# Chaotic Balanced State in a Model of Cortical Circuits

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

---

## Overall Feedback

Here are some overall reactions to the document.

**Mean-Field Approximation Accuracy at Finite Connectivity**

The analytical results assume the limits N → ∞ and K → ∞ to derive Gaussian input statistics, yet numerical validations utilize a finite connectivity of K=1000. Readers might note that for sparse networks with strong coupling, finite-size correlations could deviate from the assumed Gaussian distribution, particularly at low firing rates where the condition m_k ≫ K^(-1) becomes marginal. It would be helpful to include a quantitative estimate of the deviation from Gaussianity for K=1000 or compare the theoretical rate distribution against histograms generated from multiple network realizations. Clarifying the lower bound on firing rates for valid mean-field theory would strengthen the applicability claims to biologically realistic cortical circuits.

**Characterization of Chaotic Dynamics and Lyapunov Exponents**

In Section 8, the paper characterizes the balanced state as chaotic and suggests the maximum Lyapunov exponent is 'infinitely large' due to the discreteness of the binary units. This definition diverges from standard dynamical systems theory where Lyapunov exponents are typically finite, which might confuse readers attempting to relate these results to continuous neural models. It would be helpful to clarify the mathematical definition of the exponent in the discrete limit or supplement this analysis with an alternative metric, such as the entropy rate or correlation dimension. Distinguishing between artifacts of the binary update rule and intrinsic network chaos would aid interpretation of the system's sensitivity to initial conditions.

**Biological Plausibility of Threshold Inhomogeneity**

Section 7 demonstrates that the balanced state remains temporally fluctuating with bounded threshold distributions but freezes with unbounded Gaussian distributions. However, the text does not provide a specific critical value for the ratio of threshold width to input variance that separates these regimes. Readers might note that the dynamical regime depends critically on the tail of the threshold distribution, which is not independently measured in the study. It would be helpful to derive or numerically identify this critical bound and discuss biological mechanisms that would enforce a bounded threshold distribution. Providing justification for the bounded assumption would strengthen the link to cortical physiology and ensure robustness of the asynchronous state.

**Stability Analysis and Synaptic Delay Effects**

The model formulation assumes instantaneous synaptic updates, neglecting transmission delays and finite synaptic time constants, which are known to affect network stability in cortical circuits. Additionally, the stability conditions depend on population rates that are functions of external input, yet the phase diagrams hold excitatory activity constant. Readers might note that omitting delays could significantly alter the stability regime, particularly for fast tracking claims where timing is critical. It would be helpful to extend the stability analysis to include a delay parameter and explicitly map the stable region as a function of external drive. This would ensure the balanced state remains robust across the full dynamic range of the network.

**Sensitivity to Parameter Scaling and Model Consistency**

The main results depend critically on synaptic strengths scaling as 1/√K, which differs from conventional 1/K scaling and may represent a narrow parameter regime. Furthermore, tracking comparisons between balanced and unbalanced networks use different neuronal transfer functions, which might influence the observed performance advantage. Readers might note that this raises questions about biological plausibility and whether the speed advantage is due to balance or unit nonlinearity. It would be helpful to expand the exploration of how balanced state properties degrade as the scaling exponent deviates from 1/2 and include a control simulation where the unbalanced network also uses binary units. Ensuring model consistency would clarify the source of the tracking advantage and the robustness of the mechanism.

**Finite Network Size vs. Thermodynamic Limit Assumption**

The mean-field theory is explicitly stated to be exact in the limit N → ∞ with 1 ≪ K ≪ N_k (Section 3, Equation 3.1). However, the numerical simulations use K=1000 (Figures 3, 6, 8, 13, 14, 16, 17) without specifying the actual network size N. Readers might note that without knowing N, it is not clear whether the condition K ≪ N is satisfied in the simulations. If N is only moderately larger than K (e.g., N=10,000 giving K/N=0.1), finite-size corrections could be substantial. The paper acknowledges this in Section 3 ('In reality, networks have a large fixed size and connectivity, so that the distinction between full and sparse connectivity may be problematic') but does not provide the actual N values used or demonstrate convergence as N increases. A concrete fix would be to report N explicitly and show that results are stable across different N values while maintaining K ≪ N.

