cellular systems—ii. stability of compartmental systems

Cellular Systems-II. Stability of Compartmental Systems*

G. S. LADDE

Department of Mathematics, The Slate University of New York at Potsdam,

Potsdam, New York 13676

Communicated by R. Rosen

ABSTRACT

In this work, by employing the concept of vector Lyapunov functions, and the theory

of differential inequalities, the stability analysis of compartmental systems is initiated in a

systematic and unified way. Furthermore, an attempt is made to formulate and partially

resolve the “complexity vs. stability” problem in open compartmental systems. The recent

stability results obtained for open chemical systems are applied to open compartmental

systems in a natural way. As a byproduct of our analysis, we obtain an estimate for

washout functions. Finally, it has been demonstrated that the stability analysis of

compartmental systems provides a common conceptual framework for innumerable,

apparently unrelated systems in biological, medical, and physical sciences.

1. INTRODUCTION

The concept of compartmental processes provides suitable means of description for several wide ranges of dynamic processes in biological, physical, and social sciences. A similar fact has been stated by Rosen [22] with respect to morphogenetic and two-factor processes in a systematic and unified form. Furthermore, the scope of compartmental analysis has been well documented in the recent monographs by Jacquez [7] and by Rosen [23], in an excellent paper by Hearon [4], and also in the monograph by Rescigno and Segre [18]. However, a systematic stability analysis of compartmental systems awaits development. An attempt in this direction has been made by Hearon [4] in connection with the stability of the steady state of linear compartmental systems. For nonlinear compartmental systems, Bellman [2] was the first to formulate the concentration-dependent trans-

*The research reported herein was supported by SUNY Research Foundation Faculty

Fellowship No. 0454-O l-03 1-74-O.

MATHEMATICAL BIOSCIENCES 30, l-21 (1976)

0 American Else&x Publishing Company, Inc., 1976

1

2 G. S. LADDE

portation rates and also study the stability of the steady state of closed compartmental systems. In a recent work, Jacquez [7] has used linearized approximations to show the stability of the steady state of nonlinear open compartmental systems. However, these results lack the realism and structural insight of the stability analysis. Lyapunov’s second method has been utilized successfully for the stability study of chemical systems [22, 25, 331, controlled biological systems [31, 321, ecosystems [ 121, etc.; however, this method has remained unused for the stability study of compartmental systems, even though it has been recognized that chemical systems [4, 7, 221 and ecosystems [4, 13, 221 can be viewed as compartmental systems.

Because of the fundamental importance of open compartmental systems for demonstrating analogies between systems at different levels of biological and physical organization, we propose to extend our stability analysis of chemical systems [8] to compartmental systems. This fact will support a very fundamental and important attempt made by Rosen [21, 221 to unify innumerable, structurally diverse dynamic systems in biological and physical sciences. As remarked by Rosen [22], “such a unification, with the resultant economy of thought, seems absolutely necessary in these days of explosive growth in scientific knowledge, if the student is not to be hope- lessly submerged under the torrent of specific but specialized investigations relevant to his interest. Indeed, unless expanding knowledge is accompanied by corresponding conceptual economies, a comprehensive understanding of even a small corner of science may become impossible before much longer.“.

Our present stability analysis is the continuation of our stability analysis of chemical systems [8], and it is strongly influenced by the remarkable work of Hearon [4], Jacquez [7], Rescigno and Segre [ 181, and Rosen [21, 221. Furthermore, two of the important classes of dynamical analogies, namely, morphogenetic systems [22, 231 and two-factor systems [19, 221, have been realized as hierarchic compartmental systems. It is important to note that our method of study does not demand either linearization approximations or reduction techniques for analyzing the stability properties of open compartmental systems.

For linear, nonlinear, and hierarchical open compartmental systems, our stability analysis is analogous to the corresponding stability analysis of linear, nonlinear, and hierarchical open chemical systems [8]. In fact, it is a generalization of the first part of our work [8], so as to include wide varieties of dynamic systems in life and physical sciences.

The paper is organized as follows: In Sec. 2, we recognize the fact that the transport rate matrix of the

linear compartmental system is the Metzler matrix 1141 that is used by several economists to obtain strong stability results for economic systems.

