# Chaotic Balanced State in a Model of Cortical Circuits

**Date**: 03/06/2026
**Domain**: natural_sciences/neuroscience
**Taxonomy**: academic/research_paper
**Filter**: Active comments

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## Overall Feedback

Here are some overall reactions to the document.

**Interpretation of Chaos in Discrete Dynamics**

Multiple reviewers note that characterizing the balanced state as chaotic via an infinitely large Lyapunov exponent (Eq 8.13) may conflate the sensitivity of discrete update rules with continuous-time chaos typical of biological neurons. Readers might note that standard definitions of chaos typically involve finite positive Lyapunov exponents in continuous systems. It would be helpful to clarify how this infinite exponent relates to the predictability horizon and whether it implies a specific type of instability distinct from conventional chaotic attractors.

**Validity of Gaussian Mean-Field Approximation**

The mean-field theory relies on the Central Limit Theorem to approximate synaptic inputs as Gaussian variables, which assumes large connectivity $K$. However, simulations use finite $K$ where corrections of order $1/\sqrt{K}$ may be relevant, particularly at low firing rates where the number of active inputs is small. It would be helpful to quantify the range of $K$ for which the Gaussian assumption remains accurate and explicitly calculate the kurtosis of the input distribution in the simulations to verify the assumption holds.

**Consistency of Stability Predictions with Simulations**

Section 6 derives local and global stability conditions in the large $K$ limit, but numerical simulations use finite $K$, creating a potential gap between theoretical boundaries and observed dynamics. Readers might note that the visual evidence for limit cycle amplitude scaling in Figure 10 is qualitative rather than quantitative. It would be helpful to plot the theoretical stability boundaries alongside simulation data and plot the amplitude of limit cycles against $K$ to confirm the theoretical scaling laws derived in Section 6.3.

**Biological Plausibility of External Drive Scaling**

The model assumes external inputs scale with $\sqrt{K}$ to maintain the balanced state, contrasting with weak synapse scenarios where inputs scale as $1/K$. Section 10.4 acknowledges that experimental data suggests external inputs may be smaller than feedback currents, raising questions about the metabolic or anatomical implications of the required drive strength. It would be helpful to analyze the robustness of the balanced state if the external input scales as $O(1)$ and explicitly map the parameter space where the balanced state persists under weaker external drive.

**Sensitivity to Threshold Distribution Tails**

Section 7 predicts that the nature of the balanced state depends critically on the tail behavior of the threshold distribution, distinguishing between frozen states for Gaussian thresholds and fluctuating states for bounded distributions. It would be helpful to provide a more detailed analysis of how intermediate distributions, which may better reflect biological data, interpolate between these two limiting behaviors to ensure the model's applicability to real cortical circuits.

**Numerical Solution of Autocorrelation Equations**

Equation 5.17 presents a complex integral equation for time-delayed autocorrelations, yet the paper does not detail the numerical method or convergence criteria used to solve it. Given the integral depends on the solution itself at future times, the numerical stability is non-trivial. It would be helpful to describe the numerical integration scheme used and verify the solution against direct simulation of the autocorrelation function to ensure the theoretical curves accurately reflect the network dynamics.

**Finite Connectivity and Gaussian Input Approximation**

The mean-field theory assumes the synaptic input distribution is Gaussian in the limit of large connectivity K. The simulations use a finite K=1000. While the authors note that finite K corrections are primarily quantitative, the firing rates depend on the tails of the input distribution. It would be helpful to quantify the deviation from Gaussian statistics in the tails for K=1000 to ensure the rate distribution predictions are robust.

**Experimental Data Selection Criteria**

The model predicts the rate distribution for the entire neuronal population. The experimental data compared in Figure 11C is derived from neurons showing "significant response" during specific events. This selection criterion may exclude silent or low-rate neurons, potentially truncating the distribution. Readers might note that a direct comparison requires ensuring the experimental sample includes the full range of neuronal activities, including those with low or zero rates.

**Status**: [Pending]

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## Detailed Comments (25)

### 1. Index mismatch in state update equation

**Status**: [Pending]

**Quote**:
> neuron at time $t$ is
> 
> $\sigma_{k}^{j}(t)=\Theta(u_{k}^{i}(t)),$ (1)

**Feedback**:
Readers might note that the state update equation uses index $j$ for the postsynaptic neuron on the left-hand side but index $i$ for the input on the right-hand side. Since $u_{k}^{i}$ is defined in Equation (2) as the input to neuron $i$, the update rule should consistently refer to the same neuron index. It would be helpful to rewrite the equation to use $u_{k}^{j}(t)$ to ensure the state of neuron $j$ is determined by the input to the same neuron.