**Critical Sensitivity to Threshold Distribution Shape**

Section 7 reveals that the balanced state's key property—temporal variability at low mean rates—depends critically on the assumed threshold distribution. With unbounded Gaussian threshold distributions, the state becomes frozen (q_k ≈ m_k, bimodal rate distribution, Figure 11A) at low rates. With bounded uniform distributions, temporal variability is maintained (unimodal skewed distribution, Figure 11B). This represents a qualitative difference in predictions based on an assumption that is not empirically justified. The paper states 'Real neuronal systems exhibit a substantial inhomogeneity in single neuron properties' but does not establish which distribution type is biologically appropriate. Since the main theoretical claims about irregular firing depend on maintaining temporal variability, this assumption sensitivity should be more prominently acknowledged. Readers might note that the comparison with experimental data (Figure 11C) supports the bounded distribution case, but this post-hoc validation does not establish that cortical neurons actually have bounded threshold distributions.

**Large External Input Requirement and Biological Plausibility**

The theory requires external inputs to scale as u_k^0 = E_k m_0 √K (Equation 2.3), meaning external input must be large compared to neuronal threshold (of order √K). Section 10.2 acknowledges this: 'the balanced state is robust only when the DC external input to the local network is large—of the same order as the local excitatory and inhibitory feedback, and much larger than the net synaptic input to a cell.' However, Section 10.4 notes experimental findings suggesting LGN input to cortical cells may be 'a fraction of the net input' rather than much larger. This creates tension between the theoretical requirement and available experimental evidence. The paper states 'Further experimental clarification of this issue is called for' but does not explore whether the balanced state mechanism can operate with more biologically realistic input magnitudes. A robustness check varying the external input scaling exponent (as briefly mentioned in Section 10.2 with K^(-α) scaling) would help establish whether the qualitative predictions hold under more plausible input conditions.

**Status**: [Pending]

---

## Detailed Comments (23)

### 1. Chaos Definition Requires Qualification for Discrete Systems

**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**:
Readers might note that Section 8 states 'chaos is technically inapplicable to a system with discrete degrees of freedom such as ours'. This creates tension with the abstract's definitive claim. It would be helpful to qualify this claim in the abstract to acknowledge the discrete-system limitation while maintaining the substantive finding about sensitivity.

---

### 2. Poissonian Statistics Claim Needs Additional Justification

**Status**: [Pending]

**Quote**:
> It is shown that the autocorrelations decay on a short time scale, yielding an approximate Poissonian temporal statistics. The activity levels of single cells are broadly distributed, and their distribution exhibits a skewed shape with a long power-law tail.

**Feedback**:
Readers might note that autocorrelation decay alone does not guarantee Poissonian statistics. The text mentions both 'approximate Poissonian temporal statistics' and 'skewed shape with a long power-law tail', which could appear contradictory. It would be helpful to clarify whether the Poissonian claim refers to individual spike trains or population-averaged statistics.

---

### 3. Threshold Scaling Claim Lacks Synaptic Strength Specification

**Status**: [Pending]

**Quote**:
> 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 inputs. Thus, the network will saturate unless a dynamically developed balance between the excitatory inputs and the inhibitory inputs to a cell emerges.

**Feedback**:
The claim states the threshold gap is 'of the order of $\sqrt{K}$ excitatory inputs' but does not specify the synaptic strength scaling. From Section 10.2, synaptic strengths scale as $J \sim 1/\sqrt{K}$. Without this, the saturation argument cannot be fully verified. It would be helpful to explicitly state the $1/\sqrt{K}$ synaptic scaling in this passage to clarify the scaling relationship.

---

### 4. Index Mismatch in Neuron Update Equation 2.1

**Status**: [Pending]

**Quote**:
> The network has an asynchronous dynamics where only one neuron updates its state at any given time. The updated state of the updating neuron at time $t$ is
> 
> $\sigma_{k}^{j}(t)=\Theta(u_{k}^{i}(t)),$ (2.1)

**Feedback**:
The left side of Equation 2.1 references neuron $j$ in population $k$, while the right side references the input $u_{k}^{i}$ to neuron $i$ in population $k$. This index inconsistency makes the equation mathematically ambiguous. Rewrite Equation 2.1 as '$\sigma_{k}^{i}(t)=\Theta(u_{k}^{i}(t)),$ (2.1)' to match the indexing convention used in Equation 2.2.

---

### 5. Firing Rate Formula Assumes State Independence

**Status**: [Pending]

**Quote**:
> However, if $m_{k}^{i}$, which we will call the activity rate, is small, the probability of two consecutive updates to the active state is low, and thus for small $m_{k}^{i}$, the activity rate and the firing rate are nearly equal. Indeed, if we assume that at each update the probability of being in the active state is $m_{k}^{i}$ (very nearly true in this model for low rates, as shown in section 5.3), the firing rate is given by $r_{k}^{i}=m_{k}^{i}(1-m_{k}^{i})/\tau_{k}$.

