Computational implementation

The calculation of the attractors and its basins of attraction of all the networks presented in this work was performed using BoolNet [16] and GINSIM [17]. The implementation of the theorems developed in this work was done with the object-oriented language C# using Visual Studio 2022, community version.

Canonical form of networks that only present fixed-point attractors

Studying the landscape of attractors of different types of Boolean networks allowed us to identify special properties of those systems whose solution only consists of fixed-point attractors [13]. In this sense, for such systems, their landscape of attractors behaves as a single Boolean function, as we previously demonstrated [13]. Now, we recapitulate this fact in the following lemma.

Lemma 1. Let be a Boolean network of n nodes updated synchronously, with as its state-space. If F only have fixed-point attractors as solution and if F is converted into integer numbers, then their attractor landscape is equivalent to a function.

Proof. If the solution of F only consists of fixed-point attractors, it means that such that, for every . Since F is updated synchronously, then , such that when . Redefining the state-space as , it is clear that, starting from any element in the domain, i.e., , when , it will be reached a fixed-point attractor in the codomain . Consequently, this defines a function in the Cartesian plane, whose image will be

In other words, this Lemma states that if a Boolean network only has fixed-point attractors as a solution, then, its landscape of attractors can be defined as a function that has the state space as domain, and its image corresponds to the set of fixed-point attractors. Now, if we convert the elements of the state space to integers, we can represent the solution of such a Boolean network on a Cartesian plane. Let’s illustrate this statement with the following example:

Example 1. Consider the following Boolean network:

(E. 1)

Solving the network (E. 1) by brute force, we obtain the results shown in Fig 1A. Note that, assuming that the initial configurations and attractors are binary numbers, we can convert them to integers (Fig 1A). In this way, it is easy to plot the solution of (E. 1) on a Cartesian plane as Fig B shows. Observe that the attractor landscape of (E.1) behaves as a logical function for which, from the data obtained by the brute force method (Fig 1A), we can calculate its disjunctive normal form by adding the min-terms that produce 1 for the three nodes [18], that is:

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Fig 1. Simplification of networks that only have fixed-point attractors.

In panel (A) is shown the network (E. 1) and its synchronous solution calculated with BoolNet [16] and GINSIM [17]. Note that the input and output settings are equivalent to integers. In panel (B) is shown the projection of the landscape of attractors calculated in the last panel. On the other hand, in panel (C) is shown the network (E. 2) with its respective computational solution. Note that both networks have the same landscape of attractors, which implies that both are dynamically equivalent.

https://doi.org/10.1371/journal.pone.0319240.g001

Simplifying these expressions by using Boolean algebra, we obtain:

(E. 2)

Note that the solution of (E. 2) by brute force is identical to the original network (E. 1) (Fig. 1C). This implies that, the network (E. 2) is a simplified version of the network (E.1) and share the same landscape of attractors.

Thus, the main implication of Lemma 1 is that the canonical form of a Boolean network that only has fixed-point attractors can be observed in the Cartesian plane, and its algebraic expression can be found by using classical procedures for computing the canonical form of any logical function, such as the disjunctive normal form.

Dynamical properties of cyclic attractors

The above procedure for finding the canonical form of a Boolean network can only be applied to networks whose solution consists exclusively of fixed-point attractors. This is because cyclic attractors have special characteristics, typical of Boolean relations rather than functions, as we previously demonstrated [13]. Now, we recapitulated this result in the following proposition.

Proposition 2. Let be a Boolean network of n nodes updated synchronously, with as it state-space. If C is a cyclic attractor of F, and if it is converted to integers, then C can be plotted as a relation in the Cartesian plane.

