Fibrations, input trees and the minimal balanced coloring.

A fiber partitioning relies on the notion of neighboring nodes. We begin by defining the immediate in-degree neighboring nodes of a given node v as the multi-set of nodes fulfilling the following condition (1) where the multi-set of edges connecting into node v are given by (2) where (e_G, i) is a pair given by a positive integer number i indicating the length of a walk terminating in v and an edge at the start of said walk. The definition of a walk, in graph theory terms, is a finite or infinite sequence of edges which joins a sequence of vertices such that these sequences can have edge and vertex repeats. A walk is said to be closed when the first vertex and last vertex of its sequence are the same. For completeness we define the set of walks with zero length culminating on node v to be empty and the tail nodes to be composed solely by the node v as described below (3)

Given the above the fiber partitioning is associated with the input history of a node, which includes the input edges it receives from its immediate neighboring nodes, as well as the input those nodes receive from their immediate neighboring nodes. This notion can be extended an infinite number of times, resulting in a (possibly infinite) input tree [23]. The input tree for a node v is denoted as , which is the complete set of all walks that terminate on node v, giving rise to a rooted tree graph [28]. Such a structure can be placed in mathematical terms as (4) Where N represents the largest length to be considered for the set of walks of length N culminating in v where for each length there is a layer in the input tree. As an example the first layer of the input tree of a node v is composed of all the immediate in-degree neighboring nodes of v plus their out-degree edges (connecting to v) which are all the possible walks of length 1 terminating in v.

An input tree can be represented visually, as shown in Fig 1, which illustrates the input history of nodes E and F in the network shown in Fig 2A. Each node in the network has its own input tree, which may appear different at first. However, if the labels of the nodes and edges are ignored, a symmetry can be uncovered, revealing that some nodes in the network have the same input tree structure. These equivalent structures are referred to as isomorphic input trees, which are bijective mappings from one input tree to another input tree , such that there is a one-to-one mapping of nodes and edges from the ith layer of tree to the ith layer of tree [66]. As a note, N can potentially be infinite if there are loops in the network (walks with node v at the start and end of it); although the smallest length of walks N to guarantee the complete distinguishment of unique and not isomorphic input trees is n−1, where n is the total number of nodes in a graph G [35].

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Fig 1. Input trees and their isomorphism.

The input trees of nodes E and F for the network in Fig 2A are shown. The input tree of node E is given by all the possible paths in the network in Fig 2A which terminate on node E. All paths can be overlayed together forming a rooted tree graph which is referred to as the input tree. In this example only the first three layers of the input trees of nodes E and F are shown, but it can already be seen that their input trees are identical if all the labels of the nodes and edges are ignored. Nodes with isomorphic input trees eventually synchronize regardless of synaptic delay or small variations there parameters [67, 68]. Colored lines indicate the morphism between trees. An input tree can be annotated by the number of edges between specific nodes at specified layers, the edge from B to F is shown for both input trees where it appears at different layers.

https://doi.org/10.1371/journal.pone.0297669.g001

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Fig 2. Fiber and orbits.

The inner color of a node is a representation of the fiber to which said node belongs to and the color of the ring indicates to which orbit such a node belongs to. (A) Graph with 4 fibers and 5 orbits. The separation of fiber and orbits among the 4 upper nodes occurs due to the presence of outgoing edges into nodes E and F. (B) This graph only differs from A by the removal of the edge B→F exemplifying how restrictive orbit symmetries are. There are only trivial orbits but there still non-trivial fibers. (C) Although it is more common to have a matching of the partitionings produces by fibers and orbit symmetries in a given undirected networks there can exist situations in which these two are not matching as seen in this example. In this simple example there is a breakage between the fiber and orbit symmetries due to the in-existence of a permutation action which could swap nodes X, Y and Z respectively with their fiber symmetric counterparts W, U and V meanwhile preserving its structure and adjacency matrix.

https://doi.org/10.1371/journal.pone.0297669.g002

Nodes with the same input tree are in the same fiber, which are projections of a morphism called a graph fibration [28]. A graph fibration φ between two graphs G = (N_G, E_G) and B = (N_B, E_B) is satisfied when for any e_B ∈ E_B and any v_G ∈ N_G with φ(n) = h(e_B), there is a unique e_G ∈ E_G such that φ(e_G) = e_B and h(e_G) = v_G. A fiber is a set of nodes (cluster cell) with isomorphic input trees in G, mapped via φ onto a node in B, where G is the total space and B is the base. This paper focuses on the minimal graph fibration, collapsing a graph into its minimal base, where the base contains one node for each unique input tree, resulting in a trivial partitioning of B and a minimal balanced coloring of G [28]. A partitioning with k cluster cells that obey equitable partitioning is minimal if no other partitioning with fewer than k clusters satisfies the equitable partitioning condition [31]. An example of this can be seen in Fig 3. Throughout this paper we refer to the minimal graph fibration as the minimal balanced coloring of a graph.

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Fig 3. Graph fibration φ: G → B.

