# FARCI: Fast and Robust Connectome Inference

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## Abstract

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## 1. Introduction

^{2+}indicators, either using chemical dyes or genetically encoded Ca

^{2+}indicator (GECI), to detect Ca

^{2+}transients in a neuron associated with an action potential. When combined with powerful image processing algorithms, 2p Ca

^{2+}imaging enables long-term monitoring of the activity of neuronal ensembles in awake animals, and how the activity and thus functional connectome of these neurons change over time, for example, with learning [5,7,10,11]. Note that further data processing is needed to infer neuronal action potentials from Ca

^{2+}fluorescence traces. Extracting neuronal activity from Ca

^{2+}imaging data is a non-trivial task due to significant noise, baseline fluorescence drift, and other technical constraints, such as low sampling rate and slow decay of fluorescence sensor relative to the time-scale of neuronal firing dynamics [12]. Finally, we still lack technologies that are able to record the activity of all neurons in the brain simultaneously in complex organisms such as rodents or primates (i.e., some neurons are hidden from the measurements). Thus, functional connectome inferred from neuronal activity data does not necessarily imply the existence of (actual) synaptic connections between neurons.

## 2. Materials and Methods

#### 2.1. Spike Deconvolution

^{2+}spikes from 2p fluorescence images is an active area of research with more sophisticated methods being developed in a regular fashion. In our work, we used sparse Non-Negative Deconvolution (NND) method for spike inference because of its relatively fast and robust performance, as demonstrated in a recent Spike Inference Challenge [12]. Other spike inference algorithms, such as MLSpike [20] or CASCADE [21], can also be used in place of the NND method, if desired.

^{2+}fluorescence data (dF/F0) of neuron i, $f$ represents the deconvolution function, and ${x}_{i}$ is the deconvolved neuronal spiking activity.

#### 2.2. Spike Thresholding

#### 2.3. Spike Smoothing

^{2+}fluorescence data to the final neuronal activity spikes is illustrated in Figure 1.

#### 2.4. Partial Correlation Statistics

#### 2.5. Performance Evaluation

^{2+}fluorescence datasets (dF/F0) from the Neural Connectomics Challenge and datasets generated using NAOMi simulator [18]. The NCC datasets were simulated using a mathematical model of Ca

^{2+}fluorescence signal that takes into account limitations of Ca

^{2+}imaging technology such as temporal resolution and light scattering artifacts [15,26]. In the NCC, the challenge organizers generated in silico Calcium fluorescence images for neurons that are placed randomly in a 1 mm × 1 mm area with random connections of a given average connectivity and clustering coefficient, which isthe average number of triangles a neuron forms with its neighbors over the total possible number of triangles it could form given its connectivity. A high clustering coefficient is associated with a network with tightly connected neighborhoods [18]. A model of leaky integrate and fire neuron with short term synaptic depression [26] was implemented in the NEST simulator [27,28] to generate neuronal firing dynamics with a firing rate of 0.1 Hz. The neuronal firing was then coupled with a fluorescence response model of Ca

^{2+}markers to simulate in silico Ca

^{2+}images at a rate of 50 Hz for 60 min. Ten baseline datasets for neuronal networks of size 100 (n = 6) and 1000 neurons (n = 4)—referred to as the small and normal connectomes, respectively—were generated (see Table 1). The connectivity of the small and normal connectomes has a relatively low density that ranges within $16.3\pm 1.7\%$ for the small networks and $2.1\pm 0.5\%$ for the normal networks, suggesting that these connectomes are sparse. Six additional datasets of 1000 neurons for higher and lower levels of signal-to-noise ratio, neuronal firing rate, network clustering coefficient, and average connectivity, were also available. More details of the data generation can be found in the NCC publication [17]. We also tested the performance of FARCI on a lower imaging rate of 25 Hz by downsampling the NCC datasets—keeping every other frame of the original data.

^{2+}fluorescence images for the neuronal population in this volume. The number of neurons in the volume is generated randomly, and so is the neuronal connectome, specifically using the Hawkes model [29] based on the Watts–Strogatz small-world network [30]. Neuronal activity is modeled as correlated bursting neurons, which is coupled with an optical microscopy model to produce in silico Ca

^{2+}fluorescence images. We utilized NAOMi to generate five neuronal volumes of size 50 μm × 50 μm × 150 μm with ~100 neurons and another five volumes of size 300 μm × 300 μm × 150 μm with ~1000 neurons. NAOMi simulation parameters that were used for in silico data generation are detailed in Supplementary Table S1. The generated in silico Ca

^{2+}fluorescence images were converted to time-series fluorescence traces (dF/F0) using a built-in subroutine in NAOMi (times_from_prof).

