Simple Connectome Inference from Partial Correlation Statistics in Calcium Imaging

Abstract

In this work, we propose a simple yet effective solution to the problem of connectome inference in calcium imaging data. The proposed algorithm consists of two steps. First, processing the raw signals to detect neural peak activities. Second, inferring the degree of association between neurons from partial correlation statistics. This paper summarises the methodology that led us to win the Connectomics Challenge, proposes a simplified version of our method, and finally compares our results with respect to other inference methods.

Editors: Demian Battaglia, Isabelle Guyon, Vincent Lemaire, Javier Orlandi, Bisakha Ray, Jordi Soriano

The original form of this article appears in JMLR W&CP Volume 46.

Appendices

Appendix A. Description of the “Full Method”

This section provides a detailed description of the method specifically tuned for the Connectomics Challenge. We restrict our description to the differences with respect to the simplified method presented in the main paper. Most parameters were tuned so as to maximize AUROC on the normal-1 dataset and our design choices were validated by monitoring the AUROC obtained by the 145 entries we submitted during the challenge. Although the tuned method performs better than the simplified one on the challenge dataset, we believe that the tuned method clearly overfits the simulator used to generate the challenge data and that the simplified method should work equally well on new independent datasets. We nevertheless provide the tuned method here for reference purposes. Our implementation of the tuned method is available at https://github.com/asutera/kaggle-connectomics.

This appendix is structured as follows: Sect. A.1 describes the differences in terms of signal processing. Section A.2 then provides a detailed presentation of the averaging approach. Section A.3 presents an approach to correct the $p_{i,j}$ values so as to take into account the edge directionality. Finally, Sect. A.4 presents some experimental results to validate the different steps of our proposal.

1.1 A.1 Signal Processing

In Sect. 2, we introduced four filtering functions (f, g, h, and w) that are composed in sequence (i.e. $w \circ h \circ g \circ f$) to provide the signals from which to compute partial correlation statistics. Filtering is modified as follows in the tuned method:

• In addition to $f_1$ and $f_2$ (Eqs. 1 and 2), two alternative low-pass filters $f_3$ and $f_4$ are considered:

  $$\begin{aligned} f_3(x^t_i)&= x^{t-1}_i + x^{t}_i + x^{t+1}_i + x^{t+2}_i, \end{aligned}$$

  (7)
  $$\begin{aligned} f_4(x^t_i)&= x_i^t + x^{t+1}_i + x^{t+2}_i + x^{t+3}_i. \end{aligned}$$

  (8)

• An additional filter r is applied to smoothe differences in peak magnitudes that might remain after the application of the hard-threshold filter h:

  $$\begin{aligned} r(x^t_i) = (x_i^t)^c, \end{aligned}$$

  (9)

  with $c=0.9$.

• Filter w is replaced by a more complex filter $w^*$ defined as:

  $$\begin{aligned} w^*(x^{t}_i)&= {(x^{t}_i + 1 )^{\left( 1 + \frac{1}{\sum _{j} x^{t}_j}\right) }}^{k(\sum _{j} x^{t}_j)}, \end{aligned}$$

  (10)

  where the function k is a piecewise linear function optimised separately for each filter $f_1$, $f_2$, $f_3$ and $f_4$ (see the implementation for full details). Filter w in the simplified method is a special case of $w^*$ with $k(\sum _j x_j^t)=1$.

The pre-processed time-series are then obtained by the application of the following function: $w^*\circ r \circ h \circ g \circ f_i$ (with $i=1$, 2, 3, or 4).

1.2 A.2 Weighted Average of Partial Correlation Statistics

As discussed in Sect. 3, the performance of the method (in terms of AUROC) is sensitive to the value of the parameter $\tau$ of the hard-threshold filter h (see Eq. 4), and to the choice of the low-pass filter (among $\{f_1, f_2, f_3, f_4\}$). As in the simplified method, we have averaged the partial correlation statistics obtained for all the pairs $(\tau , \text {low-pass filter}) \in \{0.100,0.101,\ldots ,0.209\}\times \{f_1, f_2, f_3, f_4\}$.

Filters $f_1$ and $f_2$ display similar performances and thus were given similar weights (i.e. resp. 0.383 and 0.345). These weights were chosen equal to the weights selected for the simplified method. In contrast, filters $f_3$ and $f_4$ turn out, individually, to be less competitive and were therefore given less importance in the weighted average (i.e. resp. 0.004 and 0.268). Yet, as further shown in Sect. A.4, combining all 4 filters proves to marginally improve performance with respect to using only $f_1$ and $f_2$.

1.3 A.3 Prediction of Edge Orientation

Partial correlation statistics is a symmetric measure, while the connectome is a directed graph. It could thus be beneficial to try to predict edge orientation. In this section, we present an heuristic that modifies the $p_{ij}$ computed by the approach described before which takes into account directionality.

