# Chaotic Balanced State in a Model of Cortical Circuits

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

---

## Overall Feedback

Here are some overall reactions to the document.

**Divergent Lyapunov exponent and non-standard chaos characterization**

Section 8 characterizes the balanced state as chaotic with an 'infinitely large' Lyapunov exponent λ_L (Eq. 8.9), where trajectory separation grows algebraically as D_k(t) ~ t² rather than exponentially. This divergence arises from the discontinuous Heaviside update rule (Eq. 2.1) rather than conventional exponential sensitivity to initial conditions. The paper frames results within deterministic chaos theory, yet the mathematical behavior represents a distinct form of sensitive dependence specific to discontinuous binary networks. It would be helpful to clarify whether this constitutes chaos in the conventional dynamical systems sense—perhaps by computing the Lyapunov spectrum for a continuous sigmoidal approximation—or to explicitly characterize this as a distinct class of discrete-state dynamics, discussing implications for extending the model to continuous-valued neurons.

**Gaussian approximation error for finite-K Poisson input statistics**

The mean-field theory approximates Poisson-distributed synaptic input counts (Eq. A.6) with a Gaussian distribution (Eq. A.7) to derive the population activation function. In the balanced state, net input cancels to O(1) precision (Eqs. 4.1–4.2), making firing rates determined by tail probabilities near threshold. For finite K (e.g., K=1000 as simulated), the Gaussian approximation systematically underestimates Poisson tail probabilities, particularly in the low-rate regime (m_k ≪ 1) where the theory is applied. This introduces unquantified errors into fixed-point rates (Eq. 3.7), stability boundaries (Eq. 6.3), and rate distributions ρ_k(m) (Eq. 5.12) where rare events dominate. It would be helpful to derive rigorous error bounds for finite K or to evaluate exact Poisson sums for the parameter regimes used in the figures to verify that the approximation does not qualitatively alter the balanced state's existence.

**Neglect of synaptic dynamics in fast tracking and stability analysis**

The model assumes instantaneous synaptic transmission (Eq. 2.2), underlying the central claim that networks track time-dependent inputs on timescales of order 1/√K (Section 9), much faster than single-neuron time constants τ_k. However, biological synapses possess finite rise and fall times (typically 1–10 ms) that introduce additional low-pass filtering. If synaptic time constant τ_s exceeds 1/√K (likely for large K), network dynamics become limited by synaptic filtering rather than the balanced mechanism, potentially invalidating the fast-tracking result. Furthermore, the stability analysis (Section 6) does not account for delays or temporal correlations from synaptic dynamics. It would be helpful to demonstrate that the balanced fixed point remains stable and tracking capability persists when mean-field equations include realistic synaptic filter dynamics, or to characterize the breakdown when τ_s becomes dominant.

**Unverified mean-field independence assumption and finite-size correlations**

The mean-field derivation assumes input activities are uncorrelated (Appendix A), requiring K ≪ N_k and yielding Gaussian input statistics. However, the chaotic state (Section 8) exhibits sensitive dependence that may dynamically generate pairwise correlations, and finite K/N ratios (biologically ~0.1) introduce O(K/N) correlations from shared inputs that modify synaptic input variance (Eq. 3.6). The paper acknowledges that 'the distinction between full and sparse connectivity may be problematic' and that finite networks exhibit residual synchrony, but does not quantify these effects. It would be helpful to calculate leading-order corrections in K/N and 1/√K to the mean-field equations, explicitly compute correlation coefficients between neurons sharing common inputs to verify they vanish as K→∞, and establish criteria for network sizes where correlations remain below biological detection thresholds.

