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Classification of 2-node Excitatory-Inhibitory Networks

Manuela Aguiar, Ana Dias, Ian Stewart

TL;DR

These classifications constitute a first step towards analysing dynamics and bifurcations of excitatory-inhibitory networks and have potential applications to biological network models, especially neuronal networks, gene regulatory networks, and synthetic gene networks.

Abstract

We classify connected 2-node excitatory-inhibitory networks under various conditions. We assume that, as well as for connections, there are two distinct node-types, excitatory and inhibitory. In our classification we consider four different types of excitatory-inhibitory networks: restricted, partially restricted, unrestricted and completely unrestricted. For each type we give two different classifications. Using results on ODE-equivalence and minimality, we classify the ODE-classes and present a minimal representative for each ODE-class. We also classify all the networks with valence $\le 2$. These classifications are up to renumbering of nodes and the interchange of `excitatory' and `inhibitory' on nodes and arrows.These classifications constitute a first step towards analysing dynamics and bifurcations of excitatory-inhibitory networks. The results have potential applications to biological network models, especially neuronal networks, gene regulatory networks, and synthetic gene networks.

Classification of 2-node Excitatory-Inhibitory Networks

TL;DR

These classifications constitute a first step towards analysing dynamics and bifurcations of excitatory-inhibitory networks and have potential applications to biological network models, especially neuronal networks, gene regulatory networks, and synthetic gene networks.

Abstract

We classify connected 2-node excitatory-inhibitory networks under various conditions. We assume that, as well as for connections, there are two distinct node-types, excitatory and inhibitory. In our classification we consider four different types of excitatory-inhibitory networks: restricted, partially restricted, unrestricted and completely unrestricted. For each type we give two different classifications. Using results on ODE-equivalence and minimality, we classify the ODE-classes and present a minimal representative for each ODE-class. We also classify all the networks with valence . These classifications are up to renumbering of nodes and the interchange of `excitatory' and `inhibitory' on nodes and arrows.These classifications constitute a first step towards analysing dynamics and bifurcations of excitatory-inhibitory networks. The results have potential applications to biological network models, especially neuronal networks, gene regulatory networks, and synthetic gene networks.
Paper Structure (25 sections, 15 theorems, 37 equations, 15 figures, 9 tables)

This paper contains 25 sections, 15 theorems, 37 equations, 15 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Motivation from Previous Work
  3. Motifs in Escherichia coli GRN
  4. Classification
  5. Remarks on Excitation and Inhibition
  6. Summary of Paper and Main Results
  7. Classes of Excitatory-Inhibitory Networks
  8. Formal Definitions
  9. Conventions
  10. Admissible ODEs
  11. Balanced Colourings and Quotient Networks
  12. Classification of Connected 2-node Excitatory-Inhibitory Networks
  13. Connected 2-node REI Networks
  14. Connected 2-node REI Networks: ODE-classes
  15. Connected 2-node REI Networks with valence up to 2
  16. ...and 10 more sections

Key Result

Proposition 2.17

For any EI network $\mathcal{G}$ having a balanced equivalence relation $\bowtie$ with $k$ classes, the quotient of $\mathcal{G}$ by $\bowtie$ is a $k$-node EI network. Moreover, the quotient of $\mathcal{G}/{\bowtie}$ is an REI (resp. UEI, CEI) network if and only if the network $G$ is REI (resp. U

Figures (15)

  • Figure 1: Eight 3-node motifs realised in E. coli.
  • Figure 2: Smolen oscillator.
  • Figure 3: The minimal $2$-node network ODE-equivalent to the Smolen oscillator in Figure \ref{['fig:Smolen']}.
  • Figure 4: Two $2$-node ODE-equivalent REI networks. Node $1$ is excitatory, node $2$ is inhibitory; the nonnegative integer arrow multiplicities are $\alpha, \beta, \gamma, \delta$. The network is connected when one of $\beta$ or $\gamma$ is nonzero.
  • Figure 5: The two ODE-classes of connected $2$-node REI networks. Table \ref{['table:ODE_general_REIV']} states the corresponding admissible ODEs. The network NH2 is the minimal network ODE-equivalent to the Smolen network of Figure \ref{['fig:Smolen']}.
  • ...and 10 more figures

Theorems & Definitions (42)

  • Definition 1.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.5
  • Definition 2.7
  • Example 2.9
  • Remark 2.10
  • Example 2.11
  • Remark 2.12
  • Example 2.13
  • ...and 32 more