“Resource-Competing Oscillator Network as a Model of Amoeba-based Neurocomputer” Unconventional Computation 2009, Lecture Notes in Computer Science 5715, Springer, 56-69 (2009) Supplementary Materials

Unconventional Computation 2009, Lecture Notes in Computer Science 5715, Springer, 56-69 (2009) Supplementary Materials

Masashi Aono¹,^∗ Yoshito Hirata², Masahiko Hara¹, and Kazuyuki Aihara²

1 Advanced Science Institute, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

2 Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505, Japan

I. NOTATIONS Graph Representations

set of all nodes: Q={0,1,· · · , n}(n= 3) set of all edges: E={{0,1},{0,2},{0,3}}

set of all nodes connected to nodeq∈Q: Pq

State Variables

volume of nodeq∈Q: vq(t)∈R change of volume (activator): xq(t)∈R hidden variable (inhibitor): y_q(t)∈R growth variable: gq(t)∈R stimulation status: sq(t)∈[0.0,1.0]

Flow Variables

resource current forxflowing fromptoq: Ip,q(t)∈R resource current foryflowing fromptoq: Jp,q(t)∈R force transmission forgflowing fromptoq: K_p,q(t)∈R

FIG. 1:

∗Electronic address:[email protected]

Parameters modulator of oscillation frequency: f (= 1)

modulator of oscillation amplitude for nodeq: rq(amplitude=√rq) adjustor of hub amplitude√

r0: ρ∈(0.0,3.0]

(√

r0= 3ρ,√ r1=√

r2=√ r3= 3) coupling strength for edge{p, q}: Wp,q(= 1)

balancer between intra- and inter-node demands: λ(= 0.2) regulator of stimulation effect: µ(=−15)

regulator of diffusion effect δ(= 10) reference volume (normalizer): L(= 40)

adjustors of sigmoid functionsgm: b(= 35; steepness) (sgm(v) = 1/(1 +Exp{−b(v−θ)})) θ(= 0.35; critical level)

adjustors of sigmoid functionSgm: B(= 1000; steepness) (Sgm(V) = 1/(1 +Exp{−B(V −Θ)})) Θ(=−0.5; critical level)

weight matrix: Up,q





U0,0 U0,1 U0,2 U0,3

U1,0 U1,1 U1,2 U1,3

U2,0 U2,1 U2,2 U2,3

U_3,0 U_3,1 U_3,2 U_3,3



=





0 0 0 0

0 0 −0.3 −0.3 0 −0.4 0 −0.3 0 −0.2 −0.3 0





II. GENERAL FORM Dynamics ofv:

˙

v_q =x_q. (1)

Dynamics ofxandy:

˙

x_q = ∑

p∈P_q

I_p,q, (2)

˙ yq = ∑

p∈Pq

Jp,q. (3)

CurrentsIp,qandJp,qare determined by minimizing the object functions:

HI =λ∑

q∈Q

( ˙x^∗_q− ∑

p∈Pq

Ip,q)²+ (1−λ) ∑

(p,q)∈E

(Wp,q( ˙x^∗_q−x˙^∗_p)−Ip,q)², (4)

H_J =λ∑

q∈Q

( ˙y^∗_q−g_q− ∑

p∈P_q

J_p,q)²+ (1−λ) ∑

(p,q)∈E

(W_p,q( ˙y^∗_q−y˙^∗_p)−J_p,q)², (5)

wherex˙^∗_qandy˙_q^∗are given by oscillatory dynamics (“intra-node demands”)

x˙^∗_q =f(rqxq−yq−xq(x²_q+y²_q)), (6) y˙^∗_q =f(xq+rqyq−yq(x²_q+y²_q)). (7) Simultaneously solving equations_∂I^∂HI

0,1 = 0,_∂I^∂HI

0,2 = 0, and_∂I^∂HI

0,3 = 0, we get the optimal currentsI0,1,I0,2, andI0,3as follows:

I_0,1/2/3= _1+3λ¹ (

W_0,1/2/3( ˙x^∗_1/2/3−x˙^∗₀) +λ²(

2 ˙x^∗_1/2/3−x˙^∗_2/1/2−x˙^∗_3/3/1−2W_0,1/2/3( ˙x^∗_1/2/3−x˙^∗₀) +W_0,2/1/2( ˙x^∗_2/1/2−x˙^∗₀) +W_0,3/3/1( ˙x^∗_3/3/1−x˙^∗₀)) +λ(