Proof. If C is a cyclic attractor of F, it implies that , , for , and . Since each attractor has a basin of attraction , when every element of B reaches each element of C, which means that the elements of the basin of attraction and the orbit of the cyclic attractor are related as follows:

Redefining the state-space as , occurs that for each element in the domain , when , exists more than one element in the codomain, given by all elements of , which defines a relation in the Cartesian plane

Conceptualizing cyclic attractors as Boolean relations allows us to study this type of attractors with alternative algebraic approaches. In this sense, Bañares and colleagues proposed that, it is possible to study a Boolean relation by decomposing it into compatible functions [19]. The idea is to distribute all the interactions of a relation into a set of functions, as shown in Fig 2A [19]. Taking this proposal as a direct antecedent, we stated the following theorem.

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Fig 2. Dynamical properties of cyclic attractors.

In panel (A) is shown the idea proposed by Bañares et al [21], in which a relation R is divided into compatible functions and . In panel (B) is shown the typical interactions between elements of the orbit of a cyclic attractor C. In panel (C) is shown the definition of the compatible function , and its own composition in order to define and , presented in panel (D). Note that a cyclic attractor can be obtained from a single function .

https://doi.org/10.1371/journal.pone.0319240.g002

Theorem 3. Let be a Boolean relation that describes the orbit of a cyclic attractor C and let . If element is related to element , and element is related to element , and element is related to element (i.e., ) then, exists a compatible function such that, the orbit represented by can be obtained by the union of compositions of the compatible function . In other words:

Proof. Suppose that

note that R fulfils since (Fig 2B). Then, we might propose (Fig 2C), which is a compatible function of R. Observe that and (Fig 2D) are also compatible functions of R, and therefore

The most important implication of this theorem is that a cyclic attractor can be generated by the composition of a single function, which means that a cyclic attractor also has a canonical form.

Matrix-based method to solve synchronous updated Boolean networks

An important hint to find the compatible function that originates a cyclic attractor is that, by composing such a function several times, we might reconstruct the cyclic attractor (Fig 2C). Hence, in this section we will analyze the connection between function composition and the synchronous update scheme. According to its classical definition, the synchronous update scheme starts from any initial condition in the state space , to determine the state of the network at the next time step [11]. To illustrate this concept, observe that, the zero step of the synchronous update is equivalent to the initial starting condition; that is, . On the other hand, to compute the first update of , the Boolean network function must be evaluated with the initial condition ; that is: . For the next update step, it is necessary to take the result obtained at to evaluate ; in other words, . Generalizing this process, we can say that:

Observe that applying the same function twice is the same as composing the same function, i.e., , then we can rewrite the generalization of the synchronous update scheme as follows:

Therefore, the synchronous update scheme is originated by the iterative composition of the original network F. It is important to note that, the composition of two functions is equivalent to multiply the matrices associated with each function, i.e.,. Considering that, in this case, the same function is composed as the time steps advance, we might redefine the synchronous update scheme as:

(E. 3)

Where is the matrix associated with . Remarkably, this alternative definition of the synchronous update scheme is the Chapman-Kolmogorov equation [20]].

Now, in order to find , the Boolean network must be converted into a function that has an associated matrix. In this direction, Wang and colleagues propose to use the logical rules of a Boolean network to create a function that relates each of the elements of the network state space to a corresponding image, to work on it as if it were a function defined in integers [15]. We expanded this idea by using the function defined in integers to calculate . We will illustrate this procedure with the following example.

Example 2: Let be a Boolean network given by:

(E. 4)

If we use all the states of to define a function , we obtain the result shown in Fig 3A. Converting to integers gives us the function shown in Fig 3B. Thus, we might define the matrix as follows:

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Fig 3. Solving Boolean networks with only fixed-point attractors.

