# Chaotic Balanced State in a Model of Cortical Circuits

**Date**: 03/08/2026
**Domain**: natural_sciences/neuroscience
**Taxonomy**: academic/research_paper
**Filter**: Active comments

---

## Overall Feedback

Here are some overall reactions to the document.

**Chaotic Characterization Rests on a Divergent Rather Than Finite Lyapunov Exponent**

All three reviewers flag the same core problem: Section 8 and equation 8.9 show that dD_k/dt ∝ √D_k, so D_k grows as t² rather than exponentially, and the Lyapunov exponent λ_L defined in equation 8.3 diverges as D_k → 0. The paper acknowledges this divergence and attributes it to the discrete binary state space, yet still uses it as the primary—and essentially sole—quantitative evidence for chaos in the title and abstract. A divergent Lyapunov exponent is not the standard criterion for chaos in dynamical systems theory; it reflects the infinite gain of threshold units at switching events rather than sensitive dependence in the conventional sense. It would be helpful to supplement this analysis with at least two independent quantitative signatures: (a) a finite-time or regularized effective Lyapunov exponent computed from direct simulations of two nearby trajectories in finite networks, and (b) autocorrelation decay rates or power spectra (already partially available from equation 5.17) that can serve as independent indicators of chaotic dynamics. Alternatively, the paper could reframe the instability using a metric appropriate for discrete-state systems and clearly distinguish it from the standard definition of chaos, so that readers are not misled about the nature of the instability.

**Global Stability Derivation Relies on an Approximation Whose Accuracy Is Never Bounded**

Section 6.2 derives the global stability condition (equations 6.4–6.6) by approximating H(−u_k/√α_k) ≈ Θ(u_k) on the grounds that large perturbations drive u_k to O(√K). This substitution discards the variance term α_k entirely, yet α_k is itself a function of the perturbed rates m_k, which change substantially during a large perturbation. The resulting piecewise-linear system is then solved analytically to yield τ_G (equation 6.6), but τ_G depends on m_E and m_I evaluated at the balanced fixed point—values that are no longer valid when the network has left the balanced regime. The approximation's accuracy is never bounded, and the only numerical check (Figure 8, τ = 1.3 vs. τ_G = 1.50) constitutes a single illustrative case rather than a systematic validation. It would be helpful to state explicitly the error introduced by dropping α_k, to verify numerically across a range of perturbation magnitudes that τ_G from equation 6.6 matches the actual stability boundary observed in simulations, and to clarify whether m_E and m_I in equation 6.6 refer to fixed-point or instantaneous values—since this distinction affects the quantitative stability boundary shown in Figure 9.

**Uniqueness of the Balanced Fixed Point and Its Order Parameters Is Asserted Without Proof**

Sections 4 and 5 derive the balanced fixed-point rates (equations 4.3–4.4) and the additional order parameters q_k and β_k, but the paper does not demonstrate that the full coupled system—equations 3.7, 5.12, and 5.17 taken together—has a unique physically relevant solution. The fixed-point analysis in Section 4 excludes only specific unbalanced solutions (m_E = 0 with m_I = Im_0/J_I, and m_E = m_I = 1) via equations 4.9–4.10, without systematically ruling out mixed boundary cases such as m_E ∈ (0,1) with m_I = 1 or m_I = 0 with m_E interior. Equation 5.12 is noted to have two solutions (a frozen branch and a fluctuating branch), but uniqueness of the fluctuating branch is asserted informally, and additional fixed points at intermediate q_k values—particularly when threshold inhomogeneity is introduced in Section 7—are not excluded. It would be helpful to add a monotonicity or contraction argument (or a numerical sweep of the solution landscape) showing that for the parameter regime defined by equations 4.9–4.10, the coupled equations for (m_k, q_k, β_k) have a unique physically relevant solution, since multiplicity of solutions would complicate the interpretation of rate distributions and stability boundaries throughout the paper.

**Temporal Tracking Claims Are Supported Only by Illustrative Examples Without Systematic Quantitative Validation**

Section 9 argues that the balanced network tracks time-varying inputs with deviations of order 1/√K—√K times smaller than in an unbalanced network—and derives the bound on dm_0/dt in equation 9.8. However, the O(1/√K) ansatz in equation 9.5 is postulated rather than derived from the dynamics, and no condition on the rate of change dm_0/dt is given that ensures the perturbative expansion remains self-consistent. The supporting evidence consists of figures showing responses to a single ramp rate v_0 and a single sinusoidal frequency at fixed K = 1000, with no cross-correlation coefficients, signal-to-noise ratios, or mean-squared tracking error reported. The comparison with the unbalanced threshold-linear network uses a specific parameter choice whose generality is not established. It would be helpful to plot tracking fidelity (e.g., normalized cross-correlation between m_0(t) and m_E(t)) as a function of input frequency for several values of K, to verify that the O(1/√K) tracking error predicted analytically matches simulation results quantitatively, and to derive an explicit condition on dm_0/dt under which the ansatz of equation 9.5 is self-consistent.