**Inconsistency between deterministic updates and second-order statistics**

Appendix B shows deterministic and stochastic update rules yield identical mean-field equations for population rates (Eq. B.4), but the derivation of temporal autocorrelations (Section 5.2, Eq. 5.17) and delayed correlations (Appendix A.2, Eq. A.10) assumes Poisson statistics for update intervals. The integral over exponential distributions in Eq. 5.17 does not apply to deterministic models with fixed intervals and uniform time-since-update distributions. This creates ambiguity for the chaos analysis (Section 8), which relies on deterministic trajectory divergence and is not well-defined for the stochastic Poisson model where random update times introduce external variability. It would be helpful to verify autocorrelation structure for the deterministic case explicitly, clarify whether Lyapunov exponent calculations apply only to deterministic updates, and demonstrate that the stochastic model exhibits chaotic sensitivity to initial conditions with identical random seeds.

**Finite connectivity index ($K$) violates asymptotic scaling assumptions in low-rate regime**

**(a) Assumption:** The balanced state theory assumes the limit $K \to \infty$ such that the net synaptic inputs $u_k$ remain $O(1)$ (equations 4.1–4.2). This yields the prediction that population rates scale linearly with external drive, $m_k = A_k m_0$ (equations 4.3–4.4), with corrections vanishing as $O(1/\sqrt{K})$.

**(b) Violation:** The numerical simulations use a finite connectivity index $K = 1000$ (Figures 3, 6, 8, 13, 16). When the external drive $m_0$ becomes comparable to $1/\sqrt{K} \approx 0.032$, the $O(1/\sqrt{K})$ corrections to the balance conditions become significant. As shown in Figure 3 (thick vs. thin lines), the population rates deviate substantially from the asymptotic linear prediction when $m_0$ is small, exhibiting nonlinear thresholding behavior.

**(c) Consequence:** In this regime, the network activity is no longer accurately described by the simple linear balanced state, and the strong nonlinearity of single-neuron dynamics dominates (Section 4.2). This limits the validity of the analytical predictions for rate distributions and autocorrelations in the low-activity regime.

**(d) Suggested fix:** It would be helpful to employ the finite-$K$ equations (equations 4.17–4.18) provided in Section 4.2 for quantitative comparisons when $m_0 < 0.1$, or to restrict the asymptotic theory's application to parameter regimes where $m_0 \gg 1/\sqrt{K}$.

**Gaussian input statistics assumption may fail for sparse activity in finite networks**

**(a) Assumption:** The mean-field derivation assumes that synaptic inputs obey Gaussian statistics (equations 3.3–3.6) based on the Central Limit Theorem for large $K$, with variance $\alpha_k = \sum_l J_{kl}^2 m_l$ (equation 3.6).

**(b) Violation:** For the simulated value $K = 1000$ and low activity rates ($m_k \approx 0.01$ as used in Figure 4), the number of active presynaptic inputs follows a Poisson-binomial distribution with mean $Km_k = 10$ and standard deviation $\approx 3.16$. The skewness of this distribution ($\approx 0.32$) creates non-Gaussian tails that are not captured by the Gaussian approximation.

**(c) Consequence:** Because the firing probability depends on the tail of the input distribution above threshold (via the error function $H$), the Gaussian approximation may introduce quantitative errors in the calculated rate distributions $\rho_k(m)$ (Figure 4) and interspike interval statistics (Figure 7), particularly for neurons operating near the threshold boundary.

**(d) Suggested fix:** It would be helpful to verify the robustness of the results using exact binomial or Poisson statistics for the synaptic input counts rather than the Gaussian approximation, especially for simulations with $Km_k < 100$. Alternatively, increasing $K$ to $10^4$ or higher would strengthen the validity of the CLT-based derivation for low-rate regimes.

**Status**: [Pending]

---

## Detailed Comments (19)

### 1. Connection probability scaling with postsynaptic population size

**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 $k,l=1,2$.

**Feedback**:
It would be helpful to verify the connection probability scaling. If $N_l$ denotes the size of the presynaptic population $l$, the probability that a specific presynaptic neuron connects to a specific postsynaptic neuron should be $K/N_l$ to ensure that each postsynaptic neuron receives exactly $K$ inputs from population $l$ in expectation. With the stated probability $K/N_k$, the expected number of inputs from population $l$ to a neuron in population $k$ is $N_l \times (K/N_k) = K(N_l/N_k)$, which equals $K$ only if $N_1 = N_2$. Readers might note that this affects the $O(1/\sqrt{K})$ synaptic scaling and the resulting mean-field input statistics.