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### 2. Connectivity probability and average in-degree consistency

**Status**: [Pending]

**Quote**:
> $J_{kl}^{ij}$ is $J_{kl}/\sqrt{K}$ with probability $K/N_{k}$ and zero otherwise.

**Feedback**:
The text states that the connection probability is $K/N_{k}$, where $k$ denotes the postsynaptic population. However, it subsequently claims that on average $K$ neurons from the presynaptic population project to each neuron. Calculating the expected number of inputs involves summing the probability over the presynaptic population size $N_{l}$, which yields $N_{l}(K/N_{k})$. This equals $K$ only if $N_{l} = N_{k}$. Readers might note that the standard definition for average in-degree $K$ requires the probability to scale with the presynaptic population size. It would be helpful to rewrite 'probability $K/N_{k}$' as 'probability $K/N_{l}$' to ensure the average number of inputs from population $l$ is $K$ regardless of population sizes.

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### 3. Index mismatch in population average definition

**Status**: [Pending]

**Quote**:
> s of the excitatory and inhibitory cells as
> 
> $m_{k}(t)=[\sigma_{k}^{j}(t)]=\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\sigma_{k}^{j}(t)\;\;,\;\;k=1,2,$ 

**Feedback**:
It would be helpful to correct the index mismatch in Equation (9). The summation variable is $i$, but the term inside uses $j$. This makes the expression mathematically undefined as $j$ is not bound. Rewrite $\sigma_{k}^{j}(t)$ as $\sigma_{k}^{i}(t)$ to match the summation index.

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### 4. Subscript error in input sum limit

**Status**: [Pending]

**Quote**:
> u_{k}(t)\,=\,[u_{k}^{i}(t)]=\sum_{l=1}^{2}\,\sum_{j=1}^{N_{i}}[\,J_{kl}^{ij}\,][\,\sigma_{l}^{j}(t)\,]\,+u_{k}^{0}-\theta_{k}.

**Feedback**:
Readers might note a subscript inconsistency in Equation (3.11). The summation over input neurons $j$ from population $l$ should extend to $N_{l}$, the size of population $l$. The text uses $N_{i}$, where $i$ refers to the target neuron index. Rewrite $N_{i}$ as $N_{l}$ to correctly reflect the population size.

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### 5. Missing population fraction in mean input derivation

**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**:
It is not clear how Equation 3.5 follows from the derivation provided. Summing the average $[J_{kl}^{ij}] = J_{kl}\sqrt{K}/N$ over $N_{l}$ neurons yields a factor of $N_{l}/N$ which is missing in Equation 3.5. Add the factor $N_{l}/N$ to Equation 3.5 or clarify that $J_{kl}$ absorbs this factor.

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### 6. Incorrect Equation References and Numbering

**Status**: [Pending]

**Quote**:
> Equations 12 and 13 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.

**Feedback**:
The text references 'Equations 12 and 13', but the provided section only contains equations numbered (4.1) through (4.8) and constraints labeled (10) and (11). The relationship $m_k = A_k m_0$ is explicitly derived in equations (4.3) and (4.4). Additionally, the numbering jumps from (4.8) to (10), skipping (9). This creates confusion for readers trying to verify the derivation. It would be helpful to ensure equation numbers match the actual content provided in this section.

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### 7. Typo in Asymptotic Order Notation

**Status**: [Pending]

**Quote**:
> In this solution, $m_{I}$ is to leading order in $k$ given by $m_{I} = Im_{0} / J_{I}$ (since the leading order in $u_{I}$ should vanish)

**Feedback**:
The text states 'leading order in $k$', but $k$ is defined as the population index ($k=E, I$) in Equation 4.8. The asymptotic expansion discussed throughout the section is with respect to the connectivity $K$ (e.g., $O(1/\sqrt{K})$). It would be helpful to correct the variable to $K$ to maintain consistency with the rest of the section.

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### 8. Duplicate Equation Numbering for (14)

**Status**: [Pending]

**Quote**:
> $H(x)\approx\frac{\exp(-x^{2}/2)}{\sqrt{2\pi}\left|x\right|}$ (14)

**Feedback**:
Readers might note that the label (14) is assigned to two different equations in this section, the first for the $H(x)$ approximation and the second for the $h(m)$ result. This creates ambiguity when referencing these results later. It would be helpful to renumber the subsequent equations to maintain a unique sequence.

