Modular organization enhances the robustness of attractor network dynamics

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by  Neeraj Pradhan

Modular organization enhances the robustness of attractor network dynamics
Neeraj Pradhan1,2, Subinay Dasgupta3 and Sitabhra Sinha1
1The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India.
2Department of Physics, Birla Institute of Technology & Science, Pilani 333031, India.
3Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India.
Modular organization characterizes many complex networks occurring in nature, including the
brain. In this paper we show that modular structure may be responsible for increasing the robust-
nessofcertaindynamicalstatesofsuchsystems. Inaneuralnetworkmodelwiththreshold-activated
1 binaryelements,weobservethatthebasinsofattractors, correspondingtopatternsthathavebeen
1 embeddedusingalearningrule,occupymaximumvolumeinphasespaceatanoptimalmodularity.
0 Simultaneously,theconvergencetimetotheseattractorsdecreasesasaresultofcooperativedynam-
2 ics between the modules. The role of modularity in increasing global stability of certain desirable
attractors of a system may providea clue to its evolution and ubiquityin natural systems.
n
a
J PACSnumbers: 87.18.Sn,75.10.Nr,89.75.Fb,64.60.Cn

1
3
An ubiquitous property of complex systems is their signals within subnetworks [19]. However,of more inter-

modular organization [1], characterized by communities est is the possibility that modularity may play a crucial
]
n of densely connected elements with sparser connections role in the principal function of the system, viz., infor-
n
between the different communities [2]. In the biologi- mation processing in the case of brain networks. This
-
s cal world, modules are seen to occur across many length possibilityhasbeeninvestigatedindetailforthesomatic
i
d scales,fromtheintra-cellularnetworksofprotein-protein nervous system of the nematode C. elegans [20]. It is
. interactions [3, 4] and signaling pathways [5] to food therefore intriguing to speculate whether modularity is
t
a webs comprising multiple species populations [6]. Al- responsible for efficient information processing in brains
m
though such groupings are primarily defined in terms of ofmoreevolvedorganisms,themammaliancortexinpar-
- thestructuralfeaturesofthenetworktopology,inseveral ticular. To explore this idea further we can study the
d
instancesdistinctmoduleshavealsobeenassociatedwith effect of modular structure on the dynamics of attractor
n
o specific functions. Indeed, in the case of the brain, mod- network models with threshold-activated nodes, which
c ular organization at the anatomical level has long been exhibit multiple stable states or “memories” [21, 22].
[
thought to be paralleled at the functional level of cogni- These models were originally developed to understand

1 tion [7]. By observing the effects of isolating or discon- howthenervoussystemcommunicatesamongitscompo-
v necting different brainareasonthe behavior ofsubjects, nentpartsandlearnsassociationsbetweendifferentstim-
3 thefunctionalspecializationofspatiallydistinctmodules uli so that a memorized pattern can be retrieved in its
5
havebeenestablishedatdifferentlengthscales[8]- from entiretyfromasmallpartoranoise-corruptedversionof
8
hemispheric specialization to minicolumns comprising a it givenas input (“associativememory”). Indeed, recent
5
. few hundred cells which havebeen proposedas the basic experiments indicate that the spatiotemporal activation
1
informationprocessingunitsofthecerebralcortex[9,10]. dynamicsinneocorticalnetworksconvergeto oneofsev-
0
1 More recently, the analysis of neurobiologicaldata using eral different persistent, stable patterns which resemble
1 graph theoretic techniques [11] has further established the behavior observedin such models [23]. However,the
: the modular nature of inter-connections between differ- properties of attractor networks are of more general in-
v
i entareasofthemammaliancortex. Thestructuralmod- terestandhavebeenusedtounderstandsystemsoutside
X
ules revealed by tracing the anatomical connections in the domain of neurobiology,as for example, the network
r mammalian brains [12, 13] are complemented by the ob- involved in intracellular signaling where communication
a
servation of functionally defined networks having mod- between molecules within a cell take place through mul-
ular character [14, 15]. Such functional networks have tiple interacting pathways [24, 25]. In the attractor net-
been reconstructed from MRI and fMRI experiments on works,desiredpatternsarestoredbyusingalearningrule
both human [16] and non-human [17] subjects, by con- to determine the connection weights between the nodes.
