nature neuroscience: doi:10.1038/nn€¦ · however, the log-normal distribution may reflect the...

Supplementary Figure 1

Intrinsic properties of cortical neurons in culture vs density and age.

Intrinsic properties vs. neuronal density (a) or days in vitro (DIV, b). Data presented as mean ± STD (n = 491 neurons in 165 preparations). To quantify the relation between the measured parameters and density or DIV, Spearman Correlation and p-values (*: p < 0.01; **: p < 0.001) were calculated (green). To determine whether there were significant changes in the parameters with density or DIV, we compared the values at the lowest and highest densities and DIV (50 points from each end of the plots) and calculated Mann-Whitney U-test p-values (red). Note that for density there were no statistically significant differences in the measured properties (a). Several parameters (input resistance, rheobase, F0.5) changed with DIV (b) but all had reached steady-state by 2 weeks, when the experiments were performed.

Abbreviations: Vm, membrane potential; Rm, input resistance; , membrane time constant; HW, half width of action potential; AHP,

afterhyperpolarization; FI: slope of the firing vs current plot; F0.5, firing rate evoked when 0.5 nA current step was injected. Rheobase is minimum current to evoke a spike. Spike threshold is measured from resting membrane potential.

Nature Neuroscience: doi:10.1038/nn.4415


Connection probability spatial profiles.

Spatial profiles of connection probability for E-E (a), I-E (b), E-I (c), I-I (d) neurons vs Euclidian distance between neurons (first letter is presynaptic neuron). The number of connections tested is shown in grey. Data points were fitted with a Gaussian function

with parameters in Supplementary Table 2. (e) The probability of reciprocal connection between E-E

neurons (red), E-I (blue) or regardless of their types (black) is not significantly different from the expected probability (dotted lines) that was predicted from the product of unidirectional connections (a-c). (f) Combined connection probability between neurons. (g) Peak

connection probability (= connection probability for neurons <350 m apart) between E neurons (red) and between E and I neurons (E-I

and I-E, blue) vs neuronal density. (h) Synaptic delay as a function of distance for excitatory (red) and inhibitory (blue) connections. All

data (EPSPs and IPSPs) were pooled and fitted by a linear relation of slope 183 ± 34 m/ms and intercept 3.55 ± 0.41 ms (95%

confidence interval).


p.chinnadurai

Stamp


Log-normal distribution of postsynaptic potentials (PSPs).

(a) Distribution of peak PSP magnitudes. EPSP and IPSP magnitudes were not statistically different (see Supplementary Table 3) and were combined. (b) Normal probability plot of PSPs. Deviation from a straight line indicates departure from normality. (c) Distribution of the (base 10) logarithm of PSPs values. (d) Normal probability plot of the logarithm of PSPs values. This shows that PSPs sizes follow a log-normal distribution. However, the log-normal distribution may reflect the fact that the PSP magnitude varies with network density (Supplementary Table 3). (e-h) Same as a-d but for normalized values of PSPs. To remove PSPs dependence on density, PSPs were divided by the values expected from the fit of PSPs size as a function of number of connections (Fig. 1d). Altogether, this shows that PSPs at any given density follow a log-normal distribution.



Modulating correlations of light pulse trains delivered to ChR2-expressing neurons.

For reasons given in Methods, the sequences of light pulses delivered to ChR2 neurons were generated using leaky-integrate-and-fire (LIF) units. Each unit was driven with noisy inputs and the resultant spikes were used to time the light pulses. (a) To introduce

correlation in the light pulses, the current input to the LIF unit k (here k = [1, 2], Eq. 4 of the Methods) was a sum of an

independent ( ) and a common ( contribution that were weighted by and respectively (Eq. 5 of the

Methods, see also ref. 31). By construction, the correlation between two input currents and was equal to Cstim, which

varied from 0 (no correlation) to 1 (perfect correlation). (b) In the main text (Fig. 5), correlations were presented as Cstim rather than by the calculated spike count correlation for the following reason. Unlike Cstim, which is calculated from the continuous current, the

correlations in the spikes (and hence light pulses) depend on the time window ( t) used for counting spikes (see Methods). Plotting

spike correlation vs Cstim shows different relations for t = 50 ms (red) and t = 10 ms (black). Therefore, to avoid potential confusion, it

was more convenient to use Cstim.


p.chinnadurai

Stamp

p.chinnadurai

Stamp

p.chinnadurai

Stamp

p.chinnadurai

Stamp

p.chinnadurai

Stamp

p.chinnadurai

Stamp

p.chinnadurai

Stamp


Effectiveness of light stimulus in evoking spikes in ChR2-expressing neurons.

