554 
STABILITY RESULTS FOR NEURAL NETWORKS 
A. N. Michel  , J. A. Farrell  , and W. Porod 2 
Department of Electrical and Computer Engineering 
University of Notre Dame 
Notre Dame, IN 46556 
ABSTRACT 
In the present paper we survey mad utilize results from the qualitative theory of large 
scale interconnected dynamical systems in order to develop a qualitative theory for the 
Hop field model of neural networks. In our approach we view such networks as an inter- 
connection of many single neurons. Our results are phrased in terms of the qualitative 
properties of the individual neurons and in terms of the properties of the interconnecting 
structure of the neural networks. Aspects of neural networks which we address include 
asymptotic stability, exponential stability, and instability of an equilibrium; estimates 
of trajectory bounds; estimates of the domain of attraction of an asymptotically stable 
equilibrium; and stability of neural networks under structural perturbations. 
INTRODUCTION 
In recent years, neural networks have attracted considerable attention as candidates 
for novel computational systems -3. These types of large-scale dynamical systems, in 
analogy to biological structures, take advantage of distributed information processing 
and their inherent potential for parallel computation 4,5. Clearly, the design of such 
neural-network-based computational systems entails a detailed understanding of the 
dynamics of large-scale dynamical systems. In particular, the stability and instability 
properties of the various equilibrium points in such networks are of interest, as well 
as the extent of associated domains of attraction (basins of attraction) and trajectory 
bounds. 
In the present paper, we apply and survey results from the qualitative theory of large 
scale interconnected dynamical systems 6-9 in order to develop a qualitative theory for 
neural networks. We will concentrate here on the popular Hopfield model 3, however, 
this type of analysis may also be applied to other models. In particular, we will address 
the following problems: (i) determine the stability properties of a given equilibrium 
point; (ii) given that a specific equilibrium point of a neural network is asymptotically 
stable, establish an estimate for its domain of attraction; (iii) given a set of initial condi- 
tions and external inputs, establish estimates for corresponding trajectory bounds; (iv) 
give conditions for the instability of a given equilibrium point; (v) investigate stability 
properties under structural perturbations. The present paper contains local results. A 
more detailed treatment of local stability results can be found in Ref. 10, whereas global 
results are contained in Ref. 11. 
In arriving at the results of the present paper, we make use of the method of anal- 
ysis advanced in Ref. 6. Specifically, we view high dimensional neural network as an 
XThe work of A. N. Michel and J. A. Farrell was supported by NSF under grant ECS84-19918. 
2The work of W. Porod was supported by ONR under grant N00014-86-K-0506. 
American Institute of Physics 1988 
555 
interconnection of individual subsystems (neurons). This interconnected systems view- 
point makes our results distinct from others derived in the literature L2. Our results 
are phrased in terms of the qualitative properties of the free subsystems (individual 
neurons, disconnected from the network) and in terms of the properties of the intercon- 
necting structure of the neural network. As such, these results may constitute useful 
design tools. This approach makes possible the systematic analysis of high dimensional 
complex systems and it frequently enables one to circumvent difficulties encountered in 
the analysis of such systems by conventional methods. 
The structure of this paper is as follows. We start out by defining the Hop field 
model and we then introduce the interconnected systems viewpoint. We then present 
representative stability results, including estimates of trajectory bounds and of domains 
of attraction, results for instability, and conditions for stability under structural pertur- 
bations. Finally, we present concluding remarks. 
THE HOPFIELD MODEL FOR NEURAL NETWORKS 
In the present paper we consider neural networks of the Hopfield type 3. Such systems 
can be represented by equations of the form 
N 
= '--biu + &j Cj(uj) + lot i = ,:v, 
j=l 
where A,j T Ui(t) = (t) and bi =  As usual, Ci > O,Tij = 
= c,, c c,' Pdj ' RijeR = 
__1 ._ 1 
(-oc, oc),r,  + Z= Ijl, Ri > O,Ii : R + = [0,)  R,Ii is continuous, 
ai =  Gi : R  (-1, 1),Gi is continuously differentiable and strictly monotoni- 
dt  
t 
cMly increasing (i.e., Ci(uti) > Ci(uti t) if and only if u i > utf),uiCi(ui) > 0 for all ui  0, 
and Gi(O) = 0. In (1), Ci denotes capacitance, R 0 denotes resistance (possibly includ- 
ing a sign inversion due to an inverter), Gi(.) denotes an amplifier nonlinearity, and 
denotes an externM input. 
