Dynamics and Effective Topology Underlying Synchronization in Networks of Cortical Neurons

a, b, Color-coded representations of the number of different electrodes participating in a network spike (x-axis) as a function of threshold for detection of network spikes (y-axis). Colored scale bar (on the right) depicts number of occurrences. The two panels show distributions obtained over a 2 h period in two different networks that faithfully represent the behavior of the entire set of ∼40 networks served for this study. Threshold is expressed in terms of the number of electrodes that are required to be active (within a 3 ms time bin) for time-stamping an NS. Time bin width was also subjected to systematic variations (from 3 to 30 ms) with no qualitative effect (data not shown). c, Approximately 900 NSs obtained over 1 h of recording from the network shown in b, using a very low threshold (4 action potentials in 10 ms time bin). The identity of the electrodes participating in each NS is shown (black). Although in the majority of the NSs all active electrodes appear, there exists a subpopulation of aborted NSs in which only a subset of the active electrodes participate. Note that the NSs are not depicted according to the chronology of their appearance; rather, for purposes of clarity, they are sorted using a hierarchical clustering algorithm.

a, Inset, Grand average network spike, calculated by averaging 21 average NSs similar to those shown in Figure 1d (total of 5796 NSs). Marked in brown is the initial segment, enlarged (brown dots) in the main figure. An exponential growth equation, A(t) = a + b × e^(σ−1)t, was fitted to this initial segment; the resulting function (a = 0.05, b = 0.01, σ = 1.045) is depicted by a continuous black line. b, Top, Distribution of NS peak activity for one network that demonstrated a relatively broad spectrum of NS amplitudes. A very low threshold for NS detection was used here (4 action potentials in 10 ms bin). Bottom, Early recruitment phases for 207 low-amplitude NSs (≤6 action potentials/ms; depicted in black) and 208 high-amplitude NSs (≥10 action potentials/ms; depicted in gray), fitted with exponential growth equations, yielding σ values of 1.04 and 1.05, respectively.

a, Network spikes recorded over a period of 2 h in the absence (left) and presence (right) of 5 μm bicuculline. These NSs were obtained using a threshold of three action potentials within a 10 ms time bin. Each black dot marks an action potential detected in any of the electrodes during the ±250 ms surrounding the NS threshold. NSs are ordered using a clustering algorithm to enhance visualization of bicuculline effects: a reduced number of aborted NSs and a more vigorous activity within each NS. b, An average NS obtained in the presence of 5 μm bicuculline (brown). Average NS in control solution is shown in black. Top right inset, Fitted fast declining phase in the presence of bicuculline, representing the timescale of cellular-level (intrinsic) restoring forces. Bottom right inset, Fitted slow declining phase of the control NS, representing the timescale of restoring force acting through the inhibitory subnetwork. Note the timescale separation between the two types of restoring forces involved (6.5 ms for the effect of cellular-level forces; 33.7 ms for the effect of the inhibitory subnetwork). c, Averaged phases of recruitments extracted from 16 experiments before (black) and after (brown) the addition of 5 μm bicuculline to the bathing solution. Each recording episode lasted 1–2 h. Broken lines depict 1 SD (one-sided for clarity). Thick lines depict segments for which a single-exponential recruitment function could be reliably fitted for both conditions; σ values obtained were 1.06 and 1.22 for control and bicuculline conditions, respectively.

a, b, Firing rate of a relatively active neuron; the rate increases during an NS, decreases dramatically after the NS, and then gradually recovers (arrows in b). These kinetics are used for estimation of recovery from a network spike. c, Fitted recoveries, such as those depicted by arrows in b, in control (black) and bicuculline (brown) solutions. [Note that the inter-NS intervals in bicuculline solution are short; points at which the average firing rate during recovery was “contaminated” by the presence of NSs were omitted (bins 7–9, 11, 14 s)].