The network at the bottom (G) shows an example for a graph where the color of its nodes represents the fiber to which they belong. Notice that all nodes with the same (e.g. {3, 4, 5}) color are collapsed on to one representative node (y) in the network above called the base. This collapsing process is known as the lifting property of a graph fibration φ. This property also applies to edges, notice that the edges in the total space (e.g. {}) which are lifted to one representative edge in the base which is situated between nodes of the same color as in the total space ().

https://doi.org/10.1371/journal.pone.0297669.g003

Based on theory nodes in the same fiber synchronize, meaning that the associated admissible ODEs have the same value at the same time [28, 36]. Also based on theory, there are no restrictions on the differences in behavior between two nodes in two different fibers, and it is possible for two or more fibers to synchronize [27], such is dictated by the set of equilibrium solutions of an ODE system. Connected nodes in the same fiber may have phase shifts under the same frequency or an integer multiple of a base frequency, but only if the nodes have rotational symmetry, and the phase shift is a rational fraction of 2π [31]. Although the natural occurring biological variations from neuron to neuron can be expressed as variations in the same set of parameters among the admissible ODEs and the physical propagation of signals between neurons can be incorporated through different time retardation terms. All these modifications do not necessarily destroy the expected synchronization of nodes in the same fiber, but do indeed reduce the parameter space of the equilibrium solution where such is attainable [67, 68]. Therefore we forgo integrating these for the sake of simplicity of our models.

We find the fiber partitioning algorithmically by initially coloring all nodes and arrows with a unique color. After that, the algorithm recolors all nodes with the same number of colored inputs with a new color, including their outgoing arrows. These procedures continue until no recoloring is possible. This is the algorithm developed by Kamei & Cock [69], which is a generalization of the algorithms developed by Belykh and Hasler [70] and by Aldis [66] and implemented through the recreated code in [38].

Automorphism and their orbit partitioning.

The second method relies on the set of automorphisms of a graph, which is a bijective function σ from a graph onto itself preserving the structure of the graph, specifically its adjacency matrix A. The set of all automorphisms of graph G along with the operation of composition forms the automorphism group Aut(G) [71]. Specifically, an automorphism is a permutation of the vertices of the graph that preserves the adjacency relationships between them. If G is a graph with vertices {v₁, v₂, …, v_n}∈N_G and with edge set E_G, then an automorphism of G is a permutation σ of the node set such that if (v_i, v_j) is an edge in G so too is (σ(v_i), σ(v_j)) an edge in G. An automorphism function σ can take the form of a symmetric permutation matrix P where matrix is considered to be an automorphism of an adjacency matrix A if the following holds: (5) where P⁻¹ is the inverse of P. Additionally it is invertible meaning that P = P⁻¹ or PP⁻¹ = 1 allowing Eq (5) to be rewritten as: (6)

The permutations of the automorphism group can be grouped into sets that form normal subgroups meaning that a permutation from one set can commute, as in Eq (6), with the permutation of another set (disjoint unions) [34]. In other words, applying one of these permutations to a graph it only permutes a closed set of nodes, leaving the rest of the graph’s nodes intact.

If when σ is applied on a graph’s adjacency matrix A as in Eq (7) and it doesn’t change it, then multiple consecutive applications won’t either. (7)

Although an automorphism’s consecutive application preserves a graph’s adjacency matrix, it does shuffles node labels. So, a node’s label always returns to its original after visiting other nodes. Given a graph G and σ, the orbit of node v under σ is the set of all vertices that can be reached by applying σ repeatedly. That is, the orbit of v is the set nodes (8) where denotes the integers numbers. The orbit of node v includes all vertices equivalent to v under the action of σ, transformed by applying σ k times. The set of nodes belonging to an orbit are said to have the same orbit coloring [32, 72] were studies in dynamical systems show that network automorphisms lead to the synchronization nodes in an orbit [31, 69, 73, 74].

Orbit partitioning results from analyzing the complete set of automorphisms of a graph Aut(G). Orbits can be revealed not only by repetitively applying a single automorphism but also by applying the set of automorphisms Aut(G) consecutively in all possible orders and multiplicities (e.g., P₃P₁P₂P₂). This approach allows for a wider range of transformable nodes to be detected. The procedure of how this is explicitly done is out of the context of this paper, but the process of uncovering orbit coloring partitions is done using SageMath [75], an open-source mathematics software that utilizes various Python packages, including Nauty [76].

The orbit partitionings that emerge from this are equitable partitioning which are cluster cells, stressing that an orbit coloring is always an equitable partition, but the reserve is generally not true [31, 32]. This is because, by definition, orbit partitioning requires the conservation of both in-degree, requested for the equitable partition, and out-degree. As such, an equitable partition may not be an orbit partition since the out-degree condition may not be met.

Fibers vs orbits.

As discussed above the equitable partitionings of fibers and orbits are related to the concept of symmetry. The orbits obtained from automorphisms are analogous to the fibers obtained from symmetry fibrations. Orbits, however, are a more restrictive form of graph symmetry as these must satisfy in-degree and out-degree edge constraints. In contrast, fibers must only satisfy in-degree constraints.

