!"#$#%&'()"&#*(+,*&-#%.

/&*$0(1(!2#&'34(

56!7(8&)9(!:;<!5=9(>"?0*$#*&

(<*($20(@"03#4A.(-#''0**#A-9($4?0$20"(B#$2(C0"(D&E9(B0(@"4@4.0+($2&$($20(%"#$#%&'#$,(#.($20(+,*&-#%.(

#*(B2#%2($20()"&#*(&.(&(%4''0%$#30(%&*(@"4+A%0(&+&@$#30()02&3#4"9()&.0+(4*($20($"#3#&'(F&%$($2&$(*0&"(

&*(4"+0"(G(+#.4"+0"(@2&.0($"&*.#$#4*9($20"0(#.(@'0*$,(4F(-0$&.$&)'0(*0A"4*&'(%4*F#?A"&$#4*.9(#H0H9($20(

-#%"4.%4@#%(.$&$0.(@"4+A%#*?(2A-&*()02&3#4"H(=2#.(#+0&(@"4@&?&$0+(*4($0""&#*(#*($20(#*#$#&'(,0&".(

&*+(%4''0%$0+(-4.$',(*4*A.0FA'(%"#$#%.9(.E0@$#%#.-9(.&"%&.-(A@($4(F"&*E',(.#'',(IA0.$#4*.(.A%2(A.(

JB2&$ ( A.0FA' ( %&* ()0 ( F4" (-, ()"&#* ( $4 (2&30 ( &* ( &3&'&*%20KL(( !'0&"', ( $20 ( .#$A&$#4* ( %2&*?0+ ( &*+(

&'$24A?2 (-&*, ( @&@0". ( &* (-4+0'. (B#'' ( %4*$#*A0 ( $4 ( )0 (B"#$$0* (4* (24B ( $20 ()"&#* ( 4.%#''&$0 ( &*+(

.,*%2"4*#M09($20(03#+0*%0(&'"0&+,(.24B.($2&$($20(-&N4"#$,(4F($20()"&#*.(.@0*+.(-4.$(4F(#$.($#-0(

&B&,(F"4-(.,*%2"4*#%(4.%#''&$#4*.9(4*(&(+,*&-#%&'(.$&$0(NA.$(#*()0$B00*(J*0A"4*&'(4.%#''&$#4*.L(&*+(

J"&*+4-(.@#E#*?LH(<*($2#.($&'E(B0(B#''()"#0F',(+#.%A..($20(-4$#3&$#4*(&*+("0%0*$(@"4?"0..("0'03&*$($4(

$20(%"#$#%&'#$,(%4*%0@$(&$($2"00('030'.(#*%'A+#*?O('&"?0(.%&'0(F61<(2A-&*(+&$&()4$2(#*(20&'$2(&*+(

+#.0&.09(.-&''(.%&'0(*0A"4*&'(&3&'&*%20.(+,*&-#%.(&*+(F#*&'',()02&3#4"&'(+&$&H

!"#$%&'($)&*+%,*"-(./%0#%&'($#1")#.)&/

23+4*.%5*"06"),'0%7"*.*-*%8.)9("/)04:%;<*).

5'($)&*+%,*"-(./%*"(%0'(%1(/0%(=*$<+(%#>%*%/4.0'(0)&:%.#.(?3)+)1")3$%<"#&(//%0'*0%&"(*0(/%&#$<+(=%

$*0(")*+/@%A'()"%/0"3&03"(/%/<*.%*0%+(*/0%/)=%#"-("/%#>%$*,.)03-(%).%+(.,0'%B.$%0#%$$C@%D(4%0#%0')/%

$*,)&%*"(%/(+>E<"#<*,*0).,%"(*&0)#.%F#.(/%0'*0%$*).0*).%/0((<%&#.&(.0"*0)#.%,"*-)(.0/G%#/$#/)/%*.-%

13#4*.&4E-")9(. %&#.9(&0)#. % *"( %-")9)., % >#"&(/@ %5'($)&*+ % ,*"-(./ %'*9( %1((. % /03-)(- % >"#$% 0'(%

*+&'($)/0/%#.6*"-/:%130%.#6%).%0'(%HI/0%&(.03"4%6(%1(,).%0#%3.-("/0*.-%'#6%0'(4%&*.%+(*-%3/%0#%*%

.(6%>)(+-%#>%/(+>E#",*.)F(-%/0"3&03"(/%<"#-3&(-%*0%0'(%).0("/(&0)#.%#>%&'($)/0"4:%>+3)-%-4.*$)&/:%*.-%

$*0(")*+/%/&)(.&(@%J(%<"#<#/(%0#%&*++%0')/%>)(+-%&'($#1")#.)&/@

!"#$%&'(")*+,-./+(0+1%".2$3.03(2$%4+%'(.0(,5.0.,$5(

0+1%"',.+0,+6(.0'.37-'(8%"2(908"%2$-."0(!7+"%:($0&(

;"2<5+=(>+-#"%4'(?0$5:'.'

@+'1'(AB(;"%-+'

94+%)$'C1+D(!7+(E$'C1+(F"10&$-."0(8"%(G,.+0,+D(E.",%1,+'(H+$5-7(I+'+$%,7(90'-.-1-+D(E.5)$"D(G<$.0B

9(#.55(&.',1''(%+,+0-(%+'15-'(.0(,5.0.,$5(0+1%"',.+0,+(.0(<$-.+0-'(#.-7(&+8.,.-("8(,"0',."1'0+''($8-+%(

-%$12$-.,()%$.0(.0*1%:B(90(,"0,%+-+D(9(#.55(%+<"%-("0(%+,+0-(%+'15-'(#7.,7($%+()$'+&("0(-7+(810,-."0$5(

2$30+-.,(%+'"0$0,+(.2$3.03(J8AI9K($-(-7+(%+'-.03('-$-+(J%'K("8(-7"'+(<$-.+0-'B(L+(7$/+($0$5:M+&(-7+(

%'N8AI9(0+-#"%4'():(1'.03(&.88+%+0-(2+-7"&'()$'+&("0(908"%2$-."0(!7+"%:($0&(;"2<5+=(>+-#"%4'(

?0$5:'.'B(L+($%+(<%"<"'.03(<"''.)5+(,$0&.&$-+(2$%4+%'(-7$-(,$0($,,"10-(8"%(-7+(5+/+5("8(&+8.,.-("8(

,"0',."1'0+''(8"10&("0(-7"'+(<$-.+0-'B(!7+(2$.0(2"-./$-."0("8(-7.'(%+'+$%,7(.'(-7+(0++&(8"%(0+#(

2$%4+%' ( -7$- ( )$'+& ( "0 ( -7+ ( %'N8AI9(&:0$2.,' ( 2.37-(.2<%"/+ ( 0"- ( "05: ( -7+(&.$30"'.' ( )1- ( -7+(

<%"30"'.' ( "8 ( -7"'+ ( <$-.+0-'B

L"%&(&"0+(.0(,"55$)"%$-."0(#.-7(O+%P0.,$(AQ4.NA$%--10+0D(9)$.(R.+MD(R$0-+(IB(;7.$5/"($0&(A.%-$(

FB(O.55$%%+$5

A Novel Polynomial Memristor ModelA. Ascoli, V. Senger and R. Tetzlaff

Faculty of Electrical and Computer Engineering,Technische Universitat Dresden, Dresden, Germany

F. CorintoDepartment of Electronics and Telecommunications

Politecnico di Torino, Torino, Italy

Abstract

Recent studies [1] on a comparison among few of the most important memristor models proposed in literature highlighted themodel-dependence of the dynamic behaviors observed in simulations of memristor-based electronic systems. However, in most ofthese cases, it is not possible to determine which of the models is correctly capturing what would happen in practice, because realmemristors are still not commercially available. Thus, since memristor technology is not yet mature, mathematical models mayonly be validated by showing qualitative agreement between their predictions and published experimental results, which, however,do not cover all possible scenarios regarding driving conditions, circuit topologies, and parameter settings. The requirements for agood model include generality, simplicity and accuracy. In our attempt to fulfill these requirements, last year we introduced a novelmodeling approach based upon the crucial role boundary conditions may have on the emergence of memristor state dynamics.The Boundary Condition Model was recently generalized even further to include neural synapses within its modeling domain. Inthis work we propose yet another modeling approach which takes inspiration from Prof. Chua’s idea of unfolding the memristorby expanding the state evolution function, also named morphing function (let us call it g(i, w)), in series of polynomials involvingthe input i and the state variable w. The state equations of the proposed model, which we name Polynomial Memristor (PM)model are thus expressed as w = g(i, w) =

Nn=0 an in +

Mm=0 bm wm +

Pp=1(cp i + dp)

p(ep w + fp)p, for |i| > imin

and 0 for −imin ≤ i ≤ imin. Regarding the input-output algebraic equation, we cast it in the general form v = R(i, w)i, soas to have the opportunity to choose the most suitable memristance expression on the basis of the memristor under modeling.Possible choices for the input-output equation include but are not limited to the state-dependent Ohm’s law in the BCM model,the implicit relationship in the Pickett’s model and the two expressions proposed in the TEAM model. One further choice wouldbe the expansion of the memristance function into yet another series of polynomials involving input and state variable. As Prof.Chua demonstrated, massaging the state evolution function and the memristance function one may qualitatively reproduce a largenumber of current-voltage pinched hysteretic loops observed in experimental observations of memristor behavior, irrespective ofthe physical mechanisms underlying the dynamics. The PM model may thus be tuned so as to mimic the behavior of a largeclass of physical memristors. This is done by choosing a suitable memristance function and by optimizing the coefficients ofthe polynomial series so that the resulting state equation may reproduce the dynamics of the system under model. As Proof-of-Concept, we fitted the morphing function to the Pickett’s model. The first investigations focused on the case of i < −imin, wherethe variation intervals for input current i and state w were chosen so as to match experimental results from Hewlett PackardLabs. Plots (a) and (b) in Table I show the switching characteristics of the Pickett’s model, the TEAM model and the proposedPM model. It should be noted that in this investigation imin was set to 0.05mA. The values of the parameters for each of thethree models shall be reported in the final camera-ready manuscript (note that in the PM model N , M , and P were respectivelyset to 10, 13 and 4). Plots (a) and (b) in Table I reveal the added value a polynomial series expansion of the state evolution

Table ICOMPARISON OF SWITCHING CHARACTERISTICS OF PICKETT’S, TEAM AND PM MODELS.

(a) The w − i plots of Pickett’s (solid), TEAM (dotted) (b) The w − w plots of Pickett’s (solid), TEAM (dotted)and PM (dashed) models for w = 1.1nm (blue), and PM (dashed) models for i = 0.06mA (blue),w = 1.475nm (green) and w = 1.85nm (red). i = 1mA (green) and i = 2mA (red).

function may bring to the current search for a simplified and at the same time accurate representation of the Pickett’s model. Inboth pictures the PM model expectations outperform the TEAM model predictions. The PM model is able to track the dynamicsof the Pickett’s model even in the other two possible input intervals, i.e. i > imin and |i| ≤ imin. This will be reported in thecamera-ready paper, where the model shall be also tested in a couple of simple memristor circuits to demonstrate its ability tocapture expected dynamical behaviors.

REFERENCES

[1] A. Ascoli, F. Corinto, V. Senger and R. Tetzlaff: “Memristor model comparison,” IEEE Circuits and Systems Magazine, under press, 2013

!"#"$%&'$(')(*+,-(!./(01&$%234"5%#4(637$'2%8"(7.%'9

!"#"$%&'()'*+$,-.-$/"0"$/1)*&2+3,-$4"0"$56&738(9*+.$,$:2(*;8'<2+$=8>('(6(*$21$53?'2$/8@'8**)'8@$38?$/;*A()28'A>$21$5!#-.$B2>A2C$=8>('(6(*$21$DE7>'A>$38?$F*AE82;2@7$G#(3(*$H8'+*)>'(7I

AE32>JAK;')*")6

B'A)2C3+*$AE32('A$>'@83;>$3)*$A28>'?*)*?$3>$*11*A('+*$'812)&3('28$A3))'*)$12)$6;()3C'?*L38?$

GHMNI$C')*;*>> $A2&&68'A3('28>" $FE*7$3)*$ '8()2?6A*?$ '8 $>(38?3)?> $ =///$OP.",Q"R3$ S,T $38?$

=///$OP.",Q"U$S.T"$FE*$>(38?3)?$=///$OP.",Q"U$'>$21$>K*A'3;$'8(*)*>(-$L*A36>*$'($)*@6;3(*>$(E*$6>*$

21$HMN$>'@83;>$12)$&3>>$K*)>283;$3KK;'A3('28>"$!;>2-$>6AE$3KK;'A3('28>$21$HMN$?783&'A$AE32>$

3>$>E2)(V)38@*$)3?'2;2A3('28$38?$)3?'2$+'>'28$3)*$21$@)*3($'8(*)*>("

N7$82C-$>K*A()3;$&3><>$12)$68;'A*8>*?$6>*$21$HMN$>'@83;>$3)*$3?2K(*?$'8$&387$A268()'*>$21$

(E*$C2);?"$!>$3$)6;*-$(E*$1)*W6*8A7$)38@*$X$(2$,P$YZ9$'>$K*)&'((*?"$%*>'@8$21$AE32('A$>26)A*>$12)$

(E'>$)38@*$'>$3$AE3;;*8@*"$=($'>$8*A*>>3)7$(2$82($28;7$K)2+'?*$@*8*)3('28$21$AE32('A$>'@83;$C'(E'8$

(E'> $ 1)*W6*8A7 $ L38?- $ L6( $ (2 $ >78(E*>'9* $ 38 $ 2>A';;3(2) $ (E3( $ K)2+'?*> $ K)*3>>'@8*? $ >K*A()3;$

AE3)3A(*)'>('A>$21$(E*$>'@83;"

=8$ (E'> $C2)<- $ (E*$K)2L;*&$21$?*>'@8'8@$&'A)2C3+*$AE32('A $>26)A*$ (E3( $A2+*)> $ (E*$*8(')*$

1)*W6*8A7$L38?$X$ (2 $,P$YZ9$C'(E$3> $>&22(E$K2C*)$>K*A()6&$*8+*;2K*$C'(E'8 $ (E'> $L38?$3>$

K2>>'L;* $ '> $ A28>'?*)*?" $ FE6>- $ C* $ (3;< $ 3L26( $ 2>A';;3(2) $ (E3( $ @*8*)3(*> $ AE32('A $ >'@83; $ C'(E$

L38?C'?(E$*[A**?'8@$(E)**$2A(3+*>\

N$]$1Z^1_$`X"X-$CE*)*$1Z$],P$YZ9$38?$1_$]$X$YZ9$3)*$6KK*)$38?$;2C*)$1)*W6*8A'*>$21$AE32('A$

>'@83;-$)*>K*A('+*;7"

FE*$(E*2)7$21$@*8*)3('8@$AE32('A $>'@83;>$C'(E$K)*V3>>'@8*?$12)&$21$K2C*)$>K*A()6&$C3>$

?*+*;2K*?$ 12) $ (E* $ 1')>( $ ('&* $ 12) $ )'8@V>()6A(6)* $2>A';;3(2)> $C'(E $3 $828;'8*3) $*;*&*8( $E3+'8@$3$

?)2KK'8@ $ >*A('28 $ SXT" $ Z2C*+*)- $ (E*7 $ C*)* $ *>>*8('3;;7 $ &2?*; $ >7>(*&>- $ (E3( $ A26;? $ 82( $ L*$

'&K;*&*8(*? $ '8 $&'A)2C3+*$ 1)*W6*8A7$ )38@*" $FE*)*12)*- $K)'8A'K;*> $21 $@*8*)3('28- $ (E*2)7 $38?$

A2&K6(3('283;$&*(E2?>$C*)*$?*+*;2K*?$;3(*)$12)$AE32('A$@*8*)3(2)>-$'8A;6?'8@$(E2>*$C'(E$K)*V

3>>'@8*?$>K*A()3;$AE3)3A(*)'>('A>-$'8$CE'AE$&'A)2C3+*$()38>'>(2)>$C*)*$6>*?$3>$(E*$3A('+*$*;*&*8($

SR-$QT"$=8$(E*>*$K3K*)>-$*[3&K;*>$21$AE32('A$@*8*)3(2)$?*>'@8$C'(E$)3(E*)$68'12)&$>K*A()3;$?*8>'(7$

'8$K)*>A)'L*?$1)*W6*8A7$L38?$3)*$K)*>*8(*?"

%*>'@8'8@$AE32('A$>26)A*>$C'(E$(E*$L38?$C'?*)$(E38$X$2A(3+*>$?*&38?>$?*;'A3(*$>2;6('28$21$

(E*$>7>(*&$>78(E*>'> $K)2L;*&"$B2)*2+*)- $C'(E$A28>'?*)3L;* $ '8A)*3>* $21 $ (E*$6KK*) $@*8*)3('28$

L268?3)7- $ (C2$13A(2)> $ (E3( $C*)* $82( $ >2$ '&K2)(38( $3( $ 1)*W6*8A'*> $L*;2C$U$YZ9$82C$>(3)( $ (2$

'81;6*8A*\$83&*;7-$(E*$>'9*$21$(E*$*;*&*8(>$>(3)(>$(2$'81;6*8A*$(E*$A288*A('28$;*8@(E$'8$(E*$?*+'A*-$

38? $ ;2>>*> $ '8 $ (E* $ >6L>()3(* $ a5VR $ '8A)*3>* $ G12)&3;;7- $ a5VR $ A38 $ L* $ >6AA*>>16;;7 $ 6>*? $ 3( $ (E*$

1)*W6*8A'*>$6K$(2$.$YZ9I"

=8$(E'>$C2)<-$3$K2>>'L';'(7$21$AE32('A$>'@83;$@*8*)3('28$C'(E$K)*>A)'L*?$AE3)3A(*)'>('A>$3($(E*$

K)*>*8($A28>()3'8(>$'>$'8+*>('@3(*?"

S,T =///$#(38?3)?$12)$=812)&3('28$(*AE82;2@7bF*;*A2&&68'A3('28>$38?$'812)&3('28$*[AE38@*$L*(C**8$

>7>(*&>b_2A3; $ 38? $ &*()2K2;'(38 $ 3)*3 $ 8*(C2)<>b#K*A'1'A $ )*W6')*&*8(>c $ D3)( $ ,Q"R\ $ M')*;*>>$

B*?'6&$!AA*>> $d28()2; $ GB!dI38?$DE7>'A3; $_37*) $GDZeI$#K*A'1'A3('28> $12) $_2CV53(*$M')*;*>>$

D*)>283;$!)*3$4*(C2)<>$GMD!4>Ic$!&*8?&*8($,\$!??$!;(*)83(*$DZe>-$.PPf"

S.T =/// $ OP.",Q"UgV.P,."$=/// $ #(38?3)? $ 12) $ _2A3; $ 38? $&*()2K2;'(38 $ 3)*3 $ 8*(C2)<> $ V $ D3)( $ ,Q"U\$

M')*;*>>$N2?7$!)*3$4*(C2)<>$.P,."$E((K\^^>(38?3)?>"'***"2)@^3L26(^@*(^OP.^OP.",Q"E(&;

SXT !"#$%#&' ()* (+*, (-./.0 ()* ( 1*, (+$.%23' (+* (4*$ 5'8@ $2>A';;3('8@ $ >7>(*&>$38?$ (E*') $3KK;'A3('28 $ (2 $ (E*$

>78(E*>'>$21$AE32>$@*8*)3(2)>$^^$=8("$h"$21$N'16)A3('28$38?$dE32>"$,iiU"$0"$U"$42"$Q"$D"$OQ,VOUQ

SRT %&'()'*+$!"#"- $/1)*&2+3$/"0-$:E';'8><7$!"%"$j#78(E*>'>$21 $>'8@;*V()38>'>(2)$AE32('A$2>A';;3(2)>j- $

-%35*(6!7+89::;-$B37$iV,X-$/+2)3-$D2)(6@3;-$.PPR-$KK"$,XXV,XU"$

SQT %&'()'*+ $ !"#"- $ /1)*&2+3 $ /"0- $B3<>'&2+ $ 4"!"- $ D383> $ !"=" $ Y*8*)3('28 $ 21 $ dE32> $ ^^ $B2>A2C\$

F*AE82>KE*)3-$.P,.-$R.R$K"$G#/(<=00#./I"

Sound Creation Based on Nonlinear Cell CircuitsNaruno Fujii and Hiroshi Yamamoto

Faculty of Science and Technology, Sophia University7–1, Kioi-cho, Chiyoda-ku, Tokyo 102–8554, Japan

Email: nanno0917@sophia.ac.jp

Yoshifumi NishioFaculty of Electrical and Electronic Engineering, Tokushima University

2–1, Minami-Josanjima, Tokushima 770–8506, JapanEmail: nishio@ee.tokushima-u.ac.jp

Masaki Bandai and Mamoru TanakaFaculty of Science and Technology, Sophia University

7–1, Kioi-cho, Chiyoda-ku, Tokyo 102–8554, JapanEmail: mamoru.tanaka@gmail.com

I. ABSTRACT

The sound has three elements. They are ”Sound volume”, ”Musicalpitch”, and ”Tone color”. First, ”Sound volume” changes by theamplitude of the waveform. When the amplitude is large, the soundwill be also large. And the other hand, when the amplitude will besmall, the sound will be also small. Next, about ”Musical pitch”, Itchanges depend on its frequency. The sounds with high frequenciesbecome high-pitched sounds, and the sounds with low frequenciesbecome low-pitched sounds. The last, ”Tone color” can be decided byits waveforms.Sound is classified into three kinds by the waveform.They are ”Pure tone”, ”Musical tone”, and ”Unpitched sound”. ”Puretone” is a sound that has only one sine wave. And, ”Musical tone”is the lasting sounds that have regular vibrations except for thepure tone. Then, ”Unpitched sound” is a sound that has almost noregular vibrations. In this research, we focused on the musical tonewaveform. Now, the signal processing is mainly used in order togenerate the musical tone waveform. Then, we propose the methodfor generating the musical tone waveform from the state equation asa new method. We calculate the state equation from the nonlinearcircuit that is modified Nishio circuit. In this method, the frequencyand the musical tone waveform can be changed easily by changingthe value of capacitance of the nonlinear circuit.

II. SIMULATION RESULTS

I1 I2 I3 I4

v1 i2 v3 i3 v4 i4

RL1 L1 L1 L1

L2 L2 L2 L2

C C C Cv2i1

f(v ) f(v ) f(v ) f(v )1 2 3 4

Fig. 1. Circuit model. L1 = 100.7[mH], L2 = 10.31[mH], C =12.69[µF],and R = 26.72[Ω].

The nonlinear circuit which used this research is shown in Figure1. The state equation obtained from this circuit is shown in (1). In thisresearch, we calculated the voltage waveform of the capacitor fromeach node electrical potential. We considered as the waveform tobe a musical tone waveform, and generated the musical tone. Thus,

we showed to be able to generate the musical tone from the stateequation.

L1dIk

dt= vk − R

4∑

j=1

I j

L2dikdt = vk

C dvkdt = −Ik − ik − f (vk)

f (vk) = −α |vk+1|−|vk−1|2

α = 0.012

(1)

The information of music is usually written in its score. In thisresearch, MIDI information plays the role of score. MIDI informationshows the property of each music such as length and pitch ofeach note. In this MIDI information, quarter note are defined asone second. Actually to play the musics by using this method, itis necessary to combine such data into this system. To keep the”Sustain” of each notes which is decided by MIDI information, andto realize the smooth ”Release” , changing the volume of resistanceα is needed. It can be expected that resistance α works as like ainclination when sound is releasing. Then it leads smoothly shift tothe next note.

III. CONCLUSION

In this paper, we showed that the musical tone can be generatedby using the state equation that had been obtained from the nonlinearcircuit that the Nishio circuit had been modified. And to realizenot only generate the musical tone but also generate the comfort-able musical tone from state equations, what we focused on was”Enverope”. ”Enverope” are consisted of four elements as follows,”Attack”, ”Decay”, ”Sustain” and ”Release”. In order to improvethe quality of sound, ”Release” can be the very important elementin particular. Smooth ”Release” makes the sounds more artistic andnatural. Resistance α works as like a inclination of ”Release”. Then,it can be expected that resistance α is the key point of ”Release”.

REFERENCES

[1] Y.Nishio and A.Ushida, ”Chaotic Wandering and its Analysis in Sim-ple Coupled Chaotic Circuits ”, IEICE Transactions on Fundamentals,vol.E85-A,no.1,pp.248-255,Jan.2002.