---

### 2. Mismatched neuron indices in state update equation

**Status**: [Pending]

**Quote**:
> $\sigma_{k}^{j}(t)=\Theta(u_{k}^{i}(t)),$ (2.1)

**Feedback**:
It would be helpful to clarify how the indices $i$ and $j$ relate in equation (2.1). The left-hand side $\sigma_{k}^{j}$ denotes the state of neuron $j$, while the right-hand side $u_{k}^{i}$ represents the total synaptic input to neuron $i$ as defined in equation (2.2). For the update rule to describe the state transition of the updating neuron, the indices must refer to the same cell. Readers might note that replacing $u_{k}^{i}$ with $u_{k}^{j}$ would maintain consistency with the left-hand side.

---

### 3. Mean synaptic weight averaging over total network size

**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**:
Readers might note that the average connectivity weight should scale with the presynaptic population size $N_l$, not the total network size $N$. In sparse connectivity, each neuron receives $K$ connections from population $l$ of size $N_l$, giving connection probability $K/N_l$. With synaptic weights scaled as $J_{kl}/\sqrt{K}$, the quenched average is $[J_{kl}^{ij}] = (J_{kl}/\sqrt{K}) \cdot (K/N_l) = J_{kl}\sqrt{K}/N_l$. Substituting this into equation 3.11 yields $\sum_{j=1}^{N_l} (J_{kl}\sqrt{K}/N_l) m_l = \sqrt{K}J_{kl}m_l$, matching equation 3.5 only if $N_l$ is used. It would be helpful to replace $J_{kl}\sqrt{K}/N$ with $J_{kl}\sqrt{K}/N_l$.

---

### 4. Factor of K in second moment calculation of synaptic input

**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**:
It would be helpful to verify the factor in the second moment expression. With weights $J_{kl}/\sqrt{K}$ and connection probability $K/N_l$, the second moment of the weight is $[(J_{kl}^{ij})^2] = (J_{kl}^2/K) \cdot (K/N_l) = J_{kl}^2/N_l$. Thus $[(J_{kl}^{ij}\sigma_l^j)^2] = J_{kl}^2 m_l/N_l$ (not $J_{kl}^2 m_l K/N$). Summing over $N_l$ presynaptic neurons gives $\sum_{j=1}^{N_l} J_{kl}^2 m_l/N_l = J_{kl}^2 m_l$, which correctly yields equation 3.6 as $\alpha_k = \sum_l J_{kl}^2 m_l$. The factor $K/N$ in the text would incorrectly produce a variance of order $K$ after summation. It would be helpful to replace $[(J_{kl}^{ij}\sigma_{l}^{j}(t))^{2}]=J_{kl}^{2}m_{l}K/N$ with $[(J_{kl}^{ij}\sigma_{l}^{j}(t))^{2}]=J_{kl}^{2}m_{l}/N_{l}$.

---

### 5. Double summation notation in variance expression

**Status**: [Pending]

**Quote**:
> e such correlations are weak in the large $N$ limit. Similarly, the variance $\alpha_{k}$ of the input is
> 
> $\alpha_{k}(t)=[(\delta u_{k}^{i}(t))^{2}]=\sum_{l,l^{\prime}=1}^{2}\,\sum_{j,j^{\prime}=1}^{N_{l}}[\,(\,\delta(J_{kl}^{ij}\sigma_{l}^{j}(t))\,)^{2}\,],$ (3.12)
> 
> where $\delta X\equiv X-[X]$. Obse

