# Inferring Excitatory and Inhibitory Connections in Neuronal Networks

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Neuronal Network Modeling

#### 2.2. Transfer Entropy

## 3. Results

#### 3.1. Increasing the Delay Improves Inhibitory Network Reconstruction

#### 3.2. The Ratio ${g}_{\mathrm{E}}/{g}_{\mathrm{I}}$ Does Not Affect Network Reconstruction

#### 3.3. Down-Sampled Bin Size Effect on Network Reconstruction

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Chialvo, D.R. Emergent complex neural dynamics. Nat. Phys.
**2010**, 6, 744–750. [Google Scholar] [CrossRef] [Green Version] - Avena-Koenigsberger, A.; Misic, B.; Sporns, O. Communication dynamics in complex brain networks. Nat. Rev. Neurosci.
**2018**, 19, 17–33. [Google Scholar] [CrossRef] - Park, H.J.; Friston, K. Structural and functional brain networks: From connections to cognition. Science
**2013**, 342, 6195. [Google Scholar] [CrossRef] [Green Version] - Bullmore, E.; Sporns, O. Complex brain networks: Graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci.
**2009**, 10, 186–198. [Google Scholar] [CrossRef] - Tibau, E.; Valencia, M.; Soriano, J. Identification of neuronal network properties from the spectral analysis of calcium imaging signals in neuronal cultures. Front. Neural Circuits
**2013**, 7, 199. [Google Scholar] [CrossRef] - Gobel, W.; Helmchen, F. In Vivo calcium imaging of neural network function. Physiology
**2007**, 22, 358–365. [Google Scholar] [CrossRef] [Green Version] - Yang, W.; Yuste, R. In Vivo imaging of neural activity. Nat. Methods
**2017**, 14, 349–359. [Google Scholar] [CrossRef] [PubMed] - Ruz, I.D.; Schultz, S.R. Localising and classifying neurons from high density MEA recordings. J. Neurosci. Methods
**2014**, 233, 115–128. [Google Scholar] - Ito, S.; Yeh, F.C.; Hiolski, E.; Rydygier, P.; Gunning, D.E.; Hottowy, P.; Timme, N.; Litke, A.M.; Beggs, J.M. Large-scale, high-resolution multielectrode-array recording depicts functional network differences of cortical and hippocampal cultures. PLoS ONE
**2014**, 9, e105324. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Odawara, A.; Katoh, H.; Matsuda, N.; Suzuki, I. Physiological maturation and drug responses of human induced pluripotent stem cell-derived cortical neuronal networks in long-term culture. Sci. Rep.
**2016**, 6, 26181. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Eichler, M.; Dahlhaus, R.; Sandkühler, J. Partial correlation analysis for the identification of synaptic connections. Biol. Cybern.
**2003**, 89, 289–302. [Google Scholar] [CrossRef] - Sheikhattar, A.; Miran, S.; Liu, J.; Fritz, J.B.; Shamma, S.A.; Kanold, P.O.; Babadi, B. Extracting neuronal functional network dynamics via adaptive Granger causality analysis. Proc. Natl. Acad. Sci. USA
**2018**, 115, E3869–E3878. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Poli, D.; Pastore, V.P.; Martinoia, S.; Massobrio, P. From functional to structural connectivity using partial correlation in neuronal assemblies. J. Neural Eng.
**2016**, 13, 026023. [Google Scholar] [CrossRef] [PubMed] - Friston, K.; Moran, R.; Seth, A.K. Analysing connectivity with Granger causality and dynamic causal modelling. Curr. Opin. Neurobiol.
**2013**, 23, 172–178. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stetter, O.; Battaglia, D.; Soriano, J.; Geisel, T. Model-free reconstruction of excitatory neuronal connectivity from calcium imaging signals. PLoS Comput. Biol.
**2012**, 8, e1002653. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ito, S.; Hansen, M.E.; Heiland, R.; Lumsdaine, A.; Litke, A.M.; Beggs, J.M. Extending transfer entropy improves identification of effective connectivity in a spiking cortical network model. PLoS ONE
**2011**, 6, e27431. [Google Scholar] [CrossRef] [PubMed] - Pastore, V.P.; Massobrio, P.; Godjoski, A.; Martinoia, S. Identification of excitatory-inhibitory links and network topology in large-scale neuronal assemblies from multi-electrode recordings. PLoS Comput. Biol.
**2018**, 14, e1006381. [Google Scholar] [CrossRef] [PubMed] - Orlandi, J.G.; Stetter, O.; Soriano, J.; Geisel, T.; Battaglia, D. Transfer entropy reconstruction and labeling of neuronal connections from simulated calcium imaging. PLoS ONE
**2014**, 9, e98842. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Novelli, L.; Wollstadt, P.; Mediano, P.; Wibral, M.; Lizier, J.T. Large-scale directed network inference with multivariate transfer entropy and hierarchical statistical testing. Netw. Neurosci.
**2019**, 3, 827–847. [Google Scholar] [CrossRef] - Huang, C.S.; Pal, N.R.; Chuang, C.H.; Lin, C.T. Identifying changes in EEG information transfer during drowsy driving by transfer entropy. Front. Hum. Neurosci.
**2015**, 9, 570. [Google Scholar] [CrossRef] [Green Version] - Wibral, M.; Rahm, B.; Rieder, M.; Lindner, M.; Vicente, R.; Kaiser, J. Transfer entropy in magnetoencephalographic data: Quantifying information flow in cortical and cerebellar networks. Prog. Biophys. Mol. Biol.
**2011**, 105, 80–97. [Google Scholar] [CrossRef] - Kajiwara, M.; Nomura, R.; Goetze, F.; Kawabata, M.; Isomura, Y.; Akutsu, T.; Shimono, M. Inhibitory neurons exhibit high controlling ability in the cortical microconnectome. PLoS Comput. Biol.
**2021**, 17, e1008846. [Google Scholar] [CrossRef] [PubMed] - Brunel, N. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci.
**2000**, 8, 183–208. [Google Scholar] [CrossRef] [PubMed] - Izhikevich, E.M. Simple model of spiking neurons. IEEE Trans. Neural Netw.
**2003**, 14, 1569–1572. [Google Scholar] [CrossRef] [Green Version] - Alvarez-Lacalle, E.; Moses, E. Slow and fast pulses in 1-D cultures of excitatory neurons. J. Comput. Neurosci.
**2009**, 26, 475–493. [Google Scholar] [CrossRef] [PubMed] - Orlandi, J.G.; Soriano, J.; Alvarez-Lacalle, E.; Teller, S.; Casademunt, J. Noise focusing and the emergence of coherent activity in neuronal cultures. Nat. Phys.
**2013**, 9, 582–590. [Google Scholar] [CrossRef] - Markram, H.; Lübke, J.; Frotscher, M.; Roth, A.; Sakmann, B. Physiology and anatomy of synaptic connections between thick tufted pyramidal neurones in the developing rat neocortex. J. Physiol.
**1997**, 500, 409–440. [Google Scholar] [CrossRef] - Thomson, A.M.; West, D.C.; Wang, Y.; Bannister, A.P. Synaptic connections and small circuits involving excitatory and inhibitory neurons in layers 2–5 of adult rat and cat neocortex: Triple intracellular recordings and biocytin labelling in vitro. Cereb. Cortex
**2002**, 12, 936–953. [Google Scholar] [CrossRef] [Green Version] - Holmgren, C.; Harkany, T.; Svennenfors, B.; Zilberter, Y. Pyramidal cell communication within local networks in layer 2/3 of rat neocortex. J. Physiol.
**2003**, 551, 139–153. [Google Scholar] [CrossRef] - Soriano, J.; Martínez, M.R.; Tlusty, T.; Moses, E. Development of input connections in neural cultures. Proc. Natl. Acad. Sci. USA
**2008**, 105, 13758–13763. [Google Scholar] [CrossRef] [Green Version] - Kerr, J.N.; Greenberg, D.; Helmchen, F. Imaging input and output of neocortical networks in vivo. Proc. Natl. Acad. Sci. USA
**2005**, 102, 14063–14068. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Grewe, B.F.; Langer, D.; Kasper, H.; Kampa, B.M.; Helmchen, F. High-speed in vivo calcium imaging reveals neuronal network activity with near-millisecond precision. Nat. Methods
**2010**, 7, 399–405. [Google Scholar] [CrossRef] [PubMed] - Palazzolo, G.; Moroni, M.; Soloperto, A.; Aletti, G.; Naldi, G.; Vassalli, M.; Nieus, T.; Difato, F. Fast wide-volume functional imaging of engineered in vitro brain tissues. Sci. Rep.
**2017**, 7, 8499. [Google Scholar] [CrossRef] [Green Version] - Schreiber, T. Measuring information transfer. Phys. Rev. Lett.
**2000**, 85, 461. [Google Scholar] [CrossRef] [Green Version] - MacKay, D.J.; Mac Kay, D.J. Information Theory, Inference and Learning Algorithms; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Wollstadt, P.; Martínez-Zarzuela, M.; Vicente, R.; Díaz-Pernas, F.J.; Wibral, M. Efficient transfer entropy analysis of non-stationary neural time series. PLoS ONE
**2014**, 9, e102833. [Google Scholar] [CrossRef] [Green Version] - Shovon, M.H.I.; Nandagopal, N.; Vijayalakshmi, R.; Du, J.T.; Cocks, B. Directed connectivity analysis of functional brain networks during cognitive activity using transfer entropy. Neural Process. Lett.
**2017**, 45, 807–824. [Google Scholar] [CrossRef] - Thivierge, J.P. Scale-free and economical features of functional connectivity in neuronal networks. Phys. Rev. E
**2014**, 90, 022721. [Google Scholar] [CrossRef] [PubMed] - Shimono, M.; Beggs, J.M. Functional clusters, hubs, and communities in the cortical microconnectome. Cereb. Cortex
**2015**, 25, 3743–3757. [Google Scholar] [CrossRef] [Green Version] - Isaacson, J.S.; Scanziani, M. How inhibition shapes cortical activity. Neuron
**2011**, 72, 231–243. [Google Scholar] [CrossRef] [Green Version] - Sanchez-Vives, M.V.; Mattia, M.; Compte, A.; Perez-Zabalza, M.; Winograd, M.; Descalzo, V.F.; Reig, R. Inhibitory modulation of cortical up states. J. Neurophysiol.
**2010**, 104, 1314–1324. [Google Scholar] [CrossRef] - Tahvildari, B.; Wölfel, M.; Duque, A.; McCormick, D.A. Selective functional interactions between excitatory and inhibitory cortical neurons and differential contribution to persistent activity of the slow oscillation. J. Neurosci.
**2012**, 32, 12165–12179. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Vogels, T.P.; Sprekeler, H.; Zenke, F.; Clopath, C.; Gerstner, W. Inhibitory plasticity balances excitation and inhibition in sensory pathways and memory networks. Science
**2011**, 334, 1569–1573. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lizier, J.T.; Prokopenko, M.; Zomaya, A.Y. Local information transfer as a spatiotemporal filter for complex systems. Phys. Rev. E
**2008**, 77, 026110. [Google Scholar] [CrossRef] [Green Version] - Dunning, D.; Hoover, C.; Soltesz, I.; Smith, M.; O’Dowd, D.K. GABAA receptor–mediated miniature postsynaptic currents and α-subunit expression in developing cortical neurons. J. Neurophysiol.
**1999**, 82, 3286–3297. [Google Scholar] [CrossRef] [PubMed] - Ropert, N.; Miles, R.; Korn, H. Characteristics of miniature inhibitory postsynaptic currents in CA1 pyramidal neurones of rat hippocampus. J. Physiol.
**1990**, 428, 707–722. [Google Scholar] [CrossRef] [Green Version] - Collingridge, G.L.; Gage, P.W.; Robertson, B. Inhibitory post-synaptic currents in rat hippocampal CA1 neurones. J. Physiol.
**1984**, 356, 551–564. [Google Scholar] [CrossRef] - Xiang, Z.; Huguenard, J.R.; Prince, D.A. GABAA receptor-mediated currents in interneurons and pyramidal cells of rat visual cortex. J. Physiol.
**1998**, 506, 715–730. [Google Scholar] [CrossRef] [PubMed] - Gardner, D. Variations in amplitude and time course of inhibitory postsynaptic currents. J. Neurophysiol.