In panel (A) is shown the Boolean function originated by the network (E. 4). In panel (B) is shown transformed into integers. Blue lines indicate all trajectories from the domain of to the attractor found in 2. Pink lines indicate the trajectory from the domain of to attractor 3. Black dashed lines indicate other interactions excluded from trajectories originated by the iterative composition of . In panel (C) is shown de Cartesian plane that represents the solution of , while in panel (D) is shown the computational solution of the original network. Note that panels (C) and (D) represent the landscape of attractors of (E. 4).

https://doi.org/10.1371/journal.pone.0319240.g003

Where the columns indicate the outputs and the rows the inputs of the function . In consequence, the matrix expression equivalent to the synchronous method for the network is given by:

(E. 5)

Note that (E. 5) is the temporal transition matrix of jump described by the Chapman-Kolmogorov equation [22]. This interesting observation allowed us to obtain the stationary solution of this system () as a classical Markov chain, i.e.:

(E. 6)

Then,

Interestingly, this matrix corresponds to the landscape of attractors represented by the Cartesian plane shown in Fig 3C, and it is the solution of the network (E. 5) as shown in Fig 3D. Therefore, the synchronous update scheme for Boolean networks is a particular case of Markov chains.

Special properties of the landscape of attractors

The unexpected connection of the landscape of attractors with Markov chains allowed us to extrapolate interesting properties, such as, the existence of a stationary solution and the existence of periodic solutions [21]. The Example 2, showed that, for fixed-point attractors, the image of the landscape of attractors () is simply the stationary solution of , in other words:

(E. 7)

Concerning to periodic solutions of , we noted that can be obtained by adding the matrix calculated for each time-step contained in the period of a cyclic attractor. Let’s see this point in the following example.

Example 3: Let a Boolean network given by:

(E. 8)

Using the logic rules of (E. 8) to generate a Boolean function (Fig 4A), and then transforming it into integers (Fig 4B) we obtain . Note that this function fulfils Theorem 3, since , which implies that it generates a cyclic attractor . Calculating the matrix expression of the synchronous method for the network , we have:

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Fig 4. Solving Boolean networks with periodic behavior.

In panel (A) is shown the function originated by the Boolean network (E. 8). In panel (B) is shown transformed into integers. Blue arrows indicate all trajectories from the domain of to the attractor found in 0. Pink arrows indicate the trajectory from the domain of to the cyclic attractor {1, 2}. Yellow arrows indicate the trajectory from the domain of to the fixed-point attractor 3. In panel (C) is shown de Cartesian plane that represents the landscape of attractor of , and in panel (D) is shown the computational solution of the original network.

https://doi.org/10.1371/journal.pone.0319240.g004

(E. 9)

Evaluating at , we obtain:

Note that , and consequently, to calculate (Fig 4C) we need to add all periodic images that correspond to the cyclic attractor, such as:

Observe that, such a matrix corresponds exactly to the computer-calculated landscape of attractors presented in Fig 4D. Therefore, the logical addition of the matrices for the time steps belonging to the period () of a cyclic attractor is equal to the image of the landscape of attractors seen in the Cartesian plane ), i.e.,:

(E. 10)

Fixed-point attractors and its basins of attraction are easy to characterize

In Example 3 it can be observed that the function allows to identify at a glance the fixed-point attractors, since they only interact with themselves and in consequence, they are reflexive elements, i.e., elements related with themselves. Formally, we shall say:

Theorem 4: Let be a synchronously updated Boolean network of n nodes and as it states space. If is converted into and it has reflexive elements, then these elements are fixed-point attractors of .

Proof: If has reflexive elements, it implies that , such that . Considering that, the synchronous update scheme can be calculated by making composition of as a function of the time-steps, we might see that , and and so on for any time-step (Figs 5A and 5B). This implies that and always arrives to their own values for any time-step, which is the definition of a fixed-point attractor

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Fig 5. Finding fixed-point attractors and its basins of attraction.

In panel (A) is defined and the trajectories originated from the iterative composition of this function. Blue lines represent the trajectories to and black lines represent all trajectories to . In panel (B) is presented the formal definition of and . Note that both composed functions are the same as the original, which proves that and remain fixed despite the iterative composition of . In panel (C) is defined , where purple lines represent all trajectories from the domain of to the attractor . Black dashed lines represent all interactions that are excluded from these trajectories. In panel (D) is presented , and note that and eventually converge to , which illustrates the concept of basin of attraction.

https://doi.org/10.1371/journal.pone.0319240.g005

Concerning to the basins of attraction, these can be identified using the approach described above to obtain the orbit of a cyclic attractor. This principle is stated in the following proposition.