**Gaussian Input Approximation and Its Two-Time Extension Require More Rigorous Justification**

The entire mean-field framework rests on the Gaussian approximation for total synaptic inputs (equations 3.3–3.6 and Appendix A), justified by a central-limit-theorem argument that assumes independence of presynaptic activities. This independence holds rigorously only when K ≪ N_k, and the O(K/N) corrections to independence are never explicitly bounded, leaving the regime of validity unquantified. More critically, the autocorrelation analysis in Section 5.2 and Appendix A.2 introduces a two-time factorization (equations A.12–A.14) that additionally assumes temporal fluctuations are independent of the quenched spatial fluctuations encoded in q_k. In a network with fixed connectivity, temporal fluctuations at a given cell are filtered through the same synaptic weights that produce spatial inhomogeneity, creating correlations between the two sources of variability that are asserted—but not derived—to vanish in the large-N limit. It would be helpful to provide an explicit bound on the error introduced by the independence assumption as a function of K/N, to include a formal justification (analogous to a cavity argument) that the cross-correlations between temporal and spatial variability vanish before the large-K limit is taken, and to clarify whether the Gaussian approximation for two-time statistics introduces systematic errors in the autocorrelation decay times reported in Figure 5.

**Finite-K Convergence Is Unquantified and the Mapping to Spiking Neurons Is Asserted Without Derivation**

Two related gaps concern the practical scope of the theory. First, all main results—stability boundaries τ_L and τ_G, autocorrelation decay, and tracking speed—are derived in the K → ∞ limit and applied to finite-K simulations without error bounds. For biologically plausible values of K (roughly 100–1000), O(1/√K) corrections to the balance condition are 3–10% of the leading term, which could shift stability boundaries appreciably, yet only a single comparison at K = 1000 (Figure 3) is provided. Second, Section 10.3 claims that the mean-field predictions apply to integrate-and-fire networks with the same connectivity scaling, supported only by qualitative analogy and brief citations. Key features absent from the binary model—absolute refractoriness, spike-timing correlations, and conductance-based synapses—could alter the mean-field equations and stability conditions in non-trivial ways, and the ISI formula (equation 5.18) derived under the assumption of independent consecutive updates may not generalize. It would be helpful to provide at least one systematic convergence plot showing how deviations from the large-K predictions scale with K, and to either derive explicit corrections to the mean-field equations for leaky integrate-and-fire neurons or present direct simulation results from a spiking implementation with the same N and K scaling, clearly identifying which qualitative predictions are robust and which are model-specific.

**Binary-unit approximation and its mapping to spiking neurons**

Assumption involved: The paper's entire mean-field framework is built on binary threshold units (σ ∈ {0,1}) with asynchronous Heaviside updates (Eq. 2.1–2.2). The analytical results—Poisson-like ISIs, the autocorrelation decay, the fast-tracking time constant of O(1/√K), and the chaotic Lyapunov analysis—are all derived for this specific dynamics.

Data/implementation structure: The paper's own Section 10.3 acknowledges that 'constructing an exact mean-field theory for the integrate-and-fire dynamics similar to the one presented here for binary units is much more difficult,' and that a direct comparison with integrate-and-fire results (Tsodyks & Sejnowski 1995; Amit & Brunel 1997a,b) is difficult. The binary unit conflates the membrane time constant with the update interval τ_k, and refractoriness is only partially captured (a cell must first return to 0 before it can spike again, as noted in Section 2). Spike-timing correlations arising from absolute refractoriness, sub-threshold membrane dynamics, and conductance-based nonlinearities are absent.

Consequence: Quantitative predictions—particularly the O(1/√K) tracking time scale and the precise shape of the ISI distribution (Eq. 5.18)—may not transfer directly to biologically realistic neurons. The paper notes the ISI rises linearly from zero (capturing a refractory-like effect) but does not verify that the autocorrelation structure or the Lyapunov divergence result survives in spiking models.

Suggested check: It would be helpful to include even a brief numerical comparison with a leaky integrate-and-fire network using the same E/I architecture and √K synaptic scaling, to assess whether the qualitative predictions (asynchronous chaos, fast tracking) are robust to the choice of neuron model, or to more explicitly bound the quantitative error introduced by the binary approximation.