Synchronization and Wave Phenomenain Coupled Simultaneous Oscillators

Saori Fujioka†, Yang Yang‡, Yoko Uwate† and Yoshifumi Nishio††Dept. Electrical and Electronic Eng.,

Tokushima University, JapanEmail: saori, uwate, nishio@ee.tokushima-u.ac.jp

‡ The Institute of Artificial Intelligence and Robotics., Xi’an Jiaotong UniversityEmail: yyang@mail.xjtu.edu.cn

SUMMARYIn 1954, Schaffner reported that an oscillator with two degrees of freedom could oscillate simultaneously at two different

frequencies when the nonlinear characteristics are described by a fifth-power polynomial function [1]. Datardina also investigatethe fifth-power nonlinear characteristics theoretically and it has been experimentally verified using both analog simulationsand electronic circuitry [2]. However, after their pioneering works, as far as the authors know, there have not been manyresearches clarifying the theoretical analysis of the simultaneous oscillations. In our past study, we have reported synchronizationphenomena in a ladder of simultaneous oscillators with three or four LC resonators coupled by inductors [3][4]. We confirmedthe various waves in a ladder of coupled simultaneous oscillators. The circuit model is shown in Fig. 1(a). In the circuit, twohard oscillators with three LC resonators are coupled by inductors LC . Fig. 1(b) shows time waves of each resonators. Fromthis result, we can say that generation of simultaneous oscillation were confirmed. Upper resonators shows in-phase, middleresonators shows anti-phase, and bottom resonators shows double-mode, respectively.In this study, we investigate the theoretical analysis and circuit experiments in the two inductively coupled simultaneous

oscillators. In addition, we also investigate the phenomena in the N inductively coupled simultaneous oscillators in detail. Weanalyze phenomena of double-mode and simultaneous oscillation shown in Fig. 1(b) and various propagation waves shown inpast study to use averaging method. From these results, we expect progress to the practical applications.

(a) (b)

Fig. 1. (a) Ladder of coupled simultaneous oscillators with three resonators. (b) Time waves of 6 resonators

REFERENCES[1] J. Schaffner, “Simultaneous oscillations in oscillators,” IRE Transactions on Circuit Theory, Vol.1, pp.2-81, Jun. 1954.[2] S. Datardina and D.A. Linkens, “Multimode Oscillations in Mutually Coupled Van der Pol Type Oscillators with Fifth-Power Nonlinear Characteristics,”

IEEE Transactions on Circuits and Systems, Vol.25, pp.308-315, May. 1978.[3] S. Fujioka, Y. Yang, Y. Uwate and Y. Nishio, “Two Kinds of Waves in a Ladder of Coupled Simultaneous Oscillators,” Proceedings of NDES’12,

pp. 232-236, Jul. 2012.[4] S. Fujioka, Y. Yang, Y. Uwate and Y. Nishio, “Propagation Waves in a Ladder of Coupled Simultaneous Oscillators,” Proceedings of NOLTA’12,

pp. 907-910, Oct. 2012.

!"#$%&'()*#+,-+).*'./$0'*+"'1$(.'#$#+2"3

42$/)3%'5$0"6'3%&'78"&)'!*$$('9%-*)*8*"'$.':"8/$)%.$/03*)#-';<1=>?1<'<@/)#+

9*'+3-'A""%'B%$C%'.$/'3'2$%D'*)0"'*+3*'*+"'("/#")E"&'()*#+'$.'3'#$0(2"F'+3/0$%)#'-$8%&'#+3%D"-').'

*+"'(3/*)32-'$.'*+"'-$8%&'3/"'-+).*"&')%'./"G8"%#H'AH'3'.)F"&'30$8%*I'J3-"&'$%'-)0(2"'%$%2)%"3/'

0$&"2)%DK'3((/$F)03*"'/82"-'.$/'*+"'-+).*'$.'*+"'()*#+'-+).*'#$82&'A"'D)E"%'L.)/-*'()*#+'-+).*'23CMI'9%'

(-H#+$3#$8-*)# ' "F("/)0"%*-K ' +$C"E"/K ' #2"3/ ' &"E)3*)$%- ' ./$0 ' *+"-" ' (/"&)#*)$%- 'C"/" ' $A-"/E"&'

L-"#$%&'()*#+,-+).* '".."#*-MI '?+)-' /3)-"-' *+"'G8"-*)$%'$.'C+"*+"/ ' *+"-"'&"E)3*)$%- '3/"'&8"' *$' *+"'

A)$(+H-)#-'$.'*+"'%$%2)%"3/'+"3/)%D'-"%-$/K'*+"'#$#+2"3K'$/'3%'3/*).3#*'D"%"/3*"&'+)D+"/'8(')%'*+"'

38&)*$/H'(3*+C3HI'N"'&"0$%-*/3*"'*+3*'*+"'-"#$%&'()*#+,-+).*')-'D"%"/3*"&')%'*+"'#$#+2"3K'3%&'*+3*'

A$*+'#$0A)%3*)$%,*$%"'D"%"/3*)$% '3%&' 2$C,(3-- '.)2*"/)%D '3/"' *+"'B"H'.3#*$/- ' /"-($%-)A2" '.$/ ' *+"'

(+"%$0"%$%I'9%'(3/*)#823/K'C"'.)%&'*+3*'*+"'-#32)%D'23C-'$.'1$(.'#$#+2"3'#$0A)%3*)$%'*$%"-'"F(23)%'

*+"'#23--)#32K'*$'&3*"'($$/2H'"F(23)%"&'(-H#+$3#$8-*)#32'&3*3'$.'5I4I'!0$$/"%A8/D'LOPQRMI

!"#$%$&'(")$*+,$&-.*+/.0+"1&.,"&23+*$&$3&2.4+"4*+"4"%5)2)+./+&!.+

&56$)+./+)%$$6+)624*%$)+24+$$7

#8#8+79:;<=>?@?+$85:8+)ABCAD<=EF

>!"#"$%&'!$"$(')*+&(#,+$-.'!"#"$%&.'/0,,+"@!"#"$%&'!$"$('1(23*+2"4')*+&(#,+$-.'!"#"$%&.'/0,,+"

F5*,$+$0$('%6'$3('7+83(#'9(#&%0,':2$+&+$-'"*;'9(0#%<3-,+%4%8-.'=%,2%>.'/0,,+"

==G9:;<=HGIEAJ8K<I

4<LEMENO+BPQ9Q+AO+E+RJQCBN+<S+QSSQKBA=Q+C<CJACQE9+IQBP<MO+S<9+ECEJNOAO+ECM+MAEGC<OBAKO+<S+BPQ+

K<IRJQT +<OKAJJEBA<CO? +LABP +IECN+<S + BPQI+;QACG+:OQM + AC + BPQ +MASSQ9QCB + SAQJMO +<S + OKAQCKQ +O:KP+EO+

IQMAKACQ +ECM+RPNOA<J<GN8 +"RRJAKEBA<C +<S + BPQOQ +IQBP<MO +S<9 +ECEJNOAO +<S + 9PNBPIAK +QJQKB9AK +;9EAC+

EKBA=ABN+AO+QORQKAEJJN+EKB:EJ8+):KP+EKBA=ABN+AO+E+9QO:JB+<S+ONCKP9<C<:O+L<9D+<S+E+G9QEB+C:I;Q9+<S+

<OKAJJEB<9N+QJQIQCBO+U+CQ:9<CO+LPAKP+K<COBAB:BQ+K<IRJQT+;9EAC+<OKAJJEB<9N+CQBL<9D+V>W8

$JQKB9<QCKQRPEJ<G9EI+ X$$7Y + AO + E + K<II<C +O<:9KQ +<S + ACS<9IEBA<C +E;<:B + ;9EAC + EKBA=ABN + AC+

CQ:9<RPNOA<J<GAKEJ+9QOQE9KP8+$$7+AO+EC+E=Q9EGQM+O:I+<S+QJQKB9AK+K:99QCBO+S9<I+E+G9<:R+<S+CQ:9<CO?+

J<KEBQM +CQE9 + BPQ +9QK<9MACG+QJQKB9<MQ8 +$$7+OAGCEJ + AO +:O:EJJN +MA=AMQM+ ACB<+ SQL+BNRAKEJ + S9QZ:QCKN+

9ECGQO+XEJRPE?+;QBE+QBKY+V>W8+&PQ9Q+AO+BPQ+Q=AMQCKQ+BPEB+E+OB9<CG+K<99QJEBA<C+ERRQE9O+;QBLQQC+BPQ+S<9I+

<S+9PNBPIAK+$$7+EKBA=ABN+AC+E+RE9BAK:JE9+S9QZ:QCKN+9ECGQ+XO<'KEJJQM+<OKAJJEB<9N+REBBQ9CO+V>?@WY+ECM+

BPQ+S:CKBA<CEJ+OBEBQ+<S+E+JA=ACG+<9GECAOI+V@W8+)<?+BPQ+OB:MN+<S+RE9BAK:JE9+<OKAJJEB<9N+REBBQ9CO+ECM+

9QG:JE9ABN +<S+BPQA9 +ERRQE9ECKQ+<C+$$7+AO+<CQ+<S+BPQ+I<OB+AIR<9BECB+B<RAKO+AC+BPQ+;9EAC+EKBA=ABN +

9QOQE9KP8+&PQ+OB:MN+<S+O:KP+9PNBPIAK+K<IR<CQCBO+AO+QORQKAEJJN+AIR<9BECB+AC+9QOQE9KPQO+<S+KQCB9EJ+

CQ9=<:O + ONOBQI + REBP<J<GAQO? + ;QKE:OQ + O<IQ + <OKAJJEB<9N + K<IR<CQCBO + IEN + ERRQE9 + EO+

QJQKB9<QCKQRPEJ<G9ERPAK+PEJJIE9DO+<S+E+MAOQEOQ8

)JQQR+ORACMJQ+AO+<CQ+<S+BNRQO+<S+OJQQR+<OKAJJEB<9N+EKBA=ABN+<C+$$78+)JQQR+ORACMJQO+E9Q+;9AQS+

QRAO<MQO+<S+<OKAJJEB<9N+EKBA=ABN+LABP+S9QZ:QCKN+<S+['\+ECM+>]'>^+-_+ECM+KPE9EKBQ9AOBAK+ORACMJQ'JADQ+

S<9I8+)JQQR+ORACMJQO+E9Q+<S+ACBQ9QOB+M:Q+B<+BPQ+R9<;E;JQ+9QJEBA<COPAR+;QBLQQC+OJQQR+ORACMJQO+ECM+

QRAJQRON+VF?`W8+2BaO+LQJJ+DC<LC+BPEB+BPEJEI<'K<9BAKEJ+CQ:9EJ+CQBL<9D+BPEB+C<9IEJJN+GQCQ9EBQO+OJQQR+

ORACMJQO?+KEC+EJO<+GQCQ9EBQ+ORADQ'LE=Q+MAOKPE9GQO?+LPAKP+E9Q+QJQKB9<QCKQRPEJ<G9ERPAK+PEJJIE9DO+<S+

E;OQCKQ + QRAJQRON + V[W8 + "JBP<:GP + OJQQR + ORACMJQO + ECM + ORADQ'LE=Q + MAOKPE9GQO + E9Q + K<COAMQ9QM + EO+

BPEJEI<K<9BAKEJ+<OKAJJEBA<CO?+S:CKBA<CEJ+9QJEBA<COPAR+;QBLQQC+BPQI+AO+C<B+<;=A<:O8

2C +BPQ +R9QOQCB + 9QR<9B +LQ +<SSQ9 + E +LE=QJQB';EOQM +IQBP<M + S<9 + E:B<IEBAK + MQBQKBA<C +<S + OJQQR+

ORACMJQO+ECM+['\+-_+<OKAJJEBA<CO+<C+OJQQR+$$78+&PQ+R9<R<OQM+IQBP<M+AO+;EOQM+<C+BPQ+KEJK:JEBA<C+<S+

I<IQCBE9N + QCQ9GN + <S + BPQ +LE=QJQB + ORQKB9:I + <S + $$7 + BAIQ + OQ9AQO + E=Q9EGQM + <=Q9 + O<IQ + ORQKAEJ+

S9QZ:QCKN + 9ECGQO8 +!Q+ERRJAQM + BPAO +IQBP<M + B< +$$7+9QK<9MO +<S + 9EBO8 +!Q+OB:MAQM +!"7b0Ac + 9EBO+

;QKE:OQ +<S + BPQA9 + AC;<9C +R9QMAOR<OABA<C + B< +E;OQCKQ +QRAJQRON + V^W8 +!ABP + BPQ +PQJR +<S + BPQ +R9<R<OQM+

IQBP<M+LQ+RQ9S<9IQM+E:B<IEBAK+MAEGC<OBAKO+<S+$$7+9QK<9MO+ECM+K9QEBQM+IE9DACGO+LABP+;<BP+BNRQO+

<S+OJQQR+ORACMJQO8+%EBQ9+BPQOQ+9QO:JBO+LQ9Q+:OQM+B<+OB:MN+BPQ+C<CJACQE9+MNCEIAKO+<S+OJQQR+ORACMJQ+

ERRQE9ECKQ+<C+$$78+2C+RE9BAK:JE9?+LQ+PE=Q+ECEJN_QM+ACBQ9IABBQCB+;QPE=A<9+AC+BPQ+OJQQR+$$7+:OACG+

C<=QJ+ERR9<EKP+<S+<OKAJJEB<9N+REBBQ9CO+MQBQKBA<C+VdW8

&PAO+L<9D+PEO+;QQC+O:RR<9BQM+;N+BPQ+,ACAOB9N+<S+QM:KEBA<C+ECM+OKAQCKQ+<S+0:OOAE+XR9<cQKBO+

>`8(Fd8@>8]d[>? +>`8(Fd8@>8][^\?+>`8(Fd8@>8]][\?+G<=Q9CIQCB+RJEC+<C+@]>F+ECM+RJEC+RQ9A<M+<S+

@]>`+ECM+@]>[+NQE9O?+)7&1'd\Y?+0:OOAEC+/<:CMEBA<C+S<9+(EOAK+0QOQE9KP+XR9<cQKBO+>@']@']]@@>?+

>@']@']]Fdd+ECM+>F']`']]]e`Y8

>8 78+(:_OEDA?+"8+*9EG:PC+bb+)KAQCKQ8+@]]`8+#8+F]`8+68+>\@^8

@8 "848+6E=J<=+QB+EJ8bb+6PNOAKO'1ORQDPA8+@]>@8+#8+[[8+68+e`[

F8 %8+*Q+7QCCE9<?+,8+/Q99E9Ebb+)JQQR+,QM8+0Q=8+@]]F8+#8+d?+68+`@F

`8 78+f8+f<OB<R<:J<O+bb+3JAC8+4Q:9<RPNO8+@]]]8+):RRJ8+@8+)@d'Fe8

[8 "8 +*QOBQTPQ? + &8g8 + )QcC<LODA? + &PEJEI<K<9BAKEJ + "OOQI;JAQO8+ .TS<9M +1CA=Q9OABN + 69QOO?+.TS<9M?+@]]>8

^8 $8%8g8,8 + =EC + %:AcBQJEE9? + )RADQ'LE=Q + MAOKPE9GQO + ECM + OJQQR+ ORACMJQO + AC + 9EBO + bb + "KBE+4Q:9<;A<J8+$TR8+X!E9O_Y+>\\d8+#8+[d?+68+>>F8

d8 $85:8+)ABCAD<=E+QB+EJ8+bb+(9EAC+9QOQE9KP8+@]>@8+#8+>`@^8+68+>`d'>[^8

!"#$%&'()*+),-."'/*0)#1"*/#/%!)+*&)

!".%2.,,-)#1%#/3#3)#,#$%&*/2$)!-!%#(!

.4#4)56789:;).4.4)<969=9:>?@@;)0424)(7A@8B=?9;)*424)(9>?7CB=?9

!"#"$%&'!$"$(')*+&(#,+$-.'!"#"$%&'!$"$('/(01*+0"2')*+&(#,+$-.'!"#"$%&.'34,,+"

.4D4)E7C7=9:;)<4/4).CB?>BB:

5%4617%#%461')*+&(#,+$-.'5%4617%#%461.')*+$(8'9+*68%:

7CBABF4?969=9:>?@@GH87@C4I98

/9=C@=B76 ) JF=78@I> ) 7=J ) IK79> ) @= )>L7M@7CCF ) BAMB=JBJ ) BCBIM69=@I ) >F>MB8> )N@MK ) @=MB67IM@=H)

BCBIM69=)7=J)BCBIM6987H=BM@I)N7:B>)76B)9OMB=)9P>B6:BJ4)#4H4;)MKBF)M7?B)LC7IB)@=)P9Q=JBJ)LC7>87>)

7=J)BCBIM69=)7=J)@9=)>M6B78>;)NKB6B)MKB)=9=C@=B76@MF)7=J)LC7>87)@=>M7P@C@M@B>)76B)I98P@=BJ)N@MK)

J@>>@L7M@9=)I7Q>BJ)PF)IQ66B=M)OC9N)M9)MKB)P9Q=J76F)BCBIM69JB>4)%KB6BO96B;)MKB)JB:BC9L8B=M)9O)MKB)

8BMK9J>)9O)7=7CF>@>)9O)=9=C@=B76)JF=78@I>)@=)MKB)>L7M@7CCF)BAMB=JBJ)BCBIM69=@I)>F>MB8>)@>)7):B6F)

@8L96M7=M )7=J)7IMQ7C )L69PCB8)JQB)M9)MKB)L9>>@PCB)7LLC@I7M@9=> )9O)IK79>)MKB96F)O96)>QIK)?@=J)9O)

>F>MB8>;)B4H4;)O96)MKB)@JB=M@O@I7M@9=)9O)IK79M@I)>L7M@9RMB8L967C)9>I@CC7M@9=>)@=)8@I69N7:B)BCBIM69=@I)

7=J ) LC7>87 ) JB:@IB> ) STU; ) IK79> ) I9=M69C ) 7=J ) >QLL6B>>@9= ) 9O ) MQ6PQCB=IB ) SVU; ) 7=J ) IK79M@I)

>F=IK69=@W7M@9=)@=)MKB)LC7>87)7=J)8@I69N7:B)BCBIM69=@I)>F>MB8>)SXU4

%KB),F7LQ=9:)BAL9=B=M>)76B)L9NB6OQC)7=J)Q>BOQC)M99C)O96)MKB)I98LCBA)>F>MB8)JF=78@I>)M9)PB)

7=7CFWBJ4 )%KBF )76B )Q>BJ )N@JBCF ) M9 ) IK767IMB6@WB ) MKB ) >F>MB8> )N@MK ) >87CC )=Q8PB6 )9O )JBH6BB> )9O)

O6BBJ98)YMK7M) @> )>F>MB8>)N@MK)O@=@MB )LK7>B)>L7IBZ) @=) MKB)J@OOB6B=M )O@BCJ>)9O)>I@B=IB)S[U;)>QIK)7>)

LKF>@I>;)LKF>@9C9HF)7=J)8BJ@I@=B;)89CBIQC76)JF=78@I>;)7>M69=98F;)BMI4)'=O96MQ=7MBCF;)MKB)J@6BIM)

7LLC@I7M@9=)9O)MKB),F7LQ=9:)BAL9=B=M)I7CIQC7M@9=)MBIK=@\QB)YJB:BC9LBJ)O96)MKB)>F>MB8>)N@MK)MKB)

>87CC)=Q8PB6)9O)JBH6BB>)9O)O6BBJ98Z)M9)MKB)>L7M@7CCF)BAMB=JBJ)>F>MB8>)@>)L69PCB87M@I4)%KB)87@=)

I7Q>B>)9O)MK@>)L69PCB8)76B)MKB)O9CC9N@=H])Y@Z)MKB)LK7>B)>L7IB)9O)MKB)>L7M@7CCF)BAMB=JBJ)>F>MB8)@>)

@=O@=@MB^)Y@@Z)MKB)=Q8PB6)9O)MKB),F7LQ=9:)BAL9=B=M>)@>)7C>9)@=O@=@MB^)Y@@@Z)MKB)>F>MB8)>M7MB)YJBO@=BJ)O96)

B7IK)MFLB)9O)MKB)>F>MB8)Q=JB6)>MQJF)@=)@M>)9N=)N7FZ)8Q>M)PB)Q>BJ)@=>MB7J)9O)MKB)O@=@MBRJ@8B=>@9=7C )

:BIM96^)Y@:Z)MKB)96MK9H9=7C@W7@M9=)7=J)=9687C@W7M@9=)L69IBJQ6B>)8Q>M)PB)89J@O@BJ)M9)PB)7LLC@I7PCB)

O96)MKB)>M7MB>4

/B:B6MKBCB>>; ) MKB )>I@B=M@>M> ) M6F ) M9 )Q>B) MKB ),F7LQ=9:)BAL9=B=M> ) O96) MKB )>L7M@7CCF )BAMB=JBJ)

>F>MB8>)JQB)M9)MKB)H6B7M)BOO@I@B=IF)9O)MK@>)M99C4)(7@=CF;)7CC)7MMB8LM>)76B)6BJQIBJ)M9)MKB)Q>B)9O)MKB)

89J@O@I7M@9=> )9O ) MKB ) >M7=J76J )8BMK9J> )JB:BC9LBJ ) O96 ) MKB ) O@=@MB ) J@8B=>@9=7C ) >F>MB8> ) YB4H4; ) MKB)

B>M@87M@9=)9O)MKB)K@HKB>M),F7LQ=9:)BAL9=B=M)O698)M@8B)>B6@B>)96)MKB)6BL6B>B=M7M@9=)9O)MKB)>L7M@7CCF)

BAMB=JBJ ) >F>MB8 ) 7> ) MKB ) O@=@MB ) J@8B=>@9=7C ) 9=B ) N@MK ) MKB ) K@HK ) J@8B=>@9= ) PF ) 8B7=> ) 9O ) MKB)

J@>I6BM@W7M@9=Z4)'=O96MQ=7MBCF;)MKB>B)7LL697IKB>)J9)=9M)M7?B)@=M9)7II9Q=M)MKB)J@>M@=HQ@>K@=H)OB7MQ6B>)

9O)MKB)>L7M@7CCF)BAMB=JBJ)>F>MB8>)7=J)K7:B)>B6@9Q>)C@8@M7M@9=>)I9==BIMBJ)N@MK)MKB)K@HK)J@8B=>@9=)9O)

MKB)LK7>B)>L7IB)9O)MKB)J@>I6BM@WBJ)>F>MB8)S_U4

2=)MKB)L6B>B=M)N96?)NB)JB:BC9L)MKB)MBIK=@\QB) 9O)MKB)>LBIM6Q8)9O)MKB),F7LQ=9:)BAL9=B=M)

I7CIQC7M@9=)O96)MKB)>L7M@7CCF)BAMB=JBJ)BCBIM69=@I)>F>MB8>4)%KB)I7CIQC7M@9=)MBIK=@\QB)@>)7LLC@BJ)M9)MKB)

7=7CF>@>)9O)IK79M@I)>L7M@9RMB8L967C)9>I@CC7M@9=>)@=)J@OOB6B=M)>L7M@7CCF)BAMB=JBJ)BCBIM69=@I)>F>MB8>])

YTZ)LC7>87)"@B6IB)J@9JB;)YVZ)BCBIM69=RN7:B)>F>MB8)N@MK)P7I?N76J)BCBIM6987H=BM@I)N7:B)7=J)YXZ)

>B8@I9=JQIM96)>QLB6C7MM@IB4)`B)O@=J)7=)BAIBCCB=M)7H6BB8B=M)PBMNBB=)MKB)>F>MB8)JF=78@I>)7=J)MKB)

PBK7:@96)9O)MKB)>LBIM6Q8)9O)MKB),F7LQ=9:)BAL9=B=M>4).C9=H)N@MK)MKB)L69L9>BJ)8BMK9J;)MKB)L9>>@PCB)

L69PCB8>)9O)!,#>)I7CIQC7M@9=)76B)7C>9)J@>IQ>>BJ4)2M)@>)>K9N=)MK7M)O96)MKB)N@JB)IC7>>)9O)MKB)>L7M@7CCF)

BAMB=JBJ)>F>MB8>)MKB)>BM)9O)\Q7=M@M@B>)@=ICQJBJ)@=)MKB)>F>MB8)>M7MB)O96)!,#>)I7CIQC7M@9=)I7=)PB)

6BJQIBJ)Q>@=H)MKB)7LL69L6@7MB)OB7MQ6B)9O)MKB)>F>MB8>)Q=JB6)>MQJF4

%K@>)N96?)K7>)PBB=)>QLL96MBJ)PF)MKB)(@=@>M6F)9O)BJQI7M@9=)7=J)>I@B=IB)9O)&Q>>@7)YL69aBIM>;)

T[4EXb4VT4cc_d;T[4EXb4VT4TVed; )H9:B6=8B=M )LC7= )9= )VcTX )7=J )LC7= )LB6@9J )9O )VcT[ )7=J )VcT_)

FB76>Z;)&Q>>@7=)+9Q=J7M@9=)O96)E7>@I)&B>B76IK)YL69aBIM> )TVRcVRccVVT)7=J)TVRcVRXXcbTZ)7=J)PF)

3F=7>MF)+9Q=J7M@9=4

T4 -4)"4)EC@9?K)7=J)D4)!4)/Q>@=9:@IK;)"KF>4)"C7>87>)!";)cVXTcV)YVccfZ

V4 .4)#4)56789:;).4).4)<969=9:>?@@;)7=J)24)!4)&B8LB=4)$K79>)!#;)cTXTVX)YVccfZ4

X4 $4)(4)%@I9>)BM)7C4)"KF>4)&B:4),BMM4)$%;)VdVd)YVcccZ4

[4 &4)"96IKB6)7=J)D4)%K987>;)"KF>4)&B:4)#)#&;)cTcdcVY&Z)YVccTZ4_4 "4)04)<QLM>9:4).LLC4)/9=C@=B76)3F=4)Te;)ecgdV)YVcTcZ4

!"#$%&'(%$#)*$+#,-.&*)&#&/0$0(#1&2%#3(0*)4/055670*)&8+*3%77*+

!"#$%&'()*+'(),$-"./*0123"45-2*67(,)58/*92:'$%"*0'")./*"45*;"'":&*;&:&,$-".