Proposition 5: Let a synchronously updated Boolean network of n nodes redefined as a function of integers, and let . If the element is an attractor and , then and belongs to the basin of attraction of .

Proof: If is an attractor, it implies that . Suppose that we defined the function (Fig 5C). Note that (Fig 5D), which implies that, while , and converge to the attractor . Thus, and belongs to the basin of attraction of by definition

An important implication of this theory is that the basin of attraction, and the attractors in general, can be identified by a systematic search in a function defined on integers. Remarkably, this idea also applies to cyclic attractors.

Canonical form of synchronous updated networks

Now, we have all elements required to state a method for computing the canonical form of any synchronously updated Boolean network. Our proposal is based on the fact that, by the definition presented in this paper, the synchronous update scheme uses a single function to generate the entire dynamics of the network. Thus, if we evaluate when is large, i.e., . Then, we might use this function to compute the canonical form with any of the valid approaches for logical functions. Importantly, if the network converted into a function initially has a cycle, the image of as must still contain the same cycle to calculate its canonical form. Let’s illustrate this procedure with the following example.

Example 4: Let be a Boolean network given by:

(E. 11)

After using the network (E. 11) to create a Boolean function (Fig 6A), and then transforming it into integers (Fig 6B), the expression , we obtain:

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Fig 6. Simplification of networks that only have cyclic attractors.

In panel (A) is shown the result of using the network (E. 11) to generate a Boolean function . In panel (B) is shown the conversion of to integers, where pink lines indicate all trajectories that converge to cyclic attractor {2, 3}. In panel (C) is presented the network (E. 11) together with its computational solution. In panel (D) is presented the network (E. 12) together with its computational solution. Note that both networks have an identical solution, which proves that (E. 12) is a simplified version of (E. 11).

https://doi.org/10.1371/journal.pone.0319240.g006

Note that does not have any reflexive element, and by Theorem 4, this network does not have fixed-point attractors. On the other hand, by Theorem 3, the network contains a cycle given by , since (Fig 6B). Now, to approximate when , we could evaluate , then:

Reconfiguring the Boolean function equivalent to , we obtain:

Note that, the cycle is still present, since . Then, calculating the disjunctive normal form for (E.11), we have:

Simplifying this canonical form, we obtain:

(E. 12)

Which is the simplified version of the network (E. 11), as can be corroborated by comparing their computer-calculated image of the landscape of attractors (Figs 6C and 6D).

Simplified method for calculating the canonical form of synchronous updated networks

The method described in the previous section can be complicated for large networks, since it requires matrices. Considering this point, we now propose an alternative method for large networks, in which we used the above theorems to solve synchronously updated networks and to find their canonical form. Regarding to solving Boolean networks synchronously, it must be first created an integer-defined function equivalent to a Boolean network. Then, by performing a search considering Theorems 3 and 4, the cyclic and fixed-point attractors can be detected. Finally, applying Proposition 5, one must connect to all the elements that converge to a cyclic or fixed-point attractor, that is, the elements of its basin of attraction. To find the canonical form of a Boolean network, once its solution is found, the trajectories within the basins of attraction must be compacted so that all its elements converge on the attractor, in order to apply any procedure for obtaining canonical form for Boolean functions. Let’s illustrate the above procedure with the following example.

Example 5: Consider the following Boolean network:

(E. 13)

Which computational solution is shown in Fig 7A. Converting the network (E. 13) into a function (Fig 7B), we obtain:

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Fig 7. Simplification of Boolean networks in general.