(1 +W_0,1/2/3)( ˙x^∗_1/2/3−x˙^∗₀)−W_0,2/1/2( ˙x^∗_2/1/2−x˙^∗₀)

−W_0,3/3/1( ˙x^∗_3/3/1−x˙^∗₀))) ,

(8)

where the subscript expression “1/2/3” indicates that three equations are collectively written. Similarly, we obtain the optimal currentsJ0,1,J0,2, andJ0,3:

J_0,1/2/3= _1+3λ¹ (

W_0,1/2/3( ˙y^∗_1/2/3−y˙₀^∗) +λ²(

2 ˙y^∗_1/2/3−y˙^∗_2/1/2−y˙^∗_3/3/1−2W_0,1/2/3( ˙y^∗_1/2/3−y˙₀^∗) +W_0,2/1/2( ˙y^∗_2/1/2−y˙₀^∗) +W_0,3/3/1( ˙y^∗_3/3/1−y˙₀^∗)

−2g_1/2/3+g_2/1/2+g_3/3/1) +λ(

(1 +W_0,1/2/3)( ˙y^∗_1/2/3−y˙₀^∗)−W_0,2/1/2( ˙y^∗_2/1/2−y˙₀^∗)

−W_0,3/3/1( ˙y^∗_3/3/1−y˙₀^∗) +g₀−g_1/2/3)) .

(9)

Definition ofg:

gq= ∑

p∈Pq

Kp,q. (10)

Flow variablesK_p,qare determined by minimizing the object function:

HK=∑

q∈Q

(µ sqˆvq− ∑

p∈Pq

Kp,q)²+ ∑

(p,q)∈E

(δ Wp,q(ˆvp−vˆq)−Kp,q)², (11)

whereˆv_q = v_q/Lis afraction of volumedivided by thereference volumeL, andstimulations_q is applied according to the neural-net-based inhibitory coupling

sq = 1−Sgm(∑

p∈Q

Up,qsgm(ˆvp)). (12)

Solving_∂K^∂HK

0,1 = 0,_∂K^∂HK

0,2 = 0, and_∂K^∂HK

0,3 = 0, the optimal transmissionsKp,qcan be obtained analytically as follows:

K_0,1/2/3= _10L¹ (−µ(

2s0v0−4s_1/2/3v_1/2/3+s_2/1/2v_2/1/2+s_3/3/1v_3/3/1) +δ(

−4W_0,1/2/3(v_1/2/3−v₀) +W_0,2/1/2(v_2/1/2−v₀) +W0,3/3/1(v3/3/1−v0)))

.

(13)

There are dynamicsv˙q,x˙q, andy˙q given by Eqs. (1), (2), and (3) (♯({v, x, y})×♯(nodes) = 3×4 = 12equations), and we can rewrite these dynamics as ordinary differential equations by substituting analytic solutions ofI_p,q,J_p,q,K_p,q obtained as Eqs. (8), (9), and (13). Thus, numerical solutions forv_q,x_q, andy_q are obtained by routine procedures. This means that we can also calculate time series ofIp,q,Jp,q,Kp,q,gq, andsq.

In general for any star network consisting of a hub andnterminals (in this studyn= 3), the dynamics Eqs. (2) and (3) of the hub node0and the terminal nodesq(̸= 0) can be written separately as follows:

˙

x₀= −_1+nλ¹ ∑n

p=1(λ−λW_0,p+W_0,p)( ˙x^∗_p−x˙^∗₀),

˙

xq = _1+nλ¹ (

λ²(nx˙^∗_q −∑n

p=1x˙^∗_p)−(

(λ−1)(nλ+ 1)W0,q−λ)

( ˙x^∗_q−x˙^∗₀) + (λ²−λ)∑n

p=1W0,p( ˙x^∗_p−x˙^∗₀) )

, (14)