In the terature it is frequently assumed that Tij = Tji for M1 i,j = 1,... ,N and 
that i = 0 for M1 i = 1,..., N. We will me these sumptions only when explicitly 
stated. 
We are interested in the quMitative behavior of solutions of (1) near equibrium 
points (rest positions where ai  O, for i = 1,..., N). By setting the extemM inputs 
Ui(t), i = 1,... ,N, equM to zero, we define u* = [u,... ,u]TeR N to be an equilibrium 
* N 
for (1) provided that -biu i + j= Aij Gj(u) = O, for i = 1,...,N. The locations 
of such equibria in R N are determined by the interconnection pattern of the neural 
network (i.e., by the pameters Aij, i,j = 1,..., N)  well as by the parameters bi and 
the nature of the nonlinearities Gi('), i = 1,..., N. 
Throughout, we will sume that a given equibrium u* being anMyzed is an isolated 
equilibrium for (1), i.e., there ests an r > 0 such that in the neighborhood B(u*, r) = 
{(u - u*)eR N '1 u - u* I < r} no equilibrium for (1), other than u = u*, effists. 
When analyzing the stability properties of a given equilibrium point, we will be able 
to assume, without loss of generMity, that this equibrium is located at the origin u = 0 
of R N. If this is not the ce, a triviM trsformation can be employed which shifts the 
equilibrium point to the origin and which leaves the structure of (1) the same. 
556 
INTERCONNECTED SYSTEMS VIEWPOINT 
We will find it convenient to view system (1) as an interconnection of N free sub- 
systems (or isolated subsystems) described by equations of the form 
Pi = -bipi + Zii Gi(pi) + Ui(t). (2) 
Under this viewpoint, the interconnecting structure of the system (1) is given by 
N 
i#j 
AijGj(xj), i= 1,...,N. (3) 
Following the method of analysis advanced in 6, we will establish stability results 
which are phrased in terms of the quMitative properties of the free subsystems (2) and 
in terms of the properties of the interconnecting structure given in (3). This method 
of aaalysis makes it often possible to circumvent difficulties that arise in the analysis 
of complex high-dimensional systems. Furthermore, results obtained in this manner 
frequently yield insight into the dynamic behavior of systems in terms of system com- 
ponents and interconnections. 
GENERAL STABILITY CONDITIONS 
We demonstrate below an example of a result for exponential stability of an equi- 
librium point. The principal Lyapunov stability results for such systems are presented, 
e.g., in Chapter 5 of Ref. 7. 
We will utilize the following hypotheses in our first result. 
(A-l) For system (1), the external inputs are all zero, i.e., 
(A-2) 
Ui(t) = O, i=l,...,N. 
For system (1), the interconnections satisfy the estimate 
xiAij Gj(xj) < xi aijxj 
for all ]xi[ < ri, [xj[ < rj, i,j = 1,...,N, where the aij are real constants. 
(A-a) There exists an N-vector a > 0 (i.e., a T = (a,...,aN) and a > 0, for all i: 
1,..., N) such that the test matrix S = [sj] 
{ ai(-bi +aii), i = j 
Sij = (oti aij + otj aji)/2, i  j 
is negative definite, where the bi are defined in (1) and the aij are given in (A-2). 
557 
We are now in a position to state and prove the following result. 
Theorem 1 The equilibrium x = 0 of the neural network (1) is exponentially stable 
if hypotheses (A-l), (A-2) and (A-3) are satisfied. 
Proof. For (1) we choose the Lyanpunov function 
v(x)= . i 2 
i=1 "igi 
where the ai are given in (A-3). This function is clearly positive definite. 
derivative of v along the solutions of (1) is given by 
N i N 
Dvo)(x ) =  .i(2xi)[-bixi +  Aij 
i=1 j=l 
where (A-l) has been invoked. In view of (A-2) we have 
(4) 
The time 
Dv()(x) 
N N 
<_ + 
i--1 j=l 
= TRx for U 112 < r 
2 /2 
where r - min(r), I1 - E=  J , and the matrix R = [ro] is given by 
"i(-bi + aii), i = j 
rij -- "i aij, i  j. 