**Stationarity assumption in the autocorrelation mean-field equations**

Assumption involved: The derivation of the autocorrelation equation (Eq. 5.17) and the overlap equation for the Lyapunov analysis (Eq. 8.5) both rely on averaging over absolute time t (explicitly stated below Eq. A.11: 'the averaging is also over absolute time t'). This is a stationarity assumption: the joint statistics of σ_k^i(t) and σ_k^i(t+τ) are assumed to depend only on the lag τ, not on t itself.

Data/implementation structure: The paper also studies time-varying external inputs m_0(t) in Section 9 (tracking of time-dependent input). During the transient tracking phase—when m_0 is ramping (Eq. 9.13) or oscillating (Figure 15)—the population rates m_k(t) are explicitly non-stationary. The autocorrelation framework of Section 5.2 is derived under the stationary balanced fixed point and is then implicitly carried over to interpret the fluctuations during tracking.

Consequence: The claim that autocorrelations decay on a short time scale (yielding approximate Poissonian statistics) is established only for the stationary balanced state. During input tracking, the temporal statistics of single-cell activity are non-stationary, and the self-consistent integral equation (Eq. 5.17) no longer strictly applies. The paper does not derive or verify an analogous result for the non-stationary regime, so the characterization of firing irregularity during tracking rests on an extrapolation.

Suggested check: It would be helpful to either (a) explicitly restrict the Poissonian/autocorrelation claims to the stationary regime and note that they are approximations during tracking, or (b) derive a time-dependent generalization of Eq. 5.17 for slowly varying m_0(t) and verify that the autocorrelation time scale remains short relative to the tracking time scale 1/√K.

**Status**: [Pending]

---

## Detailed Comments (25)

### 1. Divergent Lyapunov Exponent Misrepresented as Standard Chaos in Introduction and Abstract

**Status**: [Pending]

**Quote**:
> In section 8, we evaluate the sensitivity of the temporal fluctuations in the local instantaneous activities to a small change in the initial condition. We conclude that a small change in the initial condition leads rapidly to a complete loss of memory of the unperturbed initial conditions. Thus, our network shows the main characteristics of chaotic systems.

**Feedback**:
The introduction asserts that the network 'shows the main characteristics of chaotic systems,' but Section 8 itself acknowledges that the standard Lyapunov criterion is inapplicable to discrete-state systems. Equation 8.9 shows dD_k/dt ∝ √D_k, so D_k grows as t² (polynomial) rather than exponentially, and equation 8.3 gives λ_L → ∞ as D_k → 0. Polynomial divergence of nearby trajectories does not satisfy the standard criterion for chaos (positive, finite Lyapunov exponent). It would be helpful to rewrite 'Thus, our network shows the main characteristics of chaotic systems' as 'Thus, our network shows a form of sensitive dependence on initial conditions; however, because the state space is discrete, the standard Lyapunov exponent diverges (Section 8), and this instability differs from chaos in the conventional sense.'

---

### 2. Power-Law Tail Claim in Abstract Overstates What the Mean-Field Framework Derives

**Status**: [Pending]

**Quote**:
> The activity levels of single cells are broadly distributed, and their distribution exhibits a skewed shape with a long power-law tail.

**Feedback**:
The mean-field framework (equations 3.3–3.6) produces a rate distribution determined by the complementary error function H(·) applied to Gaussian inputs. For homogeneous thresholds, Section 5.1 shows the distribution is bounded on [0,1] with no generic power-law tail. A power-law tail emerges only for specific threshold distributions: equation 7.12 gives ρ_k(m) ∝ 1/(m√|log m|) for a bounded uniform threshold distribution, which is not a pure power law, and equation 7.8 gives a bimodal (frozen) distribution for a Gaussian threshold distribution. The abstract's unqualified 'long power-law tail' is stronger than what any single derived result supports. It would be helpful to rewrite this as 'their distribution exhibits a skewed shape with a tail whose form depends on the distribution of neuronal thresholds' to accurately reflect the parameter-dependent results of Section 7.

---

### 3. Exactness Claim for Mean-Field Solution Overstates Rigor Given Unverified Independence Assumption

**Status**: [Pending]

**Quote**:
> We present an analytical solution of the mean-field theory of this model, which is exact in the limit of large network size.