sidering two brain areas to be connected if they are si- This ensures that the update (or recall) dynamics of the
multaneouslyactivewhenthesubjectperformsaspecific networkmakes it convergeto these pre-specified dynam-
behavioral task. ical states when an input initial state of the system is
transformed into an output state defined over the same
Thewide-spreadoccurrenceofmodularitypromptsthe
set of nodes by the collective dynamics of the network.
questionastowhythisstructuralorganizationissoubiq-
Usingsuchsimplifiedmodelshavetheadvantageofmak-
uitous [18]. One possible reasonis that it enhances com-
ing the observedphenomena simpler to analyze and also
munication efficiency by decreasing the average network
to obtain results that are independent of specific bio-
pathlengthwhileallowinghighclusteringtohelplocalize
2
logical details of different types of neurons and synaptic (a) r = 0 (b) r = 0.1 (c) r = 1
connections.
In this paper we show that if we want to store p (say)
patterns in a network with a given number of nodes and
links,thentheconvergencetoanattractorcorresponding
toanyofthe storedpatterns(i.e.,recall)willbemostef-
(d)
ficient when the network has an optimal modular struc-
ture, provided the number of patterns is not too large
(p < p ). If the degree of modularity is increased or
max
decreased from the optimum value, the reliability with
which the patterns are recalled decreases. This optimal
efficiency of recalloriginates from the network dynamics
itself. Some of the modules converge quickly to attrac-
FIG.1: (a-c)AdjacencymatricesAdefiningthenetworkcon-
tors corresponding to parts of stored patterns and then
nectionsatdifferentvaluesofthemodularityparameterr for
help other modules to reach the attractor correspond-
N = 256 nodes arranged into nm = 4 modules (average de-
ing to the entire stored pattern through interactions via gree hki = 60). Starting from a system of isolated clusters
intermodular links. If the modularity is increased (i.e., (a, r = 0), by increasing r we obtain modular networks (b,
if the number of intra-modular links is increased while r = 0.1) eventually arriving at a homogeneous network (c,
reducing the number of inter-modular links to keep the r=1). Theconnection structureof modular networksin the
intermediate range 0 < r < 1 is shown schematically in (d).
average degree fixed), the modules cannot interact with
The connection weights havedifferent magnitudes and signs.
each other strongly enough due to fewer number of in-
termodular links and the performance of the network is
less efficient. On the other hand, if the modularity is where, W is the connection strength between neurons
ij
decreased,the modules themselves become sparsely con- i and j. The function sign(z) = 1, if z > 0, = −1, if
nected and cannot reach an attractor rapidly. Also, if z < 0 and randomly chosen to be ±1 if z = 0. The
we try to store a larger number of patterns (p ≥ pmax), weight associated with each link is evaluated using the
the advantage of modularity disappears because of the Hebbian learning rule [22] for storing p random patterns
generation of a large number of spin-glass states which in an associative network:
correspond to spurious patterns.
1
The attractor network model we have used to investi- Wij = hkiΣµξiµξjµ, Wii =0, (2)
gate the role of modularity is constructed such that the
N nodescomprisingitaredividedintonm modules,each ξiµ being the i-th component of the µ-th pattern vector
havingn(=N/nm)nodes[19]. Theconnectionprobabil- (µ = 1,...p). Each of the stored patterns are gener-
ity between a pair of nodes belonging to the same mod- ated randomly by choosing each component to be +1 or
ule is ρi, while thatbetweennodesbelonging todifferent −1 with equal probability. Starting from an arbitrary
modules is ρo. The modular nature of the network can initial state, the network eventually convergesto a time-
be varied continuously by altering the ratio of inter- to invariant stable state or attractor. The overlapof an at-
intra-modular connectivity, r = ρo ∈ [0,1], keeping the tractor of the network dynamics S∗ = {σ∗} with any of
average degree hki fixed (Fig. 1)ρ.i For r = 0, the net- the stored patterns can be measuredas miµ = N1Σiσi∗ξiµ.
