T ansien Ac i a ion in a Ne wo k o Coupled Map Neu ons
J. M. Casado*
A
´ eadeFı
´sica Teo
´ ica, Uni e sidad de Se illa, Apa ado Co eos 1065, 41080, Se illa, Spain
(Recei ed 23 June 2003; published 10 No embe 2003)
The ocus o his Le e is on he ac i i y o a ne wo k o neu ons pai wise coupled by inhibi o y
connec ions. Each neu on is ep esen ed by a wo-dimensional map capable, when isola ed, o a ich
a ie y o complex dynamical egimes. I is shown ha he ne wo k exhibi s a s imulus-dependen
sequen ial ac i a ion and inac i a ion o subg oups o neu ons. This complex beha io is a he simila
o some spa io empo al ea u es obse ed in he i s s ages o he ol ac ion p ocess in some insec s and
sugges s he possibili y o la ge scale simula ion o hese p ocesses by using easonable compu a ional
capabili ies.
DOI: 10.1103/PhysRe Le .91.208102 PACS numbe s: 87.18.Sn, 05.45.Ra, 89.75.Hc
The unde s anding o how he senso y wo ld is ep e-
sen ed in he elec ical ac i i y o he b ain is one o he
undamen al asks o neu oscience [1,2]. By now, i is
well known ha he in o ma ion coming om senso y
signals can be encoded in o complex pa e ns o neu onal
ac i i y: each s imulus is ep esen ed by a speci ic and
highly ep oducible sequence o i ing ac oss some spe-
ci ic neu ons [3,4]. To analyze he neu onal dynamics ha
gi es ise o he odo -encoding capabili ies o some in-
e eb a es, ne wo ks o conduc ance-based model neu-
ons ha e been de eloped which show such pa e ns o
ansien synch oniza ion [5]. The aim o hese s udies
has been o ep oduce as close as possible he expe imen-
al esul s and so, sys ems o a g ea numbe o di e -
en ial equa ions ha e been used o model each one o he
neu ons o he ne wo k. A p ice mus be paid, howe e ,
o he use o such a de ailed model. The high dimension-
ali y o he dynamical sys em as well as he s ong non-
linea cha ac e o i s equa ions a e signi ican obs acles
o he unde s anding o he collec i e beha io o he
ne wo k.
In his and o he simila cases, he use o a simpli ied
model o a neu on is ad isable in o de o be able o
iden i y possible dynamical mechanisms behind he
complex beha io o he whole ne wo k. Recen ly, he
concep o winne less compe i ion ne wo ks has been
in oduced o in es iga e spa io empo al encoding by an
assembly o neu ons [6]. To explo e his concep , a ne -
wo k o Fi zHugh-Nagumo spiking neu ons coupled by
ime-dependen inhibi o y in e ac ions has been in es i-
ga ed o e eal how inpu in o ma ion can be e icien ly
ans o med in o spa io empo al i ing pa e ns [6].
In his Le e he ocus is on he beha io o a ne wo k
made o simple neu onal uni s. Each neu on is ep e-
sen ed by a wo-dimensional map o he kind s udied
ecen ly by Rulko [7]. Ob iously, om he poin o
iew o neu oscience, his dynamical sys em is me ely a
oy model bu , ne e heless, i shows enough dynamical
complexi y o mimic he cha ac e is ic beha io o mo e
in ol ed neu on models. A he same ime, i is simple
enough o pe mi a de ailed analysis o he mechanism
behind i s ema kable encoding capabili ies. By using his
model we ha e ound spa io empo al pa e ns o ac i a-
ion and inac i a ion o neu onal subg oups ha a e simi-
la o he pa e ns ob ained by using mo e complex
models o neu onal beha io [6]. When isola ed,
each neu on in he ne wo k is desc ibed by he wo-
dimensional map
xk1 xk;y
k;(1)
yk1ykxk10;(2)
whe e is a pa ame e se ing he ime scale o he y
a iable. In wha ollows, we will ake 0:001 so ha
he cha ac e is ic ime scales o bo h a iables a e widely
sepa a ed. The pa ame e 0 ep esen s he ex e nal inpu
ac ing on he neu on. In Eq. (1), x; yis a unc ion o he
o m
x; y(y1x1;x0;
y; 0<x<y;
1;xy;
(3)
whe e is a cha ac e is ic pa ame e . In a neu obiolog-
ical con ex , xand ya e he as and slow dynamical
a iables desc ibing he beha io o he neu on and, in
pa icula , he as a iable ep esen s he ins an aneous
ol age ac oss he neu on’s memb ane.