˙

y₀= −_1+nλ¹ (∑n

p=1(λ−λW_0,p+W_0,p)( ˙y_p^∗−y˙₀^∗)−^(n+1)λ_(n+2)L( µ∑n

p=1(v_ps_p−v₀s₀)−δ∑n

p=1W_0,p(v_p−v₀))) ,

˙

yq= _1+nλ¹ (

λ²(ny˙_q^∗−∑n

p=1y˙^∗_p)−(

(λ−1)(nλ+ 1)W0,q−λ)

( ˙y_q^∗−y˙₀^∗) + (λ²−λ)∑n

p=1W0,p( ˙y_p^∗−y˙₀^∗)

−_(n+2)L^λµ ((n+2)(nλ+1)

2 sqvq−(n+ 1)s0v0−^(n+2)λ₂ ⁻¹∑n p=1spvp

) +_2(n+2)L^λδ (

(n+ 2)(nλ+ 1)W0,q(vq−v0)−((n+ 2)λ−1)∑n

p=1W0,p(vp−v0))) .

(15)

III. FULLY SUBSTITUTED FORM Dynamics ofv:

˙ v0=x0,

˙ v1=x1,

˙ v₂=x₂,

˙ v₃=x₃.

(16)

Fully substituted dynamics ofx:

˙

x0= −_1+3λ¹ (

(λ−λW_0,1+W_0,1)(

f(r₁x₁−y₁−x₁(x²₁+y₁²))−f(r₀x₀−y₀−x₀(x²₀+y₀²))) +(λ−λW_0,2+W_0,2)(

f(r₂x₂−y₂−x₂(x²₂+y₂²))−f(r₀x₀−y₀−x₀(x²₀+y²₀))) +(λ−λW0,3+W0,3)(

f(r3x3−y3−x3(x²₃+y₃²))−f(r0x0−y0−x0(x²₀+y²₀)))) ,

˙

x1= _1+3λ¹ ( λ²(

2f(r1x1−y1−x1(x²₁+y₁²))−f(r2x2−y2−x2(x²₂+y²₂))−f(r3x3−y3−x3(x²₃+y²₃))) +(

(−2λ²+λ+ 1)W0,1+λ)(

f(r1x1−y1−x1(x²₁+y²₁))−f(r0x0−y0−x0(x²₀+y₀²))) +(λ²−λ)W0,2

(f(r2x2−y2−x2(x²₂+y₂²))−f(r0x0−y0−x0(x²₀+y²₀))) +(λ²−λ)W_0,3(

f(r₃x₃−y₃−x₃(x²₃+y₃²))−f(r₀x₀−y₀−x₀(x²₀+y²₀)))) ,

˙

x₂= _1+3λ¹ ( λ²(

2f(r2x2−y2−x2(x²₂+y₂²))−f(r1x1−y1−x1(x²₁+y²₁))−f(r3x3−y3−x3(x²₃+y²₃))) +(

(−2λ²+λ+ 1)W0,2+λ)(

f(r2x2−y2−x2(x²₂+y²₂))−f(r0x0−y0−x0(x²₀+y₀²))) +(λ²−λ)W0,1

(f(r1x1−y1−x1(x²₁+y₁²))−f(r0x0−y0−x0(x²₀+y²₀))) +(λ²−λ)W0,3

(f(r3x3−y3−x3(x²₃+y₃²))−f(r0x0−y0−x0(x²₀+y²₀)))) ,

˙

x3= _1+3λ¹ ( λ²(

2f(r₃x₃−y₃−x₃(x²₃+y₃²))−f(r₁x₁−y₁−x₁(x²₁+y²₁))−f(r₂x₂−y₂−x₂(x²₂+y²₂))) +(

(−2λ²+λ+ 1)W_0,3+λ)(

f(r₃x₃−y₃−x₃(x²₃+y²₃))−f(r₀x₀−y₀−x₀(x²₀+y₀²))) +(λ²−λ)W_0,1(

f(r₁x₁−y₁−x₁(x²₁+y₁²))−f(r₀x₀−y₀−x₀(x²₀+y²₀))) +(λ²−λ)W0,2

(f(r2x2−y2−x2(x²₂+y₂²))−f(r0x0−y0−x0(x²₀+y²₀)))) .

