A network of phase oscillators as a device for sequential pattern generation

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by  Pablo Kaluza

A network of phase oscillators as a device for sequential pattern generation
Pablo Kaluza1,2 and Hildegard Meyer-Ortmanns2
1)Abteilung Physikalische Chemie, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin,
Germany.
2)School of Engineering and Science, Jacobs University, P.O.Box 750561, 28725 Bremen,
Germany.
(Dated: 18 January 2012)
We design a system of phase oscillators that is able to produce temporally periodic sequences of patterns.
Patterns are cluster partitions which encode information as phase differences between phase oscillators. The
architectureofoursystemconsistsofaretrievalnetworkwithNgloballycoupledphaseoscillators,apacemaker
that controlsthe sequence retrieval,anda setofpatterns storedin the couplingsbetweenthe pacemakerand
the retrievalnetwork. The system performs in analogy to a centralpattern generatorof neural networksand
2
is very robust against perturbations in the retrieval process.
1
0
2 PACS numbers: 05.45.Xt, 89.75.-k, 89.75.Kd
Keywords: phase oscillators, pattern retrieval, cluster partition
n
a
J
Sequential generation of patterns is an important endogenouslyproducerhythmicallypatternedoutputlike
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issue in the context of dynamical systems that breathing, walking, or heartbeat. Apart from improv-
1
are designed for applications to artificial neural inganunderstandingofthe biologicalaspects,nowadays

] networks, gait models of animals, robot control- the engineering aspect of networks of oscillators plays a
D ling and the like. In this paper, we construct a prominentrole3. Forexample,complexdynamicalstruc-
C system of phase oscillators that is able to gen- tures in populations of phase oscillators were engineered
. erate periodic sequences of patterns via imple- by means of nonlinear time-delayed feedback that is im-
n
mented couplingsand an inherentclock. The sys- plemented in the interactions of the oscillators4.
i
l tem consists of a pacemaker providing the clock, In this work we address the problem of generation of
n
[ a retrieval network providing the patterns, and a periodic sequences of patterns by a dynamical system of
set of couplings between the pacemaker and the phase oscillators. We consider the case where a set of P
2 retrieval network, coding the patterns. The pace- patterns ξ should be presented periodically in a certain
v
maker activates the stored patterns one by one as time order. To solve this problem we design a system
4
a function of its own phase, and the retrieval net- which consists of a pacemaker, a retrieval network, and
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2 work selects the cluster partition corresponding P storedpatternsbetweentheseelements. Theproposed
2 to the activated pattern. The process of select- system can be seen as an associative memory driven by
. ing the cluster partition is realized by ensuring a pacemaker. Here, we use a particular case of an asso-
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0 that the energy function of the retrieval network ciative memory (retrieval network) with time-dependent
1 has only a single minimum while the pattern is couplings controlled by the pacemaker through the pat-
1 activated. The gradient dynamics of this energy terns ξ. Associative memory models based on the use of
: function then leads to a phase-locked motion of phase oscillators are alternatives to the Hopfield model5
v
i the retrieval network, such that the fixed phase with spins replaced by phase oscillators. These models
X relationsbetween the oscillatorsrepresent the ac- are generalizations of the Kuramoto model6 (for a re-
r tivated pattern. Thus the pattern amounts to a view see Acebron7). Examples for such generalizations
a
partition of oscillators into clusters characterized were considered by Aoyagi8 and further improved by
by a fixed phase relation (therefore called cluster Nishikawa et al.9 with the goal to achieve a storing ca-
partition). The system then performs similarly pacity similar to that of the Hopfield model along with
to a central pattern generator of biological neural error-free retrieval.