Fibration symmetries in a graph have fewer constraints than automorphisms, resulting in fewer partitionings in the graph. Orbits are a subset of fibration symmetries, which additionally conserve the out-degree structure of the adjacency matrix. This means that every orbit is also a fiber, and the in-degree constraint present in the orbit partitioning, which is the main constraint in fiber partitioning, leads the nodes in the same orbit (cluster cell) to become synchronous [33]. However, note that a fiber is not necessarily an orbit.

In Fig 2A, an example of a graph with three fibers and four orbits is shown. The existence of more fibration symmetries than automorphisms generally leads to having fewer (or equal) fibers than orbits. In other words, having more symmetries implies having fewer cluster cells. The green nodes have the same number of in-degree edges (belonging, therefore, to the same fiber) but some have different number of out-degree edges. Nodes B and D each have an out-degree equal to 2. Another way to conserve the structure of this graph other than applying the identity transformations is by rotating A → B → C → D → A twice in this order, and in concurrence reflecting horizontally all the nodes below the green nodes. Without applying this reflection, node B would have one outgoing edge into node E, therefore not preserving the original graph’s structure. Notice that any permutation for the upper four nodes would still preserve its fiber symmetry as these would continue to only receive one green in-degree edge.

As an additional example, the removal of edge B → F as seen in Fig 2B destroys the orbit symmetry of the graph. No rotation between the upper four nodes would allow node D to have an outgoing edge to node E other than its original position. There also does not exist any permutation of the lower five nodes that would keep the structure intact all due to the removal of D → E.

Circuits.

In this paper we show the fiber building blocks of the chemical networks. In order to do so and explain it in a more detailed manner each of the networks will be broken into multiple sub-graphs. Where we refer to these sub-graphs as circuits and for a graph with k fibers there will be k circuits. These circuits are produced by collapsing any of these chemical network graphs (total space) into their base graph representation and retaining the unique fibers present in the input tree of a fiber under consideration. Take the Backward Chemical network (a.k.a. B-Chem) as seen as a graph in Fig 4A as an example. In this graph the color of the nodes represents the fiber to which they belong (more on this particular network’s partitioning ahead). Neurons DA05, DA08, DA09, VA06 and VA11 are all the nodes contained in what we can call fiber Blue5 standing for the 5 blue nodes composing it. The input tree of fiber Blue5 contains nodes belonging to, using the previous nomenclature, fibers Orange4, Pink2, Cyan2, Mustard2 and Gray4. The circuit associated with fiber Blue5, call this the main fiber or the fiber under consideration, can be observed at the left of Fig 5. It is produced by collapsing the B-Chem network and retaining the fibers with their outgoing edges that compose the input tree of fiber B5, call these other fibers the induce fibers.

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Fig 4. Backward chemical synapses network partitioning results.

The partitioning of nodes into different synchronous groups is depicted by the color of the nodes. (A) The minimal balanced coloring results show 10 distinct colors/partitions for the 30 neurons involved. (B) The orbit coloring results for this network where the 30 neurons have been partitioned into 13 distinct partitions. The colored shaded areas show the normal subgroups that form the automorphism group of this graph () with their respective symmetry group represented by the same colored symbol. (C) An example of neurons with different minimal balanced coloring and orbit coloring. The latter has a permutation symmetry S2 swapping VA02 with VA03 and VA05 with VA04 simultaneous which preserves the structure of this network.

https://doi.org/10.1371/journal.pone.0297669.g004

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Fig 5. Fiber building block decomposition.

Present here at the most left is the base of the circuit induced by the input tree for the fiber containing neurons DA05, DA08, DA09, VA09, and VA11 (blue node) present in the backward chemical network as seen in Fig 4A with the same color scheme. The first input into this fiber are provided from the orange (VA12) and cyan (AVD) nodes where these form the first layer of the multilayered block. The input tree of the network formed by these nodes give a fiber number of |0, 2〉 (ℓ = 2 as multiplicity of edges are ignored). The next layer is composed by the elementary fiber building blocks of the input fibers in the previous layer. This notion is carried out at every layer until all fibers have been shown only once. Layer 2 already shows a complex elementary FBB for the cyan node with fiber numbers |3.372, 6〉. Layer 3 too contains elementary FBB with a similar fiber number as the cyan node in layer 2; the only difference here is that these two contain different number of inputs therefore the ℓ number is different in these two. Additionally in layer 3, self-loops of the gray node are present for the upper elementary FBB as the nodes forming this fiber are composed by neurons AIB and RIM which form a square network and which all send signals to the pink fiber. As for the gray fiber in the elementary FBB at the bottom it is only formed by neurons AIB which have no connections between them and therefore their collapse does not produce self-loops. The fiber building block induced by the input tree of the blue fiber concludes with a 4th layer which contains 4 neurons connect in a square. These have no external input which leads to it having fibers numbers |2, 0〉 which captures the duplication of nodes at each layer in their input tree.

https://doi.org/10.1371/journal.pone.0297669.g005

Fiber building block synthesis.

