Super-selective reconstruction of causal and direct connectivity with
application to in-vitro iPSC neuronal networks
Francesca Puppo
a,∗
, Deborah Pr´e
b,∗
, Anne Bang
b,∗∗
, Gabriel A. Silva
c,∗∗
a
BioCircuits Institute, Center for Engineered Natural Intelligence, University of California, San Diego, La Jolla, 92093 CA,
USA
b
Conrad Prebys Center for Chemical Genomics, Sanford Burnham Prebys Medical Discovery Institute, La Jolla, CA 92037,
USA
c
Department of Bioengineering, Department of Neurosciences, BioCircuits Institute, Center for Engineered Natural
Intelligence, University of California, San Diego, La Jolla, 92093 CA, USA
Abstract
Despite advancements in the development of cell-based in-vitro neuronal network models, the lack of ap-
propriate computational tools limits their analyses. Methods aimed at deciphering the effective connections
between neurons from extracellular spike recordings would increase utility of in-vitro local 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 fun-
damentally unable to correctly identify indirect and apparent connections between neurons, generating re-
dundant 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.
Keywords: neuronal network, effective connectivity, functional connectivity, apparent connectivity,
correlation, MEA, iPSC, development
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]. The emer-
gent functional states exhibited by these neuronal ensembles have been the focus of attention for many
years [15, 83] as they can be used to investigate principles that govern their development and maintenance
∗
These two authors contributed equally
∗∗
Corresponding authors
Email addresses: email: abang@SBPdiscovery.org (Anne Bang), email: gsilva@ucsd.edu (Gabriel A. Silva)
Preprint submitted to Elsevier January 25, 2021
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[35, 44] and to produce biological correlates for neural network modeling [11, 66]. The introduction of hu-
man induced-pluripotent stem cell (hiPSC) technologies [69, 82] opened the possibility to generate in-vitro
neuronal networks in typical [30, 42], as well as patient-specific genetic backgrounds [57, 5, 75, 78, 8, 39],
demonstrating the potential to reproduce key molecular and pathophysiological processes in highly con-
trolled, 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 [72, 16, 61].
One common approach to obtain information from in vitro neuronal networks is to record their activity
via multi-electrode array (MEA) or calcium fluorescence imaging and then use network activity features to
describe their physiology. One main limitation, however, is that these high-dimensional data, which report
about the information representation in the network, do not translate into a clear understanding of how this
representation was produced and how it emerged based on neuronal connectivity [14]. The synchronization
of spontaneous spike trains among different MEA sites or neurons, also referred to as network bursting, is an
example of observed neural behaviors widely reported in the literature. The generation of network bursting
in an in vitro neuronal culture is evidence that the neurons are synaptically connected. However, the extra-
cellular nature of the MEA recording does not provide information about how neurons are connected and
how signals propagate between them, such that computational analyses are necessary to reconstruct their
complex dynamic patterns and relate their emergence to the underlying wiring diagram [66]. However, this
kind of analysis presents several challenges as it requires not only identification of functional relationships
between cells, but also reconstruction of the dynamic causality (i.e., the knowledge of which neuron fires first
and affects another one) between directly linked neurons that are simultaneously involved in several differ-
ent signaling pathways. This defines the difference between functional and effective connectivity inference:
the first only reports about statistical dependencies between cells’ activities without giving any information
about specific causal and direct effects existing between two neurons [76]; the second attempts to capture a
network of effective - direct and causal - effects between neural elements [65].
Only model-based approaches [14, 34] have been proposed for inference of effective connectivity. Among
them, dynamic causal modeling (DCM) [18] and structural equation modeling [36] variants have shown best
performances. However, 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].
Because of this limitation, descriptive, model-free approaches are usually preferred as they are easy to
implement, rely on a limited number of assumptions that are directly related to the investigated neuronal
network, and can be more easily validated [14, 34]. A number of model-free methods proposed for recon-
structing the connectivity of in vitro neuronal networks [19] have been previously reviewed [48] and tested
2
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[76]. However, because they rely on purely statistical indicators, they can only infer how neurons are func-
tionally coupled, but lack the ability to identify the network of effective interactions between neurons by
either missing the directionality or confounding indirect and apparent links from direct ones. Directional-
ity conveys the causality of signaling in the network, i.e. which neural element has causal influences over
another (Figure 1A(i)). However, causality does not imply a direct connection between two neurons. In
fact, a functional coupling between two neurons can be causal even though the two neurons are not directly
connected, and this may occur if there is a multi-neurons pathway between the two cells (indirect connec-
tion, Figure 1A(ii)), or if the connection detected between the two neurons is simply a mathematical artifact
resulting from the correlation of correlations generated by common inputs from other participating neurons
(apparent connection, Figure 1A(iii)) [17].