networks. Asa result,oursystemis abletoproducesequencesof
patterns inthe retrievalnetwork. Thesepatterns areen-
coded as phase differences between the oscillators. The
associated dynamical states correspond to phase-locked
I. INTRODUCTION synchronization. Suchstatesamounttoclusterpartitions
of the oscillators’ set, in which oscillators sharing the
A sequential generation of patterns by a system of os- same phase are gathered in one cluster. The system can
cillatorsis motivatedby centralpatterngeneratorsofbi- beseenasastylizedversionofthecentralpatterngener-
ological neural networks1,2. Central pattern generators atoroftheleechheartbeat1,thatiscomposedoftwosets,
usually are considered in the context of neuroscience as the rhythmgenerator(correspondingtoourpacemaker),
an explanation of how nervous systems produce move- and the pattern generator (the retrieval network).
ments. They are autonomous neural networks that can The paper is organized as follows. In section II we
2
presentthemodelforourdeviceanditsdifferentbuilding a time-dependent energy landscape that is controlled by
blocks. In section III we illustrate its performance and the pacemaker. In addition, we impose the constraint
pointonthe analogytoa centralpatterngenerator. The that only one minimum of L may exist at a time, when
conclusions are drawn in section IV. a pattern is selected by the pacemaker. The advantage
then is that the gradient dynamics will lead to the re-
quired minimum starting from any initial condition due
II. THE MODEL OF OUR DEVICE to the absence of competing basins of attraction.
In the following we shall describe the building blocks
of our device in detail, before we illustrate how it works
The system is composed of a phase oscillator, which
for retrieving a cyclic sequence of patterns.
plays the role of pacemaker and is characterized by a
phase variable ψ, a retrieval network with N globally
coupled phase oscillators with phases φ , i = 1,...,N,
i
and P patterns ξ stored in the couplings between the A. The retrieval network
pacemaker and the retrieval network.
The functionofthe pacemakeristo controlthe timing The dynamics of this network is characterized by an
and activation of the stored patterns ξ as a function of energy function L (”L” shallremind to its role as a Lya-
its phase. In this way, the pacemaker works as a clock punov function) whose gradient determines the phase
that points to the pattern that should be activated in evolutionoftheoscillators. Ingeneral,thesystemevolves
the sequence. The retrieval network encodes the stored tothedifferentminimaofLdependingontheinitialcon-
patterns ξ as phase differences between its N phase os- dition and on the basis of attraction of each minimum.
cillators, one by one as a function of time. We call it For this retrieval network we use a particular version of
retrieval network since its function consists in retrieving the model proposed by Nishikawa et al.9. We shall show
theinformationstoredinthecouplingsbetweenthepace- howwetunethecouplingsinordertohaveonlyonemin-
maker and the network, and translating the patterns to imum in L when a pattern is selected for retrieval. The
the corresponding cluster partitions of the phase oscilla- functionLthendependsonN oscillatorphasesΦ inthe
i
tors. following way:
The operation of the retrieval network is achieved by
designing a suitable energy function L. The minima of K N 2
this function are approached by a gradient dynamics in L=− cos(Φj −Φi)−fij (1)
4N
which the energy function plays the role of the poten- i,j=X1,i6=j(cid:16) (cid:17)
tial. The minima correspond to specific cluster parti-
tions, for which the phase differences take only values of with Φi ∈ [0,2π[,i = 1,...,N the phase variables and K
zero or π. These configurations are easily mapped to bi- the coupling strength that finally determines the speed
nary sequences. In the set of N oscillators only N −1 of convergence of the dynamics towards the stationary
phasesareindependent,thereforewecanhave2N−1clus- state, see Eq. 3 below. As noted before, out of the
terpartitions. Apatternξ isthendescribedasanN−1- maximally N(N −1)/2 different phase differences, only
dimensional vector with components ξi being the phase N −1 are independent, so we choose ∆ΦiN =Φi−ΦN,
differences between the oscillators i and N. i=1,...,N−1asindependentvariables,usingthe phase
Now, in order to retrieve a selected pattern out of the of the Nth oscillator as reference. With fij we denote
2N−1 ones that are in principle available, we implement the couplings, fij ∈R,i,j ∈{1,...,N}.
