Synchronous regulatory networks in biology are formed when a collection of elements, such as molecules, neurons, or individuals, interact with one another to determine a dynamic system in discrete time steps. The overall behavior of a synchronous network is governed by a specific transition function. Detailed experimental investigations into the workings of synchronous regulatory networks can be enhanced by mathematical models that replicate the network's behavior, aiding in the development of new hypotheses and experiments. In this article, we present a quantitative method based on Boolean algebra to replicate behaviors and identify missing elements within the network.

Regulatory networks are typically studied by illustrating interactions among elements in the network using an excitation-inhibition graph [1–9]. For example, a and b represent two interacting elements, each having a binary Boolean behavior. We may now suppose that activation of a turns b on. Such activation, or excitation, can be described as a → b. If by contrast, a inhibits b, the expression will be a ⊣ b. That simple description can now be enriched in Boolean terms by using 1 to indicate an active state and 0 to indicate an inactive or inhibited state. However, this notation alone is insufficient to explain more sophisticated interactions. For example, element a may be activated when element b is inhibited. Alternatively, element a may remain unchanged upon activation of element b. Such naturally occurring interactions cannot be described solely by the excitatory and inhibitory connections. An alternative approach to describing networks and circuits in biology is to use the logical electrical diagrams commonly employed in engineering. However, these diagrams act as black boxes, where the precise interactions between the elements remain concealed from sight [5].

To overcome these limitations, we propose a digraph (directed graph) that we will refer to as a logical digraph of the biological system. The logical digraph is constructed using eight logical connectives and their combinations, representing every possible interaction between any two elements. When combined with appropriate Boolean functions, the digraph accurately defines the dynamics of the biological regulatory system. In addition to being algorithmic, our method reports on the system's attractors and limit cycles, which correspond to the Perron-Frobenius eigenvectors of the state matrix that describes the transition from one state to another over time. Therefore, the state matrix contains information about the dynamics of the biological network [10]. As new concepts arise throughout the paper, they are applied to the simple neural circuit that controls swimming in the mollusk Tritonia to exemplify the use of the logical digraph.

A regulatory biological system and its dynamics can be described by the quartet of symbols (G, S, F, η). The symbol G ={g₁,…,g_n} represents a finite assembly of n elements (genes, neurons, cells, or nodes), each of which can acquire either the Boolean active state (on, true) with a value of 1, or the inactive state (off, false) with a value of 0. We will denote the set of Boolean values as Z₂ = {0,1}, along with their usual Boolean operations: addition (⊕) and multiplication (⊗), which are described in Table 1. The glossary contains definitions of the terms.

In the set ℤ2n = {(a₁, …, a_n): a_i ∈ ℤ₂ for each j=1,…,n} the j-th coordinate corresponds to the element g_j of G. This set contains all the possible states (0,1) of the elements of G, which we will refer to as state-vectors. All the state-vectors of biological system are contained in S, which is a subset of ℤ2n.

The symbol η represents the regulatory network as a digraph whose vertices are the state vectors (the elements of S). η informs about the time-evolution of the regulatory cycle. In the expression (a₁, …, a_n) → (b₁, …, b_n), the arrow means that the state (a₁, …, a_n) changes to the state (b₁, …, b_n) in a time unit. These changes are described by the transfer function F= (f₁,…, f_n) of the system, and can be defined by n Boolean functions f_j: ℤ2n → ℤ2 that describe how the elements of G act on element g_j. In other words, the transfer function F goes from ℤ2n  to ℤ2n and the arrow (a₁, …, a_n) → (b₁, …, b_n) in η means the transfer function (b₁, …, b_n)= F(a₁, …, a_n).

We can now proceed by employing logical connectives that indicate excitation and inhibition. To begin, we will again use the excitation connective to clarify the essentials of utilizing connectives:

1. Excitation. An active a (a = 1) induces the transition of b from inactive (b = 0) to active (b = 1). The binary value 1 represents excitation from a to b, and is represented as a → b. However, the excitation value depends on the initial states of either component. As shown in Table 2, connective a → b, if a is off (a = 0), b will remain inactive (b = 0). Note, however, that if b is initially active, the activation of a also provides a 1 value. In contrast, the initial value of b remains unchanged when a = 0.

