\delta m_{k}(0)$ where $|\delta m_{k}(0)|\ll 1/\sqrt{K}$. In this case, the perturbation of the total mean input $u_{k}$ is also small; hence we can linearize the dynamic equations 3.3 around their fixed point.

**Feedback**:
From equation 3.3, δu_k = √K Σ_l J_{kl} δm_l. With |δm_l| ≪ 1/√K, this gives |δu_k| ≪ √K·(1/√K) = O(1). An O(1) perturbation to u_k is not 'small' in the sense required to linearize H(−u_k/√α_k): linearization requires δu_k ≪ 1, which demands |δm_k| ≪ 1/K (not 1/√K). As stated, the perturbation condition is off by a factor of √K from what is needed to rigorously justify the linearization. Rewrite the bound as |δm_k(0)| ≪ 1/K so that δu_k = O(1/√K) ≪ 1, which genuinely justifies linearizing H around its fixed-point value.

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### 12. Width of Rate Distribution Claimed to Scale as m_k^{3/2} but Equation 5.14 Implies m_k/√|log m_k|

**Status**: [Pending]

**Quote**:
> Equation 5.14 implies that when the mean activity $m_{k}$ decreases, the width of the distribution is proportional to $(m_{k})^{3/2}$; it decreases faster than the mean $m_{k}$.

**Feedback**:
From equation 5.14, q_k − m_k² = O(m_k²/|log m_k|), so the standard deviation σ_k = √(q_k − m_k²) = O(m_k/√|log m_k|). This scales as m_k (up to a logarithmic correction), not as m_k^{3/2}. For the width to scale as m_k^{3/2} one would need q_k − m_k² = O(m_k³), which is not what equation 5.14 states. The logarithmic correction means the width decreases only slightly faster than m_k, a much weaker statement than m_k^{3/2}. It would be helpful to add a brief derivation showing how the m_k^{3/2} scaling is obtained, or to correct the scaling claim to σ_k = O(m_k/√|log m_k|).

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### 13. Equation 7.1 Sign Convention Inconsistent with Frozen-State Limit

**Status**: [Pending]

**Quote**:
> m _ {k} (\theta) = H \left(\frac {\theta - u _ {k}}{\sqrt {\alpha_ {k}}}\right), \tag {7.1}

**Feedback**:
In the homogeneous case (equation 3.7), m_k = H(−u_k/√α_k), where u_k is the mean net input relative to threshold. Setting θ = 0 in equation 7.1 gives H(−u_k/√α_k), which is consistent. However, the frozen-state limit (equation 7.7) states m_k(θ) ≈ Θ(u_k − θ), meaning the neuron fires when u_k > θ. This is the α_k → 0 limit of H((u_k − θ)/√α_k), not H((θ − u_k)/√α_k). The two forms differ by a sign flip in the argument, and H((θ − u_k)/√α_k) → Θ(θ − u_k) = 1 − Θ(u_k − θ) as α_k → 0, which is the complement of equation 7.7. It would be helpful to rewrite equation 7.1 as m_k(θ) = H((u_k − θ)/√α_k) to be consistent with both the homogeneous limit and the frozen-state limit, and to propagate the correction to equations 7.2, 7.5, and 7.9.

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### 14. Equation 9.13 Ramp Formula Inconsistent with Stated Protocol

**Status**: [Pending]

**Quote**:
> In Figure 14 the external activity is ramped between $t = 1$ and $t = 2$,
> 
> $$
> m _ {0} (t) = m _ {0} + v _ {0} t, \tag {9.13}
> $$
> 
> and after $t = 2$ $m_0$ is kept constant again.

**Feedback**:
Equation 9.13 gives m_0(t) = m_0 + v_0 t for all t, but the protocol states the network is at equilibrium from t = 0 to t = 1 and the ramp begins at t = 1. Continuity at t = 1 requires m_0(1) = m_0, which equation 9.13 satisfies only if v_0 = 0 during [0,1]. The correct expression for the ramp phase is m_0(t) = m_0 + v_0(t−1) for t ∈ [1,2], so that m_0(1) = m_0 and m_0(2) = m_0 + v_0. As written, equation 9.13 implies the ramp starts from t = 0, which contradicts the stated equilibrium condition. Figure 16 compares the rate of change dv_E/dt against v_0, and the quantitative comparison with the large-K bound (equation 9.8) depends on the correct identification of the ramp slope. Rewrite equation 9.13 as m_0(t) = m_0 + v_0(t−1) for t ∈ [1,2].

