# Chaotic Balanced State in a Model of Cortical Circuits

**Date**: 03/07/2026
**Domain**: natural_sciences/neuroscience
**Taxonomy**: academic/research_paper
**Filter**: Active comments

---

## Overall Feedback

Here are some overall reactions to the document.

**Interpretation of Discrete Chaos Metrics**

Reviewers note the Lyapunov exponent diverges to infinity due to binary unit discreteness, differing from standard continuous dynamical systems where finite positive exponents are the norm. It would be helpful to clarify whether this infinite divergence is an artifact of the binary approximation or a robust feature of the balanced state mechanism. Comparing trajectory divergence rates with continuous rate models would aid biological interpretation of the chaotic label.

**Consistency of Stability Analysis**

The global stability analysis approximates the sigmoidal activation as a Heaviside step function, which is valid for large inputs but may not align with the balanced state definition of O(1) inputs. It would be helpful to perform a rigorous stability analysis using the full sigmoidal function to ensure theoretical predictions align with the model's core assumptions. This would strengthen the validity of the derived stability boundaries in the relevant parameter regime.

**Empirical Justification for Scaling Laws**

The model relies on synaptic strengths scaling as 1/sqrt(K) and external inputs scaling as sqrt(K) to maintain the balanced state, yet experimental constraints on these specific scaling laws are not provided. It would be helpful to include a more rigorous comparison with physiological data regarding synaptic strength distributions to establish biological grounding. Exploring how deviations from this scaling affect the existence of the balanced state would also clarify the mechanism's generality.

**Robustness to Finite Connectivity and Heterogeneity**

Theoretical results assume infinite network size and homogeneous thresholds, yet biological networks operate at finite K with variability that can cause transitions to frozen states. It would be helpful to quantify the range of threshold heterogeneity and finite connectivity over which the chaotic state remains robust. Addressing this would clarify the applicability of the theory to real cortical circuits with typical connectivity indices.

**Quantitative Comparison with Experimental Data**

The comparison between theoretical rate distributions and experimental data from monkey prefrontal cortex is currently qualitative, noting only a distinct unimodal skewed shape. It would be helpful to include a quantitative goodness-of-fit metric, such as a KL divergence or Kolmogorov-Smirnov test, between the theoretical distribution and the experimental histogram. This would provide stronger evidence for the claim that the balanced state explains observed neuronal rate variability.

**Specification of Network Size in Simulations**

The mean-field theory is derived in the limit of large network size N with sparse connectivity 1 << K << N (Equation 3.1). While simulation results explicitly state K=1000 (e.g., Figure 3 caption), the total number of neurons N is not reported. It would be helpful to specify N in the simulation descriptions to verify that the sparse connectivity assumption holds, as the validity of the mean-field approximation depends on N being sufficiently larger than K.

**Consistency of Input Assumptions in Experimental Comparison**

The theoretical predictions for rate distributions (e.g., Figure 11B) assume a constant external input m0 (Section 3). The experimental data used for comparison (Figure 11C) is derived from behaving monkeys attending to stimuli and executing movements (Section 7.3), which implies time-varying inputs. Readers might note that the stationary state assumption may not directly map to the dynamic experimental conditions. It would be helpful to clarify how the model's stationary predictions are validated against non-stationary experimental data.

**Characterization of Chaos in Finite Discrete Systems**

The paper characterizes the balanced state as chaotic based on sensitivity to initial conditions and Lyapunov exponents (Section 8). However, the model uses binary neurons with discrete updates, where finite systems technically exhibit periodic rather than chaotic dynamics. While the authors note this limitation, the claim of an infinite Lyapunov exponent relies on the N -> infinity limit. It would be helpful to discuss how the chaotic properties observed in the finite N simulations (e.g., Figure 12) relate to the theoretical infinite limit definition.

**Status**: [Pending]

---

## Detailed Comments (23)

### 1. Abstract Power Law Claim

**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**:
Readers might note that the claim of a 'power-law tail' is a specific mathematical assertion. Standard balanced state models often predict exponential tails. It would be helpful to qualify the tail behavior if not rigorously derived here, or rewrite 'long power-law tail' as 'long tail' to avoid specifying a functional form without derivation.