This approach is based on the following observation. The rise of fluorescence of a neuron indicates its activation. If another neuron is activated after a slight delay, this could be a consequence of the activation of the first neuron and therefore indicates a directed link in the connectome from the first to the second neuron. Given this observation, we have computed the following term for every pair (i, j):

$$\begin{aligned} s_{i,j} = \sum _{t=1}^{T - 1} \mathbbm {1}((x_j^{t+1} - x_i^t) \in \left[ \phi _1, \phi _2\right] ), \end{aligned}$$

(11)

that could be interpreted as an image of the number of times that neuron i activates neuron j. $\phi _1$ and $\phi _2$ are parameters whose values have been chosen in our experiments equal to 0.2 and 0.5, respectively. Their role is to define when the difference between $x_j^{t+1}$ and $x_i^t$ can indeed be assimilated to an event for which neuron i activates neuron j.

Afterwards, we have computed the difference between $s_{i,j}$ and $s_{j,i}$, that we call $z_{i,j}$, and used this difference to modify $p_{i,j}$ and $p_{j,i}$ so as to take into account directionality. Naturally, if $z_{i,j}$ is greater (smaller) than 0, we may conclude that should there be an edge between i and j, then this edge would have to be oriented from i to j (j to i).

This suggests the new association matrix r:

$$\begin{aligned} r_{i,j} = \mathbbm {1}(z_{i,j} > \phi _3) * p_{i,j} \end{aligned}$$

(12)

where $\phi _3 >0$ is another parameter. We discovered that this new matrix r was not providing good results, probably due to the fact that directivity was not rewarded well enough in the challenge.

This has lead us to investigate other ways for exploiting the information about directionality contained in the matrix z. One of those ways that gave good performance was to use as an association matrix:

$$\begin{aligned} q_{i,j} = weight * p_{i,j} + (1-weight) * z_{i,j} \end{aligned}$$

(13)

with weight chosen close to 1 ($weight=0.997$). Note that with values for weight close to 1, matrix q only uses the information to a minimum about directivity contained in z to modify the partial correlation matrix p. We tried smaller values for weight but those provided poorer results.

It was this association matrix $q_{i,j}$ that actually led to the best results of the challenge, as shown in Table 3 of Sect. A.4.

1.4 A.4 Experiments

On the interest of low-pass filters $\varvec{f_3}$ and $\varvec{f_4}$. As reported in Table 2, averaging over all low-pass filters leads to better AUROC scores than averaging over only two low-pass filters, i.e. $f_1$ and $f_2$. However this slightly reduces AUPRC.

Table 2 Performance on normal –1, 2, 3, or 4 with partial correlation with different averaging approaches

On the interest of using matrix $\varvec{q}$ rather than $\varvec{p}$ to take into account directivity. Table 3 compares AUROC and AUPRC with or without correcting the $p_{i,j}$ values according to Eq. 13. Both AUROC and AUPRC are (very slightly) improved by using information about directivity.

Table 3 Performance on normal –1, 2, 3, 4 of “Full Method” with and without using information about directivity

Appendix B. Supplementary Results

In this appendix we report the performance of the different methods compared in the paper on 6 additional datasets provided by the Challenge organisers. These datasets, corresponding each to networks of 1,000 neurons, are similar to the normal datasets except for one feature:

 

lowcon::

Similar network but on average with a lower number of connections per neuron.

highcon::

Similar network but on average with a higher number of connections per neuron.

lowcc::

Similar network but on average with a lower clustering coefficient.

highcc::

Similar network but on average with a higher clustering coefficient.

normal-3-highrate::

Same topology as normal-3 but with a higher firing frequency, i.e. with highly active neurons.

normal-4-lownoise::

Same topology as normal-4 but with a better signal-to-noise ratio.

 

The results of several methods applied to these 6 datasets are provided in Table 4. They confirm what we observed on the normal datasets. Average partial correlation and its tuned variant, i.e. “Full method”, clearly outperform other network inference methods on all datasets. PC is close to GENIE3 and GTE, but still slightly worse. GENIE3 performs better than GTE most of the time. Note that the"Full method" reported in this table does not use Eq. 13 to slightly correct the values of $p_{i,j}$ to take into account directivity.

Table 4 Performance (top: AUROC, bottom: AUPRC) on specific datasets with different methods

Appendix C. On the Selection of the Number of Principal Components

The (true) network, seen as a matrix, can be decomposed through a singular value decomposition (SVD) or principal component analysis (PCA), so as to respectively determine a set of independent linear combinations of the variable (Alter et al. 2000), or a reduced set of linear combinations combine, which then maximize the explained variance of the data (Jolliffe 2005). Since SVD and PCA are related, they can be defined by the same goal: both aim at finding a reduced set of neurons, known as components, whose activity can explain the rest of the network.

The distribution of component eigen values obtained from PCA and SVD decompositions can be studied by sorting them in descending order of magnitude, as illustrated in Fig. 3. It can be seen that some component eigen values are zero, implying that the behaviour of the network could be explained by a subset of neurons because of the redundancy and relations between the neurons. For all datasets, the eigen value distribution is exactly the same.

Fig. 3

Explained variance ratio by number of principal components (left) and singular value ratio by number of principal components (right) for all networks

In the context of the challenge, we observe that only 800 components seem to be necessary and we exploit this when computing partial correlation statistics. Therefore, the value of the parameter M is immediate and should be clearly set to 800 (${=}0.8p$).

Note that if the true network is not available, similar decomposition analysis could be carried on the inferred network, or on the data directly.

Appendix D. Summary Table

See Table 5.

Table 5 Connectomics challenge summary