A circuit is further broken down into its fundamental building block which here and as in [26] we refer to as fiber building blocks. A circuit has as many FBBs as is has induce fibers plus its main fiber, 6 FBBs for the B-Chem as it is seen in Fig 5. As mentioned for each fiber in a graph there exists a fiber building block (FBB) associated with it which is a sub-graph S ⊆ G induced by the following nodes [26]:

• all the nodes in the fiber (nodes with the same minimal balanced coloring),
• immediate in-degree neighboring nodes of all the nodes in the fiber,
• nodes belonging to the shortest loop that include a node in the fiber which are not a self-loop,
• if all points above lead to two or more disconnected graphs, the nodes composing the shortest path connecting each pair of disconnected graphs are included. These types of FBBs are dubbed composite fiber building blocks which are further explained below.

For all nodes that are considered in the second point above we refer to these as regulator nodes. If the nodes in the main fiber are found to send information back into the fiber then these too are considered as regulatory nodes.

Fiber numbers.

The FBBs that make up a circuit, are graphs in their own right and by such can also be represented by an input tree. These input trees are the bases of finding the building blocks of biological networks such as in the chemical synapses connectome of the C. elegans. Our approach is to categorize these FBBs through their structures and through such capture the fibrational composition of these networks. Below we explain how these can be decomposed into two features that numerically quantify the structure of the FBB’s input tree. We first introduce and explain the feature followed by an example using the FBB of the cyan node found in Fig 5 (top sub-graph of Layer 2).

A feature of FBB that can be use to classify them is related to the number of nodes that are present at every layer of the FBB’s input tree. Specifically how the number of nodes grow or remain the same relative to consecutive layers. This structure for a given input tree can be represented by a numerical sequence a_i that denotes the number of nodes for every layer i. This sequence also corresponds to the number of walks terminating on the single node v at the very top of the input tree, a.k.a. the root of the input tree. The exact number of nodes at every layer does not matter as much as how the number of nodes change per consecutive layer. This can be captured by the branching ratio, or common ratio, which can be found by taking the limit of a_i+1/a_i as i → ∞. This ratio is denoted by |n〉 which captures the number of loops in the input tree of a fiber. The symbol |n〉 is called a fiber number and can be an integer or an irrational number, with the latter indicating complex branching due to nested loops which are referred to as “Fibonacci fibers”.

Using our example, the first layer has only one node as is the case for any input tree, in this case the cyan node. The second layer is composed by 3 nodes (as the cyan node has 3 in-degree edges). We obtain the number of nodes in the third layer by calculating the total number of in-degree edges for all the nodes present in layer 2. For this case one of the nodes in layer 2 receives 3 in-degree edges, while the remaining two nodes (repeats) receive 4 in-degree edges making for a total of 11 nodes. Using the third and second layer we obtain a branching ratio of , calculating this branching ratio for layers evermore faraway from this input trees’ root it reaches a limit value of 3.372….

Another property that is used to characterize a FBB’s input tree for a node v is the number of unique trails with no repeating edges which terminate on node v. Where a trail is a walk in which all directed edges are distinct (transversed once). The number of trails is calculated in the base graph induced by the nodes that compose the input tree disregarding edge multiplicity between nodes. This property is symbolized by |ℓ〉 which is another fiber number and together with |n〉 are enough to categorize the topology of input trees where the two can be agglomerated into |n, ℓ〉 which are called fiber numbers.

Using the cyan FBB in Fig 5 we explain how to calculate the numerical fiber number |ℓ〉. Before anything lets call the node above and to the left of cyan node the M node (for the color mustard), and the node above and to the right the P node (for the color pink). All the unique trails terminating in the C node are: {M → C; P → C; M → P → C; P → M → C; P → M → P → C; M → P → M → C}. The previous fiber number |n〉 is ultimately another quantification of the trails present in a FBB’s input tree, particularly of closed trails that result in loops, where loops of different length can be “nested” in each other resulting in Fibonacci fibers. Using our cyan fiber building block it can be seen that it has loops of length 2, 3, and 4 of which each is nested in the other. Where an example of a loop of size two is P → M → P, a loop of size 3 can be the closed walk C → M → P → C and an example of a loop of size 4 is C → M → P → M → C.

Multilayering.

Simpler fiber building blocks can be combined to create multilayered composite blocks needed when the topology of a circuit or FBB cannot be fully captured by one fiber number pair |n, ℓ〉. In such cases, the |n, ℓ〉 of the input tree of each regulator of an FBB is added to the |n, ℓ〉 of the input tree of the main FBB itself, creating an addition between two adjacent layers. This addition is performed in a left-to-right ordered sequential manner, resulting in the following notation: (9)

The symbol ⊕ represents an addition of FBBs between adjacent layers and the symbol + is used to indicate that the two FBBs next to this symbol are in the same layer. In the example above the fiber number |n₁, ℓ₁〉 represents the FBB at the first layer. Fig 5 is an example of multilayered FBB composed of regular and Fibonacci fiber building blocks. Overall these types of combination of fiber building blocks aid in determining the order of complexity among fibers.

