Boolean analysis of lateral inhibition

We study Boolean networks which are simple spatial models of the highly conserved Delta–Notch system. The models assume the inhibition of Delta in each cell by Notch in the same cell, and the activation of Notch in presence of Delta in surrounding cells. We consider fully asynchronous dynamics over undirected graphs representing the neighbour relation between cells. In this framework, one can show that all attractors are fixed points for the system, independently of the neighbour relation, for instance by using known properties of simplified versions of the models, where only one species per cell is defined. The fixed points correspond to the so-called fine-grained “patterns” that emerge in discrete and continuous modelling of lateral inhibition. We study the reachability of fixed points, giving a characterisation of the trap spaces and the basins of attraction for both the full and the simplified models. In addition, we use a characterisation of the trap spaces to investigate the robustness of patterns to perturbations. The results of this qualitative analysis can complement and guide simulation-based approaches, and serve as a basis for the investigation of more complex mechanisms.


Introduction
Lateral inhibition is a signalling mechanism that can induce the differentiation of cells in developing tissues Sternberg [1993], Collier et al. [1996]. Transmembrane receptors of the Notch family, and the product of the Delta gene acting as ligand, have been identified as possible actors in this spatial differentiation phenomenon. In its simplest form, lateral signalling causes cells to experience two different types of fate, a primary and a secondary fate, corresponding to low and high levels of Notch. The stimulation of Notch by the ligand Delta from adjacent cells induces the cell to assume the secondary fate; high Notch activity, on its part, causes inhibition of Delta, which promotes the lateral differentiation to the primary fate. The result of this feedback is the emergence of spatial patterns of cells of primary and secondary type.
Several mathematical models have been proposed for the investigation of the Delta-Notch pattern-generating mechanism (e.g., Collier et al. [1996], Webb and Owen [2004], Gössler [2011]). In Collier et al. [1996], the authors choose a spatially-discretised model, with dynamics described by systems of differential equations. Their analysis highlights in particular that, when the feedback between cells is strong enough, patters of alternating high and low levels of Notch emerge, that do not depend on specific forms for the regulations of species production, and on the parameters. It is therefore natural to investigate whether the basic principles underlying the Delta-Notch system can be identified also in a purely qualitative, Boolean framework. Discrete models can often capture "rules" that govern properties of larger classes of systems (see for instance Thomas and d'Ari [1990], Thomas and Kaufman [2001], Albert and Othmer [2003]). In this work we consider the simplest Boolean model possible, where only two variables, representing Notch and Delta, are defined in each cell. The level of Delta in a cell is uniquely determined by the level of Notch in the same cell, whereas multiple formulations for the dependence of Notch on the levels of Delta in neighbour cells can be considered. In this work we focus on the assumption that the presence of one neighbour cell with high level of Delta is sufficient for the activation of Notch. The model we consider has already been analysed with computational approaches for some specific network geometries Mendes et al. [2013], Varela et al. [2018a]. Here we investigate properties that hold independently of the neighbour structure of the cells.
We first focus on characterising the fixed points, and show that all attractors for the asynchronous dynamics are fixed points (Section 3). These stable configurations or patterns that emerge from the simple spatial interaction structure we consider exhibit the same alternation of cells with low and high Notch level observed in the ODE models of Collier et al. [1996]. The alternation requires each cell with low Notch to be surrounded by cells with high Notch, and all cells with high Notch to have at least one neighbour with high Delta. In other words, the Delta-Notch patterns are defined by the minimal vertex covers, or maximal independent vertex sets, of the graph describing the neighbour relations (Theorem 3.1). We ask which patterns can be reached under fully asynchronous dynamics from homogeneous initial conditions, and show that all of them can be obtained (Theorem 4.2). We then provide a characterisation of the trap spaces (Theorem 4.6) of the systems, that is, subspaces that the dynamics can not leave. Determining the trap spaces allows us to study how patterns respond to perturbations. In particular, we show that changes in single cell levels can not propagate beyond cells at distance two (Section 4.3). The spatial interaction structure consisting of internal inhibition and neighbour activation can be thought as a core model for lateral inhibition, and it is not straightforward to determine which of the simple properties we present here are preserved in larger or more complex models. We discuss additional open questions in Section 5.