**Feedback**:
It would be helpful to clarify the summation structure in equation (3.12). The variance of the total synaptic input decomposes as a sum of variances from each presynaptic source under the mean-field assumption of uncorrelated activities. The expression should contain a single sum over the presynaptic population index $l$ and neuron index $j$. The current notation suggests a double sum over $l, l'$ and $j, j'$, and further confuses the notation by having $j'$ sum to $N_l$ rather than $N_{l'}$. The term inside the expectation depends only on indices $l$ and $j$, confirming this is meant to be a single sum. It would be helpful to replace $\sum_{l,l^{\prime}=1}^{2}\,\sum_{j,j^{\prime}=1}^{N_{l}}$ with $\sum_{l=1}^{2}\,\sum_{j=1}^{N_{l}}$ to correctly represent the variance calculation.

---

### 6. Undefined coupling ratio $J_k$ in global stability equation

**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**:
It would be helpful to define the notation $J_k$ appearing in equation (6.4). Readers might expect the argument of the Heaviside function to involve the difference of excitatory and inhibitory synaptic inputs, proportional to $J_{kE}\delta m_{E}-J_{kI}\delta m_{I}$. The notation $J_{k}$ appears to represent the ratio $J_{kI}/J_{kE}$, but this is not defined in this section. Without this definition, the dimensional consistency of equation 6.4 and the subsequent derivation of the global stability boundary $\tau_{G}$ (equation 6.6) are difficult to verify. It would be helpful to add a sentence immediately before equation 6.4 defining $J_{k}\equiv J_{kI}/J_{kE}$ as the ratio of inhibitory to excitatory coupling strengths for population $k$.

---

### 7. Algebraic scaling in low-rate threshold equation

**Status**: [Pending]

**Quote**:
> on for $u_{k}$ is
> 
> $$
> u _ {k} + \Delta / 2 = O (\sqrt {m _ {k}}). \tag {7.10}
> $$<!-- 

**Feedback**:
It would be helpful to verify the scaling relation for the gap between mean synaptic input and the minimal threshold. For a uniform threshold distribution on $[-\Delta/2, \Delta/2]$ and low rates $m_k \ll 1$, the population rate scales linearly with the gap $\epsilon = u_k + \Delta/2$ when $\epsilon \ll \sqrt{\alpha_k}$. Specifically, $m_k \approx \epsilon/(2\Delta\sqrt{\alpha_k})$, which implies $\epsilon = 2\Delta\sqrt{\alpha_k} \cdot m_k = O(m_k)$, not $O(\sqrt{m_k})$. The square-root scaling would require $H(x) \sim x^{-2}$ behavior, which does not occur for the error function. It would be helpful to replace $O(\sqrt{m_k})$ with $O(m_k)$ to correctly represent the linear proportionality.

---

### 8. Dimensional inconsistency in rate distribution width scaling

**Status**: [Pending]

**Quote**:
>  of Cortical Circuits
> 
> $\sqrt{q_{k}}$, where
> 
> $$
> q_k \propto \Delta \alpha_k^{3/2}. \tag{7.11}
> $$
> 
> Thi

**Feedback**:
It would be helpful to verify the scaling relation in equation (7.11) using dimensional analysis. The quantity $q_k$ is the mean squared firing rate, which is dimensionless (rate squared, with rates normalized to maximum 1). The right-hand side $\Delta \alpha_k^{3/2}$ has dimensions: $\Delta$ is threshold standard deviation (current), and $\alpha_k$ is variance (current squared), so $\alpha_k^{3/2}$ is current cubed. Thus $\Delta \alpha_k^{3/2}$ has dimensions of current to the fourth power, not dimensionless. From equations 7.12-7.14, the characteristic rates scale as $m_+ \propto \Delta\sqrt{\alpha_k}$, suggesting the width of the rate distribution scales as $\Delta\sqrt{\alpha_k}$ (with logarithmic corrections), and thus $q_k$ should scale as $\Delta^2 \alpha_k$. It would be helpful to verify the derivation and correct the exponent to ensure dimensional consistency.

