1
Explo a ion o spa ial- empo al dynamic phenomena in a 32×32-cells
s o ed p og am 2-laye CNN Uni e sal Machine Chip P o o ype
Is án Pe ás1, Csaba Rekeczky1, Tamás Roska1, Rica do Ca mona2, F ancisco Jiménez-Ga ido2,
Angel Rod íguez-Vázquez2
1 Analogical and Neu al Compu ing Labo a o y, Compu e and Au oma ion Resea ch Ins i u e, Hunga ian Academy o
Sciences, Kende u.11, Budapes , 1111 - Hunga y
2Ins i u o de Mic oelec ónica de Se illa-CNM-CSIC
A da. Reina Me cedes s/n, 41012 Se illa (SPAIN)
Tel. +34955056666 Fax. +34955056686
Abs ac . This pape desc ibes a ull-cus om mixed-signal chip ha embeds digi ally p og ammable analog
pa allel p ocessing and dis ibu ed image memo y on a common silicon subs a e. The chip was designed
and ab ica ed in a s anda d 0.5µm CMOS echnology and con ains app oxima ely 500,000 ansis o s. I
consis s o 1024 p ocessing uni s a anged in o a 32×32 g id. Each p ocessing elemen con ains wo cou-
pled CNN co es, hus, cons i u ing wo pa allel laye s o 32×32 nodes. The unc ional ea u es o he chip
a e in acco dance wi h he 2nd O de Complex Cell CNN-UM a chi ec u e. I is composed o wo CNN
laye s wi h p og ammable in e - and in a-laye connec ions be ween cells. O he ea u es a e: cellula , spa-
ial-in a ian a ay a chi ec u e; andomly selec able memo y o ins uc ions; andom s o age and e ie al
o in e media e images. The chip is capable o comple ing algo i hmic image p ocessing asks con olled by
he use -selec ed s o ed ins uc ions. The in e nal analog ci cui y is designed o ope a e wi h 7-bi s equi a-
len accu acy. The physical implemen a ion o a CNN con aining second o de cells allows eal ime ex-
pe imen s o complex dynamics and ac i e wa e phenomena. Such well-known phenomena om he eac-
ion-di usion equa ions a e a eling wa es, au owa es, and spi al-wa es. All o hese ac i e wa es a e
demons a ed on-chip. Mo eo e his chip was speci ically designed o be sui able o he compu a ion o
biologically inspi ed e ina models. These compu a ional expe imen s ha e been ca ied ou in a de elop-
men en i onmen designed o es ing and p og amming he analogic (analog-and-logic) p og ammable a ay
p ocesso s.
1 In oduc ion
Va ious phenomena o he wo ld a ound us a e he esul o local in e ac ions o pa i-
cles. This is ue o bo h mic o and mac o wo ld, o molecules, cells and e en o
membe s o a popula ion. Two la ge classes o such in ini ely many phenomena a e he
pa e n o ma ion in biology (e.g. spo s o animals o plan s) o in chemis y and ac i e
wa e p opaga ion (e.g. ac ion po en ial in he ne ous sys em, p opaga ion o ca diac
muscle exci a ion, blood coagula ion p ocess). Fig. 1 shows examples o pa e n o ma-
ion and ac i e wa e phenomena.
The exp ession “ac i e wa e” means ha he wa e p opaga ion occu s in an “ene ge i-
cally ac i e” medium, i.e. he ene gy conse a ion law does no hold because he media
can injec ene gy in o he sys em, he e o e he ampli ude and he wa e o m a e p e-
