J. Nig. Soc. Phys. Sci. 4 (2022) 812 Journal of the Nigerian Society of Physical Sciences Characterisation of Singular Domains in Threshold Dependent Biological Networks Benitho A. Ngwua,∗, Godwin C. E. Mbahb, Chika O. Mmaduakora, Sunday Isienyic, Oghenekevwe R. Ajewolea, Felix D. Ajibadea aDepartment of Mathematics, Federal University Oye-Ekiti, Oye/Ekiti, Nigeria bDepartment of Mathematics, University of Nigeria, Nsukka/Enugu, Nigeria cDepartment of Mathematics & Statistics,Akanu Ibiam Federal Polytechnic Unwana, Afikpo/Ebonyi, Nigeria Abstract Threshold-dependent networks which behave like piecewise smooth systems and belong to a class of systems with discontinuous right hand side are studied with piecewise linear differential equations. At threshold values and their intersections, known as switching boundaries and surfaces, the state of such networks is not defined because of singularity at such points. This study characterises, in terms of number, singular domains of any order in a network and the total number of such domains; and also proposes new definitions for walls, using Filippov’s First Order Theory on characterisation of (sliding) wall. The finding of this study is presented as propositions I, II and III respectively. In particular, using proposition II the study identified two white walls previously considered transparent. Introducing monotonicity to definition of transparent wall is also seen to affect qualitative dynamics like source, sink and cycles. DOI:10.46481/jnsps.2022.812 Keywords: Discontinuous System, Singular Domain, Threshold-Dependent Network, Biological System, Piecewise linear Models Article History : Received: 12 May 2022 Received in revised form: 16 July 2022 Accepted for publication: 20 July 2022 Published: 20 August 2022 c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: T. Latunde 1. Introduction Differential equations with discontinuous right-hand-side are applied in different areas of study [1, 2]. They include but are not limited to control problems [3], neural networks [4] and gene regulatory networks [5, 6]. Due to its importance, dif- ferent types of solutions exist for this class of problems [5] of which Filippov’s is the most widely invoked [6-12]. Threshold- dependent networks from biology, such as gene regulatory net- ∗Corresponding author tel. no: +2348067876254 Email address: benitho.ngwu@fuoye.edu.ng; benamaobi@gmail.com (Benitho A. Ngwu ) works can be considered as piecewise linear differential model with discontinuous right hand side [13]. The discontinuity ob- served in this class of threshold-dependent networks results from threshold boundaries known as the switching boundaries. Switching boundaries can be of different orders depending on the number of variables at play in the network. Intersection of two or more different switching boundaries is known as switch- ing surfaces. Due to the wide application and acceptance of Fil- ippov’s approach to this class of problems, the threshold bound- aries (or surfaces) are called Filippov’s boundary (or surface). In biological networks, they represent variables’ concentration levels (and may be referred to as concentration thresholds) and 1 Mbah et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 812 2 their intersections respectively. At these boundaries, the state of the variable is assumed to change. For instance, above a cer- tain concentration level (that is, threshold value) a variable may be activated or deactivated depending on the effect of control of the activity of all other variables regulating its function [14, 15]. These switching boundaries, which can also be called (switch- ing) hyperplanes, partition the state space of the network or sys- tems into rectangular boxes [16] and segments [17] when step functions are used to describe the regulatory control resulting from the threshold. Within these boxes, the state of the network is known and can be defined as a linear function because of the use of step functions. On the contrary, the state of the network is considered not known at the threshold boundaries yielding the discontinuity observed in networks of such nature. One of the major challenges of this class of network is defin- ing a solution along the switching boundary. Filippov’s ap- proach is used to study the behaviour of this class of network [7, 9, 10, 12], especially at the threshold boundary. One of the results of Filippov is defining what is called sliding vector field at a (sliding) threshold boundary, where discontinuity exists. This provides an insight into the behaviour of solutions of the system in the neighbourhood of the threshold boundary. Within the vicinity of