.<-"5$":2*67(&&1*&=*+4=&-,":)&4*67)2472*"45*927(4&1&>%/*?&@@")5&*A4)B2-'):%

!):"*.C/*D)'()*E/*!):"F@$/*6"GG&-&*HIHFHJ.C/*K"G"48;)7-&2127:-&4)7*6%':2,'*L"M&-":&-%/*6N)''*O252-"1*+4':):$:2*&=*927(4&1&>%*PQROLS

L"$'"442/*T?F.H.U*6N):#2-1"45

QF,")1V*)'(),$-"W1"1')2X)':X(&@$5")X"7XYG

9:7(+#3(

Z2*52,&4':-":2*!"#$%*&'%$#()$*#+,-*&4*5)>):"1*7)-7$):'*),G12,24:)4>*"*-2"7:)&4F5)==$')&4*P[\S*7211$1"-*

"$:&,":" * PT0SX *[2724:1%/ *R"12B)7)$' *%' .#/X * G-&G&'25 *" *4&B21 * ':2>"4&>-"G(% *"1>&-):(,*$:)1)#)4> *'21=F

&->"4)#)4>*'G":)"1*G"::2-4'*&=*"*G-2%FG-25":&-*[\*,&521*].^X*9&*),G12,24:*:(2*[\*':2>"4&>-"G(%*,&521*

&4*G-"7:)7"1*("-5N"-2/*N2*2,G1&%25*"*("-5N"-2F&-)24:25*[\*T0*,&521*]8^*)4':2"5*&=*2,G1&%)4>*:(2*

G-2%FG-25":&- *,&521/ * "45 * '(&N25 * :(": * :(2 *("-5N"-2F&-)24:25 *,&521 *("5 *"1,&': * :(2 * '",2*5%4",)7"1*

7("-"7:2-)':)7' * =&- * ':2>"4&>-"G(% *"GG1)7":)&4' * ]_^X * +4 * :()' * -2G&-:/ *N2 *G-2'24: * " *5)>):"1 *[\*G-&72''&- *

"-7():27:$-2*=&-*:(2*("-5N"-2F&-)24:25*,&521X

9(2*G-&G&'25*"-7():27:$-2*7&4')':'*&=*,2,&-)2'*0*"45*`*PJFM):*=-",2*,2,&-)2'S/*"*=-",2*M$==2-/*"45*"*

LA9X*;2,&-)2'*0*"45*`*G-2'2-B2*:N&*5)==$'25*5":"*&=* :(2*)4):)"1 * )4G$:* ),">2/*"45*:(2*=-",2*M$==2-*

G-2'2-B2' * ",G1)=)25 * 5)==2-24:)"1 * B"1$2' * M2:N224 *0* "45 *` * PB)" *LA9SX *?2-2 * 5)==$')&4 * 7&,G$:":)&4 * )'*

G2-=&-,25*M%*7&4B24:)&4"1*5)>):"1*=)1:2-'*N):(*:N&*1)42*M$==2-'X*0*=)B2*,")4F':":2'*7&4:-&112-*>242-":2'*"11*

')>4"1'X*O)>$-2*8*'(&N'*[9L*'),$1":)&4*-2'$1:'*=&-*:(2*G-&G&'25*"-7():27:$-2a*P"S*'2452-b'*),">2*Pc9dF

'("G25*7("-"7:2-S/*PMS*247-%G:25*),">2*&4*"*7&,,$4)7":)&4*7("4421/*"45*P7S*527&525*c9dF1)@2*),">2*": *

:(2*-272)B2-b' * ')52/ *N()7( *712"-1%*52,&4':-":2' * :(": * :(2*G-&G&'25*"-7():27:$-2 * )' *"M12* :& *G2-=&-,* :(2*

':2>"4&>-"G(%*7&,G$:":)&4X

2%5%+%)3%7

].^ *RX *R"12B)7)$'/ *LX *6"$4&-)242/ *"45 *;X *[">$1'@)'/ *c0*'27$-2 *7&,,$4)7":)&4 *'%':2,*M"'25 *&4 * '21=F&->"4)#)4> *

G"::2-4'/d*)4*R-&7X*8H.8*+4:X*T&4=X*627$-):%*"45*;"4">2,24:*P60;e.8S/*GX*C8./*8H.8X

]8^*fX*6$#$@)/*9X*9"@"%","/*+XDX*;&:&)@2/*"45*9X*0'")*c6:-)G25*"45*'G&::25*G"::2-4*>242-":)&4*&4*-2"7:)&4F5)==$')&4*

7211$1"-*"$:&,":"V*9(2&-%*"45*L6+*),G12,24:":)&4/d*+4:X*KX*A47&4BX*T&,G$:X/*B&1X*_/*GGX*.F._/*8HHgX

]_^*!X*+'(),$-"/*0X*67(,)5/*9X*0'")/*"45*;X*;&:&,$-"*c+,">2*':2>"4&>-"G(%*M"'25*&4*("-5N"-2F&-)24:25*-2"7:)&4F

5)==$')&4*,&521'/d*R-&7225)4>'*&=*8H._*+4:2-4":)&4"1*6%,G&')$,*&4*D&41)42"-*9(2&-%*"45*):'*0GG1)7":)&4'/*6"4:" *

O2/*A60/*8H._X

Interesting bifurcation of canard under extremelyweak periodic perturbation

Kaoru Itohand Tetsuro Endo

Department of Electronicsand Bioinformatics,

Meiji University,Kawasaki, Japan 214–8571

Email: ce31006@.meiji.ac.jp

Naohiko InabaOrganization for the Strategic Coordination

of Research and Intellectual Property,Meiji University,

Kawasaki, Japan 214–8571

Munehisa SekikawaInstitute of Industrial Science,

The University of Tokyo,Tokyo, Japan 153–8505

I. ABSTRACT

Canards are interesting bifurcation phenomena foundin 1984[1], and are drawing much attention recently [2].Let us consider the van der Pol equation in the followingform:

εx = y + x(1− x2), y = −x+B0 +B sinωτ (1)

where 0 < ε " 1 is assumed. Canards are thephenomena observed when B = 0 that follows theattracting slow manifold and the repelling slow manifold.According to the analysis in non-standard analysis, anamplitude of the oscillation changes on the order of onewhen B0 is changed on the order of exp(−1/ε). Themagnitude of exp(−1/ε) approximates 10−5 if ε is 0.1and 10−44 if ε is 0.01. The amplitude of the oscillationis surprisingly sensitive to the parameter B0.

How the dynamics of Eq. (1) is influenced by anextremely periodic perturbation when B is slightly pos-itive. In this study, we set ε = 0.1. Such order ofε is appropriate in numerical simulation because weconduct the numerical calculation with double precision.We are interested in the case where B is chosen onthe order of exp(−1/ε). We investigate the bifurcationstructure of the 1/2−subharmonic entrainment region.In we discovered an interesting bifurcation structure inthis entrainment region. The first saddle-node bifurcationcurves form an entrainment region. Note that the secondsaddle-node bifurcation occur in the entrainment region.We find that two stable canards exist. Here, stablecanards are define as a phenomenon that follows theattracting and repelling slow manifold. Furthermore, wedetect two unstable periodic orbits exist. Note that theyare saddles and are also canard shapes.

We have noticed that the stable canard that is born bythe first saddle-node bifurcation resembles the unstablecanard that is born by the second bifurcation quitewell although their phases are considerably different.Furthermore, the saddle canard that is born by the firstsaddle-node bifurcation resembles the stable canard thatis born by the first one quite well. We cannot verifythe differences of the two pair of stable canards andsaddle canards from the projection onto the state space.The correlation coefficients are calculated and confirmthat each pair of stable canards and saddle canardsresembles on the order of 0.999999 · · ·. The remarkableresemblances are noteworthy.

Fig. 1. Four solutions. Solid circles are points on Poincare section.

REFERENCES

[1] M. Diener, “The Canard unchained or how fast/slow dynamicalsystems bifurcate,” Math. intelligencer, Vol.6, No.3, pp.38–49,1984.

[2] V.I Arnol’d (ed.), “Dynamical systems V,” Encyclopedia ofMathematical Sciences, Vol.5, Springer, 1994.

Photon Assisted Tunneling in MOM Diodes

Christian Jirauschek and Peter RusserInstitute for Nanoelectronics,

Technische Universität München, Germany

We investigate photon assisted tunneling in metal-oxide-metal (MOM) tunnel diodes. Nanoscale MOM tunnel diodes can provide rectification for IR and even optical radiation [1]. Asymmetrical MOM diodes formed by metal combinations with different work functions exhibit asymmetrical I-V curves and a rectifying effect at zero bias. In addition to the energetically horizontal tunnel transitions, the electron-photon interaction accompanying the tunneling process yields also transitions with single or multiple photon emissions and absorptions. We show that compared with classical tunneling, photon assisted tunneling yields a considerable enhancement of the detector sensitivity.

We refer to measured DC characteristics of a MOM tunnel diode [2]. According to Tucker and Feldman [3] the applied voltage will modulate adiabatically the potential energy of every quasiparticle state at the ungrounded side of the tunnel barrier. The time dependence of every quasiparticle wave function will therefore be modified following a time--dependent perturbation approach according to

References [1] C. Fumeaux, G. D. Boreman, W. Herrmann, H. Rothuizen, and F. K. Kneubühl, “Polarization response of asymmetric-spiral infrared antennas,” Applied Optics, vol. 36, no. 25, pp. 6485–6490, 1997.[2] M. Bareiß, D. Kälblein, C. Jirauschek, A. Exner, I. Pavlichenko, B. Lotsch, U. Zschieschang, H. Klauk, G. Scarpa, B. Fabel, W. Porod, and P. Lugli, “Ultra-thin titanium oxide,” Applied Physics Letters, vol. 101, no. 8, pp. 083 113–083 113–4, Aug. 2012.[3] J. R. Tucker and M. J. Feldman, “Quantum detection at millimeter wavelengths,” Reviews of Modern Physics, vol. 57, no. 4, pp. 1055–1113, Oct. 1985.

!"#$%&'()*+,$-%$".(/,*%-*0123'45$",6*7.(6'3*52)%$'38"9*:;*<,.2'='()*!6,.22;*>'93$':13,6*>;(."'/9?%"%8'$%*@.3%A!.(6 0'B'%*C.9,).D.A

A?%B;%*E('&,$9'3;*%-*F/',(/,G*H4IHJK4I*L''M1B1G*@.39198'B.4B1G*?%B;%G*NKO4JJONG*P.#.(Q

R4".'2S**AB.3%T8.9,2.:Q,,QB.)1Q319Q./QM#

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

!! ]\]\ !"" "# G * IK+$=! ]*.$,*,--,/3'&,*-%$*$,61/'()*38,*

"131.2 * '(3,$-,$,(/, * Y^ZQ * !3 *8.9 *:,,(*".38,".3'/.22; * 98%D(* 38.3 * 38, */$%994/%$$,2.3'%(*."%()*38,*3'",*9,$',9*/.(*:,*"'('"'=,6*:;*19'()*38%9,*D'38*91/8*.*(,).3'&,*.13%/%$$,2.3'%(Q*!(*<,-Q*YOZG*38,*_,:,9)1,*F#,/3$1"*W'23,$*\_FW]*8.9*:,,(*.##2',6*3%*.6M193 *38,*.13%/%$$,2.3'%(*%-*38,*3'",*9,$',9*3%*38,*(,).3'&,*&.21,9Q*?8,*_FW*8.9*.29%*:,,(*.##2',6*3%*9,&,$.2*%#3'"'=.3'%(*",38%69G*91/8*.9*38,*(,1$.2*(,3D%$B*.(6*38,*K4%#3*8,1$'93'/*",38%6*-%$*`5+*.(6*?F+G*.(6*'39*,--,/3'&,(,99*8.9*:,,(*/2,.$2;*98%D(*'(*<,-Q*YHZQ*!(*38'9*#.#,$G*D,*.##2;*38,*_FW*3%*.(*.2)%$'38"*-%$*38,*"123'4.$",6*:.(6'3*#$%:2,"*YaZG*D8'/8*8.9*#$./3'/.2*.##2'/.3'%(9G*91/8*.9*93%/8.93'/*9,2,/3'%(9*3%*".V'"'=,*.*$,D.$6Q*X,*.(.2;=,*38,*#,$-%$".(/,*%-*38,*#$%#%9,6*.##$%./8*:;*,&.21.3'()*'"#$%&,",(3*%-*38,*#,$-%$".(/,*%-*38,*E[7N*.2)%$'38"*YbZ*-%$*NJG * NO * .(6 * KJ * .$",6 * :.(6'3 * #$%:2,"9Q * X, * .29% * .(.2;=, * 38, * :,93 *.13%/%$$,2.3'%(*-%$*38,*"123'4.$",6*:.(6'3*#$%:2,"9*:;*31('()*38,*#.$.",3,$*%-*38,*_FWQ

K"0123'4.$",6*:.(6'3*#$%:2,"?8,*"123'4.$",6*:.(6'3*#$%:2,"*'9*.*#$%:2,"*3%*".V'"'=,*.*3%3.2*$,D.$6*:;*:,93*9,2,/3'%(*%-*38,*93%/8.93'/*9;93,"Q*c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d4)$,,6;*.2)%$'38"G*38,*F%-3".V*.2)%$'38"G*38,*E[7N*.2)%$'38" * .(6 * 9% * %(Q * !( * 38'9 * #.#,$G *D, * 9,2,/3 * 38, *E[7N * .2)%$'38"G * .(6*'"#$%&,*'39*#,$-%$".(/,*:;*38,*_FWQ

IQ*<,.2'='()*(,).3'&,*.13%/%$$,2.3'%(*'(*"123'4.$",6*:.(6'3*

.2)%$'38"

X,*.##2;*38,*_FW*3%*38,*E[7N*.2)%$'38"G*.(6*$,.2'=, *'6,.22;*6'93$':13'&,*6;(."'/9*:;*2%D,93*/$%994/%$$,2.3'%(*."%()*38,*9,2,/3'%(9*%-*,./8*.$"Q*?8,*6,/'9'%(9*%-*38,*9,2,/3'%(9*.$,*6%(,*19'() *#$%\&]G*D8'/8*'9*38,*%13#13*%-*38,*_FW*.##2',6*3%*38,*,93'".3,6*$,D.$6*#$%:.:'2'3;*#%\&]G*

]\]N\e]\e &#&#!&# %%+$&=

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

!(* 38'9 *#.#,$G * 38,*#,$-%$".(/,9*%-* 38, *.2)%$'38"9*.$,*,&.21.3,6*19'()*KJ*.$",6*:.(6'3*#$%:2,"9Q*R./8*$,9123*'(*38,*-%22%D'()9*'9*.&,$.),*&.21,*%&,$*NJJJ*$1(9Q *W')QN* 98%D9*(,).3'&, *.13%/%$$,2.3'%(*./31.22; *"'('"'=,9*/$%994/%$$,2.3'%(Q*?8,*$,91239*.3*!*f*JG*D8'/8*/%$$,9#%(69*3%*38,*%$')'(.2*E[7N*D.9*aK#Q*59*98%D(*'(*W')Q*NG*38,*:,93*#,$-%$".(/,*%-*38,*#$%#%9,6*'"#$%&,",(3*

9/8,",*D.9*bN#G*.(6*%1$*.2)%$'38"*D.9*8')82;*'"#$%&,6*91/8*.(*,--,/3'&,*"123'4.$",6 *:.(6'3 * .2)%$'38" *:; * 19'() * (,).3'&, * .13%/%$$,2.3'%( *6;(."'/9*D8'/8*"'('"'=,9*/$%994/%$$,2.3'%(*.(6*),(,$.3,*'6,.22;*6'93$':13,6*6;(."'/9Q

W')Q*NS!?8,*$,2.3'%(*:,3D,,(*.13%/%$$,2.3'%(*.(6*/$%994/%$$,2.3'%(*.(6*38,*$.3,*%-*9,2,/3'%(9*%-*38,*%#3'".2*.$"Q

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c1$*$,91239*98%D*38.3*38,*_FW*/.(*.29%*'"#$%&,*38,*#,$-%$".(/,*%-*38,*:.(6'3*.2)%$'38"9G*D8'/8*8.&,*&.$'%19*.(6*D'6,*.##2'/.3'%(*-',269Q*

OQ*<,-,$,(/,

"#$ CQ*L%=.D.G*[8.%9G*&%2Q*KG*(%Q*IG*##Q*IIagIbHG*NhhKQ

"%$ 0Q*C.9,).D.G*?Q*!B,)1/8'*.(6*@Q*5'8.$.G*+8;9'/.2*<,&',D*_,33,$9G*&%2Q*

ahG*(%Q*NKG*##Q*KI^^gKI^aG*NhhaQ

"&$ 0Q*C.9,).D.*,3*.2QG*!R![R*?$.(9Q*W1(6.",(3.29G*&%2Q*RbJ45G*(%Q*NG*##Q*

KJHgKNKG*NhhaQ

"'$ ?Q * @%86. * ,3 * .2QG * !RRR * ?$.(9./3'%(9 * %( * ['$/1'39 * .(6 * F;93,"9 * !4

W1(6.",(3.2*?8,%$;*.(6*5##2'/.3'%(9G*&%2Q*OJG*##QbJJgbJOG*KJJIQ

"($ 0Q*C.9,).D.*,3*.2QG*_,/31$,*L%3,9*'(*[%"#13,$*F/',(/,*\!(3,$(.3'%(.2*

[%(-,$,(/,*%(*L,1$.2*!(-%$".3'%(*+$%/,99'()]G*&%2Q*^hb^G*##Q*HhIg

aJKG*KJJbQ

")$ 0Q*C.9,).D.G*_,/31$,*L%3,9*'(*[%"#13,$*F/',(/,*\!(3,$(.3'%(.2*

[%(-,$,(/,*%(*L,1$.2*!(-%$".3'%(*+$%/,99'()]G*&%2Q*H^^IG*##Q*HHgaIG*

KJNJQ

"*$ PQ*i,$"%$,2*,3*.2QG*NH38*R1$%#,.(*[%(-,$,(/,*%(*0./8'(,*_,.$('()Q*

_,/31$,*L%3,9*'(*5$3'-'/'.2*!(3,22'),(/,G*F#$'(),$*IaKJG*^Iag^^bG*KJJOQ*

"+$ +Q*51,$*,3*.2QG*0./8'(,*2,.$('()G*&%2Q^aG*!991,*K4IG*##QKIOgKOHG*KJJKQ

!"#$"%&'()*%'&+#,%)(+#-'&#*.!',#%&!$"$"/#0'!&(#,1&$1,

!"#$%&'()(* +%,"-&'.!'/01 23%4&56+07

!"#$%&'"(&)*+),-".&%/.$-0),-".&%*(/.1),(2/(""%/(23

,%./4"1)5(/6"%1/&43)7897:3);$41"%/<=5>;,?

!"#"$%&'$"(&)*&+,*-'

@()$AA/&/*()&*)&B")&B"*%"&/.$-)$(A)#%$.&/.$-)1&CA/"1)*().B$*&/.)./%.C/&1)$(A)141&"'1)$%")

6"%4)B"-#+C-3)&B")-$D*%$&*%4)&**-1)A"1/2("A)/()$)141&"'$&/.)E$4)$%")%"FC/%"A)+*%)/(&%*AC./(2)

.B$*&/.)./%.C/&1)$(A)141&"'1)/()2%$AC$&")$(A)C(A"%2%$AC$&")%"1"$%.B)$(A)"AC.$&/*()#%*2%$'1G)

=*)'""&)&B/1)%"FC/%"'"(&3)E")B$A)A"1/2("A)$(A)/'#-"'"(&"A)&E*)-$D*%$&*%4)&**-13)($'"-4)

H!"#$%&'(#)*)*+&,$#(-./&#*-&//I)+*%)1&CA4/(2).B$*&/.)./%.C/&1)$(A)141&"'1G)JB$*1)=%$/(/(2)

K*$%AL@)E$1)D$1"A)*( )0/M"AL0*A" )!B$*&/. )!/%.C/& )$(A)/& )$--*E1)&B")C1"%1)&*)/(6"1&/2$&")

$C&*(*'*C13 ) (*($C&*(*'*C1 ) $(A ) '/M"AL'*A" ) .B$*&/. ) A4($'/. ) &4#"1G ) N-1*3 ) $-&"%($&/6")

A"1/2() /A"$1 ) +*% )JBC$O1 )A/*A")$(A) /(AC.&*%-"11 ) %"$-/P$&/*( )*+ )JBC$O1 )A/*A"LD$1"A).B$*&/.)

./%.C/&1 ).$( )D" )"$1/-4 )$##-/"A )*( ) &B/1 ) &%$/(/(2 )D*$%AG )JB$*1 )=%$/(/(2 )K*$%AL@@ )#%"1"(&1 )$)

.*--".&/*()*+)1/M).B$*&/.)./%.C/&1G)=B"1")./%.C/&1)E"%")1"-".&"A)+*%)/--C1&%$&/(2)$)6$%/"&4)*+)E$41)

/()EB/.B).B$*1).$()$%/1")/()1/'#-")$($-*2)*1./--$&*%)1&%C.&C%"1).*(&$/(/(2)$.&/6")"-"'"(&13)

1#"./+/.$--4)KQ=)$(A)R#L$'#G)

=*)"M&"(A)$(A).*'#-"&")&B"1")141&"'$&/.)1&CA/"1)+*%)#%$.&/.$-)"M#"%/'"(&/(2).B$*&/.)

./%.C/&1)$(A)141&"'13)E")B$6")A"1/2("A)$(A)/'#-"'"(&"A)&E*)("E)-$D*%$&*%4)&**-13)($'"-4)

H!"#$% &'(#)*)*+ &,$#(-./// &#*- &/1IG )JB$*1 )=%$/(/(2 )K*$%AL@@@ ) /1 )D$1"A )*( )$ ).*''*()

.B$*&/.)141&"')C1/(2)Q"%S)"FC$&/*(1)$(A)/&).$()D").*(+/2C%"A)E/&B)&B%"")*#&/*($-)(*(-/("$%)

+C(.&/*()D-*.S1)*() &B")D*$%AG )R()&B" )*&B"% )B$(A3)JB$*1)=%$/(/(2 )K*$%AL@T).*(1/1&1 )*+)$)

.*'#%"B"(1/6").*'D/($&/*()*+)&B%"").B$*&/.)141&"'1)D$1"A)*()U*%"(P)+$'/-4G)

ON THE FLEXIBLE HARDWARE SOLUTIONS OF BIOLOGICAL NEURON

MODELS AND SYNCHRONIZATION APPLICATIONS

Nimet KORKMAZ and Recai KILIÇ

Department of Electrical & Electronic Engineering, Erciyes University, 38039, Kayseri, Turkey

nimetdahasert@erciyes.edu.tr kilic@erciyes.edu.tr

ABSTRACT

Numerical modeling and hardware implementations are alternative approaches to observe the

fire patterns of biological neuron models or synchronizations of neurons. The software

examinations of the biological neuron models simulate the behavior of the neurons. On the

other hand, hardware realizations are able to emulate the behavior of an individual biological

neuron or coupled neurons with real time adaptability. This study presents experimental

realizations of FitzHugh-Nagumo, Izhikevich, and Hindmarsh-Rose Neuron Models with a

programmable analog hardware. And also, the synchronizations of two Hindmarsh-Rose

neurons are implemented with this device namely FPAA (Field Programmable Analog Array)

by using both electrical and chemical coupling methods. FPAA device provides flexible

design possibilities such as reducing the complexity of design, real-time modification,

software control and adjustment within the system. Hardware realization of the synchronous

neural system is very complex and hard to implement in terms of the circuit network

structure, parameter adjustability and especially nonlinear synaptic coupling function. The

systems, which include the nonlinear expressions defined by different mathematical functions

such as quadratic functions in neuron models or exponential expression in the synaptic

coupling function, are affected the adjustable parameters within the nonlinear functions. It can

be say that FPAA hardware is a solution for implementations of the synchronous neural

system and biological neuron models, because nonlinear expressions are realized with FPAA

in a precise way. In!this study, firstly the biological neuron models are tested via numerical

simulation tools, and then they are implemented with FPAA (Field Programmable Analog

Array) in a programmable manner. And it is also examined experimentally that how the

electrical and chemical coupling of two HR neurons are able to achieve by using FPAA

device.