---

### 2. Index Mismatch in Neuron Update Equation

**Status**: [Pending]

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

**Feedback**:
The equation defines the updated state of neuron $j$ using the input $u$ of neuron $i$. However, Eq. 2.2 defines $u_{k}^{i}$ as the input to neuron $i$. This creates an inconsistency where the state of neuron $j$ depends on the input of neuron $i$. It would be helpful to align the indices to indicate the state update of the neuron receiving the input, e.g., rewrite as '$\sigma_{k}^{i}(t)=\Theta(u_{k}^{i}(t)),$ (2.1)'.

---

### 3. Connectivity Probability vs Average Connectivity Claim

**Status**: [Pending]

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

**Feedback**:
The text states connections exist with probability $K/N_{k}$, where $N_k$ is the size of the postsynaptic population. The expected number of inputs from presynaptic population $l$ (size $N_l$) to a neuron in population $k$ is $N_l \times (K/N_k)$. The text later claims 'on average, $K$ excitatory and $K$ inhibitory neurons project to each neuron,' which requires $N_l = N_k$. It would be helpful to clarify the assumption or correct the probability definition to depend on $N_l$.

---

### 4. Variance Derivation Scaling Inconsistency

**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**:
The derivation of the input variance contains a scaling inconsistency. Given the mean input derivation where $[J_{kl}^{ij}] = J_{kl}\sqrt{K}/N$, the synaptic weight magnitude must scale as $J_{kl}/\sqrt{K}$ with connection probability $K/N$. Consequently, the mean square of the weight is $[ (J_{kl}^{ij})^2 ] = (K/N) \cdot (J_{kl}^2/K) = J_{kl}^2/N$. The text claims $J_{kl}^2 m_l K/N$, which appears to include an extra factor of $K$. It would be helpful to correct the term to $J_{kl}^2 m_l / N$ to maintain consistency with the $O(1)$ variance in Equation 3.6.

---

### 5. Missing Constraint to Eliminate Inhibitory Saturation

**Status**: [Pending]

**Quote**:
> 4.8}
> $$
> 
> so $u_{k}$ is of order $\sqrt{K}$ and positive.<!-- PAGE BREAK -->Thus if we require that there be no stationary solutions with $m_{E}=0, 1$ or $m_{I}=0, 1$ for small $m_{0}$, the following constraints have to be satisfied:
> 
> $\frac{E}{I} &gt; \frac{J_{E}}{J_{I}} &gt; 1.$ (4.9)
> 
> $J_{E} &gt; 1.$ (4.10)
> 
> It is straightforward to

**Feedback**:
Readers might note that the listed constraints are insufficient to eliminate all unbalanced states. Specifically, the condition $J_{E} > 1$ prevents excitatory saturation ($m_E=1$), but it does not guarantee $J_{I} > 1$. If $J_{I} < 1$ while $J_{E} > 1$, a solution with $m_{I}=1$ remains possible. It would be helpful to include the condition $J_{I} > 1$ alongside $J_{E} > 1$ to fully prevent saturation in both populations.

---

### 6. Inconsistent Width Scaling Claim

**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**:
Equation 5.14 states that the variance of the rate distribution, $q_k - m_k^2$, scales as $O(m_k^2 / |\log m_k|)$. The width (standard deviation) is the square root of the variance, which would scale as $m_k / \sqrt{|\log m_k|}$. The text claims the width scales as $m_k^{3/2}$, which would imply a variance scaling of $m_k^3$. It would be helpful to correct the scaling claim to align with Equation 5.14.

---

### 7. Inconsistent Rate Notation in Distribution

**Status**: [Pending]

**Quote**:
> \rho_{k}(m) \equiv N_{k}^{-1} \sum_{i=1}^{N_{k}} \delta(m - m_{i}^{k}). \tag{5.10}

**Feedback**:
The notation for the time-averaged rate of the $i$-th cell in population $k$ is consistently defined as $m_{k}^{i}$ in Equations 5.3 and 5.4. However, Equation 5.10 uses the notation $m_{i}^{k}$, which swaps the population and cell indices. It would be helpful to rewrite $m_{i}^{k}$ as $m_{k}^{i}$ in Equation 5.10 to maintain consistency with Equations 5.3 and 5.4.

