The spatial structure of correlated neuronal variability

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1 r t i c l e s The sptil structure of correlted neuronl vriility Roert Rosenum,, Mtthew A Smith 3 5, Adm Kohn 6,7, Jonthn E Ruin 5,8 & Brent Doiron 5,8 6 Nture Americ, Inc., prt of Springer Nture. All rights reserved. Shred neurl vriility is uiquitous in corticl popultions. While this vriility is presumed to rise from overlpping synptic input, its precise reltionship to locl circuit rchitecture remins uncler. We comine computtionl models nd in vivo recordings to study the reltionship etween the sptil structure of connectivity nd correlted vriility in neurl circuits. Extending the theory of networks with lnced excittion nd inhiition, we find tht sptilly loclized lterl projections promote wekly correlted spiking, ut roder lterl projections produce distinctive sptil correltion structure: nery neuron pirs re positively correlted, pirs t intermedite distnces re negtively correlted nd distnt pirs re wekly correlted. This non-monotonic dependence of correltion on distnce is reveled in new nlysis of recordings from superficil lyers of mcque primry visul cortex. Our findings show tht incorporting distnce-dependent connectivity improves the extent to which lnced network theory cn explin correlted neurl vriility. The spiking ctivity of corticl neurons is often chrcterized y their verge response over lrge numer of trils, prompting welth of theoreticl studies relting the structure of neuronl networks to their tril-verged firing rte dynmics. However, tril verges do not cpture the stochstic nd irregulr dynmics chrcteristic of corticl popultions nd the nervous system in generl. Indeed, tril-to-tril fluctutions re centrl to contemporry theories of corticl computtion 3,4. A deep mechnistic understnding of neuronl vriility remins n open chllenge. Erly theoreticl studies deduced tht vrile spiking ctivity could rise through lncing of strong, yet opposing, excittory nd inhiitory synptic inputs 5,6. Expnding on this conjecture, vn Vreeswijk nd Sompolinsky 7 showed tht networks of recurrently coupled model neurons roustly crete stte where strong excittion is pproximtely lnced y inhiition, creting push-pull dynmic tht genertes irregulr spiking ctivity. More recently, lnced networks hve een implicted in theories of optiml coding 8, working memory 9 nd stimulus tuning. Numerous experimentl studies hve estlished tht excittion is often pproximtely lnced y inhiition in corticl circuits 7. In sum, lnced networks provide prsimonious model of the irregulr spiking ctivity oserved in corticl circuits. Erly lnced network models produced synchronous ctivity through sprse connectivity 7,8. However, severl experimentl studies revel tht locl corticl networks re densely connected, with connection proilities etween nery neurons sometimes exceeding 4 percent 9. These dt imply sustntil overlp etween locl synptic inputs, which could, in principle, synchronize corticl networks. However, counter to intuition, lnced networks with dense connectivity show wek spike trin correltions 3. This synchronous stte results from the correlted excittory (e) or inhiitory (i) fferents to neuron pirs eing ctively cncelled y strong negtive e i correltion, estlishing wek correltions even when connectivity is not sprse 3. Consistent with the predicted synchronous stte, some multiunit extrcellulr recordings show noise correltions tht re nerly zero on verge 4. However, mjority of popultion recordings in cortex revel comprtively lrge correltions 5,6. Severl studies suggest tht the mgnitude of noise correltions is dependent on mny fctors 7, including rousl 8, ttention 9, nesthetic stte 3,4,3,3 nd corticl lyer 3,33. Finlly, while in vivo whole-cell recordings revel strong positive e e nd i i correltions coexisting with strong e i correltions 3, these correltion sources do not lwys perfectly cncel s predicted y some theoreticl models 8. Tken together, these studies show tht corticl circuits cn exhiit oth wek nd moderte noise correltions, t odds with predictions from the current theory of lnced networks 3. In this study, we generlize the theory of correltions in densely connected, lnced networks to include the widely oserved dependence of synptic connection proility on distnce,34. We show tht sptilly rod recurrent projections disrupt the synchronous stte, producing signture sptil correltion structure: nery pirs of neurons re positively correlted on verge, pirs t intermedite distnces re negtively correlted nd distnt pirs re wekly correlted. These positive nd negtive correltions cncel so tht the verge correltion etween pirs smpled rndomly over lrge rnge of distnces is nerly zero. We uncover this non-monotonic dependence of correltion on distnce in recordings from superficil lyers of mcque primry visul cortex, ut only fter correcting for ltent source of shred fluctutions. Our findings decouple lnced excittion nd inhiition from synchronous network Deprtment of Applied nd Computtionl Mthemtics nd Sttistics, University of Notre Dme, Notre Dme, Indin, USA. Interdisciplinry Center for Network Science nd Applictions, University of Notre Dme, Notre Dme, Indin, USA. 3 Deprtment of Ophthlmology, University of Pittsurgh, Pittsurgh, Pennsylvni, USA. 4 Fox Center for Vision Restortion, University of Pittsurgh, Pittsurgh, Pennsylvni, USA. 5 Center for the Neurl Bsis of Cognition, Pittsurgh, Pennsylvni, USA. 6 Deprtment of Ophthlmology nd Visul Sciences, Alert Einstein College of Medicine, Yeshiv University, Bronx, New York, USA. 7 Dominick Purpur Deprtment of Neuroscience, Alert Einstein College of Medicine, Yeshiv University, Bronx, New York, USA. 8 Deprtment of Mthemtics, University of Pittsurgh, Pittsurgh, Pennsylvni, USA. Correspondence should e ddressed to R.R. (roert.rosenum@nd.edu) or B.D. (doiron@pitt.edu). Received 8 Mrch; ccepted 8 Septemer; pulished online 3 Octoer 6; doi:.38/nn.4433 nture NEUROSCIENCE dvnce online puliction