#### 2.6. Performance Comparison

^{2+}fluorescence data: the best performing method in the NCC by Sutera et al. 2015 [13], the baseline method in the NCC called Generalized Transfer Entropy [15], and a widely-used Calcium fluorescence analysis toolbox called FluoroSNNAP [19]. Sutera et al. algorithm comprises a four-step signal processing pipeline (low-pass filter, high-pass filter, hard thresholding, and weighting), and similar to FARCI, also produces partial correlation networks. We implemented Sutera et al. algorithm in MATLAB with the aid of the original developer [13], and were able to reproduce the results of the algorithm independently. Like FARCI, we used partial correlation coefficients generated by Sutera et al. algorithm to give the ranked list of neuronal connections (descending order) for performance scoring.

^{2+}fluorescence images. FluoroSNAPP relies on the temporal synchrony of spiking events to establish connectivity between neurons. Here, we used the MATLAB subroutine PhaseSpike in FluoroSNNAP package to evaluate the phase difference ${\Psi}_{X,Y}\left(t\right)$ of pairs of neurons X and Y, using the thresholded spike times of the neurons (i.e., the output of spike thresholding step in FARCI) as inputs. We then applied the subroutine FC_phase in FluoroSNNAP to perform 100 repeated runs of Kolmogorov–Smirnov (KS) tests comparing the phase difference ${\Psi}_{X,Y}\left(t\right)$ against a random sample $\widehat{{\Psi}_{X,Y}}\left(t\right)$ taken from a null distribution. In FluoroSNNAP, the functional connection between neurons X and Y is determined based on the 95th percentile of the p-values from the KS tests above. Correspondingly, for performance scoring, we used the 95th percentile of the p-values to rank neuronal connections (in ascending order).

## 3. Results

^{2+}fluorescence data. In FARCI, the functional neuronal connectome is represented by the partial correlation network among the neurons. Figure 2 illustrates the workflow of the functional connectome inference in FARCI, which comprises the following key steps: (1) deconvolution of spiking activity from Ca

^{2+}fluorescence data, (2) spike thresholding, (3) spike smoothing, and (4) evaluation of partial correlations. The details of the individual steps can be found in Materials and Methods. We benchmarked FARCI using the Ca

^{2+}fluorescence datasets from the Neural Connectomics Challenge (NCC) [17]. We also compared the performance of FARCI with that of the best performing method in the NCC, the inference algorithm by Sutera et al. [13].

#### 3.1. Neuronal Spike Deconvolution

^{2+}fluorescence imaging data give only indirect measurements of neuronal activity, and thus, require data pre-processing to extract the underlying neuronal action potential spikes. We employed the OASIS deconvolution algorithm [22] from the MATLAB package Suite2P [23] that uses a non-negative deconvolution strategy to provide estimates for timing and amplitude of spiking activity. Table 2 gives the AUROC and AUPR values for using the partial correlations of the deconvolved spikes to infer neuronal connectomes (see Supplementary Tables S2–S4 for more detailed results). While the AUROCs were generally good (>0.78), the AUPRs were as low as 0.23.

#### 3.2. Binarization of Neuronal Spikes

#### 3.3. Neuronal Spike Thresholding

#### 3.4. Neuronal Spike Smoothing

#### 3.5. FARCI Performance

#### 3.6. Missing Neurons

^{2+}imaging plane), the two neurons may have a high partial correlation due to the lack of data for the presynapse.

^{2+}fluorescence data from the connectomes with 1000 neurons, with the following sizes: 50, 200, 400, 600, and 800 neurons. For each random sample, we applied FARCI and the other algorithms, and for each network size, we evaluated the average of AUROC and AUPR and the computational runtime.