In the following, neurons expressing ChR2 are shown in light green. Recorded neurons are demarcated by electrodes. Light-stimulated neurons have coloured contours. (a) A train of light pulses (black dots) was applied first to one neuron (blue, top) and then to another (magenta, bottom). The pulses evoked spikes reliably over several trials (red and magenta dot rasters). In either case, no action potentials were evoked in the non-stimulated neuron. (b) The network was activated by stimulating an additional 11 ChR2-expressing neurons. The pulses delivered to the neurons were uncorrelated (Cstim = 0). The blue and the magenta neurons were recorded simultaneously. Whereas the blue neuron responded only when stimulated by the light pulses, the magenta neuron also fired action potentials due to recurrent network activity. (c) Proportion of successful spikes elicited by a light pulse of 5 ms at increasing

frequencies of stimulation ( stim = 5-20 Hz at 10 mW/mm2 in a region of interest around the neuron soma). Data were collected when

neurons were stimulated independently and are represented as mean ± SEM. The spike delay was measured between the centre of the light pulse and the peak of the spike and was 5.7 ± 1.6 ms (mean ± SD, n°=°18 cells). These measures are in good agreement with previously published work.

Mattis, J. et al. Principles for applying optogenetic tools derived from direct comparative analysis of microbial opsins. Nat Methods 9, 159-172, doi:10.1038/nmeth.1808 (2012).



Optical stimulation where ChR2 is expressed in E and I cells via viral transfection.

Networks where ChR2 is expressed virally differ from those where ChR2 expressed with transgenic mice in two ways. First, expression is sparse so that neurons can be driven individually with each ROI compared to 1-4 activated neurons per ROI with transgenic expression. Second, ChR2 was expressed in both E and I cells. (a, b) Repeating the experiments in Figs. 4-5 of the main text with the virally transfected network produced qualitatively similar firing patterns (compare also with Supplementary Fig. 7): 1) there was variability between simultaneously recorded cells (1 additional cells in a and 2 in b did not fire) and 2) variability within cells but with repeatable spikes across trials (more pronounced in b). (c) left, Trial averaged subthreshold potentials in 4 cells show: 1) variability in isolated E (red) and I (blue) inputs and compound PSP at rest (cPSP, black) across cells; 2) E-I balance so that the cPSP was less than the isolated EPSP; and 3) E-I tracking where the isolated EPSP mirrored the isolated IPSPs. These effects occurred when the pulse correlation (Cstim) was increased (middle (Cstim = 0.5), right (Cstim = 1) columns). Note that the cPSP in Neuron 3 was hyperpolarizing. This was likely due to the fact that ChR2 was also expressed in I cells, which were directly stimulated with the light pulses. Stimuli were with 25 ROIs delivered at an average rate of 5 Hz.



Optical stimulation where ChR2 is expressed only in E cells via transgenic mice.

(a, b) Two additional examples showing variability in spikes across cells (2 cells in a and b did not fire) and reliable spikes evoked across trials (more obvious in a than in b). (c, d) Two additional examples showing recordings of spikes (in cell-attached mode) and

cPSPs (in whole-cell mode with spikes inhibited) in the same cell. The stimulus has an input rate stim = 5 Hz and a stimulus correlation

Cstim = 0 as in Fig. 3a of the main text.



Fano factors and firing rates of spontaneous and evoked activities.

(a) Fano factor calculated for the data combined across densities (spontaneous: black, n = 58 neurons in 18 preparations; evoked: red, n = 113 neurons in 44 preparations). Note that the Fano factors for the evoked responses were close to 1 for increasing window size, indicating a near Poisson process. (b, c) Histogram of firing rates during spontaneous (b; average spontaneous rate 0.87 ± 1.03 Hz,

mean ± STD) and evoked (c, for an input rate stim = 5 Hz and a stimulus correlation Cstim = 0; average evoked rate 1.91 ± 1.70 Hz,

mean ± STD) activity regardless of network density. Inset: same data represented in linear scale. Note that the distribution is long-tailed in both the spontaneous and evoked conditions but during the evoked condition, the firing rate seems to saturate. (d-f) Histogram of firing rates during stimulation at 2 Hz, (d), 5 Hz (e), and 10 Hz (f) (60 neurons in 20 preparations). The saturation of firing rates is more pronounced at higher stimulation rates which may result from the exploration of a more nonlinear or even saturating portion of the neuronal transfer function and is in line with theoretical results

7,8,37.