**Feedback**:
The formula assumes the probability of a 0→1 transition equals the product of state probabilities, which requires independence between consecutive update states. However, the deterministic update rule creates temporal correlations that violate this independence assumption. Add a clarification stating 'This approximation holds when temporal correlations between consecutive updates are negligible'.

---

### 6. Variance Equation 3.12 Has Index Mismatch in Summation

**Status**: [Pending]

**Quote**:
> tified since 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 $\delta X\equiv X-[X]$. Observing th

**Feedback**:
The left-hand side of equation 3.12 shows a double sum over indices $l,l'$ and $j,j'$, but the expression inside the brackets only contains single indices $l$ and $j$. This is mathematically inconsistent. For independent inputs, the correct variance should not have primed indices. Rewrite equation 3.12 to remove the primed indices in the summation to match the actual calculation.

---

### 7. Unclear Scaling of Mean Input u_k in Balanced State

**Status**: [Pending]

**Quote**:
> The quantities $u_{k}(t)$ and $\alpha_{k}(t)$ are
> 
> $$
> u _ {k} (t) = \sqrt {K} \left(\sum_ {l = 1} ^ {2} J _ {k l} m _ {l} (t) + E _ {k} m _ {0}\right) - \theta_ {k} \tag {3.5}
> $$

**Feedback**:
Equation 3.5 shows $u_k(t)$ scaling as $O(\sqrt{K})$ explicitly, but equation 3.3 requires $u_k/\sqrt{\alpha_k}$ to be $O(1)$ for nontrivial firing rates. In the balanced state, the $\sqrt{K}$ terms must cancel. Add after equation 3.5: 'In the balanced state, the $O(\sqrt{K})$ terms cancel, leaving $u_k = O(1)$, which requires $\theta_k = O(\sqrt{K})$ and $(\sum_{l} J_{kl}m_l + E_k m_0) = O(1/\sqrt{K})$.'

---

### 8. Equation 3.10 Notation J_k Ambiguous Without Definition

**Status**: [Pending]

**Quote**:
> The variance of the inputs is
> 
> $\alpha_{k}\,=\,m_{E}+J_{k}^{2}m_{I}.$ (3.10)

**Feedback**:
Equation 3.10 introduces $J_k$ without explicit definition in this section. Comparing to equation 3.6, we can infer $J_k$ represents $J_{kI}$ and $J_{kE}^2 = 1$. However, this specialization is not explained. Add after equation 3.10: 'where $J_k \equiv J_{kI}$ denotes the inhibitory coupling strength, and we have set $J_{kE} = 1$ for excitatory connections.'

---

### 9. Unclear Attribution of Linearity to Synaptic Summation

**Status**: [Pending]

**Quote**:
> Equations 4.3 and 4.4 imply that the network activity rates grow linearly with the external rate, $m_{k}=A_{k}m_{0}$, even though microscopic dynamics is highly nonlinear. This is because the network dynamically finds an operating point at which the net input in both populations is balanced. Thus, the linearity in the network rates reflects the linearity of the synaptic summation underlying our model.

**Feedback**:
The claim that linearity 'reflects the linearity of the synaptic summation' is potentially misleading. The binary neuron model has nonlinear thresholding dynamics, yet linear input-output relations emerge at the population level. This linearity is an emergent property of the balanced state mechanism. Rewrite the final sentence as: 'Thus, the linearity in the network rates emerges from the dynamic balance mechanism that cancels large opposing inputs, despite the nonlinear microscopic dynamics.'

---

### 10. Width Scaling Claim Inconsistent With Eq 5.14 Asymptotics

**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**:
Equation 5.14 states $q_{k}=m_{k}^{2}+O(m_{k}^{2}/|\log m_{k}|)$. The width (standard deviation) is $\sqrt{q_k - m_k^2}$. From Eq 5.14, this gives width $\sim O(m_k/\sqrt{|\log m_k|})$, not $(m_k)^{3/2}$. Rewrite as: 'Equation 5.14 implies that when the mean activity $m_{k}$ decreases, the width of the distribution is proportional to $m_k/\sqrt{|\log m_k|}$; it decreases slightly faster than the mean $m_{k}$ due to the logarithmic correction.'

---

### 11. Distribution Claim Contradicts Nonlinear Transformation in Eq 5.11

**Status**: [Pending]

**Quote**:
> Note, however, that according to equation 5.11 the distribution of the time-averaged inputs $u_{i}^{k}$ to the cells is gaussian for all values of $m_{k}$.