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### 15. Tracking Timescale Expression τ_k/δm_0 Appears Inverted

**Status**: [Pending]

**Quote**:
> This initial large response causes a fast rate of increase in the population rates, since $\delta m_k \approx \tau_k^{-1} dt \Delta P_k$, implying that $\delta m_k$ reaches the value $A_k \delta m_0$ in time of order $\tau_k / \delta m_0 \approx \tau_k / \sqrt{K}$

**Feedback**:
If δm_k grows at rate τ_k^{-1} ΔP_k per unit time and ΔP_k is O(1), then δm_k ~ A_k δm_0 is reached in time ~ τ_k A_k δm_0, not τ_k/δm_0. With δm_0 ~ 1/√K, the former gives a timescale τ_k A_k/√K while the latter gives τ_k√K — these differ by a factor of K. The stated result τ_k/√K matches the former (up to the A_k prefactor), so the written expression τ_k/δm_0 appears to be a typographical inversion. Rewrite 'τ_k/δm_0 ≈ τ_k/√K' as 'τ_k δm_0 ≈ τ_k/√K' because the timescale to accumulate a change A_k δm_0 at rate ~ τ_k^{-1} is τ_k A_k δm_0 ~ τ_k/√K.

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### 16. Multinomial Distribution in A.13 Merges Independent Poisson Factors Ambiguously

**Status**: [Pending]

**Quote**:
> p_l(n_1, n_2, n_3) = \frac{(q_l K)^{n_1}}{n_1!} \frac{((m_l - q_l) K)^{n_2 + n_3}}{n_2! n_3!} e^{-(2m_l - q_l)K}.

**Feedback**:
In the Poisson approximation, n_1, n_2, and n_3 are independent Poisson variates with means q_l K, (m_l−q_l)K, and (m_l−q_l)K respectively. The correct factored form is [e^{−q_l K}(q_l K)^{n_1}/n_1!] × [e^{−(m_l−q_l)K}((m_l−q_l)K)^{n_2}/n_2!] × [e^{−(m_l−q_l)K}((m_l−q_l)K)^{n_3}/n_3!]. The written expression merges the n_2 and n_3 factors into a single term ((m_l−q_l)K)^{n_2+n_3}/(n_2! n_3!), which is correct only if n_2 and n_3 are treated as independent — but the merged form is missing the combinatorial factor (n_2+n_3)!/(n_2! n_3!) that would be needed if the sum n_2+n_3 were treated as a single Poisson variable. Since the derivation in A.14 proceeds by treating x_2 and x_3 as independent standard Gaussians, the intended distribution must keep n_2 and n_3 separate. Rewrite A.13 as the product of three independent Poisson factors to make the independence assumption explicit.

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### 17. Condition 4.9 Is a Restatement of 4.5, Not an Independent Constraint

**Status**: [Pending]

**Quote**:
> 4.8}
> $$
> 
> so $u_{k}$ is of order $\sqrt{K}$ and positive.<!-- PAGE BREAK -->Thus if we require that there be no stationary solutions with $m_{E}=0, 1$ or $m_{I}=0, 1$ for small $m_{0}$, the following constraints have to be satisfied:
> 
> $\frac{E}{I} &gt; \frac{J_{E}}{J_{I}} &gt; 1.$ (4.9)
> 
> $J_{E} &gt; 1.$ (4.10)
> 
> It is straightforward to sh

**Feedback**:
Equation 4.9 is algebraically identical to equation 4.5, which was already derived as the condition for A_E and A_I to be positive. Presenting it as a new constraint derived from the unbalanced-state analysis creates the impression that two independent lines of reasoning converge on the same inequality, whereas the genuinely new content is entirely in equation 4.10 (J_E > 1), which rules out the all-active solution. The logical structure would be clearer if the text explicitly acknowledged that equation 4.9 restates equation 4.5 and that the new constraint from the unbalanced-state analysis is only equation 4.10. Additionally, the case analysis covers only m_E = 0 and m_E = m_I = 1; the cases m_I = 0 and m_E = 1 alone are not explicitly treated, leaving the claim that 'these constraints eliminate all possible unbalanced states' without a complete enumeration.

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### 18. Eigenvalue Order Claim Needs Verification from Characteristic Polynomial

**Status**: [Pending]

**Quote**:
> Solving equations 6.1 one obtains $\delta m_{k}(t)=\delta m_{k,1}\exp(\lambda_{1}t)+\delta m_{k,2}\exp(\lambda_{2}t)$ where the eigenvalues $\lambda_{1}$ and $\lambda_{2}$ of the 2 by 2 equations (see equations 6.1) are both of order $\sqrt{K}$.