Firing rate and membrane potential vs. ‘true’ stimulus rate.

(a) Evoked firing rate vs stimulus rate of light pulses delivered to each ROI in low (n = 20 neurons in 8 preparations, magenta), medium (n = 22 neurons in 6 preparations, green), and high (n = 18 in 6 preparations, orange) density networks. Data are the same as in Fig. 4g but are here plotted as a function of the true stimulus rate which has been corrected by the expected ratio of missed spikes (Supplementary Fig. 5). (b) Average magnitude and standard deviation (inset) of EPSPs (red), IPSPs (blue), and composite PSPs (black) vs stimulus rate (n = 45 neurons in 12 preparations). Data are the same as in Fig. 4h but are here plotted as a function of the true stimulus rate. Data presented as mean ± SEM.



Decorrelation of signal and noise.

(a) Signal correlation (for an input rate stim = 5 Hz and a stimulus correlation Cstim = 0) of membrane potential vs holding potential.

Isolated EPSPs (at -80 mV holding potential) or IPSPs (at 0 mV holding potential) were significantly more correlated than cPSPs (at -60 mV holding potential; n = 253 pairs in 56 preparations, paired Mann-Whitney U-test with a p-value 7×10

-36 or 2×10

-24 for EPSPs

vs cPSPs or IPSPs vs cPSPs, respectively). (b) Correlation of inhibitory potential vs correlation of excitatory potentials. The grey line denotes the slope of unity passing through the origin (Pearson correlation coefficient between the two measures is 0.58, p = 6×10

-24).

(c) Same as a but for noise correlation. Isolated EPSPs or IPSPs were significantly more correlated than cPSPs (n = 253 pairs in 46 preparations, paired Mann-Whitney U-test with a p-value 4×10

-20 or 2×10

-7 for EPSPs vs cPSPs or IPSPs vs cPSPs, respectively). (d)

Same as b but for noise correlation (the Pearson correlation coefficient is here 0.66, p = 8×10-33

).



Mean and standard deviation of subthreshold potentials vs. Cstim when only E cells were stimulated.

(a) Average magnitudes of evoked EPSPs (red), IPSPs (blue), and composite PSP (black) as a function of stimulus correlation Cstim (n = 38 cells in 12 preparations). (b) Standard deviation vs Cstim. Data are represented as mean ± SEM.



Modulation of signal and noise correlation by Cstim when both E and I neurons were stimulated.

(a) Spike correlation as a function of the stimulus correlation (n = 10 pairs in 7 preparations). (b, c) Correlations between isolated EPSPs (red), isolated IPSPs (blue) and composite PSPs (black) at resting potential for signal (b) and noise (c) correlations (n = 99 pairs in 23 preparations). (d) Average magnitudes of evoked EPSPs (red), IPSPs (blue), and composite PSP (black) as a function of input correlation Cstim (n = 78 cells in 23 preparations). (e) Standard deviation vs Cstim. Data are represented as mean ± SEM.


Supplementary Text: Mean field equations of synaptic inputs

In the mean field framework, general variables such as the firing rate or the correlation between neurons can be analytically predicted

from network parameters7,8,17. To simplify analyses only a few key parameters are considered: the number (N) of excitatory and

inhibitory neurons; the strength (J) and number (K) of synaptic connections; and the connection probability (Pc) between cells. To

maintain in vivo-like variability in spiking, one scheme posits that K is large and may increase with, but remains much smaller than N

(1 << K << N) so that neurons receive independent inputs. The mean (𝜇) and variance (𝜎2) of the synaptic inputs, which includes

recurrent excitation (E) and inhibition (I) and external excitatory component (X), to each neuron are given by8:

𝜇(𝑡) = 𝐾 ∙ 𝐽𝐸 ∙ 𝑚𝐸(𝑡) − 𝐾 ∙ 𝐽𝐼 ∙ 𝑚𝐼(𝑡) + 𝐾 ∙ 𝐽𝑋 ∙ 𝑚𝑋(𝑡) [1]

𝜎2(𝑡) = 𝐾 ∙ 𝐽𝐸2 ∙ 𝑚𝐸(𝑡) + 𝐾 ∙ 𝐽𝐼2 ∙ 𝑚𝐼(𝑡) + 𝐾 ∙ 𝐽𝑋2 ∙ 𝑚𝑋(𝑡) [2]

where 𝑚𝛼 is the average population activity (α = E, I, X).