se ed du ing he p opaga ion. These a e he mos impo an di e ences when compa ed
wi h classical wa es. To make any easonable complex pa e n o ma ion, we need he
local ac i i y o he cells [34].
Up o now he ypical way o ca y ou expe imen s ela ed o hese phenomena was
ei he labo a o y wo k (e.g. wi h chemicals) o h ough ime-consuming compu e simu-
la ion. O he op ion is o build dedica ed ha dwa e ci cui y. Al hough i is as , i s disad-
an age is he lack o easy p og ammabili y.
The Complex Cell CNN p og ammable a ay compu e [5][11] is an ex ension o he
CNN Uni e sal Machine [1]-[4]. The mul ilaye o highe o de elemen a y cells a e o -
2
ganized in o a 32×32 squa e g id. Each cell has second o de dynamics and local in e -
connec ions o i s neighbo s. This s uc u e is especially sui able o compu ing a ce ain
se o o dinal di e en ial equa ions. Simple PDEs can be ans o med so ha hey can be
easily p og ammed on he a ay compu e . Wi h he p og ammable Complex Cell CNN
Uni e sal Machine scien is s a e p o ided wi h a unique oppo uni y o s udy ac i e-wa e
p opaga ion and o he eac ion-di usion di e en ial equa ion based phenomena in a
p og ammable manne . Exploi ing he inhe en p og ammabili y o he CNN a chi ec-
u e, complex wa e-compu ing analogic algo i hms can be designed.
In his pape we show how o ans o m he ma hema ical model o he phenomena
ou lined abo e in o a o m sui able o pa allel p ocessing on he Complex Cell CNN
chip and we p esen he i s measu emen esul s.
Fig. 1: spi al- and au owa es and pa e ns measu ed on he CACE1k chip (image size: 32×32)
The ollowing sec ion desc ibes he ma hema ical model o he co e o he complex cell
chip. In Sec ion 3 he p o o ype chip is in oduced. Sec ion 4 desc ibes he ela ion be-
ween he eac ion-di usion equa ions and he s uc u e o he complex cell chip. Finally,
Sec ion 5 con ains he eal- ime chip measu emen s. We should emphasize ha no simu-
la ion esul s a e p esen ed he e. This chip was designed also o compu ing biologically
inspi ed e ina modeling. The implemen a ion o he main e ec o a e ina model will be
published in he nea u u e elsewhe e.
2 Compu a ional a chi ec u e
The a chi ec u e o he chip ollows he design o he i s o de CNN-UM chip [10], bu
i s i s o de cell co e is eplaced by a second o de one [5]. Wi h his change we a e able
o ep oduce basic ac i e wa e phenomena (See Fig. 1). Fig. 2 shows he unc ional
a chi ec u e o he 2nd o de , wo laye CACE1k co e.
in
p
u
la
y
e 1
la
y
e 2
a12 a21y2A1x1
A2x2x2
x1
τ
ττ
τ1
τ
ττ
τ2
z1
b2u2
z2
b1u1
τ
ττ
τ1 ≥
≥≥
≥ τ
ττ
τ2 ou
p
u
Fig. 2: Func ional diag am o he CACE1k chip
The e olu ion law o he complex cell CNN is he ollowing di e en ial equa ion sys em:
3
()
()
10,10,1,1
)()()(
)(
)()()(
)(
,22,121,2,2,2
,2
2
,11,212,1,1,1
,1
1
1
1
≤≤≤≤≤≤≤≤
++++−=
++++−=
∑
∑
∈
∈
ijij
ijijij
Nkl
klklij
ij
ijijij
Nkl
klklij
ij
xuMjMi
zub xa xA xg
d
dx
zub xa xA xg
d
dx
τ
τ
(2.1)
Whe e:





−<−
≤
>
=∞→
1
1
1
lim)(
,,
,,
,,
,
ijnijn
ijnijn
ijnijn
m
ijn
xi mx
xi x
xi mx
xg (2.2)
Func ion g(.) egula es he s a e so ha i s ays wi hin he +1..-1 in e al. Va iables u1, u2
a e he independen inpu s, b1, b2 a e hei weigh ac o s espec i ely; z1,ij, z2,ij a e space
a ian bias maps.. Va iables x1, x2 deno es he s a e a iables o he laye s. Each xij co -
esponds o one cell; i is one pixel o a M by N image i we conside he sys em’s ou pu
as a pic u e. A1, A2 a e he weigh s o he in a-laye couplings, a1, a2 a e he in e -laye
weigh s. Equa ion (2.1) u ilizes he so-called ull- ange model (FSR)[7], whe e he ol age o
he s a e a iable is always he same as he ou pu .
3 Chip desc ip ion
3.1 A chi ec u e o he chip and basic p ocesso s uc u e
The p o o ype chip consis s o an analog p og ammable a ay p ocesso o 32×32 iden i-
cal cells (Fig. 3), su ounded by he bounda y condi ions o he CNN dynamics. The e is
also an I/O in e ace, a iming and con ol uni and a p og am memo y. The in e ace
consis s o a se ializing-dese ializing analog mul iplexo . The p og am memo y is com-
posed o he analog p og am memo y, and he logic p og am memo y and he swi ch
con igu a ion egis e s. The analog ins uc ions and e e ence signals need o be ansmi -
ed o e e y cell in he ne wo k in he o m o analog ol ages. Thus, a bank o D/A
con e e s in e aces he analog p og am memo y wi h he p ocessing a ay. Dis ibu ing
analog e e ences ac oss la ge dis ances wi hin a chip is no a i ial ask. Apa om he
p oblems caused by elec omagne ic in e e ence, ol age d ops in long me al lines ca y-
ing cu en s can be qui e no iceable. Thus, in o de o a oid his signal bu e ing and
low- esis ance pa hs mus be p o ided. Finally, he iming uni is made o an in e nal
clock/coun e and a se o FSMs ha gene a e he in e nal signals which con ol he
p ocesses o images up/downloading and p og am memo y accesses.
4
Fig. 3: A chi ec u e o he chip
E e y elemen a y p ocesso o he Complex Cell CNN p og ammable a ay chip in-
cludes wo coupled con inuous- ime CNN co es (Fig. 4(a)) belonging o each o he wo
di e en laye s o he ne wo k. The synap ic connec ions be ween p ocessing elemen s
o he same o di e en laye a e ep esen ed by a ows in he diag am. The basic p oc-
esso also con ains a p og ammable local logic uni (LLU) and local analog and logic
memo ies (LAMs and LLMs) o s o e in e media e esul s. All he blocks in he cell
communica e ia an in a-cell da a bus, which is mul iplexed o he a ay I/O in e ace.
Con ol bi s and swi ch con igu a ion a e passed o he cell di ec ly om he global p o-
g amming uni .
(a) (b)
Fig. 4: Concep ual diag am o he (a) basic cell and (b) CNN laye node
The in e nal s uc u e o each CNN co e is depic ed in Fig. 4(b). Each co e ecei es con-
ibu ions om he es o he p ocessing nodes in he neighbo hood which a e summed
and in eg a ed in he s a e capaci o . The wo laye s di e in ha he i s laye has a scal-
able ime cons an , con olled by he app op ia e bina y code, while he second laye has
a ixed ime cons an . The e olu ion o he s a e a iable is also d i en by sel - eedback
and by he eed o wa d ac ion o he s o ed inpu and bias pa e ns. The e is a ol age
5
limi e o implemen ing he FSR CNN model [7]. The s a e a iable is ansmi ed in
ol age o m o he synap ic blocks, in he pe iphe y o he cell, whe e weigh ed con-
ibu ions o he neighbo s’ a e gene a ed. The e is also a cu en memo y ha will be
employed o cancella ion o he o se o he synap ic blocks. Ini ializa ion o he s a e,
inpu and/o bias ol ages is done h ough a mesh o mul iplexing analog swi ches ha
connec o he cell’s in e nal da a bus.
3.2 P o o ype chip da a
The p o o ype chip has been designed and ab ica ed using 0.5µm single-poly iple-me al
CMOS echnology. I s dimensions a e 9.27 x 8.45 mm2 (Fig. 5). The cell densi y achie ed
is 29.24cells/mm2. The ime cons an o he laye s is a ound 100ns (unscaled). The p o-
g ammable dynamics o he chip pe mi he obse a ion o di e en phenomena o he
ype o p opaga ion o wa es, pa e n gene a ion, e c (See Sec ion 5 o examples). Table
1 summa izes he mos ele an da a o he p o o ype chip.
Technology 0.5
µ
m CMOS 1-P 3-M
Numbe o cells 32 x 32
Die a ea 9.27 x 8.45 mm2
Die a ea (w/o pads) 8.77 x 7.94 mm2
A ay a ea 5.98 x 5.93 mm2
Package ce amic PGA – 100
Powe supply ol age 3.3V
Logic "0" / Logic "1" 0V / 3.3V
Accu acy on he weigh s 8b
Image samples esolu ion 7-8b
I/O a es 10Ms/s
CNN ime cons an below 100ns
Table 1: Cha ac e is ic pa ame e s o he CACE1k p o o ype chip
Fig. 5: Mic opho og aph o he p o o ype chip