**Feedback**:
Appendix A states explicitly that the independence assumption underlying the mean-field theory 'holds rigorously provided that K ≪ N_k.' The O(K/N) corrections to independence are never bounded, and the two-time extension in Appendix A.2 introduces an additional factorization (temporal and spatial fluctuations are independent) that is asserted but not derived. The word 'exact' therefore overstates the rigor: the solution is exact given these assumptions, but the assumptions themselves are not proven to hold exactly in the large-N limit for two-time statistics. It would be helpful to rewrite as 'which becomes exact in the limit 1 ≪ K ≪ min(N_E, N_I) under the assumption that inputs to a given cell are asymptotically uncorrelated.'

---

### 4. Normalization J_EE = J_IE = 1 Is a Symmetry Assumption, Not Merely a Scale Convention

**Status**: [Pending]

**Quote**:
> Since the model neurons are threshold elements, the absolute scale of $u_{i}^{k}$ is irrelevant. We therefore set
> 
> $J_{EE}=J_{IE}=1,$ (2.6)

**Feedback**:
The threshold-element argument justifies fixing one absolute scale per population, but with two populations there are in principle two independent scales. Setting J_EE = J_IE = 1 simultaneously fixes the ratio J_EE/J_IE = 1, which is an additional symmetry constraint beyond what the threshold argument alone permits. The ratio J_EE/J_IE controls the relative excitatory drive to the two populations and affects the quantitative location of the balanced fixed point (equations 4.3–4.4) and the stability boundaries τ_L and τ_G. The claim that 'the only connection parameters from the network are the inhibitory and external ones' is only true after this additional assumption. It would be helpful to add a sentence after equation 2.6 stating explicitly that J_EE = J_IE = 1 is a symmetry assumption whose effect on stability boundaries is not explored.

---

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

**Status**: [Pending]

**Quote**:
> $\alpha_{k}(t)=[(\delta u_{k}^{i}(t))^{2}]=\sum_{l,l^{\prime}=1}^{2}\,\sum_{j,j^{\prime}=1}^{N_{l}}[\,(\,\delta(J_{kl}^{ij}\sigma_{l}^{j}(t))\,)^{2}\,],$ (3.12)
> 
> where $\delta X\equiv X-[X]$. 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**:
The double sum over (l,j) and (l′,j′) in equation 3.12 generates both diagonal terms (j=j′) and off-diagonal cross terms (j≠j′). The text evaluates only the diagonal contribution without explaining why cross terms vanish. For j≠j′, independence of the random connectivity variables J_{kl}^{ij} and J_{kl}^{ij′} and independence of σ_l^j and σ_l^{j′} (the mean-field independence assumption, valid when K ≪ N_k) together make these cross terms zero. This argument is standard but is not stated. It would be helpful to add a sentence after equation 3.12 explaining that cross terms vanish because J_{kl}^{ij} and J_{kl}^{ij′} are independent for j≠j′ (random connectivity) and neuronal activities are uncorrelated in the large-N mean-field limit.

---

### 6. Unbalanced m_I = 0 Case Not Analyzed; Claim That Constraints 4.9–4.10 Eliminate All Unbalanced States Is Undemonstrated

**Status**: [Pending]

**Quote**:
> It is straightforward to show that these constraints eliminate all possible unbalanced states.

**Feedback**:
The text analyzes two specific unbalanced cases—m_E = 0 (equation 4.7) and m_E = m_I = 1 (equation 4.8)—but never addresses m_I = 0 or m_E = 1 with m_I interior. For m_I = 0: u_I = √K(Im_0 + m_E) > 0 for m_E > 0, so m_I = 0 is self-inconsistent under equation 4.9 (since A_I m_0 > 0). For m_E = 1: u_E = √K(Em_0 + 1 − J_E m_I); positivity requires J_E m_I < 1 + Em_0, but J_E > 1 (equation 4.10) means m_I < 1/J_E < 1, and the consistency of u_I > 0 must also be checked. The assertion 'It is straightforward to show' substitutes for an actual demonstration. It would be helpful to enumerate all four boundary cases (m_E ∈ {0,1}, m_I ∈ {0,1}) with the sign of u_k shown explicitly for each.

---

### 7. Approximation for H(x) in Equation 4.13 Drops a Logarithmic Sub-Leading Term That Is Not Negligible at Biological Rates

**Status**: [Pending]

**Quote**:
> we can use the approximation
> 
> $H(x)\approx\frac{\exp(-x^{2}/2)}{\sqrt{2\pi}|x|}$ (4.13)
> 
> to obtain
> 
> $h(m)\approx-\sqrt{2|\log m|}.$ (4.14)

**Feedback**:
Starting from m = H(−h) ≈ exp(−h²/2)/(√(2π)|h|), taking logarithms gives log m ≈ −h²/2 − log(√(2π)|h|). Solving for h yields h ≈ −√(2|log m| − 2 log(√(2π)|h|)). The sub-leading term 2 log(√(2π)|h|) ≈ log(2π) + log(2|log m|) is not negligible compared to 2|log m| at biologically plausible rates: at m = 0.1, 2|log m| ≈ 4.6 while log(2π) ≈ 1.8, a ~40% correction. Equation 4.14 is used to compute the rate distribution in Section 5.1, so the accuracy matters. It would be helpful to state the condition |h| ≫ 1 (equivalently m ≪ 0.24) and note that equation 4.14 holds up to corrections of order log(log(1/m))/log(1/m).

