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Model-free reconstruction of neuronal network connectivity from calcium imaging signals

Olav Stetter, Demian Battaglia, Jordi Soriano, Theo Geisel

arXiv:1201.0732v3q-bio.NCcond-mat.dis-nn

TL;DR

Mapping connectivity across large neuronal ensembles remains difficult, so the paper develops a model-independent Transfer Entropy method for reconstructing network topology from calcium imaging. Conditioning on network activity improves reconstruction, which outperforms standard approaches and reveals clustered, non-local structure in cultured cortical networks.

  • Problem

    Detailed connectivity features remain poorly characterized in large neuronal ensembles, including dissociated cultures where mapping circuits is difficult.

  • Method

    The paper uses a model-independent, generalized Transfer Entropy approach to infer network topology from calcium fluorescence recordings without requiring precise spike times or parametric activity models.

  • Results

    Transfer Entropy outperforms cross-correlation, Granger Causality, and Mutual Information on synthetic data, while real cultures show non-local connections and moderately elevated clustering.

  • Takeaways & Limitations

    The method identifies cultured networks with non-random organization, including high-connectivity hubs, elevated clustering, and markedly non-local connectivity.

  • Takeaways & Limitations

    Good reconstruction may require long recordings because conditioning discards a substantial fraction of data, while recordings are limited by photo-damage.

Abstract

from arXiv · show

A systematic assessment of global neural network connectivity through direct electrophysiological assays has remained technically unfeasible even in dissociated neuronal cultures. We introduce an improved algorithmic approach based on Transfer Entropy to reconstruct approximations to network structural connectivities from network activity monitored through calcium fluorescence imaging. Based on information theory, our method requires no prior assumptions on the statistics of neuronal firing and neuronal connections. The performance of our algorithm is benchmarked on surrogate time-series of calcium fluorescence generated by the simulated dynamics of a network with known ground-truth topology. We find that the effective network topology revealed by Transfer Entropy depends qualitatively on the time-dependent dynamic state of the network (e.g., bursting or non-bursting). We thus demonstrate how conditioning with respect to the global mean activity improves the performance of our method. [...] Compared to other reconstruction strategies such as cross-correlation or Granger Causality methods, our method based on improved Transfer Entropy is remarkably more accurate. In particular, it provides a good reconstruction of the network clustering coefficient, allowing to discriminate between weakly or strongly clustered topologies, whereas on the other hand an approach based on cross-correlations would invariantly detect artificially high levels of clustering. Finally, we present the applicability of our method to real recordings of in vitro cortical cultures. We demonstrate that these networks are characterized by an elevated level of clustering compared to a random graph (although not extreme) and by a markedly non-local connectivity.

Author Summary

The study addresses the challenge of assessing global excitatory connectivity in self-organized neuronal cultures in vitro. Applying its method to real data reveals non-random cultured-network features, including high-connectivity hub excitatory neurons and elevated but not extreme clustering.

  • Assessing global excitatory connectivity remains challenging even in self-organized neuronal cultures in vitro.
  • Applying the method to real data reveals non-random features in cultured networks.
  • Cultured networks contain high-connectivity hub excitatory neurons and elevated, but not extreme, clustering.

Introduction

Mapping neuronal connectivity is difficult at large scale, and calcium imaging provides broad access to activity but suffers from slow temporal resolution and poor signal quality. The paper introduces a model-independent, information-theoretic Transfer Entropy approach, validates it on realistic synthetic data, and applies it to cultured cortical networks.

  • Large-scale neuronal circuit mapping remains impractical, even in dissociated cultures, despite progress in neuroanatomy, electrophysiology, and imaging.
  • Calcium imaging simultaneously monitors hundreds to thousands of cultured neurons but typically samples an order of magnitude slower than cellular firing dynamics.
  • The proposed method extends Transfer Entropy to extract directed functional connectivity, with edges representing operationally defined causal influence as improved predictability.
  • The data-driven approach is model-independent, avoids requiring precise spike times, and is not restricted to linear interactions between neurons.
  • Conditioning and correction for imaging time resolution enable accurate topology reconstruction from synthetic data without inferring exact spike times, while Transfer Entropy outperforms cross-correlation, Granger Causality, and Mutual Information.
  • Applied to cortical cultures, generalized Transfer Entropy identifies characteristically non-local connectivity and moderate clustering significantly above random-graph expectations.

