Mathematical representation of the structure of neuron-glia networks

Traditionally, it was commonly accepted that only neurons are involved in the brain information processing while glial cells are relegated to support and protect them. A more updated view suggests that glial cells are also involved in the information processing. In fact, one type of glia cell - astrocytes - couples to single synapses between neurons, forming tripartite synapses [15]. Astrocytes have an active role at tripartite synapses. On one hand, they can directly readout signals from the presynaptic neuron (e.g. they express receptors for GABA, dopamine, glutamate, and glycine)[18, 36]. On the other hand, they can modulate the activity of the postsynaptic neuron via the release of neuroactive substances called gliotransmitters [15, 29].

Neuron-driven activation of astrocytes induces an increase in cytosolic calcium [18, 37]. This increase in calcium, in turn, triggers the release of gliotransmitters [8]. More importantly, local increases in calcium spreads through the astrocyte syncytium, i.e. astrocyte network, mainly via IP3 diffusion [11, 30]. Interestingly, recent studies suggest that the astrocyte network plays a role in neural network phenomena such as -burst synchronization, bursting behavior, barrage firing, and gamma waves- and in cognitive capabilities (e.g. recognition memory) [6, 10, 21]. Therefore, studying neuron-glia networks could be critical to understand the brain dynamics of large populations of neurons.

Although neuron-glia networks have been modelled before, finding that astrocytes modulate dynamic coordination, facilitate ultra-slow oscillation, and increase the occurrence of bursting-like spikes, they lack a formal network representation [5, 34, 39]. Neural networks are typically represented using graphs or digraphs [2, 26, 32]. These representations allow the study of features such as connectivity, modularity, and centrality. [2]. So far, graph-like structures has guided the study of connectomes at all scales [2, 20, 33]. However, neuron-glia networks have tripartite synapses that relate three cells at once, thus they can not be represented by edges of a graph nor a digraph.

In this paper, we addressed this problem by defining a representations of neuron-glia networks. In order to achieve this goal, first, we defined a synaptic configuration as a set of ordered pairs that represent the information flux between cells in a synapse. Then, we represented a collection of synapses as a $\Upsilon$-family that relates synaptic configurations to indexes (single synapses), in order to distinguish synapses that share a synaptic configuration. To simplify this representation, we defined $\Upsilon$-multisets that contain identical copies of repeated synaptic configuration. A set of neuron-like cells (neurons and astrocytes) together with an $\Upsilon$-multiset form an $\Upsilon$-graph which represent a neuron-glia network. This representation allowed us to adapt the concept of multidigraph isomorphism to neuron-glia networks. Then, we defined the $\phi$ function that maps $\Upsilon$-graphs with their unnested forms (multidigraphs) which contain only ordered pairs. Using this function, we found network motifs (propositions 2-5) where the isomorphism between unnested forms does not guarantee the isomorphism between $\Upsilon$-graphs, meaning misrepresentation by the multidigraphs. However, we also found a mathematical condition (proposition 6) that allows the unnested forms isomorphism to guarantee the $\Upsilon$-graphs isomorphisms. Finally, we introduced a novel approach to model neuron-glia network shape, using $\Upsilon$-families, natural structure theory, and topology theory [3, 4]. Future research could use our definition of $\Upsilon$-graph isomorphism as part of an algorithm that find neuron-glia network motifs in a connectome that includes astrocytes such as the connectome reconstructed by Sizis et al., 2021 [42] (see [23] for examples of motif analysis of a connectome). Another research avenue would be extending graph measures (e.g. centrality, modularity, and distance) to $\Upsilon$-graphs, in order to analyze connectomics data (see [23] for examples of use of network metrics to study a connectome) or to study how these measures correlates with different predicted networks dynamics in dynamical models.

We follow the definitions and notations of [13] and [14] about multisets (msets) and related objects. We also follow the concepts of [1] about digraphs and related objects. In the following formal definitions, we summarize the concepts about multisets and graph theory that are necessary to understand our theoretical framework.

The following definition formalizes the concept of directed pseudograph [1] to create our multidigraph definition, using multiset theory [14, 13].

