# Super-selective reconstruction of causal and direct connectivity with application to in-vitro iPSC neuronal networks

TL;DR: A novel mathematically rigorous, model-free method to map effective - direct and causal - connectivity of neuronal networks from multi-electrode array data and is shown to offer important insights into the functional development of in-vitro iPSC-derived neuronal cultures.

Abstract: Despite advancements in the development of cell-based in-vitro neuronal network models, the lack of appropriate computational tools limits their analyses. Methods aimed at deciphering the effective connections between neurons from extracellular spike recordings would increase utility of in-vitro lo- cal neural circuits, especially for studies of human neural development and disease based on induced pluripotent stem cells (hiPSC). Current techniques allow statistical inference of functional couplings in the network but are fundamentally unable to correctly identify indirect and apparent connections between neurons, generating redundant maps with limited ability to model the causal dynamics of the network. In this paper, we describe a novel mathematically rigorous, model-free method to map effective - direct and causal - connectivity of neuronal networks from multi-electrode array data. The inference algorithm uses a combination of statistical and deterministic indicators which, first, enables identification of all existing functional links in the network and then, reconstructs the directed and causal connection diagram via a super-selective rule enabling highly accurate classification of direct, indirect and apparent links. Our method can be generally applied to the functional characterization of any in-vitro neuronal networks. Here, we show that, given its accuracy, it can offer important insights into the functional development of in-vitro iPSC-derived neuronal cultures by reconstructing their effective connectivity, thus facilitating future efforts to generate predictive models for neurological disorders, drug testing and neuronal network modeling.

## Summary (4 min read)

### 1. Introduction

- In-vitro cultures of primary neurons can self-organize into networks that generate spontaneous patterns of activity [10, 74, 58], in some cases resembling aspects of developing brain circuits [37, 24].
- These methods estimate the effective connectivity of a measured neuronal network by explicitly modeling the data generation process, i.e. only the connectivity of a simulated network model is inferred without any theoretical guarantee about its accuracy and its ability to correctly estimate the connectivity of the biological network [14, 76].
- Moreover, as the partial correlation matrix is symmetric, this method is not useful for detecting the causal direction of neuronal links.
- The authors show an experimental application of their approach to spontaneously generated in vitro networks of human iPSC-derived neurons cultured on MEAs providing an analysis and interpretation of the physiology not possible otherwise.

### 2. Materials and Methods

- The core of their methodology is graphically described in Figure 2 and includes three main phases: 1. statistical, correlationbased reconstruction of functional connectivity; 2. mathematically-rigorous super-selection of direct links via identification of peaks related to indirect and apparent links and 3.
- The algorithm constructs correlation triangles by considering all possible combinations of correlation delays for any possible triplet of neurons .
- The super-selection rule is formally presented later.
- Importantly, the algorithm removes correlations from the analysis, but does not remove inferred physical connections.

### Temporal correlations

- (2) Using the Fast Fourier Transform (FFT) and the correlation theorem, computing correlations in Eq. (1) can be efficiently performed in O(S log(S)) with S the number of samples composing the signals sj .
- Their amplitude can be regarded as a measure of the level of correlation between the spikes in their registered firing activities.
- The higher the amplitude of a peak in Rjk(τ) (if any), the higher the probability that there is a statistical dependency between neuron j and neuron k.
- As explained earlier in the text, the existence of a functional coupling between two neurons does not necessarily imply that there is an effective connection between them.
- The location of each correlation peak with respect to the origin indicates the temporal delay τpeakjk between the activity of neuron j and neuron k .

### Peak detection

- The authors implemented a peak detection algorithm applied to Rjk(τ) to identify all existing functional correlations between any pair of neurons (j, k) in the network and to discern the directional dependency between their spiking activities.
- As introduced above, the authors assume that a correlation peak in Rjk(τ) represents a potential connection between j and k and that τpeakjk is the signal propagation delay between them.
- The definition of “near-perfect match” is based on the choice for the parameter which represents the acceptable degree of temporal approximation when computing Eq. 5.
- The algorithm aims to construct njk × nkm × nmj correlations triangles given by all possible combinations of correlation peaks of any triplet of neurons’ pairs .
- Within this framework, each correlation triangle shares up to three edges with G, and the union E′ of all τhjk defines the direct graph G ′ = (V,E′).