6
4 Complex cell CNN-UM chip as a “PDE sol e ” – ypical wa es
4.1 Gene al PDE o mula ion
Le us conside he ollowing PDE o mula ion o a coupled ec o alued nonlinea
eac ion-di usion sys em (examples o nonlinea PDEs could be ound in [26]- [32]):
11 11111012
22 22222021
10 10 1 1
(,) [ (,) ( (,))] ( (,)) ( (, )) ( (,))
(,) [ (,) ( (,))] ( (,)) ( (, )) ( (,))
(, ) (); (,) (| ( (,
dx di c x g ad x x x x
d
dx di c x g ad x x x x
d
x x c x g g ad x
φφαφβφγφ
φφαφβφγφ
φφ φ
−=++
−=++
==
&& & & & &
&& & & & &
&&&
*
20 20 2 2
)) | )
(, ) (); (,) (| ( (,))|)y x c y g g ad x
φφ φ
==
&&&
*
(4.1)
whe e 12
[(,) (,)]x x
φφ

can be in e p e ed as he ime e olu ion o a ec o alued
image in ensi y ( 10 20
[() ()]xx
φφ

a e he ini ial images), he ec o

x
ep esen s he spa ial
coo dina es, he ime a iable can also be in e p e ed as he scaling pa ame e and c is
he conduc ance pa ame e in he di usion e m. The igh hand side o he equa ions con-
sis s o h ee eac ion e ms: (i)
α
(.) is he “sel - eac ing e m”; (ii)
β
(.) is a “spa ial con-
s ain ” (calcula ed om he ini ial da a); and (iii)
γ
(.) is he “c oss-coupling” e m. Some
p oposed unc ions o g(.) a e:
()
()
gg adIx KK
gg adIx K K
1
22
2
11
0
100
=− >
=+




>>
+−
exp ( ( , )) / ,
((,)) / , ,

αα
(4.2)
Decoupling he wo equa ions (γ1 = γ2 = 0) and choosing
α
(ξ) = - ξ he PDE o mula-
ion is gi en in he o m o he so-called “biased” nonlinea aniso opic di usion equa-
ion (No ds öm [27]) an ex ended e sion o he Pe ona-Malik o mula ion [26]. Ge ig
e . al [29] i s p oposed he applica ion o coupled nonlinea di usion sys ems o ec-
o - alued image p ocessing ha has been u he s udied by many o he s [32]. Fo ou
pu poses – mo i a ed by silicon implemen a ion – we ocus on a simpli ied e sion o
(4.1) ixing he di usion pa ame e o a cons an alue.
4.2 Spa ial disc e isa ion
In wo spa ial dimensions and assuming c = cons . (4.1) and (4.2) educes o:
11 1 11 11012
22 2 22 22021
( , , ) [ ( ( , , ))] ( ( , , )) ( ( , , )) ( ( , , ))
( , , ) [ ( ( , , ))] ( ( , , )) ( ( , , )) ( ( , , ))
dxy cdi g ad xy xy xy xy
d
dxy c di g ad xy xy xy xy
d
φφαφβφγφ
φφαφβφγφ
−=++
−=++
(4.3)
A e spa ial disc e isa ion using he ini e di e ence app oach one ob ains:
1, 1
1 2, 1 1, 1 1, 1, 1 1, 1 1, 1 1, 1 1 10,
2, 2
2 1, 2 2, 2 2, 2, 1 2, 1 2, 1 2,
() ( ( )) ( ( )) ( ) ( ( ) ( ) ( ) ( )) ( )
4
() (()) (()) () ( () () ()
4
ij
ij ij ij i j i j ij ij ij
ij
ij ij ij i j i j ij ij
d c
c
d
d c
c
d
φ
γφαφ φ φφφφ βφ
φγφαφ φ φφφφ
−+ −+
−+ −+
=+−+ ++++
= + −+ +++
1220,
()) ( )
ij
βφ
+
(4.4)
Finally, wi h
α
(.)=
α
0 and
β
(.) =
β
0 a simpli ied o m (wi h spa ial symme y and iso -
opy) o he CNN complex cell equa ion is de i ed:
7
1, 1
1 2, 1 1 1, 1, 1 1, 1 1, 1 1, 1 1,
2, 2
2 1, 2 2 2, 2, 1 2, 1 2, 1 2, 1 2,
1, 1 10,
() () ( ) () ( () () () ())
4
() () ( ) () ( () () () ())
4
ij
ij ij i j i j ij ij ij
ij
ij ij i j i j ij ij ij
ij ij
d c
c z
d
d c
c z
d
z
φ
γφ α φ φ φ φ φ
φγφ α φ φ φ φ φ
βφ
−+ −+
−+ −+
=+− + ++++
=+− + ++++
=12,220,2
;ij ij
zz z
βφ
+=+
(4.5)
Making
φ
ij explici ly depend on a s a e a iable
ξ
ij such as
φ
ij= (
ξ
ij) = sigm(
ξ
ij) leads a
good app oxima ion o he ull- ange CNN ci cui model ([7], sigm(.) could be a piece-
wise linea o mono onic con inuous smoo h unc ion playing he ole o a signal-
limi e ). The o m o he co esponding 2nd o de CNN empla e is as ollows:
1
11 1 1 1 1 12 1 11 1 1 1
1
2
22 2 2 2 2 21 2 22 2 2 2
2
0/40
/4 /4 ; ; ; 1; 0
0/40
0/40
/4 /4 ; ; ; 1; 0
0/40
c
Ac cc A B z
c
c
Ac cc A B z
c
αγβτ
αγβτ