Results

Transfer Entropy reconstruction is strongly state-dependent: it performs best in inter-burst fluorescence ranges just above the Gaussian noise regime and poorly during noise-dominated or fully synchronized bursting states. Conditioning on population activity improves topology recovery, including clustering discrimination, with high accuracy for non-locally clustered networks.

  • State-dependent reconstruction: Reconstruction accuracy peaks near 70% true positives in the lowest fluorescence range, declines to about 45% in intermediate-high ranges, and reaches the 10% random baseline at the highest ranges.Using equal sample sizes sharply reduces range I performance because its available sample length shrinks most severely.
  • State-dependent reconstruction: The best reconstruction quality occurs in range II, just above the Gaussian portion of the fluorescence histogram, whereas noise-dominated range I and fully developed burst ranges V–VII yield poor structural overlap.Range II corresponds to the inter-burst regime; ranges III and IV represent early burst development.
  • State-dependent reconstruction: Conditioning on population-average fluorescence below a threshold just right of the histogram’s Gaussian region excludes highly synchronized activity and defines the optimal regime for reconstruction.This conditioning level is designed to retain the most informative dynamical states while excluding ranges III–VII.
  • Real-data validation: Functional connectivity in simulated and real cultures is more strongly clustered during burst development than during inter-burst periods.This state-dependent clustering pattern is observed in both synthetic and real neuronal-culture data.
  • Topology reconstruction: Generalized Transfer Entropy reaches up to 75% true positives at 10% false positives for non-locally clustered topologies and typically 60% true positives at 10% false positives for locally clustered topologies.The method remains effective despite light-scattering artifacts.
  • Topology reconstruction: The reconstructed average clustering coefficient correlates linearly with the ground truth, with Pearson’s r = 0.92, despite a bias toward underestimating clustering in highly clustered networks.This relationship enables reliable discrimination between networks with different clustering levels and similar bursting dynamics.

Discussion

The study presents a model-independent, conditioned Transfer Entropy method that reconstructs excitatory connectivity from calcium-imaging time series, including under slow acquisition and light-scattering artifacts. It reliably recovers selected topological properties and reveals that functional connectivity, clustering, and hub correspondence depend on network dynamics, while higher-order and directional links remain challenging.

  • Methodological contribution: The model-independent algorithm infers excitatory connectivity directly from calcium-fluorescence time series and can be adapted to spike trains or voltage traces.Its model independence avoids artifacts from choosing an incorrect neuronal firing or network model.
  • Methodological contribution: The method operates efficiently on slow calcium-fluorescence acquisitions, making it useful for noisy imaging data without requiring difficult-to-access information.It operates directly on imaging time series.
  • Reconstruction performance: Only generalized TE with Markov order k = 2 or higher and proper conditioning reliably distinguished random from clustered and local from long-range topologies under light scattering.Light scattering reduced linear XC and GC measures to random-level performance by creating spurious interactions.
  • Reconstruction performance: Conditioning on the appropriate dynamical regime produced strong linear correlations between reconstructed and ground-truth average connection distance and clustering coefficient.The study optimized simulations to capture irregular bursting and time-dependent synchronization, using short-term depressing synapses to reproduce realistic spontaneous activity.
  • Reconstruction performance: Approximately 60% of ground-truth bidirectional links were reconstructed bidirectionally, but bidirectional links were severely overestimated and only 4% of unidirectional links ranked in the top 10% of TE scores.These results show that directional connectivity is substantially harder to recover than bidirectional connectivity.
  • Dynamics and functional connectivity: Synchronous bursting increased functional-connectivity clustering and reduced overlap with structural topology, while functional-community synchronization was stronger within than between hub-centered communities for all but noise-dominated range I.Structural and functional hubs corresponded tightly only in dynamic ranges II and III.