It is important that the variance 𝜎2 ∝ 𝐾 ∙ 𝐽2 ∙ 𝑚 does not depend on K and will not vanish (order 1), ensuring sufficient membrane

potential fluctuations to generate variability in spiking regardless of network size. This condition is satisfied when the excitatory and

inhibitory synaptic strengths obey 𝐽𝛼 = 𝑗𝛼 √𝐾⁄ (α = E, I, X; 𝑗𝛼 is a constant) so that:

𝜇(𝑡) = √𝐾 ∙ 𝑗𝐸 ∙ 𝑚𝐸(𝑡) − 𝑗𝐼 ∙ 𝑚𝐼(𝑡) + 𝑗𝑋 ∙ 𝑚𝑋(𝑡) [3]

𝜎2(𝑡) = 𝑗𝐸2 ∙ 𝑚𝐸(𝑡) + 𝑗𝐼2 ∙ 𝑚𝐼(𝑡) + 𝑗𝑋2 ∙ 𝑚𝑋(𝑡) [4]

Thus, an immediate consequence of 𝐽~ 1 √𝐾⁄ is that the mean activity may diverge with K. However, if the excitation (𝑗 ∙ 𝑚𝐸(𝑡) +

𝑗𝑋 ∙ 𝑚𝑋(𝑡)) is counter-balanced by inhibition (𝑗𝐼 ∙ 𝑚𝐼(𝑡)) of comparable magnitude (the difference needs to be order 1 √𝐾⁄ ), the mean

𝜇 to each neuron can be maintained near rheobase (order 1) so that the network activity is neither at a saturated level nor completely

silent as K grows large. This balanced regime can be achieved over a wide range of parameters assuming that inhibition is sufficiently

strong.


All Vm (mV) Rm (MΩ) 𝜏 (ms) Vt (mV) AHP (mV) HW (ms) F0.5 (Hz)

E (n = 378) -61.4±6.7 192 ± 101 23.8 ± 12.8 -42.9 ± 6.2 -12.8 ± 5.8 2.2 ± 0.7 34 ± 20

I (n = 113) -59.6 ± 5.7 135 ± 90 15.0 ± 9.3 -45.7 ± 6.4 -14.8 ± 5.7 1.6 ± 0.5 51 ± 28

Connected Vm (mV) Rm (MΩ) 𝜏 (ms) Vt (mV) AHP (mV) HW (ms) Fmax (Hz)

E (n = 198) -62.0 ± 6.5 178 ± 103 27.1 ± 12.6 -42.6 ± 6.1 -12.1 ± 5.8 2.3 ± 0.7 30 ± 23

I (n = 65) -59.7 ± 6.0 129 ± 78 15.5 ± 10.7 -46.1 ± 7.3 -14.6 ± 5.9 1.7 ± 0.5 46 ± 24

Non-Connected Vm (mV) Rm (MΩ) 𝜏 (ms) Vt (mV) AHP (mV) HW (ms) Fmax (Hz)

E (n = 177) -60.9 ± 7.0 205 ± 119 19.7 ± 11.4 -43.3 ± 6.5 -13.4 ± 5.7 2.0 ± 0.9 40 ± 24

I (n = 48) -59.6 ± 5.2 143 ± 104 14.3 ± 7.0 -45.2 ± 5.0 -15.0 ± 5.7 1.5± 0.4 58 ± 31

Supplementary Table 1

Intrinsic properties of neuron in culture.

Intrinsic properties of cortical neurons in culture for all recorded neurons (All, top), for neurons that evoked a PSP in a post-synaptic neurons (Connected, middle) and for neurons that were putatively attributed to excitatory or inhibitory types base on their electrophysiological properties (Non-Connected, bottom). Values correspond to mean ± SD.