**Feedback**:
Equation 5.11 states $m_{k}^{i} = H \left(\frac{-u_{k} + \sqrt{\beta_{k}} x_{i}}{\sqrt{\alpha_{k} - \beta_{k}}}\right)$, where $H$ is a nonlinear threshold function. A nonlinear transformation of a Gaussian variable does not yield a Gaussian distribution. The text appears to conflate the distribution of inputs with the distribution of rates. Rewrite as: 'Note, however, that according to equation 5.11 the distribution of the time-averaged inputs $u_{i}^{k}$ to the cells is gaussian for all values of $m_{k}$, while the distribution of rates $m_k^i$ is obtained by applying the nonlinear function H to these Gaussian inputs.'

---

### 12. Missing Cross-Correlation Terms in Input Autocorrelation Equation 5.15

**Status**: [Pending]

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

**Feedback**:
Equation 5.15 expresses the input autocorrelation as a sum of excitatory and inhibitory contributions but omits cross-correlation terms between E and I populations. If u_i^k(t) = u_E(t) + J_k u_I(t), then the full expansion should include 2J_k⟨δu_E(t)δu_I(t+τ)⟩. Add text after equation 5.15 stating 'assuming negligible cross-correlations between excitatory and inhibitory populations' or include the cross-term with justification.

---

### 13. Refractoriness Claim Lacks Mechanistic Explanation in Binary Model

**Status**: [Pending]

**Quote**:
> This enhancement of short-time correlations reflect the refractoriness in the activities of the cells that project the cell.

**Feedback**:
The text claims enhanced short-time correlations reflect refractoriness, but the model uses binary units without explicit refractory periods. The refractoriness emerges from the update dynamics rather than being built into individual cells. Rewrite as: 'This enhancement of short-time correlations reflects effective refractoriness arising from the binary update dynamics, where cells cannot switch states between update times' to accurately describe the source of the correlation structure.

---

### 14. Eigenvalue Scaling Inconsistent With Balanced State Synaptic Scaling

**Status**: [Pending]

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

**Feedback**:
There is a mathematical inconsistency between the claimed eigenvalue scaling and the balanced state synaptic scaling. In the balanced state, synaptic strengths scale as $J_{kl} \sim 1/\sqrt{K}$, so $f_{kl} \sim 1/\sqrt{K}$. This means $\sqrt{K}f_{kl} \sim O(1)$, making the eigenvalues $\lambda \sim O(1)$, not $O(\sqrt{K})$. Rewrite the eigenvalue scaling claim as 'of order 1' to maintain consistency with the balanced state synaptic scaling.

---

### 15. Perturbation Equation Includes Baseline Rate Term

**Status**: [Pending]

**Quote**:
> In fact, since the perturbation destroys the balance between excitation and inhibition, $H(-u_{k}/\sqrt{\alpha_{k}})$ of equation 3.3 can be approximated by $\Theta(u_{k})$; hence the evolution of the perturbations is described by
> 
> $\tau_{k}\frac{d}{dt}\delta m_{k}(t)=-\delta m_{k}(t)+\Theta(\delta m_{E}-J_{k}\delta m_{I})-m_{k}.$ (6.4)

**Feedback**:
Equation 6.4 is described as describing 'the evolution of the perturbations' (δm_k), yet it contains the term -m_k (the baseline population rate). For a perturbation equation, all terms should describe deviations from the fixed point. Clarify whether δm_k represents the full rate or the deviation, and ensure equation 6.4 is consistent with this definition.

---

### 16. Figure 9 Caption Upper Bound Expression Unclear

**Status**: [Pending]

**Quote**:
> Figure 9: Critical time constants $\tau_{G}$ (solid line) and $\tau_{L}$ (dashed line) as a function of $I / E$. The external rate was adjusted to keep the excitatory activity level constant at $m_{E} = 0.1$. $I / E$ was varied from 0 to $(J_{I} - m_{E}) / (J_{E}

**Feedback**:
The upper bound expression (J_I - m_E) / (J_E - m_E) for the I/E ratio is unclear without additional context. If these are synaptic coupling strengths, then subtracting a rate m_E from a coupling strength J is dimensionally inconsistent. Either define J_I and J_E explicitly in this context, or provide the derivation showing why this particular upper bound ensures m_I remains in the valid range.