Materials and Methods · Network construction and topologies · Simulation of the dynamics of cultured networks

The study simulated sparse directed neuronal networks with tunable local or non-local clustering, then modeled their spontaneous activity using excitatory leaky integrate-and-fire neurons with depressing synapses. Network parameters were chosen to reproduce cultured-network bursting while excluding autaptic links from the adjacency matrix.

  • Network construction and topologies: Networks contained N = 100 neurons randomly distributed over a squared area of 0.5mm lateral size, with connection probability p = 0.12 and non-periodic boundaries.This produced sparse connectivities resembling local cortical circuits while retaining boundary edge effects.
  • Network construction and topologies: Two network ensembles were constructed: locally clustered networks with distance-dependent connection probabilities and non-locally clustered networks with engineered clustering.The ensembles separated spatially dependent connectivity from clustering imposed independently of distance.
  • Network construction and topologies: Non-local clustering was tuned by randomly rewiring pairs of links, accepting crossings that shifted the clustering index toward a desired target.The reference random network had a full clustering index of 0.120 ± 0.004 across 6 networks, while higher-target deviations were below 0.1%.
  • Network construction and topologies: The adjacency matrix encoded directed links without diagonal entries, so the analysis excluded autaptic connections and was structurally unfit to detect them.A link j →i was represented by Aji = 1; diagonal entries would have represented self-connections.
  • Network construction and topologies: Local networks used a Gaussian distance-dependent connection probability, p0(r) = exp(−(r/λ)2), rescaled to maintain a constant average number of links C.The characteristic length scale was λ, and rescaling preserved the specified average connectivity.
  • Simulation of the dynamics of cultured networks: Network dynamics were simulated in NEST using leaky integrate-and-fire neurons with leak conductance gl = 50pS, membrane time-constant τm = 20ms, and firing threshold Vthr = 20mV.Recurrent synaptic inputs drove membrane-potential fluctuations toward threshold.
  • Simulation of the dynamics of cultured networks: Spontaneous firing was induced by independent Poisson spike trains with static coupling conductance αext = 4.0pA and stationary rate νext = 1.6Hz.The model also included a conduction delay td = 2ms and recurrent synaptic strengths that depended on network firing history.
  • Simulation of the dynamics of cultured networks: Simulations used purely excitatory synapses with short-term depression and were calibrated to produce network bursting at 0.10±0.01Hz across local and non-local realizations.A burst was defined operationally as more than 40% of neurons active within 50ms, solely for automated network generation.

Model of calcium fluorescence signals

Surrogate calcium fluorescence signals were generated from simulated spiking using a calcium model with rapid activation, slow decay, saturation, and additive Gaussian noise. The simulations used τCa = 1s, ACa = 50µM, Kd = 300µM, and noise standard deviation 0.03.

  • Calcium dynamics: Simulated spiking was converted into surrogate calcium fluorescence using a model with rapid activation and slow decay.The calcium decay time constant was τCa = 1s, and concentration increased by ACa = 50µM for each action.
  • Fluorescence transformation: The neuronal fluorescence signal combined calcium concentration with a saturating static non-linearity and additive zero-mean Gaussian noise.This fluorescence construction was applied to the activity of neuron i.
  • Simulation parameters: The simulations used a saturation concentration of Kd = 300µM and Gaussian noise with standard deviation 0.03.These parameters specified the fluorescence saturation and noise amplitude.

Modeling of light scattering · Generalized Transfer Entropy

The model incorporates light scattering from surrounding neurons using a distance-dependent scattering scale and strength factor. Generalized Transfer Entropy adapts the connectivity measure to same-bin interactions, switching dynamical states, fluorescence preprocessing, and discrete probability estimation.