In panel (A) is presented the Boolean network (E. 13) with its computational solution calculated with BoolNet [16] and GINSIM [17]. In panel (B) is presented the Boolean function , obtained by using the logic rules of (E. 13). In panel (C) is shown the transformation of to integers. In panel (D) is presented the function with all their interactions sent directly to the attractor to which they naturally converge, and in panel (E) is shown the conversion of this function to binary numbers. Finally, in panel (F) is shown the network generated from , which corresponds to the network (E. 14). Note that (E. 14) is a simplified version of (E. 13).

https://doi.org/10.1371/journal.pone.0319240.g007

Observe that is graphically represented in Fig 7C. Applying Theorems 3 and 4 we found a cyclic attractor in , and a fixed-point attractor in , since and . Now, as , this implies that the basin of attraction of is composed by . Similarly, note that the basin of attraction of 7 is defined by . To calculate the canonical form of (E.13), we just need to redefine the function by compacting the interactions that converge on the attractors as follows:

This function is graphically represented in Fig 7D. Next, we rewrote this function in binary numbers (Fig 7E) to apply the formalism of the disjunctive normal form calculation. As a result of the above, we obtain:

Simplifying this form, we obtain:

(E. 14)

Note that, the computational solution of both networks is identical (Figs 7A and 7F). Therefore, solving Boolean networks using the synchronous update scheme and finding their canonical form can be summarized in a classical search algorithm.

Matrix formalism of asynchronous update scheme

Since its conceptualization by René Thomas to study biological systems [22,23], the asynchronous update scheme has generated multiple studies to describe and characterize its mathematical properties. Some interesting properties of this update scheme include the fact that negative feedback loops can generate homeostasis with or without periodicity, while positive feedback loops generate multiple alternative stationary states [22–24]. Later studies characterized new properties of the asynchronous update scheme, such as that some cyclic attractors could not be recovered using this update scheme, or that some elements of the basins of attraction changed depending on the update order of the nodes [25]. These studies also showed that fixed-point attractors are invariant to the update scheme used to analyze Boolean networks [25]. Motivated to understand the origin of these disjoint properties and to expand the analytical focus of studying the asynchronous update scheme, we decided to apply our matrix approach to study it.

Interestingly, we found that the asynchronous update scheme is capable of generating a set of functions that iterate based on the gene update order. Consequently, it is not possible to obtain a unique canonical form for this update scheme, since its matrix definition changes depending on the gene update order. Let’s illustrate this point with the following example:

Example 6: Consider the Boolean network (E. 8) presented in the Example 3. There, we obtained as the function associated with (E. 8) (Fig 8A), and its computational solution is presented in Fig 8B. Nevertheless, if the node is updated every two time-steps, and the node is updated every three time-steps, we will notice that several functions can be generated when we evaluate the system from to (Fig 8B). Individually, such functions are given by:

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Fig 8. Exploration of asynchronous update scheme.

Panel (A) shows the synchronous update sequence of the (E. 8) network, and panel (B) shows the computationally obtained attractor landscape of (E. 8). Blue and pink arrows indicate the trajectory from the domain of to the fixed-point attractors 0 and 3, respectively. Purple arrows indicate the trajectory from the domain of to the cyclic attractor {1, 2}. Panel (C) shows the update sequence of the (E. 8) network if the update time for is two time-steps, and for it is three-time steps. The dashed arrows show the interactions that are not part of the composition of functions that underlies the trajectory from the domain of . Panel (D) shows the landscape of attractors calculated for the asynchronous updated described for panel (C). Panel (E) shows the result of inverting the asynchronous update order and panel (F) shows its landscape of attractors. Note that the update order generates functions that do not exist in the synchronous scheme, decouples the cyclic attractors, and alters the basins of attraction of the fixed-point attractors.

https://doi.org/10.1371/journal.pone.0319240.g008

Note that, by for the first three time-steps, the system image converges to fixed-point attractors (Fig 8C), fact that explains the change observed on its computational solution presented in Fig 8D. Interestingly, if we change the update order (Fig 8E), we will notice that, functions , , , and reappear but in a different order than that observed with the first asynchronous update scheme, producing notable differences in its landscape of attractors (Fig 8F).