Ordinary differential equation parameters and implementation methods.

The parameters of these ODEs shown above are taken from Kunert et al. [85]. The parameter α^leak is a time constant with a value of 10s⁻¹ and is associated with the rate of decay of the neuron’s membrane potential. This is derived from the leakage conductance per surface area g_L times the average surface area of an inter-neuron S (based on Varshney [8]g_L ⋅ S is equal to 0.01nS) divided by the membrane capacitance of a neuron (Varsheny et al. [8] defines C to be 1pF). The time constants for chemical synapses and gap junctions have the same value of g_L ⋅ S/C = 0.1nS/1pF = 100s⁻¹ where these fall within the value range from Koch’s book on biophysics [88].

The constant γ is associated with the steepness of the sigmoid function which is set to 2ln(0.1/0.9)/36mV in accordance to Wicks et. al [53], which defines this constant to be representative of a change of the value of the sigmoid function between 0.1 to 0.9 within a range of 36mV. This value is close to that reported by Kunert [85] so it is rounded to 125V⁻¹ to match Kunert’s value.

The synaptic activity variable depicts the activity magnitude at a synaptic junction, and its behavior is affected by the neuron’s membrane potential as well as the parameters determining the increase and decrease of synaptic activity. The terms a_r = 1s⁻¹ and a_d = 5s⁻¹ associated with the Chem type II model correspond to the synaptic variability’s rise and decay times [85]. The equilibrium value for the synaptic variability term comes from setting Eq (15) to zero and having the sigmoid equal to 0.5; this leads to a value of s_eq = ar/(ar + 2*ad) = 0.09. Finally α^ext is simply set to 1/C and V_s,j takes two values: 0mV for excitatory vs −70mV for inhibitory presynaptic neurons [89]. For this study, we considered all chemical synapse connections to be excitatory (although our code accepts networks with inhibitory connections). The units of I^ext are pA to match the units (V/s). All neurons in the simulation conducted in this paper have the same parameters as the different families of motor-neurons are known to be the most similar among all the differently distinct electrophysiological group of neurons [49]. We extend this notion to the parameters of the inter-neurons to have a simpler system to work with, as well due to the nature of our sub-networks being mainly composed by motor-neurons.

We set the threshold voltage V^threshold for each neuron to the corresponding solution obtained by setting Eqs (12), (13) and (15) to zero and Eq (16) to 0.5; where the solution is the stable point solution for this ODE model. This procedure (Gaussian elimination) is similar to that in Wick et al. [53] where in such is set to 0 where as we set it to I_drive (the mean value of an undulatory or of a noise stimuli is 0, therefore, these are not included). This recipe finds the solution of the threshold voltage to be the one below (18) such that A is given by (19) and b by (20) where the term c is equal to s_eq for the Chem type II model or 0.5 for the Chem type I model. It is important to point out that threshold voltage is respectful of the in-degree nature of Eq (10). Further in the paper V^threshold is used in conjunction with the Jacobian to determine the stability of the solutions.

The ODE dynamics were evolved and computed using the Runge-Kutta-4th method with a time step dt of 0.1 milliseconds. Noise dynamics were implemented through a modified stochastic Runge-Kutta method as proposed in [90], where the update time step for the noise is dt^1/2. A link to a user-friendly Matlab app developed in-house can be found in the Data Availability section. This app can reproduce any of the dynamics presented in this paper with the appropriate inputs. It can measure the level of synchronization (explained in the next section) and has additional tools for further experimentation.

Simulation test 1 setup.

We set the initial voltages to be normally distributed around the resting state potential of −37mV and without external stimuli, which can be representative of a C. elegans during a state named lethargus. During this state, neurons such as ALM have shown low spontaneous activity and remain near their resting state [95]. The initial voltages are normally distributed around the resting state, with the standard deviation varied from 0 to 0.1. We applied this procedure to both types of chemical synapse models. We do not show results for the gap junction networks without external stimuli as they become globally synchronous, with nodes having a voltage equal to the resting state, due to nodes being able to be reached by any other node in the graph. For the Chem type II model, the standard deviation of the synaptic variable varied from 0 to 0.1 with a mean of s_eq for every distribution of initial voltages. This setting allowed us to study the effects of the initial synaptic variables on the outcomes of synchronicity and inspect its stability.

Simulation test 2 setup.

For the second simulation test, we look into the synchronizations that arise in the networks under external stimuli as modeled by the ODEs found in the Admissible ordinary differential equations section. Contrary to the previous case, this setting allows us to explore if these symmetrical neural models of the C. elegans with gap junction connections synchronize to the predicted partitionings found via fiber partitionings, and if the B-Chem network dynamics separate into orbit or fiber partitionings when in an active state. Before all this, a stability analysis is conducted to determine the range at which the external stimuli do not induce any instabilities in these networks + ODE models.

Simulation test 3 setup.