  • Modeling of light scattering: Light scattering is modeled from surrounding cells within a simulated region of interest using neuron-pair distance and scattering length λsc = 0.15 mm.The scattering length scale is determined by typical light deflection in the medium and optical apparatus.
  • Modeling of light scattering: The scaling factor Asc controls the overall strength of the simulated scattering artifact.Although deconvolution could correct the artifact for sufficiently large fields of view and accurately known scattering length, real setups have relatively small fields of view.
  • Generalized Transfer Entropy: Transfer Entropy measures directed functional connectivity by comparing transition dynamics with and without the source node’s past.TE is zero when the transition matrices are identical and positive when target transitions statistically depend on source history.
  • Generalized Transfer Entropy: Calcium fluorescence signals are discretely differentiated before Transfer Entropy estimation to isolate potential spike events and improve signal-to-noise ratio.The resulting representation samples probability distributions more effectively with limited data points.
  • Generalized Transfer Entropy: The generalized method includes same-bin causal interactions because synaptic time constants are ∼1 ms whereas acquisition times are ∼10 ms.Longer-lag interactions remain accessible by evaluating Transfer Entropy with Markov order greater than one in time-bin units.
  • Generalized Transfer Entropy: Transfer Entropy is restricted to time ranges representing a single dynamical state because bursting and inter-bursting regimes can have different activity rates and transition matrices.A variable gt representing average fluorescence separates these regimes.
  • Generalized Transfer Entropy: The standard state-selection rule retains time points satisfying {t : gt < ˜g}, with simulations of figures 3 and S2 instead using ˜glow < gt < ˜ghigh.This conditioning implements the dynamical-state restriction in the generalized Transfer Entropy calculation.
  • Generalized Transfer Entropy: Fluorescence values are quantized into B discrete histogram levels, typically using B = 3 because the resulting bin width approximates twice the signal’s standard deviation.This coarse quantization still captures large fluctuations most likely associated with spiking events.

Network reconstruction

The reconstruction retains directed links whose rescaled Transfer Entropy scores exceed a threshold. Quality is assessed with ROC analysis and TP10%, with Positive Prediction Curves providing an alternative true-false comparison.

  • Reconstruction procedure: Directed node pairs are ranked by generalized Transfer Entropy, rescaled to [0,1], and retained when scores exceed TEthr.The threshold is applied after ranking and unit-range rescaling.
  • Evaluation: ROC analysis evaluates true-positive and false-positive reconstructed links across different TEthr values.The highest threshold yields no reconstructed links, whereas the lowest provides 100% of true positives and false positives.
  • Evaluation: TP10% is defined as the fraction of true positives obtained at 10% false positives and is used as a reconstruction-quality indicator.This provides a simple basis for comparing different reconstructions.
  • Alternative evaluation: Positive Prediction Curves plot the true-false ratio against the number of reconstructed links, called the true-false sum.TFR represents the fraction of true positives relative to false positives.

Alternative reconstruction methods

The study compares Transfer Entropy reconstruction with cross-correlation, mutual information, and Granger causality. These alternative analyses use the same preprocessing and can apply the same conditioning on average fluorescence as Transfer Entropy.

  • Comparison design: Transfer Entropy reconstructions are compared with cross-correlation, mutual information, and Granger causality strategies.The comparison evaluates four reconstruction approaches in total.
  • Cross-correlation: Cross-correlation assigns each potential link the largest Pearson cross-correlogram peak over lags from 0 to tmax = 60ms.The score is based on standard Pearson cross-correlation.
  • Mutual information: Mutual information reconstruction scores are evaluated from the joint probability matrix, with the sum taken over all its entries.The probability terms are conditioned on the global activity criterion gn < ˜g.
  • Granger causality: Granger causality first fits a univariate autoregressive model, then a bivariate model including the potential source signal, and compares their residual covariance traces.The bivariate scheme includes same-bin interactions, and analyses used order k = 2; k = 1 produced analogous performance.
  • Shared preprocessing and conditioning: All alternative analyses use the same preprocessing as Transfer Entropy, and conditioning on average fluorescence retains only samples with gt < ˜g.The conditioning is applied by selecting the subset of samples meeting that fluorescence criterion.

