Table Of ContentPersistentmemoriesintransientnetworks
Andrey Babichev1,2 and Yuri Dabaghian1,2∗
1Jan and Dan Duncan Neurological Research Institute,
Baylor College of Medicine, Houston, TX 77030,
2Department of Computational and Applied Mathematics,
Rice University, Houston, TX 77005
∗e-mail: [email protected]
(Dated: February3,2016)
6 Spatialawarenessinmammalsisbasedonaninternalizedrepresentationoftheenvironment,en-
1 codedbylargenetworksofspikingneurons. Whilesuchrepresentationscanlastforalongtime,the
0 underlyingneuronalnetworkistransient: neuronalcellsdieeveryday,synapticconnectionsappear
2
and disappear, the networks constantly change their architecture due to various forms of synaptic
n and structural plasticity. How can a network with a dynamic architecture encode a stable map of
a
space? Weaddressthisquestionusingaphysiologicalmodelofa“flickering”neuronalnetworkand
J
demonstratethatitcanmaintainarobusttopologicalrepresentationofspace.
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I. INTRODUCTION
Itisbelievedthatmammalianabilitytonavigate,toescapefrompredators,tofinditsnest,toplanhunting
strategiesandsoforth,isbasedonaninternalizedneuronalrepresentationofspace—acognitivemapofthe
environment [1, 3]. Neurophysiologically, this map is encoded via timed sequences of quick electrical
discharges of the neurons—neuronal spikes—in various parts of the brain. A particularly important role
in producing this map is played by the hippocampus—one of the oldest parts of the mammalian brain
in evolutionary terms. The spike times of the hippocampal principal neurons—so called place cells—are
defined by the animal’s spatial location: in rodents, each place cell fires when the animal visits a specific
location—this cell’s place field [3] (Fig. 1A). Why a given place cell fires only when an animal is “here”
ratherthan“there”isstillamystery, andhowtheensembleofplacecellsformsahippocampalmapofthe
environmentisequallymysterious. Inparticular,itremainslargelyunknownhowthespiketrainsproduced
bytheplacecellsareprocesseddownstreamfromthehippocampus.
Experimental evidence points out that groups of coactive place cells form functionally interconnected
“assemblies” [4, 5] that together drive their respective “read-classifier” or “readout” neurons in the down-
stream networks (Fig. 1B). Since coactivity of the place cells marks the overlap of their respective place
fields (Fig. 1C), the activity of a readout neuron actualizes a connectivity relationship between the regions
encoded by the individual place cells. This suggests that the hippocampus provides a qualitative represen-
tation of space, based on connectivity, adjacency and containment relationships, i.e., that the hippocampal
cognitive map is topological in nature—a hypothesis that receives an increasing amount of experimental
support[6–10].
In [11, 12] we proposed a theoretical model that showed that hippocampal cell assembly network can
indeed capture the spatial connectivity of the environment in a biologically plausible time, given that the
place cells operate within biological parameters of firing rate and place field size. However, the approach
of [11, 12] was based on analyses of the ever-growing pool of spike trains, i.e., it ignored that, in physio-
logical networks, the connectivity information may not only accumulate, but also decay. In particular, the
physiological cell assemblies may not only form, but also disband as a result of deterioration of synaptic
connections[13],thenreappearasaresultofsubsequentlearning,thendisbandagainandsoforth. Electro-
physiologicalstudiessuggestthatthelifetimeofthecellassembliesrangesbetweentensofmillisecondsto
minutesorlonger[4,5,14,15],whereasspatialmemoriesinratscanlastfordaysandmonths[16–18]. How
canthelarge-scalespatialrepresentationoftheenvironmentbestableiftheneuronalstratumthatcomputes
thisrepresentationchangesonamuchfastertimescale? Belowwediscussamodelofatransienthippocam-
pal network and use methods of Algebraic Topology to demonstrate that the topological characteristics of
thelarge-scalespatialrepresentationoftheenvironmentencodedbythisnetworkcanremainstable.
II. MODEL
The way place fields cover an environment E (Fig. 1A) calls to mind a basic theorem of algebraic
topology due to P. Alexandrov [19] and E. Cˇech [20]: it is possible to reconstruct the topology of a space
X from the pattern of overlaps between the regions that cover it. The model proposed in [11, 12, 21]
is based on the observation that this theorem can not only be applied to the place fields themselves, but
also implemented via spiking signals of cells that represent these place fields. In this approach, groups of
coactiveplacecells,c ,c ,...,c ,areviewedasabstractsimplexes,σ = [c ,c ,...,c ],whichtogetherform
1 2 n 1 2 n
asimplicial“cellassemblycomplex,”T (fordefinitionsanddetailssee[22,23]andMethodsin[11]).
