Patterns of damage in neural networks: 
The effects of lesion area, shape and 
number 
Eytan Ruppin and James A. Reggia 
Department of Computer Science 
A.V. Williams Bldg. 
University of Maryland 
College Park, MD 20742 
ruppin@cs.umd.edu reggia@cs.umd.edu 
Abstract 
Current understanding of the effects of damage on neural networks 
is rudimentary, even though such understanding could lead to im- 
portant insights concerning neurological and psychiatric disorders. 
Motivated by this consideration, we present a simple analytical 
framework for estimating the functional damage resulting from fo- 
cal structural lesions to a neural network. The effects of focal le- 
sions of varying area, shape and number on the retrieval capacities 
of a spatially-organized associative memory. Although our analyti- 
cal results are based on some approximations, they correspond well 
with simulation results. This study sheds light on some important 
features characterizing the clinical manifestations of multi-infarct 
dementia, including the strong association between the number of 
infarcts and the prevalence of dementia after stroke, and the 'mul- 
tiplicative' interaction that has been postulated to occur between 
Alzheimer's disease and multi-infarct dementia. 
*Dr. Reggia is also with the Department of Neurology and the Institute of Advanced 
Computer Studies at the University of Maryland. 
36 Eytan Ruppin, James A. Reggia 
1 Introduction 
Understanding the response of neural nets to structural/functional damage is im- 
portant for a variety of reasons, e.g., in assessing the performance of neural network 
hardware, and in gaining understanding of the mechanisms underlying neurologi- 
cal and psychiatric disorders. Recently, there has been a growing interest in con- 
structing neural models to study how specific pathological neuroanatomical and 
neurophysiological changes can result in various clinical manifestations, and to in- 
vestigate the functional organization of the symptoms that result from specific brain 
pathologies (reviewed in [1, 2]). In the area of associative memory models specifi- 
cally, early studies found an increase in memory impairment with increasing lesion 
severity (in accordance with Lashley's classical 'mass action' principle), and showed 
that slowly developing lesions have less pronounced effects than equivalent acute 
lesions [3]. Recently, it was shown that the gradual pattern of clinical deterioration 
manifested in the majority of Alzheimer's patients can be accounted for, and that 
different synaptic compensation rates can account for the observed variation in the 
severity and progression rate of this disease [4]. However, this past work is limited 
in that model elements have no spatial relationships to one another (all elements 
are conceptually equidistant). Thus, as there is no way to represent focal (localized) 
damage in such networks, it has not been possible to study the functional effect of 
focal lesions and to compare them with that caused by diffuse lesions. 
The limitations of past work led us to use spattally-organized neural network for 
studying the effects of different types of lesions (we use the term lesion to mean 
any type of structural and functional damage inflicted on an initially intact neural 
network). The elements in our model, which can be thought of as representing 
neurons, or micro-columnar units, form a 2-dimensional array (whose edges are 
connected, forming a torus to eliminate edge effects), and each unit is connected 
primarily to its nearby neighbors, as is the case in the cortex [5]. It has recently been 
shown that such spattally-organized attractor networks can function reasonably well 
as associative memory devices [6]. This paper presents the first detailed analysis of 
the effects of lesions of various size, form and number on the memory performance 
of spattally-organized attractor neural networks. Assuming that these networks are 
a plausible model of some frontal and associative cortical areas (see, e.g., [7]), our 
results shed light on the clinical progress of disorders such as stroke and dementia. 
In the next section, we derive a theoretical framework that characterizes the effects 
of focal lesions on an associative network's performance. This framework, which 
is formulated in very general terms, is then examined via simulations in Section 3, 
which show a remarkable quantitative fit with the theoretical predictions, and are 
compared with simulations examining performance with diffuse damage. Finally, 
the clinical significance of our results is discussed in Section 4. 
