( ) No. 4-69 71 5 (5-5) *1 A Coupled Nonlinear Oscillator Model for Emergent Systems (2nd Report, Spatiotemporal Coupled Lorenz Model-based Subsystem) Tetsuji EMURA *2 *2 College of Human Sciences, Kinjo Gakuin University 2-1723 Omori, Moriyama-ku, Nagoya-shi, Aichi, 463-8521 Japan The author has proposed a new Lorenz model with an excitatory-excitatory connection matrix (EEC model) or an excitatory-inhibitory connection matrix (EIC model) which consists of the three temporal coupling coefficients and three spatial coupling coefficients in a previous paper. In this paper, the author introduces an abstract coincidence detector model (ACD model) to evaluate the spartial synchronization of neurons, and a Hopfield model to decide the three spatial coupling coefficients which govern emergent ability. The paper shows that boundary regions of each phase of the self-organized phase transition phenomena which appear in the proposed model have information processing ability, and claims that a proposed model is useful to an architecture for the emergent subsystems for emergent systems. Key Words: Systems Engineering, Design Engineering, Design, Emergence, Self-Organization, Coupled Oscillator, On-Off Intermittency, Synchronization, Lorenz, Hopfield, Chaos, Coincidence Detector, Neuron, Brain 1 Clark (1) Newell-Simon PSSH Mindware McCulloch-Pitts Biological Architecture van Gelder (2) Dynamical Systems Hard Biological Dynamical Emergent c d 4 9 28. *1, 463-8521 2-1723. E-mail: emura@kinjo-u.ac.jp c d EEC EIC c d (3) 2.1 336 2 (1) x 1,4 σ(x 2,5 x 1,4 ) x 4 x 1 x 2,5 = x 1,4 (r x 3,6 ) x 2,5 ± D * x 5 x 2 L(1) x 3,6 x 1,4 x 2,5 b x 3,6 x 6 x 3 c 1 d 2 d 3 D * = D = d 1 c 2 d 3 : Excitatory - Excitatory Connection d 1 d 2 c 3 c 1 d 2 1 d 3 D * = D = 1 d 1 c 2 d 3 : Excitatory - Inhibitory Connection d 1 1 d 2 c 3
{x 1 -x 4, x 2 -x 5, x 3 -x 6 }={X, Y, Z} {X, Y, Z} EEC EIC Freeman (4) ANN Artificial Neural Network ANN Freeman mesoscopicscale 2.2 {X, Y, Z} X a (t) X b (t) (2) t t n=, 1, 2,, ( n) = x i+3 x i K(2) Δ i i=1, 2, 3 u i u i ( n) =1 if Δ i ( n) ε ( n) = if Δ i ( n) > ε L(3) (5) 1 u i ( n) = 1+ exp[ z i z ] L(4) z i = ( ε Δ i ( n) ) 1 (6) {u 1, u 2, u 3 }={X, Y, Z} {, 1} [, 1] t (4) z z 1 2.3 Table 1: Specifications EEC model EIC model r 28 28 b 8/3 8/3 c.2.4 d variable variable t.1.1.5.5 {X, Y, Z} Fujii (7) ACD Abstract Coincidence Detector Gray (8) Fujii ACD 1. 2. 3. 4. 5. 1 u i (n) w i D(n) Fig. 1: Schematic illustration of the ACD model 337
5. ACD D(n) (5) ( ) =1 if N = w i u i ( n) D n D n k = k i=1 or D = w i u i ( n) =1 i k ( ) < k i=1 or D = w i u i ( n) 1 ( ) = if N = w i u i n i L(5) n ACD ACD w io =1 (i=1,, k) k=3 N 2 {X, Y, Z} X X X {X, Y, Z} EEC EIC Fig. 3: A typical sample of the amplitude (left) and power spectrum (right) of X(t) of the EEC model (top) and EIC model (bottom), t=~[sec], d=.3. Fig. 2: A typical sample of the amplitude of X(t) and histogram of output of ACD of the EIC model, d=.3. 2.4 EEC EIC 3 {X, Y, Z} EEC EIC X {X, Y, Z} (2) (3) (5) ACD n {X, Y, Z} c d 4 4 8 9 c EEC EIC 9 4 {X, Y, Z} d d {X, Y, Z} EIC d 338