Background
In this section we set some notations and give some basic definitions. We write B for the set {0, 1}. For a ∈ B, we writeā for 1 − a, and given n ∈ N, I ⊆ {1, . . . , n} and x ∈ B n , we denote byx I the element withx I i = 1 − x i for i ∈ I, andx I i = x i otherwise. If I consists of only one element i, then we writex i forx I , and if I = {1, . . . , n}, we writex forx I . In the examples, we will simplify the notation and denote elements of B n as sequences of 0s and 1s (e.g, we will write 100011 for (1, 0, 0, 0, 1, 1)). We will also write 0 and 1 for the elements of B n with all components equal to 0 or 1 respectively.
A Boolean network on n variables, with n ∈ N, is defined by a function f : B n → B n . The set B n is also called the state space of the Boolean network. The dynamical system given by the iteration of f is called synchronous dynamics. In biological contexts, the asynchronous dynamics or asynchronous state transition graph of a Boolean network is often the object of interest. The asynchronous dynamics AD f of f is defined as the graph with vertex set B n , and edge set The interaction graph G f of a Boolean network f is the labelled multi-digraph with vertex set {1, . . . , n} and admitting an edge (j, i) with sign s Given a Boolean network f : B n → B n and x ∈ B n , we define I f (x) = {i ∈ {1, . . . , n} | f i (x) = x i }. We will simply write I(x) when the choice of Boolean network will be clear from the context. For k ≥ 1, given a path π = (x 1 , . . . , x k ) in an asynchronous state transition graph, we call strategy of the path the map ϕ : {1, . . . , k − 1} → {1, . . . , n} such that x j+1 = x j ϕ(j) for j = 1, . . . , k − 1.
Given x ∈ B n and I ⊆ {1, . . . , n}, we write In the examples, we denote a subspace x[I] using x and replacing the elements x i with i ∈ I with the symbol "⋆". For instance, 001 ⋆ ⋆1 will denote the subspace of B 6 with I = {4, 5} and x 1 = x 2 = 0, x 3 = x 6 = 1.
A set A ⊆ B n is called a trap set for a Boolean network f if, for all x ∈ A, if y is a successor for x in the asynchronous dynamics, then y ∈ A. A trap set that is also a subspace is called a trap space. For each state x ∈ B n there exists a unique minimal trap space containing x, which we denote by κ(x). Minimal trap sets are called attractors for the asynchronous dynamics. If an attractor consists of a single state, it is called fixed point or steady state, otherwise it is called a cyclic attractor.
Given an attractor A, the (weak) basin of attraction of A is the set of states x ∈ B n such that there exists a path from x to A in the asynchronous dynamics. The strong basin of attraction of A is the set of states in the basin of attraction of A that do not belong to the basin of attraction of any other attractor A ′ = A.
We will use the following result Naldi et al. [2009], Paulevé and Richard [2012], which relates properties of Boolean maps to properties of maps with a smaller number of variables.
for each x ∈ B n−1 , i = 1, . . . , n − 1. If G f does not admit an edge from n to itself, then: ADf has a path from x to y if and only if AD f has a path from (x, f n (x, 0)) to (y, f n (y, 0)).

Composition of modules
In this work we are interested in some Boolean networks that can be interpreted as arising from the combination of multiple instances of a given Boolean function. This approach is formalised for instance in Mendes et al. [2013], Varela et al. [2018a and called composition of modules.
Here we use a different definition that can be recasted in terms of compositions of modules.
We fix L ∈ N and consider an un-directed connected graph G with vertex set C = {1, . . . , L} and without loops. We call the vertices cells and G the cell graph underlying the system, as it represents a network of L cells with some neighbouring relation. For each i ∈ C, we write ) is an edge in G, we say that i and j are neighbours.
The system in each cell is described by some Boolean variables, whose behaviour can depend on the variables in the same cell or in neighbouring cells. Mendes et al. [2013], Varela et al. [2018a] also distinguish between input components and internal components, the former being variables that can only depend on variables in neighbouring cells, and the latter being variables that can only depend on other variables from the same cell. For the system studied in this work, we consider only 2 Boolean variables in each cell. We therefore do not introduce a general notation, but rather focus on special systems with 2L variables.

A Boolean Delta-Notch model
Given a cell graph G, for each cell i we consider a variable Notch and a variable Delta, that we denote n i and d i , respectively, with i = 1, . . . , L. The space we consider is therefore B 2L , and the network we study is a function F : B 2L → B 2L . Sometimes it will be convenient to denote an element x ∈ B 2L as x = (n, d) = (n 1 , . . . , n L , d 1 , . . . , d L ), so that x i = n i and x i+L = d i for i = 1, . . . , L. Given J ⊆ C, we will write J + L for the set {i + L |i ∈ J}, and for I ⊆ {1, . . . , 2L} we define I N = I ∩ C, I D = {i − L | i ∈ I ∩ (C + L)}. In addition, given I ⊆ {1, . . . , 2L}, we define S(I) = i∈I∩C S(i) ∪ i∈I∩(C+L) S(i − L).
In the simple model we consider, in each cell, Notch inhibits the production of Delta, with no other interaction taking place. The logical function that encodes the regulation of Delta in cell i is therefore defined by (n, d) →n i . Notch instead is activated by the presence of Delta in neighbouring cells. Here we consider the following two possibilities: either the presence of Delta in any of the neighbouring cells is sufficient for the activation of Notch, or the presence of Delta in all of the neighbouring cells is required. This leads to the definition of two possible Boolean functions for component i, that we denote F ∧ and F ∨ respectively: Note however that F ∧ and F ∨ verify i.e., F ∧ and F ∨ are conjugated under the function x →x, and hence admit isomorphic asynchronous state transition graphs (see Tonello et al. [2019], Proposition 2.2). It is therefore sufficient to limit our analysis to the function F = F ∨ . We call F a Boolean Delta-Notch system over the graph G.
Example 2.2. For L = 1, we have F (n 1 , d 1 ) = (0, ¬n 1 ), and the system has only one attractor, the fixed point 01, i.e., the dynamics converges to the state with low Notch and high Delta. The trap spaces for the system are ⋆⋆, 0⋆ and 01, and ⋆⋆ is the basin of attraction of 01.

