Classification of 3-node Restricted Excitatory-Inhibitory Networks

The 3-node network motif (f) of Figure  1 is an example of an REI network. The black shaded nodes are of one type (say, inhibitory) and the third node is of different type (say, excitatory). Both inhibitory nodes send two inhibitory outputs, which in this network, are directed to the two inhibitory nodes; the excitatory node sends an excitatory signal to one of the inhibitory nodes. In the coupled cell network formalism the main features we retain from this particular network are that it has 3 nodes, two of them are of one type and the third one is of different type. The equal type nodes output arrows of the same type. Different node types output different arrow types. See Figure 2 left. A general admissible system of ODEs consistent with this network has the form

where $f,g,h$ are smooth functions. Each such function captures how the evolution of each node depends on the other nodes. The overbar notation over two variables in the functions $f$ and $g$ denotes their invariance under permutation of the two variables, which occurs because the corresponding input arrows have the same type. Assuming nodes $1,2$ have internal phase space $\mbox{$\mathbb{R}$}^{k}$ and node $3$ has internal phase space $\mbox{$\mathbb{R}$}^{l}$, then $f:\,(\mbox{$\mathbb{R}$}^{k})^{3}\to\mbox{$\mathbb{R}$}^{k}$, $g:\,(\mbox{$\mathbb{R}$}^{k})^{3}\times\mbox{$\mathbb{R}$}^{l}\to\mbox{$\mathbb{R}$}^{k}$ and $h:\,\mbox{$\mathbb{R}$}^{l}\to\mbox{$\mathbb{R}$}^{l}$.

Interpreting the network as a 3-gene Escherichia coli GRN, we may assume that the variable $x_{i}=(x_{i}^{R},x_{i}^{P})\in\mbox{$\mathbb{R}$}^{2}$ is associated with gene $i$, for $i=1,2,3$. Here, $x_{i}^{R}$ is the concentration of mRNA in gene $i$ and $x_{i}^{P}$ is the concentration of protein in gene $i$. We also assume that the time evolution of the cellular concentration of proteins and mRNA molecules is determined by an ODE.(We use this term for a single ODE and for a system.) Moreover, there must be the constraint that a concentration cannot be negative. In this modeling approach, two components of the ODE are associated with each gene $i$. The equation for $x_{i}^{R}$ determines the rate of change of the concentration of the transcribed mRNA; the equation for $x_{i}^{P}$ describes the rate of change of the concentration of its corresponding translated protein. As in [14], a simple example of an admissible system of the form (1.1), where all 3 genes have 2-dimensional node spaces (that is, $k=l=2$), arises by choosing the following functions $f,g,h$:

Here, as genes $1,2$ are of the same type, we take $\beta_{1}=\beta_{2}$, $\alpha_{1}=\alpha_{2}$ and $\delta_{1}=\delta_{2}$. Also, $\delta_{i},\alpha_{i}$ represent, respectively, degradation of mRNA and protein for gene $i$, and are assumed to be independent of the concentrations of the other molecules in the cell. The function $H_{i}^{-}(x_{j}^{P})$ (resp. $H_{i}^{+}(x_{j}^{P})$) in the equation for gene $i$ describes how protein $j$ inhibits (resp. activates) mRNA $i$. In this model equation we assume that the effects of the proteins are additive; an alternative typical modeling assumption is that they are multiplicative. See for example [19]. These functions $H_{i}^{-}$ and $H_{i}^{+}$ are generally nonlinear. Typical choices for $H_{i}^{-}$ are the Hill functions:

where $n_{i}$ is a positive integer. Assuming $z\geq 0$, since it represents a concentration, $H_{i}^{-}(z)$ converges to $0$ as $z$ converges to $+\infty$ and $H_{i}^{-}(0)=1$. This property encodes inhibition into the equations. Assuming the inhibitory edges to be of the same type corresponds to taking $H_{1}^{-}=H_{2}^{-}=H_{3}^{-}$. A choice for excitation is the function

where $n^{*}_{i}$ is not necessarily equal to $n_{i}$. With the functions $f,g,h$ as in (1.2), and taking into account the structure of network on the left of Figure 2 (or the 3-gene GRN motif in Figure 1 (f)), equations (1.1) take the form:

Thus, for $i=1,2$, the rate of change of the concentration of the transcribed mRNA $i$, given by $x_{i}^{R}$, is the difference between the ‘synthesis term’ ($H_{1}^{-}(x_{1}^{P})+H_{1}^{-}(x_{2}^{P})$ for $i=1$ and $H_{1}^{-}(x_{1}^{P})+H_{1}^{-}(x_{2}^{P})+H_{2}^{+}(x_{3}^{P})$ for $i=2$), and the ‘degradation term’ $\delta_{1}x_{i}^{R}$. In fact, we can think that the evolution of gene $i$ given by $x_{i}=(x_{i}^{R},x_{i}^{P})$ is a sum of two parts: one determines the internal dynamics of the gene $i$ and the other determines the coupling effect. For $i=1$, we can consider the internal dynamics to be determined by

and the coupling part by

Alternatively, we can consider the internal dynamics to be determined by

and the coupling part by

This can be interpreted as considering different gene internal dynamics of gene $1$. Similarly, we have two analogous options for the internal dynamics of gene $2$. Taking the second option for the internal dynamics of genes $1$ and $2$, we may rewrite (1.2) as

In the coupled cell network formalism, the vector field (1.4) determines an admissible coupled cell system for the network on the right of Figure 2, which has the general form

where

In the coupled cell network formalism, we say that the two networks of Figure 2 are ODE-equivalent, precisely because every admissible ODE for the network on the right of Figure 2 can be seen as an admissible ODE for the network on the left of Figure 2, and conversely, assuming the node phase spaces of the two networks are the same. Moreover, the network on the right of Figure 2 is the minimal network in terms of number of edges among all 3-node networks that are ODE-equivalent to the networks in Figure 2. See Subsection 2.2 for formal definitions and main results on network admissible ODEs, ODE-equivalence and minimality. In this paper, we use results on network ODE-equivalence and minimality to classify the set of 3-node REI networks into ODE-classes and present minimal representatives for each ODE-class.

Our classification of 3-node REI networks is made under a variety of extra conditions, summarized in Table 1. This classification, together with that for connected 2-node EI networks in [3], are a preparatory step towards a systematic analysis of dynamics and bifurcations in EI networks.

We characterize and classify connected 3-node REI networks. We give a classification under the relation of ODE-equivalence, where two networks are ODE-equivalent if they have the same space of admissible ODEs. Sometimes we consider a restriction on the valence of the nodes. To organize and summarize these results, Table 1 lists the main classifications obtained in this paper, with columns for type of network, bounds on the valence, number of networks in the classification, plus references to associated Figures, Tables and Theorems.

Section 2 discusses REI networks from the point of view of the general network formalism of [9, 10, 20]. Subsection 2.1 gives a formal definition of ‘restricted excitatory-inhibitory’ (REI) networks. Subsection 2.2 defines the class of admissible ODEs associated with an REI network. Adjacency matrices are also discussed.

Section 3 characterizes connected 3-node REI networks and classifies them up to ODE-equivalence. Corresponding admissible ODEs are not listed, for reasons of space, but can be deduced algorithmically from the network diagrams. Subsection 3.2 classifies the connected $3$-node REI networks with valence $\leq 2$ and also classifies their ODE-classes. Subsection 3.3 classifies connected $3$-node REI networks with valence $2$ under four different conditions: (i) every node receives one arrow of each type; (ii) only the two excitatory nodes receive one arrow of each type; (iii) only the inhibitory node and one excitatory node receive one arrow of each type; (iv) given any two nodes there is no arrow-type preserving bijection between their input sets.

We define restricted excitatory-inhibitory (REI) networks, state our conventions for representing them in diagrams, and give examples.

We adopt the general form of admissible ODEs for a network as defined in [9, 10, 20] with the assumption in this paper that all nodes have the same state space, say $P=\mbox{$\mathbb{R}$}^{m}$ for some $m>0$. Given an EI network with a finite set of nodes, node $i$ is represented in the ODE system by the variable $x_{i}$ which is governed by a system of ordinary differential equations. The word ‘admissible’ is used in the sense that the ODE system encodes information about the node and arrow types. Specifically, when two input equivalent nodes have the same numbers, say $n_{e}$, of excitatory arrows and $n_{i}$ of inhibitory arrows, targeting the two nodes, we specify their dynamics by the same smooth function, say $f:P^{k+1}\to P$, evaluated at the node and at the corresponding tail nodes of the arrows targeting the node. We follow [3, Definition 2.8]:

An $n$-node network can be represented by its adjacency matrix, which is the $n\times n$ matrix $A=(a_{ij})$ such that $a_{ij}$ is the number of arrows from node $j$ to node $i$. (In the graph-theoretic literature the opposite convention is often used, which gives the transpose of the adjacency matrix defined here.) For an REI network, conditions (c)-(d) of Definition 2.1 allow us to deduce the arrow-types from its adjacency matrix, provided we know the node-types of nodes $i$ and $j$. In fact, we consider two node-type $n\times n$ matrices, which are both diagonal: given one node-type matrix, the diagonal entry $ii$ is 1 if node $i$ is of that type and zero otherwise. When we need to distinguish the different arrow-types, as is the case in this paper when classifying networks using ODE-equivalence, see Subsection 2.3, we consider arrow-type adjacency matrices, one for each arrow type. For example, for REI-networks, we will consider two arrow-type adjacency matrices, one for excitatory arrows and the other for inhibitory arrows.

As mentioned and exemplified in the Introduction, different networks with the same number of nodes are said to be ODE-equivalent if they have the same set of admissible ODEs, for any choice of node state spaces, when their nodes are identified by a suitable bijection that preserves node state spaces. See [5, 9, 10].

Consider the 3-node REI network on the left of Figure 4. The two excitatory nodes $1,2$ are input equivalent as both receive only one inhibitory arrow. A general admissible ODE-system associated with this network has the form

where $f_{1},f_{3}:\,\mbox{$\mathbb{R}$}^{l}\times\mbox{$\mathbb{R}$}^{l}\to\mbox{$\mathbb{R}$}^{l}$ are smooth functions. We see that any solution $(x_{1}^{+}(t),x_{2}^{+}(t),x_{3}^{-}(t))$ of (2.7) with initial condition satisfying say $x_{1}^{+}(0)=x_{2}^{+}(0)$ has nodes $1,2$ synchronized for all time, that is,

This property does not depend on the choices of the functions $f_{1},f_{3}$ neither the internal node phase spaces $\mbox{$\mathbb{R}$}^{l}$. It is determined only by the structure of the network on the left of Figure 4; concretely, the two nodes $1,2$ are of the same node type and each receives one inhibitory arrow from the inhibitory node $3$, which in this example is the unique inhibitory node. Equivalently, we see that the vector field $F(x_{1};x_{2};x_{3})=(f_{1}(x_{1};x_{3}),f_{1}(x_{2};x_{3});f_{3}(x_{3};x_{1}))$ leaves invariant the space $\Delta=\{(x_{1},x_{1},x_{3})\}$, that is,

In this case, we say that $\Delta$ is a robust network synchrony space. Restricting (2.7) to $\Delta$, we obtain the system

which is admissible for the 2-node network on the right of Figure 4 which is also an REI network. In the terminology of [20], the 2-node network on the right of Figure 4 is the quotient of the network on the left of Figure 4 by the equivalence relation on the node set of the 3-node network with equivalence classes $\{1,2\}$ and $\{3\}$. This relation is said to be balanced, which is equivalent to the invariance of $\Delta$ under the node and arrow adjacency matrices.

These ideas generalize to $n$-node networks and it is proved in [20] that the admissible vector fields for a network leave invariant a linear subspace defined in terms of equalities of certain node coordinates if and only if the equivalence relation on the network node set with classes given by the clusters of nodes whose coordinates are identified is balanced. See [20, Definition 6.4] for the definition of network balanced relation, [9, Proposition 10.20 ] or [10, Section 5], and [20, Theorem 6.5] for the definition of quotient network by a balanced equivalence relation.

For REI networks, it is trivial to show that the restriction of any admissible ODE for an REI network to a robust synchrony subspace is admissible for a smaller network, which is also an REI network. That is, the quotient of an REI network by a balanced equivalence relation on the network node set is also an REI network.

In this section we characterize the connected $3$-node REI networks, without imposing any restrictions, and classify them up to ODE-equivalence.

Up to duality any $3$-node REI network is as shown in Figure 5, for a suitable choice of nonnegative integer arrow multiplicities $\alpha$, $\delta$, $\tau$, $\beta_{i}$, $\gamma_{j}$, where $i=1,2,3,4$, $j=1,2$.

The adjacency matrices are

In this section we classify the connected $3$-node REI networks with valence $\leq 2$. We start by classifying them up to ODE-equivalence.