But it follows that 
= - ,  = xrs < A4(s) Il (5) 
where S is the matrix given in (A-3) and AM(S) denotes the largest eigenvalue of 
the real symmetric matrix S. Since S is by assumption negative definite, we have 
AM(S) < 0. It follows from (4) and (5) that in some neighborhood of the origin x = 0, 
we have c1122 < v() < c21xl22 and Dvo)(x ) <_ -calxl, where c = -} mini-/ > 0, 
c2 =  maxi "i > 0, and ca = --AM(S) > 0. Hence, the equilibrium x = 0 of the neural 
network (1) is exponentially stable (c.f. Theorem 9.10 in Ref. 7). 
Consistent with the philosophy of viewing the neural network (1) as an intercon- 
nection of N free subsystems (2), we think of the Lyapunov function (4) as consisting 
of a weighted sum of Lyapunov functions for eax:h free subsystem (2) (with Ui(t) -- 0). 
The weighting vector , > 0 provides flexibility to emphasize the relative importance 
of the qualitative properties of the various individual subsystems. Hypothesis (A - 2) 
provides a measure of interaction between the various subsystems (3). Furthermore, it 
is emphasized that Theorem 1 does not require that the parameters Aij in (1) form a 
symmetric matrix. 
558 
WEAK COUPLING CONDITIONS 
The test matrix S given in hypothesis (A - 3) has off-diagonal terms which may be 
positive or nonpositive. For the special case where the off-diagonal terms of the test 
matrix $ = [sij] are non-negative, equivalent stability results may be obtained which are 
much easier to apply than Theorem 1. Such results are called weak-coupling conditions 
in the literature 6,9. The conditions sij > 0 for all i  j may reflect properties of the 
system (1) or they may be the consequence of a majorization process. 
In the proof of the subsequent result, we will make use of some of the properties 
of M- matrices (see, for example, Chapter 2 in Ref. 6). In addition we will use the 
following assumptions. 
(A-4) For system (1), the nonlinearity Gi(xi) satisfies the sector condition 
O < eri _< Gi(xi) 5 ai:, for all ]xil < ri, i=l,...,N. 
xi 
(A-5) The successive principal minors of the N x N test matrix D = [dij] 
bi-Aii, i=j 
dij = hi2 
are all positive where, the bi and Aij are defined in (1) and eri2 is defined in (A-4). 
Theorem 2 The equilibrium x = 0 of the neural network (1) is asymptotically sta- 
ble if hypotheses (A-l), (A-d) and (A-5) are true. 
Proof. The proof proceeds  along lines similar to the one for Theorem 1, this time 
with the following Lyapunov function 
N 
v(x) = lxil. (6) 
i=1 
The above Lyapunov function again reflects the interconnected nature of the whole 
system. Note that this Lyapunov function may be viewed as a generalized Hamming 
distance of the state vector from the origin. 
ESTIMATES OF TRAJECTORY BOUNDS 
In general, one is not only interested in questions concerning the stability of an 
equilibrium of the system (1), but also in performance. One way of assessing the qual- 
itative properties of the neural system (1) is by investigating solution bounds near an 
equilibrium of interest. We present here such a result by assuming that the hypotheses 
of Theorem 2 are satisfied. 
In the following, we will not require that the external inputs Ui(t), i = 1,..., N be 
zero. However, we will need to make the additional assumptions enumerated below. 
559 
(A-6) Assume that there exist Ai > 0, for i = 1,... ,N, and a e > 0 such that 
b'-'Li -Aii -  IAjil _ e > O, i=l,...,N 
(i2 
j=l 
i#j 
where bi ad Aij axe defined in (1) and ai2 is defined in (A-4). 
(A-7) Assume that for system (1), 
N 
for all t>0 
for some constant k > 0 where the Ai, i -- 1,..., N are defined in (A-6). 
In the proof of our next theorem, we will make use of a comparison result. We 
consider a scalar comparison equation of the form  = G(y) where yeR, G: B(r) - R 
for some r > 0, and G is continuous on B(r) = {xeR: Ixl < r}. We can then prove the 
following auxiliary theorem: Let p(t) denote the maximal solution of the comparison 
equation with p(to) = yoeB(r), t _ to > O. If r(t), t _ to _ 0 is a continuous 
function such that r(to) _ Yo, and if r(t) satisfies the differential inequality Dr(t) = 
limk_,0+ - sup[r(t + k) - r(t)] _ G(r(t)) almost everywhere, then r(t) _ p(t) for t _ 
to _ 0, for as long as both r(t) emd p(t) exist. For the proof of this result, as well as 
other comparison theorems, see e.g., Refs. 6 ad 7. 