---

### 9. Prefactor dependence in rate distribution equation

**Status**: [Pending]

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

**Feedback**:
It would be helpful to verify the prefactor in equation (7.12). From equation 7.17 with a uniform distribution $P(\theta) = 1/\Delta$, the distribution is proportional to $P(-\sqrt{\alpha_k}(\sqrt{2\log(1/m)}-\tilde{h}_k))/(m\sqrt{\log(1/m)})$. For the relevant range of rates, $P = 1/\Delta$, yielding a prefactor of $1/\Delta$ (with numerical constants from the asymptotic expansion of $H$). The factor $\sqrt{\alpha_k/2}$ in the numerator of equation 7.12 introduces an additional dependence on the input variance that does not arise from the Jacobian transformation between $\theta$ and $m$. It would be helpful to verify the derivation and remove the $\sqrt{\alpha_k}$ dependence from the prefactor, replacing $\sqrt{\alpha_k / 2 \Delta^2}$ with $1/(\Delta\sqrt{2})$ or the appropriate constant.

---

### 10. Proportionality in fast tracking time scale derivation

**Status**: [Pending]

**Quote**:
> 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**:
It would be helpful to verify the time scale expression in the fast tracking analysis. From the differential relation $\delta m_k \approx \tau_k^{-1} \Delta P_k \, dt$, the time required for a change $\delta m_k = A_k \delta m_0$ is obtained by integration: $\Delta t \approx \tau_k \delta m_k / \Delta P_k = \tau_k A_k \delta m_0 / \Delta P_k$. Thus the time scale is proportional to $\delta m_0$, not inversely proportional. Since the text assumes $\delta m_0 \sim 1/\sqrt{K}$ (stating that $\delta m_0 \sqrt{K}$ is of order 1), the correct expression $\tau_k \delta m_0$ yields $\tau_k/\sqrt{K}$, which matches the final approximation given. However, the intermediate formula $\tau_k / \delta m_0$ would incorrectly give $\tau_k \sqrt{K}$. It would be helpful to replace $\tau_k / \delta m_0$ with $\tau_k \delta m_0$ to correctly represent the linear proportionality.

---

### 11. Internal contradiction regarding fast tracking and step response

**Status**: [Pending]

**Quote**:
>  is capable of “fast switching” in response to a fast change in the external stimulus. This may be related to the fast tracking predicted in our model. The fact that our model does not respond quickly to a sudden switching of the stimulus (see Figure 13) is probably a result of the dynamics of binary neurons. However,<!-- PAGE BREAK -->C. van Vreeswijk and H. Sompolinsky
> 
> the switching time constants observed in Tsodyks and Sejnowski (1995) is of the same order as the single-cell integration time constant, while the fast tracking should occur on a much shorter time constant. In recent numerical simulations of integrate-and-fire networks, Amit and Brunel (1997b) showed that the strength of the average cross-correlations decreases as $N$ increases (keeping the connectivity index constant). However, they do not show whether as $N$ 

**Feedback**:
It would be helpful to reconcile the apparent contradiction regarding the network's dynamic response capabilities. Section 9 predicts that the network tracks time-dependent inputs on timescales of order $1/\sqrt{K}$, which is much shorter than single-neuron integration times. However, the text explicitly states that 'our model does not respond quickly to a sudden switching of the stimulus' (referencing Figure 13), attributing this limitation to binary neuron dynamics. If the model cannot respond quickly to sudden switching, it cannot exhibit the predicted fast tracking on timescales much shorter than the single-cell constant. Furthermore, the text suggests Tsodyks and Sejnowski's 'fast switching' (which occurs on single-cell timescales) may be related to the authors' fast tracking (which should be much faster), yet these operate on fundamentally different timescales. It would be helpful to clarify the specific conditions under which fast tracking is achieved versus when it fails, or to reconcile these claims.

