- Email: [email protected]
Periodic behaviour of one and two-dimensional neural networks
PERIODIC BEHAVIOUR OF ONE AND 1140-DIMENSIONAL NEURAL NETWORKS. E. Goles. Departamento de Matem~ticas y Ciencias de la Computaci6n, Escuela de Ingenier#a, Univ. ~e--~T[~le-, Casilla 170, correo 3, Santiago, Chile. E-mail: Egoles at UCHCECVM ( b i t n e t ) . The periodic behaviour of neural networks has been studied by several authors: Caianello, De Luca and Ricciardi (1), Tchuente (8), Legendre (5), Shingai (6,7), etc. In previous work we proved that symmetric neural networks admit only fixed points or two-cycles (2,3), here we analize one and two-dimensional uniform neural arrays. In the one-dimensional case we take the following bounded linear array: a
a
a
a
° OzZ)zZ): ....... Go CbCbCb
b
c
where a, b, c, @ ~ R and the local t r a n s i t i o n function is given by f: {0,1} 3 + {0,1} with
f(xi_l,Xi,Xi+l)
=
0
if
axi_ I + cx i + bxi+ I - @ < 0
I
otherwise
0 < i < n
and constant-boundary elements; i . e . x_ 1 = x n = O, The global t r a n s i t i o n functions, F, is an application from the n-hypercube into i t s e l f such th~ (Fx)] = f ( x i _ l , X i , X i + l ) for 0 < i < n - l . In this context, we prove that the only admissible periods are i , 2 or 4 depending on the sign d i s t r i b u t i o n of synaptic weights a, b, c. Clearly i f a = b we have a s!m~metric neural network with the two-cycle behaviour (2,3,4). In the two-dimensional case; i . e . a rectangle of cells with Von Neumann neighbors and constant boundary: +b a ÷
(i,j) e
Von Neuman neighborhood a, b, c, d, e ~ R
c
+d
with : f(xi_lj,Xi+lj,Xij,xij_1,xij+1) where ~(u) = ~ 0
if
= ~ ( a x i j _ l + b x i _ i j + e x i j + cxij+1 + d x i + i j
e),O s i , J
sn-1
u < 0
L I otherwise We prove that, for non-negative synaptic weights (a,b,c,d,e ~ R+) the only admisible periods are 1 or 2. Periodic structure of previous networks was f i r s t studied by Shingai (6,7) using exhaustive analysis of the d i f f e r e n t local t r a n s i t i o n functions f. Our approach consist to associate Lyapunov functions driven the network dynamics which permit us to obtain d i r e c t l y the periodic structure and also give bounds on the transient length (O(N),N being the nun~)er of cells in the network). References ( I ) E.R. Caianello, De Luca, L.M. Ricciardi (1967), Reverberations and control of neural networks, Kybernetic 4, 10-18. (2) E. Goles, J. Olivos (1981), Comportement periodique des fornctions ~ seuil binaires et applications, Disc. App. Maths. 3, 93-105. (3) E. Goles (1985), Comportement dynamique de r~seaux d'automates, Th~se d ' E t a t , IMAG, U. Grenobl~ Francia. (4) E. Goles (1985), Dynamics of positive automata networks, Theor. Comp. Sci. 41, 19-32. (5) M. Legendre (1982), Analyse et simulation de r~seaux d'automates, Th~se, IMAG, U. Grenoble, Francia. (6) R. Shingai (1978), The maximum period realized in l-dimensional uniform neural networks, Trans IECE, Japan E61, 804-808. (7) R. Shingai (1979), Maximum period of 2-din~nsional uniform neural networks, Information and Control 41, 324-341. (8) M. Tchuente (1982), Contribution ~ l'~tude des m~thodes de calcul pour des system, s de type cooperatif, Th~se d ' E t a t , IMAG, U. Grenoble, Francia.
97
a
a
a
° OzZ)zZ): ....... Go CbCbCb
b
c
where a, b, c, @ ~ R and the local t r a n s i t i o n function is given by f: {0,1} 3 + {0,1} with
f(xi_l,Xi,Xi+l)
=
0
if
axi_ I + cx i + bxi+ I - @ < 0
I
otherwise
0 < i < n
and constant-boundary elements; i . e . x_ 1 = x n = O, The global t r a n s i t i o n functions, F, is an application from the n-hypercube into i t s e l f such th~ (Fx)] = f ( x i _ l , X i , X i + l ) for 0 < i < n - l . In this context, we prove that the only admissible periods are i , 2 or 4 depending on the sign d i s t r i b u t i o n of synaptic weights a, b, c. Clearly i f a = b we have a s!m~metric neural network with the two-cycle behaviour (2,3,4). In the two-dimensional case; i . e . a rectangle of cells with Von Neumann neighbors and constant boundary: +b a ÷
(i,j) e
Von Neuman neighborhood a, b, c, d, e ~ R
c
+d
with : f(xi_lj,Xi+lj,Xij,xij_1,xij+1) where ~(u) = ~ 0
if
= ~ ( a x i j _ l + b x i _ i j + e x i j + cxij+1 + d x i + i j
e),O s i , J
sn-1
u < 0
L I otherwise We prove that, for non-negative synaptic weights (a,b,c,d,e ~ R+) the only admisible periods are 1 or 2. Periodic structure of previous networks was f i r s t studied by Shingai (6,7) using exhaustive analysis of the d i f f e r e n t local t r a n s i t i o n functions f. Our approach consist to associate Lyapunov functions driven the network dynamics which permit us to obtain d i r e c t l y the periodic structure and also give bounds on the transient length (O(N),N being the nun~)er of cells in the network). References ( I ) E.R. Caianello, De Luca, L.M. Ricciardi (1967), Reverberations and control of neural networks, Kybernetic 4, 10-18. (2) E. Goles, J. Olivos (1981), Comportement periodique des fornctions ~ seuil binaires et applications, Disc. App. Maths. 3, 93-105. (3) E. Goles (1985), Comportement dynamique de r~seaux d'automates, Th~se d ' E t a t , IMAG, U. Grenobl~ Francia. (4) E. Goles (1985), Dynamics of positive automata networks, Theor. Comp. Sci. 41, 19-32. (5) M. Legendre (1982), Analyse et simulation de r~seaux d'automates, Th~se, IMAG, U. Grenoble, Francia. (6) R. Shingai (1978), The maximum period realized in l-dimensional uniform neural networks, Trans IECE, Japan E61, 804-808. (7) R. Shingai (1979), Maximum period of 2-din~nsional uniform neural networks, Information and Control 41, 324-341. (8) M. Tchuente (1982), Contribution ~ l'~tude des m~thodes de calcul pour des system, s de type cooperatif, Th~se d ' E t a t , IMAG, U. Grenoble, Francia.
97