2 r t i c l e s 6 Nture Americ, Inc., prt of Springer Nture. All rights reserved. ctivity, gretly extending the pplicility of lnced network theory to explining corticl dynmics. RESULTS We consider network of excittory nd inhiitory exponentil integrte-nd-fire model neurons. Neurons provide recurrent, lterl synptic input to one nother nd receive feedforwrd synptic input from nonlocl presynptic popultion. A detiled mthemticl nlysis of correltions in the limit of lrge network size is provided in Supplementry Note nd Supplementry Figures 3. Below we provide n outline of these theoreticl results nd confirm their predictions using computer simultions. We first use simplified network model to demonstrte how the synchronous stte considered in previous theoreticl work 3 is roken y heterogeneous input correltions 35. We then consider more relistic model where neurons elong to continuous sptil domin nd connection proility depends on distnce. Homogeneous input correltions re cncelled y lnced networks To demonstrte the mechnisms ffecting correltions in recurrent networks, we first simulted simplified network of N =, neurons, hlf excittory nd hlf inhiitory, tht ll receive the sme fluctuting feedforwrd input nd re ech connected with proility.5 (Fig. ). Despite the fct tht neuron pirs shre ll of their feedforwrd input nd 5% of their recurrent synptic input on verge, spiking ctivity ws synchronous, with n verge pirwise spike count correltion of (Fig.,c). This smll verge correltion is defining chrcteristic of the synchronous stte. Mthemticlly, this stte is relized when spike count covrinces in the network stisfy 3 Normlized count 3 C SS ~ O( / N) Rec syn. current Ffwd syn. current Homogeneous inputs c d Neuron index Normlized shred current 5 Time (s) Rec Ffwd Totl Time (s) where C SS denotes the verge spike count covrince etween pirs of neurons in the recurrent network, N is the numer of neurons in the network nd ~ O(/N) denotes symptotic proportionlity to /N for lrge N. Note tht covrince nd correltion scle identiclly with network size in lnced networks, so we discuss them interchngely 3. Our theoreticl nlysis proceeds y noting tht spike count covrince is inherited from synptic input covrince 6 nd therefore the two scle similrly with N in the synchronous stte, C II ~ O( / N) where C II denotes the verge covrince etween neurons synptic inputs. Synptic inputs cn e decomposed into their feedforwrd nd recurrent sources, I = F + R, so tht neurons input covrinces decompose s CII = CFF + CRR +CRF where C FF is the verge covrince etween neurons feedforwrd input currents, C RR etween their recurrent inputs nd C RF etween one neuron s recurrent nd the other neuron s feedforwrd synptic input. Recurrent synptic input, R, is composed of positive contriutions from lterl excittory synptic inputs nd negtive contriutions from inhiitory (R = e i). Shred input fluctutions re visulized y verging the inputs to severl neurons, so tht the unshred contriutions verge out (Fig. d). Overlpping inputs cuse C FF nd C RR to e positive (Fig. d). If feedforwrd input correltion is moderte, C FF ~ O(), then recurrent input trcks the feedforwrd input so tht C RF is negtive nd nerly perfectly cncels the positive sources of correltions (i.e., C RF = (C FF + C RR ) + O(/N); see Supplementry Note ). As result, the covrince etween the totl synptic inputs is wek, C II ~ O(/N) (Fig. d). This cncelltion rises nturlly in e f Normlized count Opposite pop. Heterogeneous Inputs Sme pop. Rec syn. current Ffwd syn. current h g Normlized shred current Pop. Pop. Time (s) Rec Ffwd Totl Time (s) Figure Heterogeneous feedforwrd input reks synchrony in lnced recurrent networks. () Homogeneous network digrm. A popultion of, recurrently connected excittory nd inhiitory neurons receives glolly correlted feedforwrd input. () Normlized histogrm of pirwise spike count correltions etween, rndomly selected neurons. All histogrms re normlized y their integrl. (c) Rster plot of 5 rndomly chosen neurons plotted over s. (d) Shred fluctutions in the feedforwrd (lue) nd recurrent (red) synptic inputs cncel so tht shred fluctutions in the totl synptic currents (lck) re wek. Curves were computed y verging the synptic input currents to 5 neurons, convolving with Gussin-shped kernel (σ = 5 ms), sutrcting the men nd dividing y the neurons rheose. (e h) Sme s d except neurons were seprted into two popultions with seprte feedforwrd inputs. Currents in h re from neurons in popultion. Histogrms in f show correltions from neuron pirs rndomly selected from oth popultions (lck), from the sme popultion (purple) nd from opposite popultions (green). Rec, recurrent; ffwd, feedforwrd; syn, synptic; corr., correltion; pop., popultion. dvnce online puliction nture NEUROSCIENCE

