860 
A METHOD FOR THE DESIGN OF STABLE LATERAL INHIBITION 
NETWORKS THAT IS ROBUST IN THE PRESENCE 
OF CIRCUIT PARASITICS 
J.L. WYATT, Jr and D.L. STANDLEY 
Department of Electrical Engineering and Computer Science 
Massachusetts Institute of Technology 
cambridge, Massachusetts 02139 
ABSTRACT 
In the analog VLSI implementation of neural systems, it is 
sometimes convenient to build lateral inhibition networks by using 
a locally connected on-chip resistive grid. A serious problem 
of unwanted spontaneous oscillation often arises with these 
circuits and renders them unusable in practice. This paper reports 
a design approach that guarantees such a system will be stable, 
even though the values of designed elements and parasitic elements 
in the resistive grid may be unknown. The method is based on a 
rigorous, somewhat novel mathematical analysis using Tellegen's 
theorem and the idea of Popov multipliers from control theory. It 
is thoroughly practical because the criteria are local in the sense 
that no overall analysis of the interconnected system is required, 
empirical in the sense that they involve only measurable frequency 
response data on the individual cells, and robust in the sense that 
unmodelled parasitic resistances and capacitances in the inter- 
connection network cannot affect the analysis. 
I. INTRODUCT ION 
The term "lateral inhibition" first arose in neurophysiology to 
describe a common form of neural circuitry in which the output of 
each neuron in some population is used to inhibit the response of 
each of its neighbors. Perhaps the best understood example is the 
horizontal cell layer in the vertebrate retina, in which lateral 
inhibition simultaneously enhances intensity edges and acts as an 
automatic ain control to extend the dynamic range of the retina 
as a whole . The principle has been used in the design of artificial 
neural system algorithms by Kohonen 2 and others and in the electronic 
design of neural chips by Carver Mead et. al.3, 4. 
In the VLSI implementation of neural systems, it is convenient 
to build lateral inhibition networks by using a locally connected 
on-chip resistive grid. Linear resistors fabricated in, e.g., 
polysilicon, yield a very compact realization, and nonlinear 
resistive grids, made from MOS transistors, have been found useful 
for image segmentation.4, 5 Networks of this type can be divided into 
two classes: feedback systems and feedforward-only systems. In the 
feedforward case one set of amplifiers imposes signal voltages or 
American Institute of Physics 1988 
861 
currents on the grid and another set reads out the resulting response 
for subsequent processing, while the same amplifiers both "write" to 
the grid and "read" from it in a feedback arrangement. Feedforward 
networks of this type are inherently stable, but feedback networks 
need not be. 
A practical example is one of Carver Mead's retina chips 3 that 
achieves edge enhancement by means of lateral inhibition through a 
resistive grid. Figure 1 shows a single cell in a continuous-time 
version of this chip. Note that the capacitor voltage is affected 
both by the local light intensity incident on that cell and by the 
capacitor voltages on neighboring cells of identical design. Any 
cell drives its neighbors, which drive both their distant neighbors 
and the original cell in turn. Thus the necessary ingredients for 
instability--active elements and signal feedback--are both present 
in this system, and in fact the continuous-time version oscillates 
so badly that the original design is scarcely usable in practice 
with the lateral inhibition paths enabled. 6 Such oscillations can 
 ototrans istor 
incident 
1 igh t 
!similar 
ils 
rout 
Figure 1. This photoreceptor and signal processor circuit, using two 
MOS transconductance amplifiers, realizes lateral inhibition by 
communicating with similar units through a resistive grid. 
readily occur in any resistive grid circuit with active elements and 
feedback, even when each individual cell is quite stable. Analysis 
of the conditions of instability by straightforward methods appears 
hopeless, since any repeated array contains many cells, each of 
which influences many others directly or indirectly and is influenced 
by them in turn, so that the number of simultaneously active feed- 
back loops is enormous. 
This paper reports a practical design approach that rigorously 
guarantees such a system will be stable. The very simplest version 
of the idea is intuitively obvious: design each individual cell so 
that, although internally active, it acts like a passive system as 
seen from the resistive grid. In circuit theory language, the 
design goal here is that each cell's output impedance should be a 
positive-real 7 function. This is sometimes not too difficult in 
practice; we will show that the original network in Fig. 1 satisfies 
this condition in the absence of certain parasitic elements. More 
important, perhaps, it is a condition one can verify experimentally 
862 
by frequency-response measurements. 
It is physically apparent that a collection of cells that 
appear passive at their terminals will form a stable system when 
interconnected through a passive medium such as a resistive grid. 
The research contributions, reported here in summary form, are 
i) a demonstration that this passivity or positive-real condition 
is much stronger than we actually need and that weaker conditions, 
more easily achieved in practice, suffice to guarantee stability of 
the linear network model, and ii) an extension of i) to the nonlinear 
domain that furthermore rules out large-signal oscillations under 
certain conditions. 