Model reduction
The model we described has 2L variables, none of which is autoregulated. It will be convenient to work with the reduced network N : B L → B L obtained from F by elimination of the variables d 1 , . . . , d L as delineated in Theorem 2.1. For each i = 1, . . . , L we have By application of Theorem 2.1 (i), the functions F and N have the same number of fixed points. To a fixed point n * corresponds the fixed point (n * , n * ) for F . In addition, from Theorem 2.1 (ii), given n, n ′ ∈ B L , there exists a path from from n to n ′ in AD N if and only if there exists a path from (n,n) to (n ′ ,n ′ ) in AD F .

Example of network composition
Before we start the investigation of Boolean Delta-Notch systems, in this section we briefly consider a different class of models, with the purpose of illustrating the type of analysis that one can attempt for systems obtained as composition of modules over a graph of cells. The asynchronous dynamics for the models we are about to introduce are in many respects simpler than the dynamics of Boolean Delta-Notch models of the same dimension, and hence serve as good examples for our analysis. Consider models defined as in Section 2.2 for the first L variables, and assuming activation of Delta from Notch in place of the inhibition, resulting in the Boolean functions It is easy to verify that H ∧ (n, d) = H ∨ (n,d), so that the dynamics defined by the two Boolean networks are isomorphic, and one can focus on studying the dynamics of H = H ∨ , for instance. Since all edges in the interaction graph of H have positive sign, the graph G H does not admit any negative cycle, and, as a consequence, all attractors of AD H are fixed points Richard [2010].
. . , L. Hence, if n * i = 0 for some i, then n * j = 0 for any neighbour cell j of i; and since G is assumed connected, we must have n * k = 0 for k = 1, . . . , L. Vice versa, if n * i = 1 for some i, then n * j = 1 for any neighbour cell j of i, and n * k = 1 for k = 1, . . . , L. For L = 1, the system has only the fixed point 0, whereas for L ≥ 2 H admits two fixed points, 0 and 1.
Using the following observation, we give characterisations of the trap spaces and basins of attraction.
Remark 2.4. For L ≥ 2, given x ∈ B 2L with x i = 1 for some i ∈ {1, . . . , 2L}, for each j ∈ {1, . . . , 2L} there exists a path in AD H from x to a state y with y j = 1 and y k ≥ x k for all k ∈ {1, . . . , 2L}. To see this, observe that, since G is connected, G H is strongly connected. Suppose that x j = 1 and consider a simple path π in G H from i to j. Remove any index h in π such that x h = 1, obtaining a sequence i 1 , i 2 , . . . , i k = j. Then (x i 1 ,x {i 1 ,i 2 } , . . . ,x {i 1 ,...,i k } ) is a path in AD H from x to a state y =x {i 1 ,...,i k } with the required properties. . , 2L} such that y j = 1. Given i / ∈ I, by Remark 2.4 there exists a path from y to a state z with z i = 1, hence x satisfies x i = 1. For any k ∈ I N , since H k+L (x) = x k , we have k ∈ I D , and, in addition, if h ≤ L and h / ∈ I N , since x h = 1, the set S(h)\I D must be non-empty. Vice versa, if x[I] is a subspace satisfying the hypotheses and y ∈ x[I], then any successor for y in AD H is necessarily in , then we show that there is a path from y to 0. It is sufficient to show that there is a path from y to a state z with z i+L = 0 for i = 1, . . . , L.
Define J 1 = {i ∈ C | y i = 0, y i+L = 1}. Then there is a path from y to u =ȳ J 1 +L . Now take J 2 = {i ∈ C | u i = y i = 1, u j+L = 0 for all j ∈ S(i)}. Again, there is a path from y to v =ū J 2 . It is sufficient to show that v i+L = 0 for all i ∈ C such that v i = 1, so that we can take Suppose that v i = 1 = v i+L for some i ∈ C. Then, since y j ≥ v j for all j, we have y i = y i+L = 1. This implies that there exists a j ∈ S(i) such that u j+L = 1, which, by definition of u, gives y j = y j+L = 1. Take J = {1, . . . , 2L} \ {i, j, i + L, j + L}. Then by Proposition 2.5 the subspace y[J] is a non-trivial trap space containing y, which is a contradiction.
The second part of the statement is a direct consequence of the first.
We can in addition observe that, for L ≥ 2, the asynchronous dynamics admits cyclic paths. For instance, take any x ∈ B 2L such that x i =x i+L = 1 and x j =x j+L = 0 for some i ∈ C and j ∈ S(i), and x k = x L+k = 0 for k / ∈ {i, j}. Then the asynchronous dynamics admits the cycle In the remainder of the paper we will carry out a similar qualitative analysis of Boolean Delta-Notch systems, to elucidate their asymptotic behaviour, understand whether cyclic paths are possible in the asynchronous state transition graph and characterise the reachability of attractors.