The complexity of biological interactions extends well beyond mere excitation or inhibition. The various ways in which a may affect b depend on the current state of each interactive element. Furthermore, one must consider global activity even when a is inactive, a fact not always considered in this type of analysis. For this reason, six additional connectives must be included to represent the dynamics of real biological regulation networks accurately. Although some have not been biologically described, we consider all the possible interactions between connectives. Since regulatory networks commonly have undefined components, such connectives may serve to hypothesize identities and interactions.

3. Negation of excitation . A first case occurs if activation of a inhibits an initially active b, that is (1 1) = 0. In a second case, b remains inactive during the activity of a, (1 0) = 0. Both cases resemble the inhibition seen above. However, here the inactivity of a produces inactivation of b, (0 b) = 0. Alternatively, the initial inactivity of both a and b produces activation of b, (0 0)= 1. Note that this connective differs from inhibition. As an example, the identity of gen Sp8 in Giacomantonio and Goodhill [11], p. 3, can be described as [Emx2 Fgf8] Sp8 .

4. Negation of inhibition . Unlike the previous case, any difference in the initial activity between a and b leads to a change in the b value. However, inactivity in both a and b, keeps b inactive.

5. Implication . The activity in both a and b, keeps b active. In contrast, the inactivity of a produces spontaneous activation of b; in the other cases, b will remain inactive.

6. Negation of implication . Unlike the implication, the only case where b is activated is when a is active and b is inactive. As an example, in Giacomantonio and Goodhill [11] F(t) ==> E(t), which is equivalent to F(t) & NOT E(t).

7. Disjunction . The activity of b requires the simultaneous activity of a and b. This connective, also read as a AND b, is the most abundant in literature. For example, [(NOT Emx2) Fgf8] Sp8 indicates that the value of Sp8 is the disjunction of (NOT Emx2) and Fgf8 ([11], p. 3).

8. Negation of disjunction . b becomes inactive when both a and b are simultaneously active.

So far, the interactions between the two elements have been characterized by a single symbol. The following eight complex interactions will be defined by a combination of two or more logical connectives connected by the logical connective “AND” (or disjunction).

9. Double implication or If and only if. It is represented as a b AND a b or simply as a b. Means that b will be active if a and b have the same value (see 3.1).

10. Negation of the double implication . Its representation is a b AND a b. This means that alternating the values of a and b will activate b.

11. “a” as an action from a to b (ab). Means that regardless of its value, b takes the value of a. See also Section 3.1.

12. “NOT a” (as an action from a to b) or Negation of a . Regardless of the value of a, b takes the opposite, as shown in the logical diagram of the regulatory system of Tritonia (Figure 3).

13. “b” (as an action from a to b) (a b). That is, the value of b remains invariant. This connective may be useful to test whether an element does not influence a biological system or if any other connective is needed. The appearance of this connective suggests that removing a might be possible because it is redundant. In other case, when we know that a interacts with b, the appearance of this connective indicates the presence of another element acting from a to b. See also Section 1.1.

14. “NOT b” (as an action from a to b) (a b). Means that b takes on a value that is opposite to its original value, independently of the value of a. For example, b b means autoinhibition of b.

15. Tautology (as an action from a to b) (a ), b will always be activated. As will be seen in the example below, this connective is used for autoexcitation of b as an alternative to the filled circles used in Cessac and Samuelides ([12], Figure 63).

16. Contradiction (as an action from a to b) (a b). Means that b will always be inactivated. One use of this connective is the autoinhibition of b [see also Cessac and Samuelides [12], Figure 63].

Table 2 summarizes the connectives, their logical propositions, Boolean functions, and values. The strategy now is to define the logical proposition for each Boolean function in n variables, along with the corresponding logical digraph and vice versa.