Keywords: Biological Neuron Models, Synchronization, FPAA.

!"!#$%&'()*$$!%'(+&%,*$-&'(-'(%!"*$-.-/$-#(!"!#$%&'(0!*,(1-$2(

.-%$3*"(#*$2&4!(-'(!5$!%'*"(*'4(/!"+6,*7'!$-#(+-!"4/

/8*8(9:;<=>?(*8!8(2;@ABC?(*8*8(9B;B>BCD<==

!"#"$%&'!$"$(')*+&(#,+$-.'!"#"$%&'!$"$('/(01*+0"2')*+&(#,+$-.'!"#"$%&.'34,,+"

9:;<=>/*EFA@=G8HBA

$IJ ( @>@GKD=D ( BL ( >B>G=>J@; ( A=H;BM@CJ ( BDH=GG@N=B>D ( @>O( HBAPGJQ ( DN;:HN:;J ( LB;A@N=B>(

AJHI@>=DAD(=>(DP@N=@GGK(JQNJ>OJO(DKDNJAD(M=NI(=>NJ>D=CJ(RJ@AD(BL(HI@;FJO(P@;N=HGJD(=>(NIJ(;JF=AJD(

BL(NIJ(C=;N:@G(H@NIBOJ(S.#T(B>DJN(@NN;@HND(F;J@N(@NNJ>N=B>(BL(DH=J>N=L=H(HBAA:>=NK(UV6WX8(-N(=D(MJGG(

<>BM>(NI@N(NIJ(DKDNJAD(M=NI(.#(@;J(HI@;@HNJ;=YJO(RK(NIJ(HBAPGJQ(OK>@A=HD(@>O(H@>(OJAB>DN;@NJ(@(

M=OJ ( ;@>FJ ( BL ( >B>G=>J@; ( PIJ>BAJ>@? ( =>HG:O=>F ( OK>@A=H@G ( HI@BD ( @>O ( P@NNJ;> ( LB;A@N=B> ( UV6ZX8(

,=H;BM@CJ(FJ>J;@NB;D(:D=>F(JGJHN;B>(RJ@AD(M=NI(.#(SC=;H@NB;D(U[?\XT(@;J(PJ;DPJHN=CJ(OJC=HJD(BL(

I=FI6PBMJ; (A=H;BM@CJ(JGJHN;B>=HD ( LB; ( NIJ (FJ>J;@N=B> (BL ( NIJ ( =AP:GDJD (BL (M=OJ6R@>O(A=H;BM@CJ(

;@O=@N=B>(O:J(NB(=ND(I=FI(B:NP:N(A=H;BM@CJ(PBMJ;?(@(D=APGJ(HB>DN;:HN=B>(SP@;N=H:G@;GK(C=;H@NB;D(H@>(

BPJ;@NJ(M=NIB:N(JQNJ;>@G(LBH:D=>F(A@F>JN=H(L=JGOT?(@(PBDD=R=G=NK(BL(@(D=APGJ(L;J]:J>HK(N:>=>F(@>O(

;JF=AJ(DM=NHI=>F(UV?[X8(*GG(NIJDJ(H=;H:ADN@>HJD(=>H;J@DJ(NIJ(L:>O@AJ>N@G(@>O(@PPG=JO(=APB;N@>HJ(BL(

DN:O=JD(BL(NIJ(>B>G=>J@;(OK>@A=HD(BL(NIJ(JGJHN;B>(RJ@AD(M=NI(.#8

$IJ ( =APB;N@>N (P;BRGJA( => ( NI=D ( L=JGO ( =D ( NIJ (@>@GKD=D (BL ( NIJ (.#( LB;A@N=B> ( HB>O=N=B>D (@>O(

>B>G=>J@;(OK>@A=HD(BL(JGJHN;B>(DN;:HN:;JD(=>(;JG@N=C=DN=H(JGJHN;B>(RJ@AD(S%!0T(=>(NIJ(P;JDJ>HJ(BL(

L=>=NJ(JQNJ;>@G(A@F>JN=H(L=JGO(B;(JCJ>(M=NIB:N(JQNJ;>@G(A@F>JN=H(L=JGO(MI=HI(LBH:DJD(NIJ(%!08(->(

P@;N=H:G@;? ( NIJ ( DN:O=JD ( BL ( %!0D ( M=NI ( BCJ;H;=N=H@G ( H:;;J>ND ( @;J ( >JHJDD@;K ( LB; ( NIJ ( @>@GKD=D ( BL(

HB>NJAPB;@;K(I=FI6PBMJ;(OJC=HJD(M=NI(.#(^(;JG@N=C=DN=H(C=;H@NB;D(@>O(=B>(@HHJGJ;@N=B>(DKDNJAD8

*>@GKY=>F(%!0D?(=N(=D(>JHJDD@;K(NB(N@<J(=>NB(@HHB:>N(JLLJHND(RJ=>F( =>D=F>=L=H@>N(LB;(MJ@<GK(

;JG@N=C=DN=H (RJ@AD?(=>(P@;N=H:G@;? (NIJ(=>LG:J>HJ(BL(NIJ(DJGL6A@F>JN=H (L=JGO(BL(NIJ(%!0(NI@N(JLLJHND(

HB>D=OJ;@RGK(B>(NIJ(DKDNJA(OK>@A=HD(=>(H@DJ(BL(:GN;@6;JG@N=C=DN=H(JGJHN;B>(RJ@AD8($IJ;JLB;J?(NIJ(_4(

L:GGK(JGJHN;BA@F>JN=H(DJGL6HB>D=DNJ>N(ABOJG(BL(%!0(OK>@A=HD(=D(;J]:=;JO(LB;(@HH:;@NJ(@>O(HB;;JHN(

@>@GKD=D(=>(NI=D(H@DJ8

->(NIJ(P;JDJ>N(;JPB;N(MJ(@>@GKYJ(>:AJ;=H@GGK(NIJ(JCBG:N=B>(BL(NIJ(OK>@A=HD(BL(NIJ(%!0(=>(NIJ(

;JF=AJ(BL(.#(LB;A@N=B>(M=NI(NIJ(HI@>FJ(BL(NIJ(JQNJ;>@G(A@F>JN=H(L=JGO8($IJ(@Y=A:NI@G(=>DN@R=G=NK(BL(

NIJ(;JG@N=C=DN=H(JGJHN;B>(RJ@A(GJ@O=>F(NB(NIJ(LB;A@N=B>(BL(NIJ(CB;NJQ(JGJHN;B>(DN;:HN:;J(=>(NIJ(DKDNJA(

M@D(LB:>O(B:N8($I=D(=>DN@R=G=NK(=D(OJNJ;A=>JO(RK(NIJ(=>LG:J>HJ(BL(NIJ(JQNJ;>@G(@>O(DJGL6A@F>JN=H(

L=JGOD(BL(NIJ(%!0(@>O(=N(GJ@OD(NB(NIJ(OJH;J@DJ(BL(NIJ(H;=N=H@G(C@G:J(BL(NIJ(JGJHN;B>(RJ@A(H:;;J>N(

SH:;;J>N(MIJ>(NIJ(>B>6DN@N=B>@;K(.#(=D(LB;AJO(=>(NIJ(O;=LN(DP@HJT8(-N(M@D(O=DHBCJ;JO(NI@N(NIJ(C@G:J(

BL(NIJ(JQNJ;>@G(A@F>JN=H(L=JGO(OJNJ;A=>JD(NIJ(DN;:HN:;J(BL(NIJ(JQH=N=>F(;BN@N=B>@G(ABOJ(=>(NIJ(RJ@A(

@>O?(IJ>HJ?(NIJ(DI@PJ(BL(NIJ(LB;AJO(JGJHN;B>(CB;NJQ(=>(NIJ(DKDNJA8(/B?(OJPJ>O=>F(B>(NIJ(JQNJ;>@G(

A@F>JN=H(L=JGO(C@G:J(NIJ(OB:RGJ?(N;=PGJ(B;(D=>FGJ(CB;NJQ(JGJHN;B>(DN;:HN:;J(=D(LB;AJO(=>(NIJ(%!0(MIJ>(

NIJ(JQNJ;>@G(A@F>JN=H(L=JGO(=D(GJDD(NI@>(@(HJ;N@=>(NI;JDIBGO(C@G:J8($IJ(BRN@=>JO(;JD:GND(JQPG@=>(DBAJ(

LJ@N:;JD(BL(NIJ(OJPJ>OJ>H=JD(BL(NIJ(C=;H@NB;(B:NP:N(PBMJ;(B>(NIJ(JQNJ;>@G(A@F>JN=H(L=JGO(C@G:J(NI@N(

MJ;J(BRN@=>JO(JQPJ;=AJ>N@GGK(=>(U`X8

$I=D(MB;<(I@D(RJJ>(D:PPB;NJO(RK(NIJ(%:DD=@>(+B:>O@N=B>(LB;(0@D=H(%JDJ@;HI(SP;BaJHND(V[6b[6

bb_Wc?(V[6b[6dbb[[?(V[6b[6__b\V(@>O(V[6b[6_VVb[T?(@>O(NIJ();JD=OJ>NeD(P;BF;@A(SP;BaJHND(,46

_Wc8[bV_8[(@>O(,96`V`8[bV_8[T8

V8 *8!8(4:R=>BC?(.848(/JGJA=;(ff(g8(#BAA8($JHI8(!G8([bb[8(.8(W\8('B8(Z8()8(c\c8

[8 g8 (0J>LB;O?(g8*8(/MJFGJ?(!8(/HI@A=GBFG:8(2=FI()BMJ;(,=H;BM@CJD8(#%#();JDD?($@KGB;(@>O(+;@>H=D?([bb\8

_8 %8*8(+=G@NBC(JN(@G8(ff()IKD=HD(BL()G@DA@D8([bbd8(.8(VZ8('B8(_8()8(b__VbZ8

W8 *8!8(2;@ABC(JN(@G8(ff()IKD=HD(BL()G@DA@D8([bV[8(.8(Vd8('B8(VV8()8(VV[VbV8

c8 *8!8(2;@ABC(JN(@G8(ff()IKD8("JNN8(*8([bVb8(.8(_\W8()8(_bc\8

Z8 *8!8(2;@ABC(JN(@G8(ff(->N8(g8(!GJHN;B>=HD8([bVV8(.8(d`8('B8(VV8()8(VcWd8\8 48g8(/:GG=C@>?(g8!8(1@GDI?(!8*8(#B:ND=@D8(.=;N:@G(H@NIBOJ(BDH=GG@NB;(SC=;H@NB;T(NIJB;K8(

*;NJHI(2B:DJ(,=H;BM@CJ("=R;@;K?(Vd`\8(#I8(V_8

`8 '8'8(7@OJND<==(JN(@G8(ff()G@DA@()IKD8(%JP8(Vdd_8(.8(Vd8()8([\_8

!"#$%"&'())"*+",-*'*.-&('"#$/%*01")/02&/20$3

4*5"6*0-5$0"(#7"82$7-"9/**:";#)/-/2/$"*+"<$20*-#+*05(/-&)"=>?@A4?>">B0-&C

D2&C":0$E-*2)"%*01"-#"#(/20(''F"*&&200-#."#$/%*01)"C()"+*&2))$7"*#".'*,('":0*:$0/-$)"*+"$G:'-&-/"#$/%*01)H",()$7"*#"

)-#.'$I#*7$H "*0 "*:$#I:(/C "5$()20$) " J$K.K " 7$.0$$ " (#7 ":(/C " '$#./C "7-)/0-,2/-*#) " LMH " NOPK "?$0$H "%$ " -#E$)/-.(/$ " /C$)$"

)/(#7(07"5$()20$)"*#"-5:'-&-/",-*'*.-&('"#$/%*01)",()$7"*#"&'*)$7I:(/C",$C(E-*20)"J*0,-/)PK"4C$)$".0(:C)"(0$"7$0-E$7"

+0*5")$Q2$#&$)"*+"+2#7(5$#/('"(&/-*#)",F"R0*)*:C-'("5$'(#*.()/$0"720-#.":0$I&*:2'(/*0F"&*20/)C-:H"(+/$0"LSOK"T$"+-#7"

/C(/"(:(0/"+0*5"5-'7")5(''I%*0'7":0*:$0/-$)"(#7"7-)())*0/(/-E-/FH"/C$)$".0(:C)"$GC-,-/"("7-)/-#&/-E$")/02&/20$"-#"/C$-0 "

7$.0$$")20E-E('"+2#&/-*#)"JM"I"URVP"%C-&C"%$"&(''"72('I)&('$"-#"0$+$0$#&$"/*"-/)"E-)2('")-5-'(0-/F"/*"J)-#.'$P")&('$I+0$$"

,$C(E-*20"LMOK"T$":*)/2'(/$"/C(/"/C-)"72('I)&('$",$C(E-*20"&*2'7",$"("0$)2'/"*+"/C$".0(:C)W"&*5:*)-/-*#"+0*5"&'*)$7I

:(/C)X",$C(E-*20)"&*5:0-)$7"*+"-00$72&-,'$"2#)/(,'$":$0-*7-&"*0,-/)"J;=YZ)P"LSOK";#")2::*0/H"%$":0$)$#/"(")-5:'$".0*%/C "

:0*&$))",()$7"*#"*0,-/)"%-/C":0$+$0$#/-('"(//(&C5$#/"&(:(,'$"*+":0*72&-#.")5(''I%*0'7H"7-)())*0/(/-E$".0(:C)"%-/C"72('I

)&('$",$C(E-*20K "T$")2..$)/ " /C(/ " /C$)$".0(:C)"5-.C/",$"0$:0$)$#/(/-E$ "*+"("#$%")/02&/20(' "&'())X ",$C(E-*20(' "*0,-/"

#$/%*01)K

8$+$0$#&$)

MK !',$0/I6[)\']"^(0(,[)-"(#7"8_1("!',$0/K"A5$0.$#&$"*+")&('-#."-#"0(#7*5"#$/%*01)K"

9&-$#&$H"N`aJbcSdPXbedfbMNH"MdddK

NK RK"gK"T(//)"(#7"9K"?K"9/0*.(/\K"U*''$&/-E$"7F#(5-&)"*+"h)5(''I%*0'7i"#$/%*01)K"<(/20$H"

SdSJaa`cPXcedfMeH"Mdd`K

SK 8K"9/**:"(#7"gK"g*''$0K"D$)*)&*:-&"&*5:(0-)*#"*+"&*5:'$G"#$/%*01)",()$7"*#":$0-*7-&"

*0,-/)K"UC(*)X"!#";#/$07-)&-:'-#(0F"g*20#('"*+"<*#'-#$(0"9&-$#&$H"NMJMPXeMaMMNH"NeMMK

!"#$%&$'"("&%)*+,)"+,-(.,)/-#0&"+1)"+)(2-34.,)5"&67-1$89*1-'2)

20("44*&2#0:

!"#$"%&'&()*+,#"#$"#$-.-/-&*+,0-1-2$,3-(4).-5,6.4*&784(9+,:5;-(9+,0-1-2$8(-.

<"=&488+,%":"#-28+,>?;/)5'(,6.4*&784(9,)@,!&754.+,<&7;-.9

!" # $%&'( # '()*+,-*. # /"0,+"$ # ,) # * # $($%"+ # 12 # %31 # -1&4."' # 5,%67&089:*0&+1

)"&/1)$; # <8" # .,)"*/ # -1&4.,)0 # "="/%$ # "=-,%*%1/( # *-%,1) # &41) # 1)" # &),% # *)'

,)8,>,%,)0 # *-%,1) # &41) # *)1%8"/ # 1)"; # !" # '"+1)$%/*%" # %8*% # ,) # %8,$ # $,%&*%,1)

'"$%*>,.,6*%,1) # 12 # %8" # $%*%" # 12 # "?&,.,>/,&+ # ,) # %8" # $($%"+ # 1--&/$ # %8/1&08

%8" # '"0")"/*%" # 7142 # >,2&/-*%,1)@ # *% # %8" # -/,%,-*. # $"% # 12 # 4*/*+"%"/ # A*.&"$

%8" # B*-1>,*) # +*%/,= # 12 # %8" # .,)"*/,6"' # 2.13 # ,)-.&'"$ # %31 # ',22"/")% # 4*,/$ # 12

4&/".( # ,+*0,)*/( # ",0")A*.&"$; # C--1/',)0.(D # %8" # $($%"+ # 8*$ # %31 # -/,%,-*.

2/"?&")-,"$ # *)' # -*)D # '"4")',)0 # 1) # ,),%,*. # -1)',%,1)$D # "=8,>,% # ",%8"/ # 2*$%

1/#$.13#/"0&.*/#4"/,1',-#1$-,..*%,1)$;#

!" # /"41/% # 1) # %8" # "=4"/,+")%*. # ,)A"$%,0*%,1) # 12 # %8,$ # 48")1+")1) # ,) # *)

*)*.101&$ # "."-%/1),- # -,/-&,%; # E"8*A,1/ # 12 # %8" # -1+4&%"/ # +1'". # 12 # %8,$

-,/-&,%D # '"$,0)"' # 3,%8,) # %8" # ")A,/1)+")% # F&.%,$,+D # +*%-8"$ # 3".. # %8"

%8"1/"%,-*. # '"$-/,4%,1); # 713"A"/D # -1+4&%"/ # +1'".,)0 # '1"$ # )1% # %*G" # ,)%1

*--1&)% # -"/%*,) # 2*-%1/$ # 38,-8 # */" # 4/"$")% # ,) # /"*. # 48($,-*. # "=4"/,+")% # *)'

-*) # /"'&-" # ,%$ # *--&/*-(; # H) # /"*. # "=4"/,+")%$ # 1) # %8" # -,/-&,%D # *.1)0 # 3,%8

%8"1/"%,-*..( # 4/"',-%"' # 4"/,1',- # 1$-,..*%,1)$D # %8" # /"0,+" # 12 # >&/$%,)0 # 8*$

>"") # /"4"*%"'.( # 1>$"/A"';# :1%*>.(D # ,) # %8" # +1'". # "?&*%,1)$ # *% # %8"$"

4*/*+"%"/ # A*.&"$ # %8" # >&/$%,)0 # $1.&%,1)$ # */" # *>$")%D # *)' # %8" # 1).(

*%%/*-%1/$ # */" # %31 # .,+,% # -(-."$ # 38,-8 # -1//"$41)' # %1 # 4"/,1',- # 1$-,..*%,1)$;

713"A"/D # >&/$%,)0 # -*) # >" # 1>$"/A"' # ,) # -1+4&%*%,1)$ # *$ # 3"..D # ,2 # %8" # *--&/*-( # 12

)&+"/,-*. # ,)%"0/*%,1) # ,$ # )1% # $&22,-,")%.( # 8,08@ # 21/ # "=*+4."D # >&/$%$ # */"

/"-1A"/"' # *% # %8" # '"2*&.% # 4/"-,$,1) # 12 # %8" # 3,'".( # &$"' # ,)%"0/*%1/ # IJKL # ,)

%8" # 4*-G*0" # F*%.*>; # C$ # $11) # *$ # %8" # *--&/*-( # 12 # %8" # ,)%"0/*%,1) # ,$

,)-/"*$"' # M/"$4"-%,A".(D # %8" # ."A". # 12 # 2.&-%&*%,1)$ # ,) # %8" # "=4"/,+")% # ,$

/"'&-"'ND # %8" # /"0,+" # 12 # >&/$%,)0 # ,$ # /"4.*-"' # >( # 4"/,1',- # 1$-,..*%,1)$; # !"

-1)O"-%&/" # %8*% # %8,$ # $%*%"D # 21/+*..( # )1)9"=,$%")% # ,) # %8" # &)'"/.(,)0

"?&*%,1)$D#,$#$&$%*,)"'#>(#)1,$"#12#)*%&/*.#1/#)&+"/,-*.#1/,0,);

1

Toward memristive implementation of the

Hodgkin-Huxley model

Miroslav Mirchev, Fernando Corinto, and Mario BieyDepartment of Electronics and Telecommunications

Politecnico di Torino, Torino, ItalyEmails: miroslav.mirchev@polito.it fernando.corinto@polito.it mario.biey@polito.it

I. Abstract

The Hodgkin-Huxley model introduced in [5] resembles the behavior of the membrane potential in a giant squidaxon. The basic version of the model includes channels depicting the flow of potassium and sodium ions and additionalleakage channel where chloride and other ions flow. As described in [1] the ion channels in the axon can be representedusing memristors, which are electrical elements whose voltage and current obey the Ohms law at each instant but theirresistance is state dependent [2]. The number of internal state variables required to represent the memristor’s state iscalled the order of the memristor. In accordance with the equations structure of the HH model a second order memristoris needed for the sodium channel and a first order memristor for the potassium channel. We build a memristive circuitin PSpice by replacing the time-varying resistors from the original circuit in [5] with the memristor circuit recentlyproposed in [4] which is composed of a full-wave rectifier connected in parallel with a first order RC-filter for thepotassium channel and a second order RLC-filter for the sodium channel. By applying periodic input we demonstratethat the individual ion channels in the HH model can be modeled with the memristive circuit from [4]. Furthermore,we give indications that spiking behavior suggestive to the one occurring in real neurons could be also induced in itsmemristive implementation when a short current pulse is applied to the membrane.

References

[1] L. O. Chua, V. Sbitnev, and H. Kim, “Hodgkin-Huxley axon is made of memristors, Int. J. Bifurcation Chaos, vol. 22, no. 3, p. 1230011,2012.

[2] L. O. Chua, “Memristor: the missing circuit element,” IEEE Trans. on Circuit Theory, vol. 18, no. 5, pp. 507-519, 1971.[3] D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “The missing memristor found,”Nature, vol. 453, pp. 80-83, 2008.[4] F. Corinto and A. Ascoli, “Memristive diode bridge with LCR filter,” Electronics Letters, vol. 48, no. 14, pp. 824–825, 2012.[5] A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in

nerve, ” J. Physiol., vol. 117, no. 4, pp. 500–544, 1952.

