# 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.

**Chaos Characterization Rests on a Non-Standard Infinite Lyapunov Exponent**

All three reviewers flag the same core concern: Section 8 concludes that the balanced state is 'chaotic' on the basis that the maximum Lyapunov exponent λ_L is formally infinite, because equation 8.9 shows D_k grows as t² (proportional to √D_k) rather than exponentially from any nonzero initial separation. The authors acknowledge this divergence is tied to the discreteness of binary units, yet they continue to use the word 'chaos' throughout. An infinite Lyapunov exponent is not the standard hallmark of deterministic chaos—it is a pathology of discrete-state or non-smooth systems where the notion of an infinitesimal perturbation is ill-defined, and it does not imply exponential sensitivity to initial conditions in the classical sense. The paper does not compute a finite, positive Lyapunov exponent for any finite-K realization, any continuous-rate approximation, or any integrate-and-fire generalization of the model, nor does it provide direct numerical simulations demonstrating exponential trajectory divergence over time. It would be helpful to either (a) derive a finite Lyapunov exponent in a regularized or continuous-limit version of the model, showing it remains large and positive as K grows, or (b) reframe the chaos claim explicitly as 'extreme sensitivity to initial conditions arising from the binary threshold nonlinearity,' with a clear statement that this constitutes a qualitative indicator rather than a rigorous proof of chaos in the dynamical-systems sense, and a discussion of whether the conclusion transfers to more biologically realistic neuron models.

**Statistical Independence Assumption in the Mean-Field Derivation Is Not Rigorously Bounded in the Balanced Regime**

Multiple reviewers identify a shared concern about the foundational mean-field step in Appendix A (equation A.4 and surrounding text), which assumes that activities of different input cells to a given neuron are uncorrelated, justified by K ≪ N_k. In the balanced state, however, strong recurrent feedback generates self-consistent spatial inhomogeneity in firing rates (Section 5, equations 5.5–5.7), so neurons sharing a common postsynaptic partner have correlated quenched inputs and therefore correlated time-averaged activities. The cross-term between δJ_{kl}^{ij} and δm_l^j is dropped in equation 5.7 without explicit justification, and the cumulative effect of K such correlated inputs on the variance α_k (equation 3.6) could produce O(1) corrections that are not accounted for. The paper states that pairwise cross-correlations are O(1/√K) for directly connected pairs (Section 10.1), but does not show that their aggregate contribution to the variance calculation remains sub-leading in the large-K limit when the network is in the balanced state. It would be helpful to provide an explicit order-of-magnitude bound on the correlation-induced error in equations 3.5–3.6 as a function of K/N, and to verify numerically that finite-N simulations of the autocorrelation function (equation 5.17) and population rates converge to the mean-field predictions at the rates implied by the theory.

**Self-Consistency of the Autocorrelation Equation Is Not Validated Against Direct Network Simulations**

Equation 5.17 is the central self-consistent integral equation for the temporal autocorrelation q_k(τ), and its derivation in Appendix A.2 (equations A.10–A.14) assumes that the joint statistics of inputs at two different times can be factored using the same Gaussian closure as for equal-time statistics—an assumption that is exact only when temporal correlations in the inputs are themselves weak. In the balanced state, where temporal fluctuations are the primary driver of activity, this assumption deserves explicit verification. The paper presents only a single numerical solution of the integral equation (Figure 5), compared against a Poisson reference, but does not show a direct comparison with autocorrelation functions measured from finite-N network simulations. Figure 5 demonstrates qualitative agreement with Poisson-like statistics but does not constitute a systematic validation of equation 5.17 itself. It would be helpful to show, for at least one or two parameter sets spanning the biologically relevant range of K, that q_k(τ) computed from equation 5.17 matches the empirical autocorrelation from a direct simulation, and to quantify the discrepancy as a function of K to establish the quantitative range of validity of the large-K Gaussian closure.