the threshold, a trajectory approaching a discon- tinuity boundary can either glide through, move away from, or slide on, it. This is how a threshold boundary is classified. Us- ing Filippov approach, differential models with discontinuous right hand side have been investigated for different qualitative properties such as orbits [18-20], limit cycles [1] and sliding motions and solutions on threshold boundaries [11, 13]. As noted earlier, discontinuous boundaries have been selectively investigated for existence of solutions due to a conventional definition that is based on the flow of trajectory towards such domains. For instance, Sari and Gouze in [13] conclude that transparent walls are not candidate for sliding vector motion. Their conclusion is premised on the direction of flow of solu- tion trajectories towards the wall. However, they failed to in- vestigate if such walls are actually transparent let all providing a criterion for testing the transparency of a wall beyond direc- tion of trajectories of focal points. In recent times this class of problem was considered as fast- slow systems with delay [22-25]. To analyse the behaviour of solution at the switching boundary regularisation approach was adopted. In [21]the qualitative behaviour of the system is analysed in the neighbourhood of the threshold boundary. In the use of regularisation approach, two types of hyperplane are considered, sewing and sliding manifolds. These definitions were made using singular solution of the switching manifold [23, 24]. From these studies, it is obvious that transparent walls are not studied for sling vector field or motions because vector field cannot be defined on them. Noted also is the fact that the definition of the nature of these walls have not been reviewed since its introductions. Encouraged by the definition of sewing and sliding manifolds in [23, 24] with respect to singular so- lution an attempt is made to redefine the nature of these walls using focal points (as a limit solution) in the network. If transparent walls are not candidates for sliding motion and there is no concrete criterion for determining transparency as noted earlier, then the need arises to provide for such. Sec- ondly, if switching manifolds (walls) a more robust classifica- tion of the nature of walls can be made using focal points which are limit solutions. Within the limit of our knowledge, only reg- ular domains have been characterised in terms of number in this class of network; so, a need arises to characterise singular do- mains in terms number. The objective of the study is to use Fil- ippov’s first order theory to provide a criterion for transparency and in addition introduce monotonicity to check behaviour of variables at such transparent walls; derive new definitions that shall capture some intrinsic behaviour in threshold-dependent systems, especially biological systems where relapse is known to occur and characterise singular domains in terms of num- ber. To achieve its objective, the paper is organized as follows: Introduction is contained in section 1 while the system under study and the method of study are presented in section 2. In section 3, the result of the network is presented as propositions. Analysis of the results is verified in this section too to demon- strate the usefulness and applicability of the study. Discussion of the result of the study is done in section 4 whereas conclusion of the work is in section 5. 2. Materials and Methods This section is to introduce discontinuous system of the na- ture to be considered in this work. Let Σ = {x ∈ 0} (4) H2 = { x ∈ θ0 if x < θ (8) where θ is a threshold for the variable x. With equation (8), the discontinuity at the threshold boundary becomes clear. It then follows that the hyperplane of interest coincides with the threshold value of the variable of interest. Definition 2 [8]: Let Ω = Ω1 ×Ω2 × ...×Ωn be the partitioning of phase space of (7) by the threshold hyperplanes where Ωi = {x ∈ <+ : 0 ≤ x ≤ maxi}. A domain D is defined to be a set D = D1 × D2 × ...