Note that if all edges appear once, $A(Q)$ is equivalent to a set, and the multidigraph $Q$ is equivalent to a digraph (loop allowed). Note that $A(Q)^{*}$ is a binary relation on $V(Q)$. We also formalize the multidigraphs homomorphism in the context of multiset theory as follows.

If $Q$ is homomorphic to $P$, and $P$ is homomorphic to $Q$, $Q$ and $P$ are isomorphic ($Q\cong P$).

Like neurons, astrocytes receive and release neurotransmitters, so we will refer to both as neuron-like cells. In the following subsection, $N$ and $A$ will represent a set of neuron and a set of astrocytes, respectively. Both $N$ and $A$ are sets of neuron-like cells (NL-sets). Additionally, we can define a NL-set $X$ as the union of $N$ and $A$ ($X=N\cup A$). Since astrocyte and neurons are different cellular types [15], a set of neurons and a set of astrocytes are always disjoint sets (i.e., $N\cap A=\varnothing$).

In this section, we will prove that the existence of isomorphism between the unnested forms of $\Upsilon$-graphs does not imply that the networks they represent share the same pattern. To do so, we found pairs of non-isomorphic $\Upsilon$-graphs with isomorphic unnested forms. In the following, we will show four scenarios where isomorphism of the unnested form does not implies isomorphism of the $\Upsilon$-graphs. Keep in mind that $\mathbf{P}$ is a non-empty set of $\Upsilon$-graphs. At the end of the section (proposition 6), we will prove, in the general case, that the isomorphism between the unnested forms of $\Upsilon$-graphs implies the isomorphism of their $\Upsilon$-graphs under some specific conditions.

1. 1.

   All types of synaptic configuration can be present (proposition 2).

2. 2.

   There are only tripartite configurations (proposition 3).

3. 3.

   There are only tripartite configurations and one-one-directed configurations (proposition 4).

4. 4.

   There are only tripartite configurations, and symmetric configurations (gap junctions) (proposition 5).

To prove propositions 2-5, we use theoretical examples where the isomorphism between the unnested forms does not imply the isomorphism between $\Upsilon$-graphs. For example, to prove proposition 2, we present two small network motifs (represented by matrices $A_{5\times 2}$ and $B_{5\times 3}$) formed by two neurons and one astrocyte linked by one tripartite synapse (note that matrix $A$ and $B$ have 1 as entry of first three rows of the first column, representing a tripartite synapse) together with two one-one-directed synapses (two columns with 1 at one position in $A$) or one gap junction(1 column with 1 at two positions in $B$). We prove that the unnested form of of these different motifs is isomorphic, meaning that it is possible to find non-isomorphic neuron-glia networks with isomorphic unnested forms. We provided a formal explanation in the in the following proof. In this proof, we establish basic mathematical notation for the following proofs because it is easier to explain them in a simple example rather than in more complicated examples or in the general case (proposition 6). Keep in mind that all examples showed bellow are theoretical examples and when we say “there are”, it means “it is theoretical possible” since the examples do not come from a database.

The previous instance ($C_{1}$) reflects the lack of “resolution power” of models that only use ordered pairs as elements such as digraphs or multidigraphs. Indeed, at a physiological, the neuron-glia networks represented by $a$ and $b$ will not show the same behaviour since two one-one-directed synapses are not equivalent to a gap junction. However, due to the relative difficulty to find gap junctions in networks due to technical limitations, we present the following instance of $C$ that does not posses them. To prove proposition 3, we find that two solutions for an $\Upsilon$-gm equation (defined in 5), proving that there can be non-isomorphic $\Upsilon$-graphs with isomorphic unnested forms even when gap junctions are not allowed. We follow the same approach to prove proposition 3 - 5.