---

### 8. Width of Rate Distribution Claimed Proportional to (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 variance of the rate distribution is q_k − (m_k)² = O(m_k²/|log m_k|), giving a standard deviation (width) of 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_k)² ∝ m_k³, which is not what equation 5.14 gives. It would be helpful to rewrite 'the width of the distribution is proportional to $(m_{k})^{3/2}$' as 'the width of the distribution is proportional to $m_k/\sqrt{|\log m_k|}$, derived from the variance $q_k - (m_k)^2 = O(m_k^2/|\log m_k|)$ given by equation 5.14.'

---

### 9. Autocorrelation Equation 5.17 Integrates Over Future Lag Values, Creating a Non-Causal Dependence

**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) with t ≥ 0 and τ ≥ 0, so the argument t + τ ≥ τ. This means the equation for dq_k/dτ at lag τ depends on β_k evaluated at lags strictly larger than τ, creating a forward-in-τ dependence. Since β_k(τ) is determined by q_k(τ) through equation 5.15, solving for q_k(τ) would require q_k at all larger lags first, making the equation non-causal as an ODE in τ. The paper does not explain the numerical solution method used to produce Figure 5. It would be helpful to clarify whether the argument should be β_k(|t − τ|) or β_k(τ − t) with adjusted integration limits, and to describe the numerical procedure used.

---

### 10. Enhancement of Short-Time Autocorrelations Attributed to Refractoriness of Projecting Cells Rather Than the Cell Itself

**Status**: [Pending]

**Quote**:
> As can be seen, the autocorrelations are larger than those predicted by Poisson statistics. This enhancement of short-time correlations reflect the refractoriness in the activities of the cells that project the cell.

**Feedback**:
The quantity q_k(τ) in equation 5.16 is the population-averaged autocorrelation of the postsynaptic cell's own activity σ_i^k(t). The refractory-like effect that produces the linear rise of the ISI from zero (equation 5.18) is a property of the cell itself: a binary unit must transition from active (1) to inactive (0) before it can fire again, so consecutive spikes are separated by at least one silent period. This is a property of the target cell's update rule, not of the presynaptic population. It would be helpful to rewrite 'the refractoriness in the activities of the cells that project the cell' as 'the refractory-like constraint of the binary update rule, which prevents the cell from firing in consecutive time steps without an intervening silent period.'

---

### 11. ISI Formula 5.18 Long-Time Decay Rate Claim Requires Condition m_i < 1/2

**Status**: [Pending]

**Quote**:
> $$
> I _ {i} (t) = \frac {m _ {i} \left(1 - m _ {i}\right)}{\tau_ {E} \left(1 - 2 m _ {i}\right)} \left(\exp \left(- m _ {i} t / \tau_ {E}\right) - \exp \left(- \left(1 - m _ {i}\right) t / \tau_ {E}\right)\right), t \geq 0. \tag {5.18}
> $$

**Feedback**:
The paper states 'I(t) decays purely exponentially with a decay constant m_i/τ_E.' The two exponential rates are m_i/τ_E and (1−m_i)/τ_E. The dominant long-time decay is the slower rate, which is m_i/τ_E only when m_i < 1−m_i, i.e., m_i < 1/2. For m_i > 1/2 the dominant long-time decay is (1−m_i)/τ_E, not m_i/τ_E. The normalization check confirms the formula integrates to 1 for all m_i ∈ (0,1), so the formula itself is correct, but the stated decay constant is only valid for m_i < 1/2. It would be helpful to rewrite 'I(t) decays purely exponentially with a decay constant m_i/τ_E' as 'I(t) decays purely exponentially with a decay constant m_i/τ_E (for m_i < 1/2, where this is the slower of the two exponential rates).'