3 r t i c l e s 6 Nture Americ, Inc., prt of Springer Nture. All rights reserved. the lnced stte nd does not require precise tuning of model prmeters 3. Since spiking correltions re inherited from synptic input covrince 6, this cncelltion of input covrinces leds to smll, O(/N), spike count correltions. Heterogeneous input correltions cn disrupt the synchronous stte To study the impct of heterogeneity on correltions in lnced networks, we modified the ove model y dividing the neurons into two popultions. Ech popultion received seprte feedforwrd input (Fig. e). The two feedforwrd input sources were sttisticlly identicl ut uncorrelted. Recurrent connectivity ws not chnged: neurons were rndomly connected without respect to popultion memership (identiclly to Fig. ). This input heterogeneity drmticlly chnged the structure of correltions in the network. Pirs of neurons in the sme popultion hd strongly positive spike count correltions on verge (.34), while neuron pirs from opposite popultions were negtively correlted with nerly identicl correltion mgnitude (.34), nd the verge correltion etween ll pirs ws nerly zero (4. 4 ; Fig. f,g). The mechnism responsile for this chnge in correltions cn e understood y gin seprting the synptic input covrince into its recurrent nd feedforwrd sources, ut generlizing the decomposition to ccount for neuron distnce to otin CII( d) = CFF( d) + CRR( d) + CRF ( d) Here C II (d) is the verge covrince etween input currents to pirs of neurons seprted y distnce d, where d = for neurons in the sme popultion nd d = for opposite popultion pirs, nd similrly for the other terms. Feedforwrd input is only correlted etween neurons in the sme popultion, so C FF () >, ut C FF () =. In contrst, recurrent connections do not respect popultion memership nd thus neither do the sttistics of recurrent input, CRR( ) = CRR( ) nd CRF( ) = CRF( ) () Since covrinces C RR (d) nd C RF (d) do not depend on d ut C FF (d) does, cncelltion cnnot e chieved in eqution () for oth d = nd d = simultneously. In other words, the one copy of shred recurrent synptic input cnnot cncel oth versions of the feedforwrd synptic inputs. The loss of cncelltion cuses the totl synptic current shred y neurons in the sme popultion to inherit shred fluctutions from their feedforwrd inputs (Fig. h), giving rise to positive O() correltions etween sme-popultion pirs. A competitive dynmic introduces negtive correltions etween neurons in opposite popultions (Fig. f,g). A similr mechnism ws considered in recent theoreticl study 35. For illustrtive purposes, we considered simplified network model with discrete supopultions, ut correltions nd connectivity in mny corticl circuits depend on continuous quntities such s physicl distnce or tuning similrity 9,,36. Next, we generlize these findings to more iologiclly relistic networks with connection proilities tht depend on neuron distnce. A sptilly extended network model We next considered network of N e = 4, excittory nd N i =, inhiitory model neurons rrnged on squre-shped domin modeling portion of corticl lyer. The neurons receive feedforwrd synptic input from seprte lyer of Poisson-spiking Recurrent lyer Ffwd lyer α rec α ffwd α rec neurons nd re connected with proility tht decys with distnce (Fig. ). Specificlly, the proility of connection etween two neurons in the recurrent network oeys Pr( connection) g( d; rec ) σ FF = α ffwd σ RR = σ SS + α rec Figure Correltion nd projection widths in sptilly extended networks. () Network schemtic. Blck tringles nd circles represent excittory nd inhiitory neurons. Red disks indicte recurrent synptic projections. Recurrent connection proility decys with distnce with width prmeter α rec. Blue cone denotes feedforwrd (ffwd) synptic projections from seprte lyer, with width prmeter α ffwd. () Correltions introduced y overlpping feedforwrd input to neurons in the recurrent lyer (shred lue input to red tringles) decy with distnce twice s slowly s feedforwrd connection proility (σ FF = α ffwd ). (c) The sptil width of correltions etween two neurons recurrent inputs (σ RR; input from lck tringles to red tringles) is equl to width of spike trin correltions (σ SS; dshed line) plus twice the width of recurrent projections (α rec ; solid lines). where d is the distnce etween the neurons mesured long the two-dimensionl network, g(d;σ ) exp( d /(σ )) is Gussinshped function nd α rec pproximtely represents the verge length of recurrent synptic projection. Similrly, the proility of synptic projection from neuron in the feedforwrd lyer to neuron in the recurrent lyer decys with distnce similrly to Gussin with width prmeter α ffwd, where distnce is mesured prllel to the corticl surfce (Fig. ). We next show tht the synchronous stte is relized when α rec < α ffwd, then show tht the synchronous stte cnnot e relized when α rec > α ffwd. The synchronous stte in sptilly extended corticl circuits As ove, synchronous spiking requires cncelltion etween input covrinces (cf. eqution ()), except tht d now represents continuous insted of inry distnce. Therefore, conditions on synchrony require first n understnding of how input covrinces depend on pirwise neuron distnce. Overlpping feedforwrd synptic projections introduce O() correltions etween the feedforwrd inputs to neuron pirs. Since nery pirs shre more feedforwrd inputs, these correltions re distnce dependent. Specificlly, synptic divergence cuses feedforwrd input correltions to e O() nd twice s rod s synptic projection widths (Fig. ), CFF( d) g( d; ffwd ). The fct tht C FF (d) ~ O() t first seems to preclude the possiility of n synchronous stte ecuse C FF (d) is one component of C II (d) in eqution () nd the synchronous stte requires C II (d) ~ O(/N). However, the synchronous stte is relized under cncelltion etween positive (C FF nd C RR ) nd negtive (C RF ) sources of correltions in eqution (). Cncelltion t ll distnces requires c nture NEUROSCIENCE dvnce online puliction