STABILITY OF COMPARTMENTAL SYSTEMS 3

In the present work, it is shown that for the linear compartmental system,

the quasidominant diagonal property of the corresponding transport rate matrix is necessary and sufficient condition for stability of the steady state of the linear compartmental system. Based on our study, we conclude that the dominant diagonal property enables us to draw conclusions about the tolerance for increased complexity of stable open compartmental systems. Furthermore, we also recognize a “complexity vs. stability” problem similar to that in open chemical systems [8] and show that the dominant diagonal property is a suitable mechanism for partially resolving this problem. However, for more details, we refer to [8]. In Sec. 3, nonlinear compartmental systems, and in Sec. 4, hierarchic compartmental systems are briefly formulated, and further to avoid tedium, detailed mathematical analysis,

remarks, and conclusions are summarized by reference to the first part of this work [8]. Furthermore. our analysis suggests a notion of quasi-washout function which includes the usual notion of washout function [5], and it gives us an estimate of it in a natural way. In Sec. 5, the scope of our stability analysis of hierarchical compartmental systems is exhibited by identifying classes of dynamical analogies-for example, morphogenetic systems [22, 231 and abstract two-factor systems [19, 22]-as hierarchic compartmental systems. In addition to this, the stability study of several important network problems [19, 20, 221, such as the epigenetic network, simple series network, discrimination network, and learning network, is given as a special case of the stability study of compartmental networks or hierarchical compartmental systems.

2. LINEAR COMPARTMENTAL SYSTEMS

Let us consider a compartmental system consisting of n compartments described by a linear differential equation

i=Ax+a.

where x E R” is a state variable n-vector x =(x1,x2,. .,x,)‘, and T stands for transpose of a vector. For i~1. X, represents the concentration of a labeled observable quantity (e.g., a drug), or the strength of a labeled observable quality (excitatory, or inhibitory, etc.) in the ith compartment, where 1={1,2,..., n}. A = (ati) is the n X n constant observable quantity (or quality) transport rate matrix, and a E R” is the constant input rate vector from external sources. In view of the physical significance, the elements of the matrix A must satisfy the conditions:

ai, < 0, iEI (2) -

4 G. S. LADDE

and

ay > 0, i#j, iJ E I. (3)

From (2) and (3) we immediately observe that the constant observable quantity (or quality) transport rate matrix A is a Metzler matrix [14]. Throughout our discussion, we assume that the linear compartmental system (1) is open. See [7, 18, 231.

It has been observed [ 1, 21 that the solution x(t, to,x,,) of (1) is non- negative for all tat,, (t,,x,)ER+XR;, where R+=[O,cs), R;={xE R” :x, > 0 Vi E I}. Furthermore, the steady state x* of (1) is equal to -A -]a whenever A is nonsingular. Therefore, the stability of x* = -A -‘a

of (1) is equivalent to the stability of x* = 0 of

.k=Ax. (4)

For more details, see [8]. By following the argument used in [8], we conclude that the steady state

x* = 0 of (4) is stable [22] if and only if the Metzler matrix A is a Hicks matrix [ 141.

The main objective of the present work is to study the effects of the strength of the interactions among the compartments on the stability of the steady state of an open compartmental system under structural perturbations. In order to study the effects of the strength of interactions among compartments on the stability of the steady state x* in (1) let us assume that the elements av of the matrix A have the following form:

(5)

In (5) eG are elements of the n x n interconnection matrix E =(e,), and they can take on values between zero and one. By eV we represent the strength of connection of the compartment x, with the compartment xi, ranging from disconnection (eV = 0) to full connection (eU = 1). Furthermore, the interconnection matrix E represents the structural changes in the interactions among the compartments in (1). In particular, the shutdown of a pathway from ith compartment to the jth compartment in the compartmental system is represented by e,, = 0. The disappearance of the pathways between the ith and jth compartments in the system is shown by eU = eji = 0.


The disconnection or disappearance of the pth compartment in the compartmental system (1) is represented by eiP = ePj = 0 for all ij E I. If ePi = 0 for all in Z, then the pth compartment is called a pure source compartment [24]; if eip =0 for all in I, then the pth compartment is called a pure sink compartment [24]. If e,P=ePi=O forp#i-l,i+l,pEZ, 2<i<n--1, and

elP = ePi = 0, p # 2, p E I, and e,,, = 0, then the compartmental system is a

catenary system [7, 18, 231. If ePi = eiP =0 for p # 1 and p E I, 2 Q i < n, then the open compartmental system (1) corresponds to a mammillary system of n + 1 compartments [7, 18, 231. If e,, = 1 for all i,j E I, then the compartmental system is a strongly connected system [18, 231. The diagrammatic discussion of these structural changes is given in the Figs. l-9.