---

### 12. Linearization in Equation 6.2 Omits the ∂H/∂α_k · ∂α_k/∂m_l Term

**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**:
The full derivative of H(−u_k/√α_k) with respect to m_l has two contributions: (∂H/∂u_k)(∂u_k/∂m_l) and (∂H/∂α_k)(∂α_k/∂m_l). Equation 6.2 captures only the first term. The second term equals [u_k/(2α_k^{3/2})] · φ(u_k/√α_k) · (∂α_k/∂m_l), where φ is the standard normal density and ∂α_k/∂m_l = J_{kl}². This is O(1) and thus the same order as the term retained in equation 6.2. Omitting it makes f_{kl} incomplete and could shift the stability boundary τ_L. It would be helpful to either include the ∂α_k/∂m_l contribution in equation 6.2 or add a sentence explicitly justifying why this term is negligible at the balanced fixed point.

---

### 13. Global Stability Equation 6.4 Omits the Balanced Fixed-Point Input Inside the Heaviside

**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**:
When H(−u_k/√α_k) ≈ Θ(u_k) and u_k = u_k^* + J_{kE}δm_E − J_{kI}δm_I, the argument of Θ should be u_k^* + J_{kE}δm_E − J_{kI}δm_I, where u_k^* is the O(1) balanced fixed-point input. Equation 6.4 omits u_k^* inside Θ, which shifts the piecewise-linear regions and therefore alters the analytic solution and the resulting τ_G formula (equation 6.6). Additionally, the right-hand side subtracts m_k but does not add back the fixed-point Θ(u_k^*) = m_k term, so the equation is not a pure perturbation equation. It would be helpful to rewrite equation 6.4 as τ_k (d/dt)δm_k = −δm_k + Θ(u_k^* + J_{kE}δm_E − J_{kI}δm_I) − m_k, making explicit that the Heaviside argument includes the O(1) balanced-state input.

---

### 14. τ_G Formula Arguments Inside min Have Inconsistent Dependence on Synaptic Weights

**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 inside the min have different dependences on synaptic weights: the first involves both J_E and J_I, while the second and third are pure rate ratios with no synaptic weight factors. If the derivation follows from the piecewise-linear system in equation 6.4, each linear region's stability boundary should involve the open-loop gain J_E · J_I, so one would expect factors of J_I/J_E or √(J_I/J_E) in the second and third terms as well. The paper provides no derivation showing which piecewise-linear regime of equation 6.4 yields each of the three candidate expressions, making the formula unverifiable. It would be helpful to add an inline annotation or appendix entry identifying which dynamical regime each of the three min-arguments corresponds to.

---

### 15. Instability of Q_k = m_k Fixed Point 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 text states the instability of Q_k = m_k 'will be shown below,' but no subsequent derivation appears in Section 8. The instability is actually established by equation 8.9: since dD_k/dt ∝ √D_k > 0 for any D_k > 0, any infinitesimal perturbation away from the synchronized state grows. This argument is never made explicit. It would be helpful to rewrite 'This solution is unstable, as will be shown below.' as 'This solution is unstable: equation 8.9 shows that dD_k/dt > 0 for any D_k > 0, so any infinitesimal perturbation away from the synchronized state grows.'

---

### 16. Proportionality α_k − γ_k ∝ D_k Stated Without the Explicit Constant

**Status**: [Pending]

**Quote**:
> Since $\alpha_{k}-\gamma_{k} \propto D_{k}$, equation 8.9 has a growing solution even if $D_{k}(0)=0$.

**Feedback**:
From the definitions, α_k = Σ_l J_{kl}² m_l and γ_k = Σ_l J_{kl}² Q_l, so α_k − γ_k = Σ_l J_{kl}²(m_l − Q_l) = (1/2)Σ_l J_{kl}² D_l. This explicit proportionality constant is needed to verify the coefficient in equation 8.9 and to confirm that the √D_k growth rate follows correctly. Without it, readers cannot check whether the prefactor (2/π)(e^{−u_k²/2α_k}/√α_k) in equation 8.9 is correct. It would be helpful to rewrite 'Since $\alpha_{k}-\gamma_{k} \propto D_{k}$' as 'Since $\alpha_{k}-\gamma_{k} = \frac{1}{2}\sum_l J_{kl}^2 D_l$, which is proportional to $D_k$.'

---

### 17. Derivation of Equation 8.9 from Equation 8.5 Is Absent from Both Main Text and Appendix

**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
> 
> $\tau_{k}\frac{dD_{k}}{dt}=\frac{2}{\pi}\frac{e^{-u_{k}^{2}/2\alpha_{k}}}{\sqrt{\alpha_{k}}}\sqrt{\alpha_{k}-\gamma_{k}}.$ (8.9)

**Feedback**:
The expansion of the integral in equation 8.5 around Q_k = q_k (i.e., small D_k) requires evaluating d/dε[∫Dx H²((−u_k − √γ_k x)/√ε)] at ε = α_k − γ_k → 0. This is a singular integral whose evaluation yields the coefficient (2/π)(e^{−u_k²/2α_k}/√α_k), but the calculation is shown neither in the main text nor in Appendix A (which is cited for equation 8.5 but not for equation 8.9). This is the key step establishing the √D_k growth rate and the divergent Lyapunov exponent—the paper's central claim about chaos. It would be helpful to add a derivation of equation 8.9 from equation 8.5 in Appendix A, showing explicitly how the expansion of the integral in powers of √(α_k − γ_k) yields the stated coefficient.