couplings between the pacemaker and the phase oscil- For vanishing couplings fij we get back the Kuramoto
lators which modulate the energy landscape in a way model with twice the usual frequency in the interaction
that the selected pattern becomes the only minimum at betweentheoscillators. Then,Lhasobviously2N−1min-
a given time. So the proposed system can be seen as an ima given by the vectors ξ(k),k = 1,...,2N−1 of equal
associative memory. The associative memory is particu- height with components ξ(k) ∈ {0,π}. In the configu-
i
lar in the sense that the couplings are time-dependent, ration space of phase differences {∆Φ }, these minima
iN
driven and controlled by the pacemaker through the are located at the corners of a hypercube of linear size π
choice of patterns ξ. For the selection of another pat- with one corner in the origin of coordinates. It is these
tern, another set of couplings is addressed by the pace- 2N−1 local minima that are our candidates for retrieval.
maker. This way it becomes possible to retrieve a whole Thelabelk ofthestateξ(k) isdeterminedbythepattern
temporal sequence. of 0s and πs interpreted as binary sequence in decimal
Iftheretrievalnetworkhadonlystaticsetsofcouplings representation.
betweentheoscillators,severalminimacouldcoexistand In general, the f are couplings appropriately chosen
ij
would be reached by the gradient dynamics depending to modulate the energy function L in order to retrieve
on their basins of attraction. This case corresponds to a selected pattern ξ(s). A sufficient condition for a local
an associative memory network as proposed by Aogagi8 minimum reads that the Hessian matrix of L is positive
and Nishikawa et al.9. Our aim is, however, different. definite, i.e., all eigenvalues being larger than zero, due
We consider time-dependent couplings in order to have to an appropriate choice of f .
ij
3
Now let us select one pattern ξ(s) with s ∈ The pacemaker is a phase oscillator with constant fre-
{1,...,2N−1}. Let the couplings modulate the interac- quency ω and phase ψ, whose time derivative is given
R
tion between oscillator pairs (ij) according to by
f (α,s)=α 2|ξ(s)−ξ(s)|−1 (2) ψ˙ =ωR. (4)
ij π i j
(cid:18) (cid:19) It is the phase ψ that selects and activates a pattern
r among the P stored ones over a duration B. This ac-
(s)
for i,j ∈{1,...,N,i6= j} with ξ ≡0, α any real num- tivation period lies in the phase (time) interval between
N
ber with α > 1 and s the index of the selected pattern. ψ = ψ and ψ = ψ . For simplicity we have chosen
r r+1
Sothe couplingsfij takevaluesof±α,depending onthe ψr = 2Pπ(r−1) for all r ∈ {1,...,P}, and B = 2Pπ. This
pattern. This choice is guided by the postulate that the means, when the instantaneous phase ψ comes to the
selected configuration remains the only local minimum valueψ ,couplingsf areswitchedonthatguaranteethe
r ij
when these couplings are applied, while all other former retrieval of the pattern rξs(r). Here the pre-superscript
local minima become saddles in at least one direction or r indicates the label of the pattern within the time se-
local maxima. The conjecture is that a choice according quence, the post-superscript s(r) stands for the decimal
to Eq.2 satisfies this postulate. In the appendix we shall label of the pattern r out of the selected subset. The
show that for this choice of couplings the Hessianis pos- corresponding dynamical equations read:
itive definite for each configuration that is selected from
the 2N−1 patterns, and not positive definite for all other K N
2N−1−1 configurations, which were formerly also min- Φ˙i = − sin(∆Φji) cos(∆Φji)−fji(ψ) , (5)
N
ima for vanishing couplings. (For α<1 it can be shown j=X1,j6=i (cid:16) (cid:17)
that alllocalminima remainstable,but the selected one
where
becomes the deepest.) The gradient dynamics will then
retrieve the local minimum from any initial condition, P 2
not necessarily close to the selected minimum. fij(ψ)= α π|rξis(r)− rξjs(r)| − 1 gr(ψ) (6)
Itshouldbenoticedthatthechoiceofcouplingsfij for Xr=1 (cid:18) (cid:19)
aselectedpatternξ byEq. 2canbemappedtotheHeb- and
bian rule in the particular case where only one pattern
is memorized by the associative memory of Nishikawa et gr(ψ)=Θ(ψ−ψr)−Θ(ψ−ψr−B). (7)
al.9. However, this restriction to the case of only one
Here Θ denotes the Heavyside function. This means
minimum of the energy function (ensured by the choice
thatthefunctiong (ψ)controlsthecouplingstobegiven
of α > 1) has the advantage that it leads to an error- r
free retrievalstartingfromanarbitraryinitial condition, as α 2|rξs(r)− rξs(r)|−1 over the phase interval B,
π i j
possibly far away from the final configuration. starti(cid:16)ng from ψ =ψr on. (cid:17)
Explicitlythegradientdynamicsoftheoscillatorsthen Obviously we should ensure that ψ −ψ ≥ B ≥
r+1 r
reads: ψ , that is, the time interval between two initiations
trans
of pattern retrievals and the time of application of the
N
∂L K constant couplings (to reach the new pattern) should be
Φ˙ =− =− sin(∆Φ ) cos(∆Φ )−f .