---

### 8. Inconsistent Definition of Input Autocorrelation

**Status**: [Pending]

**Quote**:
> d by the mean rate. Parameter values as in Figure 3.
> 
> 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 $q_{k}(\tau)$ is the time-delayed autocorrelations of the 

**Feedback**:
Equation 5.15 defines $\beta_k(\tau)$ as the autocorrelation of input fluctuations ($\delta u$), but the text immediately following states '$\beta_{k}(0) = \alpha_{k}$' and 'variance given by $\alpha_{k}-\beta_{k}$'. If $\beta_k(\tau)$ were the fluctuation autocorrelation, $\beta_k(0)$ would equal the variance, not the second moment $\alpha_k$. It would be helpful to clarify whether $\beta_k(\tau)$ represents the autocorrelation of the total input $u$ or the fluctuations.

---

### 9. Potential Singularity in Autocorrelation Equation

**Status**: [Pending]

**Quote**:
> a_{k}(t + \tau)} x}{\sqrt{\alpha_{k} - \beta_{k}(t + \tau)}}\rig

**Feedback**:
In Equation 5.17, the term $\sqrt{\alpha_{k} - \beta_{k}(t + \tau)}$ appears in the denominator. Given the boundary condition $\beta_k(0) = \alpha_k$, this term vanishes when $t+\tau=0$, creating a mathematical singularity. If $\beta_k(\tau)$ represents the total input autocorrelation, the variance of the input fluctuations is the constant $\alpha_k - \beta_k$, not the time-dependent difference. It would be helpful to correct the denominator to reflect the constant variance.

---

### 10. Stability Condition Dependence on Time Constant

**Status**: [Pending]

**Quote**:
> {1}$ and $\lambda_{2}$ of the 2 by 2 equations (see equations 6.1) are both of order $\sqrt{K}$. Requiring that their real part be negative yields a condition on $\tau$ of the form
> 
> $\tau&lt;\tau_{L},$ (6.3)
> 
> where $\tau_{L}$ is of order 

**Feedback**:
In the linearized equation (6.1), the time constant $\tau$ appears only as a multiplicative factor on the derivative term. The eigenvalues of the dynamical system are proportional to $1/\tau$, meaning their sign depends on the connectivity matrix, not the magnitude of $\tau$. A condition $\tau < \tau_L$ implies stability changes with $\tau$, which appears inconsistent with the structure of the first-order differential equation provided. It would be helpful to verify if $\tau$ appears in the feedback term or if the condition should apply to the connectivity parameters.

---

### 11. Discrepancy in Critical Time Constant Calculation

**Status**: [Pending]

**Quote**:
> For the parameters used in this figure, the large $K$ critical inhibitory time constants are $\tau_{L}=1.61$ and $\tau_{G}=1.50$.

**Feedback**:
Readers might note that the reported value $\tau_{G}=1.50$ appears inconsistent with Equation 6.6 and the parameters provided in Figure 9 ($J_{E}=2.0, m_{E}=0.1$). Assuming the balanced state condition $J_{E}m_{E} \approx J_{I}m_{I}$ holds, $m_{I} \approx 0.111$. Substituting these into Equation 6.6 yields $\tau_{G} \approx 1.98$, not 1.50. It would be helpful to verify the parameter values used for Figure 8 or correct the reported critical constants to ensure internal consistency.

---

### 12. Notation Ambiguity in Perturbation Equation

**Status**: [Pending]

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

**Feedback**:
The equation uses $\delta m_{k}$ on the left-hand side but includes the constant term $-m_{k}$ on the right-hand side. While $\frac{d}{dt}\delta m_{k} = \frac{d}{dt}m_{k}$, this notation may confuse readers regarding whether $m_{k}$ represents the instantaneous rate or the fixed point value. It would be helpful to explicitly define $m_{k}$ in this context as the fixed point rate $m_{k}^{*}$ to distinguish it from the perturbation variable.

