Renart (2010): Asynchronous Irregular State

The following appendices document each panel of Figure 2. They adapt the diagnostics discussed by Renart et al. (2010) to this simulation. All panels use the 30-second recording after 50 ms burn-in. Cell selection, counting windows and plotting choices below describe this implementation, rather than an exact copy of the paper.

Appendix A — Panel A: irregular firing

1. Select neurons and measure intervals.

   Use all recorded E and I spike trains after burn-in. Keep cells with at least 20 spikes. For each retained neuron, measure gaps between consecutive spikes; do not include gaps crossing the start or end of the measurement period.

   $$d_{i,k}=t_{i,k+1}-t_{i,k},\qquad k=1,\ldots,m_i. \tag{A1}$$

   Here $i$ identifies a neuron, $t_{i,k}$ is its $k$th spike time in ms, $d_{i,k}$ is an interspike interval, and $m_i$ is its number of intervals (one fewer than its spike count).

2. Calculate variability.

   Calculate each neuron’s mean interval and standard deviation, then divide them to obtain its coefficient of variation (CV). The implementation uses the population standard deviation, dividing by $m_i$ rather than $m_i-1$.

   $$\mu_i=\frac{1}{m_i}\sum_{k=1}^{m_i}d_{i,k},\qquad \sigma_i=\sqrt{\frac{1}{m_i}\sum_{k=1}^{m_i}(d_{i,k}-\mu_i)^2},\qquad \mathrm{CV}_i=\frac{\sigma_i}{\mu_i}. \tag{A2}$$

   Here $\mu_i$ and $\sigma_i$ are in ms, so CV has no units. Cells without a finite CV are excluded from the histogram. No smoothing or fitted distribution is applied.

3. Plot the distributions.

   Use 25 equal-width bins shared by E and I. They run from zero to the larger of 2.5 or 1.05 times the largest eligible CV. Plot each population as a separately normalised density histogram:

   $$H_p(b)=\frac{N_p(b)}{M_p h}. \tag{A3}$$

   Here $p$ denotes E or I, $b$ identifies a histogram bin, $N_p(b)$ counts eligible neurons in that bin, $M_p$ is the population’s total number of eligible neurons, and $h$ is the bin width in CV units. Thus each histogram has area one. The x-axis is CV and the y-axis is density, not a neuron count. Black shows E and red shows I. Annotations give medians and eligibility counts; the vertical dashed line marks CV = 1. Values near zero indicate regular firing; values near one support irregular firing, without proving a Poisson process.

Appendix B — Panel B: weak average synchrony

1. Choose cells and slide the counting window.

   Select up to 200 evenly spaced neuron indices from each population. This selection is fixed, rather than randomly redrawn. Count spikes in complete 5, 20 and 50 ms windows, each advanced by 1 ms. No window extends into burn-in or beyond the recording.

   $$n_i^{\Delta}(m)=\sum_k \mathbf{1}\!\left[m\delta\leq t_{i,k}<m\delta+\Delta\right]. \tag{B1}$$

   Here $n_i^{\Delta}(m)$ is neuron $i$‘s count in window $m$, $t_{i,k}$ is a post-burn-in spike time, $\Delta$ is the window width, $\delta$ is the slide step (all times in ms), and $\mathbf{1}$ is one when its condition holds and zero otherwise. The sum runs over that neuron’s spikes. Windows start at $m\delta$ for nonnegative integers $m$ satisfying $m\delta+\Delta\leq T$, where $T$ is the measurement duration. The code bins at the slide step and uses cumulative sums to obtain these same counts efficiently.

2. Calculate each pair’s correlation.

   For each window width, subtract each cell’s mean count across windows and calculate Pearson correlation. Exclude cells whose counts have zero variance at that width; the 20-spike threshold from Panel A does not apply here.

   $$x_i(m)=n_i^{\Delta}(m)-\bar n_i^{\Delta},\qquad r_{ij}(\Delta)=\frac{\sum_m x_i(m)x_j(m)} {\sqrt{\sum_m x_i(m)^2\sum_m x_j(m)^2}}. \tag{B2}$$

   Here $\bar n_i^{\Delta}$ is the mean count over complete windows, $x_i(m)$ is its centred count, $i,j$ identify distinct neurons, and the sums cover the same windows for both cells. Correlation $r_{ij}$ has no units and lies between −1 and +1. It measures same-window association, rather than a lagged correlation.