i ∂Φi N ji ji ji larger than the transient time ψtrans which the retrieval
j=X1,j6=i (cid:16) (cid:17) network needs to go from one pattern to the next (the
(3)
speed of the internal dynamics is controlled by K).
SimilarlytotheKuramotodynamicstheinteractionof
The information for the sequence generation is stored
the oscillators depends only on phase differences ∆Φ ,
ij in the phase values ψ ’s through the functions g (ψ)
andduetothechoiceoftrigonometricfunctionstheinter- r r
which “switch on” the appropriate couplings. Since the
actionterms are bounded asin the Kuramotodynamics.
stored patterns ξ are required at different times, they
Due to the gradient dynamics the phase differences will
must be stored outside the retrieval network. There-
evolve to a fixed point which is the minimum of the en-
fore we say that the patterns generated in the sequence
ergy function L that is closest to the initial conditions.
are memorized in the couplings between the pacemaker
Note that the second order Fourier term was related to
andthe retrievalnetwork. Thiswaythe architecturecan
the formation of clusters by Mato10. In our case, this
modulate the couplingsf betweenthe oscillatorsofthe
term appears due to the very construction of the energy ij
retrieval network.
function.
In summary,the pacemakertransformsthe static con-
tents, memorized in the couplings, into a temporal se-
quence of patterns, a feature that is in common with a
B. The role of the pacemaker
central pattern generator.
Tocompleteoursetofequations,wehavetointroduce
Weextendourdynamicstosequentialpatternretrieval noise in the retrieval dynamics. The reason is the fol-
viatimedependentcouplingscontrolledbyapacemaker. lowing. As we have seen, once a pattern is selected for
4
retrieval, the system evolves according to the dynamics by the coupling parameter K, large K accelerates the
of Eq. 3 to the only stable fixed point of L of the re- convergence to a new pattern, once the time-dependent
trieval network. When the next pattern of the sequence couplingsarechanged;alsothestrengthofthecouplings,
is activated by the pacemaker, the system is still in the parameterizedbyα,determinesthespeedofconvergence.