We studied four networks and their binary/integer edge-weighted versions. entries in Eq (10) were altered by subtracting a normally distributed value with increasing standard deviation and zero mean. Initial values were set to equilibrium/resting state to assess partitioning method reliability when edge weights are distorted. During our analyses, we kept the σ of the LoS function equal to 0.1mV.

Simulation test 1 results.

Applying the LoS measurement to the last second of simulation time for each case produces blocks of synchronicity as seen in Figs 11 and 12. For all the 4 cases pertaining to the backward chemical network, all of the expected fiber partitionings are present (diagonal boxes delineated by red). We find that for the binary version of the backward chemical network, neurons within the same fiber became synchronous with nodes on other fibers, as an example, the 1st and 6th of these diagonal blocks starting from the top in Fig 11. We confirmed that these synchronizations are still present for longer simulation times or with a more restrictive σ parameter. These were only confirmed to be significantly diminished when the networks were driven through an external sinusoidal or Gaussian random walk stimulus, as shown in the simulation test 2.

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Fig 11. Backward chemical model type I simulations with no external stimuli.

Initial voltages set to −35mV. (A) Left side pertains to the binary version of this symmetric network where an example of two synchronous fibers can be observed (AVDL+AVDR with AVEL+AVER) besides another example with 3 synchronous fibers. An equitable partitioning alone can not predict which fibers become synchronous with one another. (B) the right is the weighted version of this system which shows a perfect separation of the values of the nodes into the expected those from fiber partitioning. The block system shows the result of the LoS measurement done on the last second of the simulations. The red-lined boxes are visual indicators for the expected cluster synchronization of neurons predicted by the fiberation partitioning method.

https://doi.org/10.1371/journal.pone.0297669.g011

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Fig 12. Backward chemical model type II simulations with no external stimuli.

Synaptic variables initially set to s_eq and initial voltages set to −35mV. (A) Left side pertains to the binary version of this symmetric network. (B) is the weighted version of this system which shows a perfect separation of the values of the nodes into the expected those from fiber partitioning. The block system shows the result of the LoS measurement done on the last second of the simulations. The red-lined boxes are visual indicators for the expected cluster synchronization of neurons predicted by the fiberation partitioning method.

https://doi.org/10.1371/journal.pone.0297669.g012

Results for the forward chemical model were similar and can be seen in Fig 13. The forward chemical networks synchronize to the predictions made by the fiber and orbit coloring. In the binary Chem type II case inter-neurons PVC and motor-neurons VB07 and VB09 synchronize as tested by more sensitive simulations.

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Fig 13. Forward chemical type II model simulations with no external stimuli.

Synaptic variables initially centered around s_eq with a standard deviation of 0.1 and initial voltages set to a mean of −35mV with a 10mV standard deviation. (A) The left portion pertains to the network with binary edges. (B) The right portion is associated to the integer weight case. The block system shows the result of the LoS measurement done on the last second of the simulations. The right LoS matrix is an example of an ideal case. The red-lined boxes are visual indicators for the cluster synchronization of neurons under their expected balanced coloring partitioning.

https://doi.org/10.1371/journal.pone.0297669.g013

In Fig 14 it can be observed how these neurons belonging to different fibers become synchronous. Curiously one can notice that PVC neuron synchronize with each other and so do neuron VB07 and VB09 with each other before these two fibers synchronize.

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Fig 14. Binary forward chemical type II model simulations with no external stimuli.

Synaptic variables initially centered around s_eq with a standard deviation of 0.1 and initial voltages set to a mean of −35mV with a 10mV standard deviation. Lines are color coded based on their fiber partitioning where it can be seen that all same colored neurons eventually synchronize with each other. Neurons PVC belonging to one fiber become synchronous before synchronizing with neurons VA08 and VA09 belonging to another fiber. The LoS associated with this simulation can be observed in Fig 13A where the latter example is captured by the only off diagonal blocks.

https://doi.org/10.1371/journal.pone.0297669.g014

Simulation test 2 results.

In this section, we aim to investigate the predictive power of fiber and orbit partitionings by simulating neural activity. To achieve this, we drive the four networks by delivering the same stimulus through left-right inter-neuron pairs. These left-right inter-neuron pairs belong to the same partition in all four networks, therefore not breaking the fiber partitioning.

The external stimuli in any of these networks ultimately changes the equilibrium solution of its respective ODE due the nature of these equations. The sinusoidal and noise stimuli will lead the voltages of the neurons away and towards equilibrium periodically and aperiodically respectively. By such these two stimuli can be regarded as perturbations if kept relatively small compared to the constant term stimuli of Eq (17) which dictates the equilibrium point solutions. Additionally, it is important to ensure that the external stimuli do not induce any instabilities that could complicate the interpretation of results such as bifurcations or chaos which a fiber analysis of a graph will not be able to predict. Therefore, we conduct a stability analysis of the ODEs to determine the range within which the external stimuli does not cause any instabilities. As a common practice in the study of dynamical systems, we focus on the Jacobian of the model under study. For the Gap junction it’s Jacobian can be written as: (23) where I is the identity matrix and L is the Laplacian of the gap junction adjacency matrix. Notice that Eq (23) is independent of any external stimuli. The Jacobian matrix is a useful tool for investigating the stability of a system’s equilibrium point. By analyzing the eigenvalues of the Jacobian matrix, we can gain insight into the local behavior of the system near it’s equilibrium point. According to the established principles of linear stability analysis, if all eigenvalues are real and negative, then the system is stable, meaning that any small perturbations around the equilibrium point will dampen over time, and the system will return to its stable state. If one or more eigenvalues have positive real parts, then the system is unstable, and any small perturbations will cause the system to diverge from the equilibrium point [96, 97].