Hubs of (causal) connectivity

The method identifies causal sink nodes as hubs with above-average incoming Transfer Entropy, selecting the top 20 nodes per network. It then characterizes each hub’s local neighborhood using fluorescence cross-correlograms and related synchrony measures.

  • Hubs of (causal) connectivity: Causal sink nodes are defined as nodes receiving more total Transfer Entropy than average, with the 20 highest-in-degree nodes selected as hubs.The in-degree is based on the sum of Transfer Entropy from all other nodes to each node.
  • Hubs of (causal) connectivity: For each hub, the analyzed neighborhood C comprises the hub and its first neighbors, whose mean fluorescence is compared with the culture-wide mean.The comparison uses cross-correlograms of the group and whole-culture average fluorescence.
  • Hubs of (causal) connectivity: Differentiated fluorescence cross-correlograms are modeled with a Gaussian form because averaged fluorescence changes slowly relative to the sampling rate.The analysis extracts cross-correlation amplitude AC, peak lag τC, and standard deviation σC.
  • Hubs of (causal) connectivity: The peak lag τC indicates whether neurons in a hub neighborhood fire earlier or later on average than the rest of the network.Within-neighborhood synchrony is additionally assessed by comparing XCs for links inside C with peak XCs across the network.

Experimental preparation · Analysis of experimental recordings

Primary cortical neuron cultures were prepared, calcium-imaged during spontaneous activity, and analyzed to reconstruct network topology. Experimental reconstructions used the simulated-data pipeline, conditioning to reduce bursting effects and comparing inferred features with randomized networks.

  • Experimental preparation: Cortical neurons from embryonic day-19 Sprague-Dawley rats were dissociated and plated on poly-L-lysine-coated 13 mm glass coverslips.Cultures were maintained at 37 °C with medium refreshed every 3 days.
  • Experimental preparation: Cultures typically contained 500–700 neurons/mm2, developed connections within 24 hours, and showed spontaneous activity by DIV 3–4.The GABA switch occurred at DIV 6–7.
  • Experimental preparation: Neuronal activity was recorded at DIV 9–12 after 60 minutes of incubation with 0.4% Fluo-4-AM in pH-stable recording medium.The recording solution contained defined concentrations of NaCl, KCl, CaCl2, MgCl2, glucose, and HEPES at pH 7.4 and 320 mOsm.
  • Experimental preparation: Inhibitory synapses were blocked with 40 µM bicuculline, leaving activity driven by excitatory neurons and increasing fluorescence amplitude for firing detection.This blockade was applied after the GABA switch.
  • Experimental preparation: Regions of interest were identified for active neurons, and their average grey-levels yielded fluorescence time series; sequences typically contained about one hundred bursts.Recorded individual-neuron fluorescence examples are shown in Fig. 1B.
  • Analysis of experimental recordings: Experimental fluorescence data followed the simulated-data analysis pipeline, including discrete differentiation, metric evaluation, ranking, and thresholding to retain the top 10% of connections.The pipeline could evaluate TE or other metrics before final thresholding.
  • Analysis of experimental recordings: Because ground-truth topology was unavailable, conditioning was selected from experimental–simulated fluorescence similarity to exclude fully developed bursting transients while retaining as many data points as possible.The selected conditioning level was approximately two standard deviations above the relevant fluorescence distribution threshold.
  • Analysis of experimental recordings: Alternative conditioning levels showed that inferred average clustering coefficient and connection distance remained stable around the selected value.Robustness was assessed across a range approximately two fluorescence standard deviations wide for both experimental datasets.

Figure Legends

The figure legends identify vertical lines as distribution means and use insets to illustrate clustering through simple networks matched to the simulations’ clustering coefficients.