**Global Stability Analysis Employs a Piecewise-Linear Approximation Whose Validity Range Is Not Established or Numerically Verified**

Section 6.2 derives the global stability condition (equations 6.4–6.6) by replacing H(−u_k/√α_k) with the Heaviside step function Θ(u_k), justified by the argument that large perturbations destroy balance and drive u_k to O(√K). This approximation is valid only when |u_k| ≫ √α_k, but the paper does not specify the perturbation amplitude above which this regime is entered, nor does it analyze the intermediate regime where u_k is neither O(1) nor O(√K). During transient recovery toward the balanced state, the trajectory passes through this intermediate regime, and the full nonlinear dynamics of equation 3.3 governs the path; the paper does not verify that the piecewise-linear system faithfully captures these transients or that it does not introduce spurious slow manifolds or additional fixed points near the boundary τ ≈ τ_G. The expression for τ_G (equation 6.6) also depends on m_E and m_I, which are themselves only O(1/√K)-accurate, potentially shifting the stability boundary non-negligibly. Figure 8 shows convergence for a single parameter set (τ = 1.3), which is insufficient for systematic validation. It would be helpful to verify equation 6.6 against direct simulations of the nonlinear dynamics for several parameter combinations, to state explicitly the perturbation amplitude threshold above which the piecewise-linear approximation holds, and to clarify whether τ < τ_G is sufficient or merely necessary for global stability when the two populations have different time constants.

**Threshold Inhomogeneity Analysis Lacks a Phase Diagram and Its Biological Comparison Is Only Qualitative**

Section 7 shows that the qualitative character of the rate distribution ρ_k(m) depends critically on the form of the threshold distribution P(θ): a Gaussian (unbounded) distribution leads to a frozen bimodal state at low rates (equations 7.6–7.8), while a bounded distribution preserves temporal variability with a power-law tail (equation 7.12). The paper treats these as two separate cases without characterizing the transition between them as the tail weight of P(θ) is varied continuously, and the condition 'falls off faster than a Gaussian' is a qualitative criterion that does not specify how rapidly the transition occurs. Equation 7.15 gives a general formula stated to hold 'for small m_k' without specifying how small m_k must be relative to Δ and α_k. The biological comparison in Section 7.3 uses the bounded-distribution prediction (Figure 11B) to match a single experimental histogram from prefrontal cortex (Figure 11C, Abeles et al., 1988, unpublished data), but the experimental data pool neurons across multiple task conditions, no quantitative goodness-of-fit is provided, and alternative distributions (e.g., log-normal) are not considered. It would be helpful to provide a phase diagram in the (m_k, Δ/√α_k) plane delineating the fluctuating, frozen, and intermediate regimes; to validate equation 7.12 numerically for the bounded uniform distribution before using it for biological interpretation; and to either fit equation 7.12 quantitatively to the experimental data with explicit parameter estimates or acknowledge more explicitly that the comparison is qualitative.

**Finite-K Corrections Are Quantified Only for Population Rates, Leaving Autocorrelation, Chaos, and Tracking Claims Unvalidated at Realistic Connectivity**

Section 4.2 carefully derives finite-K corrections to population rates (equations 4.17–4.18) and validates them in Figure 3, but the autocorrelation analysis (Section 5.2, equation 5.17), the rate distribution (Section 5.1, equations 5.12–5.14), the chaos characterization (Section 8, equations 8.5–8.9), and the fast-tracking result (Section 9, equation 9.5) are all presented exclusively in the large-K limit. For biologically realistic values (K ~ 10²–10³, N ~ 10⁴–10⁵), the condition 1 ≪ K ≪ N is only marginally satisfied, and the Gaussian approximation for input statistics may introduce non-negligible errors in the tails of the distribution that govern rare firing events and the shape of the autocorrelation decay. The O(1) residual net inputs u_k that appear in equations 5.17 and 8.5 are precisely the quantities determined by the finite-K self-consistency equations 4.17–4.18, which the paper states have 'small corrections' without solving them analytically or showing uniqueness of the solution. Additionally, the fast-tracking comparison with an unbalanced network (equations 9.10–9.12) uses a different synaptic scaling (J_kl/K vs. J_kl/√K), making the O(√K) tracking advantage partly a consequence of scaling choice rather than balance per se. It would be helpful to include at least one direct simulation of the full binary network at finite K and N comparing empirical autocorrelation functions and rate distributions to theoretical predictions, to solve equations 4.17–4.18 perturbatively for explicit expressions for u_k, and to isolate the contribution of balance from synaptic scaling in the tracking comparison.

**Statistical Independence of Inputs Assumed but Recurrent Correlations Are Non-Negligible at Finite K**

Assumption involved: The mean-field derivation of the population-averaged input (Eq. 3.11) and its variance (Eq. 3.12) explicitly neglects correlations between a neuron's output connectivity and its own activity, justifying this only 'in the large N limit.' More specifically, the variance calculation in Eq. 3.12 drops the cross-terms between inputs from different presynaptic neurons, treating them as uncorrelated. This is the standard statistical independence assumption for mean-field theory.