We find that all eigenvalues of Eq (23) are negative for the values of α^leak and α^gap as described in the Ordinary differential equation parameters and implementation methods section. Therefore the gap junction model is stable under any external stimuli as it does not depend on I^ext.

Moving on, the individual terms of the Jacobian for the Chem type I model takes the form of Eq (24). At equilibrium, the sigmoid functions have a value of 0.5 (as per how V^threshold is constructed in this model [85]). The Jacobian for this chemical model depends on the external stimuli indirectly through V_i (Eq (20)) situated in the off-diagonal terms of the Jacobian. (24)

Analyzing the eigenvalues of the Jacobian matrix at the stable point solutions when , determined by Eq (18), shows which neurons will have unstable voltages for a given constant value. For the F-Chem networks with integer weights under the Chem type I model, most of these eigenvalues remain real and negative as the external stimuli applied to neurons PVC or AVB are varied from −500pA to 500pA. An exception occurs for the eigenvalues associated with the neurons that receive external stimuli. For PVC stimulation, the real part of its eigenvalue linearly crosses into the positive regime for external stimuli greater than 3.08pA or less than −2.38pA (Fig 15A). This eigenvalue is accompanied by another that mirrors its behavior. These two eigenvalues have the same value at 0.35pA of external stimuli (dashed black line in Fig 15). This occurs because when the external stimuli take this value, the leaking potential term in Eqs (12) and (13) approach zero. Simulations of this model for the respective network show a bifurcation between the left and right PVC inter-neurons values at values higher than 3.08pA of external stimuli. A similar result can be observed for the stimulating neurons AVA in the backward chemical network with binary weights as seen in Fig 15B. No simulation with negative external stimuli produced bifurcations.

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Fig 15. Stability analysis results.

(A) Eigenvalues of the Jacobian for the integer edge weighted F-Chem network under the Chem type I model as the external stimuli through inter-neurons PVC is incremented from −6pA to 6pA. The first positive eigenvalue happens at 3.08pA of external stimuli. (B) Eigenvalues of the Jacobian for the integer edge weighted B-Chem network under the Chem type I model as a function of constant external stimuli input through AVA inter-neurons. First positive eigenvalue happens at 34.1pA of external stimuli.

https://doi.org/10.1371/journal.pone.0297669.g015

A look into the Jacobian of the Chem type II model Eq (25) also indicates that it is stable under external stimuli which do not exceed specific values. Below one can find its Jacobian. (25)

At equilibrium, the voltage values of the neurons V_i equal that of the V^threshold as in Eq (18) [85]. Therefore, the sigmoid function takes a value of 0.5, and the synaptic variable term equals s_eq. This Jacobian is indirectly affected by the external stimuli through the V_i terms, which only appear in the off-diagonal term as it can be seen in Eq (26). (26)

This and other stability analysis were carried out for all the combinations of ODE models, networks, and edge types with external stimuli varying from 0pA to 500pA. In Table 1, we show the results where the values of this table indicate the strength of the external stimuli (into the indicated inter-neuron pairs) at which instability and the first positive eigenvalue arise. For all cases, only one positive eigenvalue (value underscored in Table 1) was above a reasonable strength based on multiple electrophysiological studies done on C. elegans [45, 46, 98].

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Table 1. Instability analysis.

Results for all combinations of chemical ODE models, networks and edge types. An external stimuli drove each of these networks through increasing strength from 0 up to 500pA until a positive eigenvalue emerged. These neurons are shown in the third column. Results are given in pico-Amperes.

https://doi.org/10.1371/journal.pone.0297669.t001

Knowing the maximum strength of the external stimuli at which no instabilities appear, we stimulate their respective network + ODE model with a constant input equal to 90% of the external stimuli necessary to cause instability in the network plus a sinusoidal (or noise) stimulus with an amplitude 5% of the instability strength. To test through simulations that the neurons of these graphs synchronize to those stipulated by fiber and/or orbit partitioning we stimulate some of the neuron in these network. Gap junction networks were driven through inter-neurons AVB and motor-neuron DB04 for the forward network and inter-neurons AVA and motor-neuron DA05 for the backward network. This additional driving signal applied to a motor-neuron is needed for the Gap junction models as these networks globally synchronize if there is only one driving signal irrespective of the network. The introduction of a second signal with different characteristics (frequency and phase) aids in breaking the global synchronization revealing the expected cluster synchronizations. For the Chemical synapses networks we stimulate PVC neurons for the forward network and AVA neurons for the backward network. We set the inter-neurons frequency to 2Hz and that of the motor neurons to 1Hz. We calculated the LoS matrix for each of these simulations and rounded entries lower than 1 to 0.