CA
Thisconstructionprovidesaconnectionbetweenthelocal(cellular)andtheglobal(system-level)scales:
the individual cell assemblies, just like simplexes, provide local information about the space, but together,
as a neuronal ensemble, they represent space as whole—as the simplicial complex. Thus, T provides
CA
a schematic representation of the cell assembly network and of its rewiring dynamics: formation of new
3
FIG. 1: Place cells and cell assemblies. (A). Simulated place fields in a small planar environment containing a
square hole: dots of different color mark the locations where the corresponding place cells produced spikes. The
dots of a particular color form spatial clusters—the place fields. (B). Schematic representation of four overlapping
third-ordercellassembliesbyfourthirdordersimplexes(triangles). Thevertexes(shownbysmallcircles)represent
placecellssynapticallyconnectedtotheirrespectivereadoutneurons(pentagons). Acellassemblyσ“ignites”(red
triangle)whenallofitsplacecellsbecomesimultaneouslyactive,thuselicitingaresponsefromthereadoutneuron
n (active cells have colored centers). (C). A coactivity of the place cells c , c and c occurs when the rat crosses
σ 1 2 3
throughtheoverlapofthecorrespondingplacefields—thedomain123.
placecellassembliesanddisbandingofsomeoldonesarerepresented,respectively,bytheappearanceand
disappearanceoftheircounterpart(maximal)simplexesinT .
CA
On the other hand, the cell assembly complexT provides semantics for describing the global spatial
CA
memory map in topological terms [24]. For example, a sequence of cell assemblies ignited along a path
γ navigated by the rat corresponds to a chain of simplexes Γ ∈ T —a “simplicial path” (Fig. 2). The
CA
structure ofT allows to establishcertain qualitative properties ofthe simplicial paths andto relate them
CA
tothepropertiesofthephysicalpathsinE. Forexample,asimplicialpaththatclosesontoitselfinT may
CA
represent a closed physical path, a pair of topologically equivalent simplicial paths Γ ∼ Γ may represent
1 2
physical paths γ and γ that can be deformed into one another, a hole in T may represent a physical
1 2 CA
obstacleandsoforth. However,establishingthesecorrespondencesrequireslearning: astheanimalbegins
to explore the environment, only a few place cells would have time to (co)activate, and only a few cell
assemblies would have time to form; as a result, a newly developing complex T consists of only a few
CA
maximalsimplexesandcontainsmanyholes,someofwhichcorrespondtophysicalobstaclesortoregions
thathavenotyetbeenvisitedbytheanimal,andothersare“spurious”,i.e.,reflectingtransientinformation
about place cell coactivity. As shown in [11, 12], the spurious holes tend to disappear as more spatial
informationaccumulates.
In[25]wesuggestedtwomethodsforconstructingthecellassemblynetworkandhenceproducingthe
simplicialcellassemblycomplexT thatrepresentsthisnetwork,byselectingthemostfrequentcombina-
CA
tionsofspikingplacecells. Thekeyobservationof[25]wasthatinboundedenvironments,thecoactivecell
combinationseventuallybecomerepetitive,anditisthereforepossibletoidentifythecellassembliesfrom
themostfrequentcoactivitypatterns. However,thefrequencyofagivencellassembly’sactivation, f ,was
σ
evaluated from the total number of its appearances over the entire navigation period, and then the selected
cellassemblieswerepresumedtohaveexistedforaslongasthenavigationcontinued. Inotherwords,the
cellassemblieswere“perennial”byconstruction. Inthefollowing,wemodelatransienthippocampalnet-
workbylimitingthetimeduringwhichacellassemblyσcanformtoasmallertimeinterval—“amemory
window” W(σ). Physiologically, W(σ) defines the period during which readout neuron n can identify the
σ
combination σ of coactive place cells, connect to it synaptically, retain these connections and respond to
subsequentignitionsofσ.
To simplify the approach, we consider the case in which the entire ensemble of readout neurons is
characterizedbyasingleparameterW(σ) = W,andproceedasfollows. First,weidentifythecellassemblies
4
FIG.2: Scehamtic representation of
place cell coactivity in the cell assembly
complexT . Thetopologicalstructureof
CA
T , induced from the place field layout
CA
inE, providesatopologicalrepresentation
of E. For example, the hole in the middle
of T , which produces non-contractible
CA
simplicial paths (topological loops) corre-
sponds to the central hole in the environ-
ment E, which produces non-contractible
navigation paths γ (e.g., a sample path is
shownatthebottom).Astheanimaltravels
alongγ,thehippocampalplacecellassem-
blies ignite in a certain order, which cor-
responds to a chain of maximal simplexes
inT —asimplicialpathΓ,shownbyred
CA
trianglesandtetrahedrons.
that emerge within the first W-period after the onset of the navigation, W , and then repeat the algorithm
1
for the subsequent windows W , W ,..., obtained by shifting the W over small time steps. As a result, a
2 3 1
cellassemblyσthatwasidentifiedinthewindowW ,basedonthelocalplacecellcoactivityrate f (W ),
t1 σ t1
may disappear at a certain step W , then reappear in a later window, W , disappear again and so forth.
t2 t3
The ensemble of appearing and disappearing cell assemblies can then be schematically represented by a
“flickering” simplicial complex, F , with appearing and disappearing maximal simplexes. Our task is to
W
studythenettopologicalpropertiesofF ,e.g.,whetherthelifetimesofitstopologicalloopscanbelonger
W
thanthelifetimesofitstypicalmaximalsimplex.