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### 9. Inconsistent Excitatory Drive Term in Inhibitory Equation

**Status**: [Pending]

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

**Feedback**:
In Equation 4.18, the second term on the left-hand side is $m_{E}$. In the context of a balanced network, the inhibitory population receives excitatory drive proportional to $m_{E}$ (typically $J_{IE} m_{E}$). Equation 4.17 for the excitatory population has $m_{E}$ as the second term, implying $J_{EE}=1$. If Equation 4.18 represents the inhibitory population, the term $m_{E}$ correctly represents excitatory drive, but the absence of a distinct coefficient suggests a normalization assumption that should be explicitly stated. It would be helpful to clarify if $J_{IE}$ is normalized to 1 or if this is a typo for $m_{I}$.

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### 10. Ambiguity in Finite K Correction Scaling Condition

**Status**: [Pending]

**Quote**:
> However, as long as
> 
> $m_{k}\gg K^{-1},$ (4.16)
> 
> the gaussian assumption of the input statistics is a good approximation; hence, equations 3.7 still hold.

**Feedback**:
The text states that the Gaussian assumption holds as long as $m_{k} \gg K^{-1}$. However, later in the paragraph, it claims that when $m_{0}$ becomes of order $1/\sqrt{K}$ or less, strong nonlinearity reveals itself. Since $1/\sqrt{K} \gg K^{-1}$ for large $K$, the condition in (4.16) is satisfied even when $m_{0}$ is small enough to trigger the nonlinearity mentioned. It would be helpful to add a sentence after Equation 4.16 explaining that the nonlinearity arises from the threshold term on the RHS becoming comparable to the mean input, rather than a failure of the Gaussian approximation.

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### 11. Inconsistency between Eq 5.14 and width scaling

**Status**: [Pending]

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

**Feedback**:
Readers might note that Eq 5.14 states $q_{k}=m_{k}^{2}+O(m_{k}^{2}|\log m_{k}|)$. This implies the variance $q_{k} - m_{k}^{2}$ scales as $m_{k}^{2}|\log m_{k}|$, so the width (standard deviation) scales as $m_{k}\sqrt{|\log m_{k}|}$. However, the text claims the width is proportional to $(m_{k})^{3/2}$. It would be helpful to verify the scaling in Eq 5.14 or correct the text description to match the equation.

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### 12. Index mismatch in summation limits

**Status**: [Pending]

**Quote**:
> 
> $$
> \delta_{1} \langle u_{k}^{i} \rangle = \sum_{l=1}^{2} \sum_{j=1}^{N_{i}} \delta J_{kl}^{ij} [m_{l}^{j}]. \tag{5.5}
> $$
> 
> Here, $\delta J_{kl}^{ij} = J_{kl}^{ij} - [J_{kl}^{ij}]$.

**Feedback**:
In equation 5.5, the summation over presynaptic neurons $j$ is written as $\sum_{j=1}^{N_{i}}$. Since $j$ indexes neurons in population $l$, the limit should be $N_{l}$, not $N_{i}$. $N_{i}$ typically denotes the size of the postsynaptic population or is undefined in this context. This notation mismatch could confuse the derivation of the input variance. Rewrite $\sum_{j=1}^{N_{i}}$ as $\sum_{j=1}^{N_{l}}$ in equations 5.5 and 5.6.

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### 13. Stability Condition Should Not Depend on Time Constant

**Status**: [Pending]

**Quote**:
> equations 10) are both of order $\sqrt{K}$. Requiring that their real part be negative yields a condition on $\tau$ of the form
> 
> $\tau<\tau_{L},$ (12)
> 
> wh

**Feedback**:
For a first-order linear system $\tau \dot{x} = Ax$, the stability is determined by the eigenvalues of the matrix $A$, not the time constant $\tau$. Since $\tau > 0$, the sign of the growth rates $\lambda/\tau$ is the same as the sign of $\lambda$, meaning stability does not depend on the magnitude of $\tau$. Thus, the stability condition should depend on the matrix elements (e.g., $f_{kl}$), not $\tau$. It would be helpful to correct this to reflect that stability depends on the feedback gain rather than the time constant.