**1986**, 56, 1424–1438. [Google Scholar] [CrossRef] - Liu, G.; Choi, S.; Tsien, R.W. Variability of neurotransmitter concentration and nonsaturation of postsynaptic AMPA receptors at synapses in hippocampal cultures and slices. Neuron
**1999**, 22, 395–409. [Google Scholar] [CrossRef] [Green Version] - Wyllie, D.J.; Manabe, T.; Nicoll, R.A. A rise in postsynaptic Ca
^{2+}potentiates miniature excitatory postsynaptic currents and AMPA responses in hippocampal neurons. Neuron**1994**, 12, 127–138. [Google Scholar] [CrossRef] - Sah, P.; Hestrin, S.; Nicoll, R. Properties of excitatory postsynaptic currents recorded in vitro from rat hippocampal interneurones. J. Physiol.
**1990**, 430, 605–616. [Google Scholar] [CrossRef] - Pospischil, M.; Toledo-Rodriguez, M.; Monier, C.; Piwkowska, Z.; Bal, T.; Frégnac, Y.; Markram, H.; Destexhe, A. Minimal Hodgkin–Huxley type models for different classes of cortical and thalamic neurons. Biol. Cybern.
**2008**, 99, 427–441. [Google Scholar] [CrossRef] - Nicoletti, M.; Loppini, A.; Chiodo, L.; Folli, V.; Ruocco, G.; Filippi, S. Biophysical modeling of C. elegans neurons: Single ion currents and whole-cell dynamics of AWCon and RMD. PLoS ONE
**2019**, 14, e0218738. [Google Scholar] [CrossRef] - Esser, S.K.; Hill, S.L.; Tononi, G. Modeling the effects of transcranial magnetic stimulation on cortical circuits. J. Neurophysiol.
**2005**, 94, 622–639. [Google Scholar] [CrossRef] [Green Version] - Rusu, C.V.; Murakami, M.; Ziemann, U.; Triesch, J. A model of TMS-induced I-waves in motor cortex. Brain Stimul.
**2014**, 7, 401–414. [Google Scholar] [CrossRef] [PubMed] - Louis, S.G.; Gerstein, G.L.; Grün, S.; Diesmann, M. Surrogate spike train generation through dithering in operational time. Front. Comput. Neurosci.
**2010**, 4, 127. [Google Scholar] [CrossRef] [Green Version] - Timme, N.M.; Lapish, C. A tutorial for information theory in neuroscience. eNeuro
**2018**, 5. [Google Scholar] [CrossRef] [PubMed] - Grun, S. Data-driven significance estimation for precise spike correlation. J. Neurophysiol.
**2009**, 101, 1126–1140. [Google Scholar] [CrossRef] [Green Version] - Harmah, D.J.; Li, C.; Li, F.; Liao, Y.; Wang, J.; Ayedh, W.; Bore, J.C.; Yao, D.; Dong, W.; Xu, P. Measuring the non-linear directed information flow in schizophrenia by multivariate transfer entropy. Front. Comput. Neurosci.
**2020**, 13, 85. [Google Scholar] [CrossRef] [PubMed] - Zhong, Y.; Huang, L.; Cai, S.; Zhang, Y.; von Deneen, K.M.; Ren, A.; Ren, J. Altered effective connectivity patterns of the default mode network in Alzheimer’s disease: An fMRI study. Neurosci. Lett.
**2014**, 578, 171–175. [Google Scholar] [CrossRef] [PubMed] - Inman, C.S.; James, G.A.; Hamann, S.; Rajendra, J.K.; Pagnoni, G.; Butler, A.J. Altered resting-state effective connectivity of fronto-parietal motor control systems on the primary motor network following stroke. Neuroimage
**2012**, 59, 227–237. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wang, M.Y.; Wang, J.; Zhou, J.; Guan, Y.G.; Zhai, F.; Liu, C.Q.; Xu, F.F.; Han, Y.X.; Yan, Z.F.; Luan, G.M. Identification of the epileptogenic zone of temporal lobe epilepsy from stereo-electroencephalography signals: A phase transfer entropy and graph theory approach. Neuroimage Clin.
**2017**, 16, 184–195. [Google Scholar] [CrossRef] [PubMed] - Parker, C.S.; Clayden, J.D.; Cardoso, M.J.; Rodionov, R.; Duncan, J.S.; Scott, C.; Diehl, B.; Ourselin, S. Structural and effective connectivity in focal epilepsy. Neuroimage Clin.
**2018**, 17, 943–952. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Modeled networks, simulated data and TE decomposition.(