=− = = = =





=− = = = =



(4.
6)
Analysis: I should be no ed ha he diagonal e ms could also be added esul ing in a
much be e spa ial iso opy du ing he di usion p ocess. Symme y and spa ial iso opy
o hese empla es a e due o he ac ha he e a e no con ec ion e ms in he o iginal o -
mula ion (4.1)–(4.2). Wi h hese modi ica ions he sign and magni ude o he empla e
en ies in (4.6) could be changed in any spa ial di ec ions. The ime cons an is also ixed
and equal o he wo laye s. Ha ing wo di e en ime a iables (say '' ≠ ') in (4.1)
leads o a "double ime-scale" in he desc i ized sys em '' / ' =
τ
2 /
τ
1 ≠ 1 which has
p ac ical ele ance when looking a a ious second o de models.
4.3 Typical wa e classes
The e a e a numbe o pa ame e se ings o special in e es ha lead o e y di e en
quali a i e beha io s in his symme ic second o de sys em. We ha e examined and ex-
plo ed p ima ily he ollowing simple cases (see also Sec ions 5.2):
(i) igge -wa e gene a ion (di usion p ocess in "sa u a ion" a a ious speeds)
c1 > 0 ; c2 > 0 ;
α
1-(c1+1)>0;
α
2-(c2+1)>0;
γ
1 =
γ
2 = 0;
β
1 =
β
2 = 0;
τ
2 /
τ
1 ≠ 1
(ii) a eling-wa e, spi al-wa e and au o-wa e gene a ion (spa ially in e ac ing ig-
ge -wa es)
c1 > 0 ; c2 > 0 ;
α
1-(c1+1)>0;
α
2-(c2+1)>0;
γ
1 ≠ 0;
γ
2 ≠ 0;
β
1 ≠ 0;
β
2 = 0;
τ
2 /
τ
1 ≠ 1
(iii) e ina e ec s (spa ially in e ac ing ecep i e ields)
α
1 ≈ c1 >0;
α
2 ≈ c2 > 0;
γ
1 > 0;
γ
2 < 0;
β
1 ≠ 0;
β
2 ≠ 0;
τ
2 /
τ
1 ≠ 1
5 Measu emen and compu a ion examples
In he ollowing we p esen chip measu emen o he di e en ac i e wa e phenomena:
igge wa e, a eling wa e, au owa e, and spa io- empo al edge de ec ion. The mo ie
iles o he measu emen s a e a ailable a [33]
5.1 S o ed p og ammabili y
The CACE1k chip can be p og ammed using he so-called AMC (ANALOGIC MACHINE
CODE) language. This is simila o he assemble language bu i con ains addi ional high-
le el ins uc ions, such as: image loading, ame g abbe ins uc ions, e c. and buil in im-
8
age p ocessing unc ions. In he Appendix is a simple AMC p og am which compu es
he snapsho s o he au owa e example (Subsec ion 5.2.4). The ac ual compu a ion is
pe o med in line 40. Be o e his line is he p epa a ion o he compu a ion: se ing up
he pa ame e s. A e he compu a ion is done, he esul s a e ead ou and displayed
om he chip’s local analog memo ies.
The elemen a y p og am o he chip is he empla e. I con ains he weigh ac o s o
he coupling be ween he cells and weigh s o he inpu and he bias map.
16..1,1:τ
2121
1221
2
1,1
2
0,1
2
1,1
2
1,0
2
0,0
2
1,0
2
1,1
2
0,1
2
1,1
2
1
1,1
1
0,1
1
1,1
1
1,0
1
0,0
1
1,0
1
1,1
1
0,1
1
1,1
1
==










