AACIMP 2010 Summer School lecture by Alexander Makarenko. "Applied Mathematics" stream. "General Tasks and Problems of Modelling of Social Systems. Problems and Models in Sustainable Development" course. Part 6. More info at http://summerschool.ssa.org.ua

1. NEURAL NETWORKS WITH
ANTICIPATION:
PROBLEMS AND
PROSPECTS
Alexander MAKARENKO
Institute for Applied System Analysis at
National Technical University of Ukraine
(KPI)

2. INTRODUCTION
Nonlinear networks science
(problems and effects):
stability
bifurcations
chaos
sinchronisation
turbulence
chimera states

3. MODELS AND SYSTEMS
(RECENTLY):
coupled oscillators
coupled maps
neural networks
cellular automata
o.d.e. system
………
MAINLY of NEUTRAL or
WITH DELAY

4. ANTICIPATION
 Neural network learning (Sutton, Barto,
1982)
 Control theory (Pyragas, 2000?)
 Neuroscience (1970-1980,… , 2009)
 Traffic investigations and models (1980,
…, 2008)
 Biology (R. Rosen, 1950- 60- ….)
 Informatics, physics, cellular automata,
etc. (D. Dubois, 1982 - ….)
 Models of society (Makarenko, 1998 - …)

5. ANTICIPATION
 The anticipation property is that the individual
makes a decision accounting the future states of
the system [1].
 One of the consequences is that the accounting
for an anticipatory property leads to advanced
mathematical models. Since 1992 starting from
cellular automata the incursive relation had been
introduced by D. Dubois for the case when
 „the values of of state X(t+1) at time t+1 depends
on values X(t-i) at time t-i, i=1,2,…, the value X(t)
at time t and the value X(t+j) at time t+j, j=1,2,…
as the function of command vector p‟ [1].

6. ANTICIPATION
 In the simplest cases of discrete systems this
leads to the formal dynamic equations (for the
case of discrete time t=0, 1, ..., n, ... and finite
number of elements M):
si (t 1) Gi ({si (t )},...,{si (t g (i))}, R),

 where R is the set of external parameters
(environment, control), {si(t)} the state of the
system at a moment of time t (i=1, 2, …, M), g(i)
horizon of forecasting, {G} set of nonlinear
functions for evolution of the elements states.

7. “In the same way, the hyperincursion is an
extension of the hyper recursion in which several
different solutions can be generated at each time
step” [1, p.98].
According [1] the anticipation may be of „weak‟ type
(with predictive model for future states of system,
the case which had been considered by R. Rosen)
and of „strong‟ type when the system cannot make
predictions.

8. HOPFIELD TYPE NETWORK
WITH ANTICIPATION

9. SOME EXAMPLES OF
MODELS
x j (n 1) f w ji xi (n) w ji xi (n 1)
N N
x j (n 1) f (1 ) w ji xi (n) w ji xi (n 1)
i1 i1

10. EXAMPLE OF ACTIVATION
FUNCTION
0, якщо x 0
f ( x) x, якщо x [0,1)
1, якщо x 1

11. Network with 2 coupled neurons
Single-valued periodicity

12. Neuronert with 2 neuroons
1,2 Multi-valued ciclicity
1
0,8
2 нейрон
0,6
0,4
0,2
0
-0,2 0 0,2 0,4 0,6 0,8 1 1,2
-0,2
1 нейрон

13. Netework with 6 neurons. Ciclicity

14. Network with 8 neurons

15. The influence on anticipation
parameter

16. PROBLEMS AND PROSPECTS

17. RESEARCH DIRECTIONS
 I. General investigations of abstract
mathematical objects:
 Definitions of regimes:
 Periodicity;
 Chaos;
 Solitons;
 Chimera states;
 Bifurcations;
 Attractors;
 Etc.

18. RESEARCH DIRECTIONS
 II. Investigation of concrete models and
solutions
 In artificial neural networks
 In cellular automata
 In coupled maps
 Solitons, traveling waves
 Self-organization
 Collapses
 Etc.

19. RESEARCH DIRECTIONS
 III. Interpretations and applications
 Traffic modeling
 Crowds movement
 Socio- economical systems
 Control applications
 Neuroscience
 Conscious problem
 Physics
 IT

20. REFERENCES
1. Dubois D. Generation of fractals from incursive automata, digital
diffusion and wave equation systems. BioSystems, 43 (1997) 97-114.
2 Makarenko A., Goldengorin B. , Krushinski D. Game „Life‟ with
Anticipation Property. Proceed. ACRI 2008, Lecture Notes Computer
Science, N. 5191, Springer, Berlin-Heidelberg, 2008. p. 77-82
3. B. Goldengorin, D.Krushinski, A. Makarenko Synchronization of
Movement for Large – Scale Crowd. In: Recent Advances in Nonlinear
Dynamics and Synchronization: Theory and applications. Eds. Kyamakya
K., Halang W.A., Unger H., Chedjou J.C., Rulkov N.F.. Li Z., Springer,
Berlin/Heidelberg, 2009 277 – 303
4. Makarenko A., Stashenko A. (2006) Some two- steps discrete-time
anticipatory models with „boiling‟ multivaluedness. AIP Conference
Proceedings, vol.839, ed. Daniel M. Dubois, USA, pp.265-272.