---

### 12. Experimental external input strength relative to model scaling

**Status**: [Pending]

**Quote**:
> Recent experimental findings of Ferster, Chung, and Wheat (1996) in cat primary visual cortex suggest that the input from the lateral geniculate nucleus (LGN) to layer 4 cortical cells are in fact a fraction of the net input. Stratford, Tarczy-Hornoch, Martin, Bannister, and Jack (1996) show that the total strength of the LGN synapses is about 2.5 to 3 times smaller than the total strength of the excitatory feedback synapses from layer 4 cells; however, this study does not measure the strength of the feedback from the inhibitory interneurons, so it does not allow for the estimation of the net feedback.

**Feedback**:
It would be helpful to reconcile the cited experimental findings with the model's scaling assumptions. The model assumes the balanced state emerges when external input is $O(1)$ while recurrent synaptic inputs scale as $O(\sqrt{K})$ (Section 2, assumption that total excitatory feedback current is large compared to threshold, implying $J \sim 1/\sqrt{K}$). For the simulated $K=1000$, this requires external input to be approximately $1/\sqrt{1000} \approx 3\%$ of the excitatory recurrent input. However, the cited finding that LGN input is only $1/2.5$ to $1/3$ (40–30\%) of excitatory feedback exceeds the model's scaling assumption by an order of magnitude. If external input were truly $O(\sqrt{K})$ relative to recurrent excitation, it would dominate the $O(1)$ net input and destroy the balanced cancellation mechanism. It would be helpful to explicitly reconcile this discrepancy by noting that the effective biological $K$ may be $O(10)$ rather than $O(1000)$, or by demonstrating that the balanced state persists when external input is $O(1/\sqrt{K})$ times the total recurrent input rather than the excitatory component alone.

---

### 13. Logical consistency of synaptic depression and fast tracking timescales

**Status**: [Pending]

**Quote**:
> If the input is suddenly increased by a small amount, the network rates increase to the rate the network would have in equilibrium if the synaptic strengths were not changed in a time of order $1/\sqrt{K}$ and then, on a much slower time scale, the rates decrease due to the change in the synaptic strengths.

**Feedback**:
It would be helpful to clarify the logical consistency of the fast tracking claim in the presence of synaptic depression. The paper establishes that the balanced state's fast tracking capability relies on instantaneous synaptic transmission (Eq. 2.2, Section 9). However, the text here introduces synaptic depression/facilitation dynamics that operate 'on a time scale that is as slow as or slower than the membrane time constant.' If synaptic dynamics are slow (comparable to $\tau_k$), the initial rate jump cannot occur on the fast timescale $1/\sqrt{K}$, because the effective synaptic strengths governing the response are themselves changing dynamically. The network would need to equilibrate to quasi-static effective weights, not to 'the rate the network would have in equilibrium if the synaptic strengths were not changed.' It would be helpful to clarify that the fast tracking result applies to the immediate electrical response of the membrane, while the subsequent network rate adjustment occurs on the slower synaptic dynamics timescale.

---

### 14. Unsupported claim regarding microscopic locking to inhomogeneous input

**Status**: [Pending]

**Quote**:
> On the other hand, preliminary analysis (van Vreeswijk &amp; Sompolinsky, 1998) shows that in the case of spatially inhomogeneous input fluctuations, the microscopic state of the network will tightly lock to the stimulus temporal variations.

**Feedback**:
It would be helpful to provide supporting evidence for this claim or to remove it. The statement references 'preliminary analysis' from an unpublished 1998 work without providing any supporting derivation or mechanism in the current paper. The claim contradicts the earlier finding in the same paragraph that 'the microscopic state is not tightly locked to the changing stimulus' for homogeneous input, yet no explanation is given for why spatial inhomogeneity would enable microscopic locking. The chaotic nature of the balanced state (Section 8) implies exponential sensitivity of microscopic trajectories to perturbations. For homogeneous input, the macroscopic rate tracks but microscopic states diverge. The claim that spatial inhomogeneity enables microscopic locking requires demonstrating that spatial structure suppresses this chaotic divergence, but no such mechanism is outlined. Without the referenced 1998 work being available to readers, this claim cannot be verified.