In propositions 2-5, we realized that for a pair of representations of a neuron-glia network $a$ and $\phi(a)$, where $a$ is a $\Upsilon$-graph and $\phi(a)$, its unnested form, it is safer to work with $a$ rather than $\phi(a)$ since it is theoretically possible to lose information during the unnesting of the $\Upsilon$-graph. It is important to mention that some of the showed examples may not exist in real networks but they prove that it is not safe to assume that the isomorphism between the unnested forms ($\phi(a)\cong_{\alpha}\phi(b)$) implies the isomorphism between $\Upsilon$-graphs ($a\cong b$). However, we conjectured that

holds under some specific conditions. The conditions in which 7 is true are defined in the following paragraph and, in proposition 6, we prove that these conditions are sufficient for 7 to be true.

Nowadays, a huge amount of information about neural shape and network connectivity is available in open databases [7, 38, 41]. However, the mathematical tools to study neuronal shape are still under development [19, 22]. Natural structure theory could contribute to this research field by providing a theoretical framework to study the shape of neurons, astrocytes, or even networks [3, 4]. In this section, we use a mathematical formalism (definition 15) that represents the concept of a natural structure [3, 4] to model the morphology of a neuron-glia network.

A connected neuron-glia network is evidently a natural structure, where its elements are neural-like cells and single synapses are NE-connections [3, 4]. In general, a single synapse is composed of a presynaptic membrane, an astrocyte membrane (if it is tripartite), a postsynaptic membrane, and a perisynaptic matrix (if it is chemical). The synaptic cleft is a region of the space filled of perisynaptic matrix which is composed of proteoglycans and proteins [25, 17, 27]. This matrix mediates trans-synaptic adhesion (NE-connection role) and regulates the synaptic transmission (communication role) [17, 40]. Thus, a single synapse can be thought of as an structure. In this section, we roughly model a neuron-glia network, applying part of the mathematical definition of natural structures (definition 15) from Cabrera-Febola, 2023-in preparation for publication.

In our case, the natural structure model of a connected neuron-glia network (connected $\Upsilon$-graph) whose synaptic collection is represented by a $\Upsilon$-family $F:I\mapsto S_{X}$ is $(X,\mathbb{D},I)$ where $X$ is a set of neuron-like cells (astrocytes and neurons) and $I$ is a set of single synapses (NE-connections). In order to define $\mathbb{D}$ which models the distribution of elements, we define a distribution function $g:X\cup I\to P$ where $P$ is a family of subsets of $\mathbb{R}^{3}$ such that any element $x$ of the domain of $g$ is distributed in all the points of $g(x)$ and nowhere else. Hence, we can define $\mathbb{D}$ as $(\mathbb{R}^{3},g)$ to model the spatial distribution of the network.

Neuron-like cells have not cavities, so we can assume that $g(x)$ with $x\in X$ is homeomorphic to a solid ball. On the other hand, $g(s)$ with $s\in C$ is the space where neurotransmission occurs, thus it can be assumed to be also homeomorphic to a solid ball. Additionally, we can state that synapses are partially composed of membranes from all implied neuron-like cells, so $g(s)\cap g(x_{i})\cap g(x_{j})\cap...\cap g(x_{z})\neq\varnothing$ if and only if $s\in I$ and $F(s)$ is a relation on $\{x_{i},x_{j},...,x_{z}\}$. Therefore, since the $\Upsilon$-graph is connected, it is possible to conjecture that the $\{\bigcup g(e):e\in X\cup I\}$ is a solid. There should be more topological and geometrical properties of neuron-glial networks; however, this topic is out of the scope of our article.

Neurons has been the focus of two hundreds of years of neuroscience research [33]. However, experimental evidence suggests that astrocytes play an important role in neuronal dynamics, making it necessary to deepen our understanding of how astrocytes network influences neural phenomena [6, 10, 21]. Although there were some attempts to model neuron-glia network dynamics, they lack of a formal network representation [5, 34, 39]. To fill this lack, in this study, we present a formal representations of a neuron-glia network, the $\Upsilon$-graph.

An $\Upsilon$-graph is defined as a tuple $(X,S_{X})$ where $X$ is a set of neural-like cells and $S_{X}$ a multiset of synaptic configurations. In our framework, synaptic configurations are sets of ordered pairs that represent the information flux inside a synapse. Since neurons can establish many synapses with the same information flux, i.e. the same synaptic configuration, $S_{X}$ is a multiset whose counting function allows us to know the number of occurrence of a synaptic configurations in the network.