Methods such as correlation [53], coherence [25, 23, 12], mutual information [19, 22, 51], phase and
generalized synchronization [51, 2], and joint entropy [19] describe only statistical dependencies between
recorded neurons without carrying any information of causality or discriminating direct and indirect effects.
Techniques such as cross-correlation [19, 28], directed and partial directed coherence [1, 56], transfer en-
tropy [19, 28, 33] and Granger causality [21, 59] 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.
However, because these operators rely only on pairwise statistical comparisons and treats pairs of neurons
independently, they show the same limitations when dealing with indirect connections and external inputs.
Only a few techniques can compete in the challenge of inferring the effective connectivity of a network.
Partial-correlation [19, 68], which takes into account all neurons in the network, showed best performance
in detecting direct associations between neurons and filtering out spurious ones [45]. The most significant
limitation of this solution is its high computational cost. Moreover, as the partial correlation matrix is
symmetric, this method is not useful for detecting the causal direction of neuronal links. It also does not
attempt to infer self-connections [14]. A combination of correlation and network deconvolution was used
by Magrans and Nowe [13] to infer a network of undirected connections with elimination of arbitrary path
lengths caused by indirect effects. However, this method also can not identify directions of connections and
the singular value decomposition of network deconvolution has an extremely high computational complexity
[45]. A convolutional neural network approach [54] showed the same limitations in computational complexity
and undetected self- and causal connections. Figure 1B graphically summarizes the inference capabilities of
the state-of-the-art connectivity methods as reported in [14, 76, 17, 2].
In this work, we propose a novel, mathematically rigorous method that uses a model-free approach (i.e.
does not depend on a set of underlying assumptions about the biology of participating cells) to decompose
the complex neural activity of a network into a set of numerically validated direct, causal dependencies
between the active component neurons that make up the network. First, the inference power of statistical
3
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approaches (signal-, network- and information theory- based) allows mapping the functional connectivity
of the network. Then, we propose a mathematically rigorous selection scheme that distinguishes between
apparent or non-direct links and direct ones, therefore enabling inference of direct causal relationships be-
tween connected neurons which more realistically describe the effective connectivity of the network (Figure 2).
We evaluate the performances of the proposed method on synthetic datasets generated through simulation
of an integrate-and-fire neuronal network mimicking the activity of in vitro cultures of neurons, and demon-
strate important improvements, relative to the state-of-the-art connectivity methods, to network inference
accuracy due to a deterministic component of our method capable of identifying false positive connections.
We show an experimental application of our 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. We describe the temporal evolution associated with the connectivity and dynamic
signaling of developing hiPSC-derived neuronal networks, including increasing synchronized activity and
the formation of small numbers of hyper-connected hub-like nodes, as similarly reported by others [30, 8].
These results further support the performance quality of our approach and provide an example of how this
connectivity method can be used to characterize network formation and dynamics, thus facilitating efforts
to generate predictive models for neurological disease, drug discovery and neural network modeling.
2. Materials and Methods
Theoretical framework for connectivity reconstruction
The central contribution of this manuscript is in providing an innovative, computationally efficient,
and easy-to-apply method for decomposing the collective firing properties stored in the electrophysiological
recordings from neuronal networks on MEAs into direct (one-to-one) and causal (directional) relationships
between all participating neurons. We propose a multi-phase approach that identifies and discards any
correlation link that does not directly relate to a direct interaction between two cells. The core of our
methodology is graphically described in Figure 2 and includes three main phases: 1. statistical, correlation-
based reconstruction of functional connectivity; 2. mathematically-rigorous super-selection of direct links
via identification of peaks related to indirect and apparent links and 3. reconstruction of directed causal
connectivity between neurons.
