Noise focusing and the emergence of coherent activity in neuronal cultures

Journal name:
Nature Physics
Volume:
9,
Pages:
582–590
Year published:
DOI:
doi:10.1038/nphys2686
Received
Accepted
Published online

Abstract

At early stages of development, neuronal cultures in vitro spontaneously reach a coherent state of collective firing in a pattern of nearly periodic global bursts. Although understanding the spontaneous activity of neuronal networks is of chief importance in neuroscience, the origin and nature of that pulsation has remained elusive. By combining high-resolution calcium imaging with modelling in silico, we show that this behaviour is controlled by the propagation of waves that nucleate randomly in a set of points that is specific to each culture and is selected by a non-trivial interplay between dynamics and topology. The phenomenon is explained by the noise focusing effect—a strong spatio-temporal localization of the noise dynamics that originates in the complex structure of avalanches of spontaneous activity. Results are relevant to neuronal tissues and to complex networks with integrate-and-fire dynamics and metric correlations, for instance, in rumour spreading on social networks.

At a glance

Figures

  1. Experimental observation of nucleation and propagation.
    Figure 1: Experimental observation of nucleation and propagation.

    a, High-contrast bright-field image of a neuronal culture at DIV 10, grown on glass and confined within a circular cavity 3mm in diameter. The culture contains 3,000 neurons. b, Bright-field image showing a detail of the culture and the distribution of neurons. The circle identifies a single neuron. c, Corresponding fluorescence image during a spontaneous activity event. Bright spots are firing neurons. The resolution of the image is the same as the actual measurements, and illustrates our ability to monitor network activity at a single-neuron level. d, Fluorescence signal from 30min recording of spontaneous activity in the culture shown in a, averaged over the 500 brightest neurons. The top plot corresponds to measurements with both excitation and inhibition active (E+I network); the bottom one corresponds to excitation-only measurements (E network), with inhibitory synapses blocked with 40μM bicuculline. Fluorescence peaks identify network bursts. The symbols below each burst identify its initiation in a specific area of the culture. e, Distribution of IBIs (top) and burst propagation velocities v (bottom) for E+I and E networks. Statistics is based on six cultures of identical size and density, at DIV 9–10. On average, E networks are characterized by larger IBIs and propagation velocities. fh, Examples of the propagation of spontaneously generated bursts in cultures of different sizes and developmental stages. The analysis of the onset times of neuronal firing provides the average velocity of the advancing front and its initiation point (white circle). E+I and E networks show a qualitatively similar behaviour, with a propagation in the form of a circular wave. g also depicts, for the data shown in d, the approximate initiation point of each burst. For clarity, nearby initiations are grouped defining the nucleation sites. Three main initiation sites are identified. The size of the white circles is proportional to the relative occurrence of nucleation events at each site. Localized nucleation occurs both in E+I and E networks. The location of the nucleation sites and its relative importance is different in the two networks, and illustrates the sensitivity of nucleation to network details.

  2. Spatial distribution of nucleation sites in experiments.
    Figure 2: Spatial distribution of nucleation sites in experiments.

    a,b, Nucleation probability density functions (PDFs) for a circular culture (a) and a rectangular one (b) at DIV 9. The top plots correspond to measurements in untreated cultures (E+I networks); the bottom ones correspond to measurements in the same cultures after the blockade of inhibitory synapses with 40μM bicuculline (E networks). Data are obtained by coarse-graining the burst initiation points of a given measurement. The number of bursts n observed in each measurement is indicated at the bottom right corner of each plot. The small circles in a are the neurons in the network. Neurons are not shown in b for clarity. The grid lines are a guide to the eye. Nucleation is highly localized in specific regions for both E+I and E measurements. For the circular culture and E+I measurements, nucleation is peaked at the bottom centre of the network. The blockade of inhibition reconfigures the distribution of nucleation probability, but its degree of localization is similar. The larger rectangular culture accommodates a higher number of nucleation sites. The blockade of inhibition changes their location and relative weight while maintaining their strong localization. c, Lorenz curves for different culture sizes, and comparing E+I (top) and E networks (bottom). The curves are obtained by plotting the accumulated nucleation probability as a function of the area fraction of the culture, and after averaging over cultures of similar area and developmental stage. For both E+I and E measurements, data correspond to cultures at DIV 9–11 and areas A±1mm2, with A=2.5mm2 (red, N=4); 6.1mm2(blue, N=8); and 15.5mm2 (black, N=5). The grey area depicts the 95% confidence interval for the data with the highest standard deviation (A=15.5mm2). The insets provide a detail of the plots. The sharp increase to 1 of the Lorenz curves illustrates the strong localization of nucleation probability. The different curves collapse within experimental error, evidencing the scaling of nucleation with system size.

  3. Statistics of background avalanches.
    Figure 3: Statistics of background avalanches.

    a, Spatial representation of the background avalanche activity in a circular culture of 2.5mm in radius and density 300neuronsmm−2, with an average connectivity of left fencekright fence~70. Only the top 1% of most active connections that participate in the background activity are shown. Different colours identify communities according to community detection algorithms50. The background activity forms a subnetwork clustered in specific regions of the culture containing only a 25% of the total population. b, Statistics of background avalanches for networks with different connection probability α and different density ρ (αρ=200mm−2) thus keeping the mean connectivity fixed at left fencekright fence~70. The avalanche size distribution shows power-law statistics for almost three decades. c, Relationship between avalanche duration and avalanche size for the same networks as in b. Inset: the curves collapse into a single one when rescaled with the connection probability α, although deviations start to appear at larger sizes. In all cases the calculated exponent is below 1. Each curve is averaged over 5 different network realizations and over 3h of simulated activity.