For the next theorem, we adopt the following notation. We let 5 = mini ail 
where aix is defined in (A- 4), we let c = e5 , where e is given in (A-6), and 
we let 6(t,to, xo) = [ql(t, to,Xo),...,qN(t, to,xo)] T denote the solution of (1) with 
()(tO, tO, XO) = X 0 = (X10,... , XNO) T for some tO >_ O. 
We are now in a position to prove the following result, which provides bounds for 
the solution of (1). 
Theorem 3 Assume that hypotheses (A-6) and (A-7) are satisfied. Then 
N k 
II(t, to, xo)11   ili(t, t0, x0)l _ (- k)e-(-) + 
i=1 C C 
t>_to>_O 
provided that  > k/c and IIx011 = E Alxi01 <_ , where the hi, i= 1,...,N are 
given in (A-6) and k is given in (A-7). 
Proof. For (1) we choose the Lyapunov function 
N 
(7) 
i=1 
560 
Along the solutions of (1), we obtain 
N 
Dvo)(x) < xrz>w + xilu(t)l (s) 
i=1 
where w T = [G'-lx[,... Gr(xr)lXN[ ] A = (A,.. AN) T, and D = [dii] is the test 
matrix ven in (A-5). Note that when (A-6) is satisfied, as in the present theorem 
then (A-5) s automaticM]y satisfied. Note Mso that w  0 (i.e., w{  0 { = 1... N) 
and w = 0 if d only if  = 0. 
Using manipulations ivolviug (A-6), (A- 7) and (8), it is ey to show that D(,) () S 
-c() + }. This inequMity yields now the comparison equatiou  = -cy + } whose 
uique solution is given by 
p(t, to,Po) = (Po-- 
e -c(t-t) + -, for all t > to. 
If we let r - v, then we obtain from the comparison result 
N 
p(t) > r(t) = v(qb(t,to, xo)) = y] xilr)dt, to,xo)l = II(t, to,xo)l[, 
i.-ml 
i.e., the desired estimate is true, provided that I(to)[ = = I1oll _< a and 
a > k/c. 
ESTIMATES OF DOMAINS OF ATTRACTION 
Neural networks of the type considered herein have many equilibrium points. If 
a given equilibrium is asymptotically stable, or exponentially stable, then the extent 
of this stability is of interest. As usual, we assume that x = 0 is the equilibrium of 
interest. If qb(t, to, x0) denotes a solution of the network (1) with qb(t0, to, x0) = x0, then 
we would like to know for which points x0 it is true that qb(t, to, x0) tends to the origin 
as t - o. The set of all such points x0 makes up the domain of attraction (the basin of 
attraction) of the equilibrium x = 0. In general, one cannot determine such a domain 
in its entirety. However, several techniques have been devised to estimate subsets of 
a domain of attraction. We apply one such method to neural networks, making use 
of Theorem 1. This technique is applicable to our other results as well, by making 
appropriate modifications. 
We assume that the hypotheses (A-i), (A-2) and (A-3) are satisfied and for the free 
subsystem (2) we choose the Lyapunov function 
i 2 
vi(pd = (0) 
Then Dvi(2)(pi) _< (-bi + aii)p, Ipil < ri for some ri > 0. If (A-3) is satisfied, we 
must have (-bi +ail) < 0 and Dvi(2)(pi) is negative definite over B(ri). 
Let Cvo, = {pieR: vi(pi) = p < r = v0/}. Then Cvo, is contained in the domain 
of attraction of the equilibrium Pl = 0 for the free subsystem (2). 
To obtain an estimate for the domain of attraction of x = 0 for the whole neural 
network (1), we use the Lyapunov function 
561 
It is now an easy matter to show that the set 
(10) 
N 
i-'l 
will be a subset of the domain of attraction of x = 0 for the neural network (1), where 
 = min,(aivoi)= min air i . 
1<i<3/ l_<i_<N 
In order to obtain the best estimate of the domain of attraction of x = 0 by the 
present method, we must choose the ai in an optimal fashion. The reader is referred to 
the literature 9,3,4 where several methods to accomplish this are discussed. 
INSTABILITY RESULTS 
Some of the equilibrium points in a neural network may be unstable. We present 
here a sample instability theorem which may be viewed as a counterpart to Theorem 
2. Instabihty results, formulated as counterparts to other stability results of the type 
considered herein may be obtained by making appropriate modifications. 