Data/implementation structure: The network has sparse random connectivity with K presynaptic inputs per neuron, where 1 ≪ K ≪ N. Because the same recurrent network generates both the inputs and the activity being averaged, pairs of neurons that share common presynaptic partners will have correlated inputs. The paper acknowledges (Section 5, final paragraph) that 'because of the sparsity of the connectivity, different cells receive input from different subpopulations, so that the fluctuations in their input will be only very weakly correlated,' but this argument applies to cross-neuron correlations, not to the within-neuron temporal correlations that enter the variance calculation.

Consequence: At finite K, the O(1/K) corrections to the variance from pairwise input correlations are not explicitly bounded. The paper addresses finite-K corrections to the mean rates (Section 4.2, Eqs. 4.17–4.18) but does not carry through a corresponding finite-K correction to α_k that would account for residual input correlations. Since the rate distribution shape (Section 5.1) and the Lyapunov exponent calculation (Section 8) both depend sensitively on α_k − β_k, it would be helpful to quantify how large K must be before these correlation corrections become negligible relative to the leading-order variance.

Suggested check: It would be helpful to include a direct network simulation (rather than the Gaussian-process surrogate of Section 5.3) that measures the empirical variance of inputs to individual neurons and compares it to α_k from Eq. 3.6, across a range of K values, to verify that the independence approximation is adequate for the K = 1000 used in the figures.

**Stationarity Assumption in Mean-Field Equations May Not Hold During Time-Dependent Tracking (Section 9)**

Assumption involved: The mean-field dynamic equations (Eq. 3.3) are derived under the assumption that the Gaussian statistics of inputs across the population hold at each instant in time, with variance α_k(t) determined self-consistently by the instantaneous population rates m_k(t) (Eq. 3.6). The autocorrelation analysis of Section 5.2 further assumes that the system is in a stationary balanced state when computing q_k(τ).

Data/implementation structure: Section 9 applies the same mean-field framework to a time-varying external input m_0(t), including ramped (Eq. 9.13) and sinusoidal inputs (Figure 15). The tracking analysis assumes the deviation from perfect tracking is O(1/√K) (Eq. 9.5), which relies on the balanced-state mean-field equations remaining valid throughout the transient. However, the paper itself notes (Section 9, discussion of Eq. 9.8) that if m_0 changes too rapidly, 'the network will not stay in the balanced state during the rate change, so u_k is of order √K.' This is precisely the regime where the Gaussian input statistics and the stationarity of α_k break down.

Consequence: The bound in Eq. 9.8 is derived from the tracking ansatz (Eq. 9.5) that presupposes stationarity, yet the bound itself defines the boundary of that stationarity. Near this boundary, the mean-field equations used to derive the bound may not be self-consistent. The figures (Figures 14–16) use the finite-K solution but do not explore inputs near the tracking bound where the approximation is most likely to fail.

Suggested check: It would be helpful to verify numerically (via direct simulation of the binary network) that the population rates and input variance remain close to their instantaneous balanced-state values throughout the ramp and sinusoidal protocols, particularly as the rate of change of m_0 approaches the bound of Eq. 9.8. This would confirm that the stationarity assumption is not materially violated in the parameter regimes shown.

**Status**: [Pending]

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## Detailed Comments (23)

### 1. Abstract Overstates Poissonian Statistics Claim

**Status**: [Pending]

**Quote**:
> It is shown that the autocorrelations decay on a short time scale, yielding an approximate Poissonian temporal statistics.

**Feedback**:
The derivation in Section 5.2 establishes only that the ISI coefficient of variation approaches 1 in the large-K limit—a necessary but not sufficient condition for Poisson statistics. A Poisson process additionally requires exponential ISI distributions and the absence of higher-order temporal correlations. The paper's own binary-spike model introduces a refractory-like two-step transition (equation 5.18) that deviates from Poisson at short lags, and Figure 5 shows only a qualitative comparison to a Poisson reference. It would be helpful to rewrite 'yielding an approximate Poissonian temporal statistics' as 'yielding an approximate Poissonian coefficient of variation (CV ≈ 1), consistent with but not fully establishing Poisson statistics.'

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### 2. Abstract Chaos Claim Contradicts Section 8 Caveat

**Status**: [Pending]

**Quote**:
> The chaotic nature of the balanced state is revealed by showing that the evolution of the microscopic state of the network is extremely sensitive to small deviations in its initial conditions. The balanced state generated by the sparse, strong connections is an asynchronous chaotic state.

