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P169: Entrainment of competitive threshold-linear networks
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** ****https://meet.google.com/jyn-vuez-qhx**

**Andrea Bel**, **Horacio Rotstein**, **Walter Reartes**

Neuronal oscillations are ubiquitous in the brain and emerge from the combined activity of the participating neurons (or nodes), the connectivity and the network topology. Recent neurotechnological advances have made it possible to interrogate neuronal circuits by perturbing one or more of its nodes. The response to periodic inputs has been used as a tool to identify the oscillatory properties of circuits and the flow of information in networks. However, a general theory that explains the underlying mechanisms and allows to make predictions is lacking beyond the single neuron level. Threshold-linear network (TLN) models describe the activity of connected nodes where the contribution of the connectivity terms is linear above some threshold value (typically zero), while the network is disconnected below it. In their simplest description, the dynamics of the individual nodes are one- dimensional and linear. When the nodes in the network are neurons or neuronal populations, their activity can be interpreted as the firing rate, and therefore the TLNs represent firing rate models [4]. Competitive threshold-linear networks (CTLNs) are a class of TLNs where the connectivity weights are all negative and there are no self-connections [3,6]. Inhibitory networks arise in many neuronal systems and have been shown to underlie the generation of rhythmic activity in cognition and motor behavior [1,2]. Despite their simplicity, TLNs and CTLNs produce complex behavior including multistability, periodic, quasi-periodic and chaotic solutions [3,5,6]. In this work, we consider CTLNs with three or more nodes and cyclic symmetry in which oscillatory solutions are observed. We first assume that an external oscillatory input is added to one of the nodes and, by defining a Poincaré map, we numerically study the response properties of the CTLN networks. We determine the ranges of input amplitude and frequency in which the CTLN is able to follow the input (1:1 entrainment). For this we define local and global entrainment measures that convey different information. We then study how the entrainment properties of the CTLNs is affected by changes in (i) the time scale of each node, (ii) the number of nodes in the network, and (iii) the strength of the inhibitory connections. Finally, we extend our results to include other entrainment scenarios (e.g., 2:1) and other network topologies.

Acknowledgments: This work was supported by the Universidad Nacional del Sur grant PGI 24/L113-2019 (AB, WR) and the National Science Foundation grant DMS-1608077 (HGR).

**References**:

1. M. Arriaga and E. B. Han, Dedicated hippocampal inhibitory networks for locomotion and immobility, Journal of Neuroscience, 37 (2017), pp. 9222–9238.

2. D. A. Burke, H. G. Rotstein, and V. A. Alvarez, Striatal local circuitry: a new framework for lateral inhibition, Neuron, 96 (2017), pp. 267–284.

3. C. Curto and K. Morrison. Pattern completion in symmetric threshold-linear networks. Neural Comput.,28(12):2825–2852, 2016.

4. P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, The MIT Press, 2001.

5. R. L. T. Hahnloser, On the piecewise analysis of networks of linear threshold neurons, Neural Networks, 11 (1998), pp. 691–697.

6. K. Morrison, A. Degeratu, V. Itskov, and C. Curto, Diversity of emergent dynamics in competitive threshold-linear networks: a preliminary report, arXiv, (2016), p. 12 pp.

**Speakers**

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Neuronal oscillations are ubiquitous in the brain and emerge from the combined activity of the participating neurons (or nodes), the connectivity and the network topology. Recent neurotechnological advances have made it possible to interrogate neuronal circuits by perturbing one or more of its nodes. The response to periodic inputs has been used as a tool to identify the oscillatory properties of circuits and the flow of information in networks. However, a general theory that explains the underlying mechanisms and allows to make predictions is lacking beyond the single neuron level. Threshold-linear network (TLN) models describe the activity of connected nodes where the contribution of the connectivity terms is linear above some threshold value (typically zero), while the network is disconnected below it. In their simplest description, the dynamics of the individual nodes are one- dimensional and linear. When the nodes in the network are neurons or neuronal populations, their activity can be interpreted as the firing rate, and therefore the TLNs represent firing rate models [4]. Competitive threshold-linear networks (CTLNs) are a class of TLNs where the connectivity weights are all negative and there are no self-connections [3,6]. Inhibitory networks arise in many neuronal systems and have been shown to underlie the generation of rhythmic activity in cognition and motor behavior [1,2]. Despite their simplicity, TLNs and CTLNs produce complex behavior including multistability, periodic, quasi-periodic and chaotic solutions [3,5,6]. In this work, we consider CTLNs with three or more nodes and cyclic symmetry in which oscillatory solutions are observed. We first assume that an external oscillatory input is added to one of the nodes and, by defining a Poincaré map, we numerically study the response properties of the CTLN networks. We determine the ranges of input amplitude and frequency in which the CTLN is able to follow the input (1:1 entrainment). For this we define local and global entrainment measures that convey different information. We then study how the entrainment properties of the CTLNs is affected by changes in (i) the time scale of each node, (ii) the number of nodes in the network, and (iii) the strength of the inhibitory connections. Finally, we extend our results to include other entrainment scenarios (e.g., 2:1) and other network topologies.

Acknowledgments: This work was supported by the Universidad Nacional del Sur grant PGI 24/L113-2019 (AB, WR) and the National Science Foundation grant DMS-1608077 (HGR).

1. M. Arriaga and E. B. Han, Dedicated hippocampal inhibitory networks for locomotion and immobility, Journal of Neuroscience, 37 (2017), pp. 9222–9238.

2. D. A. Burke, H. G. Rotstein, and V. A. Alvarez, Striatal local circuitry: a new framework for lateral inhibition, Neuron, 96 (2017), pp. 267–284.

3. C. Curto and K. Morrison. Pattern completion in symmetric threshold-linear networks. Neural Comput.,28(12):2825–2852, 2016.

4. P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, The MIT Press, 2001.

5. R. L. T. Hahnloser, On the piecewise analysis of networks of linear threshold neurons, Neural Networks, 11 (1998), pp. 691–697.

6. K. Morrison, A. Degeratu, V. Itskov, and C. Curto, Diversity of emergent dynamics in competitive threshold-linear networks: a preliminary report, arXiv, (2016), p. 12 pp.

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Monday July 20, 2020 8:00pm - 9:00pm CEST

Slot 04

Slot 04