2 Analyzing the effects of focal lesions 
Our analysis pertains to the case where in the pre-damaged network, all units 
have an approximately similar average level of activity  A focal structural lesion 
1This is true in general for associative memory networks, when the activity of each unit 
is averaged over a sufficiently long time span. 
Patterns of Damage in Neural Networks 3 7 
(anatomical lesion), denoting the area of damage and neuronal death, is modeled 
by clamping the activity of the lesioned units to zero. As a result of this primary 
lesion, the activity of surrounding units may be decreased, resulting in a secondary 
area of functional lesion, as illustrated in Figure 1. We are primarily interested in 
large focal lesions, where the area s of the lesion is significantly greater than the 
local neighborhood region from which each unit receives its inputs. Throughout our 
analysis we shall hold the working assumption that, traversing from the border of 
the lesion outwards, the activity of units gradually rises from zero until it reaches 
its normal, predamaged levels, at some distance d from the lesion's border (see 
Figure 1). As s is large and d is determined by local interactions on the borders of 
the structural lesion, we may reasonably assume that the value of d is independent 
of the lesion size, and depends primarily on the specific network characteristics, 
such as it architecture, dynamics, and memory load. 
Figure 1: A sketch of a structural (dark shading) and surrounding functional (light 
shading) rectangular lesion. 
Let the intact baseline performance level of the network be denoted as P(0), and 
let the network size be A. The network's performance denotes how accurately it 
retrieves the correct memorized patterns given a set of input cues, and is defined 
formally below. A structural lesion of area s (dark shading in Figure 1), causing 
a functional lesion of area As (light shading in Figure 1), will then result in a 
performance level of approximately 
P(s) : P(O) [A - (s + As)] + PaAs = P(0) - (APAs)/(A - s) (1) 
.g- $ ' 
where Pa denotes the average level of performance over As and AP = P(0) - PA. 
P(s) hence reflects the performance level over the remaining viable parts of the 
network, discarding the structurally damaged region. Bearing these definitions in 
mind, a simple analysis shows that the effect of focal lesions is governed by the 
following rules. 
Consider a symmetric, circular structural lesion of size s = 7rr 2. As, the area of 
functional damage following such a lesion is then (assuming large lesions and hence 
Rule 1: 
(2) 
38 Eytan Ruppin, James A. Reggia 
1.0 
0.8 
0.6 
0.4 
0.2 
--k=l ,,. 
----- k=5 '-. 
k=25 ' 
0'00.0 500.0 1000.0 1500.0 
Lesion size 
Figure 2' Theoretically predicted network performance as a function of a single 
focal structural lesion's size (area): analytic curves obtained for different k values; 
A = 1600. 
and 
e(s)  e(0) A- s ' (3) 
for some constant k = v/dAP. Thus, the area of functional damage surrounding 
a single focal structural lesion is proportional to the square root of the structural 
lesion's area. Some analytic performance/lesioning curves (for various k values) are 
illustrated in Figure 2. Note the different qualitative shape of these curves as a 
function of k. Letting x = s/A be the fraction of structural damage, we have 
kv 1 
P(x)  P(O) 1- x x/ ' (4) 
that is, the same fraction x of damage results in less performance decrease in larger 
networks. This surprising result testifies to the possible protective value of having 
functional 'modular' cortical networks of large size. 
Expressions 3 and 4 are valid also when the structural lesion has a square shape. 
To study the effect of the structural lesion's shape, we consider the area As[hi of 
a functional lesion resulting from a rectangular focal lesion of size s = a  b (see 
Figure 1), where, without loss of generality, n = a/b _ 1. Then, for large n, we find 
that the functional damage of a rectangular structural lesion of fixed size increases 
as its shape is more elongated, following 
Rule 2' 
a il ~ v-/4a 
$  -- $ , 
(5) 
and 
P(s) -_- P(O) 2(A- s) ' (6) 
Patterns of Damage in Neural Networks 39 
To study the effect of the number of lesions, consider the area A, " of a functional 
lesion composed of m focal rectangular structural lesions (with sides a = n. b), each 
of area s/to. We find that the functional damage increases with the number of focal 
sub-lesions (while total structural lesion area is held constant), according to 
tule 3: 
(7) 
and 
P(s) P(O) a(A- s) ' (8) 
While Rule 3 presents a lower bound on the functional damage which may actually 
be significantly larger and involves no approximations, Rule 2 presents an upper 
bound on the actual functional damage. As we shall show in the next section, the 
number of lesions actually affects the network performance significantly more than 
its precise shape. 