**Feedback**:
Section 8 explicitly acknowledges that 'the concept of chaos is technically inapplicable to a system with discrete degrees of freedom such as ours, since in this case the size of a perturbation of the system state is bounded by the discreteness of the system.' The abstract's unqualified assertion that chaos is 'revealed' by sensitivity to initial conditions is internally inconsistent with this caveat. It would be helpful to rewrite the abstract to reflect the same qualification that appears in Section 8, noting that the sensitivity is analogous to chaotic behavior rather than constituting a rigorous proof of chaos in the dynamical-systems sense.

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### 3. Connection Probability Uses Postsynaptic Index, Likely Should Be Presynaptic

**Status**: [Pending]

**Quote**:
> The connection between the $i$th postsynaptic neuron of the $k$th population and the $j$th presynaptic neuron of the $l$th population, denoted $J_{kl}^{ij}$ is $J_{kl} / \sqrt{K}$ with probability $K/N_{k}$ and zero otherwise. Here $k,l=1,2$. The synaptic constants $J_{k1}$ are positive and $J_{k2}$ negative. Thus, on average, $K$ excitatory and $K$ inhibitory neurons project to each neuron.

**Feedback**:
The expected number of presynaptic inputs from population l to a given neuron i in population k is N_l × (K/N_k). For this to equal K, one needs N_l = N_k, which is not generally true. If the connection probability were K/N_l (using the presynaptic population size), the expected in-degree would be N_l × (K/N_l) = K regardless of population sizes. The use of N_k (postsynaptic) rather than N_l (presynaptic) appears to be an error. It would be helpful to rewrite 'with probability $K/N_{k}$' as 'with probability $K/N_{l}$' to correctly recover the stated average in-degree of K.

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### 4. Population Average in Equation 3.2 Uses Inconsistent Index

**Status**: [Pending]

**Quote**:
> $m_{k}(t)=[\sigma_{k}^{j}(t)]=\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\sigma_{k}^{j}(t)\;\;,\;\;k=1,2,$

**Feedback**:
The summation runs over index i from 1 to N_k, but the summand uses σ_k^j(t) with a different free index j. As written, the right-hand side equals N_k × σ_k^j(t) / N_k = σ_k^j(t), which is the activity of a single neuron j rather than the population average. The summand should use σ_k^i(t) to match the summation variable. It would be helpful to rewrite equation 3.2 as m_k(t) = [σ_k^i(t)] = (1/N_k) Σ_{i=1}^{N_k} σ_k^i(t).

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### 5. Variance Derivation in Equation 3.12 Drops Cross-Population Terms Without Justification

**Status**: [Pending]

**Quote**:
> $\alpha_{k}(t)=[(\delta u_{k}^{i}(t))^{2}]=\sum_{l,l^{\prime}=1}^{2}\,\sum_{j,j^{\prime}=1}^{N_{l}}[\,(\,\delta(J_{kl}^{ij}\sigma_{l}^{j}(t))\,)^{2}\,],$

**Feedback**:
The double sum as written includes cross terms with (j,l) ≠ (j',l'), but the text silently discards all off-diagonal terms. The justification given addresses only the mean-squared diagonal term, not the cross-covariance terms. For the cross terms to vanish, one needs independence of J_{kl}^{ij} and J_{kl'}^{ij'} for distinct synapses, which holds under the random connectivity assumption but is not stated. It would be helpful to rewrite the derivation to explicitly state that cross terms vanish because distinct synaptic weights are drawn independently, so [δ(J_{kl}^{ij} σ_l^j) · δ(J_{kl'}^{ij'} σ_{l'}^{j'})] = 0 for (j,l) ≠ (j',l').

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### 6. Variance Decomposition in Equation 5.7 Drops Cross-Term Between δJ and δm

**Status**: [Pending]

**Quote**:
> Adding the two contributions yields
> 
> $$
> [ (\delta \langle u_{k}^{i} \rangle)^{2}] = \sum_{l=1}^{2} J_{kl}^{2} q_{l} = q_{E} + J_{k}^{2} q_{I} = \beta_{k}. \tag{5.7}
> $$

**Feedback**:
The total quenched deviation is δ⟨u_k^i⟩ = δ₁⟨u_k^i⟩ + δ₂⟨u_k^i⟩, so its squared average includes a cross-term 2[δ₁⟨u_k^i⟩ · δ₂⟨u_k^i⟩] involving [δJ_{kl}^{ij} · δm_l^j]. This cross-term vanishes only if the quenched connectivity fluctuations δJ_{kl}^{ij} are statistically independent of the self-consistently developed rate inhomogeneities δm_l^j, which holds for K ≪ N_l but is not stated here. It would be helpful to add a sentence stating 'The cross-term vanishes because δJ_{kl}^{ij} and δm_l^j are statistically independent for K ≪ N_l,' and to show the individual contributions [(δ₁⟨u_k^i⟩)²] and [(δ₂⟨u_k^i⟩)²] before combining.