FIG. 1. An isolated compartmental sys- FIG. 2. A strongly connected compart-

tern. mental system.

FIG. 3. The shutdown of a path- FIG. 4. The shutdown of

way from the ith to thejth compart- ways between the ith and jth

ment. partments.

path-

com-

6 G. S. LADDE

FIG. 5. The disconnection of the FIG. 6. The pth compartment is a

pth compartment. pure source.

FIG. 7. The pth compartment is a pure sink.

FIG. 8. An open catenary compartmental system.

STABILITY OF COMPARTMENTAL SYSTEMS

FIG. 9. A mammillary compartmental system of n + 1 compartments.

Further note that in (5), (Y; is the rate of excretion from the ith compartment (or rate of loss to the environment), and aij is the rate at which material moves from the jth compartment to ith compartment.

In this section, we investigate the stability analysis of (1) under structural perturbations by means of algebraic conditions. Very recently, such conditions were utilized for studying the connective stability property of chemical systems [8] and model ecosystems [lo, 27, 281. Note that we need the concept of fundamental interconnection matrix E [26] and the concept of connective stability [26], which have been utilized in our earlier work, in order to study the stability analysis of (1) under structural perturbations.

Now we present sufficient conditions for the connective stability of the system (1). Recall that the Metzler matrix is a Hicks matrix if and only if it is a quasidominant diagonal matrix [14], i.e., there exist numbers cr >O, di > 0, i E Z such that

Iu,I-q’ i dilaql>a, jEZ. (6) i=l i#j

Note that the conditions (6) include the usual diagonal dominance:

as a special case whenever 4. = 1. The condition (7) was utilized by Hearon

8 G. S. LADDE

[4] to deduce the stability of linear systems. As noted in [8], to deduce the connective stability of (1) it is enough to assume that (6) holds for a given fundamental interconnected matrix 6.

Analogously to linear chemical systems [8], we recognize the “complexity vs. stability” problem, and conclude that so long as the interaction strength satisfies (6), the stability of (1) -is assured. Thus the dominant diagonal property provides a suitable mechanism for partially resolving this problem. Furthermore, for the linear compartmental system (1) the relation (6) is a necessary and sufficient condition for the stability of the steady state x* of (1). For further details see the first part [8].

3. NONLINEAR COMPARTMENTAL SYSTEMS

Consider the nonlinear time-varying compartmental system described by the system of differential equations

i=A(t,x)x. (8)

Here x E D (0, p) is the state variable vector, where D (0, p) is defined by

D(0,p)={x~R":~x~~<p;Vi~Z}, (9)

with pI >0 VI’E 1. The n x n transport rate matrix function is A : R, x

D(O,p)+R"'. We shall suppose that A is smooth enough to assure the existence and uniqueness of solutions x( t, te, x0) of (8) for all t 2 to, (t,,x,) E R, X D (0,~); further assume that it possesses non-negative solutions whenever the initial state is non-negative.

It is well known [22] that nonlinear open systems admit a multiplicity of steady states. Therefore, it is essential to study the stability behavior in the vicinity of a given steady state. In view of the discussion in [8], it is enough to assure that x* = 0 is the only steady state of (8) in D (0, p).

The stability analysis of (8) cannot be formulated in algebraic terms as is possible for the linear constant systems (1). Therefore, we use Lyapunov’s second method [I 1, 221 to deduce the stability of the steady state x*=0 of (8). As far as I am aware, this has not yet been done for nonlinear compartmental systems. Our stability analysis confirms the stability analysis conducted by Bellman [2] for closed compartmental systems.

In order to derive the conditions for connective stability of the compartmental system (8) we assume that the elements aU(t,x) of the transport rate matrix function A (t,x) have the following form:

-+i(ttx)+ c ekr(fhki(t3x.)+ IZ ek,(t)+k,(t,x>, i=j a,(r,x) = kEI-(I) kEl(i)

e,(~)+v(fyx), i#j,

(10)


for all (t,x) E R, x D (O,p), where $I~,@~ : R, X D (O,p)dR are functions of

(t,x), and V~EZ, Z-(i)={kEZ:k#i, (~~~(f,x)<O for all (t,x)ER+x

D(O,p)}, I(i)= I\Z-(i). We note that the set Z-(i) can be empty set, and also note that +i is the rate of excretion from the ith compartment (or rate of loss to the environment), and $Q is the rate at which the observable quantity or quality moves from thejth compartment to the ith compartment (a negative value signifying transfer in the other direction); the functions eU : R + +[O, I] are elements of the n X n interconnection continuous matrix function E(t) = (e,( 2)). The usefulness of the special nature of the transport

rate matrix A (2,x) and interconnection matrix function E(t) can be described in the framework of the earlier work [8, 10, 281.