---

### 18. Ramp Input Formula 9.13 Inconsistent with Stated Boundary Conditions

**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**:
The formula m_0(t) = m_0 + v_0 t gives m_0(0) = m_0 at t = 0, but the ramp is stated to begin at t = 1. At t = 1 the formula yields m_0 + v_0, and at t = 2 it yields m_0 + 2v_0. For the ramp to start from the equilibrium value m_0 at t = 1, the correct expression should be m_0(t) = m_0 + v_0(t − 1) for 1 ≤ t ≤ 2. As written, the formula implies the ramp has been running since t = 0, contradicting the stated setup that the network is at equilibrium for 0 ≤ t ≤ 1. It would be helpful to rewrite equation 9.13 as m_0(t) = m_0 + v_0(t − 1) for 1 ≤ t ≤ 2.

---

### 19. Tracking Time Scale Argument Contains an Algebraic Inconsistency

**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**:
Integrating δm_k ≈ τ_k^{-1} T ΔP_k and setting δm_k = A_k δm_0 gives T ~ τ_k δm_0 / ΔP_k. Since δm_0 ~ 1/√K and ΔP_k ~ O(1), this gives T ~ τ_k/√K, which is the correct result. However, the text writes this as 'τ_k/δm_0,' which equals τ_k√K (since δm_0 ~ 1/√K)—the reciprocal of the intended expression. The expression 'τ_k/δm_0 ≈ τ_k/√K' is therefore algebraically wrong: τ_k/δm_0 ~ τ_k√K ≠ τ_k/√K. It would be helpful to rewrite as 'δm_k reaches A_k δm_0 in time of order τ_k · δm_0 / ΔP_k ~ τ_k/√K.'

---

### 20. Equation 9.6 Labeled 'Leading Order in K' But Retains an O(1/√K) Correction Term

**Status**: [Pending]

**Quote**:
> $\tau_{k}\frac{dm_{k}^{\infty}(t)}{dt}=-m_{k}^{\infty}(t)+H\left(-\frac{u_{k}^{\infty}(t)+\sum_{l}J_{kl}m_{l}^{1}(t)}{\sqrt{\alpha_{k}^{\infty}(t)}}\right),k=1,2.$ (9.6)

**Feedback**:
The text describes equation 9.6 as 'to leading order in K,' yet it retains the correction term Σ_l J_kl m_l^1(t) inside the argument of H. Since u_k^∞(t) is O(√K) (from equation 9.2) and Σ_l J_kl m_l^1(t) is O(1) (because J_kl ~ 1/√K and m_l^1 ~ O(1)), including the O(1) correction alongside the O(√K) leading term is a 1/√K perturbative correction, not a leading-order term. A genuine leading-order equation would drop this correction entirely, yielding an identity. The correction term must be retained to determine m_k^1, but then the equation is a first-order perturbative equation. It would be helpful to clarify that equation 9.6 is the O(1/√K) perturbative equation obtained by expanding H to first order in m_k^1/√K, rather than labeling it 'leading order in K.'

---

### 21. Net Input Scaling Factor Stated Incorrectly in Section 10.2

**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**:
From equations 4.3–4.4, the balanced net input u_E is O(1) (it must be finite to produce finite firing rates via H). The external input is Em_0√K = O(√K). Therefore the ratio u_E/(Em_0√K) = O(1/√K), not O(1/(m_0√K)). Including m_0 in the denominator makes the ratio O(1/(m_0√K)), which has units of 1/rate and is not dimensionless in the natural scaling of the model. It would be helpful to rewrite 'smaller than the external input by a factor of the order of $1/(m_0\sqrt{K})$' as 'smaller than the external input by a factor of the order of $1/\sqrt{K}$.'