4 r t i c l e s 6 Nture Americ, Inc., prt of Springer Nture. All rights reserved. tht ll correltion sources hve the sme shpe (Supplementry Note ), mening tht CRF( d), CRR( d) g( d; ffwd ) The implictions of this requirement on the sptil profile of spiking correltions re clrified y noting tht recurrent synptic input is generted y spike trins in the recurrent network. Synptic divergence cuses the correltions etween neurons recurrent synptic inputs to e roder in spce thn the correltions etween spike trins ccording to (Fig. c): srr = sss + rec Here σ RR is the width of correltions etween neurons recurrent synptic input currents nd σ SS is the width of spike trin correltions in the recurrent network. In generl, we use α to denote the widths of synptic projections nd σ to denote the widths of correltions. Correltions etween recurrent inputs re constrined y the cncelltion required in the synchronous stte. Specificlly, eqution () requires tht the width of correltions etween recurrent synptic inputs stisfy srr = ffwd Comining the two expressions for σ RR ove yields sss = ( ffwd rec ) () (3) The existence of rel solution to eqution (3) requires tht α ffwd > α rec ; in other words, the sptil width of the recurrent projections must e nrrower thn the width of feedforwrd projections for the synchronous stte to exist. Further, eqution (3) implies tht σ SS > σ FF α rec, so tht spike trin correltions re sptilly nrrower thn correltions etween feedforwrd input currents. Thus, recurrent dynmics ctively shrpen the sptil profile of correltions in the synchronous stte (compre to the shrpening of tuning curves in previous work 37 ). To test these theoreticl findings, we performed network simultions with feedforwrd synptic projections roder thn recurrent projections (Fig. 3). The simultions confirmed tht C RR (d) nd C RF (d) decyed similrly with distnce to C FF (d) (Fig. 3). This llowed cncelltion etween positive nd negtive sources of correltions, so tht correltions etween neurons totl synptic currents nd etween their spike trins were wek over ll distnces (Fig. 3 e). Despite their smll verge, spike count correltions hd lrger stndrd devition (s.d. =.; Fig. 3d), consistent with results for nonsptil networks 3 (Fig. ). Neurons in the network receive strong excittion tht is cnceled y strong inhiition on verge (Fig. 3f nd Supplementry Fig. 4,), confirming tht the network mintins lnced stte. Correltions computed from simultions greed with closed-form mthemticl predictions (Fig. 3e; see Supplementry Note for equtions). Additionl simultions confirmed tht men correltions decy towrd zero t incresing network size (Supplementry Note nd Supplementry Fig. 4c,d). Brod lterl connections produce signture sptil correltion structure As noted ove, the cncelltion etween positive nd negtive correltions necessry for the synchronous stte cnnot e relized when recurrent projections re roder thn feedforwrd (α rec > α ffwd ) Current covrince α rec < α ffwd Rec rec Ffwd ffwd Ffwd rec Totl.5.5 Distnce (.u.) c Neuron index d Norm. count 4 Rec lyer Time (s) Distnce (.u.) ecuse eqution (3) cnnot e solved in this cse. Insted, neuron pirs inherit correltions from overlpping feedforwrd inputs so tht C SS (d) ~ O(). We confirmed this prediction y numericl simultions identicl to those discussed ove, ut with recurrent projections roder thn feedforwrd (Fig. 4). As predicted, recurrent input correltions were too sptilly rod to cncel with the more shrply decying feedforwrd correltions (Fig. 4), so tht the totl input correltion etween nery neurons ws lrge (Fig. 4; compre to Fig. 3). This effect introduced modertely strong correltions etween nery spike trins (Fig. 4c e) tht did not decy to zero t incresing network size (Supplementry Note nd Supplementry Fig. 4e,f). Nevertheless, the network mintined excittory inhiitory lnce (Fig. 4f nd Supplementry Fig. 4,). Since recurrent inputs must cncel feedforwrd inputs in lnced networks, C RF (d) is negtive (Figs., 3 nd 4 nd Supplementry Note ). Moreover, rod recurrent projections cuse C RF (d) to decy slowly with distnce (Fig. 4). Through eqution (), this imprts non- monotonicity in the dependence of C II (d) on d (Fig. 4), nd spike count correltions inherit this non-monotonic shpe (Fig. 4e). Following the sme rgument mde for the homogeneous network, the spike count correltions verged over neuron pirs t ll distnces is O(/N) (Fig. 4e nd Supplementry Note ). However, s noted ove, the verge correltion over ech distnce cnnot e O(/N). Hence, there must e cncelltion etween positive nd negtive e. f Pro. dens..4. i e+i Sim. Theory Men e Synptic current Figure 3 The synchronous stte in sptilly extended network model. () Network schemtic. As in Figure with recurrent projections nrrower thn feedforwrd projections (α rec =.5α ffwd ). () Averge covrince etween different sources of synptic currents to excittory neuron pirs s function of distnce. Positive