---

### 17. Inconsistent Gap Scaling in Bounded Threshold Analysis

**Status**: [Pending]

**Quote**:
> theta H \left(\frac {- u _ {k} + \theta}{\sqrt {\alpha_ {k}}}\right). \tag {7.9}
> $$
> 
> To assess the effect of $\Delta$, we analyze equation 7.9 in the low $m_{k}$ limit. In this case, the solution for $u_{k}$ is
> 
> $$
> u _ {k} + \Delta / 2 = O (\sqrt {m _ {k}}). \tag {7.10}
> $$

**Feedback**:
For Gaussian noise, the firing rate $m_k$ depends exponentially on the gap $\delta = u_k + \Delta/2$. To achieve $m_k \to 0$, the gap must scale as $\delta \sim \sqrt{\alpha_k \log(1/m_k)}$, not $O(\sqrt{m_k})$. The claimed scaling in Eq 7.10 implies a vanishing gap, which for Gaussian noise yields a finite non-zero rate. Rewrite Eq 7.10 to reflect the logarithmic scaling required for Gaussian tails.

---

### 18. Contradiction Between Gap Scaling and Rate Distribution

**Status**: [Pending]

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

**Feedback**:
The derivation of Eq 7.12 relies on inverting the Gaussian tail of the transfer function $H$, which assumes the input is far below threshold (large gap). This contradicts Eq 7.10, which claims the gap is $O(\sqrt{m_k})$ (small). The text cannot simultaneously hold that the gap is small (7.10) and that the rate distribution follows the Gaussian tail inversion (7.12). Clarify the asymptotic regime to ensure consistency.

---

### 19. Undefined Notation J_k in Gamma Equation

**Status**: [Pending]

**Quote**:
> $\gamma_{k}(t)=\sum_{l=1}^{2}(J_{kl})^{2}Q_{l}(t)=Q_{E}(t)+J_{k}^{2}Q_{I}(t).$ (8.6)

**Feedback**:
The transition from the summation form to the explicit form introduces notation J_k without definition. For the equality to hold, we need (J_{kE})² = 1 and (J_{kI})² = J_k². This normalization assumption should be stated explicitly. Rewrite equation 8.6 to clarify the notation and normalization assumption.

---

### 20. Incorrect Time Scale Formula in Tracking Derivation

**Status**: [Pending]

**Quote**:
> implying that $\delta m_k$ reaches the value $A_k \delta m_0$ in time of order $\tau_k / \delta m_0 \approx \tau_k / \sqrt{K}$; see the dotted line in Figure 13.

**Feedback**:
The expression for the time scale contains a mathematical inconsistency. The text states that $\delta m_0 \sqrt{K}$ is of order 1, which implies $\delta m_0 \sim 1/\sqrt{K}$. Consequently, the term $\tau_k / \delta m_0$ scales as $\tau_k \sqrt{K}$, contradicting the stated result of $\tau_k / \sqrt{K}$. Rewrite the quoted text as 'in time of order $\tau_k \delta m_0 \approx \tau_k / \sqrt{K}$' to correct the scaling error.

---

### 21. Internal Contradiction on Tracking Response Speed

**Status**: [Pending]

**Quote**:
> t 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

**Feedback**:
This passage states the model does not respond quickly to stimulus switching, attributing this to binary neuron dynamics. However, Section 10.5 states 'The analysis of tracking capabilities of our network shows that the macroscopic state of the network responds fast to changing input.' Rewrite to clarify the distinction between single-cell and population-level response times to resolve the apparent contradiction.

---

### 22. Figure 11B Reference Inconsistency

**Status**: [Pending]

**Quote**:
> Further experimental clarification of this issue is called for. Measurements of the distribution of time-averaged rates within a local population of neurons and the change in its shape when the overall level of response increases, similar to those of Figure 11B, would be an interesting test of the underlying statistical characteristics of the network spatiotemporal fluctuations.

**Feedback**:
The text claims that Figure 11B demonstrates the change in rate distribution shape as the overall response level increases. However, the description of Figure 11 in Section 7 indicates that panels 11A and 11B compare different threshold distributions rather than different response levels. Rewrite the reference to accurately reflect Figure 11B's content, such as 'similar to those in Figure 11 comparing threshold distributions.'

---

### 23. External Input Notation Inconsistent Between A.12 and A.14

**Status**: [Pending]

**Quote**:
> ose that are active only in one of the times ($n_{2,l}$ 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_

**Feedback**:
In equation A.12, the external input appears explicitly as $\sqrt{K} E_k m_0$, but in equation A.14 this term is replaced by $u_k$ without explanation. In the balanced state, the $\sqrt{K}$ terms should cancel, leaving $u_k$ as the order-1 residual net input. Rewrite the text between A.12 and A.14 to explicitly state: 'In the balanced state, the $\sqrt{K}$ terms cancel, and we define $u_k$ as the order-1 net input after this cancellation, consistent with equations 3.8 and 3.9.'

---