We assume that the functions +,,G~ in (10) satisfy the constraints

+,(2,x) 2 a;aekiPki Q - eki(9+ki(Lx) G ekiakr for kEZ_ (i)

and

leki(thki(t,x)l Q ski for kEZ(i)and(t,x)ER+XD(O,p), (11)

where (Y~ > 0, Rki > 0, (Yki > 0, i E I, and

I > 2 OLki. (12) kEI(i)

For the significance as well as some implications of this condition, see [8]. By K= (aO) we represent the n x n constant matrix defined by

(13)

i#j,

where g0 are elements of the fundamental interconnection matrix E as defined in [8, 261 and (Y,,N~,R~~ are as defined in (11).

Consider the linear autonomous auxiliary or comparison system described by the differential equations

ti=Ku, (14)

where ME R:, and 2 is as defined in (13). From (12) and (13), it is obvious that K is a Metzler matrix.

10 G. S. LADDE

In the following, we state the connective stability results (local as well as global) with respect to the system (8). Their proofs follow directly by imitating the proofs of Theorems 1 and 2 in [8].

THEOREM I

Assume that the system (8) satisfies the constraints (11). Then the quasi-

dominant diagonal property of the n X n constant Metzler matrix 2 in (14) implies the exponential connective stability of the steady state x* = 0 of the

system (8).

THEOREM 2

Let the hypotheses of Theorem 1 be satisfied with pi = 00 for every i E I, so that D(O,p)=R”. Then the quasidominant diagonal property of the n ~‘n constant Metzler matrix A- in (14) implies the global exponential connective stabilip of the steady state x* = 0 of the system (8).

Further note that all remarks and observations made relative to nonlinear chemical systems [8] can be formulated analogously with respect to nonlinear compartmental systems (8).

REMARK 1

We observe that if Z(i) = {i} and cuii=O (no self-interactions), then the compartmental system (8) behaves analogously to the linear compartmental system (1).

REMARK 2

With respect to the compartmental system (8) the relations (26) and (29)

in [8] gives

~txtt,tO,xo)) G v(x,)exp[- a(t- to)], t > t,, (15)

whenever X,E RJO,p), where R,(O,p) and V(X) are defined in [8].

Now, we define the following concept:

DEFINITION I

The function Y(X): R”-tR+ defined by

Y(X)= i d;\xil=dT.IxJ ,=I

(16)

is called a quasi-washout function, where d=(d,,dz,. .,dJ’, di>Oo, and

I~I=~I~~1~/~*l~..~rl~,I~=.


REMARK 3

We observe that the concept of quasi-washout function includes the well known concept of a washout function [5, 71, g(x) = 1.1~1, as a special case whenever d, = 1 for all i E I.

REMARK 4

In view of Remark 3, the relation (15) gives the estimate for quasi- washout functions, in particular washout functions, with respect to the nonlinear compartmental system (8). This is one of the important byprod- ucts of our stability analysis.

4. HIERARCHICAL COMPARTMENTAL SYSTEMS

Several compartmental systems in biological and chemical sciences are hierarchical in nature. The analysis of such complex systems has been performed by employing different kinds of approximation techniques [4, 6, 7, 181.

In the present section, we would like to preserve to some extent the original character of hierarchical compartmental systems with respect to their stability behavior. This we achieve by dividing a large compartment into a relatively small number of groups and decomposing the compartmental system into interconnected compartmental subsystems. Then the stability of the entire compartmental system is established in terms of the stability of the compartmental subsystems and the stability of the aggregate or comparison or auxiliary compartmental system that is formed by the decomposition-aggregation technique. More details about the important features of this method have been discussed in our first part. For details, see

[gl. Let us consider a very general open compartmental system described by

the system of differential equations

.i= k(t,x) (17)

where x E D (0, p) is the state variable, and the transport rate vector function is k:R+xD(O,p)-+R”; D(O,p)={x~R”:I(x~ll<p, VieZ), X,E

RYZ:,, m, = m, with x = (x,‘.x:, . . . , xf)‘. Here T stands for the transpose of a vector or matrix; for iEZ, p, is some positive real number. We shall suppose that k is smooth enough to assure the existence and uniqueness of solutions x(t,t,,x,) of (17) for all t> to, (t,,x,)ER+ x D(O,p); further assume that x(t, t,,,xa) > 0, for all t > t,, whenever (t,, x0) > 0. For more detailed conditions, see [2, 3, 91.