Total [%] Total [%] Fig. 4: Excitatory Excitatory Connection Model.1.2.3.4.5.6 Excitatory Inhibitory Connection Model d.1.2.3.4.5.6 d Histogram of total firing ratio and synchronized ratio of {X,Y,Z} versus d of the EEC model, c=.2 (top) and EIC model, c=.4 (bottom), t=~[sec]. 3 3.2 4 {X, Y, Z} (2) (3) (5) ACD n {X, Y, Z} 5 EIC 6 EEC w ij th i = th i =-.5 th i i s i s i = Total [%] Excitatory Connection Weight on the EIC Model.1.15.2.25 +Wij 3.1 Hopfield Hopfield (9) Ising Attractor Neural Network Model (6) ( ) = sign w ij s j ( n) s i n +1 j L(6) Hopfield d i (i=1, 2, 3) (7) dynamic d i ( n) = w ij u j n j w ij = w ji w ii = ( ) + s i th i L(7) Hopfield i j j i u j (n) n j {,1} s i i th i i Total [%] Fig. 5: Inhibitory Connection Weight on the EIC Model.1.15.2.25 -Wij Histogram of total firing ratio and synchronized ratio of {X,Y,Z} versus w ij of the EIC model at excitatory connection weight (top, s i =, th i =) and inhibitory connection weight (bottom, s i =, th i =-.5), t=~[sec]. EIC 5 d i {X, Y, Z} d i EEC 6 4 339
Freeman () Strange Attractor Total [%] Excitatory Connection Weight on the EEC Model.1.15.2.25 +Wij Hopfield d i Hopfield Hopfield Total [%] Inhibitory Connection Weight on the EEC Model.1.15.2.25 -Wij Fig. 6: Histogram of total firing ratio and synchronized ratio of {X,Y,Z} versus w ij of the EEC model at excitatory connection weight (top, s i =, th i =) and inhibitory connection weight (bottom, s i =, th i =-.5), t=~[sec]. 3.3 Hopfield Hopfield (8) n E(n) ( ) = 1 2 E n w ij u i ( n)u j ( n) ( s i th i ) u i n i j i ( ) L(8) 7 EIC X u 1 ACD D(n) E(n) u 1 ACD Y Z ACD Fig. 7: Spike train of u 1 (top), output of ACD (middle) and energy of network (bottom) versus t of the EIC model, w ij =-.15, s i =, th i =-.5, t=~[sec]. ACD 4 Edge of Chaos th i 3
8 th i =-1/3 d 1 (n) E(n) ACD D(n) Hopfield 8 ACD 1 7 8 Hopfield Hopfield 4 Abstract Coincidence Detector Hopfield Edge of Chaos (1) Clark, A., Mindware, Oxford University Press (1991) (2) van Gelder, T. J., The Dynamical Hypothesis in Cognitive Science., Behavioral and Brain Sciences, 21 (1998) 1. (3). (4) Freeman, W. J., How Brains Make Up Their Mind, Columbia University Press () (5) Inoue, M. and Nagayoshi, A., A Chaos Neuro-computer, Physics Letters, A158 (1991) 373. (6) Inoue, M. and Nagayoshi, A., Solving an Optimization Problem with a Chaos Neural Network, Progress of Theoretical Physics, 88 (1992) 769. (7) Fujii, H. et al., Dynamical Cell Assembly Hypothesis, Neural Networks, 9 (1996) 13. (8) Gray, C. M. et al., W., Oscillatory Responses in Cat Visual Cortex Exhibit Inter-columnar Synchronization which Reflects Global Stimulus Properties, Nature, 338 (1989) 334. (9) Hopfield, J. J., Neural Networks and Physical Systems with Emergent Collective Computational Abilities, Proceedings of the Fig. 8: Spatial coupling coefficient d 1 (top), energy of network (middle), and output of ACD (bottom) versus t of the EIC model at unbalanced synapse weight, w 12 =1/3, w 13 =-1/3, w 23 =-1/3, th i =-1/3, s i =, t=3~[sec]. National Academy of Science of the USA, 79 (1982) 2554. () Tateno, K., Hayashi, H. and Ishizuka, S., Complexity of Spatiotemporal Activity of a Neural Network Model which Depends on the Degree of Synchronization, Neural Networks, 11 (1998) 985. 341