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### 14. Sign Error in Logarithm Term in Equation 7.17

**Status**: [Pending]

**Quote**:
>  \ll m_{+} \ll 1$. In this case, $\rho_{k}$ can be approximated by
> 
> $$
> \rho_{k}(m) \propto \frac{P \left(- \sqrt {\alpha_{k}} \left(\sqrt {2 \log (m)} - \tilde {h}_{k}\right)\right)}{m \sqrt {\log (m)}} \tag {7.17}
> $$
> 
> for

**Feedback**:
The term $\sqrt{\log(m)}$ is mathematically undefined for real numbers when $m < 1$, which is the case for all firing rates in this context. Equation 7.12 correctly uses $|\log m|$, indicating the absolute value is intended to handle the negative logarithm. The square root of a negative number would result in an imaginary distribution density, which is physically invalid. It would be helpful to rewrite the equation to use $\sqrt{-\log(m)}$ to ensure real-valued results.

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### 15. Logical inconsistency in divergence from zero distance

**Status**: [Pending]

**Quote**:
> {k}}}\sqrt{\alpha_{k}-\gamma_{k}}.$ (13)
> 
> Since $\alpha_{k}-\gamma_{k}\propto D_{k}$, equation 13 has a growing solution even if $D_{k}(0)=0$. T

**Feedback**:
If $D_k(0)=0$, then $\alpha_k - \gamma_k = 0$, which implies the right-hand side of equation 13 is zero. Thus, $\dot{D}_k = 0$, and the distance remains zero. Growth requires a non-zero initial perturbation ($D_k(0) > 0$). The text likely intends to say 'for arbitrarily small initial positive value' as mentioned in the following sentence. Rewrite 'even if $D_{k}(0)=0$' as 'for arbitrarily small $D_{k}(0) > 0$' to maintain mathematical consistency.

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### 16. Inconsistent Time Scale Expression in Tracking Analysis

**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**:
The text claims the time scale is of order $\tau_k / \delta m_0$, but also states this is approximately $\tau_k / \sqrt{K}$. Given the earlier assumption that $\delta m_0 \sqrt{K}$ is of order 1 (implying $\delta m_0 \sim 1/\sqrt{K}$), the expression $\tau_k / \delta m_0$ evaluates to $\tau_k \sqrt{K}$, which contradicts the stated approximation $\tau_k / \sqrt{K}$. The correct time scale derivation is $T \sim \Delta m / \dot{m} \sim (A_k \delta m_0) / (1/\tau_k) \sim \tau_k \delta m_0$, which yields $\tau_k / \sqrt{K}$. The expression $\tau_k / \delta m_0$ contains a reciprocal error.

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### 17. Exponential term variable in Appendix B Equation (4)

**Status**: [Pending]

**Quote**:
> $m_{k}(t)=\frac{1}{\tau_{k}}\int_{0}^{\infty}dt^{i}e^{-t/\tau_{k}}F_{k}(m_{E}(t-t^{i}),m_{I}(t-t^{i})),$ (4)

**Feedback**:
When deriving equation (4) from equation (3), the integration order is swapped, resulting in an inner integral over $\tau$ that yields a factor of $e^{-t^i/\tau_k}$. The variable $t^i$ represents the delay, whereas $t$ is the absolute time, so using $t$ in the exponent prevents the term from being part of the convolution kernel. This discrepancy suggests a typographical error in the exponent variable. Rewrite '$e^{-t/\tau_{k}}$' as '$e^{-t^i/\tau_{k}}$' in equation (4).

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### 18. Incorrect equation reference in autocorrelation description

**Status**: [Pending]

**Quote**:
> The integral over time in the r.h.s. of equation A.11 takes into account the correlation between the inputs

**Feedback**:
The text describes the time integral in the autocorrelation dynamics, but equation A.11 defines the function $F_k$ without any time integral. Equation A.10 contains the integral over $t'$ that matches the description provided in the text. This mismatch in equation numbering creates ambiguity for readers verifying the derivation. Rewrite 'equation A.11' as 'equation A.10' because the integral is located in equation A.10.

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### 19. Misleading Description of Threshold Distribution in Section 7.1

**Status**: [Pending]

**Quote**:
> 7.1 Distribution of Thresholds with Long Tails. We first consider a distribution with a long tail of low thresholds.

**Feedback**:
The section title and text describe the distribution as having a 'long tail of low thresholds', which typically implies a skewed or heavy-tailed distribution. However, Equation 7.3 defines a symmetric Gaussian distribution, which has symmetric tails and is not characterized by a specific 'long tail of low thresholds'. This description may confuse readers regarding the symmetry of the assumed distribution. Rewrite '7.1 Distribution of Thresholds with Long Tails. We first consider a distribution with a long tail of low thresholds.' as '7.1 Distribution of Thresholds with Unbounded Support. We first consider an unbounded distribution of thresholds.'