=










=
−
−
−−−−
−
−
−−−−
τ
21 τ:τ
AA
zzbb
aa
aaa
aaa
aaa
aaa
aaa
aaa
(5.1)
The ope a ion o he a ay compu e is comple ely de e mined by he 25 empla e alues,
he ini ial s a es and bounda y condi ions. In (5.1) A1 and A2 ma ices deno es he weigh s
in e laye connec ions o he slowe and he as e laye espec i ely. The s eng h o he
in luence o he second laye on he i s is con olled by a21 and a12 s ands o he e e se
case. Symbols b1, b2, z1 and z2 a e he weigh s o he independen inpu s and he space
a ian bias maps. The a io o he ime cons an s o he wo CNN laye s a e con olled
by τ1, τ2 is ixed. An analogic algo i hm is made up o combina ion o empla e execu-
ions, logic ins uc ions and spa ial a i hme ic ope a ions. The example shown in he Ap-
pendix is a e y simple one, i con ains only empla e execu ion ope a ion.
5.2 Wa e phenomenon
This subsec ion con ains he on-chip measu emen s o he ac i e wa e phenomena.
5.2.1 T igge wa e
This example shows a e y simple e ec : an ac i e (nonlinea ) wa e in he as e laye
ini ia es a second wa e in he slowe laye h ough he posi i e in e laye coupling. The
inpu is ed o he as e laye . Fig. 6 shows snapsho s o he wa e e olu ion.
1:165.16.030
03.3
6.08.06.0
8.06.08.0
6.08.06.0
111
15.11
111
2121
122121
=−====
==










−=










=
21 τ:τ
AA
zzbb
aa (5.2)
Fig. 6: T igge wa e p opaga ion in he as e and in he slowe laye . The i s ilm shows he ou pu o
he as e laye .
9
5.2.2 E asu e e ec
This example is an “enhancemen ” o he igge wa e example: a nega i e coupling is
in oduced om he slowe laye back o he as e one. As a esul , a e a while, he
igge wa e on he as e laye is e ased by he igge wa e om he slowe laye . As in
he case o he igge wa e he inpu is gi en o he as e laye . Fig. 7 shows he snap-
sho s o he wa e e olu ion on he as e laye and Fig. 8 shows he e olu ion on he
slowe laye .
1:167.26.030
4.53.3
6.08.06.0
8.06.08.0
6.08.06.0
111
15.11
111
2121
122121
=−====
−==










−=










=
21 τ:τ
AA
zzbb
aa (5.3)
Fig. 7: Re ina like wide- ield e asu e e ec , as e laye .
Fig. 8: Re ina like wide- ield e asu e e ec , slowe laye .
5.2.3 T a eling wa e
The cha ac e is ic o his wa e is ha a single wa e on a els ac oss he ac i e me-
dium. The au owa e is simply o med om he igge wa e empla e by adding a nega-
i e in a-laye coupling om he slowe o he as e laye . In Fig. 9 obse e he annihila-
ion p ope y o his wa e ype.