### Edge covering minimization

- Therefore, the challenge is to find a discrimination rule that allows us to select the correct τhjk in the correlation triangles that satisfy Eq. (4) (or Eq. (5)).
- The larger the variation of the delayed timing between the input and the output neurons, the larger the phase noise in the associated correlation peak which will have smaller amplitude and larger standard deviation than the correlation among neurons with input and output signals without phase noise and perfectly synchronized.
- To clarify this point, let’s consider the example reported in Figure 4.
- If the authors consider the triangles ACD or ABD as in Figure 4, in both cases, the delay τDA is the one detected and discarded because, as explained in Figure 3, the corresponding correlation peak has lower amplitude and larger width as it results from a signal that has propagated through three links.
- Within this picture, the authors can establish the second step in the super-selection algorithm as: Given a correlation triangle (τhjk, τ h′ km, τ h′′ mj), the delay associated with the smallest correlation peak am- plitude min(Ahjk, A h′ km, A h′′ mj) (8) estimates the false positive connection and is discarded from E′.

### Connectivity matrix reconstruction

- There are three fundamental parameters that affect the inference performance of this method: T defines the time window (−T,+T ) over which to search for correlation peaks in the correlation function Rjk. σ is important because the location of the detected correlation peaks weakly depends on it.
- Large values increase the number of detected correlation peaks and computed correlation triangles with the potential shortcoming that true positive connections are filtered out.
- This process allowed us to individuate a range of values for T and σ where the inference performances of the algorithm reached a maximum plateau (Fig. 6D), and define the reconstruction rule of their connectivity method as follows (the numerical details are discussed in the next section of Materials and Methods).
- On the other hand, the maximum value for T can be chosen as several times (for example 5-6) the average delay among neurons in order to include more correlation triangles later used to refine the selection.

### Experimental methodology

- To develop and validate their connectivity method the authors used spiking data generated via simulations of neuronal networks based on the Izhikevich model [29] (see Supplementary Material and Fig. S2).
- The original code was modified to guarantee high levels of activity in the network as well as bursting like behavior similar to that registered in their experiments .
- It is worth noting that their model is general and not restricted to sparse connectivity.
- For each network size, 20 different networks were randomly generated, simulated and then analyzed for connectivity reconstruction.

### Performance measures

- The performances were assessed based on the indicators TP/Nc (net number of true positive (TP) connections, with Nc the number of connections in the simulated neuronal network), FP/Nc (net number of false positive (FP) connections) and ∆ = (TP − FP )/Nc (confidence indicator).
- ∆ is independent on the connectivity of the network being reconstructed and, because of its definition, it better defines the level of confidence in the detection of TPs by highlighting the method capabilities in rejecting or not the false positive connections.
- For the sake of comparison with the literature, the authors also used the more standard Accuracy measure ACC = (TP + FP )/Ntot [49, 19].

### Numerical experiments

- This allowed us to isolate the activity of a few units per electrode, which resulted in the reconstruction of the activity of multiple detected neurons in the MEA well .
- In a few cases, the clustering approach did not perform efficiently and generated either too many or too few clusters which were detected by observing the increased number of outliers in the spike sorting output.
- For σ, observations on the recordings and the level of smoothing required for high-frequency noise removal led us to select σ ∈ (0.3, 0.8) ms.
- The authors calculated the average in-degree and out-degree of the vertexes of each directed graph corresponding to a reconstructed network by calculating all incoming and outgoing connections for each network vertex and then averaging over the total number of vertexes per network.

### Numerical results

- To evaluate the performances of their method, the authors built a numerical model by designing and simulating an integrate-and-fire neuronal network mimicking the activity of in vitro cultures of neurons (see Supplementary Material).
- As a result, the same false positive connections observed for a given pair of neurons are very unlikely detected by many different points, and at high frequency they are filtered out.
- For two given points p1 and p2 the algorithm detected 15% and 20% FPs, respectively; however, most of the FPs detected by point p1 did not correspond to the ones detected by point p2.
- Since the authors made the hypothesis that each node in the network model has the same average connectivity (preserved in- and out-degree) which does not increase with the network’s size, they do not expect and they are not interested in seeing connectivity dependent variations.
- The authors also studied the integration and segregation properties of cultured human iPSC-derived neural networks (9C).