3 Numerical Simulation Results 
We now turn to examine the effect of lesions on the performance of an associa- 
tive memory network via simulations. The goal of these simulations is twofold. 
First, to examine how accurately the general but approximate theoretical results 
presented above describe the actual performance degradation in a specific associa- 
tive network. Second, to compare the effects of focal lesions to those of diffuse 
ones, as the effect of diffuse damage cannot be described as a limiting case within 
the framework of our analysis. Our simulations were performed using a standard 
Tsodyks-Feigelman attractor neural network [8]. This is a Hopfield-like network 
which has several features which make it more biologically plausible [4], such as low 
activity and non-zero positive thresholds. In all the experiments, 20 sparse random 
{0, 1} memory patterns (with a fraction of p << 1 of l's) were stored in a network 
of N = 1600 units, placed on a 2-dimensional lattice. The network has spatially 
organized connectivity, where each unit has 60 incoming connections determined 
randomly with a Gaussian probability 5(z) = V/-/2rr exp(-z2/2a2), where z is the 
distance between two units in the array. When a is small, each unit is connected 
primarily to its nearby neighbors. As in [4], the cue input patterns are presented 
via an external field of magnitude e = 0.035, and the noise level is T = 0.005. The 
performance of the network is measured (over the viable, non-lesioned units) by the 
standard overlap measure which denotes the similarity between the final state $ 
the network converges to and the memory pattern  that was cued in that trial, 
defined by 
N 
1 E( -p)$i(t). (9) 
m(t) = p(1- p)N 
In all simulations we report the average overlap achieved over 100 trials. 
We first studied the network's performance at various a values. Figure 3a displays 
how the performance of the network degrades when diffuse structural lesions of 
increasing size are inflicted upon it (i.e., randomly selected units are clamped to 
zero), while Figure 3b plots the performance as a function of the size of a single 
square-shaped focal lesion. As is evident, spatially-organized connectivity enables 
40 Eytan Ruppin, James A. Reggia 
the network to maintain its memory retrieval capacities in face of focal lesions of 
considerable size. Diffuse lesions are always more detrimental than single focal 
lesions of identical size. Also plotted in Figure 3b is the analytical curve calculated 
via expression (3) (with k = 5), which shows a nice fit with the actual performance 
of the spatially-connected network parametrized by r = 1. Concentrating on the 
study of focal lesions in a spatially-connected network, we adhere to the values 
r = I and k = 5 hereafter, and compare the analytical and numerical results. With 
these values, the analytical curves describing the performance of the network as a 
function of the fraction of the network lesioned (obtained using expression 4) are 
similar to the corresponding numerical results. 
(a) 
(b) 
0'.0 .0 1 .0 1 .0 0.0 5.0 .0 15.0 
Les se Les se 
Figure 3: Network performance  a function of lesion sie: simulation results 
obtained in four differen neworks, each characterized by a disfinc disLribufion of 
spafially-organied connectivity. (a) Diffuse lesions. (b) Focal lesions. 