**A**) Example of modeled neuronal network including 100 cells, 80% excitatory and 20% inhibitory, connected via a spatial gaussian kernel. Red/blue circles and arrows denote excitatory/inhibitory neurons and directed connections, respectively. (

**B**) Simulated membrane voltage time series (continuous curves) and corresponding calcium signals (dashed curves) for two representative cells (black and gray). Calcium traces are computed by a convolution of the computed spike train with a calcium response kernel. (

**C**) Calcium traces (black and gray) after down-sampling at 10 ms bin size and noise addition. At the bottom, spike trains as extracted from the down-sampled calcium traces. The red curve shows the average calcium response of the network. (

**D**) Raster plot showing down-sampled calcium traces (top) and spike events (bottom) over the entire network in 1 min, as detected from the down-sampled calcium traces. (

**E**) 1-min down-sampled calcium traces (black and gray) for the same representative cells shown in panels (

**B**,

**C**), as computed from the convolution with the original spike train. (

**F**) TE decomposition scheme on two representative spike trains. Red and blue dots denote combinations of state encoding excitatory and inhibitory information from Y to X, respectively. The parameter d represents the delay considered on the source Y, in terms of bins.

**Figure 2.**Reconstruction of structural connections at varying delays. (

**A**) ROC curves showing the accuracy of excitatory and inhibitory components of TE (red and blue, respectively) in predicting real synaptic connections. ROC analysis is conducted over 10 realizations of the model, based on the same spatial distribution of cells but a different sorting of excitatory and inhibitory neurons. Continuous red and blue lines and corresponding light red and blue areas denote mean and standard deviation, respectively, computed over all the simulations. TE is evaluated on spike trains extracted from the calcium traces down-sampled at 10 ms. TE components are tested separately in predicting excitatory and inhibitory connections by comparing TE matrices to the corresponding structural adjacency matrices encoding solely excitatory or inhibitory links. The local excitatory component of TE decays in performance at increasing delays, while the local inhibitory component increases in performance at increasing delays. (

**B**) Comparison of local excitatory and inhibitory components of TE in a single modeled network. The excitatory component of TE at 0-bin delay is plotted versus the inhibitory component evaluated at different delays. Delays greater than 0 in the inhibitory part give rise to improved segregation and more accurate reconstruction of real spatial connections. Gray dots denote TE indices for uncoupled cells, while red and blue dots denote TE indices for excitatory and inhibitory real connections. Black lines denote the best thresholds for predictions as evaluated in the independent ROC analysis of panel (

**A**), based on the Youden’s J. Left and bottom density plots represent projections of the two-dimensional distributions of TE indices on excitatory and inhibitory components, respectively.

**Figure 3.**Reconstruction of structural connections at a varying ratio ${g}_{\mathrm{E}}:{g}_{\mathrm{I}}$, and at different delays. (

**A**) ROC curves showing the accuracy of excitatory and inhibitory components of TE (red and blue, respectively) in predicting real synaptic connections. ROC analysis is conducted over 10 realizations of the model, based on the same spatial distribution of cells but a different sorting of excitatory and inhibitory neurons. Continuous red and blue lines and corresponding light red and blue areas denote mean and standard deviation, respectively, computed over all the simulations. Dark, medium and light red/blue curves represent ROC at different delays for excitatory/inhibitory connections. TE is evaluated on spike trains extracted from the calcium traces down-sampled at 10 ms. TE components are tested separately in predicting excitatory and inhibitory connections by comparing TE matrices to the corresponding structural adjacency matrices encoding solely excitatory or inhibitory links. Left, central, and right columns show results at a varying balance between excitation and inhibition in terms of synaptic conductance (${g}_{\mathrm{E}}$:${g}_{\mathrm{I}}$): 1:1 (left), 1:2 (central, control case), 1:3 (right). In every case, the local excitatory component of TE decays in performance at increasing delays, while the local inhibitory component increases in performance at increasing delays. (

**B**) Comparison of local excitatory and inhibitory components of TE in a single modeled network. The excitatory component of TE at 0-bin delay is plotted versus the inhibitory component evaluated at 2-bin delay, giving the best segregation and more accurate reconstruction of real spatial connections. Gray dots denote TE indices for uncoupled cells, while red and blue dots denote TE indices for excitatory and inhibitory real connections. Black lines denote the best thresholds for predictions as evaluated in the independent ROC analysis of panel (

**A**), based on the Youden’s J. Left and bottom density plots represent projections of the two-dimensional distributions of TE indices on excitatory and inhibitory components, respectively.