Rulko has shown ha his map has basically h ee
dynamical egimes ha a e dependen on he alues o
he pa ame e s and 0[7]. Speci ically, he neu on can
be silen o nonspiking ( o small and 0), in a egime
o single spiking ( o 00) o in a spiking-bu s ing
egime, whe e he neu on i es bu s s o closely spaced
spikes iding a slowe wa e. Le us conside his las case.
Because o he smallness o he pa ame e , he e olu-
ion o he as a iable xcan be analyzed by conside ing
he a iable yas a slowly d i ing con ol pa ame e . I
ollows om Eqs. (2) and (3) ha he alue o y emains
unchanged (yk1yk)onlyi
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xxs10:(4)
No ice ha i x>x
s, hen he alue o yslowly inc eases
upon i e a ion o he map, whe eas i x<x
s, hen y
dec eases unde he same ci cums ances. On he o he
hand, he unc ions gi ing he ixed poin s o he as
map can be ob ained di ec ly om Eq. (2),
x1y 
1y24y
p2;x0:(5)
This exp ession de ines he s able sand uns able u
b anches o slow mo ion ac oss he phase space. These
b anches ha e been plo ed in Fig. 1. The dynamics o he
sys em in he bu s ing egime can be analyzed in e ms o
he pa ame e 0. I he ixed poin xs10is on
he s able b anch, hen he neu on is in a silen s a e. On
he o he hand, when xsis on he uns able b anch, as
shown in Fig. 1, he neu on is in he bu s ing egime.
No ice ha he slow mo ion along he s able b anch o
ixed poin s co esponds o he las pa o he in e bu s
in e als. This slow mo ion ends a he poin whe e s able
and uns able b anches o ixed poin s mee and disappea .
The e, he phase poin mus jump o he spiking b anch
(no shown in Fig. 1). In doing so, he line xxsis
c ossed om below and ywill subsequen ly dec ease
slowly as he phase poin ca ies ou he spiking pa o
he cycle. The bu s ends a a homoclinic bi u ca ion ha
akes place a he c ossing o he uns able b anch wi h he
line x1[7]. On eaching his poin , he phase poin
mus jump back o he s able b anch, whe e yslowly
inc eases wi h ime, hus leading he phase poin o
mo e along sand making he whole p ocess o es a
i sel . In Fig. 1 one o hese bu s s has been supe imposed
o he bi u ca ion diag am o he as subsys em o illus-
a e he s uc u e o he bu s ing oscilla ion.
Le us add ess he main opic o his Le e by consid-
e ing a ne wo k o neu ons desc ibed by he ollowing se
o Ncoupled maps
xn
k1 xn
k;y
n
k;(6)
yn
k1yn
kxn
k1n;(7)
nn
0X
N
m1
gn;mxm
k;(8)
whe e n1;2;...;N. No ice ha , in p inciple, he ex-
e nal inpu n
0can be di e en o di e en neu ons in
he ne wo k. Pa o he whole inpu ac ing on each
neu on comes om i s couplings o o he uni s in he
ne wo k. He e, he coe icien gn;m gi es he ac ion on
neu on ncoming om neu on m. As he pa ame e is
chosen o be smalle han he minimum alue o all he
xn, he e ec o he es o he neu ons on he dynamics
o each one o hem is pu ely inhibi o y. In Fig. 2
we p esen he a chi ec u e o a ne wo k o nine
neu ons whose synch onizing beha io is going o be
explo ed nex .
Le us conside ha he s imulus ac ing on each neu on
is iden ical, ha is, le us ake n
00 o all n.The
beha io o one coupled neu on can be analyzed by using
he same a gumen employed o he isola ed case. Le us
conside , o example, he beha io o neu on 1. I s
b anches o s able and uns able ixed poin s a e s ill gi en
by Eq. (5) bu now, he cons ancy o y1du ing he as
ime scale leads o he condi ion
^
xx 1
k10x2
k;(9)
−4.05 −4−3.95 −3.9 −3.85 −3.8
y
−3
−2
−1
0
1
2
3
x
s
uxs
FIG. 1. S able sand uns able (u) b anches o he slow
dynamics o one isola ed neu on plo ed on he phase plane
y; x. The pa ame e s a e 6and 00:25.The em-
po al e olu ion o he phase poin du ing he ime cou se o a
bu s is also depic ed o illus a e he discussion ca ied ou in
he ex .
FIG. 2. The s uc u e o a neu al ne wo k wi h some neu ons
being connec ed h ough pu ely inhibi o y synapses. No ice
ha gn;m gm;n. The di ec ional connec ions explici ly d awn
by means o solid do s co espond o gn;m 1:0. All he o he
e ms in Eq. (8) ake alues gn;m 0.