former fixed point that turns into an unstable one. To
kick the system out of this fixed point and follow the re-
quired sequence, we apply Gaussian white noise of small -200
a)
intensity T. Eq.3 is then replaced by B
-300
K N L(t)
Φ˙ = − sin(∆Φ ) cos(∆Φ )−f (ψ) +Tη (t),
i ji ji ji i -400
N
j=X1,j6=i (cid:16) (cid:17)
(8)
-500
where ηi(t) is a random variable describing the white 0 1.257 2.513 3.77 5.027 6.283 7.54 8.796
noise with zero mean, < η (t)η (t′) >= δ δ(t−t′), and Time
i j ij
T is the noise intensity. Our system is then described by 5
b)
Eq.s 4, 8, 6, and 7. 4
3
III. NUMERICAL STUDY D(t)2
1
In this section we study an example of this system
and focus on its dynamical properties. We integrate the 0
0 1.257 2.513 3.77 5.027 6.283 7.54 8.796
dynamics using a second order stochastic Runge-Kutta Time
method11 with time step ∆t = 0.01 and noise intensity 1500
c)
tTor=sin0.0th0e1.reWtreievcaolnnsiedtewroarks.eTthoifsNnet=wo1r1kpchaanstehoesrceiflolrae- kξe (t) 1000 ξ942 ξ1023
encode 210 = 1024 patterns as cluster partitions. Out at ξ672
st
of this set, we have randomly selected P = 5 patterns. st ξ477
They are 1ξ672, 2ξ0, 3ξ942, 4ξ477, 5ξ1023 in the indi- ose 500
cated order. The time dependent-couplings are changed Cl ξ0
at phase values ψ = 0, 2π/5, 4π/5, 6π/5 and 8π/5. 0
r 0 1.257 2.513 3.77 5.027 6.283 7.54 8.796
The parametersarechosenasK =10,B =2π/P, α=2 Time
and ω =1.
R
Figure 1a showsthe energyL asfunction oftime eval-
uated in the actual state of the retrieval network. L
jumpsfromitsminimalvalueatL≈−500tosomelarger
FIG.1. Sequentialpatternretrievalofasequences ={ξ672,
value around L≈−300, where it remains as long as the 1
ξ0, ξ942, ξ477, ξ1023}. (a) Energy L(t) as function of time.
system of oscillators searches the new minimum, corre-
(b) Euclidean distance of the actual state of the system to
sponding to the new choice of external fields. When the
the closest state (corresponding to patterns on the corners
new minimum is found, L drops to the minimal value of the hypercube). It vanishes for roughly half of the period
again. During such an interval of duration B, the Eu- B, so that the system then has retrieved the desired state.
clideandistance D(t) in configurationspace between the (c) Closest states to thecurrent evolving state as function of
actualstateandtheclosestpattern(cornerofthehyper- time. As seen from the figure, the closest states themselves
cube) has a peak at an intermediate time interval where vary with time. The set of closest states agrees with the set
thesystemismovingfromonetothenextselectedstate. of selected states. The states carry their decimal labels.
We see these peaks in Fig. 1b. For about half of the pe-
riod B this distance is zero, indicating that the state of
the system system corresponds to the required pattern. InFig.2weshowthreepermutationsofthesamesetof
In Fig. 1c we plot the states which are closest to the the five stored patterns. Indicated are the five plateaus
instantaneous states of the system as a function of time. in time where a certain pattern remains the closest to
Obviouslythecloseststatesarejusttheselectedones,but the current state and where this pattern agrees with the
this does not mean that the actual states (evolving with system’sstateoverroughlyhalfoftheperiod(the analo-
time) are identical with the selected ones over the whole gous figuresto Fig. 1b arenot displayedhere). It should
durationoftheplateau;asmentionedbefore,thedistance be noticed that a change in the pattern sequence from
to the selected states vanishes only for roughly half of Fig. 2a to 2b and 2c only amounts to reorder the phase
the period as it is seen from Fig. 1b. The width of the shifts ψ , no other change of the system’s structure is
r
peaksinthedistancefromthecloseststatescanbetuned needed.
5
1500 rhythmgenerator(correspondingtoourpacemaker),and
a)
kξe (t) 1000 ξ1023 ξ942 twhoerkp)a.ttTerhnegpeantetrearntogre(ncoerraretsoprotnhdeirneggteoneoruartreesttrhieevaalcntueat-l
stat ξ672 motor pattern in response to the driving input from the
st ξ477 rhythm generator (in our case in response to the driving
e 500
os input of time-dependent couplings from the pacemaker).