---

### 13. Missing Prefactor in General Rate Distribution Formula

**Status**: [Pending]

**Quote**:
> ution $P(\theta)$ will yield a rate distribution $\rho_k$ for population $k$ that is given by
> 
> $$
> \rho_k(m) = \sqrt{2\pi} P\left(-\sqrt{\alpha_k} (h(m) + \tilde{h}_k)\right) e^{h^2(m)/2}, \tag{7.15}
> $$
> 
> where $\tilde{h}_k$ is determined b

**Feedback**:
The derivation of the rate distribution $\rho_k(m)$ from the threshold distribution $P(\theta)$ requires the Jacobian factor $|d\theta/dm|$. Given $\theta = -\sqrt{\alpha_k}(h(m) + \tilde{h}_k)$, the Jacobian is $\sqrt{\alpha_k} |dh/dm|$. Since $|dh/dm| = \sqrt{2\pi} e^{h^2/2}$, the correct prefactor should include $\sqrt{\alpha_k}$. Without this factor, Equation 7.15 is inconsistent with the specific case derived in Equation 7.12. It would be helpful to include the factor $\sqrt{\alpha_k}$ in the prefactor.

---

### 14. Sign Ambiguity in Low-Rate Threshold Condition

**Status**: [Pending]

**Quote**:
> imit. In this case, the solution for $u_{k}$ is
> 
> $$
> u _ {k} + \Delta / 2 = O (\sqrt {m _ {k}}). \tag {7.10}
> $$<!-- PAGE BREAK -->C. v

**Feedback**:
For the mean rate $m_k$ to be small in the bounded threshold case, the mean input $u_k$ must be below the minimum threshold $-\Delta/2$. This implies $u_k + \Delta/2$ should be negative. However, Equation 7.10 states the magnitude is $O(\sqrt{m_k})$ without indicating the sign. A positive value would imply $u_k > -\Delta/2$, leading to higher firing rates inconsistent with the low-rate limit. It would be helpful to explicitly indicate the negative sign to reflect the condition that mean input is below the minimum threshold.

---

### 15. Lyapunov Exponent Definition Mismatch

**Status**: [Pending]

**Quote**:
> n $k$, defined by equation 8.2, should grow at least exponentially for small $D_{k}$, The maximum Lyapunov exponent $\lambda_{L}$, defined by
> 
> $\lambda_{L}\equiv\lim_{D_{k}\to 0}D_{k}^{-1}\frac{dD_{k}}{dt},$ (8.3)
> 
> should be positive. Note that in calculating $\lambda_{L}$, we will first take the large $N$ limit of $D_{k}$ 

**Feedback**:
The standard definition of the maximal Lyapunov exponent for a chaotic attractor is the time average of the local growth rate along the trajectory on the attractor. Equation 8.3 defines the local growth rate at the unstable fixed point ($D_k \to 0$), which corresponds to the locked state (Eq 8.7), not the chaotic balanced state. While the instability of the locked state drives the chaos, the Lyapunov exponent of the balanced state should be evaluated at the typical separation distance on the attractor. It would be helpful to clarify that $\lambda_L$ refers to the instability of the locked state rather than the attractor itself.

---

### 16. Scaling Inconsistency Between Probability and Rate Changes

**Status**: [Pending]

**Quote**:
> ast 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**:
The text states that the probability change $\Delta P_k$ is of order 1, yet derives that the rate change $\delta m_k$ is of order $1/\sqrt{K}$ using the relation $\delta m_k \approx \tau_k^{-1} dt \Delta P_k$. If $\Delta P_k \sim O(1)$ and the time scale is $\tau_k/\sqrt{K}$ (implying $dt \sim 1/\sqrt{K}$), then $\delta m_k \sim O(1/\sqrt{K})$ is consistent with the equation. However, in the binary neuron model, the rate $m_k$ is typically defined as the probability of being active ($m_k \approx P_k$). If $\Delta P_k \sim O(1)$, then $\delta m_k$ should also be $O(1)$, not $O(1/\sqrt{K})$. It would be helpful to clarify the relation between $m_k$ and $P_k$ or the scaling of $\Delta P_k$.

---

### 17. Cross-Correlation Scaling for Connected Pairs

**Status**: [Pending]

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

**Feedback**:
In standard balanced network theory, cross-correlations between connected pairs typically scale with the square of the synaptic weight ($J^2 \sim 1/K$) rather than the weight itself ($J \sim 1/\sqrt{K}$). It is not clear how the $O(1/\sqrt{K})$ scaling is derived in this model without further justification. Clarifying the derivation or referencing the specific assumption that leads to this scaling would help readers verify consistency with established results.