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ins ead o ha gi en by Eq. (4). Now, he alue o x1a
which y1 emains unchanged is no longe s a iona y bu
oscilla es in ime due o he mo ion o he a iable x2
and, in pa icula , i can be d i en om he uns able o
he s able b anch o ixed poin s o ice e sa. Fo some
ixed alues o 0, each bu s o neu on 2 can d i e ^
xx1 o
c oss he s able b anch sand hen, neu on 1 is pushed o
i s silen egime. On he con a y, when neu on 2 is in he
in e al be ween wo bu s s, ^
xx1is d i en o c oss he
uns able b anch uand hus neu on 1 en e s i s as spiking
egime. Indeed, as we can see in Fig. 3, neu on 1 i es
du ing he las pa o he silen phase o neu on 2, ha is,
when he phase poin o his las neu on is mo ing along
i s s able b anch o ixed poin s. No ice ha , when d i en
o exci a ion, neu on 1 mus i e he bu s en i ely e en i
he d i ing p o ided by neu on 2 e u ns o sub h eshold
alues. This a gumen explains why he successi e bu s s
o neu ons 1 and 2 a e o ced o ake place a di e en
imes. Ob iously, his same analysis applies o neu ons
d i en by wo o mo e inhibi o y inpu s al hough in hose
cases he beha io can be mo e complex due o he ex-
is ence o closed eedback loops among he neu ons.
Thus, he deg ee o o e lapping among he di e en
bu s s depends c i ically on he de ailed a chi ec u e o
he ne wo k.
As a whole, he ne wo k ac s as a dynamical sys em
ha ing di e en egimes. I 0is small enough, all neu-
ons decay o hei es ing s a e a e an ini ial ansien .
When he cons an ( onic) s imulus is inc eased, some
g oups o neu ons s a he synch onous i ing o spikes,
whe eas o he s pe o m only a small slow-wa e oscilla-
o y beha io associa ed wi h he in e bu s ime scale.
When he s imulus is u he inc eased un il i eaches a
gi en h eshold (00:85), all he neu ons in he ne -
wo k s a i ing successi e bu s s o spikes. In Fig. 4 we
p esen some bu s s p oduced by a sup a h eshold s imu-
lus ac ing on all he neu ons wi h he same in ensi y and
iming. As we can obse e, he ne wo k de elops a spa-
io empo al pa e n o i ing ha amoun s o he ansien
ac i a ion and inac i a ion o ou di e en assemblies o
neu ons. In ac , neu ons labeled 1 and 9 i e in synch ony
and will be conside ed as assembly A. The same occu s
wi h neu ons labeled 2, 3, 7, and 8 (assembly B)andwi h
neu ons labeled 4 and 6 (assembly C). Assembly Din-
cludes only he neu on labeled 5. The ac i i y o all he
neu ons belonging o he same assembly is synch onous
no only a he le el o bu s s bu also a he le el o
indi idual spikes. No ice ha , in spi e o he links con-
nec ing only neighbo ing neu ons, he di e en assem-
blies can include also he no neighbo ing uni s. Thus, he
exis ence o inhibi o y couplings induces a global s uc-
u e on he whole ne wo k.
The lowes signal depic ed in Fig. 4 is he a e age o
he spike ains i ed by all he neu ons belonging o he
ne wo k
FPkN1X
N
n1
xn
k:(10)
As we can see, his las signal seems o p esen a pe iodic
a ia ion wi h a cha ac e is ic equency. Howe e , his
4500 5000 5500 6000
n
1
2
Neu ons
a
bc
a
c
dd
FIG. 3. Sequence o bu s s gene a ed by neu ons 1 and 2
unde he ac ion o a sup a h eshold s imulus o cons an
in ensi y. In a, neu on 2 is mo ing slowly along i s s able
b anch o ixed poin s. Jus a li le ea lie , in d, i has induced
he i ing o neu on 1 h ough he inhibi o y coupling. No ice
ha , once s a ed, he i ing o neu on 1 mus pe o m he
whole bu s e en i neu on 2 e u ns quickly o sub h eshold
alues. This leads o a small o e lapping o he bu s s i ed by
bo h neu ons nea he ins an labeled b.
0 1000 2000 3000 4000 5000 6000 7000 8000
n
FP
1
2
3
4
5
6
7
8
9
Neu ons
FIG. 4. Spike ains gene a ed by he neu ons o he ne wo k
unde he ac ion o a sup a h eshold s imulus o cons an
in ensi y. The s imulus is deli e ed o all neu ons a n
2000, and i consis s in a sudden ele a ion o 0 om 2:0
o 0.0, all he neu ons being s imula ed in he same way. The
plo labeled FP has been ob ained by a e aging he ins an a-
neous alues o all he spike ains.