To represent the regulatory network in Boolean terms, we need to convert logical propositions into Boolean functions. This connection can be established if a bidirectional equivalence, also known as tautology, exists, which means that the conversion holds true under every possible interpretation. For example, the two logical propositions P and Q are equivalent if the logical proposition “P if and only if Q” is a tautology. Table 2 shows that the truth table of the connective excitation, a b, is identical to the logical proposition OR, indicating that their Boolean propositions are also the same. Therefore, a tautology exists, allowing us to substitute one for the other.

Table 3 shows a bijection between the set of logical propositions in n variables and the set of Boolean functions from ℤ2n to ℤ2, for each natural number n ≥ 1.

We will now return to the synchronous biological regulatory system (G,S,F,η) and will use η to build the transfer function F (alternatively, we can use F to build η). The Supplementary material contains a computer routine that calculates the function F from a predefined regulatory network η.

The logical digraph of a biological regulatory system (G,S,F,η) has one vertex for each of the n elements of G. Such vertices will be denoted as g₁, …, g_n (G={g1,…,gn}), and the directed connectives joining the vertex correspond to the logical connectives in Table 2. S is the set of state-vectors and F = (f₁, …, f_n) is the set of Boolean functions that define the regulatory network η. The glossary at the end defines each term in the text.

The well-known and relatively simple neural network controlling swimming of the mollusk Tritonia offers numerous advantages for testing mathematical network theory applied to animal behavior [13]. Like in other invertebrates, swimming in Tritonia is produced by a sequence of sigmoidal body waves occurring during alternate cycles of contraction of the ventral and dorsal muscles. The beauty of such neuronal circuitry lies in the combination of its simplicity and the reproducibility of the emerging behavior. Studies conducted by Getting, Katz, and others [1, 13–17] have shown that swimming is produced by a central pattern generator integrated by four types of central neurons that establish a stereotyped connectivity.

Motoneurons transmit output information that alternates the contraction of dorsal and ventral muscles. In brief, activation of the Dorsal Interneuron (DI) generates bursts of action potentials that activate the dorsal motoneurons through excitatory connections. Swimming begins in the upward direction. Simultaneously the DI neuron activates the cerebral type 2 (C2) neuron. The C2 neuron responds with a stream of impulses that produce a delayed excitation of the ventral interneurons (VI). Two types of VI neurons cooperate to the pattern; however, for this study, it suffices to condense both into a single representative neuron (see also Tamvacakis et al. [18]). The VI interneuron connects to the ventral motoneurons, which cause the ventral contraction of the animal [19].

As of now, only direct excitatory and inhibitory connections have been identified. However, the timing of the cyclic activation of the CPG necessitates additional connections. Reciprocal inhibitory connections between the DI and VI motoneurons are crucial. The DI and VI interneurons communicate with one another through reciprocal inhibitory connections (Figure 1). The activity of DI neurons suppresses VI neurons, clearing the circuit to produce a single dorsal output while prolonging the excitation period of the dorsal motoneurons. Moreover, the excitation of C2 neurons, followed by the excitation of VI neurons, inhibits DI and C2 interneurons, allowing a singular ventral output from the circuit. A third type of connection that contributes to the duration of the burst of impulses by the DI and VI motoneurons is excitatory autapses—specifically, synaptic connections that neurons form onto themselves. Activation of autapses during bursts of action potentials extends the excitation of the DI or VI neurons, thereby lengthening the duration of their firing and creating inhibition of the antagonistic neuron. The network in Figure 1A is a basic description that utilizes only the excitation and inhibition connectives to describe the interactions between the three neuron types.

Based on the information above, the dynamics of the neural network involved in Tritonia swimming, with three types of neurons firing in an orderly manner, can be illustrated as the sequence of steps in Figure 2, where the contribution of neurons follows the order (ID, IV, C2).

We are now ready to build the Boolean function F: ℤ23→ℤ23, which in this case is defined as F = (f_DI, f_VI, f_C2), and its logical digraph. We must remember that F^t represents the state of the network at time t, therefore for a sufficiently large t (t ≥ 8, in this example), the biological system will transit around the entire dicycle [20–23].

We will start by defining:

which is the set of state vectors representing the different sequential working states.