#### 3.7. Computational Speed

## 4. Discussion

^{2+}imaging data. FARCI combines a fast non-negative deconvolution algorithm OASIS [22], spike thresholding, and spike smoothing, to extract information for neuronal spike events from Ca

^{2+}fluorescence signals. FARCI produces a partial correlation network of the neurons for functional connectome inference. We benchmarked FARCI using in silico ground truth datasets from the Neural Connectomics Challenge [17] and by the state-of-the-art simulator NAOMi [18], and compared its performance with the winning algorithm in the NCC by Sutera et al. [13], Generalized Transfer Entropy [15], and Fluorescence Single Neuron and Network Analysis Package [19]. The results showed that FARCI outperforms the comparative methods in terms of connectome inference accuracy as measured by AUROC and AUPR and computational runtimes and scaling. FARCI and Sutera et al. methods provided AUROC values that are generally high (mostly above 0.8), and are better than GTE and FluoroSNNAP. Of note, FluoroSNNAP consistently gave the lowest scores that were similar to a random predictor (AUROC of 0.5). As the ground truth connectomes were sparse, we had imbalanced classes with many more negatives than positives. In this case, AUROC is known to be a poor diagnostic tool for method performance, and AUPR is the more sensitive metric for method performance. In terms of AUPR, FARCI outperformed; it had a slight advantage over the Sutera et al. algorithm and was superior to GTE and FluoroSNNAP. Also, the high performance of FARCI is robust with respect to the connectome size, data noise and sampling rate, and network densities.

^{2+}fluorescence data. First, the typical rate of data sampling for two-photon Ca

^{2+}imaging ranges between 30–100 ms (i.e., ~10–30 Hz) [35], which is much longer than the time scale of neuronal action potentials and the following refractory period between 1–5 ms [36]. Given the sampling rate of Ca

^{2+}imaging, neurons may have fired several times in between any two image frames, and thus the expected sequential timing of pre- and post-synaptic neuron firing has a low chance to be captured accurately. Besides, because of the temporal coding scheme of neurons, the most informative data for establishing causal connections may reside in brief periods of time when the relevant neurons are active. While model-free methods for establishing causal connections using Ca

^{2+}imaging data exist in the literature (e.g., using the concept of transfer entropy [15]), fundamental challenges in determining directional (causal) connectivity from time series data, like the ones mentioned above, will put a limit to the accuracy of the inferred connectome [6].

^{2+}spikes is known to lead to omission of neuronal response to inhibition [37]. Note that such an issue afflicts most of the current spike inference algorithms, but there are possible workarounds for connectomes in which inhibited neurons play a major role [37].

## 5. Conclusions

^{2+}fluorescence data. FARCI relies on multivariate partial correlation analysis of (pre-processed) neuronal Ca

^{2+}spike activity to establish connectivity among neurons. We benchmarked FARCI using in silico time-series Ca

^{2+}fluorescence datasets from the Neural Connectomics Challenge [17] and those generated by the state-of-the-art simulator NAOMi [18], against the winning method in the NCC by Sutera et al. [13], Generalized Transfer Entropy [15], and FluoroSNNAP [19]. The results demonstrated the superior performance of FARCI, both in accuracy and computational time and scaling, over the comparative methods. However, FARCI produces a partial correlation network as its output, and thus does not provide the directionality of the neuronal connections. Also, like many existing inference methods, FARCI does not account for the activity of inhibitory neurons during the reconstruction of functional connectome.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Data preprocessing pipeline of Calcium fluorescence data. (

**A**) Raw Ca

^{2+}fluorescence signal (dF/F0). (

**B**) Deconvolved spikes using OASIS. (

**C**) Thresholded spikes. (

**D**) Smoothed spikes.

**Figure 2.**Workflow of connectome inference in FARCI. FARCI combines thresholding and smoothing of neuronal spikes, the output of which is used to generate partial correlation networks.

**Figure 3.**Effect of signal thresholding on AUROC and AUPR in networks of (

**A**) 100 (n = 6) and (

**B**) 1000 neurons (n = 10). While AUROC tends to drop with increasing α, AUPR reaches a peak for values of 2 < α < 3. The shaded area denotes the standard error of the mean (SEM).

**Figure 4.**Performance comparison of FARCI with Sutera et al. GTE, and FluoroSNNAP algorithms. The accuracy of the inferred connectome is measured by (

**A**,

**C**) AUROC and (

**B**,

**D**) AUPR for (

**A**,

**B**) NCC and (

**C**,

**D**) NAOMi datasets. The complete results of the benchmarking and comparison are provided in Supplementary Tables S7 and S8. Statistical significance was assessed using two-sided paired t-test. The results for KS and F test for normality and constant variance are given in Supplementary Table S9.