Abbreviations: Vm, membrane potential; Rm, input resistance; 𝜏, membrane time constant; Vt, threshold membrane potential; AHP, after hyperpolarization potential; HW, half width of action potential; F0.5, maximum firing rate for an injected current of 0.5 nA.


Post-synaptic Presynaptic

E I

E VPSP = 2.2 ± 2.2 mV 𝑡𝑝𝑒𝑎𝑘 = 11.4 ± 7.7 ms

p0 = 0.38 𝜎 = 602 µm

VPSP = 2.4 ± 2.4 mV 𝑡𝑝𝑒𝑎𝑘 = 7.4 ± 5.4 ms

p0 = 0.59 𝜎 = 835 µm

I VPSP = -2.2 ± 2.4 mV 𝑡𝑝𝑒𝑎𝑘= 22.0 ± 5.4 ms

p0 = 0.64 𝜎 = 481 µm

VPSP = -2.7 ± 2.0 mV 𝑡𝑝𝑒𝑎𝑘= 13.7 ± 6.2 ms

p0 = 0.59 𝜎 = 412 µm


Synaptic properties of neuron in culture.

Average synaptic properties of neurons in culture and fit values of the connection probability profiles from Supplementary Fig. 2. Data are reported as mean ± SD (n = 261 EPSPs and n = 99 IPSPs).

Abbreviations: VPSP, post-synaptic potential; 𝑡𝑝𝑒𝑎𝑘, time to peak from onset (measured as the maximal slope of the post-synaptic potential); p0, peak connection probability; 𝜎, is the characteristic length of the connection probability function and denotes the spread of connectivity.


Generalized linear model Correlations

Est. SE P-value DIV Dens. PreS PostS Dist. Delay T2P Vm Rm Tau

Intercept 2.40 0.10 8.9×10-70

DIV 0.055 0.113 0.63 -0.17 -0.03 -0.02 0.079 -0.02 -0.11 -0.01 -0.31 -0.25

Dens. -1.212 0.111 9.9×10-24 -0.17 0.057 0.044 -0.05 -0.11 0.108 0.145 0.075 0.19

PreS 0.050 0.124 0.69 -0.03 0.057 -0.06 -0.11 -0.16 0.427 0.059 0.02 -0.05

PostS 0.151 0.118 0.20 -0.02 0.044 -0.06 -0.02 -0.18 -0.31 0.174 -0.31 -0.36

Distance -0.289 0.123 0.02 0.079 -0.05 -0.11 -0.02 0.479 -0.05 -0.08 -0.14 0.004

Delay -0.102 0.129 0.43 -0.02 -0.11 -0.16 -0.18 0.479 0.156 -0.01 0.147 0.24

T2P -0.111 0.138 0.42 -0.11 0.108 0.427 -0.31 -0.05 0.156 0.003 0.395 0.429

Vm 0.303 0.108 0.005 -0.01 0.145 0.059 0.174 -0.08 -0.01 0.003 0.041 -0

Rm -0.133 0.156 0.39 -0.31 0.075 0.02 -0.31 -0.14 0.147 0.395 0.041 0.702

Tau 0.323 0.162 0.05 -0.25 0.19 -0.05 -0.36 0.004 0.24 0.429 -0 0.702


Correlation between synaptic strength and network density.

left, Generalized linear regression model explaining the absolute magnitude of PSPs as a function of 10 parameters: the age of the culture (DIV), the density (in logarithmic scale) of the culture, the type (E or I) of presynaptic (PreS) and postsynaptic (PostS) neuron, the distance between neurons, the delay of PSP, the time to peak of PSP (T2P), and the resting membrane potential (Vm), the resistance (Rm) and the time constant (Tau) of postsynaptic neuron. Parameters were z-scored to enable direct comparison of coefficients. In bold is mentioned the significant parameter affecting the synaptic strength (p-value < 0.005, Bonferroni correction for multiple comparisons). Note that PSPs sizes do not depend on cell type which justifies the fact that we pooled EPSPs and IPSPs in our analysis as in Fig. 1d and Supplementary Fig. 2h.

right, Correlation matrix between input parameters of the linear model (in bold correlation with a p-value < 0.001, Bonferroni correction for multiple comparisons).


nature neuroscience: doi:10.1038/nn€¦ · however, the log-normal distribution may reflect the...

Documents