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### 7. Sign Convention in Equation 7.1 Inconsistent with H Definition Throughout Paper

**Status**: [Pending]

**Quote**:
> The mean activity rate of neurons in the $k$ th population that have a local threshold $\theta$ is
> 
> $$
> m _ {k} (\theta) = H \left(\frac {\theta - u _ {k}}{\sqrt {\alpha_ {k}}}\right), \tag {7.1}
> $$

**Feedback**:
Throughout the paper H is the complementary Gaussian CDF (H(x) = ∫_x^∞ Dz), so a neuron fires more when u_k > θ. The rate should be H((u_k − θ)/√α_k). As written, equation 7.1 gives H((θ − u_k)/√α_k), which is the lower-tail CDF convention—the opposite sign. This inconsistency propagates: taking α_k → 0 in equation 7.1 as written gives Θ(θ − u_k) = 1 − Θ(u_k − θ), the opposite of the frozen-state result in equation 7.7. Equation 7.4, derived by integrating 7.1 over a Gaussian threshold distribution, recovers m_k = H(−u_k/√(α_k + Δ²)), which is consistent with the rest of the paper only if equation 7.1 should read H((u_k − θ)/√α_k). It would be helpful to rewrite equation 7.1 as m_k(θ) = H((u_k − θ)/√α_k).

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### 8. Equation 5.15 Missing Excitatory Coupling Coefficient

**Status**: [Pending]

**Quote**:
> $$
> \beta_{k}(\tau) = [ \langle \delta u_{i}^{k}(t) \delta u_{i}^{k}(t + \tau) \rangle ] = q_{E}(\tau) + J_{k}^{2} q_{I}(\tau), \tag{5.15}
> $$

**Feedback**:
The equal-time limit must satisfy β_k(0) = α_k. From equation 3.6, α_k = J_{kE}² m_E + J_{kI}² m_I, so the τ-dependent generalization should be β_k(τ) = J_{kE}² q_E(τ) + J_{kI}² q_I(τ). In equation 5.15 the excitatory term appears as q_E(τ) without any J_{kE}² prefactor, while the inhibitory term correctly carries J_k². This asymmetry is inconsistent unless J_{kE} = 1 by convention, which is never stated in this section. It would be helpful to either rewrite equation 5.15 as β_k(τ) = J_{kE}² q_E(τ) + J_{kI}² q_I(τ), or explicitly state that J_{kE} ≡ 1 is assumed, to make the τ → 0 limit self-consistent with β_k(0) = α_k.

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### 9. Self-Consistent Equation 5.17 Uses β_k(t+τ) Where β_k(τ) Is Expected

**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 integrand involves β_k(t + τ), where t is the integration variable running from 0 to ∞. This makes equation 5.17 a functional integro-differential equation requiring knowledge of β_k at lags larger than τ to evaluate the right-hand side at any given τ—a substantially more complex object than a simple ODE. The paper does not explain how the numerical solution in Figure 5 was obtained, nor does it clarify whether β_k(t + τ) is correct or whether β_k(τ) was intended. Readers might note that if β_k(t + τ) is correct, the solution method should be described explicitly. It would be helpful to add a sentence clarifying the solution strategy (e.g., iterative substitution) and confirming that the t-dependence inside β_k(t + τ) is physically motivated by the averaging over the last update time.

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### 10. Lyapunov Exponent Definition in Equation 8.3 Is Non-Standard and Yields Infinite Value by Construction

**Status**: [Pending]

**Quote**:
> The maximum Lyapunov exponent $\lambda_{L}$, defined by
> 
> $\lambda_{L}\equiv\lim_{D_{k}\to 0}D_{k}^{-1}\frac{dD_{k}}{dt},$

**Feedback**:
The standard Lyapunov exponent is λ_L = lim_{t→∞} (1/t) ln(D_k(t)/D_k(0)), which measures exponential divergence. The definition in equation 8.3 is instead the instantaneous fractional growth rate at D_k = 0. Equation 8.9 shows dD_k/dt ∝ √D_k, so D_k^{−1} dD_k/dt ∝ D_k^{−1/2} → ∞ as D_k → 0. The infinite λ_L follows directly from the non-standard definition applied to a √D_k growth law, not from exponential divergence. Indeed, D_k ~ t² for small t, and the standard Lyapunov exponent for this power-law growth would be lim_{t→∞} (1/t) ln(t²) = 0. It would be helpful to rewrite equation 8.3 using the standard definition and note explicitly that the instantaneous growth rate D_k^{−1} dD_k/dt diverges as D_k → 0 due to the discrete-state nature of the system, rather than implying a standard positive Lyapunov exponent.