Cl
ξ0
0
0 6.2832 12.566
Time V. ACKNOWLEDGMENTS
1500
b)
kξe (t) 1000 ξ942 ξ1023 muWchefworouvladlulaikbeletodistchuasnskionAsl.exander S. Mikhailov very
at ξ672
st
st ξ477
e 500
os VI. APPENDIX
Cl ξ0
0 Consider the energy function for an all-to-all coupled
0 6.2832 12.566
Time system of N phase oscillators:
1500
c)
kξe (t) 1000 ξ1023 ξ942 L=− K N (cos(Φ −Φ )−f )2. (9)
st stat ξ672 ξ477 4N ijX,i6=j j i ij
e 500
s
Clo From now on we set the coupling strength K =1. Let
ξ0 the couplings be chosen according to
0
0 6.2832 12.566
Time 2
f (α,s)=α |ξs−ξs|−1 fori,j ∈{1,...,N,i6=j}
ij π i j
(cid:18) (cid:19)
(10)
FIG. 2. Sequential pattern retrieval of different sequences
with ∆Φ = x −x , ξs = x ≡ 0, α any real number
with the same states. (a) Time evolution of the sequence s2 ij i j N N
= {ξ0, ξ1023, ξ942, ξ672, ξ477}. (b) Time evolution of the se- with α > 1 and s the index of the selected pattern. We
quences ={ξ942,ξ1023,ξ0,ξ477,ξ672}. (c)Timeevolutionof now prove a sufficient condition that the Hessian matrix
3
thesequences ={ξ1023,ξ672,ξ942,ξ0,ξ477}. Inallcaseswe with respect to the N −1 independent phase differences
4
showonlythestateswhichareclosesttothecurrentevolving is positive definite for the selected pattern and not posi-
state. Forroughlyhalfaperiodthedistancebetweentheclos- tive definite for all other 2N−1−1 patterns, providedwe
est state and the actual state of the system vanishes, which choose the external fields f according to Eq.2. From
ij
is interpreted as pattern retrieval. the first derivative ∂L/∂x we immediately see that can-
i
didatesforextremaarex ∈{0,π},wherex wasdefined
i i
as ∆Φ = Φ −Φ , while for f > 1 the individual
iN i N iN
IV. CONCLUSIONS cos-dependent terms are different from zero. For a par-
ticular given choice of f ,f with possibly alternating
ij iN
signsthe firstderivativescanvanishalsoatintermediate
We have designed a system of phase oscillatorsthat is
values of x which we project on [0,2π[ that can lead to
able to produce periodic sequences of patterns. Patterns i
further extrema. This partwe treatnumericallyin order
are stored in the couplings of the system and retrieved
to exclude that these extrema compete with the selected
and encoded as phase differences. Due to the task divi-
minimum that shall be retrieved.
sionbetweenthe pacemaker,the storedpatternsandthe
NextletusconsidertheHessianofLasfunctionofthe
retrieval network,the system is very flexible and robust.
phase differences. Apart from the normalization factor,
Different sequences of the stored patterns can be imple-
its diagonal elements are given as:
mented without modifying the system’s structure. The
retrieval network itself operates in a robust way since it
∂2L
hasbyconstructiononlyoneminimum,thereforethedy- =−sin2(x )+cos(x ) cos(x )−f
∂x2 k k k kN
namicsconvergestothedesiredpatternindependentlyof k
the initial conditions. N−1 (cid:0) (cid:1)
− sin2(x −x )−cos(x −x ) (11)
Our device may be regardedas a very stylized version k j k j
ofthecentralpatterngeneratoroftheleeches’heartbeat. j=X1,j6=k(cid:18)
According to Hooper1, the central pattern generator of
· cos(x −x )−f ,
the leech heartbeat can be divided into two sets, the k j jk
(cid:19)
(cid:0) (cid:1)
6
its off-diagonalelements are: and in view of that we shall need α>1, see below.)