---

### 15. Description of update time integral in autocorrelation derivation

**Status**: [Pending]

**Quote**:
> -t'/\tau_{k}) F_{k}(m_{E}, m_{I}; q_{1}(t' + \tau), q_{2}(t' + \tau)), \tag{A.10}
> $$
> 
> where
> 
> $$
> F_{k} = \langle[\Theta(u_{k}^{i}(t)) \Theta(u_{k}^{i}(t + t' + \tau))]\rangle. \tag{A.11}
> $$
> 
> Here the averaging is also over absolute time $t$. The integral over time in the r.h.s. of equation A.10 takes into account the correlation between the inputs<!-- PAGE BREAK -->C. van Vreeswijk and H. Sompolinsky
> 
> to a cell that updates its state at time $t + \tau$ and its state at the last update before time $t + \tau$, which we assume occurred at time $t - t'$.

**Feedback**:
It would be helpful to clarify the description of the integral in equation (A.10). The text describes the integral over $t'$ as representing the last update before $t+\tau$ occurring at $t-t'$, but this contradicts the mathematical structure of the derivation. In the derivation, $q_k(\tau) = \langle s_i(t)s_i(t+\tau)\rangle$ satisfies the differential equation $\tau_k \frac{dq_k}{d\tau} = -q_k(\tau) + \langle s_i(t)\Theta(u_k(t+\tau))\rangle$, where the second term requires averaging over the last update time of neuron $i$ at time $t$ (not $t+\tau$). If the last update before $t$ occurred at $t-t'$, then $s_i(t) = \Theta(u_k(t-t'))$, and by stationarity $\langle \Theta(u_k(t-t'))\Theta(u_k(t+\tau))\rangle = F_k(m_E, m_I; q_1(t'+\tau), q_2(t'+\tau))$. Thus the integral averages over the history of the neuron at the earlier time $t$, not the later time $t+\tau$. It would be helpful to rewrite the sentence as: 'The integral over time in the r.h.s. of equation A.10 takes into account the correlation between the input to a cell at time $t+\tau$ and its state at the last update before time $t$, which we assume occurred at time $t-t'$'.

---

### 16. Product notation scope in equation A.12

**Status**: [Pending]

**Quote**:
> $n_{2,l}$ and $n_{3,l}$, respectively), one can write
> 
> $$
> \begin{array}{l}
> F_k = \prod_{l=1,2} \sum_{n_{1,l}n_{2,l}n_{3,l}} p_l(n_{1,l}, n_{2,l}, n_{3,l}) \Theta \left( \sqrt{K} E_{k} m_{0} + \sum_l \frac{J_{kl}}{\sqrt{K}} (n_{1,l} + n_{2,l}) - \theta_k \right) \\
> \times \Theta \left( \sqrt{K} E_{k} m_{0} + \sum_l \frac{J_{kl}}{\sqrt{K}} (n_{1,l} + n_{3,l}) - \theta_k \right), \tag{A.12}
> \end{array}
> $$
> 
> where
> 
> $$
> p_l(n_1, n_2, n_3) = \frac{(q_l K)^{n_1}}{n_1!} \frac{((m_

**Feedback**:
It would be helpful to clarify the scope of the product notation in equation (A.12). The notation $\prod_{l=1,2} \sum_{n_{1,l}n_{2,l}n_{3,l}}$ placed before the probability $p_l$ and the Heaviside functions creates ambiguity regarding the scope of the product. Because the Heaviside functions contain $\sum_l$ (summing over populations $l=1,2$ inside the argument), they couple the variables from different populations. Therefore, the product cannot be distributed across the Heaviside functions as the notation might imply. The expression should explicitly group the probability product inside the summation or use a single sum over all configuration variables. It would be helpful to rewrite equation (A.12) with explicit parentheses: $$F_k = \sum_{\{n_{1,l},n_{2,l},n_{3,l}\}_{l=1,2}} \left( \prod_{l=1,2} p_l(n_{1,l}, n_{2,l}, n_{3,l}) \right) \Theta(\dots) \Theta(\dots)$$ to clarify that the product over $l$ applies only to the probabilities $p_l$.