× Dn where Di is one of the following: Di = {xi ∈ Ωi : 0 ≤ xi < θ 1 i } (9) Di = {xi ∈ Ωi : θ j i < xi < θ j+1 i }, j = {1, 2, ..., pi − 1}(10) Di = {xi ∈ Ωi : θ pi i < xi ≤ maxi} (11) Di = {xi ∈ Ωi : xi = θ i j}, j = {1, 2, ..., pi} (12) The threshold hyperplane partitions the state space of the net- work into two different domains known as regulatory and switching [8],or regular and switching [27]. Regulatory (or reg- ular) domain refers to boxes where the state of the network is known. This is given by any of equations (9) to (11). It then follows that a domain is regulatory or regular if none of its vari- ables’ concentration is at a threshold value which can be inter- preted as Di = {xi ∈ Ωi : xi , θi, } or Di = {xi ∈: Ωi : xi < θi, xi > θi} Switching (or singular) domain, similarly, is where the state of the network cannot be determined because at least one variable assumes a threshold value. So, D is called a switching (or sin- gular) domain of order k ≤ n, denoted Dk, if exactly k variables have threshold values in Dk. xs in this case is called a switching variable. This is an intersection of a box and hyperplane and given by a combination of equation (12), with or without, any or some of equations (9) to (11). Mathematically speaking, a singular domain is defined as D = ( xs = θs, xr , θr ) where subscripts s and r denote switching (or singular) and regulatory (or regular) variables respectively. Example 1 To illustrate these domains, consider a two dimensional net- work (that is, a network with two variables only) whose vari- ables have one threshold each. The state space of the network is shown in Figure 1. There are four regular boxes in this network given as Bi i = 1, 2.3, 4 These boxes are defined as follows B1 = {x1 < θ1, x2 < θ2} B2 = {x1 > θ1 x2 < θ2} B3 = {x1 > θ1, x2 > θ2} B4 = {x1 < θ1, x2 > θ2} The switching domains for example 1 are the following D1 1 = {x1 < θ1, x2 = θ2} D2 1 = {x1 > θ1 x2 = θ2} 3 Mbah et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 812 4 D3 1 = {x1 = θ1, x2 < θ2} D4 1 = {x1 = θ1, x2 > θ2} D5 2 = {x1 = θ1, x2 = θ2} As the superscript denotes, D11 to D 1 4, known as walls, are of order 1 because only one variable switch at a time in them whereas D25 , called centre, is a switching domain of order two because the two variables switch simultaneously. Total number of regulatory domains in a network where each xi has mi thresh- olds have been obtained for this class of network [28, 8]. It is given as n∏ i=1 ( mi + 1 ) The number of switching domains in this network has not been given in literature and as such, it is one of the main result of this study shall present in section 3. Consider the piecewise linear model presented below ẋi(t) = βiS ±(x,θ) −γi xi (13) where S ±(x,θ) = 1 − S ∓(x,θ) is same as equation (8). Within a box Bi, equation of motion given by (7) becomes ẋi = fi Bi −γi xi (14) In vector form, equation (14) is given as Ẋ = F B − ΓX (15) where F B and Γ are diagonal matrices which give the produc- tion and degradation rate function in the system. Within a box, Bi, the solution of (14) is given as xi(t) = fi γi + ( x0 − fi γi ) e−γi (t−t0 ) (16) and in finite time, t →∞, the solution trajectory approaches φ Bi i = fi γi (17) which is called the focal point of the box Bi. The set of all focal points in the network is given in vector form as Φ = {φ Bi i } n i=1 It is obvious from equation (16) that solution curves of equation (14) are straight lines inside a box. These curves, which are directed towards the focal point of their respective boxes, originate from a different point on a threshold hyper- plane and might show corners [29]. As observed earlier, if the solution trajectory hits a threshold wall one may be at a loss for what to do. Sequel to this, a threshold wall is said to be trans- parent if solutions trajectory can be extended through it. That is to say, if trajectory can cross through it. A threshold wall is said Figure 1: Network Partitioned by two thresholds θ1 and θ2 for x1 and x2 respectively to be white if solution trajectory rebounds from it while a wall is said to be black if trajectories slide on it. These definitions are good but fail to capture qualities such as relapse in biological network, where a variable can regain or loss its position in the curse of discharging its duty. For black walls, Filippov proposes a way to define a vector field on it. This class of walls are con- sidered stable [13]. But white walls are never stable. Though the interest of this study is not in the stability or otherwise of these classes of domains, it is important to note that a definition that captures the actual happening within a threshold wall can really affect the stability properties of walls. For results on sta- bility (of box and wall), see Theorems 1 and 6 (respectively) in [13]. The definition of the threshold hypeprlane, depending on the behaviour of trajectory in the vicinity of the hyperplane, is of any of the following • if a threshold hyperplane is such that trajectories depart from it and enter the adjacent boxes, then it is said to be white • if a threshold hyperplane is such that trajectories ap- proach it from either adjacent box, then it is said to black • a threshold wall that is neither black nor white is said to transparent As shall be shown later, this classification in a ’conventional sense’ does not capture intrinsic behaviours in some systems, and as a result, this study shall propose new definitions that can capture such qualities. 