!"#$#!%&$'(%'!()*+)',%&$-'%%&,%)."#(&)(/,!"$*,'0#%'*,)

',)%"&).$&(&,!&)*+),*'(&

*1'1)-2345678429)-1*1)0:;<5=67=9)#1#1)>2<282=34??9)#1&1)"<5@2=

!"#"$%&'!$"$(')*+&(#,+$-.'!"#"$%&'!$"$('/(01*+0"2')*+&(#,+$-.'!"#"$%&.'34,,+"

21?1@234567842AB@5?61C2@

'8D7<@?DD78CE)?3)5);F?G;?D2;3)H:782@7828)?8)8286?875<)3C?78C71)'D)?3)2F37<=7I9)?8)H5<D?C;65<9)5D)

D<583?D?28 ) J<2@ ) H7<?2I?C ) 23C?665D?283 ) D2 ) D:7 ) C:52D?C ) 2873 ) 53 )K766 ) 53 ) 5D ) D:7 ) F2;8I5<?73 ) 2J ) D:7)

3E8C:<282;3)<7B?@7)2837D)LMN1)'8D7<@?DD78D)F7:5=?2<)H<7C7I73)56@23D)566)482K8)DEH73)2J)C:52D?C)

3E8C:<28?O5D?289 )K?D:)D:7)@7C:58?3@3)2J)D:7?< )5<?3?8B)58I)C:5<5CD7<?3D?C3 )2J) ?8D7<@?DD78CE)F7?8B)

I?JJ7<78D)J2<)=5<?2;3)3E8C:<28?O5D?28)DEH731)(7=7<56)DEH73)2J)?8D7<@?DD78D)3E8C:<282;3)F7:5=?2<)5<7)

D<5I?D?28566E ) I?3D?8B;?3:7I1 ) %:7E ) 5<7 ) ?8D7<@?DD78D ) H:537 ) 3E8C:<28?O5D?28 ) LP9)QN9 ) ?8D7<@?DD78D)

B787<56?O7I)3E8C:<28?O5D?28)LRN9)?8D7<@?DD78D)65B)3E8C:<28?O5D?28)LSN)58I)?8D7<@?DD78D)82?37)?8I;C7I)

3E8C:<28?O5D?28)LTN1

*87)2J) D:7)@23D ) ?8D7<73D?8B)DEH73)2J )?8D7<@?DD78D )3E8C:<282;3)F7:5=?2< ) ?3 ) D:7)?8D7<@?DD78D)

H:537)3E8C:<28?O5D?281)%:7)DEH7)2J)?8D7<@?DD78CE)<756?O7I)?8)D:?3)C537)I7H78I3)F2D:)28)D:7)=56;7)2J)

D:7)C28D<26)H5<5@7D7<)@?3D;8?8B)58I)D?@7)3C567)2J)2F37<=5D?281)'J)?8D7<5CD7I)3E3D7@3)5<7)36?B:D6E)

I7D;87I)J<2@)75C:)2D:7<)D:7)7E767D)?8D7<@?DD78CE)D5473)H65C7)LPN1)+2<)D:7)65<B7)=56;73)2J)H5<5@7D7<)

@?3@5DC:)<?8B)?8D7<@?DD78CE)?3)<756?O7I)LQN1)%:7)35@7)DEH7)2J)?8D7<@?DD78CE)?3)2F37<=7I)?8)D:7)H:537)

3E8C:<28?O5D?28)<7B?@7)5D)D:7)F2;8I5<E)D?@7)3C5673)2J)2F37<=5D?28)?8I7H78I78D6E)28)D:7)=56;7)2J)

C28D<26)H5<5@7D7<)@?3@5DC:)LUN9)K:7<753)28)D:7)F2;8I5<E)2J)3;C:)<7B?@7)7E767D)?8D7<@?DD78CE9)<?8B)

?8D7<@?DD78CE )2< ) C27V?3D78C7 )2J ) F2D: ) DEH73 )2J ) ?8D7<@?DD78CE9 ) ?171 ) D:7 ) 32WC5667I ) ?8D7<@?DD78CE )2J)

?8D7<@?DD78C?739)D5473)H65C7)LXN1)

&VD7<856)82?37)?3)482K8)D2)?8J6;78C7)3;JJ?C?78D6E)28)D:7)C:5<5CD7<?3D?C3)2J)?8D7<@?DD78CE1 )'8)

H5<D?C;65<9)7JJ7CD)2J)82?37)28)D:7)828W5;D282@2;3)H7<?2I?C)3E3D7@)?8I;C73)87K)DEH7)2J)?8D7<@?DD78CE)

YDEH7)')?8D7<@?DD78CE)?8)D:7)H<7378C7)2J)82?37Z)K:?C:)C:5<5CD7<?3D?C3)5<7)C62376E)C2887CD7I)K?D:)D:7)

7E767D)2873)L[N1)%:7<7J2<79)?D)?3)?8D7<73D?8B)D2)5856EO7)D:7)?8J6;78C7)2J)82?37)28)C:52D?C)3E3D7@)F7?8B)

?8)D:7)=?C?8?DE)2J)D:7)3E8C:<282;3)<7B?@71

'8)D:7)H<7378D)<7H2<D)K7)5856EO7)D:7)?8J6;78C7)2J)82?37)28)C:5<5CD7<?3D?C3)2J)?8D7<@?DD78D)H:537)

3E8C:<28?O5D?28 ) F2D: ) ?8 ) D:7 ) 3E3D7@3 ) K?D: ) 5 ) 3@566 ) 8;@F7< ) 2J ) I7B<773 ) 2J ) J<77I2@ ) YDK2)

;8?I?<7CD?28566E)C2;H67I)$\3367<)23C?665D2<3Z)58I)3H5D?566E)7VD78I7I)@7I?5)Y:EI<2IE85@?C56)@2I763)

2J).?7<C7)I?2I73)?8)D:7)C537)2J);8?I?<7CD?2856)C2;H6?8BZ1)]7)3:2K)D:5D)?8)F2D:)C283?I7<7I)C5373)D:7)

82?37)2J)3@566)?8D783?DE)I273)82D)56@23D)?8J6;78C7)28)C:5<5CD7<?3D?C3)2J)?8D7<@?DD78CE9)?171)?8)D:7)C537)

2J)D:7)3@566)=56;73)2J)H5<5@7D7<)@?3@5DC:)28)D:7)F2;8I5<E)2J)3E8C:<28?O5D?28)7E767D)?8D7<@?DD78CE)

D5473 )H65C71 )%:7)?8C<7537 )2J )82?37) ?8D783?DE )<73;6D3 ) ?8 ) D:7)B<2KD:)2J ) D:7) D:<73:26I)=56;7 )2J ) D:7)

3E8C:<282;3)<7B?@7)2837D)3D?H;65D7I)FE)D:7)6233)2J)D:7)H:537)C2:7<78C7)2J)D:7)<73H2837)3E3D7@)

5DD<5CD2<1 )!2837G;78D6E9 )28)D:7)F2;8I5<E)2J)H:537)3E8C:<28?O5D?28)?8)D:7)3;H7<C<?D?C56)<7B?28)2J)

C28D<26 ) H5<5@7D7<3 ) <?8B ) ?8D7<@?DD78CE ) C2@73 ) ?8D2 ) F7?8B ) K:7<753 ) ?8 ) D:7 ) 3;FC<?D?C56 ) 287 ) D:7)

C27V?3D78C7)2J)<?8B)58I)7E767D)?8D7<@?DD78C?73)D5473)H65C71)+2;8I)F7:5=?2<)?3)533;@7I)D2)H233733)5)

:?B:)67=76)2J)B787<56?DE1)*87)C58)7VH7CD)D:5D)3?@?65<)<7B;65<?D?73)K2;6I)F7)2F37<=7I)?8)<756)3E3D7@31)

%:?3)K2<4 ):53)F778)3;HH2<D7I)FE)D:7)-?8?3D<E)2J)7I;C5D?28)58I)3C?78C7)2J)$;33?5)YH<2^7CD3)

MR1_QU1PM1`USM9 )MR1MQP1PM1MRPT9 )B2=7<8@78D )H658 )28 )P`MQ)58I)H658 )H7<?2I )2J )P`MR)58I)P`MS)

E75<3Z9)$;33?58)+2;8I5D?28)J2<)_53?C)$7375<C:)YH<2^7CD3 )MPW`PW``PPM)58I)MPW`PW``QUUZ)58I)FE)

aE853DE)+2;8I5D?281

M1 .1)_7<B79)/1).2@75;9)!:1)b?I56)cc)de*<I<7)a583)d7)!:5231)"7<@5889).5<?31)M[XX1

P1 #1(1).?42=34E)7D)561)cc).:E31)$7=1)d7DD1)M[[U1)b1)U[1),21)M1).1)RU1

Q1 #1&1)"<5@2=)7D)561)cc).:E31)$7=1)d7DD1)P``T1)b1)[U1).1)MMRM`M1

R1 #1&1)"<5@2=9)#1#1)>2<282=34??)cc)&.d1)P``S1)b1)U`1),21)P1).1MT[1

S1 (1)_2CC567DD?9)a1d1)b5665I5<73)cc).$&1)P```1)b1)TP1),21)S1).1)UR[U1

T1 *1'1)-234567842)7D)561)cc)%.1)P`MM1)b1)ST1),21)[1).1)MQT[1

U1 -1*1)0:;<5=67=)7D)561)cc).:E31)$7=1)&1)P`MM1)b1)XQ1).1)`PUP`M1

X1 #1&1)"<5@2=)7D)561)cc)!"#*(1)P`MQ)Y3;F@?DD7IZ1

[1 #1&1)"<5@2=)7D)561)cc).:E31)d7DD1)#1)P`MM1)b1)QUS1).1)MTRT1

!"#$%#&'()*#'$+,%,)"-)."$'()*/0%1%0+

!"#$%#&'(#)*+,+-).#$#/%#&'(#)*+0+-).'1'23()0+45+-)6#789%#&'(#)*+45+

!"#$%$&$'()*(+,-'.%/'"$01(0"2(34').'$%501(647#%5#8(019:0.0;%(<0=0>4(?0$%)"01(@"%A'.#%$78(B1/0$78 (

<0=0>4#$0"

!$)25'):38;):')/8'<'$2)8'<7=2<)3>)$3$=?$'#8)#$#=&<?<)3>)2?"')1&$#"?@<)3>)<3=#8)#@2?(?2&)AB*C)

3$)25')%#<')3>)DEFGHIJDI))<3=#8)"#K$'2?@)>?'=1<)LDM)#$1)DNDNHIJDI)?$2'8$#2?3$#=)<7$</32)$7"%'8)

LIM)@5#$K')1#2#+)*//=?@#2?3$)3>)O#@;#81HP#;'$<)"'2531)>38)25')/5#<')28#Q'@238&)8'@3$<287@2?3$)

7<?$K)2?"')1'=#&)#8K7"'$2<)LR-SM)5#<)<53:$)25#2)?$)'('8&)<?$K=')"#T?"7")B*)/'8?31)#)<&<2'")

8'/8'<'$2'1)%&)25'<')2?"')<'8?'<)1?</=#&<)@5#32?@)%'5#(?38)#$1)?2<)1&$#"?@<)8'<3=('<)?2<'=>)?$23)#)

=?"?2'1 ) <'2 )3> );'& )(#8?#%='<U ) @388'=#2?3$ )1?"'$<?3$ )VI) 1'/'$1'$@' )3$ ) 25' )$7"%'8 )3> )/5#<')

(#8?#%='<)$ )<#278#2'<) #2)$)(#=7'<)WHN)#$1)$3$?$2'K'8)VI) (#=7'<)?$)25')381'8)3>)S+GHG+N+)V78?$K)

"?$?"7")B*)/'8?31< ) <7@5 )%'5#(?38 ) ?< )"7@5 ) ='<< ) /83$37$@'1 )38 ) #%<'$2+ )P5'<' ) 8'<7=2< ) #8')

@388'</3$1 ) ) 23 ) 253<' )3%2#?$'1 ) ?$ ) LGM ):5'8' )$3$=?$'#8 ) 2?"' ) <'8?'< ) #$#=&<?< )3> ) 1#?=& ) <7$</32)

$7"%'8<)?$)25')$3825'8$)#$1)<3725'8$)<3=#8)5'"?</5'8'):#<)/'8>38"'1)>38)#="3<2)@3"/='2')<3=#8)

@&@=' ) >83" ) DEEI ) 23 ) IJJD ) :5?@5 ) 'T2'$1< ) 25'" ) 23 ) # )) =#8K' ) $7"%'8 ) 3>) @&@='<) #$1 ) 325'8)

"#$?>'<2#2?3$<)3>)<3=#8)#@2?(?2&+

!2)5#<)#=<3)%''$)>37$1)25#2) 178?$K)"#T?"7")B*)/'8?31<)1?"'$<?3$)3>)#228#@238)7$1'8K3'<)

/'8?31?@)@5#$K'<):?25)@&@=')2?"')@=3<')23)GJ)&'#8<-):5'8'#<)#%$38"#==&)=3:)#228#@238)1?"'$<?3$<)

AXD+WC)#8')3%<'8('1)178?$K)K=3%#=)A?$)<'@7=#8)(#8?#2?3$C)B*)"?$?"7"+)

D+)>2/UYY>2/+$K1@+$3##+K3(YBPOYBZ[*\]V*P*YB^6BOZP]6^.0_\BY!6P_\6*P!Z6*[

I+)522/UYY:<3+<2#$>381+'17Y"'#$>=1Y2?"'<'8?'<

R+)O#@;#81)6+`+-)a872@5>?'=1)b+O+-)c#8"'8)b+V+-)B5#:)\+B+)d'3"'28&)>83")#)2?"')<'8?'<+)

))))O5&<+)\'(+)['22+-)SG-)/+)FDIHFDW-)DENJ+)

S+)P#;'$<)c+)V'2'@2?$K)<28#$K')#228#@238<)?$)278%7='$@'+)e3=+)NEN)3>)['@278')632'<)?$)

)))).#25'"#2?@<+)B/8?$K'8-)0'8=?$-)/+)RWWHRND-)DEND+)

G+)6+)b'(2?@-)b+B+-)B@5:'?29'8)#$1)a+b+)a'=7@@?)*f*-)RFE)AIJJDC-)WDDHWDG

A Sequential Classification Concept using FlawedTraining Data - An Application to Flow Cytometry

Thomas Niederberger∗, Thomas Ott∗, Carlo Albert†, Francesco Pomati†∗ ZHAW Zurich University of Applied Sciences, Switzerland Email: ottt@zhaw.ch

† Swiss Federal Institute of Aquatic Science and Technology (EAWAG), Switzerland

Abstract—Environmental scientists use high-throughputflow cytometry devices to count and measure cells inan automated procedure. In flow cytometry suspendedcells pass a single wavelength laser beam and emit aspecific optical signal depending on their structure andfluorescence. The signal is recorded and features areextracted from this signal. A cytometry device measuresthousands of individual particles very quickly. This leadsto data sets containing tens of thousands of measurements.

The Swiss Federal Institute of Aquatic Science and Tech-nology (Eawag) has recently developed a novel cytometryapproach in order to automatically measure phytoplanktonoccurrences in lakes [1] and how phytoplankton commu-nities respond to stress [3]. The collected data sets canbe analyzed manually, however this is time-consuming,cumbersome and prone to error. Therefore an automaticsolution would be preferred. A typical approach for anautomated analysis is to apply clustering methods, suchas k-means [2]. Classifiers such as naive Bayes or supportvector machines cannot be applied directly because noclassified data exists for supervised learning.

We propose a novel approach how to analyze andclassify the measured data using a sequential, multi-layerstructure where we extract a certain class in each layer.The concept relies on data sets which are dominated bycertain classes. This means that in certain data sets asingle class (phytoplankton species) makes up for a largeproportion (> 60%) of the data. Such data often occur innatural measurements. We treat such a data set as if allitems were from the dominant class only and then traina simple classifier for this dominant class. The classifiermust be robust as our training data contains a significantamount of misclassified items. When traversing throughthe classification hierarchy, all dominant classes will beremoved. The remaining data can then be further analyzedby techniques such as outlier detection or clusteringmethods in order to find rare species.

Preliminary results using artificial data show the po-tential of our approach: despite the fact that our trainingdata is heavily flawed and sometimes contains up to 40%misclassified examples (Fig 1) we are still able to achievea classification accuracy of > 80% on a separate data set

Fig. 1. Flawed training classification. Blue points denote the basicset. The big, red points are used to train a classifier for the upperright distribution.

Fig. 2. First layer classification result. Red points were classifiedas belonging to the dominant class.

(Fig 2). In our contribution we will discuss our approachin more detail. Furthermore, we will present results onlaboratory data as well as on real world data from lakes.

REFERENCES

[1] F. Pomati, J. Jokela, M. Simona, M. Veronesi, B. Ibelings.An automated platform for phytoplankton ecology and aquaticecosystem monitoring. In Environmental science & technology.45.22 (2011): 9658-9665.

[2] L. Boddy, M. Wilkins, C. Morris. Pattern recognition in flowcytometry. In Cytometry, 44.3 (2001): 195-209.

[3] F. Pomati, F., L. Nizzetto. Assessing triclosan-induced ecologicaland trans-generational effects in natural phytoplankton communi-ties: a trait-based field method. In Ecotoxicology. 2013, Publishedonline ahead of print.

Musical Generation from Chaos with Ring Attractor based onParameter Combination

Saho Osano and Mamoru TanakaDepartment of Information and Communication Science, Sophia University

7–1, Kioi-cho, Chiyoda-ku, Tokyo 102–8554, JapanEmail: osahoosaho@gmail.com

Email: mamoru.tanaka@gmail.com

Masatoshi SatoFaculty of System Design, Tokyo Metropolitan University

6–6 Asahigaoka, Hino, Tokyo 191–0065, JapanEmail: masa5899@gmail.com

ABSTRACT

In this paper, we propose a novel musical generator using the chaosof Nishio circuit. The normalized expression of the Nishio circuit wasprocessed by forward Euler’s rule. Each sound can be created bychanging the resistance value in the Nishio circuit. And furthermore,we used an envelope to make the opening and the end of the soundseem natural. In this way the proposed musical generator becomesmore smoothly.

The proposed sounds are made by the software called Csound.Csound is a computer programming language for making sound. Inthis simulation, we show that the chaos of the Nishio circuit cangenerate musical tones.

I. ALGORITHM

A. The Nishio CircuitChaos produces complicated behavior from a simple numerical

formula, which looks random. There are Lorenz, Cellular automaton,intermittent chaos, One-half integral, etc. as the chaos algorithm.Statistical characteristics are different, depending on algorithms.

The chaos of the Nishio circuit is a chaos with ring attractor, andthe original sound can be heard as a sound. The value of chaos withring attractor is limited.

The normalized expression of the Nishio circuit is given by

dxkdτ = β(xk + yk) − zk − γ

∑4j=1 x j

dykdτ = αβ(xk + yk) − zk − f (yk)dzkdτ = xk + yk

(1)

(k = 1, 2, 3, 4)

wheref (yk) = 0.5(δyk + 1 − |δyk − 1|) (2)

α, β, γ and δ are defined by

α =L1

L2, β = γ

√CL1,

γ = R√

CL1, δ = rd

√CL1

(3)

α, β, and γ are arbitrary constants, respectively and use this timethe value which the ring attractor generated.

Each note sound can be issued by changing the parameters in theequation. This time β, γ and δ are changed.

Frequency characteristic of the chaos of Nishio circuit is shown inFig. 1.

This figure shows that the frequency characteristic has 1/f char-acteristic. It is well known that the 1/f fluctuation is an irregular

Fig. 1. The frequency characteristic of the Nishio circuit

oscillation in the natural physical world and is of comfortable andagreeable sensation for human.

II. SIMULATION RESULTS

In this paper, we proposed that the chaos of the Nishio circuitcan make a sound without using complicated filters. Furthermore,by fixing the generating sounds to x, y and z respectively, the sameparameter could make a number of frequencies. There were othercases in which the parameter generated same frequency, even thoughthe sounds were fixed to x, y, and z respectively. Each sound can becreated by changing the capacitance value in the equation. Parametercombination was changed as follows. β = 0.14, γ = 0.3 and δ = 100are set as a one cell and β = 0.2, γ = 0.3 and δ = 250 are set as aone cell and so on.

III. CONCLUSION

In this paper, we proposed musical tone using the chaos of theNishio circuit. There is a limit in a parameter combination forgenerating musical tone. Therefore, musical tone was expressedby using parameter combination. For example, eight musical tonegenerated when each value was changed based on β = 0.14, γ = 0.3and δ = 100.

REFERENCES

[1] R. Boulanger, ”The Csound Book: Perspectives in Software Synthesis,Sound Design, Signal Processing,and Programming ”, The MIT PressCambridge Massachusetts London England, Mar. 2000

[2] C. H. Hansen, ”Understanding Active Noise Control”, Taylor & Francis,Dec. 2001

[3] R. N.Madan, ”CHUA’S CIRCUIT: A Paradigm for CHAOS”, WorldScientific, 1997

[4] T. Ross, ”Sound Scattering from Oceanic Turbulence - Chaos or Crit-ters?”, VDM Verlag, Dec. 2008

A Semi-Supervised Learning Systemfor Micro-Text Classification

Thomas Ott∗, Markus Christen†, Thomas Niederberger∗, Reto Aebersold‡,Suleiman Aryobsei§, Reto Hofstetter§∗ ZHAW Zurich University of Applied Sciences, Switzerland, Email: ottt@zhaw.ch

†University of Zurich, Switzerland‡ Atizo AG, Bern, Switzerland

§University of St. Gallen, Switzerland

Abstract—Modern communication platforms are sourcesof large samples of micro-texts that are in need of machinetext processing for text classification and interpretation. Onthe Swiss innovation platform Atizo, contributors submitideas or solutions to problems posed by companies. In sucha crowdsourcing process, hundreds of people contribute alarge amount of micro-text data that needs to be structuredalready during the process of idea generation in orderto avoid repetitions and to optimize the solution space.Due to low word-count and unstructured writing, micro-texts pose a challenge for automatized text processing.Technically, the goal is to partition a growing set of micro-texts T = t1, ..., tn into m topical groups Gj ⊂ T , j ∈1, ...,m, (∪j Gj = T , Gk ∩ Gl = ∅), where each text ischaracterised by a set of words ti = wi1 , ....wini

and thecorresponding word counts (bag-of-words model, e.g. [1])and each group Gj is associated with a set of terms or keywords kGj defining the topic of the group, i.e.

ti → Gj , kGj ⊂ ∪α tα with tα ∈ Gj

according to some suitable optimality criterion.In our research, we addressed the whole chain of issues

related to data pre-processing (where text enrichment playsa major role), data classification, data visualization, aswell as solution benchmarking and process improvement.Our research eventually culminated in the developmentof a complete system for machine-supported clusteringand classification of micro-texts on the innovation plat-form Atizo. The system combines a bottom-up clusteringapproach with a top-down classification approach (Fig 1).This leads to a kind of semi-supervised learning procedure,where an unsupervised learning step to identify the bestcluster(s) is combined with a supervised control step todefine a classifier. For the clustering step, we implementeda PCA-based approach (Hebbian clustering [2]) that yieldsclusters and a corresponding characterization by means ofkey words. In the supervised step, a human supervisorintervenes by selecting the most significant key wordsfor the best cluster (as assessed by a suitable criterion).These key words are then used to generate a classifier.

Fig. 1. Flowchart of the entire classification system

The entire procedure of clustering and classification is runthrough again as long as new data is entering the systemor significant clusters can be found.In our contribution, we present the technical details of

our system and discuss the results from preliminary studieson the innovation platform Atizo.Acknowledgment: This project has been supported by KTI grant 12747.1 PFES-ES.

REFERENCES[1] M. Steyvers, T. Griffiths. Probabilistic Topic Models. In LatentSemantic Analysis: A Road to Meaning. (Lawrence Erlbaum),2007.

[2] T. Niederberger, N. Stoop, M. Christen, T. Ott. Hebbian PrincipalComponent Clustering for Information Retrieval on a Crowdsourc-ing Platform. In Proceedings of NDES’12, 2012.

!"#$%&!'%$()**+'),-(.'+(,-)'/(0)/$%(*/$1%'(+)2%'!(21!0$+(

3$2$+)4"'2

!"#$%&'()*+,-.'$/0',12$%'-3435..

617$89#1/9':;'<&1298%2$&'$/0'<&1298:/%2"'</=%/118%/=>'<82%?1"'@/%A18"%9?>'-$?"18%>'BCDBE>'*F8G1?