---

### 18. Inconsistency in Rate Scaling for Alpha < 1/2

**Status**: [Pending]

**Quote**:
> Specifically for low rates, the rate distribution will become narrow.

**Feedback**:
It would be helpful to clarify the scaling of the mean input in this regime. The text states that net average inputs scale as $K^{1-\alpha}$. For $\alpha < 1/2$, the exponent $1-\alpha$ is positive, implying the mean input grows with $K$. For binary units with fixed thresholds, a growing mean input drives the firing rate toward saturation ($r \to 1$), not low rates. Thus, claiming 'low rates' in the large $K$ limit appears inconsistent with the derived input scaling. It would be helpful to clarify whether the rate distribution becomes narrow (approaching saturation) or if the scaling assumption differs.

---

### 19. Internal Consistency of Fast Tracking Prediction

**Status**: [Pending]

**Quote**:
> 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.

**Feedback**:
Readers might note that the text states 'fast tracking' is predicted in the model, yet Figure 13 shows the model does not respond quickly. Since the mean-field theory is derived for binary units, there is a potential disconnect between the theoretical prediction and the simulation results attributed to binary dynamics. It would be helpful to clarify whether the mean-field theory for binary units predicts fast tracking or if this prediction applies only to the integrate-and-fire limit.

---

### 20. Inconsistency Between Experimental Claim and Balanced State Scaling

**Status**: [Pending]

**Quote**:
> input from the lateral geniculate nucleus (LGN) to layer 4 cortical cells are in fact a fraction of the net input.

**Feedback**:
In the balanced state regime, the net input to a neuron is the sum of external and recurrent inputs, which cancel to leave a small residual ($O(1)$), while the external input remains large ($O(\sqrt{K})$). The text states that experimental findings suggest the LGN input is 'a fraction of the net input,' implying the external drive is smaller than the total net drive. This relationship appears inconsistent with the fundamental scaling of the balanced state model where external drive must exceed the net drive due to recurrent cancellation. It would be helpful to clarify whether 'net input' refers to the recurrent component or discuss how the model accommodates experimental data.

---

### 21. Ambiguity in Population Response Linearity

**Status**: [Pending]

**Quote**:
> The internal synaptic inputs act as a strong negative feedback, which linearizes the population responses to the external drive despite the strong nonlinearity of the individual cells.

**Feedback**:
The term 'population responses' is ambiguous in this context. It is not clear whether this refers to the mean firing rate, the variance, or the full temporal dynamics. In balanced state theory, the linearity typically applies to the mean rate-input relationship. It would be helpful to specify the quantity being linearized to ensure the methodological claim is precise, e.g., rewrite 'linearizes the population responses' as 'linearizes the mean population firing rate'.

---

### 22. Justification for Perturbation Scale Threshold

**Status**: [Pending]

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

**Feedback**:
Readers might note that the text asserts a necessity to distinguish perturbations based specifically on the scale $1/\sqrt{K}$, which corresponds to finite-connectivity fluctuations in balanced networks. However, standard linear stability analysis typically considers infinitesimal perturbations ($\delta r \to 0$) without a $K$-dependent threshold. It is not clear why the stability properties change specifically at the $1/\sqrt{K}$ scale rather than at a generic finite amplitude. It would be helpful to clarify whether this distinction arises from the nonlinear saturation of the activation function or finite-size noise effects.

---

### 23. Growth Rate Dependence on Initial Perturbation

**Status**: [Pending]

**Quote**:
> This should be contrasted with systems with finite positive Lyapunov exponents, where the initial rate of growth depends on the magnitude of the initial perturbation of the initial conditions.

**Feedback**:
The statement implies that for systems with finite Lyapunov exponents, the initial growth rate depends on the perturbation magnitude, whereas for the infinite exponent case, it does not. However, for Eq 8.9 ($\tau \dot{D} \propto \sqrt{D}$), the initial rate $\dot{D}$ also depends on the magnitude ($\sqrt{D(0)}$). The distinction lies in the Lyapunov exponent ($\dot{D}/D$), which is finite for standard chaos but diverges here. It would be helpful to clarify that the Lyapunov exponent ($\dot{D}/D$) is finite in standard systems but diverges here, rather than claiming the rate dependence differs.

---