(17)

Fully substituted dynamics ofy:

˙

y0= −_1+3λ¹ (

(λ−λW0,1+W0,1)(

f(x1+r1y1−y1(x²₁+y₁²))−f(x0+r0y0−y0(x²₀+y²₀))) +(λ−λW0,2+W0,2)(

f(x2+r2y2−y2(x²₂+y²₂))−f(x0+r0y0−y0(x²₀+y₀²))) +(λ−λW0,3+W0,3)(

f(x3+r3y3−y3(x²₃+y²₃))−f(x0+r0y0−y0(x²₀+y₀²)))

−^4λµ_5L(

−3v0(1−Sgm(0))

+v₁(1−Sgm(U_2,1sgm(^v_L²) +U_3,1sgm(^v_L³))) +v₂(1−Sgm(U_1,2sgm(^v_L¹) +U_3,2sgm(^v_L³))) +v₃(1−Sgm(U_1,3sgm(^v_L¹) +U_2,3sgm(^v_L²))))

−^4λδ_5L(

(W0,1+W0,2+W0,3)v0−W0,1v1−W0,2v2−W0,3v3

)),

˙

y1= _1+3λ¹ ( λ²(

2f(x1+r1y1−y1(x²₁+y²₁))−f(x2+r2y2−y2(x²₂+y₂²))−f(x3+r3y3−y3(x²₃+y₃²))) +(

(−2λ²+λ+ 1)W0,1+λ)(

f(x1+r1y1−y1(x²₁+y²₁))−f(x0+r0y0−y0(x²₀+y₀²))) +(λ²−λ)W0,2

(f(x2+r2y2−y2(x²₂+y₂²))−f(x0+r0y0−y0(x²₀+y²₀))) +(λ²−λ)W0,3

(f(x3+r3y3−y3(x²₃+y₃²))−f(x0+r0y0−y0(x²₀+y²₀))) +^4λµ_5Lv0(1−Sgm(0))

−^λµ_5L(3 + 5λ)v1(1−Sgm(U2,1sgm(^v_L²) +U3,1sgm(^v_L³))) +_10L^λµ(−1 + 5λ)(

v₂(1−Sgm(U_1,2sgm(^v_L¹) +U_3,2sgm(^v_L³))) +v₃(1−Sgm(U_1,3sgm(^v_L¹) +U_2,3sgm(^v_L²)))) +_10L^λδ(

(−1 + 5λ)(W0,2+W0,3)−(6 + 10λ)W0,1

)v0

+_5L^λδ(3 + 5λ)W0,1v1

−_10L^λδ(−1 + 5λ)(W_0,2v₂+W_0,3v₃) )

,

˙

y₂= _1+3λ¹ ( λ²(

2f(x2+r2y2−y2(x²₂+y²₂))−f(x1+r1y1−y1(x²₁+y₁²))−f(x3+r3y3−y3(x²₃+y₃²))) +(

(−2λ²+λ+ 1)W0,2+λ)(

f(x2+r2y2−y2(x²₂+y²₂))−f(x0+r0y0−y0(x²₀+y₀²))) +(λ²−λ)W_0,1(

f(x₁+r₁y₁−y₁(x²₁+y₁²))−f(x₀+r₀y₀−y₀(x²₀+y²₀))) +(λ²−λ)W_0,3(

f(x₃+r₃y₃−y₃(x²₃+y₃²))−f(x₀+r₀y₀−y₀(x²₀+y²₀))) +^4λµ_5Lv0(1−Sgm(0))

−^λµ_5L(3 + 5λ)v2(1−Sgm(U1,2sgm(^v_L¹) +U3,2sgm(^v_L³))) +_10L^λµ(−1 + 5λ)(

v1(1−Sgm(U2,1sgm(^v_L²) +U3,1sgm(^v_L³))) +v3(1−Sgm(U1,3sgm(^v_L¹) +U2,3sgm(^v_L²)))) +_10L^λδ(

(−1 + 5λ)(W0,1+W0,3)−(6 + 10λ)W0,2

)v0

+_5L^λδ(3 + 5λ)W_0,2v₂

−_10L^λδ(−1 + 5λ)(W0,1v1+W0,3v3) )