work is fragmented into n isolated clusters, whereas at As we are interested in the set of all the attractors of
m
r = 1, it is a homogeneous or Erdos-Renyi random net- stored patterns rather than one specific pattern, we fo-
work. We ensure that the resulting adjacency matrix A cus our attention on the maximum overlap with the
(i.e., A = 1 if i,j are connected, and 0, otherwise) is stored patterns, m = max |m |. To examine the global
ij µ µ
symmetric. We have explicitly verified that the results stability of the attractors corresponding to the stored
reported below do not change appreciably if A is non- patterns, we use random strings as the initial state of
symmetric (corresponding to a directed network). the network which should have almost no overlap with
The time-evolution of the system is governed by the any of the stored patterns, on average. The probabil-
dynamics of the variables associated with each node of ity vg ≡ hProb(m > mo)i that such a random initial
thenetwork. AnIsingspinσ =±1isplacedateachnode state eventually almost converges to one of the stored
i
which may represent any binary state variable, such as patterns, gives an estimate of the overall volume that
a two-state neuron (firing=1, inactive=−1). The state the basins of attraction of stored patterns occupy in the
of the spins are evaluated at discrete time-steps using N-dimensional network configuration space {S}. Here
random sequential updating according to the following mo is a threshold for the overlap of the asymptotic sta-
deterministic (or zero temperature) dynamics: ble state above which the network can be considered to
have recalled a pattern successfully and h...i indicates
σ (t+1)=sign(Σ A W σ (t)), (1) averagingovermany different networkconfigurations A,
i j ij ij j
3
1 1 terns ξµ(α). As the recall dynamics within each module
(a) (b)
isnearlyindependentoftheothermodulesforlowr,they
0.8 0.8
mayeachconvergetosub-partsofdifferentpatterns,i.e.,
p= 2
0.6 0.6 p= 3 thevalueofµwouldnotbeidenticalfortheattractorsof
vm vg p= 6 allthe nm modules. Thus,theresultingattractorforthe
0.4 p= 3 0.4 p= 8 entirenetworkcorrespondstoa“chimera”memorystate,
0.2 pp== 68 0.2 p=10 {ξµ1(1),...,ξµnm(nm)},i.e.,aspuriouspatterncompris-
p=10 ing fragments of different stored patterns [26, 27].
0 0
10−3 10−2 10−1 100 10−3 10−2 10−1 100 From the perspective of enhanced robustness of the
r r
dynamical attractors of the entire network, even more
interesting is the behavior of v and v when r is in-
g m
creased further after the modules have become inter-
FIG. 2: Fractional volume of phase space occupied by the
basins of attraction of the stored patterns in a single mod- connected appreciably. Over an intermediate range of
ule (vm) and the entire random modular network (vg). Note pmin <p<pmax,wenoticeanon-monotonicvariationof
that, when the number of stored patterns is within a crit- bothv andv withrespecttor. Fig.2showsthatboth
g m
ical range (pmin = 2 < p < pmax = 9), these quantities curves attain a maximum around r ∼ n−1 ≃ 1 ,
showanon-monotonicvariationwithr,havingapeakaround c N−n nm−1
whereaneuronhasthesamenumberofconnectionswith
rc ≃ 1/(nm −1) ∼ 0.14. Results are shown for N = 1024, nodesbelongingtoitsownmoduleasithaswithneurons
nm =8 and hki = 120. Different numbers of stored patterns
belonging to different modules. When the relative num-
p are indicated using various symbols.