Interestingly, by , the image of the system composition has already converged on fixed-point attractors in both cases (Figures 8C and 8E). Note that, for this example of asynchronous update, the cyclic attractor reported for this network has disappeared, and the size and elements of the basins of attraction have varied without altering the presence of the fixed-point attractors (Figures 8D and 8F). Finally, considering that the solution of both update schemes only consists in fixed-point attractors, we shall calculate the disjunctive normal form for the landscape of attractors recovered in Fig 8D:

(E. 15)

We also performed the same calculation for the landscape of attractors presented in Fig 8F:

(E. 16)

Simplifying (E. 15), we obtain:

(E. 17)

Doing the same for (E. 16), we obtain:

(E. 18)

Observe that none of these two simplified systems corresponds to the original form of the network (Fig 8B). Thus, it is not possible to obtain a unique canonical form for this update scheme, because asynchronous update scheme composes several functions using different orders, and the composition of functions is not commutative.

Proof of concept on real biological networks

Now, we tested the effectiveness of our theorems to deal with real biological networks. To this end, we selected four previously published networks focused on studying different aspects of the functioning of the immune system. The first network was a 12-node network that models the differentiation of CD4 + T cells in response to microenvironments of different pro- and anti-inflammatory cytokines [8]. The second network was an 18-node network that represents the differentiation of CD8 + T cells in response to metabolic changes and cytokine combinations [7]. The third network was a 20-node network that models the differentiation of hematopoietic stem cells (HSCs) to give rise to the myeloid, lymphoid, and erythroid lineages [26]. The fourth selected network was a 15-node network that models the polarization of macrophages in response to cytokines present in the extracellular environment [27]. Finally, we selected a 23-node network that models the epithelial-to-mesenchymal transition in human hepatocytes proper of hepatocellular carcinoma [9].

These networks were implemented in BoolNet [16] and GINSIM [17] using the synchronous update scheme (Supporting information). On the other hand, to implement Theorem 4 to calculate the fixed-point attractors, a list of all the initial conditions () was first created for each of the networks, that is, lists with elements, where is the number of nodes in each network. Next, all the networks were evaluated with each of their initial conditions to create a new list of output elements (). Finally, we searched in both lists elements that met the condition , since Theorem 4 mentions that this class of elements are by definition the fixed-point attractors of a Boolean network. This simple algorithm was implemented with the C# programming language in Visual Studio 2022 (Supporting information). As a result of these two procedures, we find that the implementation of Theorem 4 is able to find all the fixed-point attractors of the network, just like the BoolNet and GINSIM software (Fig 9), which demonstrates the effectiveness of the theory developed in this work.

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Fig 9. Comparative results of the search for fixed-point attractors.

In this figure we show the result of using BoolNet [16] and GINSIM [17] versus the computational implementation in C# of Theorem 4 to search for fixed point attractors in real biological networks. Panel (A) presents the attractors obtained from a 12-node network representing CD4 + T cell differentiation, obtained from [8]. Panel (B) presents the attractors obtained from an 18-node network representing CD8 + T cell differentiation, obtained from [7]. Panel (C) presents the attractors of a 20-node network representing hematopoietic stem cell differentiation, obtained from [26]. Panel (D) shows the attractors of a 15-node network representing macrophage polarization, obtained from [27]. Finally, panel (E) shows the attractors of a 23-node network that models the development of hepatocellular carcinoma through epithelial-to-mesenchymal transition [9]. In all cases, the attractors obtained were identical, which validates our theoretical approach. Note: The only network for which GINSIM could not be used was the epithelial-mesenchymal transition network due to the size of the code used in this network.