II. FIRST-ORDER LINEAR ANALYSIS OF A SINGLE CELL 
We begin with a linear analysis of an elementary model for the 
circuit in Fig. 1. For an initial approximation to the output 
admittance of the cell we simplify the topology (without loss of 
relevant information) and use a naive' model for the transconductance 
amplifiers, as sho in Fig. 2. 
e mk/// 
+ 
( 
(s) 
Figure 2. Simplified network topology and transconductance amplifier 
model for the circuit in Fig. 1. The capacitor in Fig. 1 has been 
absorbed into Co2. 
Straightforward calculations show that the output admittance is 
given by 
- 1 gmlgm2Rl 
Y(s) = [gin2 + Ro2 + s Co2] + . (1) 
(1 + s RolCol) 
This is a positive-real, i.e., passive, admittance since it can always 
be realized by a network of 'the form shown in Fig. 3, where 
-1 -1 -1 
R 1 = (gm2 + Ro2) , R2= (gmlgm2Rol) , and L = Col/gmlgm2. 
Although the original circuit contains no inductors, the 
realization has both capacitors and inductors and thus is capable 
of damped oscillations. Nonetheless, if the transamp model in 
Fig. 2 were perfectly accurate, no network created by interconnecting 
such cells through a resistive grid (with parasitic capacitances) 
could exhibit sustained oscillations. For element values that may 
be typical in practice, the model in Fig. 3 has a lightly damped 
resonance around 1 KHz with a Q = 10. This disturbingly high Q 
suggests that the cell will be highly sensitive to parasitic elements 
not captured by the simple models in Fig. 2. Our preliminary 
863 
L 2 [ 
Y(s) 
Figure 3. Passive network realization of the output admittance (eq. 
(1) of the circuit in Fig. 2. 
analysis of a much more complex model extracted from a physical 
circuit layout created in Carver Mead's laboratory indicates that 
the output impedance will not be passive for all values of the trans- 
amp bias currents. But a definite explanation of the instability 
awaits a more careful circuit modelling effort and perhaps the design 
of an on-chip impedance measuring instrument. 
III. POSITIVE-REAL FUNCTIONS, 8-POSITIVE FUNCTIONS, AND 
STABILITY OF LINEAR NETWORK MODELS 
In the following discussion s = +j is a complex variable, 
H(s) is a rational function (ratio of polynomials) in s with real 
coefficients, and we assume for simplicity that H(s) has no pure 
imaginary poles. The term closed right half plane refers to the set 
of complex numbers s with Re{s} > 0. 
Def. 1 
The function H(s) is said to be positive-real if a) it has no 
poles in the right half plane and b) Re{H(j)} > 0 for all . 
If we know at the outset that H(s) has no right half plane poles, 
then Def. 1 reduces to a simple graphical criterion: HS is positive- 
real if and only if the Nyquist diagram of H(s) (i.e. the plot of 
H(j) for  > 0, as in Fig. 4) lies entirely in the closed right half 
plane. 
Note that positive-real functions are necessarily stable since 
they have no right half plane poles, but stable functions are not 
necessarily positive-real, as Example 1 will show. 
A deep link between positive real functions, physical networks 
and passivity is established by the classical result 7 in linear 
circuit theory which states that H(s) is positive-real if and only if 
it is possible to synthesize a 2-terminal network of positive linear 
resistors, capacitors, inductors and ideal transformers that has H(s) 
as its driving-point impedance or admittance. 
864 
Def. 2 
The function H(s) is said to be 8-positive for a particular value 
of 8(8  0, 8  7), if a) H(s) has no poles in the right half plane, 
and b) the Nyquist plot of H(s) lies strictly to the right of the 
straight line passing through the origin at an angle 8 to the real 
positive axis. 
Note that every 8-positive function is stable and any function 
that is 8-positive with 8 = /2 is necessarily positive-real. 
I {G(j) } 
m 
{G (j) } 
Figure 4. Nyqist diagram for a function that is 8-positive but 
not positive-real. 
Ex.9ple 1 
The function 
G(s) = 
(s+l) (s+40) 
(s+5) (s+6) (S+7) 
(2) 
is 8-positive (for any 8 between about 18  and 68 ) and stable, but it 
is not positive-real since its Nyquist diagram, shown in Fig. 4, 
crosses into the left half plane. 
The importance of 8-positive functions lies in the following 
observations: 1) an interconnection of passive linear resistors and 
capacitors and cells with stable linear impedances can result in an 
unstable network, b) such an instability cannot result if the 
impedances are also positive-real, c) 8-positive impedances form a 
larger class than positive-real ones and hence 8-positivity is a less 
demanding synthesis goal, and d) Theorem 1 below shows that such an 
instability cannot result if the impedances are 8-positive, even if 
they are not positive-real. 
Theorem 1 
Consider a linear network of arbitrary topology, consisting of 
any number of passive 2-terminal resistors and capacitors of arbitrary 
value driven by any number of active cells. If the output impedances 
865 
of all the active cells are e-positive for some common e, 0<e<-, 
then the network is stable. 