**Figure 4.**Reconstruction of structural connections at a varying down-sampled bin size, and at different delays. (

**A**) ROC curves showing the accuracy of excitatory and inhibitory components of TE (red and blue, respectively) in predicting real synaptic connections. ROC analysis is conducted over 10 realizations of the model, based on the same spatial distribution of cells but a different sorting of excitatory and inhibitory neurons. Continuous red and blue lines and corresponding light red and blue areas denote mean and standard deviation, respectively, computed over all the simulations. Dark, medium and light red/blue curves represent ROC at different delays for excitatory/inhibitory connections. Networks dynamic is simulated at a ratio ${g}_{\mathrm{E}}$:${g}_{\mathrm{I}}$ equal to 1:2. TE components are tested separately in predicting excitatory and inhibitory connections by comparing TE matrices to the corresponding structural adjacency matrices encoding solely excitatory or inhibitory links. Left, central, and right columns show results at a varying down-sampled bin size: 5 ms (left), 10 ms (central, control case), 20 ms (right). For the majority of cases, the local excitatory component of TE decays in performance at increasing delays, while the local inhibitory component increases in performance at increasing delays. At 20 ms bin size, 1-bin delay results in higher accuracy in predicting inhibitory links. (

**B**) Comparison of local excitatory and inhibitory components of TE in a single modeled network. The excitatory component of TE at 0-bin delay is plotted versus the inhibitory component giving rise to the best segregation of real connections. Gray dots denote TE indices for uncoupled cells, while red and blue dots denote TE indices for excitatory and inhibitory real connections. Black lines denote the best thresholds for predictions as evaluated in the independent ROC analysis of panel (

**A**), based on the Youden’s J. Left and bottom density plots represent projections of the two-dimensional distributions of TE indices on excitatory and inhibitory components, respectively.

**Table 1.**Effect of noise in the identification of synaptic connections. Average AUC, Youden’s J, and corresponding sensitivity and specificity over 10 neuronal networks (see Section 3). Reconstruction starting from signals down-sampled at 10 ms bin size. Balance between excitation and inhibition ${g}_{\mathrm{E}}/{g}_{\mathrm{I}}$ is set to 1:2. Both in deterministic and noisy data, ROC analysis shows equivalent performance in links prediction.

Deterministic [bin 10 ms, ${\mathit{g}}_{\mathbf{E}}/{\mathit{g}}_{\mathbf{I}}$ 1:2] | Noise [bin 10 ms, ${\mathit{g}}_{\mathbf{E}}/{\mathit{g}}_{\mathbf{I}}$ 1:2] | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

E | I | E | I | |||||||||

$\mathit{d}$ | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 |

AUC | 0.86 | 0.84 | 0.76 | 0.58 | 0.79 | 0.90 | 0.86 | 0.84 | 0.76 | 0.58 | 0.76 | 0.89 |

J | 0.57 | 0.54 | 0.41 | 0.21 | 0.55 | 0.68 | 0.57 | 0.54 | 0.41 | 0.19 | 0.46 | 0.65 |

Sens | 0.83 | 0.76 | 0.63 | 0.30 | 0.65 | 0.81 | 0.83 | 0.79 | 0.68 | 0.29 | 0.57 | 0.80 |

Spec | 0.75 | 0.79 | 0.78 | 0.91 | 0.90 | 0.87 | 0.74 | 0.76 | 0.73 | 0.90 | 0.89 | 0.84 |

**Table 2.**Effect of excitation inhibition balance in the identification of synaptic connections. Average AUC, Youden’s J, and corresponding sensitivity and specificity over 10 neuronal networks. Reconstruction starting from signals down-sampled at $10\phantom{\rule{0.166667em}{0ex}}$ms bin size. Balance between excitation and inhibition ${g}_{\mathrm{E}}/{g}_{\mathrm{I}}$ is set to 1:1, and 1:3. Unbalance toward excitation leads to worse results in the identification of inhibitory links, however, still preserving good performance at 2-bin delays. Unbalance toward inhibition improve inhibitory links identification keeping unaltered identification accuracy for excitatory connections.

