# Chaotic Balanced State in a Model of Cortical Circuits

**Date**: 03/08/2026
**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.

**Mean-Field Approximation Validity for Finite Connectivity**

The paper claims mean-field theory is exact in the large network limit, yet simulations use finite K=1000 where conditions like $m_k \gg K^{-1}$ are borderline satisfied at low rates. Readers might note that finite K corrections could substantially affect balanced state properties, particularly for rate distributions. It would be helpful to provide explicit error bounds showing how K=1000 results deviate from K→∞ predictions, or to demonstrate convergence across multiple K values to establish the approximation's validity range.

**Lyapunov Exponent Characterization for Discrete Units**

Section 8 characterizes chaos by claiming the Lyapunov exponent is infinitely large, while acknowledging this definition is technically inapplicable to discrete systems. Readers might note that binary units cannot exhibit the continuous trajectory divergence required for conventional chaos definitions. It would be helpful to clarify what mathematical properties beyond sensitivity to initial conditions justify the 'chaotic' label and whether alternative metrics like correlation dimension would be more appropriate.

**External Input Scaling Versus Biological Measurements**

Equation 2.3 assumes external input scales as $\sqrt{K}$, which Section 10.4 acknowledges may conflict with experimental findings suggesting LGN input is a fraction of net input. This creates tension between the model's core scaling assumption and biological measurements of cortical input strengths. It would be helpful to analyze what happens to the balance conditions when external input scales as O(1) rather than O($\sqrt{K}$) and whether alternative parameter regimes could reconcile theory with measurements.

**Threshold Distribution Assumption and Low-Rate Dynamics**

Section 7 shows that gaussian threshold distributions produce frozen states at low rates while bounded distributions maintain fluctuating activity, yet the biological justification for bounded distributions remains limited. Readers might note that this choice critically determines whether the model predicts frozen or fluctuating states at low firing rates. It would be helpful to provide biological justification for the bounded distribution assumption or discuss what experimental measurements could distinguish between these threshold distribution types.

**Stability Regime Coverage and Biological Parameter Mapping**

Section 6 provides stability conditions for constant input, but the connection to time-varying input tracking and the coverage of all instability regimes (e.g., $\tau_G < \tau_L$) is incomplete. Readers might note that without exploring this parameter region or mapping experimental inhibitory time constants onto the phase diagram, the practical stability of the balanced state remains incompletely characterized. It would be helpful to show parameter regions where $\tau_G < \tau_L$ occurs and overlay experimentally measured $\tau_I$ values to demonstrate the balanced state operates in the stable regime for biologically plausible parameters.

**Infinite Network Size Limit vs. Finite-K Simulations**

The paper's mean-field theory is stated to be 'exact in the limit of large network size' (Section 3) with the technical assumption that N→∞ is taken first, then K→∞ (Section 3, equation 3.1: 1≪K≪N_k). However, the numerical results and figures consistently use finite K values (e.g., K=1000 in Figures 3, 6, 8, 13-17). The paper acknowledges this in Section 4.2, noting 'For finite K the residuals of order 1/√K in the rates are not negligible' and provides corrected equations 4.17-4.18. Readers might note that the stability analysis in Section 6 derives critical time constants τ_L and τ_G that are 'of order 1' but depend on the large-K limit where eigenvalues are 'of order √K' (equation 6.1). It would be helpful to clarify whether the finite-K simulations (K=1000) are sufficiently large for the asymptotic stability conditions to hold, particularly since the paper states 'in reality, networks have a large fixed size and connectivity, so that the distinction between full and sparse connectivity may be problematic' (Section 3). A concrete robustness check would be to show how the stability boundaries shift for different K values.

**Deterministic Chaos Claim vs. Stochastic Update Rule**

Section 8 claims 'the balanced state corresponds to a chaotic attractor' and analyzes sensitivity to initial conditions via Lyapunov exponents. However, Section 2 describes two update models: one stochastic ('each neuron updates its state at time intervals that have Poisson statistics') and one deterministic ('each neuron updates its state at equally spaced times where the time between updates is different for each neuron'). The paper states 'the two models have the same mean-field equations' (Section 2), but this equivalence is claimed without detailed derivation in the main text (referenced to Appendix B). For the chaos analysis in Section 8, it is not clear which update rule is used for the Lyapunov exponent calculation. If the stochastic update rule is used, the 'chaos' may partially reflect the stochastic update timing rather than purely deterministic chaotic dynamics. The paper acknowledges 'This definition of chaos is technically inapplicable to a system with discrete degrees of freedom such as ours' (Section 8) but proceeds to calculate λ_L anyway. It would be helpful to explicitly state which update rule was used for the chaos analysis and whether the infinite Lyapunov exponent (λ_L→∞) result holds for both update schemes.