berofinter-modularconnectionsareincreasedbeyondr ,
c
the fractionalvolume of configurationspace occupied by
as well as, pattern ensembles {ξ} and initial states. The the attractors corresponding to the stored patterns tend
value of the threshold m has been taken to be 0.95 for to decrease. This implies that the homogeneous network
o
mostoftheanalysispresentedhere;wehaveverifiedthat (r =1) is actually less robust than its modular counter-
varying it over a small range does not alter our results. part (r ≃ rc) in terms of global stability of the stored
In a similar way, we can define overlap for each module, attractors. As p increases beyond pmax, both vg and
mµ(α)= n1Σiσi∗(α)ξiµ(α) where the sum is over all spins vm decrease at the resulting high loading fraction p/hki
in the α-th module with α = 1,...,n being an index through the generation of a large number of spin-glass
m
running over the different modules. The relative size of states [22]. We have explicitly verified that the maxi-
thebasinsofattractionatthemodularscaleischaracter- mum number of stored patterns pmax beyond which the
izedbythequantityv =hhProb(m(α)>m )i i,where non-monotonic nature of the variation is lost, increases
m o α
m(α)=max |m (α)|andh...i indicatesaveragingover when the total number of neurons N is increased, keep-
µ µ α
all the modules. ing the overall density of connections, hki/(N −1), and
We firstlookathowthetotalvolumeoftheconfigura- the number of modules, nm, fixed [28].
tionspaceoccupiedbythebasinsofattractionforstored For low values of p, i.e., p ≤ p , both v and v
min g m
patterns ξµ changesasthe modularcharacterofthe net- increase with r eventually reaching 1 and becoming in-
work is alteredby varying r for a fixed hki. Fig. 2 shows dependent of r once the connectivity between the mod-
the combinedfractional volume of the phase space occu- ules become appreciable. We find from our numerical
piedbythebasinsofattractionofthestoredpatternsfor results that p = 2, independent of the system size
min
the entire network (v ) as well as for the corresponding N or other model parameters. This observation helps
g
sub-patternsinasinglemodule(v ). Differentcurvesin- in identifying the key mechanism for the non-monotonic
m
dicate various number of stored patterns p. We immedi- variationofv withr. Whileatlowr,v issmallbecause
g g
atelynoticethatwhilev hasfinitevaluesovertheentire the low connectivity among modules favor the chimera
m
range of r, v is zero at low values of r where a module states, at very large r the attractors corresponding to
g
is connectedto the restofthe networkbyveryfew links, the stored patterns have to compete with mixed states.
if at all. The value of r at which v starts rising from 0 Mixed states are spurious attractors that correspond to
g
appears to be independent of the number of stored pat- symmetriccombinationsofanoddnumberofstoredpat-
terns p. Below this value of r, the connectivity between terns (e.g., ξ1+ξ2+ξ3) which exist for all p > 2. This
the modules is insufficientto recallthe entire storedpat- is explicitly shown by the distribution of the overlap, m,
tern,eventhoughindividualmodulesmayhavecomplete of the attractors of a network with any of the p stored
overlapwithdifferentstoredpatterns. Toexplainthesit- patterns (showninFig. 3 for p=4). For low values ofr,
uation,wecandecomposeeachstoredpatternintermsof thedominanceofchimerastatesresultinlowoverlapval-
n sub-patterns defined overthe different modules, viz., ues. When the modules become highly inter-connected
m
ξµ = {ξµ(α)}, where α = 1,...,n . Starting from a as r → 1, most randomly chosen initial strings will con-
m
random initial state, a module α may convergeto an at- verge to a stored pattern resulting in a large peak at
tractor corresponding to any of the n different subpat- m =1 in the overlap distribution. However, we also no-
m
4
5
( a )
( a )
102 ps) 4 〈 τg 〉
e
st 3
C
m )100 〈τ〉 (M 12 〈 τm 〉
P ( 0
10−2
ps) 2 ( b ) ∆ τg
e
1 st
0.8 C
M 1
m0.60.40.2 10−1 r 10−2 10−3 ∆τ ( 0 ∆ τm
( b ) 0 100 10−3 10−2 10−1 100
r
1
total mixed states(cid:9)
mix 0.5 same sign FIG. 4: (a) The average convergence time τ to different at-
f different sign tractors in a random modular network, shown for individual
modules(hτmi,circles)andtheentirenetwork(hτgi,squares).