The proof of Theorem 1 relies on Lemma 1 below. 
Lal 
If H(s) is e-positive for some fixed e, then for all s o in the 
closed first quadrant of the complex plane, H(s o) lies strictly to 
the right of the straight line passing through the origin at an angle 
e to the real positive axis, i.e., Re{s o }  0 and Im{s o}  0 ----> 
8-7 <  H(So) < e. 
Proof of Lemma 1 (Outline) 
Let d be the function that assigns to each s in the closed right 
half plane the perpendicular distance d(s) from H(s) to the line 
defined in Def. 2. Note that d(s) is harmonic in the closed right 
half plane, since H is analytic there. It then follows, by application 
of the maximum modulus principle 8 for harmonic functions, that d takes 
its minimum value on the boundary of its domain, which is the 
imaginary axis. This establishes Lemma 1. 
Proof of Theorem 1 (Outline) 
The network is unstable or marginally stable if and only if it 
has a natural frequency in the closed right half plane, and s o is a 
natural frequency if and only if the network equations have a nonzero 
solution at s o . Let {I k} denote the complex branch currents of such 
a solution. By Tellegen's theorethe sum of the complex powers 
absorbed by the circuit elements must vanish at such a solution, i.e., 
[ lZk 12 + [ IZk12/SoCk + Zk(SolZk I= = 0, 
resistances capacitances cell 
terminal pairs 
(3) 
where the second term is deleted in the special case So=O, since the 
complex power into capacitors vanishes at So=0. 
If the network has a natural frequency in the closed right half 
plane, it must have one in the closed first quadrant since natural 
frequencies are either real or else occur in complex conjugate pairs. 
But (3) cannot be satisfied for any s o in the closed first quadrant, 
as we can see by dividing both sides of (3) by [ lIkl 2, where the 
sum is taken over all network branches. After this division, (3) 
asserts that zero is a convex combination of terms of the form Rk, 
terms_of thp for k (CkSo)-l, and terms of the form Zk(So). 
Visualize wnere tnese terms lie in the complex plane: the first set lies 
on the real positive axis, the second set lies in the closed 4-th 
quadrant since s o lies in the closed 1st quadrant by assumption, and 
the third set lies to the right of a line passing through the origin 
at an angle e by Lemma 1. Thus all these terms lie strictly to the 
right of this line, which implies that no convex combination of them 
can equal zero. Hence the network is stable! 
866 
IV. 
STABILITY RESULT FOR NETWORKS WITH NONLINEAR 
RESISTORS AND CAPACITORS 
The previous result for linear networks can afford some limited 
insight into the behavior of nonlinear networks. First the nonlinear 
equations are linearized about an equilibrium point and Theorem 1 is 
applied to the linear model. If the linearized model is stable, then 
the equilibrium point of the original nonlinear network is locally 
stable, i.e., the network will return to that equilibrium point if 
the initial condition is sufficiently near it. But the result in this 
section, in contrast, applies to the full nonlinear circuit model and 
allows one to conclude that in certain circumstances the network 
cannot oscillate even if the initial state is arbitrarily far from 
the equilibrium point. 
Def. 3 
A function H (s) as described in Section III is said to satisfy 
the Popov criterion 10 if there exists a real number r>0 such that 
Re{(l+jr) H(j)} > 0 for all . 
Note that positive real functions satisfy the Popov criterion 
with r=0. And the reader can easily verify that G(s) in Example 1 
satisfies the Popov criterion for a range of values of r. The important 
effect of the term (l+jr) in Def. 3 is to rotate the Nyquist plot 
counterclockwise by progressively greater amounts up to 90  as  
increases. 
Theorem 2 
Consider a network consisting of nonlinear 2-terminal resistors 
and capacitors, and cells with linear output impedances Zk(S). Suppose 
i) the resistor curves are characterized by continuously 
diffe,rentiable functions i k = gk(Vk) where gk(0) = 0 and 
0 < gk(Vk) < G <  for all values of k and Vk, 
ii) the capacitors are characterized by i k = Ck(vk)$ k with 
0 < C 1 < Ck(V k) < C 2 <  for all values of k and Vk, 
iii) the impedances Zk(S) have no poles in the closed right 
half plane and all satisfy the Popov criterion for some common 
value of r. 
If these conditions are satisfied, then the network is stable in the 
sense that, for any initial condition, 
 i (t) dt <  
0 all branches 
(4) 
The proof, based on Tellegen's theorem, is rather involved. 
will be omitted here and will appear elsewhere. 
It 
867 
ACKNOWLEDGEMENT 
We sincerely thank Professor Carver Mead of Cal Tech for 
enthusiastically supporting this work and for making it possible for 
us to present an early report on it in this conference proceedings. 
This work was supporedbyDefense Advanced Research Projects Agency 
(DoD), through the Office of Naval Research under ARPA Order No. 
3872, Contract No. N00014-80-C-0622 and Defense Advanced Research 
Projects Agency (DARPA) Contract No. N00014-87--0825. 
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