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signal is no exac ly pe iodic in all i s de ails due o
a iabili y displayed by he di e en bu s s i ed by
each neu on. As he alue o 0is inc eased, he e-
quency o he bu s s displayed by he ield po en ial (FP)
oscilla ion de ined by Eq. (10) also inc eases, hus p o id-
ing a kind o a e coding. The de ailed o m o his
oscilla o y FP signal comes om he ansien o e lap-
ping o he successi e bu s s coming om he di e en
assemblies. In Fig. 5 we can obse e a close-up iew o
he empo al de elopmen o he i ing ac oss he ne wo k.
No ice ha du ing some epochs, he only con ibu ion o
he FP oscilla ion comes om he ac i i y o a single
assembly, and hus he global ou pu o he ne wo k is
synch onized by he hy hm o one o i s subg oups. In
o he epochs he FP oscilla ion esul s om he weigh ed
con ibu ions o one, wo, o e en h ee di e en assem-
blies. I is clea ha he whole empo al s uc u e o he
ne wo k’s ou pu will also depend on he loca ion and
in ensi y o he inpu s. When s imula ed di e en ly, he
synch onous ac i i y ound unde iden ical d i ing no
longe appea s, and consequen ly he numbe o assem-
blies as well as he succession o bu s s becomes al e ed
wi h espec o he case o homogeneous s imula ion.
The use o a simpli ied, ye a om i ial, neu on
model has allowed us o build a winne less compe i ion
ne wo k in which ansien ac i a ion and inac i a ion o
some g oups o neu ons appea as a consequence o pai -
wise inhibi o y couplings be ween hem. Acco ding o
p e ious esul s ob ained by using mo e complex model
neu ons, hese ansien local pa e ns seem o be gene ic
in ne wo ks ha ing de e minis ic ajec o ies connec ing
ixed poin s and limi cycles in he s a e space o he
whole sys em. The model s udied in his Le e p o es
ha dynamical encoding appea s also in ne wo ks whe e
he dynamical complexi y o each neu on model has been
conside ably educed.
Fo hese ne wo ks, he na u e o he synch oniza ion
allows he di e en s imuli o be dynamically encoded by
speci ic and ep oducible sequences o i ing coming om
di e en assemblies o neu ons ac oss he ne wo k, a
p ope y ha can be used o pe o m disc imina ion asks.
The simplici y o he model neu on used in his wo k
allows a de ailed analysis o he mechanisms behind he
dynamical beha io o he ne wo k and, u he mo e, i
will pe mi he modeling and implemen a ion o la ge
ne wo ks (wi h hund eds o neu ons, o example) by
using a easonable compu ing capaci y.
The au ho acknowledges he Di eccio
´n Gene al de
In es igacio
´n Cien ı
´ ica y Te
´cnica (DGICYT) o Spain
o suppo (P ojec No. BFM2002-03822).
*Elec onic add ess: [email protected]
[1] A. Sco , Neu oscience. A Ma hema ical P ime
(Sp inge , New Yo k, 2002).
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W. Bialek, Spikes: Explo ing he Neu al Code (MIT
P ess, Camb idge, MA, 1997).
[3] M. Weh and G. Lau en , Na u e (London) 384,162
(1996).
[4] M. Bazheno , M. S op e , M. Rabino ich, H. D. I.
Aba banel, T. J. Sejnowski, and G. Lau en , Neu on 30,
569 (2001).
[5] M. Bazheno , M. S op e , M. Rabino ich, R. Hue a,
H. D. I. Aba banel, T. J. Sejnowski, and G. Lau en ,
Neu on 30, 553 (2001).
[6] M. Rabino ich, A. Volko skii, P. Lecanda, R. Hue a,
H. D. I. Aba banel, and G. Lau en , Phys. Re . Le . 87,
068102 (2001).
[7] N. F. Rulko , Phys. Re . E 65, 041922 (2002).
4200 4450 4700 4950 5200 5450
n
FP
A
B
C
D
Neu ons
FIG. 5. The s uc u e o he ield po en ial oscilla ion as
a consequence o he ansien ac i a ion and inac i a ion
o he ou assemblies o neu ons ollowing he sequence
D-A-B-C-D-A-.... Each spike ain in he igu e ep esen s
one assembly o neu ons i ing synch onously. The con ibu ion
o each assembly o he FP signal depends on he numbe o
neu ons ha cons i u e he assembly. The FP signal in his plo
has been scaled up by a ac o o 3 in o de o depic mo e
clea ly i s cha ac e is ic ea u es.
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