We briefly formulate the decomposition-aggregation technique relative to (17). For more details as well as comments, see [8].

12 G. S. LADDE

Let us decompose the system (17) into n interconnected subsystems described by

xi=fi(t,Xi)+hi(t,x), iE I, (18)

wherexi~D(O,p,)={xi~R”+:JJxiJJ<p,} and

x= x,r,x: )...) ( XnT) =. (19)

(See Fig. 10.) In (18), the function 4 : R, X D (O,p,)-+ R 4 represents the interactions among compartments within the ith subsystem, and the function hi : R, x D (0, p)-+Rq describes the interactions between the subsys-

tems. We assume that

hi(r,-“)=~,(t,ei,x,,ej2x2,...,ernx,), i E I, (20)

l--------------

-___-___---- .-I

FIG. 10. Hierarchical compartmental system.

STABILITYOFCOMPARTMENTALSYSTEMS 13

where eU : R++[O, 1] are coupling functions which are elements of the n x n continuous interconnection matrix function E(t).

Without loss in generality, we assume that f,(t,O)zO, h,(t,O)=O. This assures that x* =0 is the steady state of the system (18).

When E(t) -0, from (18) we get the isolated subsystems described by

&=f,(r,x;), i E I, (21)

which has the steady state XT = 0, i E I. For each compartmental subsystem in (21) we assume that there exists

a scalar function Vi : R + x D (O,pJ+ R + such that Vi E C(‘,‘)[ R + x

D (O,p,), R,]. Further assume that the scalar functions Vi(t,xi) and the interaction functions h,(t,x) satisfy the following conditions:

Di2l)K/i(ttXt)G -~,(v,(t,~i))~i3(~,(t,~,)) (23)

for (t, x,) E R + X D (0, pi); and that for (1, x) E R + X D (0, p) in addition,

[grad Vi(t,x,)I.hi(f,x) 4 - 2 I

%iPki ( ‘i (‘Yxi)) kEI_ (i)

+ 2 ?kiaki( V(ttx>) &3( vi(lTXi)) kEI(i) 1

(24) j=l Jfi

where V(t,x), G,, q,aii,&, and +i3 are as described in [8], and where for FEZ, kEZ_(i), we have Pkr~C[R+,R+].

We define the n x n matrix function A(u) : R:+ Rf12 with the coefficients

a,(u)= 1 [ - %(4) + E eki& (%> + IT2 ckiaki(U), i=j,

I iEI- (i) 1 iEI(i) (25)

i#j,

where oi and c+pki are defined in (23) and (24) respectively. We further

14 G. S. LADDE

assume that

and x(u) is a quasidominant diagonal matrix on R:.

Consider the comparison compartmental system described by the system of differential equations

ic=k(u)w(u), (27)

where u E R:, x(u) is as defined in (25) and

Now, one can imitate Theorems 3, 4, and 5 as well as the remarks and comments relative to hierarchic chemical systems [8] for hierarchic compartmental systems (17). We omit the details.

REMARK 5

A remark analogous to Remark 1 can be made with regard to (17).

Now we generalize the definition of a quasi-washout function as follows:

DEFINITION 2

The function v(x) : R”-+R+ defined by

v(x)= i: d;vi(t,Xi)=dT~V(t,X) (28) i=l

is called a generalized quasi-washout function, where d = (d,, d,, . . . , d,JT,

di > 0, and V(t,x) is defined in (24). If 4 = 1, then v(x) in (28) is said to be a generalized washout function.

REMARK 6

From (48) in [8] and (28) we have

V(X(~,c3,-%)) G i: ri(t,t,,u,), (29) i=l

where r(t, t,, uO) is the maximal solution of (27). Thus (29) gives the estimate for generalized quasi-washout or washout function. This shows the usefulness of our stability analysis.


