Fluctuation-Driven Learning Rule for Continuous-Time
Recurrent Neural Networks and Its Application to Dynamical
System Control
Kazuhisa Watanabe, Takahiro Haba, Noboru Kudo, and Takahumi Oohori
Faculty of Engineering, Hokkaido Institute of Technology, Sapporo, 006-8585 Japan
SUMMARY
Fluctuation-driven learning rule is proposed for con-
tinuous-time recurrent neural networks. In so doing, ran-
dom fluctuations
n
j
p,
t (j: neuron number, p: input pattern
number, 0 d
t
d
T
p
,
T
p
: pattern length) are superimposed on
every neurons threshold. Probability density N
j
n
j
of fluc-
tuation amplitude is treated as a time-invariant, and auxil-
iary function g
j
n
j
: dN
j
/dn
j
g
j
N
j
is introduced. For
fluctuations n
j
p, t, neuron outputs r
j
p, t and instantane-
ous error ep,
t
are probabilistic quantities. In so doing,
learning rule for synaptic weight w
ji
from i-th neuron is
R
ji
p,
t
³
0
t
g
j
r
i
d
W
/
W
j
,
'
p
w
ji
P
³
0
T
p
e R
ji
dt/T
p
(r
j
: time
constant of membrane potential, P : learning coefficient). It
is shown theoretically that expected mean error
(
³
0
T
p
edt
/
T
p
may be minimized by steepest descent. This
learning rule does not require any additional functions such
as adjoint system or sensitivity system, and can be executed
in time-forward direction by simple integrating, which is
distinctive of previous algorithms. The features of the pro-
posed method are confirmed through numerical experi-
ments with JK flip-flop, dynamical systems inverse model,
and speed control of moving object. © 2001 Scripta Tech-
nica, Syst Comp Jpn, 32(3): 1423, 2001
Key words: Fluctuation-driven learning; recurrent
neural network; JK-FF; dynamical inverse model; speed
control.
1. Introduction
Continuous-time recurrent neural networks (RNN),
with each neurons state given through differential equa-
tions, offer nonlinear dynamics with respect to variations
of internal states. Such networks offer most universal neural
circuit model, and are considered promising in such fields
as signal processing with complicated hysteresis effects, or
robot control.
Doya and Yoshizawa [1], Pearlmutter [2], and Sato
[3], almost at the same time, proposed learning algorithms
for continuous-time RNN. Analysis of learning properties
was conducted by Tokoi and colleagues [4] and Nakajima
and Ueda [5]. Doya and Yoshizawa [6], Sato and colleagues
[7], Lee and colleagues [8], and Adachi and Kotani [9]
studied recall learning of spatiotemporal patterns and pre-
dictive learning; the possibility of application of such algo-
rithms to spatiotemporal data processing is being verified
in terms of computation theory.
However, all aforementioned algorithms [13] were
developed on the same assumptions as error back-propaga-
tion algorithm for nonrecurrent (feed-forward) neural net-
works (FNN), namely: (1) teaching waveforms for each
visible neuron are given explicitly and (2) output functions
for all neurons are differentiable.
Such prerequisites restrict universality of time-con-
tinuous RNN. For example, in the case of control learning
of a dynamical system with unknown characteristics, learn-
ing-support subsystems such as forward models [10] are
© 2001 Scripta Technica
Systems and Computers in Japan, Vol. 32, No. 3, 2001
Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J83-D-II, No. 3, March 2000, pp. 10341042
14