2.2. Filippov’s First Order Theorem ([12]) Let x ∈ Σ and let n(x) be the normal to Σ at x. Let nT (x) f1(x) and nT (x) f2(x) be the projections of f1(x) and f2(x) onto the normal to the hypersurface Σ, where nT (x) is the transpose of n(x) . (a) Transversal intersection exists if, at x ∈ Σ , [nT (x) f1(x)][n T (x) f2(x)] > 0 (18) (b) Sliding mode exists if, at x ∈ Σ , [nT (x) f1(x)][n T (x) f2(x)] < 0 (19) 4 Mbah et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 812 5 From equation (18), it is evident that either [nT (x) f1(x)] > 0 and [nT (x) f2(x)] > 0 or [nT (x) f1(x)] < 0 and [nT (x) f2(x)] < 0. As explained in [14] the flow will enter H1 or H2 respectively in each of those cases. For equation (19) to be satisfied, then [nT (x) f1(x)] < 0 (20) and [nT (x) f2(x)] > 0 (21) or [nT (x) f1(x)] > 0 (22) and [nT (x) f2(x)] < 0 (23) Equations (18), (20) and (21), and (22) and (23) are used to characterise the hyperplane Σ as transparent, black and white respectively. Transparent hyperplane is same as transverse in- tersection given by equation (18), equations (20) and (21) gives the condition that guarantee a hyperplane to be black known as sliding hyperplane and equations (22) and (23) produces white wall called repulsive sliding hyperplane, see [14]. On the threshold hyperplane, Filippov’s convex function has been defined as follows [12, 13] fw = (1 −α) f j + α fi (24) where α(x) = nT (x) f j(x) nT (x)( f j(x) − fi(x)) (25) and 0 ≤ α(x) ≤ 1. w stands for wall, i : xi < θi and j : x j > θ j Filippov’s First Order Theory shall be used to propose new characterisation or definition for the nature of threshold hy- perplanes in networks that exhibit the properties under study. These definitions shall focus on the existence of convex con- stant α(x) given by equation (25) which is verifiable rather than the definitions presented earlier in subsection 2.1. 3. Result and Analysis This section presents the result of this study. The results come in the form of proposition that will give new characterisa- tion of walls and their nature. Examples of these walls are the threshold hyperplane, x1 = θ1 and x2 = θ2 shown in Figure 1 and defined as before. Sequel to the presentation shall be anal- yses of some networks (using the new results from this study) to expose the weakness in the existing definitions thereby high- lighting the significance of our study. 3.1. Proposition I: Nature of switching hyperplane A characterisation of switching hyperplane (that is wall) is presented using the vector field equation (14) in the boxes adja- cent to the dswitching hyperplane with the assumption that nT as defined in subsection 2.2 is non-negative. Let B j = {x ∈ Xn : xi < θi} and Bk = {x ∈ Xn : xi > θi} be two boxes adjacent to a threshold wall xi = θi. Given that the conditions contained in equations (24) and (25) are satisfied by (20) at the threshold hyperplane Σ where xi = θi, then the nature of the hyperplane Σ can be defined as any of the following • Σ is said to be transparent if f B j i = f Bk i (26) • Σ is said to be white if f B j i > f Bk i (27) • Σ is said to be black if f B j i < f Bk i (28) where f Bτi refers to vector field equation (14) for xi in the box τ = i, j 3.2. Proposition II: Nature of walls Another way of characterising a switching hyperplane using the focal points of boxes adjacent to the hyperplane is presented here. Let Σ be a hyperplane where only one variable, xi, is singular (that is where xi = θi) and B j = {x ∈ Xn : xi < θi} and Bk = {x ∈ Xn : xi > θi} be two boxes adjacent to Σ with φ j and φk the respective focal points of the boxes adjacent to Σ. Then • Σ is said to be transparent if xi < θi ⇒ φ j > θi and xi > θi ⇒ φk > θi (29) or xi < θi ⇒ φ j < θi and xi > θi ⇒ φk < θi (30) In the event of (29), the new definitions which this study proposes is that the wall is transparently increasing while for (30) it is said to be transparently decreasing. • Σ is said to be white if xi < θi ⇒ φ j < θi and xi > θi ⇒ φk > θi (31) • Σ is said to be black if xi < θi ⇒ φ j > θi and xi > θi ⇒ φk < θi (32) 3.3. Proposition III: Number of switching hyperplanes The stability of network of the nature considered here has been discussed [30, 31, 13]. The equilibrium of this network revolves around the focal points in the two domains - regulatory and switching respectively [13, 30]. The total number of regulatory domains has been obtained [28]. If the stability of this system can be discussed with respect to its switching domain, it will not be out of place to characterise singular domains of all orders. So, this subsection is for such result. First, number of walls is presented followed by pencils, centres and number of regulatory domains where k variables switch. 