.'%"#$%&:H9F8GI182%?1"J10FJ98..'G%&%2I182%?1"J10FJ98

)56789:7

KL$:" 'M$"10 '7"1F0:' 8$/0:#'/F#M18 '=1/18$9%:/ 'L$" 'M11/ '$998$29%/= ' %/9181"9 ' ;:8 ':A18 ' 9N:'

012$01"'M12$F"1':;'2&:"1'81&$9%:/"L%7'M19N11/'2L$:"'$/0'28?79:=8$7L?J'O8:7189%1"':;'98F1'2L$:">'"F2L'

$"'"1/"%9%A1'0171/01/21':/'%/%9%$&'A$&F1">'9:7:&:=%2$&'#%P%/='$/0'18=:0%2%9?>'"$9%";?'2:/;F"%:/'$/0'

0%;;F"%:/ ' 81QF%81#1/9" ' :; ' 28?79:=8$7L?J '*L181;:81> '#$/? ' 8$/0:# ' /F#M18 ' =1/18$9:8" ' L$A1 ' M11/'

78:7:"10'9:'M1'F"10'%/'28?79:=8$7L%2'$77&%2$9%:/"'F7'9:'/:NJ'R:"9':;'9L1#'F9%&%H1'2L$:9%2'#$7"';:8'

7"1F0: ' 8$/0:# ' /F#M18 ' =1/18$9%:/ ' M12$F"1 ' :; ' 9L1%8 ' 91#7:8$&&? ' 0%"28191 ' /$9F81 ' $/0 ' 1$"1 ' :;'

%#7&1#1/9$9%:/J'S:N1A18>'81$&'&%;1'$77&%2$9%:/"':;'9L1"1'#:01&"'L$A1'9:'M1';%/%91'%/'81":&F9%:/>'L1/21> '

9L1"1'#$7"'$81'"7$9%$&&?'0%"2819%H10J 'T7$9%$&'0%"2819%H$9%:/'2$F"1"'0?/$#%2$& '01=8$0$9%:/'$/0'"%/21'

2L$:" ' %" ' 01;%/10 ' %/ ' 2:/9%/F:F" ' "7$21> ' 78:7189%1" ' :; ' 98F1 ' 2L$:" ' 0:/U9 ' $77&? ' $/?#:81J ' T7$9%$&&? '

0%"2819%H10'2L$:9%2'#$7"'91/0'9:'1PL%M%9'718%:0%2'M1L$A%:8'$;918'":#1'98$/"%1/9'81"7:/"1'$/0'9L%"'%"'

2:#7&191&?'0%;;181/9';8:#'98F1'2L$:"'NL%2L'%"'/1A18'718%:0%2J'*L%"'M1L$A%:8'%"'/:9'N1&&'01;%/10'M?'$ '

#$9F81'9L1:819%2$&'M$2G=8:F/0'$/0'718%:0'&1/=9L"'#$?'M1'A18?'"L:89'$&9L:F=L'L%=L'81":&F9%:/'%"'F"10J'

*: ' 1&%#%/$91 ' 9L1"1 ' 08$NM$2G"> ' 0%;;181/9 ' 1/=%/118%/= ' ":&F9%:/" ' L$A1 ' M11/ ' 78:7:"10J ' V:8 ' 1P$#7&1'

7189F8M%/= ' 2L$:9%2 ' "?"91#" 'N%9L ' $/ '#W"1QF1/21 '=1/18$9:8 ' "11#" ' 9: 'M1 ' $ ' 78:#%"%/= '#19L:0> ' MF9 '

&%#%9$9%:/"':A18'7189F8M%/='%/918A$&'1;;129"'9L1'"12F8%9?':;'9L1'"?"91#J'

3/' 9L%" '7$718> '$/:9L18 '#19L:0'M$"10':/'0%"2819%H$9%:/':; '2:/9%/F:F" ' 9%#1'2L$:9%2 '"?"91#"'

N%9L:F9'7189F8M$9%:/'%"'78:7:"10J'K:/9%/F:F"'9%#1'2L$:9%2'"?"91#"'L$A1'9:'M1'L%=L18'%/'0%#1/"%:/'$/0'

9L1%8 ' 0%=%9$& ' %#7&1#1/9$9%:/" ' 81QF%81 ' M:9L ' "7$9%$& ' $/0 ' 91#7:8$& ' 0%"2819%H$9%:/ 'NL%2L ' M8%/=" '#:81'

78:M&1#"J'X1A189L1&1"">'9L1%8'718%:0"'$81'#F2L'&:/=18'9L$/'2L$:9%2'#$7"'N%9L'9L1'"$#1'81":&F9%:/'$/0'

9L1%8'7$8$#1918'2:#7&1P%9?'$&&:N'8%2L18'0?/$#%2"'9:'M1':M9$%/10J'''

1

Numerical Analysis of Cascaded Hopf Amplifiers

Marco Reit and Wolfgang MathisInstitute of Theoretical Electrical Engineering,

Leibniz Universitat Hannover, GermanyEmail: reit@tet.uni-hannover.de

Ruedi StoopInstitute of Neuroinformatics,

University of Zurich / ETH Zurich, SwitzerlandEmail: ruedi@ini.phys.ethz.ch

I. Abstract

Physiological experiments suggest that the active non-linear amplification process in the cochlea has the samequalitative behavior as a system close to a Hopf instability[1]. Thus, a Hopf-type amplifier was proposed as a basicelement in the cochlea modeling [2]. It was shown, thata chain of alternating Hopf amplifiers and low-pass filtersis eminently suitable to model the entire cochlea [3]. Thismodel shows all crucial nonlinear cochlea effects such astwo tone suppression and combination tone generation[3]. Each Hopf cell is described by a complex differentialequation of the kind

z = (µ+ j)ωchz − ωch |z|2 z − ωchF, z, F ∈ C, (1)

or its real representation

x = µωchx− ωchy − ωchx(x2 + y2)− ωchp

y = µωchy + ωchx− ωchy(x2 + y2)− ωchq,

(2)

where the output z and the external forcing F are givenby z = x+jy and F = p+jq, respectively. The parameterωch corresponds to the individual oscillation frequency ofthe Hopf amplifier. The main property of this Hopf cell isa µ-dependent nonlinear amplification of the input signalfor small negative µ-values.It is well-known that efferent connections from the brainto the outer hair cells in the cochlea, where the activeamplification occurs, exist. These connections are probablyresponsible for controlling the amplification by an auditoryneuronal feedback in order to focus on a desired soundcomponent [4] [5]. This amplification control mechanismis already given in the Hopf cochlea model through thetuning of the µ-values. Recent investigations about rulesfor µ-patterns show a remarkably fine tunability of theHopf cochlea model [5].For the development and implementation of µ-parametercontrol algorithms for an active tuning, a highly flexibledigital realization of the Hopf cochlea is advantageous.With respect to applications e.g. hearing aids or cochleaimplants, the investigation of coherent tuning by usingcomplex input stimuli requires a real-time processing.Therefore, a digital signal processor was chosen as atarget platform. Since the Hopf differential equation (2)

is the basic element in the cochlea model, a numericalintegration method is inevitable to calculate the output ona digital platform. The condition of real-time processingin accordance with a maximum number of implementedHopf cells sets several demands on the integration method.To ensure a robust and failsafe operation, the number ofnecessary calculation steps for each integration step shouldbe minimized and kept predictable. Therefore, the explicit4th-order Runge-Kutta method appears appropriate interms of accuracy and performance [6].Since the integration method is explicit and therefore notA-stable, a specific parameter-dependent stability regionexists for the numerical solution of the nonlinear differen-tial equation (2). In this work, we demonstrate the influ-ence of all possible parameters on the boundaries of thestability region. These parameters include, inter alia, thestepwidth of the Runge-Kutta method, the amplitude andfrequency of the input signal and the oscillation frequencyof the Hopf amplifier. Special attention is given to the µ-values of all linked Hopf cells within the entire cochleachain. We analyze the interval of valid µ-values for eachfeedforward coupled Hopf cell. This defines the tunabilityof the Hopf cochlea in our digital real-time realization.Our results on the stability behavior make it possible toadjust the numerical solving method to the particularproblem of a chain of Hopf cells. Thus, we present anadapted highly effective dynamic step size control for theRunge-Kutta method without the use of error estimators.Despite the enhancement of the stability by the step sizecontrol, we can avoid the computational intensity, whichan error estimator needs for real-time processing.Based on the results of this work, we discuss the possibili-ties and limitations of the considered digital Hopf cochlearealization.

References

[1] C.D. Geisler, “From sound to synapse: Physiology of the mam-malian ear”, Oxford Univ. Press, 1998.

[2] V. M. Eguıluz, M. Ospeck, Y. Choe, A. J. Hudspeth, and M. O.Magnasco, “Essential nonlinearities in hearing”, Phys. Rev. Lett.,vol. 84, pp. 5232–5235, 2000.

[3] R. Stoop, T. Jasa, Y. Uwate, and S. Martignoli, “From hearing tolistening: Design and properties of an actively tunable electronichearing sensor”, Sensors, vol. 7, pp. 3287–3298, 2007.

2

[4] A. J. Hudspeth, “Making an effort to listen: mechanical amplifi-

cation in the ear,”Neuron, vol. 59, pp. 530–545, 2008.[5] F. Gomez, and R. Stoop, “Tuning the Hopf Cochlea Towards Lis-

tening”, Proceedings of Nonlinear Dynamics of Electronic Systems(NDES), 2012.

[6] M. Reit, R. Stoop, and W. Mathis, “Time-Discrete Nonlinear

Cochlea Model Implemented on DSP for Auditory Studies”, Pro-ceedings of Nonlinear Dynamics of Electronic Systems (NDES),2012.

!"#$%#&'()*+#',%-.)"/)')01%2-3&4)5"6%#7)8%&$4)

9#4:-2%6&);"1&()<('#./&()0+.2&,

!"#$%%&'()*(+,''&-./(0$-1"(23"%3435/(0$,-"(0"%43$-6"5($%6(7&8&-(+,''&-.

.9%'838,8&(:"-(;$%"&<&18-"%31'/(=&1#%3'1#&(>%3?&-'38@8(0A%1#&%/(B&-C$%D5>%3?&-'38E(63(7&-,43$/(98$<D

)(F363-&183"%$<(C"?3%4(:3&<6(3%6,183?&(G"H&-(8-$%':&-(I0J97=K('D'8&C(:"-(G"H&-3%4("FL&18'(C"?3%4(

$<"%4($(G$8#*(9%6,183?&(G"H&-(8-$%':&-('D'8&C'(#$?&(F&&%(6&'1-3F&6(3%(M.N*(=#&(0J97=('D'8&C(3'(

F$'&6("%('8$83"%$-D(G-3C$-D(1"3<'(<3%&$-<D($--$%4&6($<"%4($(G$8#($%6($('&1"%6$-D(1"3<(3%($%("FL&18(

C"?3%4($<"%4(8#&(G$8#(M5N*(="(C3%3C3O&(G"H&-(<"''("%<D(8#&(G-3C$-D(1"3<'(,%6&-(8#&(C"?3%4("FL&18'(

$-&($183?$8&6*(=#&(G-3C$-D(1"3<'($-&("G&-$8&6($'($('H381#&6(-&'"%$%8(1"%?&-8&-*(P#&%('H381#3%4(

:-"C("%&(G-3C$-D(1"3<(8"(8#&(',F'&Q,&%8(G-3C$-D(1"3<("%(8#&(G$8#(3%("-6&-(8"(:"<<"H(8#&(C"?3%4(

"FL&18(8#&(&<&18-31($%6(C$4%&831(&%&-43&'('8"-&6(3%(8#&(1"3<'($%6(1$G$138"-'($-&(G$''&6("%(8"(8#&(

',F'&Q,&%8 ( 1"3<' ( $%6 ( 1$G$138"-' ( 3% ( "-6&- ( 8" ( $?"36 ( <"'' ( ": ( 8#& ( '8"-&6 ( &%&-4D* ( 9% ( 8#3' (H"-R ( $(

F363-&183"%$<('H381#&6(3%?&-8&-(MSN(3'(3%8-"6,1&6(8#$8($<<"H'(G"H&-(8-$%':&-(3%%(F"8#(63-&183"%'*

=#&(0J97=('D'8&C(&T#3F38'(8#&('8$83"%$-D(G-3C$-D(<""G'( (!"#( $%6(8#&(C"?3%4('&1"%6$-D(<""G (!$*(

=#&('H381#&'($!#($%6($%#/(-&'G&183?&<D/((1"%%&18(8#&(G-3C$-D(<""G'(!"#(&38#&-(H38#(8#&(1"CG&%'$83"%(

1$G$138"-' (&"#( "-(H38# (&"#U.*(="(:"<<"H(8#&(C"?3%4("FL&18(8#&(G-3C$-D(1"CG&%'$83"%(1$G$138"-(3'(

-&G<$1&6(FD(8#&(',F'&Q,&%8("%&(H#&%(8#&(1$G$138"-(?"<8$4&(3'(O&-"($%6(8#&((G-3C$-D(<""G(3'(-&G<$1&6(

FD(38'(:"<<"H&-(H#&%(8#&(<""G(1,--&%8(3'(O&-"*(=#3'(G$''3%4("?&-(3'(3%'8$%8$%&",'/(3--&'G&183?&(":(8#&(

8-$%'3&%8(83C&'(":(8#&(-&'"%$%8(13-1,38'*

=#&('H381#&' ($'/ ($(( $%6 ($&/ ($)( $-&(1"%8-"<<&6(FD(8#&(1,--&%8' (#.($%6 (#5/(-&'G&183?&<D/(',1#(8#$8(8#&(

13-1,38(3'('&<:V"'13<<$83%4($%6(8#&(G"H&-((:<"H(1$%(F&(1"%8-"<<&6($%6(63-&18&6(:-"C(8#&(<&:8(8"(8#&(-34#8(

"-(3%("GG"'38&(63-&183"%*

)%(3%8&-&'83%4($GG<31$83"%(":(0J97=('D'8&C'(H3<<(F&(8#&("%(8#&(-"$6(G"H&-3%4(":(&<&18-31(?&#31<&'*(

>%6&-(F-&$R3%4(1"%6383"%'(G"H&-(1$%(F&(:&6(F$1R(8"(8#&(',GG<D*

+&:&-&%1&'(

M.N(0*(23"%343/()*(W"'8$%O"/($%6(0*(0"%43$-6"/(X;&8H"-R(0&8#"6'(:"-()%$<D'3'($%6(2&'34%(":(+&'"%$%8(

P3-&<&''(7"H&-(=-$%':&-(YD'8&C'/Z(&*+,-./0120-*.03114056#/.7.880"19./0:/+;82./<"/#;=#,7.80+;>0?;@#;../#;@ 0

?A,71/+-#1;8BC0*--,DEE999F0#;-.=*9.3F0G/@/(5[..*

M5N(!*()*(+,''&-($%6(7*(+,''&-/(X2&'34%(1"%'36&-$83"%'(:"-($(C"?3%4(:3&<6(3%6,183?&(G"H&-(8-$%':&-('D'8&C/Z(H???0

H;-./;F06#/.7.80"19./(:/+;82./0&1;2./.;=.0"./I@#+06":&/(0$D(.\(V(.](5[.S*

MSN(^*(_$3($%6(W*(03/(X`<3C3%$8&(-&$183?&(G"H&-($%6(3%1-&$'&('D'8&C(&::313&%1D(":(3'"<$8&6(F363-&183"%$<(6,$<V

$183?&VF-364&(2WV2W(1"%?&-8&-'(,'3%4(%"?&<(6,$<VG#$'&V'#3:8(1"%8-"</Z(H???0:/+;8+=-#1;801;("19./0?7.=-/1;#=8/(

?"<*(5S/(%"*(]/(GG*(5a[\b5a.c/(5[[d*

Maximum-Flow Neural Network for Minimum Cut Problemon an Undirected Graph

Masatoshi SatoFaculty of System Design, Tokyo Metropolitan University

6–6 Asahigaoka, Hino, Tokyo 191–0065, JapanEmail: masa5899@gmail.com

Mamoru TanakaDepartment of Electrical and Electronics Engineering, Sophia University

7–1, Kioi-cho, Chiyoda-ku, Tokyo 102–8554, JapanEmail: mamoru.tanaka@gmail.com

I. INTRODUCTION

This paper present a new approach to finding minimum s-t cutin an undirected graph. In our previous research, we proposed theMaximum Flow Neural Network (MF-NN) which is possible to solveany maximum flow problems [1]. Since MF-NN can be realized byusing a nonlinear resistive circuit, the maximum flow problem can besolved with analog high-speed parallel processing. In other words, thenetwork analysis boil down to the question of obtaining the currentdistribution by nonlinear resistive circuit analysis. Therefore, if MF-NN can be designed by the integrated circuit which is definableand changeable internal logic circuit like analog type ProgrammableLogic Device (PLD), a solution can be obtained in an instant. In thispaper, we propose a novel minimum s-t cut solution based on MF-NNin an undirected graph. Since a minimum cut solution using MF-NNis obtained by applying the process of Ford-Fulkerson’s minimumcut algorithm, the flow chart of the proposed method is almost thesame as Ford-Fulkerson’s minimum cut algorithm. However, it isimportant that the proposed method does not require the process ofgetting the maximum flow. Moreover, since this research indicatesthat the minimum cut algorithm can be realized by circuit analysis,speed improvement of the minimum cut algorithm can be expected.

II. MINIMUM-CUT ALGORITHM BASED ON MF-NN

Start

Network Analysis by MF-NN

Decomposition of Strongly Connected Components

Hasse Diagram

Minimum Cut

Fig. 1. Flow Chart of Minimum Cut Algorithm by MF-NN

The process of minimum cut algorithm based on MF-NN is shownin Fig. 1. First, a given undirected flow network is analyzed by MF-NN. The analyzed network is generally called the residual networkwhich shows the difference between each branch capacity and flow.Next, the residual network is decomposed into strongly connectedcomponents Hi (i = 1, 2, · · · , n) which is a local maximum stronglyconnected subgraph in the graph which is not necessarily the strongconnected. Then, the Hasse diagram which is a type of mathematicaldiagram used to represent a finite partially ordered set is obtainedfrom the graph decomposed into strongly connected components.Finally, The minimum cut is obtained from the cut which divideshasse diagram.

a

b

c

d

e

f

g

h

60

10

40

20

20

30

20

40

10

20

10

10

30

30

20

50

30

40

10 10

10 10

10 10

Fig. 2. Flow Network

a

b

c

d

e

f

g

h

9.17

8.60

8.60

8.45

0.875

0.993

0.664

8.60

10 H1

H2

minimum cut

Fig. 3. Experimental Result

III. SIMULATION RESULTSAn undirected flow network for an experiment is shown in Fig. 2.

Each value on branches shows the branch capacity. From this simu-lation, the graph decomposed into strongly connected components isobtained as shown in Fig. 3. The number on each node shows the nodevoltage. The red dot-line shows the branch that the branch capacity isnot filled with the flow. The areas of H1,H2 show strongly connectedcomponents. From this analysis result, the Hasse diagram is given byH1 ⇒ H2, and the minimum cut is minκc = 120, where κc is thecut capacity which is the sum of the branch capacities contained inthe cut.

IV. CONCLUSIONIn this paper, we proposed the novel minimum cut algorithm by

using MF-NN in an undirected graph. Since MF-NN is possible tobe realized by a nonlinear resistive circuit, this research indicates thatthe minimum cut algorithm can be realized by using circuit analysis.Therefore, speed improvement of the minimum cut algorithm can beexpected.

REFERENCES[1] M. Sato, H. Aomori, and M. Tanaka, ”Maximum-Flow Neural Network :

A Novel Neural Network for the Maximum Flow Problem”, IEICE Trans-actions on Fundamentals of Electronics, Communications and ComputerSciences, Vol. E92–A, No. 4, pp. 945–951, Apr 2009.

A Laboratory Experiment of a Half-CenterOscillator Integrated-Circuit

Munehisa Sekikawa and Takashi KohnoInstitute of Industrial Science, The University of Tokyo,

4-6-1 Komaba, Meguro-ku, Tokyo 153–8505 JapanEmail: sekikawa, kohno@sat.t.u-tokyo.ac.jp

ABSTRACT

Rhythmic activities in the nerve system arises through interactions among neurons. Central pattern genera-tors (CPGs) are neural networks that autonomously generate rhythmic motor patterns [1]. One of the most elucidatedCPGs is the Leech heart interneuron network [2]. The basic rhythm in this network is produced by half-centeroscillators (HCOs), which is the simplest rhythm generator and thus widely studied as a model to understandrhythm generation in the CPG. An HCO consists of two neurons each of which has ability to generate autonomousrhythmic activity. These two neurons reciprocally inhibit each other by synaptic connections, which results inanti-phase synchronized rhythmic activities. This network was originally suggested by Brown [3].

In this work, we realized an artificial HCO that is constructed by silicon neurons and synapse circuits. Thesecircuits were designed so that they reproduce the dynamical property in neuronal cells and synapses in real-time. Two silicon neurons and four silicon synapses were integrated in single very-large-scale integrated (VLSI)circuit chip to realize an HCO circuit. This chip was fabricated through TSMC in a 0.35µm CMOS process. Amathematical-structure-based design approach [4], [5] was employed to model silicon neurons that copy essentialdynamical properties in a class of autonomous bursting cells using simple circuitry. Synapse circuits were designedbased on the kinetic model of synaptic transmission in chemical synapses [6]. In our circuits, MOSFETs operatein their sub-threshold region, and therefore, the fabricated chip is intended to consume very low power. We tunedour circuits so that each silicon neuron circuit copies the dynamical structure in the square-wave bursters [6] andeach silicon synapse circuit mimics the dynamics in inhibitory chemical synapses such as GABAA synapses [6].Our HCO circuit could produce rhythmic activities similar to those in biological HCOs. Figure 1 shows a timeseries of membrane potentials of two silicon neurons. Their bursting patterns keeps anti-phase synchronized whilebursting rhythm of each silicon neuron is fluctuated by noises.

time / sec

Mem

bra

ne

pot

enti

cal

/ m

V

-100-80-60-40-20

0 20 40

0 0.5 1 1.5 2

Fig. 1. Time series of the HCO integrated-circuit. Red is the membrane voltage of one neuron and green is that of another neuron.REFERENCES

[1] E. Marder and R.L. Calabrese, “Principles of rhythmic motor pattern production,” Physiol. Rev., vol. 76, no. *, pp. 687–717, 1996.[2] G.S. Cymbalyuk, Q. Gaudry, M.A. Masino, and R.L. Calabrese, “Bursting in leech heart interneurons: Cell-autonomous and network-

based mechanisms,” J. Neuroscience, vol. 22, no. 24, pp. 10580–10592, 2002.[3] T.G. Brown, “ On the nature of the fundamental activity of the nervous centres; together with an analysis of the conditioning of rhythmic

activity in progression, and a theory of the evolution of function in the nervous system,” J. Physiol, vol. 48, no. 1, pp. 18–46, 1914.[4] T. Kohno and K. Aihara, “A MOSFET-based model of a Class 2 nerve membrane,” IEEE Trans. Neural Networks, vol. 16, pp. 754–773,

2005.[5] T. Kohno and K. Aihara, “Bottom-up design of Class 2 silicon nerve membrane,” J. Intell. Fuzzy Syst., vol. 18, pp. 465–475, 2007.[6] C. Koch and I. Segev Eds, “Methods in neuronal modeling: From ions to networks,” (MITPress, Cambridge, MA, 1998).

!"#$%&#'()&#"*%+,(-./.)/%&+(0%/1(/$2+'3.$(.+/$&"4(2+-(

5$2+,.$()2#'2*%/46(&$%,%+'(2+-(&7.$)&8%+,

98%/$4(!8%$+&7

!2$2/&7(:$2+)1(&3(;<=<(>&/.*?+%@&7(A+'/%/#/.(&3(B2-%&(.+,%+..$%+,(2+-(C*.)/$&+%)'(&3(/1.(B#''%2+(=)2-.84(&3(

!)%.+).'D(!2$2/&7D(B#''%2

!"#$%"&'(')*)+",+-*$*.+/(.+'-/)0*1+2'23#/(')

!"#$%&&'(

!"#$%&'(")*&+,%-&./"01"2334&,5"6'&,*',-"7#829:;"<*-.&.$.,"10%"2334&,5"6&=$4>.&0*-;"9?5,*-@&4; "6@&.A,%4>*5B"

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

5,#1#0/6&,.#0%/'9#A/#1.1'%#%*/#@&./7#%&#0%3.:#%*/#@&2'*&</,/0+0 # &? # @+%&6*&,.2+1 # ?&7.0"#I+%&6*&,.2+1#12/#&2<1,/77/0#+,#%*/#/3812:&%+6#6/77#6&,0+0%+,<#&?#1,#&3%/2#@/@-21,/#%*1%#/,67&0/0#1,#+,,/2 # &,/ # &? # 6&,0+./21-7: # 712</2 # 12/1" # )*/#@+0@1%6*#&? #12/10 # 7/1.0 # %&#6*1216%/2+0%+6 #?&7.+,<#'1%%/2,0 # &? # %*/ # +,,/2 # @/@-21,/9 # A*+6* # 12/#-/7+/>/.#%&#-/#/00/,%+17#?&2#%*/#'2&'/2#?3,6%+&,+,<#&?#%*/#&2<1,/77/"#J/#0%3.:#%*/#2&7/#&?#@/6*1,+60#+,#?&7. # </,/21%+&, # -: # 0+@371%+,< # %*/ # /43+7+-2+3@#0*1'/ # &? # %*/ # +,,/2 # @/@-21,/ # 10 # +%0 # 032?16/#+,62/10/0"#J/#?+,.#1#%21,0+%+&,#?2&@#1;+0:@@/%2+6#0'*/2+617 # +,>1<+,1%+&,0 # %& # 10:@@/%2+69#0%&@1%&6:%/=7+8/ # 0*1'/0 # EF+<" # BG9 # 1,. # ?&2 # />/,#*+<*/2#12/10#1#*+/2126*:#&?#+,>1<+,1%+&,0#&,#%&'#&?#/16*#&%*/2C"#H32#?+,.+,<0#12/#03''&2%/.#-:#2/6/,%#/;'/2+@/,%0#1,.#6&22&-&21%/#%*/#+@'&2%1,%#2&7/#&?#@/6*1,+60#+,#@+%&6*&,.2+1#@&2'*&</,/0+0"

F+<32/K # E%&'G #)21,0+%+&, #&? #1, #/;6/00 #6&,/ # ?2&@#0:@@/%2+6 # +,%& # 10:@@/%2+69 # 08/A/. # 0*1'/ # 10#6+263@?/2/,%+17 # <2&A%* # +,62/10/0 # E1=6GL # E-&%%&@G#I&2'*&7&<+617 # '*10/ # .+1<21@ # &? # 1 # @/@-21,/#+,0+./ # 1 # 0'*/2+617 # 61>+%: # +, # ./'/,./,6/ # &? # %*/#/,67&0/.#>&73@/#>#1,.#%*/#@/@-21,/#032?16/#1"

456678597:59;<

• M"N"#M/22@1,,9#F"O"#J+%%/79#5,0%+%3%/#?&2#P3+7.+,<#I1%/2+1709#Q)M#R32+6*9#$A+%S/271,.