,

˙

y3= _1+3λ¹ ( λ²(

2f(x3+r3y3−y3(x²₃+y²₃))−f(x1+r1y1−y1(x²₁+y₁²))−f(x2+r2y2−y2(x²₂+y₂²))) +(

(−2λ²+λ+ 1)W0,3+λ)(

f(x3+r3y3−y3(x²₃+y²₃))−f(x0+r0y0−y0(x²₀+y₀²))) +(λ²−λ)W0,1

(f(x1+r1y1−y1(x²₁+y₁²))−f(x0+r0y0−y0(x²₀+y²₀))) +(λ²−λ)W0,2

(f(x2+r2y2−y2(x²₂+y₂²))−f(x0+r0y0−y0(x²₀+y²₀))) +^4λµ_5Lv0(1−Sgm(0))

−^λµ_5L(3 + 5λ)v₃(1−Sgm(U_1,3sgm(^v_L¹) +U_2,3sgm(^v_L²))) +_10L^λµ(−1 + 5λ)(

v₁(1−Sgm(U_2,1sgm(^v_L²) +U_3,1sgm(^v_L³))) +v₂(1−Sgm(U_1,2sgm(^v_L¹) +U_3,2sgm(^v_L³)))) +_10L^λδ(

(−1 + 5λ)(W0,1+W0,2)−(6 + 10λ)W0,3

)v0

+_5L^λδ(3 + 5λ)W0,3v3

−_10L^λδ(−1 + 5λ)(W_0,1v₁+W_0,2v₂) )

,

(18)

FIG. 2: Time series of six oscillation modes at(δ, µ) = (0,0)with initial states randomly chosen fromxq∈[−3.0,3.0]such thatP

q∈Qxq= 0, whereyq= 0andvq= 15. The red, green, light blue, and dark blue lines correspond to the nodes 0, 1, 2, and 3, respectively. (A) In-phase synchronization at(ρ, f) = (3,0.456). (B) Partial In-phase synchronization A at(ρ, f) = (2.5,0.529). (C) Partial In-phase synchronization B at(ρ, f) = (2,0.628). (D) Spontaneous mode switching at(ρ, f) = (1.5,0.746). (E) Long-term Behavior at(ρ, f) = (1,1). (F) Rotation at(ρ, f) = (1,1.139).

FIG. 3: Time series using the mode 3 at(L, θ) = (40,0.35)with the initial volumes(v0, v1, v2, v3) = (30,10,10,10). In eachv-panel, the broken line shows the critical levelLθ = 14. (A) Steady behavior at(δ, µ) = (0,0). (B) Volume diffusion at(δ, µ) = (20,0). (C) Problem-solving process at(δ, µ) = (20,−200).

FIG. 4: Volume allocation patterns. Each nodeqover the critical levelLθ(broken lines) representsvq> Lθ, otherwisevq≤Lθ. Flat-headed arrows indicate inhibition by light.

FIG. 5: Time series of problem-solving processes at(δ, µ) = (10,−15)and(L, θ) = (40,0.35)with the initial volumes(v0, v1, v2, v3) = (45,5,5,5). In eachv-panel, broken lines show the reference volumeLand critical levelLθ= 14. In eachsˆv-panel, we showed the stress levelP

q∈Qsqvˆqwhich was time-averaged after reaching a steady allocation pattern. For all modes, the parameters(ρ, f)were given as well as Fig.2.

FIG. 6: Comparison of performances in diversity production (A) and stress minimization (B) among the oscillation modes. (A) Distributions of finally reached allocation patterns. (B) Distributions of time-averaged stress levelsP

q∈Qsqvˆq.

FIG. 7: An example of spontaneous transition among several volume allocation patterns produced by the mode 4. All parameters were set as well as Fig. 5.