Asymptotic behaviour
In this section we study the fixed points of Boolean Delta-Notch systems, and we show that, for any architecture, all attractors are fixed points. We refer to the fixed points also as stable spatial patterns, or simply patterns, for the system. They are characterised by a specific alternating structure of primary fate and secondary fate cells, which is determined by the structure of the cell graph G.
Recall that a vertex cover of a graph G is a subset Q of the vertices of G such that every edge of G has an endpoint in Q (see for instance West [2001]). The following result is intuitive: at a fixed point, cells with low level of Notch are surrounded by cells with high level of Notch, and every cell with high level of Notch has a neighbour with low level of Notch.
Theorem 3.1. The fixed points of the Boolean Delta-Notch system over the graph G are in one-to-one correspondence with the minimal vertex covers of the graph G.
Proof. As seen in Section 2.2.1, the fixed points of the Boolean Delta-Notch system are in one-to-one correspondence with the fixed points of N .
Consider the bijective map h : is not a vertex cover. Vice versa, consider a minimal vertex cover Q of G, and define n = h −1 (Q). If j∈S(i)n j = 1, then n j = 0 for some j ∈ S(i), and since (i, j) is an edge in G we must have i ∈ Q and n i = 1. If instead j∈S(i)n j = 0, then n j = 1 and j ∈ Q for all j ∈ S(i). Since Q is minimal, we find i / ∈ Q and n i = 0.
Remark 3.2. It follows from Theorem 3.1 that, for any i ∈ C, there exists a fixed point x for the Boolean Delta-Notch system over G that satisfies x i =x i+L = 0, x j =x j+L = 1 for all j ∈ S(i). In particular, if L ≥ 2, then the Boolean Delta-Notch system over G admits at least two fixed points. (b) n does not admit two consecutive 0s or three consecutive 1s; (c) n 1 = 0 or n 2 = 0, and n L−1 = 0 or n L = 0.
At fixed points, Notch consists therefore of alternating 0s and 1s, possibly with pairs of consecutive cells with level 1, away from the boundary. This is consistent with simulation results obtained in the continuous case in Collier et al. [1996], where the rate of production of Notch in cell i is taken to be F i (n, d) = f 1 a+x k , for a > 0, k ≥ 1, and the rate of production of Delta in cell i is proportional to g(n i ), with g(x) = 1 1+bx h , b > 0, h ≥ 1. The authors observe in fact that two adjacent cells can display high levels of Notch, whereas cells with low level of Notch are always separated by secondary fate cells, in agreement with experimental findings. Figure 2: Levels of Notch at fixed points for the Boolean Delta-Notch systems associated to the graphs P 2 , P 3 and P 4 . A white square represents a cell with high level of Notch. Figure 3: Levels of Notch at fixed points for the Boolean Delta-Notch systems associated to the graphs C 2 , C 3 and C 4 . White represents high level of Notch.
Recall that we choose to analyse Boolean systems where the level of Notch in each cell is found as the disjunction of levels of Delta in neighbouring cells. As observed in Section 2.2, the fixed points of F ∧ , the function obtained by setting the level of Notch in each cell to the conjunction of levels of Delta in neighbouring cells, are found by negating the fixed points of F ∨ . Hence the fixed points of F ∧ admit pairs of consecutive cells with low levels of Notch. As expected, simulations of continuous models that adopt, for the rate of production of Notch, the function (n, d) → j∈S(i) f (d j ), give patterns with pairs of consecutive cells with low Notch. The function F ∨ seems therefore a more appropriate modelling choice for the biological system.
We now want to show that there are no cyclic attractors in asynchronous state transition graphs of Boolean Delta-Notch systems.
Theorem 3.6. Consider a Boolean Delta-Notch system F . For each non-fixed point x ∈ B 2L , there is a path in AD F from x to a fixed point.
Proof. Consider (n, d) ∈ B 2L . Since there exists a path from (n, d) to (n,n), by Theorem 2.1 (ii) it is sufficient to show that a fixed point for N can be reached from n.
Consider an asynchronous path starting from n obtained by concatenating any upward path π + reaching a state n ′ with I + (n ′ ) = ∅, and any downward path π − from n ′ reaching a state n ′′ with I − (n ′′ ) = ∅. Write π − as (n 1 , n 2 , . . . , n k ), with n 1 = n ′ and n k = n ′′ . We show by induction that every state n h in π − verifies I + (n h ) = ∅. Since n k = n ′′ verifies I − (n k ) = ∅, it is a fixed point, and we conclude.
By construction, the conclusion holds for n ′ . Suppose that I + (n h ) = ∅ for h < k. Consider n h+1 obtained from n h by changing one component, say n h t , from 1 to 0. We need to show that, if n h+1 s = 0 for some s ∈ C, then s / ∈ I + (n h+1 ). Suppose that n h+1 s = 0; then s / ∈ S(t) and there are two cases: (1) if s = t, it means that for all j ∈ S(s), n h+1 j = n h j = 1, hence s / ∈ I + (n h+1 ).

