Stability and Observability 
Max Garzon Fernanda Botelho 
$arzonnhernes. risc i. menst. edu bore lhof herme s. msc i. memst. edu 
Institute for Intelligent Systems Department of Mathematical Sciences 
Memphis State University 
Memphis, TN 38152 U.S.A. 
The theme was the effect of perturbations of the defining parameters of a neural net- 
work due to: 1) measurements (particularly with analog networks); 2) discretization 
due to a) digital implementation of analog nets; b) bounded-precision implementa- 
tion of digital networks; or c) inaccurate evaluation of the transfer function(s); 3) 
nose in or incomplete input and/or output of the net or individual cells (particu- 
larly with analog networks). 
The workshop presentations address these problems in various ways. Some develop 
models to understand the influence of errors/perturbation in the output, learning 
and general behavior of the net (probabilistic in Piche and Tresp; optimization in 
Rojas; dynamical systems in Botelho & (]arson). Others attempt to identify de- 
sirable properties that are to be preserved by neural network solutions (equilibria 
under faster convergence in Peterfreund & Baram; decision regions in Cohen). Of 
particular interest is to develop networks that compute robustly, in the sense that 
small perturbations of their parameters do not affect their dynamical and observ- 
able behavior (stability in biological networks in Chauvet & Chauvet; oscillation 
stability in learning in Rojas; hysterectic finite-state machine simulation in Casey). 
In particular, understand how biological networks cope with uncertainty and errors 
(Chauvet z Chauvet) through the type of stability that they exhibit. 
QUESTIONS AND ANSWERS 
Some questions served to focus the presentations and discussion. Some were (par- 
tially) answered, and others were barely touched: 
o What are the most significant errors in defining parameters with respect to output 
behavior? By evidence presented, i/o and weights seem to be the most sensitive. 
o Is there an essential difference between perturbations in weights (long-term mem- 
ory) and inputs (short-memory)? They seem to play a symmetric role in feedforward 
and, to some extent, recurrent nets. But evidence is not conclusive. 
 How can the effects o/perturbations be kept under control or eliminated altogether? 
If one is only interested in dynamical qualitative features, small enough errors of 
any kind (as incurred in digital implementations for example) are not relevant for 
most nets (What you see on the screen is what should be happening). 
o Are they architecture (in)dependent? On the other hand, they spread rapidly un- 
der iteration and exact quantification varies with the architecture. 
o Are stability and implementation based on dynamical features the only ways to 
1171 
1172 Garzon and Botelho 
cope oitk errors/pertgrbatiorf The difficulty to quantify (perhapa due to lack of 
research) seems to indicate so. Stability worth a closer look for its own sake. 
o Doez requiring robuzt computation really reztrict the capabilitiez of aeural aet- 
vorkzf Apparently not, since in all likelihood there exist universal neural nets 
which tolerate small errors (see talk by Botelho & Garson). Wide open. 
TALKS AND SHORT ABSTRACTS 
 Trajectory Control of Convergent Networks, Natan Peterfreund and Y. 
Baram. We present a class of feedback control functions which accelerate con- 
vergence rates of autonomous nonlinear dynamical systems such as neural network 
mode]s, without affecting the basic convergence properties (e.g. equilibrium points). 
natantx. techni0n. ac. il 
 Sensitivity of Neural Network to Errors, Steven Piche. Using stochastic 
mode]s, analytic expressions for the effects of such errors are derived for arbitrary 
feedforward neural networks. Both, the degree of nonllnearity and the relationship 
between input correlation and the weight vectors, are found to be important in 
determining the effects of errors. piche,cc. 
 Stability of Learning in Neural Networks, Rail Roja& Finding optimal 
combinations of learning and momentum rates for the standard backpropagation 
involves difficult tradeoffs across fractal boundaries. We show that statistic prepro- 
cessing can bring error functions under control. r0j aiinf. fu-berlin. de 
 Stability of Purkinje Cells in Cerebellar Cortex, Gilbert C'hauvet and Pierre 
Chauvet~ The cerebellar cortex (involved in learning and retrieving) is a hierarchical 
functional unit built around a Purkinje cell, which has its own functional proper- 
ties. We have shown experimentally that Purkinje dynamical systems have a unique 
solution, which is asymptotically stable. It seems possible to give a general expla- 
nation of stability in biological systems. chauvetibt.univ-angers. 
 Recall and Learning with Deficient Data, Volker Tresp, $ubutai Ahrnad, 
Ralph Neueier. Mean values and maximum likelihood estimators are not the best 
ways to cope with noisy data. See their LA:5 poster summary in these proceedings 
for an extended abstract. treipzfe. iemen.de 
 Computation Dynamics in Discrete-Time Recurrent Networks, Mike 
Casey. We consider training recurrent higher-order neural networks to recognize 
regular languages, using the cycles in their diagrams for hysterectic simulation of 
finite state machines. The latter suggests a general logical approach to solving the 
'neural code' problem for living organisms, necessary for understanding information 
processing in the nervous system. mcaseysdcc.ucsd.edu 
 Synthesis of Decision Regions in Dynamical Systems, Mike Cohen. As 
a first step toward a representation theory of decision functions via neural nets, 
he presented a method which enables the construction of a system of differential 
equations exhibiting a given finite set of decision regions and equilibria with a very 
large class of indices consistent with the Morse inequalities. mikeSpark. bu. edu 
 Observability of Discrete and Analog Networks, F. Botelho and M. Garzon. 
We show that most networks (with finitely many analog or infinitely many boolean 
neurons) are observable (i.e., all their corrupted pseudo-orbits actually reflect true 
orbits). See their DS:2 poster summary in these proceedings. 