covrince etween neurons feedforwrd input currents (lue) nd etween their recurrent input currents (red) cncel with negtive covrince etween one neuron s feedforwrd nd the other neuron s recurrent input (purple) to produce wek covrince etween their totl input (lck). Curves were computed from input currents to 4 rndomly selected excittory neurons nd were normlized y the pek feedforwrd input covrince. (c) Spike rsters of the 4 excittory neurons comprising the center squre of neurons in the recurrent lyer. (d) Normlized histogrm of pirwise spike count correltion etween 5, rndomly selected neurons. (e) Men spike count correltion etween neurons (± s.e.m.) s function of their distnce. Solid lck curve computed from the correltions etween 5, rndomly smpled neurons. Dshed red curve is from mthemticl clcultions (see Supplementry Note ). Gry dshed line shows men cross ll smpled pirs. (f) Distriution of excittory (lue), inhiitory (red) nd totl (lck) synptic currents cross the memrnes of 4 rndomly selected excittory neurons, mesured in units of the neurons rheose. Arrows indicte men vlues. Rec, recurrent; ffwd, feedforwrd; corr., correltion; sim., simultion; pro. dens., proility density; e, excittory; i, inhiitory. dvnce online puliction nture NEUROSCIENCE

5 r t i c l e s 6 Nture Americ, Inc., prt of Springer Nture. All rights reserved. Current covrince α rec < α ffwd Rec rec Ffwd ffwd Ffwd rec Totl.5.5 Distnce (.u.) c Neuron index d Norm. count 4 Rec lyer Time (s).5.5 Distnce (.u.) correltions t different distnces. As in Figure e h, competitive dynmic cuses nery neurons to e positively correlted nd more distnt neurons to e negtively correlted. This competitive dynmic does not extend eyond the rech of recurrent projections, so sufficiently distnt neurons re wekly correlted. Hence, correltion decreses nd then increses with distnce. This non-monotonicity cn e explined more precisely using mthemticl theory of correltion trnsfer (Fig. 4e nd Supplementry Note ). The heterogeneity of positive nd negtive correltions t different distnces increses the stndrd devition of pirwise correltions, ut only modestly (s.d. =.6, Fig. 4d; compre to Fig. 3d). In summry, when recurrent projections re sptilly nrrower thn feedforwrd projections (α rec < α ffwd, s in Fig. 3), correltions re wek etween pirs of neurons t ll distnces. When recurrent projections re roder thn feedforwrd (α rec > α ffwd, s in Fig. 4), nery neurons re positively correlted, neurons t moderte distnces re negtively correlted nd distnt neurons re wekly correlted. Moreover, the verge correltion etween pirs of neurons smpled rndomly t ll distnces is smll. The non-monotonic dependence of correltion on distnce is distinct signture of correltions rising from rod recurrent projections. We next investigted whether this correltion structure predicted y our theory is present in corticl recordings. Sptil correltion structure in visul corticl circuit We next sked whether our theoreticl chrcteriztion of correltions in sptilly extended networks cn explin correltions in corticl circuit. Lyers /3 nd lyer 4C of mcque primry visul cortex (L/3 nd L4C) provide n idel circuit for testing our predictions. Pirs of neurons in L/3 exhiit modertely lrge noise correltions tht decy with distnce, ut neurons in L4C, which re primry source of interlminr input to L/3, exhiit extremely wek pirwise noise correltions 3,33 (Fig. 5, with dt from previous studies 33,36 ). Neurons in L4C receive much of their feedforwrd input from thlmic projections, which form sptilly rod synptic fields, round mm in dimeter, ut lterl projections within mcque L4C form nrrower, su-millimeter synptic fields 34. Thus, our theoreticl prediction tht correltions re wek when α ffwd > α rec is consistent with the wek pirwise correltions oserved etween L4C neurons in vivo (s in Fig. 3). e f Pro. dens i e+i Sim. Theory Men e Synptic current Figure 4 Brod recurrent projections led to correlted lnced stte. ( f) Sme s Figure 3 e except recurrent projections were chnged to e roder thn feedforwrd projections (α rec =.5α ffwd ). This chnge prevents the recurrent network from cnceling positive feedforwrd input correltions (), resulting in popultion-wide spike count correltions with incresed s.d. (c,d) nd with positive correltions etween nery neurons ut negtive correltions etween more distnt neurons (e). Nevertheless, the network mintins lnce (f). Normlized count L4C L/3 (c =.7) (c =.) Spike count correltion Spike count correltion Electrode distnce (mm) Interlminr projections from L4C to L/3 hve similr sumillimeter width to excittory intrlminr projections within L4C, nd lterl projections from inhiitory sket cells in L/3 form su-millimeter synptic fields similr to those in L4C 34. Excittory neurons in L/3, however, form long-rnge lterl synptic projections with synptic fields spnning severl millimeters 34. Our theoreticl results cn e generlized to this setting, where inhiitory nd excittory projections hve different sptil profiles (Supplementry Note ). This extension predicts the sme correltion structure reported in Figure 4. However, correltions mesured in L/3 re positive on verge over rod rnge of distnces 36 (Fig. 5), in disgreement with this prediction. We hypothesized tht this inconsistency could e explined y recent studies showing tht much of the correlted vriility mesured in L/3 rises from low-dimensionl shred source of ltent vriility 3,3,38 