**Homogeneous Threshold Assumption in Core Theory**

The primary balanced state theory (Sections 3-6) assumes homogeneous neuronal thresholds (θ_k is population-averaged, same for all neurons in population k). Section 7 then addresses inhomogeneity as a separate extension, showing that 'an unbounded threshold distribution has a relatively strong qualitative effect on the balance state, in the limit of low mean rate' (Section 7.1), potentially creating a 'frozen state' with bimodal rate distribution (equation 7.8, Figure 11A). However, the introduction states the goal is to explain 'temporal irregularity in the electrical activity of cortical neurons in vivo' which exhibit known heterogeneity. The core theory's predictions about rate distributions (Figure 4, homogeneous case) differ qualitatively from the inhomogeneous case (Figure 11B) at low mean rates. Readers might note that the experimental comparison in Figure 11C is matched to the inhomogeneous theory (Figure 11B), not the homogeneous theory developed in Sections 3-6. It would be helpful to clarify whether the balanced state's key properties (fast tracking, asynchronous chaos) persist with realistic threshold heterogeneity, since Section 10.2 notes the 'weak-synapses scenario...breaks down in the presence of inhomogeneity in the local thresholds.'

**Status**: [Pending]

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## Detailed Comments (18)

### 1. Mean-field exactness claim lacks finite-size qualification

**Status**: [Pending]

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

**Feedback**:
This claim states the mean-field theory is 'exact in the limit of large network size' without specifying the required ordering of limits or connectivity conditions. Section 4.2 acknowledges 'For finite K the residuals of order 1/√K in the rates are not negligible' and Section 3 requires 1≪K≪N_k for the mean-field assumption to hold rigorously. Readers might note that the abstract's unqualified exactness claim could mislead about the theory's applicability to finite networks like the K=1000 simulations used throughout the paper. It would be helpful to add 'when N→∞ is taken before K→∞ with 1≪K≪N' after 'large network size' to specify the required limit ordering.

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### 2. Rate inhomogeneity claim conflates homogeneous and inhomogeneous threshold cases

**Status**: [Pending]

**Quote**:
> It shows that the balanced state is associated with a strong Poisson-like firing pattern and also with a broad inhomogeneity in the average rates of individual neurons.

**Feedback**:
The introduction states the balanced state shows 'a broad inhomogeneity in the average rates of individual neurons' as a general property. However, Section 7 clarifies this inhomogeneity specifically arises from threshold inhomogeneity: 'In section 7 considers the effect of inhomogeneity in the local thresholds. We show that in the presence of inhomogeneity, the distribution of rates acquires a characteristic skewed shape with a long tail.' In the homogeneous threshold case (Sections 3-6), neurons within each population are statistically identical by symmetry. It would be helpful to rewrite this to distinguish the homogeneous balanced state properties from the inhomogeneous threshold extension.

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### 3. Equation 2.1 Index Inconsistency

**Status**: [Pending]

**Quote**:
> rk 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)
> 
> where $\Theta(x)$ is th

**Feedback**:
It is not clear how the indices in equation 2.1 correspond to the neuron being updated. The left-hand side uses index $j$ ($\sigma_{k}^{j}$), while the right-hand side depends on the input to neuron $i$ ($u_{k}^{i}$). Given equation 2.2 defines the input $u_{k}^{i}$ for neuron $i$, the update rule should consistently refer to neuron $i$ to describe its own state transition. It would be helpful to rewrite equation 2.1 as '$\sigma_{k}^{i}(t)=\Theta(u_{k}^{i}(t)),$' to maintain consistent indexing for the updating neuron.

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### 4. Incomplete elimination of unbalanced m_I=0 state

**Status**: [Pending]

**Quote**:
> uad (k = E, I), \tag {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 show that these constraints eliminate all possible unbalanced states.
> 
> Throughout the article, we will assume that equations 4.

**Feedback**:
The text claims constraints 4.9-4.10 eliminate all unbalanced states, but the preceding analysis only explicitly demonstrates elimination of m_E=0 (equation 4.7) and m_E=m_I=1 (equation 4.8) states. The m_I=0 unbalanced state is not analyzed. For m_I=0, equation 4.1 gives m_E=-E m_0 (negative, unphysical) while equation 4.2 gives m_E=-I m_0 (also negative). It would be helpful to explicitly address this case to support the claim that 'all possible unbalanced states' are eliminated.