• I"I"#IT77/29#H"#O1*21@1,9#5,0%+%3%#./#U*+@+/9#V*:0+43/#/%#./0#I1%W2+13;9#X,+>/20+%W#V137=Y/271+,/9#I/%S9#F21,6/

• I"#P/,#D@129#Z1-&21%&+2/#./#V*:0+43/#$%1%+0%+43/9#Q!$9#V12+09#F21,6/

9=>=9=?4=;<+

(#I"#P/,#D@12#1,.#["#V&@/13#V2&6"#\"#$&6"#Z&,."#D#]^C9 #_B`=_^^9 #(``_L# Q" #U/2.19 #Z" #I1*1./>1,9#1,.#N"#I"#V10+,+9#V!D$#(a(9#_9#Baa]L##Q"#U/2.1#1,.#Z"#I1*1./>1,9#V*:0"#\/>"#Z/%%"#`a9#_9#BaaC

B# !" # $%&&'9 # F"O" #J+%%/79 # I" # P/, # D@129 # I"I#IT77/29 #M"N" #M/22@1,,9 #V*:0" #\/>" #Z/%%" #(a^9 #b9#Ba(a

C#H"#O1*21@1,9#!"#$%&&'9#I"I"#IT77/29#QVZ#`_9#bcaac9#Ba(B

!"#$%&$'(%)'*+"',)#-&'*%'.$"'/%)"'*+#&'%&"'0-)-&12$'/-1+*'

.$"'/%)"'*+#&'%&"'3"4#5'$4#6"

!."3-'7*%%8'#&3'!#68+'9:'7*%%8';&$*-*.*"'%('<".)%-&(%)/#*-4$'=>?@AB?>'>C)-4+

D" ' $*.3-"3 ' *% ' 0+#* ' "E*"&* ' 4%)*-4#6 ' 4%6./&$ ' 0-*+ ' *+"-) ' 8#)*-4.6#) ' 0-)-&1F ' 4#& ' ,%%$* ' &".)#6'

4%/8.*#*-%&:'=8%&'#'G#$*'$.)G"5'%('4%6./&#)'&"*0%)H$'8")(%)/-&1'G#)-%.$')"#6I0%)63'4%1&-*-G"'

*#$H$F'0"'3"*"4*'&%'$-1&$'%('"&+#&4"/"&*:';*'-$'%&'#'/"$%$4%8-4J-&*")4%6./&#)J$4#6"'*+#*'*+"'

0-)-&1'#/%&1'*+"'4%6./&$F'6#)1"65'-))"$8"4*-G"'%('*+"-)'-&&")'%)1#&-K#*-%&F'"&+#&4"$'*+"'$8""3'%('

-&(%)/#*-%&'*)#&$(")'#&3'/-&-/-K"$'*+"'*%*#6'0-)-&1'6"&1*+')"L.-)"3'*%',-&3'3-$*)-,.*"3'4%6./&#)'

4%/8.*#*-%&$'*%0#)3$'$8#*-%*"/8%)#665'4%+")"&*')"$.6*$:

D"'$.11"$*'*+#*',)#-&'"((-4-"&45'/#5',"')"6#*"3'*%'#'3%.,65'()#4*#6'4%&&"4*-G-*5'6#0F')"$.6*-&1'-&

&"*0%)H$ ' 0-*+ ' "((-4-"&45 ' 8)%8")*-"$ ' ,"5%&3 ' *+%$" ' ,5 ' $4#6"I()"" ' &"*0%)H$ ' #&3 ' 0" ' "E+-,-*'

4%))%,%)#*-&1'"G-3"&4"'(%)'*+-$'$.11"$*-%&:'M"$8-*"'*+"'4.))"&*'"/8+#$-$'%&'$-/86")F'":1:F'4)-*-4#6F'

&"*0%)H$F'&"*0%)H$'0-*+'/%)"'*+#&'%&"'4%&&"4*-G-*5'3"4#5',"+#G-%)'/#5',"'*+"').6"')#*+")'*+#&'*+"'

"E4"8*-%&:

!"(")"&4"$

N: O"5%&3'74#6"IP)""'7/#66ID%)63'<"*0%)H$Q'R%)*-4#6'R%6./&$'(%)'S.-4H'O)#-&$

!" !#68+'7*%%8F'T-4*%)'7##$"F'R6"/"&$'D#1&")F'O)-**#'7*%%8F'#&3'!."3-'7*%%8F'U!9'NVWNVX'

YZVN[\

!"#$%"&%'()$*%"(*'#%+%#,(-./'#/*#%0"&1(%"&%23#&(-$04(&#*#5&(0-(

6%4%"%&356('0"&'%0/&(*7*$5"5&&

8"90(:*2.%*9/''3%(

;$*"<-/$#=(>5$4*",

!"($5'5"#(,5*$&=(75(3*+5(&55"(*(-.006(0-(?*?5$&(&#/6,%"2(%"#$%"&%'()$*%"(*'#%+%#,(6/$%"2(7*<5-/.($5&#(

@A$5&#%"2(&#*#5AB(*"6(%#&(/&5(*&(*(#00.(#0(/"65$&#*"6()$*%"(-/"'#%0"=(*&(75..(*&(#0(65+5.0?()%04*$<5$&(

-0$(6%--5$5"#(6%&5*&5&C(!"(&?%#5(0-(4*",(%"#5$5&#%"2($5&/.#&=(#35$5(%&(.%##.5('0"&5"&/&(*)0/#(#35(*"&75$(

0-(+5$,()*&%'(D/5&#%0"&($52*$6%"2(#35&5(&?0"#*"50/&=(%"#$%"&%'()$*%"(*'#%+%#,(-./'#/*#%0"&C(E3*#(%&(

#35%$ (0$%2%"F (E3*# ( %& ( #35%$ ( -/"'#%0"F (G$5( #35, ($5.*#56 ( #0 ('0"&'%0/& (*7*$5"5&& (*"6(&?0"#*"50/&(

#30/23#F(!"(#3%&(#*.<=(75(7%..(-%$&#(?$5&5"#(4*",(6%--5$5"#(?.*/&%).5(*"&75$&(#0(#3%&(D/5&#%0"&=(*"6(75(

7%..(#35"(&307(73*#(75('*"(.5*$"(*)0/#(#354(),(#35(&#/6,(0-()$*%"(*'#%+%#,(6/$%"2(#35(6%--5$5"# (

&#*#5&(0-(+%2%.*"'5(*"6(*7*$5"5&&(75(5H?5$%5"'5(*&(75(-*..(%"#0(655?(&.55?(5*'3("%23#C(G&(*($5&/.#(0-(

#3%&(&#/6,=(75(5H?5'#(#0(2*%"(%"&%23#&("0#(0".,(0"($5&#%"2(&#*#5()$*%"(*'#%+%#,(6/$%"2(7*<5-/.($5&#=()/#(

*.&0(0"(#35(45'3*"%&4&(73%'3(/"65$.%5(.0&&(0-('0"&'%0/&"5&&(%"(#35(65&'5"#(#0(655?(&.55?C(E5(7%..(

-%"*..,(6%&'/&&(#35(?$5+*.5"'5(0-(&.55?(0"(#35($5&#%"2(&#*#5(.%#5$*#/$5=(&307%"2(5+%65"'5(#3*#(*(.*$25(

?$0?0$#%0"(0-($5&#%"2(&#*#5(&#/6%5&(*$5(*'#/*..,(*--5'#56(),(&.55?('0"-0/"6&C

!"!#$%&'$"()$##*)&+$)&!,!-(./*&

0(!1.#.2!(.$"&$+(3*&0(*!,4&0(!(*0&$+&,4"!5.'!#&040(*50

!"#$%&'%(%)#*+,+-./0&12.+3&4256"%+.+30&75%+89%&:;(#"+#$/0&<9-$%3&'%(%)#*+,+;3

!"#$%#&'()*$&+,("#%+-).'/"(',"(01)2&#,&()3"()450-%+'$)6+%&#+&-)'#7)8&+5#"$"901

:%$#%;-1).%,5;'#%'

<&*%9+#.2&6=&%8%>.+*#&?6$.96"&.#?@$+A;#3&@%*#&B##$&8#3?9+B#8&+$&"+.#9%.;9#&=69&3.%B+"+C%.+6$&6=&

;$5$6D$&%$8E69&3"6D"2&*%92+$F&3.#%82&3.%.#3&G=+H#8&>6+$.3I&6=&$6$"+$#%9&82$%(+?%"&323.#(3J&

'@# & 3+(>"#3. & 6$# & +3 & .@# & 8#9+*%.+*# & =##8B%?5 &(#.@68 & >96>63#8 & KL & 2#%93 & %F6 & MNOJ &469#&

36>@+3.+?%.#8&(#.@683&#(>"62&#+.@#9&"6DP>%33&69&@+F@P>%33&=+93.&698#9&.9%?5+$F&=+".#93&+$&.@#&

=##8B%?5&"66>&MKPQOJ&R6D#*#90&.@#&8#9+*%.+*#&.#?@$+A;#&%$8&.@#&698+$%92&=+".#9&(#.@68&%9#&%B"#&

.6&3.%B+"+C#&.@#&;$3.%B"#&3>+9%"3&%$8&;$3.%B"#&$68#3&6$"2J&'@#2&=%+"&.6&?6$.96"&.@#&3%88"#&.2>#&

3.#%82&3.%.#30&B#?%;3#&6=&.@#&36P?%""#8&S688&$;(B#9&"+(+.%.+6$TJ&'6&F#.&%96;$8&.@#&>96B"#(&

U29%F%3&%$8&?6P%;.@693&3;FF#3.#8&;3+$F&%$&;$3.%B"#&=+".#9&MVO&+$3.#%8&6=&%$&698+$%92&3.%B"#&

=+".#9J & '@# & #==+?+#$?2 & 6= & .@# & (#.@68 & D%3 & 8#(6$3.9%.#8 & +$ & % & *%9+#.2 & 6= & (%.@#(%.+?%"0&

#"#?.96?@#(+?%"0&>@23+?%"&%$8&#"#?.96$+?&323.#(3&MV0&WX&GKLLYIOJ&<&(68+=+?%.+6$&6=&.@#&(#.@68&

@%3&B##$&>96>63#8&=69&3.%B+"+C%.+6$&6=&?6$3#9*%.+*#&323.#(30&#JFJ&%&3>%?#?9%=.&+$&.@#&Z%F9%$F#&

>6+$.&ZK&6=&.@#&7;$!!%9.@&%3.9682$%(+?%"&323.#(&MWX&GKLNLIOJ&'@#&(68+=+#8&.#?@$+A;#&(%5#3&;3#&6=&B6.@0&.@#&3.%B"#&%$8&.@#&;$3.%B"#&=+".#930&D695+$F&+$&>%9%""#"J

U9%?.+?%"&+(>"#(#$.%.+6$&6=&.@#&?699#3>6$8+$F&#"#?.96$+?&?6$.96""#93&9#A;+9#3&%&"%9F#&

3#.&6=&6>#9%.+6$%"&%(>"+=+#9&B%3#8&;$+.3[&+$*#9.6930&+$*#9.+$F&+$.#F9%.6930&+$*#9.+$F&%88#93&MWOJ&

7>#?+%"&?%9#&3@6;"8&B##$&.%5#$&D@#$&;3+$F&;$3.%B"#&=+".#93J&\$&.@#&>9#3#$.&>%>#90&D#&3;FF#3.&%&

3+(>"# & %8%>.+*# & #"#?.96$+? & =##8B%?5 & ?6$.96""#90 & D@+?@ & ?6*#93 & .@# & >633+B+"+.+#3 & 6= & (%$2&

?6$.96""#93&8#*#"6>#8&36&=%9&=69&%8%>.+*#&3.%B+"+C+$F&3.#%82&3.%.#3J&\.&+$?";8#3&=6;9&;$+.3&6$"2[&

.D6&+$3.9;(#$.%.+6$&%(>"+=+#93&G#JFJ&<]VKL&=96(&.@#&<$%"6F&]#*+?#3I0&%&3.%B"#&^_&?+9?;+.&%$8&

% & 3D+.?@J &'@# & "%..#9 & %""6D3 & .6 &?@663# &B#.D##$ & 3.%B+"+C+$F & 3>+9%"3E$68#3 &%$8 & 3%88"#3J &'@#&

?6$.96""#9&+.3#"=&@%3&%$&+$.#9$%"&=##8B%?5&"66>0&D@+?@&(%5#3&%&3.%B"#&^_&?+9?;+.&.6&B#@%*#&%3&

%$&;$3.%B"#&=+".#9J&'@#&6;.>;.&*%9+%B"#&;&6=&.@#&^_&?+9?;+.&+3&F+*#$&B2&.@#&8+==#9#$.+%"&#A;%.+6$&

;" &`&<!;a6=G;!<I0&D@#9#&<&+3&%$&%??#33+B"#&6;.>;.&*%9+%B"#&6=&.@#&323.#(0&=&+3&.@#&=##8B%?5&?6#==+?+#$.&G=)bNI0&6&+3&.@#&3D+.?@&>%9%(#.#9[&6)`&L&G6>#$I0&6)`&N&G?"63#8IJ&\$&.@#&?%3#&6=&6&`&L&

.@#&^_&?+9?;+.&+3&3+(>"2&%&3.%B"#&=+".#9[ &;" &`&<!;)=69&3.%B+"+C+$F&3>+9%"3&%$8&$68#3J&c@#9#%3&=69&6&`&N&+.&#==#?.+*#"2&B#?6(#3&%$&;$3.%B"#&=+".#9[&;" &`&G=!NIG;!<I&=69&3.%B+"+C+$F&3%88"#3J

7#*#9%" & #H%(>"#3 & 6= & 3.%B+"+C+$F &' ) >(%"(%& ;$5$6D$ & 3.#%82 & 3.%.#3 & G3>+9%"30 & $68#30&

3%88"#3I&6=&*%9+6;3&%;.6$6(6;3&$6$"+$#%9&323.#(30&+$?";8+$F&.@#&8%(>#8&];==+$F!R6"(#3&63?+""%.690 & .@# & >#9+68+?%""2 & 3>+5+$F & d+.CR;F@!e%F;(6 & 63?+""%.690 & .@# & ?@%6.+? & ];==+$F! R6"(#3!Z+$8B#9F & 63?+""%.690 & %$8& .@# & ?@%6.+? & Z69#$C & 63?+""%.690 & @%*# & B##$ & +$*#3.+F%.#8&%$%"2.+?%""20&$;(#9+?%""2&%$8&#H>#9+(#$.%""2J

MNO :+#"%D35+&7J0&:6;%C%6;+&4J0&]#96C+#9&]J0&1"69+#;H&UJ&U@23J&^#*J&!670&fKWV&GNggfIJ

MKO ^;"56*&eJ&dJ0&'3+(9+$F&ZJ&7J0&<B%9B%$#"&RJ&]J&\J&U@23J&^#*J&*890&fNh&GNgghIJ

MfO e%(%i-$%3&<J0&U29%F%3&jJ0&'%(%)#*+,+;3&<J&U@23J&Z#..J&!:960&KQQ&GNggQIX&\$.J&kJ&:+=;9?%.+6$&%$8&_@%63&

70&gQW&GNggWIJ

MhO _+6=+$+&4J0&Z%B%.#&<J0&4#;??+&^J0&1%"%$.+&4J&U@23J&^#*J&*;90&fgY&GNgggIJ

MQO :9%;$&]J&kJ&U@23J&^#*J&*7<0&LNVKNf&GKLLYIJ

MVO U29%F%3&jJ0&U29%F%3&lJ0&j+33&\J&mJ0&R;836$&kJ&ZJ&U@23J&^#*J&Z#..J&<=0&KhhNLf&GKLLKIX&U@23J&^#*J&*790&

LKVKNQ&GKLLhIJ

MWO '%(%)#*+,+;3&<J0&'%(%)#*+,+-./0&4256"%+.+3&1J0 &&,)'$J&U@23J&^#*J &*7<0&LKVKLQ&GKLLYIX&U@23J&^#*J &*<:0&

LKVKLQ&GKLNLIJ

!"#$%"&&'#()*+#!,%"#+)'#)-#)-%%-+)".)!"/0&12)"*!'&&-$"%*

!"#$%&'(%)%*+,-.-/&0'123-&'42567%-3-&0'87+$%'(%)%*+,-.-#39

!"#$%#&'()*$&+,("#%+-).'/"(',"(01)2&#,&()3"()450-%+'$)6+%&#+&-)'#7)8&+5#"$"901

:%$#%;-1).%,5;'#%'

:2$;<"6$-=%3-6$ '6> ';6/?7+@ '6&;-77%36"& '-& '% ';6))6$'6A&+",%3-6$ ' -$ '% ',%"-+32 '6> ' >-+7@& ' -$'

$%3/"+0'&;-+$;+'%$@'+$B-$++"-$BC'(<+'?<+$6)+$6$'<%&'A++$'D-@+72'-$,+&3-B%3+@'-$'?<2&-;%70'

+7+;3"6$-;0 ';<+)-;%70 'A-676B-;%70 '%$@';6))/$-;%3-6$ '&2&3+)&0 'D<+"+ ' -3 '<%& 'A++$' >6/$@' 36'

6;;%&-6$%772';%$'B-,+'"-&+'36'"%3<+"'&/"?"-&-$B'+>>+;3&C'!&'%$'+E%)?7+0'D<+"+%&'-$'3<+';6$3+E3'

6>'-$>6")%3-6$'?"6;+&&-$B'-$'A"%-$'&2$;<"6$2'-$'$+/"6$%7'$+3D6"5&'-&',+"2'-)?6"3%$30'%'366'

&3"6$B'&2$;<"6$-=%3-6$'6>'$+/"6$&')%2'+$@'/?'-$'+&&+$3-%7'3"+)6"'%$@'F%"5-$&6$G@-&+%&+G7-5+'

+>>+;3&C'H$'3<+'7%33+"';%&+0'3<+'&3%$@%"@'3<+"%?2'>6"'?%3-+$3&'-&'+7+;3"-;%7'@++?'A"%-$'&3-)/7%3-6$'

D-3<'&3"6$B'"+7%3-,+72'<-B<'>"+I/+$;2'JKLMM'N=O'?/7&+'3"%-$&C'P$>6"3/$%3+720'3<-&'3"+%3)+$3'-&'

6>3+$'%;;6)?%$-+@'D-3<'&-@+'+>>+;3&C'!'7%"B+'$/)A+"'6>'%@,%$;+@'>++@A%;5'%$@'$6$G>++@A%;5'

3+;<$-I/+&'36'%,6-@'&2$;<"6$-&%3-6$'6>'-$3+"%;3-$B'6&;-77%36"&'-$'B+$+"%70'%$@')6"+'&?+;->-;%772'

D-3<'3<+'?6&&-A7+'%??7-;%3-6$'36'$+/"6$%7'%""%2&0'<%,+'A++$'@+&;"-A+@'-$'7-3+"%3/"+0'+CBC'QLGRSC

H$'3<-&'?%?+"0'D+'@+&;"-A+'@+&2$;<"6$-=%3-6$'6>'%$'%""%2'6>'3<+'%&2))+3"-;'T-3=N/B<! U%B/)6 ' J%TNUO ' 6&;-77%36"& ' /&-$B ' % ' &-)?7+ ' >++@A%;5 ' 3+;<$-I/+C ' !$ ' -$@-,-@/%7 ' %TNU'

6&;-77%36" '<%&'A++$'@+&;"-A+@'+7&+D<+"+'QVS0 'D<+"+%&'3<+'?"6?+"3-+&0 '&?+;->-;%772 ' 3<+'?<%&+'

&2$;<"6$-=%3-6$0 ' 6> ' 3<+ ' $+3D6"5 ' 6> ' )+%$G>-+7@ ' ;6/?7+@ ' %TNU ' 6&;-77%36"& ' <%,+ ' A++$'

-$,+&3-B%3+@ ' -$ ' QWSC ' ! ' 3D6G3+")-$%7 ' >++@A%;5 ' ;6$3"677+" ' -& ' %??7-+@ ' 36 ' 3<+ ' $+3D6"5C ' (<+'

;6$3"677+"',-"3/%772'$/77->-+&'3<+',673%B+'%3'3<+';6/?7-$B'$6@+'%$@'+>>+;3-,+72'@+;6/?7+&'3<+'

-$@-,-@/%7'6&;-77%36"&C'X63<0'$/)+"-;%7'&-)/7%3-6$&'%$@'<%"@D%"+'+E?+"-)+$3&'%"+'?+">6")+@C'

(<+'"+&/73&'>6"'3<+'%""%2'6>'YM')+%$G>-+7@';6/?7+@'%TNU'6&;-77%36"&'%"+'?"+&+$3+@C

Z+'+)?<%&-=+0 '3<%3'3<+'@+,+76?+@'+7+;3"6$-;'>++@A%;5';6$3"677+"0 '-)?7+)+$3-$B'3<+'

)+%$G>-+7@ '"+&+33-$B 'J$/77->2-$BO ' 3+;<$-I/+0 ' -& '%$ '+&&+$3-%772 ' 3D6G3+")-$%7 '@+,-;+C '(</&0 ' -3'

@->>+"&'>"6)')6"+';6)?7-;%3+@'>6/"G3+")-$%7'>++@A%;5'@+&-B$&0'@+,+76?+@'&6'>%"'36'&/??"+&&'

$+/"%7'&2$;<"6$2C'(<+'>++@A%;5';6$3"677+"0'?"6?6&+@'-$'3<-&'D6"50'+)?762&'3<+'&%)+'&-$B7+'

$6@+'36')+%&/"+'3<+')+%$G>-+7@',673%B+'%3'3<+';6/?7-$B'$6@+'%$@'36'>++@'A%;5'3<+'&3-)/7%3-6$'

-$36'3<+'%""%2C'F2"%B%&'&,)'$<';7%-)+@0'3<%3'3<+'&-)/73%$+6/&'"+B-&3"%3-6$'%$@'&3-)/7%3-6$'6>'3<+'

D<67+ ' %""%2 ' -& ' -)?6&&-A7+ ' QRS ' %$@0 ' 3<+"+>6"+0 ' 3<+ ' "+?/7&-,+ ' ;6/?7-$B ' QYS ' %$@ ' &6)+ '63<+"'

>++@A%;5 ' 3+;<$-I/+& ' >%-7 ' 36 ' &/??"+&& ' 3<+ ' &2$;<"6$2C ' H$ ' ;6$3"%&30 ' D+ ' @+)6$&3"%3+ ' 3<+'

+>>+;3-,+$+&&'6>'3<+'>++@A%;5'3+;<$-I/+'36'@+&3"62'?<%&+'&2$;<"6$-=%3-6$'/&-$B'%'&-$B7+'$6@+0'

A63<'>6"'"+B-&3"%3-6$'%$@'&3-)/7%3-6$C'(<+')%-$'?<2&-;%7'%$@'+$B-$++"-$B')+;<%$-&)'A+<-$@'

3<+'?6&&-A-7-32'36';6$3"67'%'@2$%)-;%7'&2&3+)',-%'%'&-$B7+'$6@+'-&'3<%3'3<+';6$3"677+"'&+$&+&'3<+'

+7+;3"-; '+;((&#,[="$,'9&0 'A/3'>++@&'A%;5'3<+'+7+;3"-; '="$,'9&[+;((&#,C '(<+"+>6"+0 ' 3<+"+'-& '$6'

-$3+">+"+$;+'A+3D++$'3<+'3D6'@->>+"+$3'+7+;3"-;%7')+%&/"+&C

:6)+'63<+"'?6&&-A-7-3-+&'36';6$3"67'&2$;<"6$2'-$'3<+'%""%2&'6>'6&;-77%36"&'/&-$B'3<+'3D6G

3+")-$%7'>++@A%;5';6$3"677+"&'%"+'@-&;/&&+@'-$'%'"+;+$3'?%?+"'Q\SC

QLS ]6&+$A7/)'4C'1C0'F-56,&52'!C':C'F<2&C']+,C'^+33C'340'LLRLM_'J_MMROC

Q_S F6?6,2;<'`C'aC0'N%/?3)%$$'bC0'(%&&'FC'!C'F<2&C']+,C'^+33C'350'LWRLM_'J_MMVOC

QYS (&-)"-$B'^C':C0']/756,'UC'TC0'^%"&+$'4C'^C0'1%AA%2'4C'F<2&C']+,C'^+33C'360'MLRLML'J_MMVOC

QRS F2"%B%&'cC0'F6?6,2;<'`C'aC0'(%&&'FC'!C0'8F^'780'RMMM_'J_MM\OC

QVS (%)%*+,-.-/&'!C0'(%)%*+,-.-#39'8C0'42567%-3-&'1C0'X/)+7-+$9':C0'c-",%-3-&']C0':366?']C'^+;3/"+'U63+&'

b6)?C':;-<'69:70'WLd'J_MMeOC

QWS (%)%*+,-.-#39'8C0'42567%-3-&'1C0'(%)%*+,-.-/&'!C'U6$7-$+%"'!$%72&-&f'46@+77-$B'%$@'b6$3"67';90'LLd'

J_ML_OC

Q\S (%)%*+,-.-/&'!C0'(%)%*+,-.-#39'8C0'42567%-3-&'1C'!??7C'F<2&C'^+33C';8;0'__Y\MY'J_ML_OC

Effectiveness of Markov Codes with Negative Autocorrelationand Gaussian Chip Waveforms in FD/S3

Tomoaki Yorozuya†, Mikio Hasegawa‡, Yoshihiko Horio†, and Kazuyuki Aihara∗

† Graduate School of Engineering, Tokyo Denki University, Tokyo 120-8551, Japan. e-mail: 13kme57@ms.dendai.ac.jp

‡ Graduate School of Engineering, Tokyo University of Science, Tokyo 125-8585, Japan.