**Figure 5.**Performance evaluation of connectome inference with missing neurons. Comparison of FARCI with Sutera et al. GTE and FluoroSNNAP in terms of (

**A**,

**C**) AUROC and (

**B**,

**D**) AUPR using subsampled datasets from (

**A**,

**B**) NCC and (

**C**,

**D**) NAOMi datasets. Error bars indicate 95% confidence interval.

**Figure 6.**Computational runtimes for the NCC datasets. (

**A**) Runtime comparison of FARCI, Sutera et al. GTE, and FluoroSNNAP algorithms using Normal-1 network. (

**B**) FARCI runtimes for different sizes of subsampled networks for 1000-neuron datasets from the NCC (n = 10).

**Table 1.**Datasets provided in the Neural Connectomics Challenge. Each dataset contains three types of information: 1. neuronal activity in the form of Ca fluorescence signals, 2. the ground truth connectome structure, and 3. the spatial coordinates of neurons.

Networks | # of Neurons | Description |
---|---|---|

small (6 datasets) | 100 | six connectomes with 100 neurons |

normal (4 datasets) | 1000 | four connectomes with 1000 neurons |

normal3-highrate | 1000 | normal-3 connectome with a higher neuronal firing frequency |

normal4-lownoise | 1000 | normal-4 connectome with a higher signal-to-noise ratio |

highcc | 1000 | a connectome of 1000 neurons with a higher clustering coefficient |

lowcc | 1000 | a connectome of 1000 neurons with a lower clustering coefficient |

highcon | 1000 | a connectome of 1000 neurons with a higher average connectivity |

lowcon | 1000 | a connectome of 1000 neurons with a lower average connectivity |

**Table 2.**Effect of different signal processing steps on connectome inference. Unlike binarization, signal thresholding and smoothing improve the accuracy of connectome inference.

AUROC | AUPR | |||||
---|---|---|---|---|---|---|

Filter | Small (n = 6) | Normal (n = 4) | Others (n = 6) | Small (n = 6) | Normal (n = 4) | Others (n = 6) |

Neuronal Spikes $x$ | 0.871 ± 0.058 | 0.892 ± 0.002 | 0.905 ± 0.030 | 0.543 ± 0.097 | 0.335 ± 0.005 | 0.347 ± 0.067 |

Spikes + Binarization $u\left(x\right)$ | 0.563 ± 0.055 | 0.653 ± 0.006 | 0.647 ± 0.069 | 0.189 ± 0.021 | 0.042 ± 0.002 | 0.050 ± 0.021 |

Spikes + Thresholding $g\left(x\right)$ | 0.876 ± 0.051 | 0.882 ± 0.003 | 0.895 ± 0.040 | 0.620 ± 0.096 | 0.408 ± 0.012 | 0.421 ± 0.114 |

Spikes + Smoothing $h\left(x\right)$ | 0.891 ± 0.035 | 0.908 ± 0.002 | 0.917 ± 0.025 | 0.538 ± 0.056 | 0.330 ± 0.004 | 0.346 ± 0.053 |

FARCI $h\left(g\left(x\right)\right)$ | 0.916 ± 0.031 | 0.908 ± 0.002 | 0.918 ± 0.032 | 0.741 ± 0.067 | 0.491 ± 0.003 | 0.497 ± 0.100 |

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Meamardoost, S.; Bhattacharya, M.; Hwang, E.J.; Komiyama, T.; Mewes, C.; Wang, L.; Zhang, Y.; Gunawan, R. FARCI: Fast and Robust Connectome Inference. *Brain Sci.* **2021**, *11*, 1556.
https://doi.org/10.3390/brainsci11121556

**AMA Style**

Meamardoost S, Bhattacharya M, Hwang EJ, Komiyama T, Mewes C, Wang L, Zhang Y, Gunawan R. FARCI: Fast and Robust Connectome Inference. *Brain Sciences*. 2021; 11(12):1556.
https://doi.org/10.3390/brainsci11121556

**Chicago/Turabian Style**

Meamardoost, Saber, Mahasweta Bhattacharya, Eun Jung Hwang, Takaki Komiyama, Claudia Mewes, Linbing Wang, Ying Zhang, and Rudiyanto Gunawan. 2021. "FARCI: Fast and Robust Connectome Inference" *Brain Sciences* 11, no. 12: 1556.
https://doi.org/10.3390/brainsci11121556