5 Mbah et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 812 6 3.3.1. Walls Let (13) be such that only one variable out of n variable switch at a time, where each variable has mi thresholds. Then the number of walls belonging to xi = θi is mi n−1∏ j=1 ( m j + 1 ) (33) where j = 1, 2, · · · , i − 1, i + 1, · · · , n The total number of walls in such network is given n∑ i=1 [ mi n−1∏ j=1 ( m j + 1 )] (34) 3.3.2. Pencils Let equation (13) be such that each variable xi has mi thresh- olds at which it interacts with other variables in the network. The number of pencils in such a network where k variables switch at a time is k∏ s=1 ms n∏ r=k+1 ( mr + 1 ) (35) The total number of such pencils is n∑ i=1 [ k∏ j=1 m j n−k∏ j=k+1 ( m j + 1 )] (36) 3.3.3. Centres The number of centres in a network with n variables is n∏ s=1 ms (37) 3.3.4. Regular Domains where k variables switch The total number of regular domains in a network where k variables switch at a time is n−k∏ i=1 ( mi + 1 ) (38) 3.4. Verification of Results Three networks to illustrate the effectiveness and advan- tage of our results over existing results are presented here. To achieve this, the following network examples from [25, 30, 32] are considered. The walls of the network from [32] agrees com- pletely with our result on the definition of walls. The second ex- ample from [25] showed a wall as decreasing instead of increas- ing which changes and affects the qualitative properties of the network. In the third example [32], two new non-transparent walls were discovered. These were obtained from the result of this study. 3.4.1. Example Networks The three example networks are as follows Example 2 [28] ẋ1 = Z1 + Z2 − 2Z1Z2 −γ1 x1 ẋ2 = 1 − Z1Z2 −γ2 x2 Example 3 [26] ẋ1 = k1Z̄2Z3 −γ1 x1 ẋ2 = k2Z2Z3 −γ2 x2 ẋ3 = k3(Z̄1 + Z2 − Z̄1Z2) −γ3 x3 Example 4 [13, 27] ẋ1 = k1Z 1 1 + k3Z 2 2 − x1 ẋ2 = k2Z 2 1 + k4Z 2 2 − x2 Example 2 is a network that has two variables with a thresh- old associated with each of the variables. Using the result of equation (33), the number of thresholds walls associated with these thresholds is 1(1 + 1) × 1(1 + 1) = 4. This is as shown in Figure 1. The network of this example has no pencil and just a centre. With the use of equation (8) in this example, the vector equation and their focal points inside the boxes Bi, i = 1, 2, 3 and 4 as defined by equations (14) and (17) respectively are as given below. B1 : ẋ1 = −γ1 x1; ẋ2 = 1 −γ2 x2 and (0,γ −1 2 ) (39) B2 : ẋ1 = 1 −γ1 x1; ẋ2 = 1 −γ2 x2 and (γ −1 1 ,γ −1 2 ) (40) B3 : ẋ1 = −γ1 x1; ẋ2 = −γ2 x2 and (0, 0) (41) B4 : ẋ1 = 1 −γ1 x1; ẋ2 = 1 −γ2 x2 and (γ −1 1 ,γ −1 2 ) (42) Threshold hyperplanes (walls) of this network as described in example 1 has the following characterisations: D1 1 = {x1 < θ1, x2 = θ2} is transparent. D2 1 = {x1 > θ1 x2 = θ2} is black. D3 1 = {x1 = θ1, x2 < θ2} is white. D4 1 = {x1 = θ1, x2 > θ2} is black. D11 is described as transparent in [32]. The focal points of the switching variable x2 in boxes adjacent to D11 ,which are B1 and B4, given by equations (40) and (41) satisfies propositions I and II. For instance, x2 has the same focal points (which is greater than zero) in each of the said boxes. Inside B1 , x2 < θ2 but φ2 > θ2 (see [32] for the value of the constant) satisfy- ing proposition II. Again, the production function for x2, the switching variable, in the two boxes are the same (1 in each case). This satisfies Proposition I as well. The same analy- sis can be carried out for all the other threshold hyperplanes, 6 Mbah et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 812 7 D21, D31 and D11 to see that the proposition is effective in char- acterizing threshold hyperplanes. Example 3 has three (3) variables with a threshold each. The use of proposition III shows that there are a total of twelve (12) threshold hyperplanes in the network. Each of the thresholds for the variable has four (4) walls associated with it. When the result of monotonicity of transparent walls is applied to these walls, it contradicts the result presented in [25].To see this, we refer to Figure 2 shown below. The use of (29) or (30) on these walls given as the edges of the cube in Figure 2 reveals that the wall be- tween the box B2 := x1 < θ1, x2 > θ2, x3 < θ2 and B3 := x1 > θ1, x2 > θ2, x3 < θ2 is not increasing in nature as shown in the flow diagram in [26]. Our result shows that the