To examine Rule 2, a rectangular structural lesion of area s = 300 was induced in 
the network. As shown in Figure 4a, as the ratio n between the sides is increased, 
the network's performance further decreases, but this effect is relatively mild. The 
markedly stronger effect of varying the lesion number (described by Rule 3) is 
demonstrated in figure 4b, which shows the effect of multiple lesions composed of 
2, 4, 8 and 16 separate focal lesions. For comparison, the performance achieved with 
a diffuse lesion of similar size is plotted on the 20tth x-ordinate. It is interesting 
to note that a sufficiently large multiple focal lesion (s = 512) can cause a larger 
performance decrease than a diffuse lesion of similar size. That is, at some point, 
when the size of each individual focal lesion becomes small in relation to the spread 
of each unit's connectivity, our analysis looses its validity, and Rule 3 ceases to hold. 
Patterns of Damage in Neural Networks 41 
(a) 
0,930 
0.910 
0.890 
0.870 
o o Simulation results 
--- Analytical results 
(b) 
0.0 
I ,, = . Simulation (s = 256) 
 ', .... Analytic (s = 256) 
0'8%.0 1 0 2.0 3 0 4 0 5.0 6.0 0.0 5.0 10.0 15.0 20.0 25.0 
Rectangular ratio n No. of sub-lesions m 
Figure 4: Network performance as a function of focal lesion shape (a) and num- 
ber (b). Both numerical and analytical results are displayed. In Figure 4b, the 
x-ordinate denotes the number of separate sub-lesions (1,2,4,8,16), and, for com- 
parison, the performance achieved with a diffuse lesion of similar size is plotted on 
the 20tth x-ordinate. 
4 Discussion 
We have presented a simple analytical flamework for studying the effects of focal 
lesions on the functioning of spatially organized neural networks. The analysis 
presented is quite general and a similar approach could be adopted to investigate 
the effect of focal lesions in other neural models. Using this analysis, specific scaling 
rules have been formulated describing the functional effects of structural focal lesions 
on memory retrieval performance in associative attractor networks. The functional 
lesion scales as the square root of the size of a single structural lesion, and the form of 
the resulting performance curve depends on the impairment span d. Surprisingly, 
the same fraction of damage results in significantly less performance decrease in 
larger networks, pointing to their relative robustness. As to the effects of shape and 
number, elongated structural lesions cause more damage than more symmetrical 
ones. However, the number of sub-lesions is the most critical factor determining 
the functional damage and performance decrease in the model. Numerical studies 
show that in some conditions multiple lesions can damage performance more than 
diffuse damage, even though the amount of lost innervation is always less in a 
multiple focal lesion than with diffuse damage. 
Beyond its computational interest, the study of the effects of focal damage on the 
performance of neural network models can lead to a better understanding of func- 
tional impairments accompanying focal brain lesions. In particular, we are inter- 
ested in multi-infarct dementia, a frequent cause of dementia (chronic deterioration 
of cognitive and memory capacities) characterized by a series of multiple, aggregat- 
42 Eytan Ruppin, James A. Reggia 
ing focal lesions. Our results indicate a significant role for the number of infarcts in 
determining the extent of functional damage and dementia in multi-infarct disease. 
In our model, multiple focal lesions cause a much larger deficit than their simple 
'sum', i.e., a single lesion of equivalent total size. This is consistent with clinical 
studies that have suggested the main factors related to the prevalence of dementia 
after stroke to be the infarct number and site, and not the overall infarct size, which 
is related to the prevalence of dementia in a significantly weaker manner [9, 10]. Our 
model also offers a possible explanation to the 'multiplicative' interaction that has 
been postulated to occur between co-existing Alzheimer and multi-infarct dementia 
[10], and to the role of cortical atrophy in increasing the prevalence of dementia after 
stroke; in accordance with our model, it is hypothesized that atrophic degenerative 
changes will lead to an increase in the value of d (and hence of k) and increase 
the functional damage caused by a lesion of given structural size. This hypothesis, 
together with a detailed study of the effects of the various network parameters on 
the value of d, are currently under further investigation. 
Acknowledgement s 
This research has been supported by a Rothschild Fellowship to Dr. Ruppin and 
by Awards NS29414 and NS16332 from NINDS. 
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