4. We conjectured tht this shred vriility increses pirwise correltions in L/3 t ll distnces, therey wshing out the negtive correltions predicted y our theory. To serch for low-dimensionl vriility in our dt, we used Gussin process fctor nlysis 3,4 (GPFA), sttisticl lgorithm tht extrcts shred fluctutions from popultion of spike trins (see Online Methods). Applying this lgorithm to our L/3 recordings reveled source of one-dimensionl covriility tht decys with distnce (Fig. 5c). This distnce dependence implies tht nery neurons re ffected similrly y the ltent vrile. To test whether one-dimensionl ltent vriility explins the discrepncy etween our theoreticl predictions nd dt, we uilt twolyer network model representing mm y mm squre of cortex (Fig. 6). The first lyer, representing L4C, ws similr to the model in Figure 3, with the profile of feedforwrd nd recurrent projections chosen to mtch experimentlly constrined thlmic nd lterl projection widths 34. The second lyer, representing L/3, ws similr to Figure 4, with feedforwrd synptic input from excittory neurons in the L4C model nd recurrent projection widths lso chosen to mtch ntomicl mesurements. To cpture ltent vriility in L/3, the feedforwrd synptic input to ech neuron in the second lyer ws modulted y time-vrying, multiplictive gin modultion. We chose multiplictive source of vriility to e consistent with the properties of lowdimensionl vriility previously reported in mcque V (ref. 3), ut n dditive source of ltent vriility would produce similr overll results. The gin modultion contriutes n O( N ) source of covrince to the feedforwrd inputs tht the recurrent network cnnot cncel 3. To cpture the distnce dependence of ltent vriility (Fig. 5c), the mgnitude of the gin modultion ws heterogeneous cross the network in such wy tht nery neurons received similr modultions nd more distnt neurons received less similr modultions. c Ltent covrince Electrode distnce (mm) Figure 5 Dependence of correltions on lyer nd distnce in mcque V. () Histogrm of pirwise correltions etween neurons in superficil (puttive L/3, lck) nd middle (puttive L4C, gry) lyers of mcque primry visul cortex. Legends give verge correltion. () Averge pirwise correltion etween puttive L/3 neurons (± s.e.m.) s function of the distnce etween the electrodes on which the neurons were recorded. (c) Averge ltent covrince etween puttive L/3 neurons (± s.e.m.). Dt points in c were normlized y the pek t mm. nture NEUROSCIENCE dvnce online puliction

6 r t i c l e s 6 Nture Americ, Inc., prt of Springer Nture. All rights reserved. c Thlmic input Shred gin mod. L/3 L4C Proility density 3 L/3 (c =.7) L4C (c = 4 ) Simultions of this two-lyer model reveled tht correltions etween neurons in L4C were extremely smll on verge (Fig. 6,c), consistent with our theoreticl predictions (Fig. 3) nd consistent with in vivo recordings (Fig. 5). Correltions in the model L/3 lyer were modertely lrge nd positive over ll distnces (Fig. 6,c), comprle to those in in vivo recordings (Fig. 5). Thus, our model recovers the corse structure of correltions in L4C nd L/3. However, our explntion of positive correltions in L/3 is unstisfying ecuse the ddition of glolly shred vriility destroys the distinct non-monotonic reltionship etween correltion nd distnce predicted y our theory (compre Fig. 4e to Fig. 6). We next sked whether this structure could e recovered y filtering out glolly shred vriility. To ccomplish this, we computed the residul correltion mtrix estimted y GPFA. Residul correltions pproximte the spike count correltions with the contriution from low-dimensionl vriility removed 3. Residul correltions computed etween the simulted L/3 spike trins exhiited the predicted non-monotonic dependence on distnce, corroorting the ility of the GPFA lgorithm to extrct lowdimensionl vriility nd leve the structure of residul correltions intct. We next computed the men residul correltion in mcque L/3 s function of electrode distnce. In doing so, we oserved the sme non-monotonic dependence of residul correltion on distnce predicted y our theory (Fig. 7; further sttisticl nlysis in Supplementry Note 3 nd Supplementry Fig. 5). In summry, comining theoreticl nlysis nd computer simultions of multilyer network revels prsimonious model of the sources of shred vriility in visul corticl circuit in vivo. Under this model, positive correltions introduced y shred thlmic inputs to L4C neurons re ctively cnceled y negtive correltions rising from recurrent circuitry so tht pirs of L4C neurons t ll distnces exhiit wek verge spike count correltions 3,33. Correltions etween neurons in L/3 re introduced y overlpping feedforwrd inputs from L4C nd low-dimensionl source of vriility. Correltions rising from overlpping L4C projections re filtered y recurrent circuitry in L/3 to promote non-monotonic dependence of correltion on distnce. This non-monotonic correltion structure is wshed out y low-dimensionl ltent vriility, ut cn e recovered using GPFA to estimte nd remove this vriility. DISCUSSION Previous theoreticl work on sptilly homogeneous lnced networks with dense connectivity shows tht they produce very..5 L/3 L4C Spike count correltion Neuron distnce (mm) Figure 6 Dependence of correltions on lyer nd distnce in sptilly extended, multilyer network model. () Network schemtic. Thlmic input to L4C is roder thn recurrent projections within L4C. Projections from L4C