---

### 22. Stationarity Assumption in Autocorrelation Framework Not Flagged as Inapplicable During Tracking

**Status**: [Pending]

**Quote**:
> Here the averaging is also over absolute time $t$.

**Feedback**:
Equation A.11 defines F_k as the expectation of Θ(u_k^i(t))Θ(u_k^i(t+t′+τ)) averaged over absolute time t, which is a stationarity assumption: joint statistics depend only on the lag, not on t. This is valid at the balanced fixed point where m_k is constant. However, Section 9 studies time-varying m_0(t) where m_k(t) is explicitly non-stationary (ramping or oscillating). The autocorrelation framework of Section 5.2 is implicitly carried over to interpret firing irregularity during tracking, but the self-consistent integral equation 5.17 no longer strictly applies in that regime. The paper does not flag this limitation anywhere in Appendix A.2 or Section 9. It would be helpful to add a parenthetical such as '(valid under the stationarity assumption that m_k is constant, as holds at the balanced fixed point)' and a cross-reference noting that this assumption is not satisfied during the tracking regime of Section 9.

---

### 23. Equation 7.6 Approximation q_k ≈ H(−u_k/Δ) Does Not Follow Straightforwardly from Dropping α_k and β_k

**Status**: [Pending]

**Quote**:
> For fixed $\Delta$, if the mean rates become sufficiently low so that $m_k \ll \Delta$, the intrinsic variances $\alpha_k$ and $\beta_k$ can be neglected compared with $\Delta$; hence one obtains
> 
> $$
> m _ {k} \approx q _ {k} \approx H \left(\frac {- u _ {k}}{\Delta}\right). \tag {7.6}
> $$

**Feedback**:
Equation 7.5 gives q_k = ∫Dx [H((−u_k − √(Δ²+β_k) x)/√(α_k−β_k))]². When α_k, β_k ≪ Δ², the ratio √(Δ²+β_k)/√(α_k−β_k) → ∞, so H((−u_k − √(Δ²+β_k) x)/√(α_k−β_k)) → Θ(−u_k − √(Δ²+β_k) x) → Θ(−x) for u_k < 0. The integral ∫Dx [Θ(−x)]² = 1/2, which does not equal [H(−u_k/Δ)]² in general. The claimed approximation q_k ≈ H(−u_k/Δ) therefore requires a more careful argument than simply neglecting α_k and β_k. It would be helpful to add an explicit derivation showing how equation 7.5 reduces to [H(−u_k/Δ)]² in the stated limit, or to replace equation 7.6 with the correct limiting expression.

---

### 24. Comparison with Experimental Rate Distribution Conflates Truncated Empirical and Untruncated Theoretical Distributions

**Status**: [Pending]

**Quote**:
> The average rate (of the neurons that showed any activity during the time of measurement) was 15.8 Hz. The observed histogram has a distinct unimodal skewed shape with a tail extending up to 80 Hz. These results are consistent with the theoretical predictions of Figure 11B.

**Feedback**:
The theoretical rate distribution ρ_k(m) in equations 7.12 and 7.17 is defined for all m ∈ (m_−, m_+) including very small m, whereas the experimental histogram from Abeles et al. (1988) is conditioned on neurons 'that showed any activity during the time of measurement,' truncating the distribution at m = 0. This conditioning removes the silent neurons, which is precisely the regime (m near 0) where the theoretical predictions differ most between the Gaussian and bounded threshold cases. The claimed consistency between Figure 11B and Figure 11C therefore rests on a comparison between a truncated empirical distribution and an untruncated theoretical one. It would be helpful to acknowledge that the relevant theoretical prediction to compare is the conditional distribution ρ_k(m | m > 0), not the full ρ_k(m), since this conditioning affects the apparent shape and mean.

---

### 25. Multinomial Factorization in Equation A.13 Treats n_2 and n_3 as Independent Without Justification

**Status**: [Pending]

**Quote**:
> $$
> p_l(n_1, n_2, n_3) = \frac{(q_l K)^{n_1}}{n_1!} \frac{((m_l - q_l) K)^{n_2 + n_3}}{n_2! n_3!} e^{-(2m_l - q_l)K}. \tag{A.13}
> $$

**Feedback**:
The factor ((m_l−q_l)K)^{n_2+n_3}/(n_2! n_3!) is the product of two independent Poisson distributions for n_2 and n_3, each with mean (m_l−q_l)K. This factorization is valid only if 'active at time t but not t+t′+τ' and 'active at t+t′+τ but not t' are independent events for a given cell. However, both events are constrained by the same cell's autocorrelation q_l(τ), so they are not independent in general. The correct joint distribution for a single cell's state at two times has three outcomes with probabilities (q_l, m_l−q_l, m_l−q_l, 1−2m_l+q_l), yielding a trinomial over K cells, not a product of two independent Poissons. It would be helpful to rewrite equation A.13 as a trinomial (or bivariate Poisson) distribution and add a sentence justifying why n_2 and n_3 are treated as independent in the large-K limit.

---