  4. Dynamics of ignition avalanches.
    Figure 4: Dynamics of ignition avalanches.

    a, Spatial representation of the ignition avalanche activity in the same network shown in Fig. 3a. Only the top 1% of most active connections are shown. b, Nucleation probability density, coarse-grained over the connectivity correlation length scale of 0.26mm. Each nucleation point is defined by the geometrical centre of the neurons that are the first to fire in a burst. Their distribution is highly localized in specific regions of the system that define the nucleation sites (three in this case). The nucleation sites are more focused than the ignition avalanche activity itself. c, Degree probability distribution p(k) of the different networks and subnetworks studied. The distribution of the ignition avalanche (IA) and background avalanche (BA) subnetworks is completely different from the structural one, and shifts from a Gaussian-like profile to one that is consistent with a scale-free network. The distribution of a randomized version of the original connectivity, with only 1% of the links taken at random, is also shown for comparison. d, Temporal profile of activity during the ignition avalanches. Three network types with different connection probability are shown. The activity growth during the ignition avalanches follows exponential distributions up to the end of the ignition avalanche that sets the temporal scale origin, the moment at which a faster growth describes the formation of the burst. e, Activity growth in the region where the ignition avalanche transforms into a burst, normalized by system size. The growth during the bursting phase becomes independent of the connectivity probability α, consistent with the fact that the bursting phase is governed by the neuronal dynamics and not by the noise focusing. Curves were averaged over five different network realizations and the solid regions show the 95% confidence interval.

  5. Nucleation statistics and noise focusing.
    Figure 5: Nucleation statistics and noise focusing.

    ae, Contour plots of different observables, coarse-grained over the connectivity correlation length scale of 0.26mm. The specific network realization contains 7,500 neurons with an average of 70 connections per neuron, on a square of 5×5mm2 (ρ=300neuronsmm−2 with a connection probability of α=2/3), with periodic boundary conditions. a, Nucleation probability density, identifying three nucleation sites (as in Fig. 4). b, Number of triangles per neuron (number of neighbours of a neuron that are themselves neighbours). c, Background activity. Average number of spikes per second and neuron between bursts. d, Streamlines of the average flow of ignition avalanches superimposed on the nucleation PDF. For every ignition avalanche we assign a unitary vector to each participating neuron pointing towards the nucleation point of that ignition avalanche. Averaging over all of the ignition avalanches results in a vector field whose streamlines define the basins of attraction of the different nucleation sites. Neurons outside the nucleation sites still have a preferred direction and contribute to the ignition avalanches. e, Local percolation fraction. For every point in the system a region of radius 0.4mm is selected. The local percolation fraction is the fraction of neurons in the selected region that needs to be activated simultaneously to generate a burst. f, Lorenz curves of the different observables. The diagonal line (black) corresponds to a flat distribution. Deviations from the diagonal give a clear image of the sharpness of a distribution. For the nucleation 20% of the system area contains 80% of the probability. Another nucleation curve for networks with α=1/3 but the same average number of connections per neuron, is shown. The ignition avalanche score, defined as the fraction of times a given neuron participates in an ignition avalanche, is much less localized owing to the large spatial extension of ignition avalanches, outside the nucleation sites. Note the spatial fluctuations of triangles per neuron are much smaller. Lines averaged over five different network realizations and the solid regions show the 95% confidence interval. Comparison of ac shows that high clustering and high background activity do not correlate strongly with high nucleation probability. The latter is much more peaked and selective, as similar values of those observables yield significantly different values of nucleation probability. Looking at the zones denoted by the square and star it is clear that local statistics cannot explain the selection of a site with high nucleation probability. Zones with high clustering (star) may not correspond to high background noise, and vice versa (square). This mismatch is also identified by the calculation of Pearson’s correlation coefficient between the nucleation map and any other network observable, giving at best values of ~ 0.5.

  6. Noise amplification mechanisms.
    Figure 6: Noise amplification mechanisms.

    a, Dynamical noise amplification. Left: schematic of the mechanism, where a subset of m (from a total of n) firing neurons projecting over neuron B induces a firing probability pm. Right: dependence of pm on the input fraction. The firing probability is greatly amplified before the quorum percolation condition m0 is met. b, Topological noise amplification. Left: schematic of the effect of feedforward loops in the induced firing probability of neuron C. Right: critical fraction fn of input neurons Ai needed to activate neuron C with a probability 1/2 as a function of the number of feedforward loops formed with the intermediate Bj neurons.

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Author information

Affiliations

  1. Departament d’ECM, Universitat de Barcelona. Av. Diagonal 645, 08028 Barcelona, Spain

    • Javier G. Orlandi,
    • Jordi Soriano,
    • Sara Teller &
    • Jaume Casademunt
  2. Departament Física Aplicada, EETAC. Universitat Politècnica de Catalunya, BarcelonaTech. Esteve Terrades 5, 08860 Castelldefels, Spain

    • Enrique Alvarez-Lacalle

Contributions

J.G.O. developed the model in silico, and performed the numerical simulations and data analysis. J.S. conceived and designed the experiments. J.G.O. and E.A-L. conceived the model in silico. J.S. and S.T. performed the experiments and analysed experimental data. J.G.O., E.A-L. and J.C. contributed to the theoretical analysis. J.C. developed analytical tools and the conceptual framework. All authors contributed to data interpretation and wrote the manuscript. J.S. and J.C. supervised the project.

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The authors declare no competing financial interests.

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