to L/3 re nrrower thn recurrent excittory (ut not inhiitory) projections within L/3. Neurons in L/3 lso receive shred gin modultion (mod.). () Histogrms of pirwise correltions etween rndomly selected neurons in ech lyer. (c) Averge pirwise correltion etween neurons in ech lyer s function of the distnce etween the neurons. Spike count correltion L/3 residul corr... Model Neuron distnce (mm) L/3 residul corr. wek spike trin correltions 3. We hve generlized this theory to ccount for heterogeneous inputs nd distnce-dependent connection proility. In this frmework we hve mde two notle discoveries. First, in greement with the originl findings, when lterl synptic projections re sptilly nrrower thn incoming feedforwrd projections, correltions re extremely wek on verge t ll distnces. This theoreticl finding cn explin the wek pirwise correltions oserved etween neurons in middle lyers of mcque primry visul cortex 3,33. However, correltions mesured in corticl recordings re not lwys wek 5. Second, networks with roder lterl thn feedforwrd projections produce correltions tht do not decy to zero t incresing network size. In previous studies of lnced networks with sptilly homogeneous or clustered connectivity 3,4, the synchrony condition C SS ~ O(/N) is stisfied nd popultion verged pirwise correltions vnish in the lrge network limit. In contrst, sptilly extended networks with rod lterl projections violte the synchrony condition, nd consequently the expected pirwise correltions t specific distnce do not vnish. Nonetheless, men excittory nd inhiitory currents lnce nd firing rtes re moderte even when the synchrony condition is violted (Supplementry Fig. 4, nd Supplementry Note ). This represents novel solution for lnced networks tht, for the first time, formlly decouples network-wide synchrony from excittory inhiitory lnce. We focused on the dependence of correltions on distnce, ut correltions lso depend on tuning similrity. Prtitioning L/3 neuron pirs y tuning similrity revels tht correltions re strongest etween similrly tuned neurons 36 (Supplementry Fig. 6,). Modifying our computtionl model to cpture tuning-dependent correltions produced non-monotonic dependence of residul correltion on tuning similrity in some prmeter regimes, ut the relevnt prmeters hve not een mesured experimentlly (Supplementry Fig. 6c e nd Supplementry Note 4). Nevertheless, the modified theory could explin negtive correltions previously oserved in computer simultions of networks with tuning-specific connectivity 3 nd the finding tht negtive correltions re more frequent etween disprtely tuned neurons in V (ref. 43). As with nerly ny computtionl model, mny of the prmeters used in our simultions my not reflect their corresponding vlues in specific corticl res of specific species. However, our theoreticl nlysis does not depend on the precise vlues of these. Mcque V Electrode distnce (mm) Figure 7 Residul correltions in mcque V nd in model. () Residul correltion (corr.) etween neurons within the model L/3 network s function of distnce. () Residul correltions etween puttive L/3 neurons in mcque primry cortex (sme dt s Fig. 5). Residul correltion pproximtes spike count correltions fter single source of shred ltent vriility is removed. Both plots show men ± s.e.m. Correltions decresed in the first two ins (P < ; unpired t-test), incresed from the third to fourth in (P =.9) nd from the third to the fifth in (P =.67). dvnce online puliction nture NEUROSCIENCE

7 r t i c l e s 6 Nture Americ, Inc., prt of Springer Nture. All rights reserved. prmeters. Our finding tht the synchronous stte requires α rec < α ffwd is fundmentl property of networks with lnced excittion nd inhiition. We used simplified model of visul corticl circuit. In relity, pyrmidl neurons in V form oth locl nd long-rnge projections, connection proility in primte V depends on oth distnce nd tuning similrity, nd these dimensions re coupled 44. Moreover, connectivity properties of inhiitory neurons depend on their sutype 45. We modeled unidirectionl connections from L4 to L/3, ut L4 lso receives indirect feedck from L/3 through deeper corticl lyers. Spike trins in our model feedforwrd lyer were modeled y homogeneous Poisson processes, in contrst to the oscilltory firing rtes evoked y drifting grting stimuli in the dt we nlyzed. Our model cn e extended to ccount for these dditionl fetures without ffecting our overll conclusions. Our findings hve importnt implictions for the interprettion of correltions in neurl recordings. The verge (residul) correltion etween cell pirs smpled cross lrge rnge of distnces could e extremely smll, even when nery pirs re positively correlted with moderte mgnitude (Figs. 4 nd 7). Hence, sutrcting low-dimensionl ltent vriility nd prtitioning neuron pirs y distnce cn revel correltion structure tht would otherwise not e pprent. A previous study 3 computed residul correltions s function of distnce in primte V, ut did not report nonmonotonic dependence. While we cnnot e certin why their findings differ from ours, the ccurte estimtion of residul correltions with GPFA depends on the mount of dt used to estimte shred vriility. Our dt re well-suited for this purpose, s they contin over 8 pirs of units per recording on verge. There is long history of computtionl models of corticl circuits tht consider either networks with sptilly dependent coupling or lnced excittion nd