∗ Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan.

Abstract

A frequency division/spread spectrum system (FD/S3) isrobust against a large shift in frequency by using frequency-domain spread spectrum (SS) codes [1]. It was shown that acombination of Gaussian chip waveforms and Markov spread-ing codes, which have negative autocorrelations, improves thebit error rate (BER) in asynchronous code division multipleaccess (CDMA) systems [2]. In this paper, we evaluate the ef-fectiveness of the combination of the Markov codes with nega-tive autocorrelations and Gaussian chip waveforms in FD/S3.In FD/S3, a signal bandwidth is divided into P ′ bands with

bandwidth of F . The data bandwidth F is further dividedinto N ′ chip bands. The frequency domain SS codes are allo-cated to each chip. The received signal rFD(t;X′) and tem-plate waveform uFD(t;X′) are defined by

rFD(t;X′) =P ′−1∑

q′=0

dFDq′ uFD(t− t0;X′)

×ej(2π(q′F+f0)t+ϕFD) + η(t) + ξ(t), (1)

uFD(t;X′) =1√N ′

N ′−1∑

m′=0

X ′m′zFDm′ (t), (2)

zFDm′ (t) = z(t)ej(2πm′Fct+φFD

m′ ), (3)

where X′ = X ′m′ (X ′

m′ ∈ 1,−1), z(t), dFDq′ ∈ 1,−1,N ′, F , Fc, η(t), ξ(t), t0, f0, and ϕFD are frequency-domainSS codes, a chip waveform, a data symbol, the spreading fac-tor in the frequency domain, the symbol bandwidth, the chipbandwidth, noise, an interference, a time delay, a carrier fre-quency, and an initial phase, respectively. In the correlatorat a receiver, correlations between the received signal and thecarrier-modulated template waveform are calculated. Assum-ing the time delay and initial phase have been obtained as t0and ϕFD, respectively, the correlator output of FD/S3 can bedefined by [3]

cFDp′ (µ) =

∫ ∞

−∞rFD(t;X′)

×uFD(t− t0;Y ′)e−j(2π(p′F+µ)t+ϕFD)dt, (4)

where µ is a variable parameter for estimating the carrierfrequency f0, and Y ′ is an estimator for X′.We evaluate through simulations the effectiveness of

Markov codes as the SS codes, which have negative auto-correlations, in combination with Gaussian chip waveformsin FD/S3. We use a minimum deviation value of the signalcomponents mDV as a performance criterion, with which wecan differentiate the signal and noise components, defined by

mDV = min0≤p′≤P ′−1

DV (p′), (5)

100

105

110

115

120

125

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

!

a) Gauss, SNR=25 dB

c) Rectangular, SNR=25 dB

b) Sinc, SNR=25 dB

mDV

Figure 1: Minimum deviation value mDV averaged over 30trials when λ is varied.

with a deviation value of the p′th signal component given by

DV (p′) = 10× A(p′)−B

C+ 50, (6)

where A(p′) is the absolute value of the p′th signal compo-nent, B is the average of the absolute value of the signal andnoise components, and C is the standard deviation of the ab-solute value of the signal and noise components. Figure 1shows mDV as a function of λ, which corresponds to the au-tocorrelation of the Markov sequence used as the SS code. Weused the Kalman’s procedure to generate the Markov randomsequences with λ %= 0, while we used Chebyshev bit sequencesfor λ = 0 (i.i.d.). In the simulations, we assume a 2.4 GHz fre-quency band and a signal bandwidth of 4.8 MHz with P ′ = 5,N ′ = 64, Fc = 15 kHz, SNR = 25 dB, and t0 = t0. We useda) Gaussian, b) Sinc, and c) rectangular chip waveforms asindicated in Fig. 1.

Figure 1 shows that the combination of the Markov codeswith negative autocorrelations together with the Gaussianchip waveforms gives the largest mDV value. Therefore, thepeak of the signal components can be easily detected. As aconsequence, we can conclude that the combination of theMarkov codes with negative λ and the Gaussian chip wave-forms are advantageous in FD/S3.

References

[1] T. Kohda, Y. Jitsumatsu, K. Fujino, and K. Aihara, “Fre-quency division (FD)-based CDMA system which permits fre-quency offset,” in Proc. of 2010 Int. Symp. on Spread SpectrumTechniques and Applications, pp. 61–66, 2010 .

[2] Y. Jitsumatsu, M. Tahir Abbas Khan, and T. Kohda, “Gaus-sian chip waveform together with Markovian spreading codesimprove BER performance in chip-asynchronous CDMA sys-tems,” in Proc. of IEEE GLOBECOM, CD-ROM, 2006.

[3] T. Kohda, Y. Jitsumatsu, and K. Aihara, “Frequency synchro-nisation using SS technique,” in Proc. of Int. Symp. on WirelessComm. Syst., pp. 855–859, 2012.

A Synchronous Exponential Chaotic Tabu Search Algorithmin Quadratic Assignment Problems for Parallel HardwareImplementation with Electronic Circuits

Akihito TOYODA1, Yoshihiko Horio1 and Kazuyuki Aihara2

1 Graduate School of Engineering, Tokyo Denki University, 5 Senjuasahi-cho, Adachi-ku, Tokyo120-8551, Japan, Email :12kme35@ms.dendai.ac.jp2 Institute of Industrial Science, The University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo153-8505, Japan

AbstractChaotic tabu search algorithms [1] implemented using chaotic neural networks are effective for

solving quadratic assignment problems (QAPs). Hardware implementations of the chaotic tabusearch algorithms are substantially faster than software implementations, allowing large-scale QAPsto be solved within reasonable time frames. A synchronous updating algorithm for a hardwareimplementation that simultaneously updates all neuronal states has been proposed [2]. However,multiple neurons may simultaneously fire during synchronous updates, preventing the 2-opt exchangefrom selecting a single neuron. To solve this problem, several neuron selection methods have beenproposed that select one from among all neurons which have fired, and the effectiveness of thesemethods has been confirmed by simulations [2, 3, 4]. The neuron selection methods in [2, 3, 4] sortneurons according to their internal state values. In the case of a size-N QAP, the total numberof neurons is N2. Therefore, sorting all neuronal states is time-consuming when N is large, whichreduces the effectiveness brought by hardware speed. Moreover, high-precision (meaning large andexpensive) hardware is needed to distinguish between the infinitesimal differences among manyneuronal states. Therefore, in [5], a neuron selection method which does not rely on neuronal statesorting was proposed. However, this synchronous exponential chaotic tabu search algorithm is stillnot suitable for hardware implementation because of restrictions in analog electronic circuits.

In this paper, we propose an improved synchronous exponential chaotic tabu search algorithmfor quadratic assignment problems suitable for hardware implementation with mixed analog/digitalelectronic circuits. First, we modify the chaotic neuron model used in the synchronous exponentialchaotic tabu search. Next, we remove global couplings among neurons, which prevent the realizationof large-scale hardware-based tabu search systems. In addition, we can easily optimize theperformance of the system by adjusting the threshold values applied externally to the neurons.Numerical simulations are employed to confirm the effectiveness of the proposed method, where wedemonstrate the performance optimization by adjusting the threshold θ. Time evolutions of the costfunction and the internal states of neurons are also shown to illustrate the chaotic search dynamicsof the synchronous exponential chaotic tabu search.

References[1] M. Hasegawa, T. Ikeguchi and K. Aihara, “A novel chaotic search for quadratic assignment

problems,” European Journal of Operations Research, vol. 139, pp. 543–556, 2002.[2] N. Yokota, Y. Horio, K. Aihara and M. Hasegawa, “A modified synchronous exponential chaotic

tabu search for quadratic assignment problems,” Tech. Rep. IEICE, vol. NLP107, no. 561, pp.49–54, 2008.

[3] A. Shirakuma, Y. Horio and K. Aihara, “An improved neuron selection technique in anexponential chaotic tabu search hardware system for large quadratic assignment problems,”in Proc. Int. Workshop on Nonlinear Circuit and Signal Processing, CD-ROM, 2009.

[4] T. Kawamura, Y. Horio and M. Hasegawa, “Mutual information analyses of chaoticneurodynamics driven by neuron selection methods in synchronous exponential chaotic tabusearch for quadratic assignment problems,” Lecture Notes in Computer Science, vol. 6443, pp.49–57, 2010.

[5] A. Toyoda, Y. Horio and K. Aihara, “An improved exponential chaotic tabu search in quadraticassignment problems for parallel hardware implementation,” Tech. Rep. IEICE, vol. 112, no. 6,NLP2012-3, pp. 13–18, 2012.

A Comparative Study of Volterra-based NonlinearS-parameters and X-parameters

Christian Widemann, Harry Weber, Sebastian Stegemann, and Wolfgang Mathis,Institute of Theoretical Electrical Engineering

Leibniz Universität Hannover, D-30167, Hanover, Germany

Email: widemann, weber, stegemann, mathis@tet.uni-hannover.de

Abstract— In this contribution, two different approaches to theanalysis of nonlinear analog circuits using scattering variables areexamined, the nonlinear scattering parameters (S-parameters)and the X-parameters After the theoretical backgrounds areshortly recapitulated, the benefits and drawbacks of each methodare outlined. Both methods are applied to simple exemplarycircuits and the obtained results are compared.

I. EXTENDED ABSTRACT

In the last two decades, more and more research effortwas carried out in order to find an equivalent to the lin-ear S-parameters for nonlinear multiports under large signalexcitation. The S-parameter description of linearized N-portshave turned out to be convenient for several applications sincethe 1960s. Different approaches for behavioral modeling ofnonlinear circuits were developed as input-output descriptions.The so-called VIOMAP approach that is based on the Volterraseries was neglected for the benefit of the black box modelapproach based on describing functions [1] since it candescribe strong nonlinear behavior. Based on the describingfunctions, two approaches were established [2], the nonlinearscattering functions and the X-parameters. The latter werebrought to market by Agilent Technologies in combinationwith the emerging nonlinear network analyzer (NVNA) andare reviewed in section I-A. In contrast to these black boxmodel approaches, the extension of the Volterra series analysisof nonlinear systems with scattering variables was proposedby Weiner in 1976 [3] (cf. section I-B).

A. X-parameters

The X-parameters are based on the polyharmonic distortion(PHD) model approach [4]. Therein, it is assumed that thesystem is excited by single tone sinusoidal large signal at portone with the fundamental frequency f0 which is referred toas a11. This large signal defines the large signal operatingpoint (LSOP) of the N-port. Additional signals that occur atthe port q are assumed to be small signals at the nth orderharmonic frequency, i.e. n·f0, whose impacts on the N-port aresuperimposed linearly on the LSOP. The resulting scatteringvariable bpm at port p and the harmonic frequency m · f0 iscalculated by

bpm =∑

qn

Spq,mnP+m−naqn +

∑

qn

Tpq,mnP+m+na∗qn. (1)

Therein, the connecting X-parameters Spq,mn and Tpq,mn

depend only on the LSOP set by a11. Due to the phasenormalization P = e+jφa11 the X-parameters are functionsof the amplitude |a11|.

B. Volterra-based Nonlinear S-parameters

In 1976, Weiner presented how it is possible to describenonlinear dynamic systems with scattering variables based onthe Volterra series [3]. It was shown how the nonlinear scat-tering transfer functions can be calculated from the nonlinear

!"#$%#&'(

)*+"(,-./!0

1./!0

-./!0 12/!0-3/!0 -)/!0

43/!0

"3/!0

#3/!0

4)/!0

")/!0

#)/!0

Fig. 1. Nonlinear 2-port with frequency domain port variables.

voltage transfer functions HpS,n that describe the nth ordervoltage component at port p depending on the source voltageUS (cf. Fig. 1). As a result, the dependencies

S11,n(f1, ... , fn) =

2H1S,1(f1)− 1 n = 1

2nZ0,Sn−1

2 H1S,n(f1, ... , fn) n > 1

S21,n(f1, ... , fn) = 2nZ0,Sn

21

√

Z0,LH2S,n(f1, ... , fn) (2)

with n ≥ 1 for the latter were found for a nonlinear 2-port.These functions that are valid for general input signals weretransformed to the nonlinear S-parameters

S(p;1n),n(f1, ... , fn) =1

2n−1

(

n

q

)

Sp1,n(f1, ... , fn) (3)

valid for sinusoidal input signals. Therein, the multinomialcoefficient

(

nq

)

represents the frequency generation predictions.These nonlinear S-parameters are especially convenient for thematched case, i.e. ZS = Z0,S and ZL = Z0,L, since only oneterm is necessary to determine the nth order component of theb-variable at port p for the frequency f1 + ...+ fn by

bp,n(f1 + ...+ fn) = S(p;1n),n(f1, ... , fn)n∏

k=1

a1,1(fk) (4)

with a1,1(f) =US(f)

2√

Z0,S

.

C. Exemplary Comparison

These methods are both used for the analysis of simpleexemplary circuits and the results are compared with respectto the accuracy of the predicted output spectra. Furthermore,limitations of each method are discussed such as that thePHD model concept that leads to (1) is not applicable if theamplitudes of the small signals increase.

REFERENCES

[1] J. Verspecht, D. Schreurs, A. Barel, and B. Nauwelaers, “Black boxmodelling of hard nonlinear behavior in the frequency domain,” inMicrowave Symposium Digest, 1996., IEEE MTT-S International, vol. 3,1996, pp. 1735–1738 vol.3.

[2] G. Sun, Y. Xu, and A. Liang, “The study of nonlinear scattering func-tions and x-parameters,” in Microwave and Millimeter Wave Technology(ICMMT), 2010 International Conference on, 2010, pp. 1086–1089.

[3] D. Weiner and G. Naditch, “A scattering variable approach to the volterraanalysis of nonlinear systems,” Microwave Theory and Techniques, IEEETransactions on, vol. 24, no. 7, pp. 422–433, 1976.

[4] J. Verspecht and D. Root, “Polyharmonic distortion modeling,” Mi-crowave Magazine, IEEE, vol. 7, no. 3, pp. 44–57, 2006.

Investigation of Characteristics ofMulti-Layer Perceptron with Neurogenesis

Yuta Yokoyama†, Yoko Uwate† and Yoshifumi Nishio††Department of Electrical and Electronic Engineering, Tokushima University

2-1 Minami-Josanjima, Tokushima-shi, Tokushima, 770-8506, JapanEmail: yuta, ikuta, uwate, nishio@ee.tokushima-u.ac.jp

SUMMARYIt is said that there are about 10 billion neurons in the human’s brain. The network is formed by connecting of more than

one neuron. However, neurons had been considered to be lost with age until several years ago. It was impossible to generatenew neuron in the adult brain. This process is called “neurogenesis”. The neurogenesis in the hippocumpus of the human brainwas discovered in the late 1990s by Erickson et al [1]-[3]. Neurogenesis is that new neurons are generated in the human brain.We focus on characteristics of the neurogenesis with biologically. In the previous study, we have proposed artificial networkmodel which was applied the neurogenesis to Recurrent Neural Network (RNN) [4] and Multi-Layer Perceptron (MLP) [5].In this study, we investigate in more detail the influences of neurogenesis. We apply the behavior of neurogenesis to Multi-

Layer Perceptron (MLP) which is one of a feed-forward neural networks. Then, we consider characteristics of extinction andgeneration of neurons. We explain how to introduce generated neurons and extinct neurons. We use that the MLP is composedof three layers (one input, one hidden, and one output layer). In this network, we choose the neuron which is smallest action andnew neurons are generated there. At the same time, new neurons are generated in the hidden layer during the learning. Then,all the weights connecting to the generated neurons are newly set small random values. In this study, we assume the processto generated neurons and connection to “neurogenesis.” After that, the connection weights are newly calculated. Figure 1 (b)shows a structure of the proposed network. We consider that the proposed network is composed of three layers. The numberof neurons in the input layer is 25, and the output layer is 3. Then, we set that the conventional MLP has 5 neurons in thehidden layer. Therefore, the proposed MLP are set that number of neurons in the hidden layer increases.From Fig 1 (c), we compare the conventional MLP and the proposed MLP. We are able to obtain the good performance by

generating new neurons in the hidden layer. We considered that the characteristic of neurogenesis can be applied well.

Fig. 1. (a) conventional network, (b) proposed network, (c) result.

REFERENCES[1] S. Becker, J. M. Wojtowicz, “A Model of Hippocampal Neurogenesis in Memory and Mood Disorders,” Cognitive Sciences, vol. 11, no. 2, pp. 70-76,

2007.[2] R. A. Chambers, M. N. Potenza, R. E. Hoffman, W. Miranker, “Simulated Apotosis/Neurogenesis Regulates Learning and Memory Capabilities of

Adaptive Neural Networks,” Neuropsychopharmacology, pp. 747-758, 2004.[3] H. Satoi, H. Tomimoto, R. Ohtani, T. Kondo, M. Watanabe, N. Oka, I. Akiguchi, S. Furuta, Y. Hirabayashi and T. Okazaki, “Astroglial Expression of

Ceramide in Alzheimer’s Disease Brains: A Role During Neuronal Apoptosis,” Neuroscience, vol. 130, pp. 657-666, 2005.[4] Y. Yokoyama, T. Shima, C. Ikuta, Y. Uwate and Y. Nishio, “Improvement of Learning Performance of Neural Network Using Neurogenesis,” Proceedings

of RISP International Workshop on Nonlinear Circuits and Signal Processing (NCSP’12), pp. 365-368, Mar. 2012.[5] Y. Yokoyama, C. Ikuta, Y. Uwate and Y. Nishio, “Performance of Multi-Layer Perceptron with Neurogenesis,” Proceedings of International Symposium

on Nonlinear Theory and its Applications (NOLTA’12), pp.715 718, Oct. 2012.

!"#$%#&'()*('+,'$)-".&$/)"0)!&1("#'$)23#'4%+/

5&%#1$$')567)56'#'8'&9:);&$."/)<7)=">6'?1$"9

!"#$%&%'( ) *%+%,- ) .%/(01%" ) 21(34&5(/67 ) %"#$%&%'( ) !341847 ) 9:7 ) !";%/6 ) <=<<>?7 ) *%+%,-5/%17)

,%+@8A,+) )B@;%("AC0;) )7)D-014E)9F9G9>H99>:9I7)$%JE)9F9G9>H)99>:9I

K1 ) /-4 )L&4541/ )L%L4& )M4)58@@45/ ) /-4 )14M)101"(14%& );0N4" ) O0& ) /-4 )N45C&(L/(01 )0O )148&%")

14/M0&,5A )P6)854)0O)08&);0N4")M4)C%1)/%,4)(1/0)%CC081/)-(4&%&C-(C%")54"O#5(;("%& )L&0L4&/(45 )0O)

148&015A)P6)854)0O)M4""#,10M1)/-40&(45)0O)O&%C/%"5)M4)C-0054);(1(;%")5C%"4)0O);4%58&4;41/)F5(+4)

0O)C4""5)C034&(1@)%1)0'Q4C/H)(1N4L41N41/"6)01)3%"84)0O)N4O(1(1@);4%58&4A)$0&)/-4)N45C&(L/(01)0O)

430"8/(01)0O);4%58&4)(1)N4L41N41C4)8L01)@(341)L%&%;4/4&)0O)0&N4&)M-(C-)(5)N4/4&;(1(1@)3%&(%'"4)0O)

%)L-65(C%")L&0C4557)M4)C-0054)/-4)5C%"4)0O);4%58&4;41/)3(%)/-(5)L%&%;4/4&)%1N)N45(&4N);4%58&4A)

R41C47)O&%C/%");4%58&4)(5)%)101"(14%&)O81C/(01)N4L41N(1@)01)/-4)L&0C455A)S4)58@@45/)O0&;8"%5)O0&)

/-4)N45C&(L/(01)0O)101"(14%&)O&%C/%" );4%58&4A )P6)854)0O)/-4)O0&;8"%5)M4)C%1)4JL"(C(/"6)N45C&('4)

L&0C45545 ) (1 );8"/("%64& )148&%" )14/M0&,5A )!"50)M4)C%1)N45C&('4)3%&(%/(015)0O)%C/(01)L0/41/(%" )0O)

148&015)N4L41N(1@)01)4J/4&1%")L4&/8&'%/(015)%1N)/-&45-0"N)L0/41/(%"A)S4)-%34)5-0M1)/-%/)C-%0/(C)

5(@1%" ) O&0;)148&015)C%1)'4)N45C&('4N)'6)&4"%/(34"6) "%&@4)3%"845)0O) O&%C/%" )N(;415(01A )T/4LM(54)

C-%1@45)0O)L0/41/(%")"4%N)/0)&4(15/%"")3%"845)0O)L-%545A)S4)L&4541/)08&)&458"/5)0O)18;4&(C%")%1%"6545)

C0&&45L01N(1@)/0)M4"")U),10M1);0N4"5)%1N)4JL4&(;41/5)01)148&%")N61%;(C5)%1N)(1N(C%/4)&%1@45)0O)

L%&%;4/4&5)O0&)C0&&45L01N41C4)'4/M441)/-40&4/(C%")%1N)4JL4&(;41/%")N%/%A)

V-%1@(1@)0O)54//(1@5)(1)/-4)5L4C(O(4N)(1/4&3%"5)/-40&4/(C%""6)%""0M5)85)/0)N45C&('4)%;L"(/8N47)

O&4W841C67 ) C-%0/(C ) O0&;7 )L-%54 )C-%1@47 ) %C/(01 )L0/41/(%" ) 0O ) 148&0157 );4%58&4N ) (1 ) /-4 ) &454%&C-A)

X%&(085 );0N4"5 ) 0O ) /-454 ) L%//4&15 ) %&4 ) 5-0M1 ) (1 ) /-4 )M0&,5A )Y-4 ) /-40&6 ) N45C&('45 ) /-4 ) 3%&(085)

L&0L4&/(45)0O)148&015E)%CC8;8"%/(01#'8&5/(1@7)%56;;4/&(C)(1/4&;(//41C67)C-%0/(C7);8"/("%64&5)(1)%)

-(4&%&C-6)%1N)54"O#5(;("%&(/67)N4L0"%&(+%/(017)L-%54)&4(15/%""7)101#(50C-&01(5;)FC-%1@(1@)/-4)L-%54)

/&%Q4C/0&(45HA