https://doi.org/10.1371/journal.pone.0319240.g009

Solving large scale networks using the canonical form

Analytical approaches to working with large networks have been developed for years [28,29]. The central idea of these approaches is to divide a large network into connected modules, to find the semi-attractors of each module, and join them to obtain the global attractors [28,29]. However, this interesting methodology does not consider how to recover the basins of attraction. As we demonstrated in previous sections, the canonical form of a Boolean network contains the relevant information to fully recover the attractor landscape of the network, avoiding unnecessary interactions. For this reason, we decided to explore whether the canonical form of Boolean networks can improve the aforementioned methodology to solve large scale networks by recovering the basins of attraction. To explain our proposal, we will use the following nine-node Boolean network:

(E. 19)

If we divide this 9-node network into 3-node modules, we can rewrite them as:

Next, by using Theorem 4 on each module, we found the semi-attractors and for each module. Then, we used the procedure performed in Example 5 to calculate the canonical form of each module. As a result of this, we obtained:

Note that for module there are only initial configurations, and according to their canonical form, the attractors and depend exclusively on the value of . Therefore, the 4 configurations with will go to the attractor and the other 4 with will go to . For modules and , it is exactly the same situation, but now the dependence is on the value of instead of . Joining the semi-attractors for each module, we obtain the configurations:

Using the full network (E. 19) with Theorem 4 to determine which of such configurations are global attractors, we found the following true attractors:

Regarding the basins of attraction of each attractor, the canonical form of each module indicates that all configurations with and will reach the attractor , and there will be a total of configurations. Likewise, all configurations with and , which will be 128, will converge towards the attractor . While 128 configurations with and will reach the attractor . Similarly, the 128 configurations with and will necessarily reach the attractor . These results were corroborated using Visual Studio Community 2022 (see Supplementary Material).

Finally, we apply this methodology to study the dynamics of a well-known 200-node network. Since it is difficult to calculate the basins of attraction of a 200-node network, given that its state space has initial configurations, we decided to expand the network (E. 19) to 200 nodes, given that this network has known dynamics. Specifically, we transformed the network (E. 19) as follows:

(E. 20)

We apply the methodology described in this section, but now we delimit 10 modules of 20 nodes each. After finding the semi-attractors for all modules, we proceeded to calculate the canonical form of each module. Subsequently, we joined the semi-attractors and found 2048 candidates for global attractors. As in the case of the network (E. 19), for the module the value of and determined that the semi-attractors , , , were reached. For the rest of the modules, only the attractors and were found, which depended entirely on the value of . Finally, we used the network (E. 20) with Theorem 4 to identify the true attractors, which were equivalent to the 4 attractors of the network (E. 19). In this case, the size of the basins of attraction of each of these attractors was configurations, and the reachability conditions also depended on the values of and . This experiment with a toy network demonstrates that the theory developed in this work can be used to calculate the reachability conditions of large-scale networks.

Glossary

Attractor Landscape: This concept refers to a set of attractors proper of a dynamical system. In this paper, we used this term to refer to the set of fixed-point and cyclic attractors of a Boolean network.

Basin of Attraction: This concept refers to the set of all initial states that converges toward a specific attractor in a Boolean network.

Canonical Form: In mathematics this concept refers to a standardized or normalized representation of an equation, which allows for easier comparison and analysis.

Cyclic attractor: This concept refers to a set of states in a dynamic system through which the system cycles indefinitely in a period.

Disjunctive Normal Form (DNF): This is a standard way of writing a Boolean expression as a sum of min-terms.

Fixed-point attractor: This concept refers to a stable state of a dynamic system where the system remains once it is reached. Formally: , where in a fixed-point attractor.

Function: This is a special type of relation that assigns exactly one output to each input from a given domain (). Formally: , where is the codomain.

Function Composition: This concept refers to the process of applying one function to the result of another. Formally: , where and are functions.

Min-terms: This concept refers to the simplest form of expression in Boolean algebra, which represents a unique combination of logic variables that yields a value of true, or numerically 1.

Reflexivity: This concept refers to a property of a binary relation in which every element is related to itself. Formally: .

Relation: This term refers to a logical or mathematical connection between elements of two or more sets, often used to define functions, mappings, or other associations.