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### 11. Equation 8.9 Derivation Not Shown; Prefactor Requires Non-Trivial Gaussian Integral

**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 derivation of equation 8.9 from equation 8.5 is not shown. Writing Q_k = m_k − D_k/2 gives α_k − γ_k = Σ_l J_{kl}²(D_l/2) ∝ D_k, confirming the proportionality claim. However, the coefficient 2/π and the factor e^{−u_k²/2α_k}/√α_k require a careful asymptotic expansion of the integrand of equation 8.5 as α_k − γ_k → 0. The result appears to come from differentiating the stationary-state equation for q_k with respect to Q_k near Q_k = m_k, using the identity ∂/∂γ ∫Dx [H(−(u+√γ x)/√(α−γ))]²|_{γ=α} = (1/π)(e^{−u²/2α}/√α), with the factor of 2 from D_k = 2(m_k − Q_k). It would be helpful to add a brief derivation in the appendix showing how equation 8.9 follows from equation 8.5, including the evaluation of the Gaussian integral that yields the prefactor 2/π · e^{−u_k²/2α_k}/√α_k.

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### 12. Sign of u_E in Equation 4.7 Contradicts the Stated Condition 4.5

**Status**: [Pending]

**Quote**:
> Equation 4.5 admits an unbalanced solution in which $m_{E} = 0$. In this solution, $m_{I}$ is to leading order in $K$ given by $m_{I} = Im_{0} / J_{I}$ (since the leading order in $u_{I}$ should vanish) so that
> 
> $$
> u _ {E} = \sqrt {K} \left(E - J _ {E} I / J _ {I}\right) m _ {0} &lt; 0. \tag {4.7}
> $$

**Feedback**:
Under condition 4.5 (E/I > J_E/J_I > 1), we have E·J_I > J_E·I, so E − J_E·I/J_I > 0, making u_E in equation 4.7 positive (not negative) for m_0 > 0. A positive u_E is inconsistent with m_E = 0, which is the correct conclusion—the m_E = 0 solution is excluded under condition 4.5. The inequality sign in equation 4.7 should be > 0, not < 0. The < 0 result holds under condition 4.6 (E/I < J_E/J_I < 1), not condition 4.5. It would be helpful to rewrite the surrounding text to state: 'Under condition 4.5, u_E = √K(E − J_E·I/J_I)m_0 > 0, which is inconsistent with m_E = 0, thereby excluding this unbalanced solution.'

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### 13. Finite-K Equations 4.17–4.18 Both Contain m_E Rather Than Population-Specific Rate

**Status**: [Pending]

**Quote**:
> $Em_{0}+m_{E}-J_{E}m_{I}=(\theta_{E}+\sqrt{\alpha_{E}}h(m_{E}))/\sqrt{K}.$ (4.17)
> $Im_{0}+m_{E}-J_{I}m_{I}=(\theta_{I}+\sqrt{\alpha_{I}}h(m_{I}))/\sqrt{K}.$ (4.18)

**Feedback**:
Both equations contain the term +m_E on the left-hand side. For the inhibitory population (equation 4.18), the self-recurrent drive should involve m_I, not m_E, consistent with the two-population structure. Comparing with the large-K balance equations 4.1–4.2, both have +m_E because the excitatory-to-excitatory and excitatory-to-inhibitory weights are both normalized to 1 (J_{EE} = J_{IE} = 1). If this normalization is the intended convention, it should be stated explicitly here, since the +m_E in equation 4.18 is not self-recurrence but rather the excitatory input to the inhibitory population. It would be helpful to add a clarifying note that both equations share +m_E because J_{IE} = 1 by the normalization of equation 2.6.