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### 5. Condition m_k >> K^(-1) insufficient for equations 3.7 validity

**Status**: [Pending]

**Quote**:
>  by equation A.5. 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. Thus, the leading finite $K$ corrections can be incorporate

**Feedback**:
The condition m_k >> K^(-1) ensures the Gaussian approximation is valid, but this does not guarantee equations 3.7 (the infinite-K limit equations) hold accurately. For equations 3.7 to hold, the finite-K corrections in equations 4.17-4.18 must be negligible, which requires m_k >> K^(-1/2), not just m_k >> K^(-1). Since K^(-1/2) >> K^(-1) for large K, these are distinct regimes. It would be helpful to rewrite the quoted passage to clarify that equations 3.7 hold only when m_k >> K^(-1/2), while the Gaussian approximation alone requires only m_k >> K^(-1).

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### 6. Ambiguity in when finite-K corrections affect qualitative predictions

**Status**: [Pending]

**Quote**:
> Except for thresholding the population rates, the finite $K$ corrections affect only the quantitative results, not the qualitative predictions of the simple large $K$ theory.

**Feedback**:
This claim appears to conflict with Section 7 and Section 10.2, which state that inhomogeneous thresholds with unbounded distributions produce qualitatively different behavior (frozen states with bimodal rate distributions) at low rates. If finite-K corrections become important at low rates (m_0 ~ O(1/√K)), and threshold heterogeneity also matters at low rates, the combination could produce qualitative changes beyond just 'thresholding the population rates.' It would be helpful to qualify this statement to specify the parameter regimes where qualitative predictions remain unchanged.

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### 7. Distribution width scaling inconsistent with equation 5.14

**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**:
The claimed scaling of the distribution width as $(m_k)^{3/2}$ does not follow from equation 5.14. From equation 5.14, $q_k = m_k^2 + O(m_k^2/|\log m_k|)$, the variance is $q_k - m_k^2 = O(m_k^2/|\log m_k|)$. The standard deviation (width) is therefore $\sqrt{q_k - m_k^2} = O(m_k/\sqrt{|\log m_k|})$, not $O(m_k^{3/2})$. It would be helpful to rewrite the sentence to reflect the correct scaling derived from equation 5.14.

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### 8. Inconsistent scope definition for perturbation analysis

**Status**: [Pending]

**Quote**:
> ery small.<!-- PAGE BREAK -->Chaotic Balanced State in a Model of Cortical Circuits
> 
> # 6 Stability of the Balanced State
> 
> To determine the stability of the balanced state, we have to study the response of the system to small perturbations in the population activity rates. However, because of the nature of the balanced state, we have to distinguish two scales of perturbations: 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 $1/\sqrt{K}$.
> 
> ### 6.1 Local Stability.
> 
> Local stability of the balanced state requires that a sufficiently small

**Feedback**:
The first sentence restricts the analysis to 'small perturbations,' but the second sentence introduces 'global perturbations' defined as 'large compared to $1/\sqrt{K}$.' Since Section 6.2 covers Global Stability for large perturbations, the initial restriction is inaccurate. This contradiction may confuse readers about whether large perturbations are actually analyzed in this section. It would be helpful to rewrite 'small perturbations' as 'perturbations' in the first sentence to accurately reflect that both local and global scales are studied.

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### 9. Stability condition incorrectly depends on time constant tau

**Status**: [Pending]

**Quote**:
> (t).$ (6.1)
> 
> Calculating $f_{kl}$ by partial differentiation of the r.h.s. of equation 3.3 yields
> 
> $f_{kl}=\frac{\exp(-u_{k}^{2}/2\alpha_{k})}{\sqrt{2\pi\alpha_{k}}}J_{kl}.$ (6.2)
> 
> 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 $\sqrt{K}$. Requiring that their real part be negative yields a condition on $\tau$ of the form
> 
> $\tau&lt;\tau_{L},$ (6.3)
> 
> wher

**Feedback**:
For the linear system $\tau \dot{\mathbf{x}} = (-\mathbf{I} + \sqrt{K}\mathbf{F})\mathbf{x}$, the eigenvalues are $\lambda = \frac{1}{\tau}(-1 + \sqrt{K}\mu)$, where $\mu$ are eigenvalues of $\mathbf{F}$. Since $\tau > 0$, the sign of $\text{Re}(\lambda)$ is determined entirely by $\text{Re}(-1 + \sqrt{K}\mu)$, which depends on the connectivity parameters in $\mathbf{F}$, not on $\tau$. The time constant $\tau$ scales the rate of growth or decay but cannot flip the sign of the real part to determine stability. It would be helpful to rewrite the text to state that stability requires conditions on the spectral radius of the connectivity matrix.

