The functional clique.

<p>(A) Cross correlation functions C() between the spike trains of two critical neurons. has been measured as the position of the maximum of the cross correlation between the time series of the two considered neurons. The panels refer to all the possible pair combinations of the critical neurons, furthermore blue (red) histograms refer to the analysis performed during the population burst build up (during periods out of the bursting activity). For more details see the subsection Functional Connectivity in Methods. The order of activation of each pair is reported on the top of the corresponding panel, whenever the cross-correlation has a significant maximum at some finite time . Note that during the PB onset, neurons activate reliably in the following order . During the out-of-burst activity, clear time-lagged activations are present only among the pairs - and -. (B) Structural connections among the four critical neurons: the black arrows denote the directed connections. The data here reported, as well in all the following figures, refer to a network with correlations of type T1 plus T2.</p

Model based reconstruction of the SNS experiment for the critical neuron .

<p>Number of emitted PBs as a function of the stimulation current applied to the neuron . The red line with dots refers to the results of the SNS experiment on (same curve as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003823#pcbi-1003823-g003" target="_blank">Fig. 3 C</a>) and the black line to the estimations obtained by measuring the PB occurrence with the simple model for SNS, described in the Methods. The measurement were performed in both cases over a time interval s.</p

Single neuron stimulation (SNS) can stop population bursting activity in presence of type T1 plus T2 correlations.

<p>A sketch of a SNS experiment for a network with type T1 plus T2 correlations is reported in (A) and (B): the neuron is stimulated with a DC step for a time interval (as shown by the red line on the top panel). Average firing rate of neuron (A) and network activity (B) as measured during the experiment. (C) and (D) refer to correlated and uncorrelated networks, respectively. Upper panels display the number of population bursts, PBs, delivered during SNS experiments versus the stimulated neuron, ordered accordingly to their average firing rates under control conditions (bottom panels). Each neuron was stimulated with a DC step (switching its excitability from to ) for an interval s. The critical neurons are signaled by red circles. The number of PBs, emitted in control conditions within an interval s, are also displayed: red dashed lines indicate their averages, while the shaded gray areas correspond to three standard deviations. The data refer to mV and neurons.</p

The critical neurons precede the population bursts in a network with correlations of type plus .

<p>(A) Raster plot of the network activity: every dot denotes a firing event. The dashed green lines and black dots refer to the four critical neurons. (B) Enlargement of a representative population burst: PBs are anticipated by the ordered firing sequence . For clarity reasons, in the raster plots, at variance with all the other figures, the neuronal labels are not ordered accordingly to their firing rates.</p

Impact of single neuron stimulation on the population activity: dependence on the injected current.

<p>Color coded rates of emission of PBs during SNS experiment performed on each single neuron for a range of injected DC currents (y-axis) in networks with correlations of type T1 plus T2 (A) and without any correlations (B). The neurons are ordered according to their functional out-degree rank (x-axis) and the PB rates during SNS are normalized to the PB rate in control conditions. (C–F) Number of PBs emitted during SNS of the critical neurons ,,, versus the stimulation current . The red arrows indicate employed for the SNS experiments in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003823#pcbi-1003823-g001" target="_blank">Fig. 1 C</a>. The blue vertical dashed lines mark the value of the intrinsic excitability and the horizontal magenta solid line the bursting activity of the network, both measured at rest. The number of PBs are measured over a time interval 84 s.</p

Effective synaptic strength during single neuron current stimulation.

<p>Average synaptic strength of the afferent (A), and efferent (B) connections of the critical neuron during SNS with mV (these data corresponds to the experiment reported in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003823#pcbi-1003823-g001" target="_blank">Fig. 1</a>). The output (input) effective synaptic strength is measured in terms of the average value of the fraction () of the synaptic transmitters in the recovered state associated to the efferent (afferent) synapses (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003823#s4" target="_blank">Methods</a>). (C) Time averaged synaptic strengths as measured during SNS experiments performed on each of the four critical neurons for various stimulation currents . The legend clarifies to which neuron corresponds the average synaptic strengths displayed in the figure, the averages have been performed over 84 s.</p

Comparison between single neuron stimulation (SNS) and deletion (SND) in a network with correlations of type plus .

<p>(A) Number of PBs emitted during SND experiments versus the label of the removed neuron. (B) Functional and structural properties of the network, as measured in control conditions, i.e. in absence of any stimulation/manipulation of the neurons. From top to bottom: functional out-degree , intrinsic excitability , and total structural connectivity . The red dashed line and the gray shaded area in (A) as well as the neuron labels are as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003823#pcbi-1003823-g001" target="_blank">Fig. 1 C</a>, the blue dashed line denotes mV. (C) Comparison between SNS and SND: the number of PBs occurring during SNS (resp. SND) is reported as a function of , and . In all panels the green (red) circles mark the critical neurons, which under SND (SNS) can silence the bursting activity of the network. The bursting activity is recorded over an interval s.</p

(A) Synapse strength and firing time delay between the neurons and .

<p>Time evolution of the effective synaptic strength (red solid line and right <i>y</i>-axis) and of the firing time delay (black line with dots and left <i>y</i>-axis). <b>(B),(C) Failures and successes in population burst ignition.</b> Spike time delay (top panel) and (bottom panel) of neuron and , respectively, referred to the last firing time of . Panels (B) and (C) clearly show that PBs (denoted by green vertical lines) can occur only when the neuron and fire within precise time windows after the firing of neuron . In (B) a clear failure is indicated by red circles, in this case fired at the right time, but was too slow; in (C) neuron fires at the right moment several times (black dots are within the gray shaded area in the top panel), but the avalanche is not initiated until does not emit a spike within a precise time interval after the firing of . In all the figures, the data refer to control conditions. The blue horizontal dashed lines refer to the average value of or at the PB onset, while the shaded gray areas indicate the corresponding standard deviations.</p

Anti-resonances observed during SNS of .

<p>The first column reports the stimulation currents for which pronounced minima (anti-resonances) are observed in the stimulated PB activity during SNS experiment on (same data as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003823#pcbi-1003823-g003" target="_blank">Fig. 3 C</a> and red curve in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003823#pcbi-1003823-g008" target="_blank">Fig. 8</a>), the second column the corresponding ratios. and are the firing periods of the LIF neurons and in isolation, namely, and . The third column refers to the anti-resonances generated by employing the simple model for SNS of introduced in the Methods (same data as the black curve in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003823#pcbi-1003823-g008" target="_blank">Fig. 8</a>). The reported values correspond to the minima in the PB activity for this model, the absence of a value means that the model did not display a corresponding minimum. The data refer to SNS experiments performed over a time interval of duration 84 s.</p><p>Anti-resonances observed during SNS of .</p

Single neurons dynamics.

<p>(A and E) Extracellular recording from a single electrode in the leech (A) and in the hippocampal network (E) showing the activity of neurons with periodic (A), Poissonian (E) and bursting (A and E) firing. (B and F) ISI distribution of identified neurons with exponential dynamics for leech (B) and hippocampal (F) network. Black dashed lines indicate exponential fit. (C and G) ISI distribution of identified neurons with bi-exponential dynamics for leech (C) and hippocampal (G) network. Black dashed lines indicate bi-exponential fit. (D and H) ISI distribution of identified neurons with periodic dynamics for leech (D) and hippocampal (G) network. Label T indicates the position of the peak, corresponding to the period of the firing.</p