Corollary 3.7. The asynchronous state transition graph of a Boolean Delta-Notch system does not admit cyclic attractors.
Asynchronous trajectories of Boolean Delta-Notch systems can not therefore lead to sustained oscillations. Observe that not every fixed point is reachable from every non-fixed point: for instance, in the Boolean Delta Notch system over P 3 (see Example 3.3) there is no path from 011100 to the fixed point 101010. In the next section, we study the basins of attraction.

Reachability of fixed points
In the following, we consider the problem of determining which patterns can be obtained from some initial states. The reachability of fixed points for Boolean Delta-Notch systems over hexagonal grids from given initial conditions has been previously studied in Mendes et al. [2013]. We start the section by showing that all the fixed points can be reached from homogeneous states, that is, states where the levels are the same in every cell, and identify other classes of states for which this property holds.

Homogeneous initial conditions
Remark 4.1. From each state (n, d), there is a path to ( j∈S(1) d j , . . . , j∈S(L) d j , d) and a path to (n,n) in AD F . Hence • if a state is reachable from (0, 0), it is reachable from (n, 0) for all n ∈ B L ; • if a state is reachable from (1, 0), it is reachable from (1, d) for all d ∈ B L .
• for L ≥ 2, if a state is reachable from (1, 1), it is reachable from (n, 1) for all n ∈ B L ; • if a state is reachable from (0, 1), it is reachable from (0, d) for all d ∈ B L .
The asynchronous dynamics of every Boolean Delta-Notch system with L ≥ 2 admits therefore a cycle that includes all homogeneous states (see Fig. 4, left). In addition, the following result shows that all fixed points are reachable from homogeneous states (see Fig. 4, right, for an example).  1). On the right, some paths in the asynchronous dynamics associated to the graph P 4 , from the homogeneous state (1, 0) to the three fixed points (see Theorem 4.2). White represents high levels.
Proof. We show that, for each fixed point x ∈ B L for N , there is a path from 1 ∈ B L to x in AD N . By Theorem 2.1 (ii), this implies that for each fixed point (x,x) of F there is a path from (1, 0) to (x,x). Remark 4.1 then allows to conclude. Consider a fixed point x for N , and define I(x) = {i ∈ C | x i = 0}, k = |I(x)|. Set x 0 = 1, choose an order i 1 , . . . , i k for the indices in I(x), and, for each h = 1, . . . , k, define the state x h =1 {i 1 ,...,i h } . Then, for each h = 0, . . . , k − 1, x h i h+1 = 1, x i h+1 = 0, and, since x is fixed, for all j ∈ S(i h+1 ) we have x j = 1, so that x h j = 1 and N i h+1 (x h ) = 0. Hence the asynchronous dynamics AD N admits an edge from x h to x h+1 , for h = 0, . . . , k − 1. In other words, there is a path in AD N from x 0 = 1 to x k = x with strategy i 1 , . . . , i k .