III. RESULTS
We studied the dynamics of the flickering complex F and the topological information encoded in it,
W
as a function of the discrete time t = n∆t, ∆t = 2.5 secs, for the memory window width W = 5 minutes.
n
Theresultsproducedbyatypicalneuronalensemblecontaining N =300simulatedplacecellsareshown
c
on Fig. 3. First, we observe that the fluctuations of the number of the maximal simplexes in F , N ,
W σ
remain confined within about 20% from the mean, 220 ≤ N ≤ 300, so that the simulated hippocampal
σ
networkcontainsabout260fluctuatingcellassembliesonaverage(Fig.3A).However,whilethesizeofthe
flickeringcomplexremainsbounded,thepoolofthemaximalsimplexeschangessignificantly: astemporal
separation∆ = |t −t |betweenthememorywindowsincreases,thecorrespondingcomplexesF (t)and
ij i j W i
F (t ) become more and more dissimilar (Fig. 3B). After about 50 time steps (∼ 120 seconds), the set of
W j
simplexes in F is essentially renewed, which implies that the cell assembly network, as described by the
W
model,iscompletelyrewired(Fig.3C).
A typical maximal simplex lasts on average about 10 discrete time steps (τ ≈ 25 seconds), which
σ
is close to the range of values established experimentally [5]. Such rapid rate of the simplex renewal in
the flickering cell assembly complex allows us to address our main question: what is the dynamics of the
topologicalcharacteristicsofF andhowdotheycorrespondtothetopologyoftheenvironment?
W
The topology of the environment can be described in terms of the Betti numbers—roughly speaking, a
Bettinumberb definesthenumberofn-dimensionaltopologicalloopsinF (i.e.,closedsurfacescounted
n W
up to topological equivalence). In the case of the environment shown on Fig. 1A, the Betti numbers are
as follows: b = 1 (i.e., the environment is connected), b = 1 (i.e., there is one 1D topological loop
0 1
that encircles the hole in the middle), while b = 0 (no topological loops in higher dimensions). Using
n>1
the methods of Persistent Homology [26–28], we evaluated these numbers for the flickering cell assembly
5
FIG. 3: Time evolution of the cell assembly complex. (A) The number of maximal simplexes (blue trace) and
thetotalnumberofallsimplexes, scaleddownbyafactorof10(red)intheflickeringsimplicialcomplexF . The
W
memory window spans over W = 5 minutes, shifting over ∆t = 2.5 secs at a time. Shown are first 200 shifts. (B)
Thematrixofsimilaritycoefficients, r , betweenthepairsofflickeringcellassemblycomplexesF (t)andF (t )
ij W i W j
definedastheproportionofthemaximalsimplexescontainedinF (t)butnotinF (t ). Forclosemomentst andt ,
W i W j i j
thedifferencesaresmall,butastimeseparationgrows,thedifferencesbecomeslarger. (C)Ateverymomentoftime
t,thebluelineshowsthefractionofthemaximalsimplexesthatwerealsopresentatthepreviousmoment,t . The
i i−1
greenlineshowsthefractionoftheoriginalmaximalsimplexes(fromF (t ))remaininginF (t). Thepopulationof
W 1 W i
simplexeschangesentirely(byabout0.95%)inabout60steps. Thedashedlinemarksthemeansimplexdecayrate,
τ =10windowshifts(about25seconds).
σ
complex for the sequence of windows, and found that F does, in fact, reliably capture the topological
W
properties of the environment over long periods, which significantly exceed both the simplexes’ lifetimes,
τ ,andthewidthofthememorywindow,W.
σ
AsillustratedonFig.4A,thefirstandthesecondBettinumbersalmostneverdeviatefromtheirrespec-
tivephysicalvalues(b = 1andb = 0). Theoccasionalchangesofb andb canbeviewedasshort-time
1 2 1 2
“topological fluctuations” in the hippocampal map. For example, an occurrence of b = 2 value indicates
1
theappearanceofanextra(non-physical)1Dloop,andatthemomentswhenb = 0,all1DloopsinF are
1 W
contractible, i.e., the central hole (Fig. 1A) is not represented in the hippocampal map. Similarly, the mo-
mentswhenb (cid:44) 0indicatetimeswhentheflickeringcomplexF containsnon-physical,non-contractible
2 W
2D bubbles (one can speculate about the cognitive effects that these fluctuations may produce). The 0-th
and the third Betti number always came out correct, b = 1, b = 0, implying that despite the fluctuations
0 3
of its simplexes, the cell assembly complex F does not disintegrate into pieces and remains contractible
W
inhigher(D > 2)dimensions,i.e.,thatthetopologicalfluctuationsinthehippocampalmapareboundedto
1Dpathsand2Dsurfaces.