Appendix: Substitution Process Semi-Fully Substituted dynamics ofy:

˙

y0= −_1+3λ¹ (

(λ−λW_0,1+W_0,1)(

f(x₁+r₁y₁−y₁(x²₁+y₁²))−f(x₀+r₀y₀−y₀(x²₀+y₀²))) +(λ−λW_0,2+W_0,2)(

f(x₂+r₂y₂−y₂(x²₂+y₂²))−f(x₀+r₀y₀−y₀(x²₀+y²₀))) +(λ−λW_0,3+W_0,3)(

f(x₃+r₃y₃−y₃(x²₃+y₃²))−f(x₀+r₀y₀−y₀(x²₀+y²₀)))

−^4λµ_5L (

−3v₀s₀+v₁s₁+v₂s₂+v₃s₃)

−^4λδ_5L(

(W0,1+W0,2+W0,3)v0−W0,1v1−W0,2v2−W0,3v3

)),

˙

y1= _1+3λ¹ ( λ²(

2f(x₁+r₁y₁−y₁(x²₁+y₁²))−f(x₂+r₂y₂−y₂(x²₂+y²₂))−f(x₃+r₃y₃−y₃(x²₃+y₃²))) +(

(−2λ²+λ+ 1)W0,1+λ)(

f(x1+r1y1−y1(x²₁+y²₁))−f(x0+r0y0−y0(x²₀+y₀²))) +(λ²−λ)W0,2

(f(x2+r2y2−y2(x²₂+y₂²))−f(x0+r0y0−y0(x²₀+y²₀))) +(λ²−λ)W0,3

(f(x3+r3y3−y3(x²₃+y₃²))−f(x0+r0y0−y0(x²₀+y²₀))) +^4λµ_5L v0s0−^λµ_5L(3 + 5λ)v1s1+_10L^λµ(−1 + 5λ)(

v2s2+v3s3

) +_10L^λδ(

(−1 + 5λ)(W_0,2+W_0,3)−(6 + 10λ)W_0,1) v₀ +_5L^λδ(3 + 5λ)W0,1v1−_10L^λδ(−1 + 5λ)(W0,2v2+W0,3v3)

) ,

˙

y2= _1+3λ¹ ( λ²(

2f(x₂+r₂y₂−y₂(x²₂+y₂²))−f(x₁+r₁y₁−y₁(x²₁+y²₁))−f(x₃+r₃y₃−y₃(x²₃+y₃²))) +(

(−2λ²+λ+ 1)W_0,2+λ)(

f(x₂+r₂y₂−y₂(x²₂+y²₂))−f(x₀+r₀y₀−y₀(x²₀+y₀²))) +(λ²−λ)W0,1

(f(x1+r1y1−y1(x²₁+y₁²))−f(x0+r0y0−y0(x²₀+y²₀))) +(λ²−λ)W0,3

(f(x3+r3y3−y3(x²₃+y₃²))−f(x0+r0y0−y0(x²₀+y²₀))) +^4λµ_5L v₀s₀−^λµ_5L(3 + 5λ)v₂s₂+_10L^λµ(−1 + 5λ)(

v₁s₁+v₃s₃) +_10L^λδ(

(−1 + 5λ)(W0,1+W0,3)−(6 + 10λ)W0,2

)v0

+_5L^λδ(3 + 5λ)W0,2v2−_10L^λδ(−1 + 5λ)(W0,1v1+W0,3v3) )

,

˙

y3= _1+3λ¹ ( λ²(

2f(x₃+r₃y₃−y₃(x²₃+y₃²))−f(x₁+r₁y₁−y₁(x²₁+y²₁))−f(x₂+r₂y₂−y₂(x²₂+y₂²))) +(

(−2λ²+λ+ 1)W_0,3+λ)(

f(x₃+r₃y₃−y₃(x²₃+y²₃))−f(x₀+r₀y₀−y₀(x²₀+y₀²))) +(λ²−λ)W_0,1(

f(x₁+r₁y₁−y₁(x²₁+y₁²))−f(x₀+r₀y₀−y₀(x²₀+y²₀))) +(λ²−λ)W0,2

(f(x2+r2y2−y2(x²₂+y₂²))−f(x0+r0y0−y0(x²₀+y²₀))) +^4λµ_5L v₀s₀−^λµ_5L(3 + 5λ)v₃s₃+_10L^λµ(−1 + 5λ)(

v₁s₁+v₂s₂) +_10L^λδ(

(−1 + 5λ)(W0,1+W0,2)−(6 + 10λ)W0,3

)v0

+_5L^λδ(3 + 5λ)W0,3v3−_10L^λδ(−1 + 5λ)(W0,1v1+W0,2v2) )

,

(19)