Trap spaces and basins of attraction
In this section, we give a characterisation of the trap spaces of Boolean Delta-Notch systems. Unlike in previous sections, we will now work with the 2L-dimensional model F , rather than its reduction N , since the reachability properties of the two systems can differ. We start with a straightforward remark. Observe that, on the other hand, i ∈ I D does not imply i ∈ I N : for instance, 01010⋆ is a trap space for the system with graph P 3 (see Example 3.3). The following lemma shows that, if the level of Delta can vary for cell j in a trap space, while the level of Notch is fixed, then the levels of Delta and Notch in neighbouring cells are fixed, and the level of Notch in cell j is zero. Proof. Consider j ∈ I D with j / ∈ I N , and take an element y ∈ x[I] with y j+L = 1. Then there exists a path from y to a state z with z k = 1 and z k+L = 0 for all k ∈ S(j), and since x[I] is a trap space, we have z ∈ x[I]. Since j / ∈ I N , we must have x j = k∈S(j) z k+L = 0. This is possible only if I D ∩ S(j) = ∅ and x k+L = 0 for all k ∈ S(j).
Before formalising the details, let us first give an intuitive description of the form that a trap space can take. Consider a subspace x [I]. Clearly, if x[I] is a trap space, F i (x) = x i for all i / ∈ I, and we can assume that x is a fixed point.
Consider first the case I N = I D . Levels of Delta in cells with index in C \ I D can only change if Notch changes. The levels of Notch can change first only at the border with I D , since, away from I D , the levels of all components are fixed. If the level of Notch at the border with I D is low, then it can change to a high level whenever a cell in I D reaches a high level of Delta. Hence the level of Notch in x at the border with I D must by high. For this high level to be always preserved, a neighbour cell must have a fixed high level of Delta. In other words, we can generate the trap spaces x[I] with I N = I D as follows: take x fixed point, and define Z(x) is the set of indices of cells with low Notch and high Delta. Then, we can identify a trap space by fixing a combination of neighbourhoods of cells in Z(x); that is, for each J ⊆ Z(x), the subspace is a trap space.
Let us now consider the general case I N ⊆ I D . As shown in Lemma 4.4, a trap space can have cells with fixed, low levels of Notch and undefined Delta, surrounded by cells with fixed levels. Therefore, for each I N defined above, for any is also a trap space. This discussion reveals in particular that the smallest trap spaces that are not fixed points are of the form x[{i + L}] for some steady state x and some i ∈ C such that x i = 0 and, for all j ∈ S(i), there is an index k ∈ S(j), k = i such that  To summarise, the trap spaces are characterised by areas of fixed Notch and Delta, each with a two-layer border, the most external one having high Notch, and the internal one having some low Notch cells -at least one neighbour cell for each cell at the outermost border -which are necessary to keep the level of Notch at the border high. In addition, isolated cells with low Notch can have undefined Delta. Examples of trap spaces for a hexagonal grid and for a linear graph are given in Fig. 5. We now formalise the characterisation outlined above. ∈ I for all j ∈ S(i), then F i (y) = j∈S(i) y j+L = j∈S(i) x j+L = x i = y i . Consider now the case of i / ∈ I, i ≤ L and I ∩ S(i) = ∅. Then (i) implies i / ∈ I D , and (ii) gives the existence of k ∈ S(i) ∩ I c D such that x k = 0, and therefore y k+L = x k+L = 1. Hence F i (y) = j∈S(i) y j+L = 1 = F i (x) = x i = y i . Viceversa, consider a trap space x[I]. Since we must have F i (x) = x i for all i / ∈ I, we can assume that x is a fixed point. The first point is proved in Lemma 4.3 and Lemma 4.4. Consider i ∈ S(I) ∩ I c D and take j ∈ I ∩ S(i). Then there exists a state y ∈ x[I] with y j+L = 1, and therefore x i = F i (x) = k∈S(i) y k+L = 1. Now take a state z ∈ x[I] with z k+L = 0 for all k ∈ S(i) ∩ I D . Then x i = 1 = k∈S(i) z k+L = k∈S(i)∩I c D z k+L . This means that there exists k ∈ S(i) ∩ I c D such that x k+L = 1, which concludes. We can now characterise the fixed points that are reachable from a given state. We show that all the attractors found in the minimal trap space containing the state are reachable. The idea of the proof is as follows. First, one observes that the analysis can be reduced to the case where the initial state x does not belong to any non-trivial trap space. In this case, a path can be exhibited from x to a state with homogeneous, low levels of Delta. The path can be obtained through the following steps: first all low levels of Delta that can increase are increased, but only if they are not completely surrounded by cells with high Notch and low Delta. Then, Notch levels are increased in all cells where it is possible. Since x does not belong to any non-trivial trap space, it is then sufficient to bring all the levels of Delta down.
Lemma 4.8. Consider x ∈ B 2L such that κ(x) = B 2L . Then there exists a path in AD F from x to (1, 0).
Proof. It is sufficient to show that there exists a path in AD F from x to a state z with z i+L = 0 for all i ∈ C (see Remark 4.1).
Define the set J = {i ∈ C | x i = 0 and x j = 1, x j+L = 0 for all j ∈ S(i)}. These are the indices of isolated cells with low Notch levels. If x i+L = 1 for some i ∈ J, then by Remark 3.2 there exists a fixed point y that coincides with x on {i} ∪ S(i). Then the subspace y[I ∪ (I + L)] with I = C \({i}∪S(i)) satisfies the conditions of Theorem 4.6 (ii) and is a trap space containing x. Since x does not belong to any non-trivial subspace, we have x i+L = 0 for all i ∈ J.
Consider the set of indices J 1 = {i ∈ C | x i = x i+L = 0}. Then J ⊆ J 1 , and there is a path in AD F from x to v =x (J 1 +L)\(J+L) . Now define J 2 = {i ∈ C | v i = 0 and v j+L = 1 for some j ∈ S(i)}. Again, there is a path in AD F from v to w =v J 2 . Note in addition that w ≥ v ≥ x, so that x i = 1 implies w i = 1. If instead x i = 0, we have: • If i / ∈ J and x j+L = 0 for all j ∈ S(i), then there exists k ∈ S(j) such that x k = 0 and v k+L = 1, so that w i = 1.
• If i / ∈ J and there exists k ∈ S(i) such that x k+L = 1, then v k+L = 1 and w i = 1.
In summary, w verifies w i = 1 for all i ∈ C \ J and w i+L = 0 for i ∈ J. As a consequence, taking J 3 = {i ∈ C \ J | w i = w i+L = 1}, we have that the state z =w J 3 +L is reachable from w and verifies z i+L = 0 for all i ∈ C, and we conclude.
The previous lemma shows that, from states that do not belong to any non-trivial subspace, any homogenous state can be reached. This result, combined with Theorem 4.2, gives that any fixed point can be reached from such initial conditions. When the initial state y belongs to some non-trivial subspace, the fixed points that can be reached are limited by the minimal subspace κ(y) containing y. To prove that all fixed points contained in κ(y) can be reached from y, we consider the projection of the dynamics on the subspace κ(y), and study it as the combination of smaller Boolean Delta-Notch subnetworks. It can be shown that, in general, in such a scenario, the full dynamics in the trap spaces can be derived from the dynamics of the isolated active subnetworks (Siebert [2009]). Here we give a self-contained proof.   Proof. Consider the subgraph of G obtained by removing all vertices outside I D and all the incident edges. Then G ′ can be decomposed into connected graphs G 1 , . . . , G k with vertex sets C 1 , . . . , C k respectively. We will now consider the projection of the dynamics on the components identified by C 1 , . . . , C k . For each h ∈ {1, . . . , k}, writing C h = {j 1 , . . . , j |C h | }, and denoting by π i : B 2L → B the projection on the i th component, consider the maps π h : B 2L → B 2|C h | , π h = (π j 1 , π j 2 , . . . , π j |C h | , π j 1 +L , π j 2 +L , . . . , π j |C h |+L ), and ι h : Since, by Theorem 4.6 (ii), x j+L = 0 for all j ∈ S(I) ∩ (C \ I D ), we have that, for each h ∈ {1, . . . , k}, i ∈ C h and y ∈ x[I], F i (y) = j∈S(i) y j+L = j∈S(i)∩C h y j+L , that is, the dynamics on each connected component C h is not influenced by variables outside C h , and F h is a Boolean Delta-Notch system on G h . Then (i) follows from the application of Theorem 4.2 to each Boolean network F h .
If y ∈ x[I] satisfies κ(y) = x[I], first observe that, if i ∈ I D and i / ∈ I N , then by Lemma 4.4 x i+L = 1, x i = y i = z i = 0, and y i+L = z i+L = 0. In addition, for each h = 1, . . . , k, π h (y) does not belong to any non-trivial trap space defined by F h . (ii) is therefore a consequence of Lemma 4.8.
To prove (iii), consider w fixed point in x[I] and i ∈ I. Since by Theorem 4.6 (ii) x j+L = 0 for all j ∈ S(i), we have w i = 0 and w i+L = 1.
Take i as in the hypothesis of (iv) and (v), and j ∈ S(i) ∩ I D . By Theorem 4.6 (i) we have that i, j ∈ I N . Call C h the connected component containing i and j. Then F h , and therefore F , have at least two fixed points by Remark 3.2. (v) is found by application of Remark 4.1 to F h .
Theorem 4.10. For every y ∈ B n and for every fixed point x ∈ κ(y) there exists a path from y to x in AD F . Proof. Take y ∈ B 2L and any x fixed point in κ(y). By Theorem 4.6, we can write κ(y) = x[I] for some I ⊆ {1, . . . , 2L}. We conclude using Proposition 4.9, (ii) and (i).
The theorem states that, for any Boolean Delta-Notch model and any state y, all attractors that are contained in the minimum trap space containing y are reachable from y. As a corollary of the theorem, the basin of attraction of a fixed point x is found by taking all the trap spaces defined starting from x as in Theorem 4.6, and removing all states found in trap spaces that do not contain the fixed point x. We can reformulate the observation as follows.
Proposition 4.11. For L ≥ 2, for each fixed point x ∈ B 2L , the basin of attraction is given by where M is the set of maximal, non-trivial trap spaces.
Proof. Write T for the set of all non-trivial trap spaces. Consider a fixed point x, and denote by B its basin of attraction. Given y ∈ B c , by Theorem 4.10 we have that x / ∈ κ(y), hence the equality B c = t∈T,x / ∈t t. It remains to show that any state y contained in a trap space that does not contain x is also contained in a maximal trap space that does not contain x. Suppose that y ∈ z[I] with z fixed point and x / ∈ z[I]. Then there exist an i / ∈ I, i ∈ C such that z i = 0 and x i = 1. The characterisation of trap spaces in Theorem 4.6 implies that {i} ∪ S(i) ⊆ I c , and by Remark 4.7 the subspace z[J ∪ (J + L)] with J = C \ ({i} ∪ S(i)) is a maximal non-trivial trap space that contains y and does not contain x.
We can also characterise the strong basins of attraction. Proof. For L = 1, the result is trivial. For L ≥ 2, first observe that, by Proposition 4.9, (iii), the trap spaces x[I] with S(i) ∩ I = ∅ for all i ∈ I are contained in the strong basin of attraction of x. It remains to show that any other state in the basin of attraction of x is also in the basin of attraction of some other fixed point.
Consider a state z in the basin of attraction of x, and suppose that the trap space κ(z) can be written as x[I] with I such that there exist i, j ∈ I with j ∈ S(i). By Proposition 4.9, (iv), there exists another fixed point y = x, y ∈ x[I]. Then by Theorem 4.10 the state z is in the basin of attraction of x and in the basin of attraction of y.
The size of the strong basins of attraction grows therefore with the number of low Notch whose neighbouring high-Notch cells have other neighbours with low Notch. For example, for the linear graphs P L and C L , the size of the strong basin of attraction is the largest for "regular" patterns, i.e., patters that do not admit two adjacent cells with high Notch. This is in agreement with the observation in Collier et al. [1996] that patterns emerging from simulations for systems associated to linear graphs display alternating high and low Notch levels, with only some occasional "defects" given by consecutive cells with high Notch.
Example 4.13. For the system in Example 4.5, the strong basin of attraction of p 1 = 101010 is given by the fixed point itself, whereas the strong basin of attraction of p 2 = 010101 is J = ⋆10 ⋆ 01 ∪ 01 ⋆ 10⋆. The basin of attraction of p 1 the set B 6 \ J, whereas the basin of attraction of p 2 is the set B 6 \ {p 1 }.