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### 10. Equation 6.4 Fixed Point Inconsistency

**Status**: [Pending]

**Quote**:
> 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)
> 
> These equations are pie

**Feedback**:
This equation describes the evolution of perturbations $\delta m_k$, which are claimed to decay to zero. However, setting the time derivative to zero to find the fixed point yields $\delta m_k = \Theta(\dots) - m_k$. For the perturbation to decay to zero ($\delta m_k=0$), the step function $\Theta$ must equal the rate $m_k$. Since $\Theta$ takes values 0 or 1 and $m_k$ is an intermediate rate (e.g., 0.1), this equality is impossible. It would be helpful to clarify if there is a typo in the constant term or if this equation describes full rates rather than perturbations.

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### 11. Dimensional Inconsistency in Equation 7.11 Scaling

**Status**: [Pending]

**Quote**:
> qrt{q_{k}}$, where
> 
> $$
> q_k \propto \Delta \alpha_k^{3/2}. \tag{7.11}
> $$
> 
> This mean

**Feedback**:
The quantity $q_k$ represents the second moment of the firing rate, which is dimensionless. However, the right-hand side $\Delta \alpha_k^{3/2}$ has dimensions of threshold$^4$ (since $\Delta$ is threshold and $\alpha_k$ is threshold$^2$). This dimensional mismatch indicates a mathematical error in the derived scaling law. It would be helpful to rewrite Eq 7.11 to reflect a dimensionally consistent scaling, for example $q_k \propto (\sqrt{\alpha_k}/\Delta)$, and verify the derivation steps leading to this exponent.

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### 12. Missing Jacobian Factor in Equation 7.15

**Status**: [Pending]

**Quote**:
> o_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 d

**Feedback**:
The transformation from threshold distribution $P(\theta)$ to rate distribution $\rho_k(m)$ requires the Jacobian factor $|d\theta/dm|$. Given $m = H((\theta - u_k)/\sqrt{\alpha_k})$, the inverse relation is $\theta = u_k + \sqrt{\alpha_k} h(m)$, yielding $d\theta/dm = \sqrt{\alpha_k} h'(m)$. While $h'(m)$ accounts for the $\sqrt{2\pi} e^{h^2/2}$ term, the prefactor $\sqrt{\alpha_k}$ from the chain rule is missing in Eq 7.15. It would be helpful to rewrite Eq 7.15 as $\rho_k(m) = \sqrt{2\pi \alpha_k} P(\dots)$ to correctly include the Jacobian scaling factor.

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### 13. Equation 8.9 growth claim contradicts D_k=0 initial condition

**Status**: [Pending]

**Quote**:
> k}^{2}/2\alpha_{k}}}{\sqrt{\alpha_{k}}}\sqrt{\alpha_{k}-\gamma_{k}}.$ (8.9)
> 
> Since $\alpha_{k}-\gamma_{k} \propto D_{k}$, equation 8.9 has a growing solution even if $D_{k}(0)=0$. This implies that the Lyapunov exponent $\lambda_{L}$ is infinitely large in the balanced state. Figure 12 shows the evoluti

**Feedback**:
If α_k - γ_k ∝ D_k, then equation 8.9 gives τ_k dD_k/dt ∝ √D_k. At D_k = 0, this yields dD_k/dt = 0, meaning no growth occurs from exactly zero initial distance. The claim 'even if D_k(0) = 0' is mathematically incorrect. The correct statement should be: 'equation 8.9 has a growing solution for arbitrarily small D_k(0) > 0.' It would be helpful to rewrite the quoted text to reflect that the Lyapunov exponent diverges as $D_{k} \to 0+$, not at $D_k=0$.

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### 14. Timescale formula has inverted scaling error

**Status**: [Pending]

**Quote**:
> nitial disruption of the balance in the inputs and the highly nonlinear dynamics of single cells. This initial large response causes a fast rate of increase in the population rates, since $\delta m_k \approx \tau_k^{-1} dt \Delta P_k$, implying that $\delta m_k$ reaches the value $A_k \delta m_0$ in time of order $\tau_k / \delta m_0 \approx \tau_k / \sqrt{K}$; see the dotted line in Figure 13. The final change in the population rates linearly follows the chang

**Feedback**:
There is a mathematical error in the timescale formula. The text states $\delta m_0 \sqrt{K}$ is of order 1, meaning $\delta m_0 \sim 1/\sqrt{K}$. If the rate of change is $dm_k/dt \sim 1/\tau_k$ (order 1), then the time to reach change $A_k \delta m_0$ should be $(A_k \delta m_0)/(1/\tau_k) \sim \tau_k \delta m_0 \sim \tau_k/\sqrt{K}$. However, the text writes $\tau_k/\delta m_0$, which would equal $\tau_k \sqrt{K}$. It would be helpful to rewrite '$\tau_k / \delta m_0 \approx \tau_k / \sqrt{K}$' as '$\tau_k \delta m_0 \approx \tau_k / \sqrt{K}$'.