Recall that η is obtained by applying the transfer function F to each vector of ℤ2n, and that there is a unique arrow from v to F(v), for every v∈ℤ2n. Therefore,

F(0, 0, 0) = (f_DI(0, 0, 0), f_VI(0, 0, 0), f_C2(0, 0, 0)) = (1, 0, 0), F(1, 0, 0) = (1, 0, 1), F(1, 0, 1) = (0, 0, 1), F(0, 0, 1) = (0, 1, 1), F(0, 1, 1) = (0, 1, 0), F(0, 1, 0) = (1, 1, 0) and F(1, 1, 0) = (1, 0, 0).

where,

f_DI(0, 0, 0) =1,

f_VI(0, 0, 0) = 0,

f_C2(0, 0, 0) = 0.

The same procedure must be repeated for every state vector of the system.

To build explicitly the Boolean function f_DI: ℤ23→ℤ2 we have to find all the state-vectors of S that under f_DI are going to give 1. They are (0,0,0), (1,0,0), (0,1,0) and (1,1,0).

By using bijection, we have:

And by utilizing the known equalities

a ⊗ a = a

a ⊕ a = 0

a⊕(a⊕1) = 1

a⊗(a⊕1) = 0,

we get that f_DI(a_DI, a_VI, a_C2) = (1⊕ a_C2) ⊗1. The constant function 1 equals the tautology that represents the excitatory autapse in Figure 1.

To build f_VI, we repeat the procedure. The state vectors with a 1 value under f_VI are (0,0,1), (0,1,1) and (0,1,0). Then,

Likewise,

Therefore,

Again, let's take f_DI(a_DI, a_VI, a_C2) = (1⊕ a_C2) ⊕1 to find the logical connectives in DI. Table 2 shows that the logical proposition corresponding to 1⊕ a_C2 is “NOT(C2)”, which corresponds to:

Let's now repeat the analysis for VI:

f_VI(a_DI, a_VI, a_C2) =(a_DI ⊕ 1) ⊗ [(a_VI ⊗ a_C2) ⊕ a_VI ⊕ a_C2]

with the corresponding logical proposition “NOT(DI) AND (VI OR C2)”. From Table 2, we obtain the logical digraph in VI:

Finally, we obtain C2:

f_C2(a_DI, a_VI, a_C2) =[(a_DI ⊗ a_C2) ⊕ a_DI ⊕ a_C2] ⊗ (a_VI ⊕ 1),

whose logical proposition is “(DI OR C2) AND NOT(VI)”, and its logical digraph is:

The assembly of all the components above illustrates the logical diagram of the neural regulatory network involved in Tritonia swimming, as shown in Figure 3.

Note that a benefit of the composition of connectives in the logical digraph, is the possibility to analyze subcompartments of the network. For example, the interactions from a to b and from b to c provide the interaction from a to c. Therefore, D1 C2 and are equivalent representations of DI to C2.

Another observation concerns the equivalence of logical propositions in Section 1.1. The effect of autapses, shown as tautologies in Figure 3, enhances the firing of VI and DI neurons. The functioning of the regulatory network could be simplified by removing the tautologies that represent these connections. However, we recommend including all components that contribute to a more accurate representation of the circuit dynamics.

The digraph in Figure 3 contrasts with the excitation-inhibition graph in Figure 1A, which not replicate the functional network.

In the first part of our paper, we went from the transfer function F: S→S of a biological system to its logical digraph. In this section, we will discuss the tools for studying the dynamics of a regulatory network of a biological system.

1. We introduce a method for detecting and illustrating various interactions among the components that control the operation of regulated biological networks.

2. A logical digraph of the regulatory biological network is built using eight logical connectives from Boolean algebra, which together accurately depict all possible interactions among the network's elements.

3. Rules are provided to convert the components of the network from logical to Boolean representation.

4. The transfer function, its orbits, and the state matrix of the network enable us to identify the topology of the network, including its limit cycles, attractors, and dynamics.

5. The spectrum of the state matrix determines the Perron-Frobenius eigenvalues, which predict the time evolution of the network.

6. An algorithm plus three software routines are provided to find the attractors and limit cycles of biological regulatory systems, to build their state matrix and to find the eigenvectors in systems with a large number of elements.