Robustness of patterns
We can use the characterisation of strong and weak basins of attraction to study the robustness of stable patterns in response to small perturbations. We want to answer the following questions:  Figure 6: Changes in levels of Notch or Delta in one cell can induce the system to attain a different pattern. Changes to low levels of Notch or high levels of Delta can propagate to neighbour cells, and changes to high levels of Notch or low levels of Delta can affect cells at distance two (see Proposition 4.14. White represents high activity. 3. in any other case, there are cyclic paths reachable from y (Proposition 4.9 (v)).
Intuitively, areas with borders of high Notch, sustained by cells with high Delta opposite the perturbation, are unaffected by the perturbation. For changes of only one variable level in one cell, we have that: • Isolated changes of low Notch to high Notch, or high Delta to low Delta can only affect direct neighbour cells.
• Isolated changes from high Notch to low Notch, or low Delta to high Delta can only affect cells at maximum distance of 2 from cell i.
The observations are proved in the proposition below. The examples in Fig. 6 show that the bounds on the distance of affected cells are the smallest possible.
Proposition 4.14. Consider x ∈ B 2L fixed point for a Boolean Delta-Notch system, and take i ∈ C. Proof. (i) When x i = 0, we have that x j = 1 for all j ∈ S(i). Define J = {j ∈ S(i) | x k = 1 ∀k ∈ S(j), k = i} and I = {i} ∪ J. We show that x[I ∪ (I + L)] is a trap space, by applying condition (ii) of Theorem 4.6. Consider h ∈ S(I) ∩ I c . We need to show that S(h) ∩ I c is non-empty and x k = 0 for some k ∈ S(h) ∩ I c .
1. if h ∈ S(i), then by definition of I there exists k ∈ S(h), k = i such that x k = 0. In addition, k / ∈ J since x j = 1 for all j ∈ J.
2. if h / ∈ S(i), then h ∈ S(j) for some j ∈ J, and therefore x h = 1. Hence there exists k ∈ S(h) such that x k = 0. In addition, k = i (since h / ∈ S(i)) and k / ∈ J (since x j = 1 for all j ∈ S(i)). Hence k ∈ S(h) ∩ I c .
(ii) For the case x i = 1, define K = {j ∈ S(i) | x j = 0}, and set J = {k ∈ S(K) | x h = 1 ∀h ∈ S(k), h / ∈ K}. Clearly, x k = 1 for all k ∈ J. We use again condition (ii) of Theorem 4.6 to show that x[I ∪ (I + L)] with I = {i} ∪ K ∪ J is a trap space. Taking h ∈ S(I) ∩ I c , we show that S(h) ∩ I c is non-empty and x k = 0 for some k ∈ S(h) ∩ I c .