3. Plot means and pairwise spread.

   Collect each distinct E–E and I–I pair once, and every selected E–I pair. For each pair type and window width, plot its mean correlation and shade one standard deviation on either side:

   $$\bar r=\frac{1}{P}\sum_{p=1}^{P}r_p,\qquad s_r=\sqrt{\frac{1}{P}\sum_{p=1}^{P}(r_p-\bar r)^2}. \tag{B3}$$

   Here $P$ is the number of valid neuron pairs, $r_p$ is one pair’s correlation, $\bar r$ is the plotted mean, and $s_r$ is the pairwise standard deviation. The x-axis is counting-window width in ms, not elapsed recording time. The y-axis is mean Pearson correlation. Black shows E–E, red I–I and cyan E–I; the horizontal dashed line marks zero. Connecting lines join the three measured widths. Shading is spread between pairs, not a confidence interval. Overlapping windows and pairs sharing neurons are not independent observations.

Appendix C — Panel C: population averaging

1. Count spikes and choose groups.

   Count all E and I spikes in disjoint 20 ms bins, separately for each population. Unlike Panel B, these bins do not slide. Include low-rate and silent neurons. Test group sizes 4, 8, 16, and successive powers of two through 2,048, capped at the population size; also include the full population and remove duplicate sizes. For each size, draw 50 groups without replacement within each group. Different groups may overlap and need not be nested. Selection uses fixed random seeds, separately for E and I.

2. Measure how much averaging reduces fluctuations.

   For each group, average its counts in each bin, then compare the variance of this average with the mean individual variance:

   $$\bar n_G(m)=\frac{1}{K}\sum_{i\in G}n_i(m),\qquad Q_G(K)=\frac{\operatorname{Var}_m[\bar n_G(m)]} {K^{-1}\sum_{i\in G}\operatorname{Var}_m[n_i(m)]}. \tag{C1}$$

   Here $G$ is a selected group of $K$ neurons, $i$ identifies a member, $m$ indexes 20 ms bins, $n_i(m)$ is its spike count, and $\bar n_G(m)$ is the group average. Variance is calculated across bins, using division by the number of bins. The ratio $Q_G$ has no units. Exclude groups whose denominator is zero. For uncorrelated count fluctuations, the reference is

   $$Q_G(K)=\frac{1}{K}. \tag{C2}$$

   This reference does not require Poisson firing. Positive shared fluctuations can keep the ratio above it and produce a plateau at large group sizes.

3. Plot groups and fit the slope.

   The x-axis is group size $K$ and the y-axis is normalised variance $Q$; both axes are logarithmic. Black shows E and red shows I. At each size, plot the median across valid groups and shade their 5th–95th percentiles. The grey dashed line is the independent-neuron reference. Fit a straight line to all finite, positive median points in log–log coordinates:

   $$\log Q_{\mathrm{median}}(K)\approx a+b\log K. \tag{C3}$$

   Here $Q_{\mathrm{median}}$ is the median group ratio, $a$ is a fitted intercept, and $b$ is the slope shown in the legend. Each group size receives equal weight; the reference slope is −1. Shading describes group selection, not uncertainty across time or network seeds. At full population size, every draw contains the same neurons. This is a pooling test within one network, not a network-size sweep.

Appendix D — Panel D: current covariance cancellation

1. Reconstruct signed currents.

   Use the 32 evenly spaced E target neurons with recorded voltages and conductances. Combine recurrent and external excitation. At each raw simulation step, reconstruct currents using that step’s conductances and the preceding voltage, matching the simulator’s held-conductance update:

   $$I_i^E[q]=(g_i^E[q]+g_i^X[q])(E_E-V_i[q-1]),\qquad I_i^I[q]=g_i^I[q](E_I-V_i[q-1]). \tag{D1}$$

   Here $q$ indexes raw simulation steps, $i$ identifies a sampled target, $g_i^E,g_i^X,g_i^I$ are recurrent excitatory, external excitatory and inhibitory conductances in µS, $E_E,E_I$ are reversal potentials in mV, and $V_i$ is membrane voltage in mV. Signed currents $I_i^E,I_i^I$ are in nA. The first raw step uses the resting voltage; reconstruction precedes burn-in removal, so the first measured step uses the final burn-in voltage. Excitation is generally positive and inhibition negative in this voltage range.