These resulting matrices were subtracted from their idealized LoS matrix obtained from fiber partitioning where these have diagonal block elements equal 1 (elements inside squares delineated with red lines in Figs 11–13) and all other entries equal zero. This subtraction was divided by 2 times the number of elements in LoS matrix without counting diagonal elements. A zero value in these indicates a perfect agreement with the idealized LoS matrix from fiber partitioning. A negative value indicates additional synchronization between two or more balanced colored groups. A positive value would indicate that the expected fiber partitionings were not found or disagree in some form.

Table 2 presents the results of our analysis. According to these results, all networks, regardless of the ODE model or type of edge, separate into distinct synchronous groups under the LoS metric. This finding confirms the partitions obtained via fiber partitioning. We did not observe any agreement in orbit coloring for the B-Chem network, which would have led to a positive entry in Table 2 by dividing diagonal boxes into two or more diagonal boxes. The negative values in this table indicate that some of the cluster cells expected from fiber partitioning synchronize with one another, but this is not a negative result because fiber partitioning only requires neurons in a fiber to synchronize and does not preclude different fibers from synchronizing.

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Table 2. Synchronization among fibers.

Synchronization difference between the LoS matrix of a driven network and its ideal LoS matrix. A value of zero indicates perfect agreement between the LoS matrix of a driven networks and its ideal LoS matrix with distinct synchronous groups. A negative value indicates that two or more minimal balanced colorings have the same value at the same time. A positive value would indicate that the dynamics of its neurons do not cluster synchronize to the partitioning of the minimal balanced coloring. The networks here were driven through inter-neurons mentioned in Sections. The best ODE model to differentiate synchronous partitionings was the Chem type II model.

https://doi.org/10.1371/journal.pone.0297669.t002

Simulation test 3 results.

We examined neuron synchronization and fiber partitioning agreement, testing robustness by adding random edge weights to simulate missing or uncertain information. We added normal distribution weights (std. dev. 0-0.1 in 0.01 steps, with 0 mean) to all weights, and stimulated the same neurons as in simulation test 2 in ten separate occasions with all the resulting LoS being averaged (see LoS matrices of Fig 16). The LoS obtained from simulation test 2 was binarized (elements below a value of 1 were zeroed) and used as a mask on the averaged results. The resulting matrices were subtracted from the masking matrix.

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Fig 16. Effects on synchronicity due to randomizing non-zero weights.

The addition of a normally distributed amount (with zero mean) to the non-zero weights of all the networks studied here (Gap and Chem type II models) is studied. Each perturbed network is simulated for 5 seconds with the last second used to calculate its LoS and subtracted from the idealized LoS (see Fig 13B for an example). The difference is only calculated on the expected minimal balanced coloring synchronizations (diagonal blocks). Examples of the LoS calculated for the F-Chem-Integer, B-Chem-Integer and B-Gap-Integer networks are shown from top to bottom respectively.

https://doi.org/10.1371/journal.pone.0297669.g016

In this simulation, the driving signals are composed of a constant input of 0.1pA plus an oscillatory term with a frequency of 2Hz for inter-neurons and 1Hz for motor-neurons with an amplitude set to 0.5pA plus a Gaussian random walk scaled between −0.01pA and 0.01pA. This gives a max amplitude of 0.61pA, below any of the values that would induce instabilities in any of our networks as seen in Table 2. All initial voltages were set to −35mV, and all initial synaptic variables to s_eq. Fig 16 shows the results. Varying weights affects LoS synchronicity of networks similarly, regardless of binary or integer edge weights (continuous and dotted lines, respectively). Backward chemical locomotion network is slightly more robust to weight changes than forward chemical locomotion network. All gap junction model synchronizations are robust to weight changes. However, most synchronicities were lost when measured through the strict LoS method, which requires signals to have the same value at the same time within a distance less than σ = 0.1mV. Different synchronicity measures may lead to different conclusions about the dynamics of stimulated networks. For instance, in the case of the integer edge-weighted F-Chem network with edge weights having a standard deviation of 0.05, its dynamics are illustrated in Fig 17. Although LoS analysis shows no synchronizations, the PLV metric demonstrates that neurons of the same minimal balanced coloring exhibit high synchronicity under phase synchronicity. As such, by relaxing the constraint of synchronicity from having the same value simultaneously to having the same instantaneous phase, it is possible to highlight the synchronicities expected from fibration symmetries. This was seen for all other networks as well.

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Fig 17. Phase synchronization results.

(A) Dynamics for the forward chemical integer edge weighted network (with an average edge weight of 2.6 before any edge alterations). This network was driven through inter-neurons PVC as explained in section: Simulation test 3 results. The alterations to it’s edges weights were done by adding to them normally distributed numbers with standard deviation 0.05 and mean 0. (B) Phase Locking Value (PLV) matrix of these dynamics.

https://doi.org/10.1371/journal.pone.0297669.g017