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### 14. Width of Rate Distribution Stated as O(m_k^{3/2}) But Derivation Is Not Shown

**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 Var(m_k^i) = q_k − m_k² = O(m_k²/|log m_k|), giving a standard deviation of O(m_k/√|log m_k|), not O(m_k^{3/2}). For the width to scale as m_k^{3/2}, one would need Var(m_k^i) ~ m_k³, which requires q_k − m_k² ~ m_k³. It is not clear how this follows from equation 5.14 as stated. It would be helpful to either derive the O(m_k^{3/2}) scaling explicitly from equation 5.14 and the definition of ρ_k, or to restate the claim with the correct scaling that follows directly from equation 5.14.

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### 15. Expression for f_{kl} in Equation 6.2 May Double-Count the √K Factor

**Status**: [Pending]

**Quote**:
> Calculating $f_{kl}$ by partial differentiation of the r.h.s. of equation 3.3 yields
> 
> $f_{kl}=\frac{\exp(-u_{k}^{2}/2\alpha_{k})}{\sqrt{2\pi\alpha_{k}}}J_{kl}.$ (6.2)

**Feedback**:
Differentiating H(−u_k/√α_k) with respect to m_l via the chain rule gives ∂H/∂m_l = H'(−u_k/√α_k) · (−1/√α_k) · ∂u_k/∂m_l. Since u_k = √K Σ_l J_{kl} m_l + u_0, we have ∂u_k/∂m_l = √K J_{kl}. With H'(x) = −(1/√(2π))e^{−x²/2}, this gives f_{kl} = [exp(−u_k²/(2α_k))/√(2πα_k)] · √K · J_{kl}. The factor of √K appears in f_{kl} itself, but equation 6.1 already has an explicit √K prefactor multiplying Σ_l f_{kl} δm_l. Together these would give a √K · √K = K prefactor, making eigenvalues O(K) rather than O(√K). It would be helpful to rewrite equations 6.1 and 6.2 together with an explicit statement of whether the √K in equation 6.1 is already incorporated into f_{kl}, to eliminate the apparent double-counting.

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### 16. Equation 6.4 Mixes Absolute and Perturbation Variables Inconsistently

**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 Θ argument contains only the perturbation variables δm_E and δm_I, while the last term −m_k involves the unperturbed fixed-point rate. For consistency, the full equation for m_k = m_k* + δm_k is τ_k d(δm_k)/dt = −δm_k + Θ(m_E* + δm_E − J_k(m_I* + δm_I)) − m_k*. As written, the Θ argument omits the background rates m_E* and m_I*, making the equation dimensionally and conceptually inconsistent. Additionally, the Θ argument should include the √K-scaled synaptic weights from u_k (equation 3.3), not just δm_E − J_k δm_I. It would be helpful to rewrite equation 6.4 to include the fixed-point rates in the Θ argument and clarify whether the √K prefactors from u_k are absorbed into the definition of δm_k.

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### 17. τ_G Formula First Term Under Square Root Is Not Derived

**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 three arguments of the min are structurally heterogeneous: the second and third terms are simple rate ratios arising naturally from the piecewise-linear analysis, while the first term contains √(J_I/J_E) and the product m_k(1−m_k) (a Bernoulli variance) whose appearance in the global analysis is not derived. The paper states the piecewise-linear equations 'can be solved explicitly' but does not show the derivation of τ_G. Without the derivation it is not possible to verify whether the square-root term is correct or whether it should instead be m_I/m_E. It would be helpful to include at least a sketch of the derivation showing which phase-plane region produces each of the three candidate expressions for τ_G.

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### 18. Ramp Input Formula in Equation 9.13 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, a ramp starting from t = 0. At t = 1 this gives m_0(1) = m_0 + v_0, already including a full unit of ramping, contradicting the stated protocol that the input is constant from t = 0 to t = 1. The correct expression for a ramp active only between t = 1 and t = 2 should be m_0(t) = m_0 + v_0·(t − 1). This matters because the tracking bound in equation 9.8 is expressed in terms of dm_0/dt = v_0, and Figure 16 plots dm_E/dt versus v_0 using this ramp; an offset error in the ramp onset shifts the transient response. It would be helpful to rewrite equation 9.13 as m_0(t) = m_0 + v_0·(t − 1) for 1 ≤ t ≤ 2.

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### 19. Tracking Advantage Conflates Balance with Synaptic Scaling Convention

**Status**: [Pending]

**Quote**:
> Thus, in the unbalanced network, the difference between the network rates and the rates in a perfectly tracking network will be of order $\sqrt{K}$ times larger than in a balanced network.