1. The functional connectivity (statistical dependencies) of a network is computed via pairwise correla-
tion studies. Functional interactions between neurons are represented by correlation peaks and their
delays τ . The algorithm constructs correlation triangles by considering all possible combinations of
correlation delays for any possible triplet of neurons (Figure 2.1). Importantly, correlation triangles do
not refer to any three-neuron physical connection, sometimes referred to as “neural triangles” in the
4
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literature [63, 55]. Here we define a correlation triangle as a mathematical object that our algorithm
uses to classify functional interactions based on all possible triplets of correlation delays that can be
formed in the network. Therefore, correlation triangles exploit the entire signal history of neurons in
order to determine the correlation peaks.
2. Correlation peaks associated with indirect or apparent links in corresponding correlation triangles are
discarded from the analysis by means of a mathematical super-selection rule which deterministically
classifies the type of dependence between each triplets of neurons (Figure 2.2). The super-selection
rule is formally presented later. Here, it can be summarized as follows. If a correlation triangle is
made up of three correlation delays that are the combination of one another, one of the component
correlation delays is either representative of an indirect link (Figure 1A(ii)) or of an apparent link
(Figure 1A(iii)); therefore, this correlation delay does not refer to an effective connection and must be
discarded. When the algorithm finds a correlation triangle which satisfies this condition, it deepens
into the classification of the involved correlation delays and selects the correlation peak to remove
based on the peak’s amplitude. For example, in Figure 2.2, the algorithm identifies an indirect link
between 1 and 3 (a multi-neuron pathway), and an apparent link between 1 and 3 (correlation due to
a common output). The correlation peaks corresponding to these links in the correlation triangles are
discarded from the analysis. Importantly, the algorithm removes correlations from the analysis, but
does not remove inferred physical connections.
3. Only when all correlation peaks between two neurons are discarded, the algorithm recognizes that a
specific interaction is only apparent and deletes the corresponding connection. For example, in Fig-
ure 2.3, there is no existing connection between neurons 1 and 3. The estimated effective connectivity
includes only direct links for (1, 2) and (2, 3): two connections with opposite directionality exist be-
tween neuron 1 and 2 because positive and negative correlation peaks are detected in R
1,2
(τ
−1
1,2
and
τ
+1
1,2
); one link connects neuron 2 to neuron 3 as a result of the positive correlation peaks in R
2,3
(τ
+1
2,3
and τ
+2
2,3
).
The following sections describe the mathematical details of the developed technique. The connectivity
reconstruction algorithm and associated functions for performance evaluation were implemented in Matlab
and code is available online at https://github.com/fpuppo/ECRtools.git.
Reconstruction of functional connectivity
Temporal correlations
To identify the temporal correlations between the activity of all pairs of N recorded neurons j, k ∈
{1, ..., N} in the network, we computed the pairwise correlation function between the corresponding signals
s
j
and s
k
R
jk
(τ) =
Z
s
j
(t)s
k
(t + τ )dt (1)
5
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![Figure 1: Definition of functional and effective connectivity. (A) Classification of causal (directional link), indirect (multi-neurons pathway) and apparent (functional coupling due to common input) connectivity. (B) Classification of most common connectivity inference methods in terms of causality and detection of direct links. On the x -axes, the graph shows a scale of causality which refers to the ability of a given connectivity method to infer or not the directionality of the functional connections between neurons. On the y-axes, the graph visually quantifies the capabilities of one approach to detect direct links between neurons by identifying and discarding multi-neurons connections and apparent ones. Indicators that are acausal and do not infer direct links can only report about functional connectivity (light yellow); indicators which contain information about direction of interaction and direct neuron-to-neuron communication are close to the inference of effective connectivity (orange color). Most common model-free techniques are indicated with black dots. Model-based methods are reported with blue triangles and their ability to infer effective connectivity is negatively weighted by the impossibility to test their performances. The super-selective correlation approach we propose is reported in black like the other model-free methods; a diamond signal is used to emphasize the fact that, although being model-free, it aims at inferring effective connections. The graph visually summarizes results of comparisons between connectivity methods from review papers [14, 76, 19] and do not contain precise quantitative information about the differences.](/figures/figure-1-definition-of-functional-and-effective-connectivity-2yx7qrh7.webp)