---

### 17. Definition of connectivity parameter K

**Status**: [Pending]

**Quote**:
> tions. An essential ingredient of our model is the introduction of strong connections among the units. A cell is connected, on the average, to $K$ other cells, and $K$ is large. However, the gap between the threshold of the cell and its resting potential is only of the order of $\sqrt{K}$ excitatory 

**Feedback**:
It would be helpful to clarify whether $K$ refers to the total number of synaptic connections from both populations combined, or the number of connections per population (e.g., $K$ excitatory and $K$ inhibitory). In the standard balanced network architecture described in this paper, each neuron typically receives $O(K)$ inputs from the excitatory population and $O(K)$ from the inhibitory population separately. If $K$ represented total connectivity, the threshold scaling argument that follows would need adjustment, since the threshold is $O(1)$ while individual synaptic weights are $O(1/\sqrt{K})$, requiring $\sqrt{K}$ inputs to reach threshold. Readers might note that Section 2 clarifies that 'on average, $K$ excitatory and $K$ inhibitory neurons project to each neuron,' but it would be helpful to state this explicitly in the Introduction to avoid confusion.

---

### 18. Definition of order parameter $\beta_k$ in Section 7

**Status**: [Pending]

**Quote**:
> on,
> 
> $$
> m _ {k} = H \left(\frac {- u _ {k}}{\sqrt {\alpha_ {k} + \Delta^ {2}}}\right), \tag {7.4}
> $$
> 
> which determines $u_{k}$, and
> 
> $$
> q _ {k} = \int D x \left[ H \left(\frac {- u _ {k} - \sqrt {\Delta^2 + \beta_ {k}} x}{\sqrt {\alpha_ {k} - \beta_ {k}}}\right) \right] ^ {2}. \tag {7.5}
> $$
> 
> Now let us consider the limit of low mean rates, 

**Feedback**:
It would be helpful to define the parameter $\beta_k$ appearing in equation (7.5). From equations 5.2 and 3.10, $\beta_k = \sum_l J_{kl}^2 q_l$ represents the variance contribution from shared inputs (quenched disorder in the connectivity). However, readers cannot be expected to remember equation numbers from Section 3 when reading Section 7. It would be helpful to add a sentence immediately after equation 7.5 defining $\beta_k = q_E + J_k^2 q_I$ (for the appropriate populations) or referencing equation 3.10 to ensure clarity.

---

### 19. Conditions for fluctuation reduction by synaptic depression

**Status**: [Pending]

**Quote**:
> Another effect of synaptic depression to take into account is that the effect of a spike is decreased if it follows shortly after the preceding spike. This will decrease the fluctuations in the input, relative to the fluctuations in the activity of the presynaptic cells.

**Feedback**:
It would be helpful to qualify the conditions under which synaptic depression decreases input fluctuations. This claim requires careful examination, as the effect depends on the relative timescales. Synaptic depression with short-term dynamics introduces history-dependence: the effective synaptic strength depends on the interspike interval. For a Poisson presynaptic process with rate $m$, the variance of the depression-modulated input depends on the covariance structure of the synaptic efficacy process. If depression is strong, spikes in rapid succession are suppressed, which could either reduce input variance by dampening burst events, or increase effective variance by introducing temporal correlations in the efficacy. The claim that fluctuations 'decrease' is not universal—it depends on the depression time constant relative to the mean ISI. It would be helpful to rewrite 'This will decrease the fluctuations' as 'This may decrease the fluctuations when the depression time constant is short compared to the mean interspike interval, though the effect depends on the relative timescales' to accurately reflect the conditional nature of this claim.

---