(A-S) For system (1), the interconnections satisfy the estimates 
where i = cri when Aii< 0 and i = cri2 when Aii > 0 for all Ixi] < ri, and for 
all I;rl < rj,i,j = 1,...,N. 
(A-9) The successive principal minors of the N x N test matrix D = [dij] given by 
di = { cri' i = j 
-IAijl, i g j 
are positive, where cri = bi _Aii when ieF, (i.e., stable subsystems) and ai = 
_bl q_ Aii when iF (i.e., unstable subsystems) with F = F, U F and F = 
{1,...,N} and F, g qb. 
We are now in a position to prove the following result. 
Theorem 4 The equilibrium x = 0 of the neural network (1) is unstable/f hypotheses 
(A-l), (A-8) and (A-9) are satisfied. If in addition, F, = q5 (q5 denotes the empty set), 
then the equilibrium x = 0 is completely unstable. 
562 
Proof. We choose the Lyapunov function 
= i(-Iil) + ilxil (11) 
where ai > 0, i: 1,... ,N. Along the solutions of (1) we have (following the proof of 
Theorem 2), Dv0)(m )  -a Dw for  mB(r), r: min r where a 
D is defined in (A-9), nd w  [' ()lml] We conclude that 
= [  
Dv(x)(x) is negative definite over B(r). Since every neighborhood of the origin m = 0 
contns t let one point x  where v(x ) < 0, it follows that the equilibrium x = 0 for 
(1) is unstable. Moreover, when F,: , then the function v(x) is negative definite d 
the equilibrium x: 0 of (1) is in fct completely unstable (c.f. Chapter 5 in Ref. 7). 
STABILITY UNDER STRUCTURAL PERTURBATIONS 
In specific applications involving adaptive schemes for learning algorithms in neural 
networks, the interconnection patterns (and external inputs) are changed to yield an 
evolution of different sets of desired asymptotically stable equilibrium points with 
propriate domains of attraction. The present diagonal dominance conditions (see, e.g., 
hypothesis (A-6)) can be used as constraints to guarantee that the desired equilibria 
always have the desired stability properties. 
To be more specific, we assume that a given neural network has been designed with 
set of interconnections whose strengths can be varied from zero to some specified values. 
We express this by writing in place of (1), 
N 
ki = -bixi +  Oij Aij Gj(xj) + Ui(t), for i = 1,...,N, (12) 
j=l 
where 0 _ tij _ 1. We also assume that in the given neural network things have been 
arranged in such a manner that for some given desired value A > O, it is true that 
A - mini (_k_ _ iiAii). From what has been said previously, it should now be clear 
that if Ui(t) -- O, i = 1,... ,N and if the diagonal dominance conditions 
A--  [OijAijl >0, for i: 1,...,N (13) 
j=l 
i#j 
are satisfied for some Ai > 0, i = 1,... ,N, then the equilibrium z = 0 for (12) will be 
asymptotically stable. It is important to recognize that condition (13) constitutes a sin- 
gle stability condition for the neural network under structural perturbations. Thus, the 
strengths of interconnections of the neural network may be rearranged in any manner 
to achieve some desired set of equilibrium points. If (13) is satisfied, then these equi- 
libria will be asymptotically stable. (Stability under structural perturbations is nicely 
surveyed in Ref. 15.) 
563 
CONCLUDING REMARKS 
In the present paper we surveyed and applied results from the qualitative theory 
of large scale interconnected dynamical systems in order to develop a qualitative the- 
ory for neural networks of the Hop field type. Our results are local and use as much 
information as possible in the analysis of a given equilibrium. In doing so, we estab- 
lished criteria for the exponential stability, asymptotic stability, and instability of an 
equilibrium in such networks. We also devised methods for estimating the domain of 
attraction of an asymptotically stable equilibrium and for estimating trajectory bounds 
for such networks. Furthermore, we showed that our stability results are applicable 
to systems under structural perturbations (e.g., as experienced in neural networks in 
adaptive learning schemes). 
In arriving at the above results, we viewed neural networks as an interconnection 
of many single neurons, and we phrased our results in terms of the qualitative proper- 
ties of the free single neurons and in terms of the network interconnecting structure. 
This viewpoint is particularly well suited for the study of hierarchical structures which 
naturally lend themselves to implementations 16 in VLSI. Furthermore, this type of 
proach makes it possible to circumvent difficulties which usually arise in the analysis 
and synthesis of complex high dimensional systems. 
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I.W. 
A.N. 
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