wall is actually transparently decreasing which produces Figure 2. From this it can be seen that the box (node), indicated as a source in [26] is not so. This node is the box B2 represented as (010) on the flow diagram shown in Figure 2. The node that is a source from the result of this study is the box B3 represented as (110) on Figure 2. To see clearer the usefulness of the results of this study, consider the focal point of these two boxes obtainable as (0, 0, k3γ−13 ). This means that the two boxes have the same focal point. From this focal point one sees that x1 < θ1 ⇒ φ1 = 0 < θ1 and x1 > θ1 ⇒ φ1 = 0 < θ1 which satisfies equation (30) and shows that the said wall is transparently decreasing. Again, the focal point of the box B2 defined as a source by the flow diagram of [25] is 0 and contradicts that box as a source because the switching variable x1 is trapped within that box at all times. The box B3 , on the other hand, has the focal points of all the variables exiting the domain in finite time as each variable switch, showing that the said box is a source indeed. The focal points of these boxes as obtained agree quite well with those reported in [13] as well. Finally, example 4 is a two-dimensional network whose variables have two thresholds each. The phase space diagram of the network is presented in Figure 3. As is evident from Fig- ure 3, this network has eight boxes denoted B1 to B9. There are twelve one-dimensional domains (that is, walls) which are the edges of the corresponding boxes bounded by the thresh- old values, θ1 and θ2 respectively. The intersection points of the thresholds, (θi1,θ i 2) where i = 1, 2, is the centre and there are four of these centres in the network presented in example 4. See Figure 3. This can be verified using equations (36) and (37) of proposition III. Each threshold has three walls associated with it while each variable has six walls associated with it. Testing these walls to characterize them as black, white or transparent, using proposition I, revealed that there are two walls previously considered transparent in [13] which are actually white in na- ture. These are the walls between B3 and B4 on one side, and B7 and B8 on the other. To see this, observe that the vector motion for x2 which switches between the boxes B3 and B4 in these boxes are respectively ẋ2 = k2 + k4 − x2; ẋ2 = k2 − x2 (43) while for the wall belonging to x1 between the boxes B7 and B8 Figure 2: Flow Diagram of Example 3 Network Figure 3: Phase Space Diagram of Example 4 Network the vector equations are respectively ẋ1 = k1 + k3 − x1; ẋ1 = k1 − x1 (44) Using proposition I shows f B42 − f B2 2 = k4 and k4 , 0, meaning that the walls is not transparent. As such α defined by equation (25) can be found such that equation (24) can be defined on the wall. Similar thing can be conducted on equation (44) to show that the wall is not transparent. This notwithstanding, Proposi- tion II can be used to characterize the said wall. For this, the focal point of x1 in the boxes of interest are φ B8 1 = k1 + k3; φ B7 1 = k1 (45) Apply proposition II to equation (45) to see that x1 < θ11 ⇒ φ B7 1 = k3 < θ 1 1 and x1 > θ 1 1 ⇒ φ B7 1 = k1 + k3 > θ 1 1 , (see [28, 27] for details on the relationship between rate constants and threshold values). Thus, a conclusion can be reached that the wall is white and not transparent. These walls though white can have Filippov vector motion de- 7 Mbah et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 812 8 fined on them using equations (24) and (25). This non-unique vector motion can be obtained as follows W1 : (ẋ1, ẋ2) = (0, k4 − x2) (46) W2 : (ẋ1, ẋ2) = (k4 − x2, 0) (47) Using the characterisation in propositions II, the flow dia- gram shown in Figure 4 is obtained. 4. Discussion A regulatory domain (that is, box) in piecewise linear mod- els of the nature discussed here can be stable or not. It is said to be asymptotically stable if its focal point is contained in the box as t →∞, otherwise it is said to be unstable. The same concept was adopted in [13] to study the stability of non-transparent walls. That is, a threshold wall is said to be stable if the fo- cal point of the vector defining motion on it belongs to the set defined by the closed convex hull of equation (6) in finite time. If it is such that it resides on the wall as t → ∞, then it is asymptotically stable. Based on this, it is expected that the qualitative properties of the network in example 3 should change due to the wrong classification of the walls described above. The stability analysis of the walls of the network which depends on the nature of walls is seen to be greatly affected. This