∂2L =sin2(x −x)−cos(x −x) cos(x −x)−f . H with one mismatch Next we evaluate the Hessian
k l k l k l lk
∂x ∂x for a configuration that differs from the selected pattern
k l
(cid:0) (12(cid:1)) inasinglephasedifference. Withoutlossofgeneralitywe
For a choice of the external fields fij according to the assume the mismatch to happen in the first coordinate,
rule (2), the matrix simplifies to affectingtheHessianinthefirstcolumnandthefirstrow
accordingto H =(N−1)(1−α), H =(α−1)=H
11 1j j1
∂2L N−1 for j =2,...,N −1, while the remaining (N −2)(N −2)
=− ∓1 ±1−f + ∓1 ±1−f
∂x2 kN jk submatrix remains circulant. The Sylvester criterion,
k ( (cid:0) (cid:1) j=X1,j6=k(cid:18) (cid:0) (cid:1)(cid:19)) applied to the positive definiteness of the overall
(13)
(N −1)×(N −1) matrix, is now violated due to the
for the diagonal elements and k =1,...,N −1, and to
first element H = (N −1)(1−α) < 0 for α > 1, so
11
∂2L that the configuration with one mismatch is no longer a
=∓1 ±1−flk (14) local minimum of the energy function L. (As necessary
∂x ∂x
k l and sufficient condition for a Hermitian matrix to be
(cid:0) (cid:1)
for the off-diagonal elements, with k,l ∈ {1,...,N − positive definite, the Sylvester criterion requires that all
1},k 6= l. The upper (lower) sign in front of the the leading principal minors of the matrix are positive.)
bracketwithf standsforthecasethatthe differenceof
lk
components |xs−xs|, readoff fromthe selectedconfigu- H with k > 1 mismatches Now the configura-
k l
ration ξ~s, is zero (π), respectively. tion has k mismatches with the selected configuration
which we arrange to occur in the first k coordinates.
Next we study the positive definiteness of this matrix
for all 2N−1 configurations which may be selected for Hereitshouldbe noticedthatfiN willhavethe“wrong”
sign with respect to ∆Φ , i = 1,..,k, but f will have
retrieval. Here it is convenient to classify the configu- iN il
the “right” sign with respect to ∆Φ for i,l ∈{1,...,k},
rations in terms of their Hamming distance from the il
since two mismatches compensate in the relative phase
selected pattern, i.e. the number of mismatches of
componentsbetweenξ~andξ~s,whichvariesbetweenzero differences (“wrong” (or “right”) refer to the feature
which prevents (or ensures) the property of becoming
and N −1. By a suitable permutation of the coordinate
a local minimum, respectively.) This explains why the
axis in configuration space we can always achieve that
components of the k×k submatrix S (H) in the upper
the k mismatches occur in the first k coordinates of k
left corner of the Hessian are given by
ξ so that the corresponding Hessian H is chosen as
representative for all patterns with k mismatches. S (ii)=(N −1)−α(N −2k+1), i=1,...,k (15)
k
H with no mismatches According to our choice for the diagonal elements and
of f , their signs are opposite to those of cos(∆Φs ),
ij lk S (ij)=−(1+α) i,j =1,...,k, ,i6=j (16)
that is ∆Φs = 0 or (π), so that cos∆Φs = 1 or (−1) k
lk lk
andflk =−α(or+α), α>1,respectively. Thediagonal for the off-diagonal elements. The submatrix Sk(H) is
elementsthensimplifyto∂2L/∂x2 =(N−1)(1+α),the again circulant and has eigenvalues λ =(N −k)(1−α)
k 1
off-diagonal elements to ∂2L/∂xk∂xl = −(1+α). The with multiplicity 1 and λ2 =N −α(N −2k) with multi-
Hessiantherefore takesthe formof an(N−1)×(N−1) plicity k−1, so that the determinant of this submatrix
dimensionalcirculantmatrix,whoseeigenvaluesturnout reads|S (H)|=λ λk−1. Nowwehavetodistinguishthe
k 1 2
to be λ1 =1+α with multiplicity 1 and λ2 =N(1+α) following cases:
with multiplicity (N-2). (Here we have used the follow- 1. kodd. Forkodd,λk−1 isalwayspositivewhileλ <0
2 1
ing: Eigenvalues of an n×n circulant matrix, specified for α > 1, so that |S (H)| < 0 for odd k and α > 1 and
k
by the vector (c0,c1,...,cn−1), are known to be given as the Sylvester criterion for H being positive definite is
c′ = n−1e2πijk/nc with j = 0,−1,−2,...,−(n−1). violated as it should be for any positive number of mis-
j k=0 k
In our case the Hessian has a particularly simple form, matches.