  • Vertical lines indicate the mean of each distribution.
  • Insets illustrate the amount of clustering in the simulated networks.
  • The insets show simple networks with the same clustering coefficients as the simulated networks.

Supporting Figure Legends

The supplementary figures characterize reconstruction behavior across thresholds, dynamic states, topology, algorithmic conditions, and experimental cultures. They show how clustering estimates, connectivity distances, and reconstruction performance vary with network structure and analysis settings.

  • Supplementary Figure 1: Degree varies linearly with the fraction of included links, while reconstructed clustering rises steeply at low thresholds and peaks near the ground-truth clustering coefficient.The clustering profile forms a broad hill between approximately 5% and 10% of reconstructed links.
  • Supplementary Figure 2: Functional connectivity hubs are identified across seven simulated dynamic intervals, with hub-neighbor cross-correlations compared against within-regime node-pair correlations.The figure highlights hub locations and compares average cross-correlation values for corresponding dynamical regimes.
  • Supplementary Figure 3: Four fluorescence-signal ranges define distinct dynamical regimes in two experiments and a simulated locally clustered network, enabling state-dependent analyses across simulated and real data.The regimes are encoded by different colors and analyzed separately in each case.
  • Supplementary Figure 4: Mutual Information and Transfer Entropy correctly estimate average connection distance in non-locally clustered topologies, whereas cross-correlation invariantly underestimates it.The comparison includes non-linear causality measures MI and TE and the linear measure XC.

Tables

Table 1 reports the mean and standard deviation of internal synaptic weights used to simulate 12 networks. The networks comprise six non-locally clustered networks ordered by ascending clustering coefficient and six locally clustered networks ordered by ascending length scale.

  • Synaptic weights: Table 1 lists the mean and standard deviation of the internal synaptic weights αint used in the simulations.The weights were used for six networks with a non-locally clustered ensemble and six with a locally clustered ensemble.
  • Network ensembles: The six non-locally clustered networks are listed in ascending order of clustering coefficient CC.
  • Network ensembles: The six locally clustered networks are listed in ascending order of length scales λ.

Figures and Supporting figures

The figures establish the simulation and experimental activity framework, then evaluate how dynamical state, topology, conditioning, algorithm choice, TE formulation, and recording length affect connectivity reconstruction. They also compare reconstructed and structural network properties across non-locally and locally clustered topologies.

  • Activity framework: Figure 1 contrasts real and simulated calcium fluorescence activity, including population averages and their distributions, in neuronal cultures and simulations.The figure combines bright-field and fluorescence images with example time series and population-level summaries.
  • Network dynamics: Figure 2 shows that networks with different clustering coefficients can exhibit similar dynamics, using spike rasters and inter-burst interval histograms.The comparison concerns non-local clustering ensembles.
  • Dynamical-state dependence: Figure 3 assesses directed functional connectivity across seven fluorescence-amplitude ranges and evaluates reconstruction quality as a function of dynamical state.The analysis compares amplitude-conditioned regimes with an analysis using the entire fluorescence range.
  • Topology reconstruction: Figures 4 and 5 evaluate TE-based reconstruction for non-locally and locally clustered topologies using generalized TE with Markov order k = 2 and fluorescence conditioning.Figure 4 compares structural and reconstructed clustering coefficients, degree distributions, and connection distances; Figure 5 evaluates connectivity after retaining the top 10% of links.
  • Algorithmic performance: Figure 6 compares cross-correlation, Granger Causality, Mutual Information, and other reconstruction algorithms across clustering coefficients, conditioning levels, and light-scattering artifacts.Overall reconstruction performance is quantified by TP10%, where black represents 0% true positives and white represents 100% true positives.
  • TE formulation and recording length: Figure 7 examines reconstruction quality across three TE formulations and different recording lengths for non-locally and locally clustered topologies.ROC curves and vertical TP10% markers support visual comparison of the formulations.
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