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### 15. Contradiction in Low-Rate Claim for Alpha Less Than Half

**Status**: [Pending]

**Quote**:
> luctuations in the input decrease with $K$ so that the network goes to the frozen state in the large $K$ limit. On the other hand, if $\alpha&lt;1/2$, the fluctuations grow with $K$, and therefore the inhomogeneity in the threshold becomes negligible in the large $K$ limit. Thus, a network with inhomogeneous thresholds will act in the same way as a network with homogeneous thresholds. Specifically for low rates, the rate distribution will become narrow. Thus, only for a network with synaptic strengths of orde

**Feedback**:
The claim that low rates are achievable in the balanced state for $\alpha < 1/2$ appears contradictory to the scaling analysis provided. In this regime, fluctuations scale as $K^{1/2-\alpha}$, which diverges as $K \to \infty$. The balance condition cancels the mean input to $O(1)$ (comparable to the threshold), but the diverging fluctuations ($\gg$ threshold) will drive the neurons to switch states frequently, resulting in a firing rate near 0.5 (saturation) rather than low rates. It would be helpful to rewrite 'Specifically for low rates, the rate distribution will become narrow' to clarify how low rates are maintained despite diverging noise.

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### 16. Contradiction on fast tracking response capability

**Status**: [Pending]

**Quote**:
> sponse 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 numeri

**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.' These claims appear contradictory without clarification on what constitutes 'fast' tracking. It would be helpful to rewrite the passage to reconcile the apparent contradiction between the fast tracking claim and the Figure 13 limitation.

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### 17. Deterministic Chaos Claim Versus Stochastic Update Rule

**Status**: [Pending]

**Quote**:
> the two models have the same mean-field equations

**Feedback**:
Section 8 claims 'the balanced state corresponds to a chaotic attractor' and analyzes sensitivity to initial conditions via Lyapunov exponents. However, Section 2 describes two update models: one stochastic and one deterministic. The paper states 'the two models have the same mean-field equations', but this equivalence is claimed without detailed derivation in the main text. For the chaos analysis in Section 8, it is not clear which update rule is used for the Lyapunov exponent calculation. It would be helpful to explicitly state which update rule was used for the chaos analysis and whether the infinite Lyapunov exponent result holds for both update schemes.

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### 18. Normalization error in probability distribution equation A.13

**Status**: [Pending]

**Quote**:
> ), \tag{A.12}
> \end{array}
> $$
> 
> where
> 
> $$
> p_l(n_1, n_2, n_3) = \frac{(q_l K)^{n_1}}{n_1!} \frac{((m_l - q_l) K)^{n_2 + n_3}}{n_2! n_3!} e^{-(2m_l - q_l)K}. \tag{A.13}
> $$
> 
> In the large $K$ limi

**Feedback**:
The exponential prefactor in equation A.13 is incorrect and the distribution does not sum to 1. For independent Poisson variables with n_1 ~ Poisson(q_l K), n_2 ~ Poisson((m_l - q_l)K), and n_3 ~ Poisson((m_l - q_l)K), the normalization requires summing over all values: Σ_{n_1} (q_l K)^{n_1}/n_1! = e^{q_l K}, Σ_{n_2} ((m_l - q_l)K)^{n_2}/n_2! = e^{(m_l - q_l)K}, and Σ_{n_3} ((m_l - q_l)K)^{n_3}/n_3! = e^{(m_l - q_l)K}. The total sum equals e^{q_l K} × e^{2(m_l - q_l)K} = e^{(2m_l - q_l)K}, so the prefactor must be e^{-(2m_l - q_l)K} × e^{(2m_l - q_l)K} = 1. However, the current prefactor e^{-(2m_l - q_l)K} gives total sum e^{q_l K} ≠ 1. It would be helpful to rewrite equation A.13 with prefactor e^{-2m_l K} instead of e^{-(2m_l - q_l)K} to ensure proper normalization.

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