0
10−2 10−1 100 It is measured in terms of Monte Carlo (MC) steps required
r toreachatime-invariantstatestartingfrom arandominitial
configuration. (b) The difference in the average convergence
times (in MC steps) to an attractor not corresponding to a
FIG. 3: (a) Distribution of the overlap of the attractors of stored pattern (m<0.95) and to one of the stored patterns,
the network dynamics with the stored patterns in a random ∆τ. The difference is shown for both an individual module
modularnetwork,atdifferentvaluesofthemodularityparam- (∆τm, circles) and the entire network (∆τg, squares). The
eter r. P(m) is the probability of having overlap m. Com- peak close to rc ∼ 0.14 corresponds to a significantly faster
pleteoverlapwiththestoredpatterns(m=1)becomesmore convergence to the stored patterns relative to the other at-
probable as r becomes larger than a threshold value. How- tractors. Results shown for N = 1024, nm = 8, hki = 120
ever, at large values of r, there is a secondary peak around and p=4.
mg ∼0.5correspondingtomixedstates(i.e.,linearcombina-
tion of odd number of stored patterns). This peak shows a
dip at rc ≃1/(nm−1)∼0.14. (b) The variation, as a func- differentsigns(e.g.,ξ1−ξ2+ξ3). Thecurvescorrespond-
tionofr,ofthefractionoftotalnumberofspuriousattractors
ing to each of these show that although the latter has a
that are mixed states, fmix. For r>rc, the mixed states ac-
higher number of possible combinations, it is the attrac-
countforalmostalltheattractorsnotcorrespondingtoanyof
thestored patterns. They can beeithercombinationshaving tors corresponding to the same sign combinations which
the same sign (square) or different signs (diamond). Results occupyalargerportionofthephasespace. Thisisacon-
shownforN =1024,nm =8,hki=120andnumberofstored sequence of the Hebbian learning rule, which provides a
patterns, p=4. bias for the same sign combinations in preference to the
different sign combinations.
So far we have discussed the long-time asymptotic
tice a smaller peak around m ≃ 0.5, which corresponds properties of the system. The dynamical aspect repre-
to3-patternmixedstates(whichhaveoverlapof0.5with sented by the time required to reach equilibrium also
eachofthe threeconstituentstoredpatterns). Note that exhibits unexpected properties. Fig. 4 shows that the
as r is gradually decreased from 1, about r ≃ rc the m network converges faster to attractors corresponding to
distribution shows a sharp dip for overlaps around 0.5. stored patterns as compared to mixed states (and other
This corresponds to an increase in the phase space vol- attractors that do not have significant overlap with any
ume occupied by the attractors of the stored patterns at of the stored patterns), at both the modular and the
theexpenseofthemixedstates. Asimilardipinthe dis- network level. Moreover, this difference is slightly en-
tribution is also observed for the corresponding overlap hanced close to r , the modular configuration where the
c
around 0.5 for each module (figure not shown). Thus, basins of the stored patterns cover the largest fraction
the cooperative interactions between the different mod- of the configuration space. The non-monotonic varia-
ulesnotonlyaffecttherecalldynamicsatthegloballevel, tion of the convergence time with decreasing modularity
but also locally within each module. arises as a result of two competing effects: increasing r
Fig. 3 (b) shows explicitly that the attractors not cor- decreasesthe intra-modularconnectivity, resulting inin-
responding to any of the stored patterns, belong almost creasing time for each module to relax to an attractor;
exclusively to mixed states at high r. In principle, these onthe other hand, this is accompaniedby anincrease in
combinations can be of same sign (e.g., ξ1+ξ2+ξ3) or theconnectionsbetweenmodules,thateventuallycauses
5
the entire system to relax faster to attractors. This dy- tems (XI Plan) Project.
namicalpictureprovidesuswithapossibleclueastothe
enhanced global stability of the attractors correspond-
ing to stored patterns close to r . As there is a distinct
c
time-scale separation between the convergence dynam-
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6
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