### Discussion

- The authors demonstrated a new model-free based approach to infer effective - active, direct and causal - connections from in vitro neuronal networks recorded on MEAs.
- In terms of scalability, their algorithm matches the scalability properties of most of the state-of-the-art connectivity methods which, being based on pairwise statistical and correlation studies, scale with a practical computational complexity of the order of O(n2), where n is the number of neurons.
- More specifically, the authors have three main routines whose complexity should be assessed: the computation of pairwise correlations, the peak detection algorithm, and the detection of false positive connections.
- When applied to biological networks, their reconstruction method assumes that the spike sorting is accurate enough to separate the most prominent contributions in the activity of the network.
- High density MEAs that include tens of thousands of electrodes at cellular and subcellular resolution [73, 80, 38, 7], and complementary voltage and calcium imaging approaches [26, 41, 3, 27] that provide useful information about the localization and classification of recorded neurons, offer optimized acquisition settings and will greatly improve the resolution and accuracy of their approach.

### 4. Conclusions

- The authors have presented an innovative approach to map the effective connectivity of neural networks from multielectrode array data.
- Notably, their connectivity algorithm succeeds in detecting direct connections between neurons through a mathematically rigorous selection scheme that distinguishes between apparent or non-direct links and direct ones, therefore enabling inference of directed causal relationships between connected neurons.
- Furthermore, it will be broadly applicable to experimental techniques for neural activation and recording, increasing its utility for the analyses of spontaneous neural activity patterns, as well as neuronal responses to pharmacological perturbations and electrical and optogenetic stimulations [70, 26, 38, 41, 3].
- The authors thank Fabio L. Traversa for fruitful and constructive discussions on the theoretical framework.
- D.P. and A.B. designed and organized the experimental studies.

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### "Super-selective reconstruction of c..." refers background in this paper

...The introduction of human induced-pluripotent stem cell (hiPSC) technologies [12, 13] opened the possibility to generate in-vitro neuronal networks in typical [14, 15], as well as patient-specific genetic backgrounds [16, 17, 18, 19, 20, 21], demonstrating the potential to reproduce key molecular and pathophysiological processes in highly controlled, reduced, experimental models that enables the study of neurological disorders and the discovery and testing of drugs, especially in the context of the individual patient [22, 23, 24]....

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### "Super-selective reconstruction of c..." refers methods in this paper

...With the use of Matlab routines, we also computed the CCo as the fraction of triangles around a node, which is equivalent to the fraction of node’s neighbors that are neighbors of each other [61]....

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...Techniques such as cross-correlation [31, 41], directed and partial directed coherence [42, 43], transfer entropy [31, 41, 44] and Granger causality [45, 46] are examples of causal indicators as they provide inference of directionality of dependence between time series based on time or phase shifts, or prediction measures....

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###### Q2. What have the authors stated for future works in "Super-selective reconstruction of causal and direct connectivity with application to in-vitro ipsc neuronal networks" ?

In addition, it has good scaling capabilities and can be further generalized to any kind of network, thus allowing to target different problems in intact neurons, synthetic models as well as in vitro and in vivo systems. Furthermore, it will be broadly applicable to experimental techniques for neural activation and recording, increasing its utility for the analyses of spontaneous neural activity patterns, as well as neuronal responses to pharmacological perturbations and electrical and optogenetic stimulations [ 70, 26, 38, 41, 3 ]. As novel electrophysiology technologies come online and are validated, the methods the authors presented here will be in an immediate position to take advantage of them, resulting in fundamental improvements in spatial resolution and reconstruction accuracy. Furthermore, their algorithm can be further extended, improved, and possibly integrated with already in-use techniques to overcome important limitations such as the detection of inhibitory connections and the inference of effective connectivity in the bursting regime.