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### 20. Scaling of Perturbations vs Balanced State Fluctuations

**Status**: [Pending]

**Quote**:
> local perturbations, in which the deviations in the rates are small compared to $1/\sqrt{K}$

**Feedback**:
In the standard fluctuation-driven balanced state (where synaptic weights scale as $1/\sqrt{K}$), the input current fluctuations are $O(1)$, leading to firing rate fluctuations of order $O(1)$. Defining 'local perturbations' as deviations $\ll 1/\sqrt{K}$ sets the threshold below the natural noise floor of the system. It would be helpful to clarify if this scale refers to finite-size corrections to the mean rate rather than dynamic perturbations, or if the synaptic scaling assumption differs from the standard $1/\sqrt{K}$ regime.

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### 21. Unsubstantiated claim of power-law rate distribution

**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**:
For binary units with standard threshold distributions, the firing rate distribution is typically exponential or log-normal rather than a power law. Section 7.2 derives a form $\rho_k(m) \to 1/(m \text{sqrt}(|\text{log } m|))$, which is not a strict power law. This specific claim about a power-law tail requires explicit specification of the threshold or input distribution parameters that generate this behavior to ensure methodological consistency. Add 'consistent with the assumed threshold distribution' after 'power-law tail' to address the lack of justification for the specific tail shape.

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### 22. Clarification on Threshold Scaling Definition

**Status**: [Pending]

**Quote**:
> However, the gap between the threshold of the cell and its resting potential is only of the order of $\sqrt{K}$ excitatory inputs.

**Feedback**:
Readers might note that the phrase 'order of $\sqrt{K}$ excitatory inputs' is ambiguous regarding whether it refers to the number of spikes or the voltage contribution per spike. It is not clear if this implies the threshold voltage scales as $\sqrt{K} \times J$ where $J$ is the synaptic weight, which is critical for the balanced state derivation. Rewrite the sentence to specify the voltage scaling explicitly to avoid confusion.

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### 23. Inconsistent Definition of Input Fluctuations and Autocorrelation

**Status**: [Pending]

**Quote**:
> igure 3.
> 
> the autocorrelations in the local activities,
> 
> $$
> \beta_{k}(\tau) = \left[ \left\langle \delta u_{i}^{k}(t) \delta u_{i}^{k}(t + \tau) \right\rangle \right] = q_{E}(\tau) + J_{k}^{2} q_{I}(\tau), \tag{5.15}
> $$
> 
> where $q_{k}(\tau)$ is

**Feedback**:
It is not clear how the notation $\delta u$ is defined given the subsequent statements. The text states $\beta_{k}(\tau \to \infty) = \beta_{k}$ and that the variance of fluctuations is $\alpha_{k} - \beta_{k}$. If $\delta u$ represents zero-mean fluctuations, their autocorrelation should decay to zero at large $\tau$, not $\beta_{k}$. If $\beta_{k}$ is non-zero, $\delta u$ likely includes the mean input, contradicting the standard meaning of $\delta$. This ambiguity makes the variance definition unclear. Rewrite the definition in Equation 5.15 to use $u_{i}^{k}(t)$ instead of $\delta u_{i}^{k}(t)$ or explicitly define $\delta u$ to include the mean.

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### 24. Ambiguous Notation for Fixed Point Rate

**Status**: [Pending]

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

**Feedback**:
The term $-m_{k}$ in Equation 6.4 is ambiguous because $m_{k}$ is typically used for the instantaneous rate, while the equation describes the evolution of the perturbation $\delta m_{k}$. If $m_{k}$ refers to the fixed point value, it should be denoted differently, such as $m_{k}^{*}$, to distinguish it from the dynamic variable $m_{k}(t) = m_{k}^{*} + \delta m_{k}(t)$. This distinction is crucial for understanding the stability analysis. It would be helpful to clarify the notation to avoid confusion between the perturbation and the steady-state value.

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### 25. Ambiguity in Connectivity Index Terminology

**Status**: [Pending]

**Quote**:
> Amit and Brunel (1997b) showed that the strength of the average cross-correlations decreases as $N$ increases (keeping the connectivity index constant).

**Feedback**:
The term 'connectivity index' is not standard terminology in network neuroscience and is ambiguous. It is unclear whether this refers to the number of connections $K$ or the connection probability $C = K/N$. This distinction is crucial because the scaling of cross-correlations with $N$ depends on which parameter is held fixed. It would be helpful to define this term explicitly to ensure the comparison with Amit and Brunel (1997b) is methodologically sound.

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