2. Average covariance over time and distinct target pairs.

   Remove burn-in and subtract each current trace’s mean over the entire remaining recording. At each lag, average products of these centred currents over every valid time sample and every ordered pair of different target neurons:

   $$\delta I_i^a[n]=I_i^a[n]-\frac{1}{N}\sum_{u=0}^{N-1}I_i^a[u],$$ $$C_{ab}(\ell\,\mathrm{d}t)= \frac{\displaystyle\sum_{i\ne j}\sum_{n\in\mathcal O_\ell} \delta I_i^a[n]\,\delta I_j^b[n+\ell]} {M(M-1)(N-|\ell|)}. \tag{D2}$$

   Here $a,b$ denote E or I current, $M$ is the number of sampled targets, $N$ is the number of post-burn-in samples, $n,u$ index those samples, $\ell$ is an integer lag, and $\mathrm{d}t$ is the simulation timestep. The overlap set $\mathcal O_\ell$ contains indices for which both $n$ and $n+\ell$ are inside the recording. Thus positive lag compares the first current with a later value of the second. The denominator uses the actual overlap length; means remain global rather than being recalculated at each lag.

   With 32 targets there are 992 ordered distinct-cell pairs. Same-cell terms are excluded. An FFT computes these sums efficiently with padding to prevent circular wraparound; it does not smooth the curves. All components use the same pair set and normalisation.

3. Sum components and calculate cancellation.

   Compute the four components and their sum, and separately check that the sum matches covariance calculated from the total current:

   $$C_{\mathrm{tot}}(\tau)=C_{EE}(\tau)+C_{II}(\tau)+C_{EI}(\tau)+C_{IE}(\tau). \tag{D3}$$

   Here $\tau=\ell\,\mathrm{d}t$ is lag in ms and $C_{\mathrm{tot}}$ is the between-cell covariance of summed E and I current. The cancellation percentage at zero lag is

   $$100\left[1-\frac{C_{\mathrm{tot}}(0)}{C_{EE}(0)+C_{II}(0)}\right]\%. \tag{D4}$$

   This compares the total with the sum of same-type components; it is reported when that denominator is positive. The in-plot annotation gives the ratio inside the subtraction, rather than the percentage. This decomposition describes coordinated fluctuations, not just a balance of mean currents.

4. Filter only the additional total-current trace.

   For the grey dotted trace, first apply a causal 20 ms exponential filter to each cell’s total current, then calculate its covariance using the same procedure:

   $$J_i[q]=I_i^E[q]+I_i^I[q],\qquad \widetilde J_i[q]=\alpha\widetilde J_i[q-1]+(1-\alpha)J_i[q],\qquad \alpha=e^{-\mathrm{d}t/\tau_f}. \tag{D5}$$

   Here $J_i$ is total current, $\widetilde J_i$ is filtered total current, $\tau_f=20$ ms is the filter time constant, and $\alpha$ is its decay factor. Initialise the filter at zero before the first raw step and filter the full record before removing burn-in. This is covariance of filtered currents, not smoothing applied to the plotted covariance curve.

5. Read the axes and traces.

   The x-axis is lag in ms, from −50 to +50, at the simulation timestep. The y-axis is mean between-cell covariance in nA², not correlation bounded by −1 and +1. Dashed black shows EE, solid red II, solid cyan EI and dashed amber IE. Solid black is their total; the grey dotted line shows the filtered total’s covariance. E and I label current types here: every target neuron is E.

   Negative cross-type components oppose positive same-type components, reducing the black total around zero lag. The main curves receive no extra smoothing or fitted shape. They look smooth because synaptic currents have finite decay times and covariance is averaged over time and many pairs. Neither the pairs nor neighbouring lag points are independent, and no confidence band is shown.

Network diagram

Parameters

The table uses the saved starting network and settings behind the displayed figures. Paired values are E / I. Synapses use a single decay time and fixed delays, approximating the paper’s model. Weight sampling compensates for SNNLab’s sparse rescaling; clamping negative samples slightly increases the surviving mean. The table distinguishes these weight definitions and includes zeros in overall statistics.

All measurements exclude burn-in. Count windows slide by 1 ms. Signed currents are reconstructed from conductances and the preceding voltage; current covariances compare distinct cells. Excitation includes recurrent and external input.