Deterministic [bin 10 ms, ${\mathit{g}}_{\mathbf{E}}/{\mathit{g}}_{\mathbf{I}}$ 1:1] | Deterministic (bin 10 ms, ${\mathit{g}}_{\mathbf{E}}/{\mathit{g}}_{\mathbf{I}}$ 1:3] | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

E | I | E | I | |||||||||

$\mathit{d}$ | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 |

AUC | 0.84 | 0.84 | 0.76 | 0.54 | 0.71 | 0.83 | 0.86 | 0.84 | 0.75 | 0.60 | 0.82 | 0.92 |

J | 0.55 | 0.54 | 0.42 | 0.12 | 0.39 | 0.56 | 0.58 | 0.54 | 0.39 | 0.26 | 0.60 | 0.71 |

Sens | 0.82 | 0.79 | 0.67 | 0.32 | 0.53 | 0.72 | 0.83 | 0.75 | 0.64 | 0.33 | 0.69 | 0.85 |

Spec | 0.73 | 0.75 | 0.75 | 0.80 | 0.86 | 0.84 | 0.75 | 0.79 | 0.75 | 0.93 | 0.91 | 0.86 |

**Table 3.**Effect of varying bin size in the identification of synaptic connections. Average AUC, Youden’s J, and corresponding sensitivity and specificity over 10 neuronal networks. Reconstruction starting from signals down-sampled at 5 and $20\phantom{\rule{0.166667em}{0ex}}$ms bin size. Down-sampling at 5 ms has minimal effect on excitatory links prediction accuracy, slightly reducing performance in the detection of inhibitory connections. Down-sampling at 20 ms significantly deteriorates excitatory links prediction at increasing bin delays and highlights a loss performance for the identification of inhibitory synapses at 2-bin delays.

Deterministic [bin 5 ms, ${\mathit{g}}_{\mathbf{E}}/{\mathit{g}}_{\mathbf{I}}$ 1:2] | Deterministic (bin 20 ms, ${\mathit{g}}_{\mathbf{E}}/{\mathit{g}}_{\mathbf{I}}$ 1:2) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

E | I | E | I | |||||||||

$\mathit{d}$ | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 |

AUC | 0.84 | 0.83 | 0.81 | 0.59 | 0.71 | 0.81 | 0.87 | 0.81 | 0.58 | 0.54 | 0.84 | 0.78 |

J | 0.54 | 0.54 | 0.50 | 0.27 | 0.46 | 0.59 | 0.61 | 0.48 | 0.13 | 0.10 | 0.57 | 0.42 |

Sens | 0.82 | 0.79 | 0.74 | 0.34 | 0.56 | 0.72 | 0.85 | 0.74 | 0.51 | 0.31 | 0.71 | 0.69 |

Spec | 0.72 | 0.75 | 0.77 | 0.93 | 0.90 | 0.88 | 0.76 | 0.75 | 0.62 | 0.79 | 0.86 | 0.73 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ghirga, S.; Chiodo, L.; Marrocchio, R.; Orlandi, J.G.; Loppini, A.
Inferring Excitatory and Inhibitory Connections in Neuronal Networks. *Entropy* **2021**, *23*, 1185.
https://doi.org/10.3390/e23091185

**AMA Style**

Ghirga S, Chiodo L, Marrocchio R, Orlandi JG, Loppini A.
Inferring Excitatory and Inhibitory Connections in Neuronal Networks. *Entropy*. 2021; 23(9):1185.
https://doi.org/10.3390/e23091185

**Chicago/Turabian Style**

Ghirga, Silvia, Letizia Chiodo, Riccardo Marrocchio, Javier G. Orlandi, and Alessandro Loppini.
2021. "Inferring Excitatory and Inhibitory Connections in Neuronal Networks" *Entropy* 23, no. 9: 1185.
https://doi.org/10.3390/e23091185