Previous works on neural networks had not considered each synapse as an individual element but the collection of synapses as the edge of the digraph or graph [26, 28, 35]. However, graph-like representations can misrepresent a neuronal network. For example, if a neuronal network has electrical synapses, each electrical synapse is equivalent to two chemical synapses with inverse directions in a digraph. In contrast, our work presents a new mathematical object, $\Upsilon$-graph which consider each single synapses (electrical, chemical, or tripartite synapse) as an element of the network.

An alternative representation that would address the issue of the multiple types of synapses in the brain would be a multilayer network [9]. In this mathematical object, a layer can represent a type of synapses, solving the misrepresentation problem discussed in the previous paragraph. However, multilayer networks like digraphs assign arcs between two vertices, thus misrepresenting the triadic nature of tripartite synapses. This misrepresentation is an issue when two different networks are represented by the same mathematical objects. In proposition 3, we found a theoretical example of multidigraph that can represent multiple different neuron-glia networks with only tripartite synapses. This misrepresentation can not be fixed by using multilayer networks because there is only one type of synapse in our example.

In addition to $\Upsilon$-graphs, there are other mathematical objects that can represent the information flow in triadic relations: directed hypergraphs [12] and abstract directed simplicial complex [26]. The problem with directed hypergraphs is that a vertex can not be part of the input and outputs of an hyperarc, meaning that, in our context, a neuron-like cell (i.e. astrocyte or neuron) could only send information but not receive information throughout the same synapse. This is not the case for tripartite synapses because the astrocyte receives information from the presynaptic neuron and sends information to the postsynaptic neurons [15, 29, 36, 18]. The case of simplicial complexes need further study to prove wether it is always possible to built a simplicial complex from a $\Upsilon$-graph or not. Simplicial complexes are used to study the algebraic topology of networks [16, 26, 31, 32] while graph-like objects are used to find network motifs and to make graph measures such as between centrality and modularity [2].

Since graph representations are widely used in neuroscience, we defined a function ($pre-\phi$, see appendix 9.2) that unnests $\Upsilon$-graphs into multidigraphs. However, using linear algebra, we proved that the isomorphism between unnested forms does not guarantee isomorphism between their respective $\Upsilon$-graphs in several scenarios (propositions 2-5). Therefore, multidigraph representations can misrepresent a neuron-glia networks. Interestingly, we also found that if the coefficient matrices and the vectors of synaptic configurations of the two $\Upsilon$-graphs are equal after interchanging rows, and the coefficient equations’ columns are linearly independent, the isomorphism between unnested forms guarantees the isomorphism between $\Upsilon$-graphs. Under these specific conditions, we can use graph theoretical tools developed for multidigraphs without misrepresentation issues.

In this work, we also sketched a novel approach to model the network shape. As $\Upsilon$-family represents each single synapse as a distinguishable element, we use this object to create a natural structure model. This natural structure model is a tuple $(X,\mathbb{D},I)$ where $X$ is a set of neural-like cells, $I$ is a set of single synapses,and $\mathbb{D}$ is a tuple $(\mathbb{R}^{3},g)$ where $g$ is the distribution function. In our model, single synapses and neuron-like cells are homeomorphic to solid balls. We conjecture that the portion of space where neuron-like cells and synapses are distributed is also a solid. This hypothetical solid could be studied applying differential geometry or algebraic topology which open new ways for studying structural connectomes.

In conclusion, we have developed a mathematical framework for working with connectivity structure of neuron-glia networks ($\Upsilon$-graphs, conectedness, isomorphism, and other constructs.) We have also connected the concept of $\Upsilon$-graphs with graph-like objects, multidigraphs, and studied their relations ($\phi$ function and matrix equation). On the other hand, we have sketched a natural structure model to represent the spatial structure of neuron-glia networks.

We thank D. A. Pacheco (Harvard University) and Leonardo Torres (Max Planck Institute for Mathematics in the Sciences) for their helpful comments and partial manuscript proofreading.