**Feedback**:
The 2×2 matrix M from equation 6.1 has diagonal entries O(1) and off-diagonal entries O(√K). The characteristic polynomial gives eigenvalues λ = (1/2)[Tr(M) ± √(Tr(M)²−4Det(M))]. If the diagonal f_{kk} terms are nonzero, Tr(M) ~ √K and Det(M) ~ K, giving both eigenvalues O(√K). But if f_{EE} = f_{II} = 0 (no self-coupling), Tr(M) = O(1) and Det(M) = O(K), giving eigenvalues with imaginary parts O(√K) and real parts O(1) — in which case the decay timescale would be O(1), not O(1/√K). The text does not verify that f_{kk} ≠ 0 at the balanced fixed point using equation 6.2. It would be helpful to evaluate f_{kk} explicitly and confirm that both eigenvalues have real parts of order √K under the stated stability condition.

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### 19. Regime 3 (τ_G < τ < τ_L) Appears Empty for the Parameters of Figure 9

**Status**: [Pending]

**Quote**:
> n excitation and inhibition. This is described schematically in Figure 10B.
> 3. Balanced fixed point with shrinking basin: $\tau_{G} &lt; \tau &lt; \tau_{L}$. In this case, perturbations of order 1 go to a global limit cycle, while perturbations much smaller than $1 / \sqrt{K}$ evolve to the fixed point. There must be an unstable limit cycle with amplitude of order $1 

**Feedback**:
The paper states in the paragraph preceding section 6.3 that 'For these parameters, τ_L is always smaller than τ_G,' referring to Figure 9. Yet regime 3 requires τ_G < τ_L — the opposite ordering. If τ_L ≤ τ_G throughout the parameter space of Figure 9, regime 3 is an empty interval for those parameters. The numerical example in Figure 8 gives τ_L = 1.61 > τ_G = 1.50, which is consistent with regime 3 existing for those parameters but contradicts the claim about Figure 9. It would be helpful to specify the parameter range (e.g., values of J_E, J_I, or I/E) for which τ_G < τ_L so that regime 3 is non-empty, or to note explicitly that regime 3 does not occur for the Figure 9 parameter sweep.

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### 20. ISI Peak Location Claim t ∝ τ_E Incorrect for Small m_i

**Status**: [Pending]

**Quote**:
> This function rises linearly from zero and peaks at $t \propto \tau_{E}$. For intervals of the order of $\tau_{E} / m_{i}$ or longer, $I(t)$ decays purely exponentially with a decay constant $m_{i} / \tau_{E}$ as in the ISI of a single Poisson process with a rate $m_{i} / \tau_{E}$.

**Feedback**:
Setting dI/dt = 0 in equation 5.18 gives t* = τ_E log((1−m_i)/m_i)/(1−2m_i). For small m_i this simplifies to t* ≈ τ_E |log m_i|, which diverges as m_i → 0 and is much larger than τ_E. For the parameter value m_i = 0.06 used in Figure 7, t* ≈ τ_E log(0.94/0.06)/0.88 ≈ 3.1 τ_E, not τ_E. The claim that the peak occurs at t ∝ τ_E (with an O(1) proportionality constant) is accurate only when m_i is not small. Rewrite the peak location as 't* = τ_E log((1−m_i)/m_i)/(1−2m_i), which is of order τ_E |log m_i| for small m_i' to accurately reflect the formula.

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### 21. Net Input Scaling Factor Inverted in External Input Comparison

**Status**: [Pending]

**Quote**:
> the external input to an excitatory cell is $Em_0\sqrt{K}$, whereas the net input to this cell, $u_E$, is smaller than the external input by a factor of the order of $1/(m_0\sqrt{K})$, where $m_0$ is the rate of an input cell and is assumed to be much larger than $1/\sqrt{K}$.

**Feedback**:
In the balanced state u_E is O(1) (comparable to the threshold θ_E, which is O(1) by construction). The external input is Em_0√K. The ratio u_E/(Em_0√K) is therefore O(1/(m_0√K)), not O(1/(m_0√K)) with an extra factor of m_0. Concretely, at m_0 = 0.5 and K = 1000, the ratio is ~1/(0.5×31.6) ≈ 1/15.8, which is O(1/√K) independent of m_0 once m_0 is absorbed into the O(1) prefactor. The correct statement is that u_E is smaller than the external input by a factor of O(1/√K), not O(1/(m_0√K)). Rewrite 'smaller than the external input by a factor of the order of 1/(m_0√K)' as 'smaller than the external input by a factor of the order of 1/√K'.