Further numerical analyses demonstrate that, as the memory window increases, the Betti numbers b
1
andb becomemorestable. Incontrast,asthememorywindowshrinks,thefluctuationsofthetopological
2
loops intensify (Fig. 4B). This implies that if the characteristic timescale of the network’s transience is
sufficientlylarge, thenthecorrespondingcoactivitycomplexremainsfixedandtopologicallyequivalentto
the“perennial”cellassemblycomplex,FW=∞ = TCA. Ontheotherhand,ifthesimplexesaretoounstable,
6
FIG.4: Stabilityofthetopologicalinformation. (A).Thelow-dimensionalBettinumbers,b ,b ,b asafunction
1 2 3
ofthediscretetime,computedusing5minutewidememorywindowW. The0thBettinumber(b =1atalltimes)is
0
notshown. (B).AsthememorywindowdecreasestoW =3minutes,thefrequencyandthenumberofthetopological
fluctuationsincreases: theflickeringcomplexrepeatedlyproducesanextra(spurious)1Dtopologicalloopandupto
twospuriousbubbles.
thenthecellassemblycomplexT failstoacquirethecorrectphysicalstructure,nomatterhowlongisthe
CA
learningperiod.
IV. DISCUSSION
Inphysicalliterature,fluctuatingsimplicialcomplexeshavebeenpreviouslyusedinthecontextofSim-
plicialQuantumGravitytheoriesfordiscretizingquantumfluctuationsofthespace-time[29,30]. Anatural
requirementforthesetheoriesisthatthequantumfluctuationsatthemicro-scaleshouldaverageoutinthe
thermodynamiclimit,yieldingasmoothspace-timecontinuumatthemacroscale.
Therearecertainparallelswithcognitiverepresentationsofspacethatemergefromthespikingactivity
ofneuronalensembles. Atthemicrolevel,theencodedspatialinformationisnaturallydiscreteatthecellu-
larlevelandallowsaschematicrepresentationintermsofsimplicialcomplexes[11,12,24,25]. Moreover,
since the various place cell assemblies that spike at different locations are transient structures, these auxil-
iary simplicial complexes fluctuate. On the macro level, the emergent hippocampal maps provide a stable
topologicalrepresentationofthephysicalenvironmentoverlongperiods[10,16–18],enablingtopological
reasoningduringanimals’spatialnavigation[8,9].
Ourmodelprovidesoneexplanationforhowthesetwoexperimentallyestablishedaspectsofhippocam-
pal neurophysiology can coexist. According to the model, while the simplexes of the flickering simplicial
complex F fluctuate at the physiological timescale, its Betti numbers can keep their physical values in-
W
definitely. Thefactthatthesimplexfluctuationscanaverageoutatthescaleoftheentirecomplexsuggests
thatrapidrewiringofthecellassembliescanpreservethestabilityofthecognitivemapasawhole.
Althoughtheseresultswereobtainedusingasimplecomputationalmodel, wehypothesizethattheob-
servedeffectreflectsamoregeneralphenomenonthatmightapplytophysiologicalmechanismsofsynaptic
andstructuralplasticityinthehippocampalnetwork.
V. METHODS
PlacecellspikingactivityismodeledasastationarytemporalPoissonprocesswithaspatiallylocalized
rate [31] that is characterized by the spatial location of the peak, r , the peak firing amplitude, f , and the
c c
place field size, s . We simulated an ensemble of N = 300 place cells with log-normally distributed peak
c c
7
firing amplitudes, with the mean (cid:104)f (cid:105) = 14 Hz, log-normally distributed place field sizes with the mean
c
(cid:104)s (cid:105) = 17cm(fordetailsseeMethodsin[11]). Spikingismodulatedbythe8Hzθ-oscillations—thebasic
c
rhythm of the extracellular field in the hippocampus [32]. Neuronal coactivity is defined as firing that oc-
curredovertwoconsecutiveθcycles—acomputationallyestablishedoptimalvalue[12],whichcorresponds
to the parameters estimated from experimental data [32, 33]. The topological analyses were implemented
usingtheJPlexpackage[34].
VI. ACKNOWLEDGMENTS
TheworkwassupportedbytheNSF1422438grantandbyHoustonBioinformaticsEndowmentFund.
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