Numerical solution of the modified logistic equation dA/dt = (s(A,t) σ − 1) i(A,t) A (1 − A) for spontaneous and evoked NSs. a, Time constants of i(A) and s(A). Exponential rate equations for i and s were devised to approximately match the timescales of activation and recovery of the restoring forces at the extremes of A = 1 and A = 0, based on data shown in Figures 3–5. The equations used the following (time in millisecond units): βs = 30e^{−20(A(t) + 0.5)}; αs = 0.00001e^10A(t); βi = 10e^{−20(A(t) + 0.5)}; αi = 0.000001e^10A(t). 0 ≤ A(t) ≤ 1; σ was set to 1 below a threshold of A(t) = 0.05. Once the threshold is crossed, σ is set to a value ranging from 1.04 to 1.06. Noise was generated using normal (Gaussian) distribution around mean A(t), with SD of 0.001 and 0.0002 for b and c, respectively. For the case of evoked network spikes (c), a fraction of the population is excited (i.e., stimulated) from around threshold and higher in discrete steps. Inset in c shows data from an experiment in which a network was excited by applying short (0.4 ms) biphasic current pulses between a pair of electrodes [stepping from 10 μA (bottom trace) to 80 μA (top trace)]. Average evoked responses to 15 presentations of stimuli at each amplitude are shown (each trace is 300 ms long). Missed responses were not included except for the case of 10 μA stimulation amplitude. We observed 15 of 15 missed responses for 10 μA stimuli, 5 of 15 for 20 μA, 4 of 15 for 30 μA, and 2 of 15 for 40 μA; beyond 40 μA stimulation, no missed responses were observed.

a, Firing probability of neurons as a function of time surrounding an NS. Color scale ranges from 0 to 1 spike per 5 ms (the few cases in which >1 spike occurred during a 5 ms bin are represented as 1 spike/bin). b, In a given network, the early-to-fire neurons are similar for spontaneous (left) and evoked (right; 3 different stimulation sources) NSs. For the case of spontaneous NS, arrows point to times for which firing probabilities are presented. Probabilities of firing 0–5 and 25–75 ms after stimulation are shown for three different stimulation sites (S1, S2, S3). Short (0.4 ms) biphasic 30 μA current pulses between each of three different pairs of electrodes were applied. Average responses to 60 presentations of stimuli for each stimulation point are shown. Horizontal lines depict four examples of early-to-fire neurons. Poststimuli time histograms for the three stimulation sites are shown at the bottom right.

A network with a relatively large number of aborted network spikes is subjected to analyses aimed at comparing early-to-fire neurons in aborted versus full-blown NSs. a, All 1087 NSs detected over 1 h of recording are shown in a raster plot, ± 250 ms surrounding a detection threshold of four action potential in 10 ms time bin. A black dot depicts an action potential recorded in any of the electrodes during the NS. Responses were reordered using a clustering algorithm to enhance visual separation between full-blown and aborted network spikes. b, The distribution of peak number of action potentials in each NS (1 ms time bin; same procedure as in Fig. 3b). Distributions around threshold for 537 NSs with a peak of ≤5 action potentials (c) and 437 NSs with a peak of ≥10 action potentials (d) are shown using color scale ranges from 0 to 0.33 action potentials per 5 ms (the cases in which >0.33 action potentials occurred during a 5 ms bin are represented as 0.33 action potentials per bin). Note that the same early-to-fire neurons are active for both subsets of NSs. The correlation between firing rates of the electrodes of both subsets at times ranging from −300 to −50 ms before NS threshold is 0.98.

a, Cumulative number of spikes emitted by six neurons in a network, as a function of time relative to the occurrence of an NS (for definition of NS timestamp, see Materials and Methods). Inset, Distribution of intervals between spontaneously occurring NSs (491 synchronous events recorded from this network). b, Firing rate distribution 100–75 ms before the NS (black) and throughout the NS (gray; 1200 neurons from 20 networks; 9400 NSs). The distribution is fitted by a power-law function with a scaling power d ≈ −2. Inset, A total of 387 different neurons (12 different networks) that were active as early as −100 to −50 ms before an NS threshold crossing point were selected. For these neurons, the coefficient of variance (CV) of instantaneous frequency (manifested by interspike interval) during 50 ms after threshold crossing was calculated and plotted as a function of their firing rate ±300 ms surrounding the NS detection threshold. Note that the coefficient of variance is negatively correlated to the neuronal firing rate (r = −0.22; p < 0.0001).

Effect of neuronal densities on synchronization time. a, Photographs of exemplar mature networks (3rd week in vitro) plated at 0.1× (left) and 2× (right) the standard density. Exemplars of the resulting network spike amplitudes are shown in b and normalized in the inset in c, demonstrating that the time it takes for the network to synchronize does not increase as the number of neurons participating in the synchrony increases. To allow analyses under conditions of low density, a 10 ms time bin for threshold detection was used for both (high and low) densities. c, A summary of 13 different networks, showing that the time-to-peak of the average network spike decreases as the network becomes more dense. Time-to-peak was calculated from 10% activity to peak activity.