P
for which one element in each row is (N − 1)(1 + α), 2. k even. For k even, both eigenvalues may be negative
while all other N − 2 elements are −(1 + α). Using so that |S (H)| > 0. In order to see that the Sylvester
k
these values and the fact that the sum over all n roots criterion is still violated, we have to distinguish the fol-
of the unit circle adds up to zero leads to our results for lowing cases:
the eigenvalues.) Now, since for α > 1 all eigenvalues (i) For α > 1 and k > N/2 we have λ < 0 and λ > 0,
1 2
are positive, the selected configuration corresponds to a so that the Sylvester criterion is violated.
local minimum in configuration space, whatever pattern (ii) For α>1 and k <N/2, λ <0 for α>N/(N−2k),
2
has been chosen for retrieval. (In order to have only a so that |S (H)|<0 only for 1<α<N/(N −2k).
k
local minimum at the selected configuration, obviously (iii) To finally see what happens for α > 1 and α >
α > −1 would be sufficient, but at the same time, the N/(N −2k) let us consider the determinant of the sub-
other patterns should become saddles or local maxima, matrixofsizel=k−1intheupperleftcornerofH.This
7
matrix has eigenvalues σ1 =(N −k+1)−α(N −k−1) of Dynamical Order: Synchronization Phenomena in Complex
andσ =N−α(N−2k)withevenalgebraicmultiplicity Systems World Scientific Lecture Notes in Complex Systems,
2
(k−2), so that again the sign of λ determines the sign Vol. 2. World Scientific,Singapore.
ofthis subdeterminant. Nowσ <01for 1< N−k+1 <α, 4KissIZ,RusinCG,KoriH,andHudsonJL(2007)Engineering
1 N−k−1 complex dynamical structures: sequential patterns and desyn-
butthisiscertainlysatisfied,sinceintheconsideredcase chronizationScience 316:1886-1889.
k ≥2andαwasevenlargerthanN/(N−2k)byassump- 5Hopfield J J (1982) Neural networks and physical systems with
tion. Sothis(k−1)×(k−1)-dimensionalsubdeterminant emergent collective computational abilities, Proc Nat Acad Sci
violates the Sylvester criterion for H to be positive defi- USA79(8):2554-2558.
6Kuramoto Y (1984) Chemical Oscillations, Waves, Turbulence,
nite.
(Springer,NewYork).
In particular, for the maximal number of mismatches 7Acebro´nJA,BonillaLL,P´erezVicenteCJ,RitortFandSpigler
k =N−1,λ =1−α<0forα>1andλ =N+α(N− R(2005)TheKuramotomodel: Asimpleparadigmforsynchro-
1 2
2)>0 for N >2, and for N =2, λk−1 =λN−2 =1>0, nizationphenomena Rev Mod Phys 77:137-185.
2 2 8AoyagiT(1995)Networkofneuraloscillatorsforretrievingphase
so that the corresponding pattern again ceases to be a
informationPhys Rev Lett 74(20):4075-4078.
local minimum of the energy function. 9NishikawaT,LaiY-CandHoppensteadt FC(2004)Capacityof
oscillatoryassociative-memorynetworkswitherror-freeretrieval.
1Hooper SL (2000) Central pattern generators Current Biology Phys. Rev. Lett.92:108101.
10(5):R176-R177. 10Hansel D, Mato G and Meunier C (1993) Clustering and slow
2BrocardF,TazerartS,andVinayL(2010)Dopacemakersdrive switching in globally coupled phase oscillators. Phys. Rev. E
the central pattern generator for locomotion inmammals?, The 48(5): 3470-3477.
Neuroscientist 16(2):139-155. 11Honeycutt R L (1992) Stochastic Runge-Kutta algorithms. I.
3ManrubiaSC,MikhailovASandZanetteDH(2004)Emergence Whitenoise.Phys. Rev.A45(2):600-603.