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### 22. Normalization of Rate Distribution ρ_k(m) Not Verified

**Status**: [Pending]

**Quote**:
> oad and skewed. The full shape of the rate distribution is given by
> 
> $$
> \rho_k(m) \approx \frac{\sqrt{\alpha_k / 2 \Delta^2}}{m \sqrt{|\log m|}}, \quad m_- &lt; m &lt; m_+ \tag{7.12}
> $$
> 
> and zero otherwise. The bounds of $m$ 

**Feedback**:
For ρ_k(m) ∝ 1/(m√|log m|), the normalization integral ∫ dm/(m√|log m|) = 2√|log m| evaluated between m_− and m_+. Using equations 7.13–7.14, |log m_−| ≈ Δ²/(2α_k) dominates over |log m_+|, so the integral ≈ 2√(Δ²/(2α_k)) = √2 Δ/√α_k. For ρ_k to integrate to 1, the prefactor must be √α_k/(√2 Δ) = √(α_k/(2Δ²)), which matches equation 7.12 only when the lower cutoff dominates. This approximation breaks down when m_+ is not negligibly small. It would be helpful to add a sentence stating that the prefactor is determined by normalization with the integral dominated by m_−, and that this approximation is valid when |log m_−| ≫ |log m_+|, i.e., Δ²/(2α_k) ≫ |log(Δ√(α_k/|log α_k|))|.

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### 23. Scaling for Synaptic Exponent α < 1/2 Leads to Diverging Net Input

**Status**: [Pending]

**Quote**:
> In this limit, the net average inputs into the populations scale as $K^{1-\alpha}$, while the quenched and temporal fluctuations in the inputs scale as $K^{1/2-\alpha}$.

**Feedback**:
For α < 1/2, the fluctuations K^{1/2−α} grow with K. After the O(K^{1−α}) mean terms cancel via the balance condition, the residual net input is O(K^{1/2−α}) → ∞. A diverging net input would drive all neurons to saturation or silence rather than a balanced chaotic state. The paper asserts 'for any α, this leads to asynchronous chaotic activity in a homogeneous network' without addressing how a balanced chaotic state can persist when the net input diverges. It would be helpful to clarify whether the balance condition itself can be maintained self-consistently when fluctuations grow, or to restrict the claim to α ≥ 1/2.

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### 24. Equation 9.10 Uses Undefined Subscript Notation for Rectification

**Status**: [Pending]

**Quote**:
> $$
> \tau_{k} \frac{d m_{k}}{d t} = - m_{k} + \left(\tilde{J}_{k0} m_{0} + \sum_{l}J_{kl} m_{l} - T_{k}\right)_{+}, \tag{9.10}
> $$
> 
> with $(x)_{s} = (x + |x|)/2$ if $s=1$.

**Feedback**:
The notation (·)_+ in equation 9.10 is the standard rectification max(x,0). However, the definition reads (x)_s = (x+|x|)/2 'if s=1', where the subscript s is never introduced or explained. Since (x+|x|)/2 = max(x,0) for all real x regardless of any parameter s, the conditional qualifier 'if s=1' is either a vestige of a more general definition that was removed or a typographical error. The conditional form implies a generality that is never used and obscures the meaning of the activation function. Rewrite the parenthetical as 'with (x)_+ = max(x,0)' and remove the undefined s qualifier.

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### 25. Approximation h(m) ≈ −√(2|log m|) Omits Subleading log|h| Term Without Quantifying Error

**Status**: [Pending]

**Quote**:
> x\frac{\exp(-x^{2}/2)}{\sqrt{2\pi}|x|}$ (4.13)
> 
> to obtain
> 
> $h(m)\approx-\sqrt{2|\log m|}.$ (4.14)<!-- PAGE BREAK -->Substituting this result in equation 4.12 yields
> 
> $u_{k}\approx-\sqrt{2\,\alpha_{k}|\log m_{k}|}\,.$ (4.15)
> 
> This relatio

**Feedback**:
Inverting m = H(−h) ≈ exp(−h²/2)/(√(2π)|h|) gives log m ≈ −h²/2 − (1/2)log(2π) − log|h|. Dropping the log|h| term yields h ≈ −√(2|log m|). The relative error from this additional approximation is O(log|log m|/|log m|): at m_k = 0.01, |log m| ≈ 4.6 and log|h| ≈ log(3.0) ≈ 1.1, giving a ~24% relative error. This compounded approximation propagates into equation 4.15 and all downstream results (rate distribution in Section 5.1, threshold distribution in Section 7). It would be helpful to add a note such as 'where we have used log m ≈ −h²/2, neglecting the subleading term (1/2)log(2|log m|), valid when |log m| ≫ 1' to make the approximation hierarchy explicit.

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