will in turn affect cycles and some other properties of the network. It then follows that the definition proposed in this net- work is better and should be preferable as it can actually bring out some hidden qualities in the networks. From these qualita- tive properties, one can study the behaviour of the variables. As seen from the result some walls were defined in a way that does not capture the intrinsic behaviour associated with bi- ological regulatory networks. Such behaviours include relapses in elements involved in the regulatory activities. This is one of the reasons the two walls in example 4 previously thought to be transparent were not but white. Our conjecture in this case is that the focal point of the switching variable could not attain the threshold boundary let alone crossing it. As it approaches the threshold vicinity, relapse occurs in the production activity which decreases rise in variable’s concentration that result in failure to stimulate switching which caused its focal point to drift away from the wall. Knowledge of such properties can help in identifying where there should be search for failures in such systems The implication of equation (26) which is one of the results of this work is that if no α exists with respect to the wall of in- terest such that equation (24) holds, then the wall is transparent, a criterion for searching for transparent walls. Also by provid- ing equation (35), one can know switching domain of any order k < n, in terms of number. The study as presented used step function as threshold func- tion which is limiting because it is a crude approximation of happenings within the vicinity of thresholds. Smooth functions such as Hill function is known to be better in studying systems with steep behaviour. We limited this study to step function be- cause of the interest of the work which is to classify the thresh- old hypeplanes. It is the reason we proposed new definition Figure 4: Flow Diagram across the Walls of Example 4 Network based on the position of focal points in adjacent boxes to thresh- old walls to take care of relapses and rebounds in the vicinity of the threshold. One of the questions that arise from this study is how the new definition affects stability of singular solutions at threshold boundaries. For instance, solutions are not expected to stay on transparent wall. The two walls identified as white here were not studied for stability in [13]. Also the qualitative properties too have to change. Currently a work on the devel- opment of the propositions into theorems that can take care of stability and related qualitative properties is under way. Fur- thermore, with knowledge of how many singular domains are obtained in a network, one can study the stability of the entire system by investigating the relationship between the stability of regular and singular domains. 5. Conclusion The work presented in this paper is piecewise linear models of threshold dependent networks. Such network is associated with rectangular regions bounded by the threshold hyperplane of the network. Dynamics of the system inside these rect- angular regions can be given by piecewise linear differential equations (PLDE). These threshold hyperplanes as defined before failed to capture the real behaviour of the network in their vicinity. By using Filippov’s method we proposed a definition which identified correctly some hyperplanes not properly defined before. The consequence of our condition for transparent walls is that the first task in walls analysis should be to test for transparency or not. By so doing, one can then know which walls to investigate for qualitative properties. Classifying transparent wall as decreasing or increasing reveals the behaviour of variable in the vicinity of threshold walls. This classification means that one can easily identify which variable is gaining or losing in function at the walls and as such take appropriate decision on what to do to obtain a better result at such places. The characterisation of nature of walls presented in propo- sition I and II is considered from the point of view of focal 8 Mbah et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 812 9 point and vector field of regulatory domains adjacent to the wall respectively. It is derived from Filippov’s first order theory presented in section two. Furthermore, the result on switching domain obtained has not been given before to the best of our knowledge. It then follows that the results presented in this work, which is novel and applicable to real life problem, can be used with ease by non-mathematics researcher. 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