5. SCOPE OF HIERARCHICAL COMPARTMENTAL SYSTEMS

In order to show the scope of the stability analysis of hierarchical compartmental systems, we present two of the most important analogous systems, namely, morphogenetic systems [20, 22, 231 and two-factor systems [19, 221. Briefly, we exhibit the applicability of our results to these systems. More detail dynamical aspects have been presented by Rosen [22] in a unified and systematic way.

A. MORPHOGENETIC SYSTEMS

In a recent study [22] of morphogenetic systems, it is observed that the two-morphogen system can be generalized in a variety of ways. For more details, see [22]. From this observation, we recognize that a hierarchical compartmental system (17) represents the generalized morphogenetic system. We substantiate this assertion in the following discussion.

We identify a cell as a compartmental subsystem. Similarly, we identify a morphogen as a compartment. From this, the compartmental system (17) represents the dynamical system for the n-cell system. In (19), x E R”

denotes the state of the morphogenetic system, and the component xi E R q of x denotes the state of the ith cell for i E I. This implies that the state of the ith cell is determined by the number m, > 1 of morphogens. We note that the cells in the system need not be identical. The decomposition of the system (17) into n interconnected subsystems is described by (18). In (18), the function j; represents the nonlinear nonstationary interactions among the morphogens within the ith cell, and the function hi describes the nonlinear and nonstationary interactions between the cells. Furthermore, hi includes any sign pattern of the interactions, thus allowing for a mixed interactions such as diffusive-active interactions. The elements eU(t) of the n x n interconnection matrix function E(t) in (20) not only represent the structural changes in the interactions among the cells but also represent the topology of the array of cells. The equations (21) can be explained analogously. Hereafter, the system (17) or (18) will be called the morphogenetic system.

Now by following the rest of the argument used in the Sec. 4, we conclude that the stability of the steady state of the morphogenetic system (17) can be analyzed.

To illustrate the usefulness of our foregoing discussion, we shall briefly analyze some well-known [22, 231 special forms of the morphogenetic systems (17) in the context of our previous stability analysis.

The Rasevsky-Turing Type of System Let us consider with Rashevsky and Turing, the case of a ring of an n-cell as shown in Fig. 11. In the

16 G. S. LADDE

FIG. 11.

two-morphogen case, without loss in generality. the system is described by

~‘i,=ux,,+bx,,+D,(x,+,,+x r-l, -2x,,>,

~,2=~x,,+~x,,+~,(~,+,*+~,-,*-xi2), 1 iEI (30)

where x,,,xi2 are the concentrations of the two morphogens in the ith cell,

and a, h, c, d. D,, and D, are parameters as described in [22, 231. Setting

we have

The interconnection matrix E (1) = ( eq( r)) will be e, = eP, = 0, p # i - 1, i +

1.i. p E I. Note that 0 and n + 1 are identified with n and 1. If the steady

state of the isolated system (21) is asymptotically stable, then there exists a

Lyapunov function V, satisfying the properties described in Sec. 4. For

details see [l I]. The rest of the analysis follows analogously.

One may also apply the analysis of linear compartmental systems di-

rectly. For example. the stability condition (7) relative to (30) reduces to

O>a+lb] and O>c+ld(.


Note that these stability conditions are algebraically simple and attractive, even though they are restrictive.

Open Chemical Systems An excellent and interesting description of open systems as morphogenetic analogs is given by Rosen [22] in a very systematic way. From our previous discussion we conclude that our results are generalizations of our earlier results [8]. This fact also supports the remarkable observation made by Hearon [4] that the chemical species can be treated as compartments with the understanding that flow from one compartment to another means chemical conversion. Moreover, as indi- cated by Rosen [22, 231, and noted by Ladde [8], genetic systems and ecosystems can be treated as chemical systems. This fact confirms the analysis of Mulholland and Keener [13] of ecosystems as compartmental models.

E. TWO-FACTOR SYSTEMS

In 1933, a two-factor theory was initiated by Rashevsky to account for phenomena of peripheral nerve excitation. Recently, an effort has been made by Rosen [ 19, 221 to develop the relationship between current ideas of cellular regulation and the two-factory theory in a systematic and unified way.