Conclusion and prospects
In this work we gave some characterisations of the dynamics of simple Boolean models of the Delta-Notch system, complementing existing computationally-costly algorithmic analyses (e.g. Mendes et al. [2013], Varela et al. [2018a]). When a network of cells is represented by an undirected graph G, the identification of the fixed points can be traced back to determining the minimal vertex covers (or the maximal independent vertex sets) of the graph G (Theorem 3.1). The emerging patterns are consistent with those obtained in the spatially-discrete continuous model of Collier et al. [1996]. We also gave a characterisation of the trap spaces (Theorem 4.6) and of the patterns that can be reached from a given state (Theorem 4.10). In particular, we saw that all patterns can be obtained from homogeneous starting points (Theorem 4.2). The effects of single-cell perturbations on patterns were discussed in Section 4.3.
Our analysis demonstrates how, in some cases, the dynamics resulting from the composition of Boolean modules can be understood to a significant degree of detail, without resorting to simulations. All our statements concern the structure of the dynamics and do not allow for quantitative results regarding, for instance, the distribution of Notch obtained with trajectories starting from a given initial condition, as considered, for example, in Varela et al. [2018b]. The study of the asynchronous dynamics as a Markov chain is used to quantify simulation results of Boolean models (Stoll et al. [2017]) and could help with the interpretation of simulation results. In addition, we only considered models where the presence of Delta in at least one neighbour cell is sufficient for the activation of Notch, or the presence of Delta in all neighbour cells is required for the activation of Notch. An extension of the approach to other scenarios can be considered, for instance by assuming Notch to be activated when a certain minimum amount of neighbour cells with high levels of Delta is reached, as in Varela et al. [2018b]. The model presented here provides a basis for the exploration of networks with more elaborate cell modules, and for the investigation of the role of the simple mechanism we considered in the generation of spatial inhomogeneity in more complex Boolean systems.

Funding
Funded by the Volkswagen Stiftung (Volkswagen Foundation) under the funding initiative Life? -A fresh scientific approach to the basic principles of life (project ID: 93063).