**Feedback**:
The O(√K) tracking advantage is derived by comparing a balanced network with synaptic weights O(1/√K) against an unbalanced network with weights O(1/K). The linearized tracking error equations 9.11–9.12 for the unbalanced network contain O(1) coefficients J_kl (not O(1/√K)), so the steady-state deviation m_k^1 is O(1) rather than O(1/√K). However, this ratio is at least partly a direct consequence of the different synaptic scaling conventions rather than of balance per se: an unbalanced network with 1/√K synapses would also have O(1/√K) deviations if its gain were comparable. It would be helpful to add a sentence clarifying that the O(√K) advantage arises from the combination of balance and the 1/√K synaptic scaling, and that the comparison with a 1/K-scaled unbalanced network does not isolate the contribution of balance alone.

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### 20. Cross-Correlation Scaling for Directly Connected Pairs May Be O(1/K) Not O(1/√K)

**Status**: [Pending]

**Quote**:
> The maximal value of the cross-correlations occurs for pairs that are directly connected, and this cross-correlation is of the order of the strength of the synapse, $O(1/\sqrt{K})$.

**Feedback**:
A single shared connection between two neurons contributes a covariance term of order (J/√K)² = J²/K to the joint input fluctuations of the postsynaptic pair. The cross-correlation of output activities—normalized by individual variances O(J²)—would then scale as (J²/K)/J² = 1/K, not 1/√K. The O(1/√K) figure would require the shared-input covariance to be O(1/√K), which would need the synaptic weight to be O(K^{−1/4}), inconsistent with the paper's J/√K scaling. It would be helpful to provide the explicit derivation showing why the O(1/√K) result holds, or to rewrite 'this cross-correlation is of the order of the strength of the synapse, $O(1/\sqrt{K})$' as 'this cross-correlation is of the order of the square of the synaptic strength, $O(1/K)$' if the naive calculation is correct.

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### 21. Multinomial Distribution in Equation A.13 Written 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**:
The three categories (active at both times, active only at t, active only at t+τ) should follow independent Poisson distributions in the large-K limit, giving p_l as the product of three independent Poisson factors. The formula as written pools n_2 and n_3 into a single power ((m_l−q_l)K)^{n_2+n_3}/(n_2!n_3!), which equals the product of two independent Poisson factors only because the two rates are identical. However, the expression looks like a single Poisson for the combined count n_2+n_3 with an extra 1/(n_2!n_3!) factor, which would be missing the binomial coefficient C(n_2+n_3, n_2). It would be helpful to rewrite equation A.13 as the explicit product of three independent Poisson factors to make the factorization transparent and prevent misreading.

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### 22. Duplicated Paragraph in Section 10.5 Appears to Be a Copy-Paste Error

**Status**: [Pending]

**Quote**:
> between cortical pyramidal cells show a marked degree of depression. It should be investigated how such dynamical synapses affect the balanced state. If one assumes synaptic depression between the excitatory-to-excitatory synapses only, and facilitation between the synapses from the inhibitory to the excitatory and from the excitatory to inhibitory populations (Thomson, West, &amp; Deuchars, 1995; Thomson, West, Hahn, &amp; Deuchars, 1996), the equilibrium rates in the network decrease, relative to those in a network without facilitation.

**Feedback**:
The entire paragraph beginning 'between cortical pyramidal cells show a marked degree of depression...' through '...the rates decrease due to the change in the synaptic strengths' appears verbatim a second time in the section, immediately following the paragraph about first and second moments of input. The duplication begins mid-sentence ('between cortical pyramidal cells...'), strongly suggesting a copy-paste error during manuscript assembly. The duplicated block also repeats the subsequent paragraph about EPSP height versus release probability and facilitation effects. It would be helpful to remove the entire duplicated passage (from 'between cortical pyramidal cells show a marked degree of depression' through the second occurrence of 'These issues warrant further study').

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### 23. Equation A.10 Is a Functional Integro-Differential Equation; Solution Method Not Described

**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 right-hand side integrand depends on q_l(t' + τ) for t' ≥ 0, requiring knowledge of the autocorrelation at lags larger than τ to evaluate the equation at any given τ. This makes A.10 a functional integro-differential equation, not a simple ODE, and it cannot be solved by forward integration from τ = 0. The paper states 'So q_k satisfies equation 5.17' without explaining how the numerical solution in Figure 5 was obtained. It would be helpful to add a sentence after equation A.10 explaining the solution strategy (e.g., iterative substitution starting from a trial function) and clarifying that the equation is a functional integro-differential equation, so readers understand how q_k(τ) is computed in practice.

---