We call a two-factor element or unit in the network a compartment. This idea of compartments in two-factor theory supports the analysis of regions of a dendritic tree as compartments [15] (as is noted by Hearon [4]), where each compartment may be subject to a different degree of inhibition or excitation. Following Rosen [22], xi = (xi,,xiJT is the state variable specify- ing the ith unit in a network, and without loss in generality, the dynamic equations of the network of n units may be put in the form

.k,=A’x,+h,(t,x), iE I, (31)

where

Ai= 4, ai

[ 1 41 a12

is a stable matrix [22], and h;(t,x) is as defined in (20). We note that the topology of the network is determined by interconnection matrix E(t). As noted in [22], (31) represents the most general dynamical system which can be built up out of n interacting units of dimension 2. A’x; represents the interaction between the state variables corresponding to ith unit, and h,(t,x) represents the interaction between units. For these interactions, A ‘xi, hi(t,x),

18 G. S. LADDE

Rosen [22] has proposed the term “generalized reaction-diffusion schemes”. From this observation, we note that (31) is indeed a special type of hierarchical compartmental system (17) that is, the system (18). Further note that A ‘x, =f;(t.x,), m, = 2 Vi E I, and m = 2n. The assumption that A is a stable matrix implies that the steady state of isolated systems (21) corresponding to (31) is asymptotically stable (221, which in turn implies that there exist functions V;(t,x,): R, x R *-+R+ satisfying the relations described in [8], Vi E I. See [ 111. Thus the stability of the steady state of (3 1) can be determined by applying one of the Theorems 3, 4, 5 in [8], depending on the nature of the constraints on h,(t,x,).

Now we will exhibit a few simple well known networks described by Rosen [19, 221 as special cases of (31).

Simple Series Networks In (31). we assume that

A!=[ -0”;’ _:;*I, a;,,a;,>o, iEI,

we a!so assume the elements e,(t) of the interconnection matrix function E(f) are such that e,,(t)=0 forj#i-1, ~EI; e,,_,(t)=1 for 2<i<n, and e,,(r)=O. Finally, we assume

h,(t,.x)= A, Y(t,x)

[ 1 B,Y(t,x)

Then the network (31) represents the simple series shown in Fig. 12. Here x,, and _Y,* represent the excitatory and inhibitory factors respectively of the ith compartment or unit. This type of network is very important in the theory of nerve conduction, and it has been called a prototype [22] of all the various phenomenological models for the conduction of excitation along a nerve fiber.

Discrimination Network In (31) we assume that


and the elements e,(t) of interconnection matrix E(t) satisfy the relation e,(l)= 1, i+j, and eii(t)= 0, for i,jE I. We also assume that

hi(t3X)= :,y(t x, ,) [, )I+ I )

noting that unit 3 is identified with unit 1. Under these conditions, the system (31) describes the simplest kind of two-factor discrimination network. See Fig. 13, where the dashed arrow represents an inhibitory interaction and the solid arrow an excitatory interaction. For more details, see [ 19, 221.

FIG. 13.

inhibitory interactions

excitatory interactions

Learning Network In (31), we assume that A i is as given in the discrimination network; that the elements e,(t) of the interconnection matrix E(t) are equal to one for all ije I= { 1,2}; and further that the function h,(t,x) is given by

h,(t,x)= A I I Y, I (4 + Pi I Y12 (~2)

-AI,YII (~1)

h,(t,x)= I --A2,Y21(~2) 1 A~,Y~,(xz)+P,IY,~(x,) ’

With this, the system (27) represents a kind of learning network. See Fig. 14, where again the dashed and solid arrows represent the inhibitory and

20 G. S. LADDE

excitatory interactions between the units. For details and systematic development, see [ 19, 221.

6. CONCLUSION

The stability behavior of complex compartmental systems under structural perturbations has been discussed with the help of the stability analysis of open chemical systems [8]. It has been shown that the stability study of morphogenetic systems and two-factor systems may be viewed as the stability study of compartmental systems, thus unifying the stability analysis for innumerable diverse hierarchical dynamical systems in the life and physical sciences. In particular, the class of ecosystems and genetic systems is “contained” (in a certain sense) in the class of chemical systems [8, 221; the class of chemical systems is “contained” in the class of morphogenetic systems [22] as well as in the class of compartmental systems [4, 7, 18, 231; and now it has been recognized that a certain class of morphogenetic systems and two-factor systems is “contained” in the class of compartmental systems. This fact strongly confirms one of the Rosen’s [21] “main and basic contributions” to theoretical biology.

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