inhiition in sptilly homogeneous networks 7,3. Only recently hs the sptil structure of corticl connectivity een included in networks with lnced excittion nd inhiition 9,37,46, nd guiding theoreticl principles re lcking. Our theory hs tken this sptil structure into ccount nd produced two core predictions for corticl circuits with long rnge lterl connections: first, nery neurons exhiit significnt positive correltions; second, the dependence of pirwise correltion on pirwise distnce is non-monotonic. These predictions re clerly flsifile nd hence represent strong tests of our theory. The superficil lyers of visul cortex hve long-rnge lterl connections 34, mking them suitle test ed for our theory of correltions. After ccounting for source of glol vriility, oth of our predictions were verified from popultion recordings in mcque V (Fig. 7). Further, similr noise correltion structure hs een reported in recordings from mouse V (ref. 47). The successful vlidtion of our predictions mrks our theory s promising frmework for studying the structure of neurl vriility in corticl circuits. Nevertheless, there re mny spects of corticl dynmics tht remin unexplined y lnced networks, such infrequent yet lrge memrne fluctutions during spontneous dynmics 5,48. Cpturing these dynmics in corticl models with lnced rchitectures remins n open chllenge. Methods Methods, including sttements of dt vilility nd ny ssocited ccession codes nd references, re ville in the online version of the pper. Note: Any Supplementry Informtion nd Source Dt files re ville in the online version of the pper. Acknowledgments We re grteful to T.S. Lee nd A. Movshon for reserch support. This work ws supported y Ntionl Science Foundtion grnts NSF-DMS-5788 (R.R.), NSF-DMS-335 (B.D.), NSF-DMS-578 (B.D.), NSF-DMS-693 (J.E.R.), NSF-DMS-5688 (J.E.R.) nd NSF-DMS-358 (J.E.R.); Ntionl Institute of Helth grnts RNS7865 (B.D., J.E.R.), CRCNS-RDC539 (B.D.), REY6774 (A.K.), REY98 (M.A.S.) nd P3EY898 (M.A.S.); two grnts from the Simons Foundtion collortion on the glol rin (SCGB#3593MC;BD, B.D. nd AK, A.K.); y the Eye nd Er Foundtion of Pittsurgh (M.A.S.); nd y Reserch to Prevent Blindness (A.K., M.A.S.). AUTHOR CONTRIBUTIONS R.R. nd B.D. conceived the project; R.R. performed the simultions, dt nlysis nd mthemticl clcultions; M.A.S. nd A.K. provided the experimentl dt. J.E.R. nd B.D. supervised the project. All uthors contriuted to writing the mnuscript. COMPETING FINANCIAL INTERESTS The uthors declre no competing finncil interests. Reprints nd permissions informtion is ville online t reprints/index.html.. Ermentrout, B. Neurl networks s sptio-temporl pttern-forming systems. Rep. 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The mechnism of orienttion selectivity in primry visul cortex without functionl mp. J. Neurosci. 3, ().. Shu, Y., Hsenstu, A. & McCormick, D.A. Turning on nd off recurrent lnced corticl ctivity. Nture 43, (3).. Hider, B., Duque, A., Hsenstu, A.R. & McCormick, D.A. Neocorticl network ctivity in vivo is generted through dynmic lnce of excittion nd inhiition. J. Neurosci. 6, (6). 3. Okun, M. & Lmpl, I. Instntneous correltion of excittion nd inhiition during ongoing nd sensory-evoked ctivities. Nt. Neurosci., (8). 4. Dorrn, A.L., Yun, K., Brker, A.J., Schreiner, C.E. & Froemke, R.C. Developmentl sensory experience lnces corticl excittion nd inhiition. Nture 465, (). 5. Grupner, M. & Reyes, A.D. Synptic input correltions leding to memrne potentil decorreltion of spontneous ctivity in cortex. J. Neurosci. 33, (3). 6. Zhou, M. et l. Scling down of lnced excittion nd inhiition y ctive ehviorl sttes in uditory cortex. Nt. Neurosci. 7, (4). 7. Xue, M., Atllh, B.V. & Scnzini, M. Equlizing excittion-inhiition rtios cross visul corticl neurons. Nture 5, (4). 8. Amit, D.J. & Brunel, N. Model of glol spontneous ctivity nd locl structured ctivity during dely periods in the cererl cortex. Cere. Cortex 7, 37 5 (997). 9. Ko, H. et l. Functionl specificity of locl synptic connections in neocorticl networks. Nture 473, 87 9 ().. Fino, E. & Yuste, R. Dense inhiitory connectivity in neocortex. Neuron 69, 88 3 ().. Levy, R.B. & Reyes, A.D. Sptil profile of excittory nd inhiitory synptic connectivity in mouse primry uditory cortex. J. Neurosci. 3, ().. Oswld, A.M., Doiron, B., Rinzel, J. & Reyes, A.D. Sptil profile nd differentil recruitment of GABAB modulte oscilltory ctivity in uditory cortex. J. Neurosci. 9, (9). 3. Renrt, A. et l. The synchronous stte in corticl circuits. Science 37, (). 4. Ecker, A.S. et l. Decorrelted neuronl firing in corticl microcircuits. Science 37, (). 5. Cohen, M.R. & Kohn, A. Mesuring nd interpreting neuronl correltions. Nt. 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8 r t i c l e s 6 Nture Americ, Inc., prt of Springer Nture. All rights reserved. 6. Doiron, B., Litwin-Kumr, R., Rosenum, R., Ocker, G.K. & Josić, K. The mechnics of stte-dependent neurl correltions. Nt. Neurosci. 9, (6). 7. Kohn, A., Zndvkili, A. & Smith, M.A. Correltions nd rin sttes: from electrophysiology to functionl imging. Curr. Opin. Neuroiol. 9, (9). 8. Poulet, J.F. & Petersen, C.C. Internl rin stte regultes memrne potentil synchrony in rrel cortex of ehving mice. Nture 454, (8). 9. Cohen, M.R. & Munsell, J.H. Attention improves performnce primrily y reducing interneuronl correltions. Nt. Neurosci., (9). 3. Mochol, G., Hermoso-Mendizl, A., Skt, S., Hrris, K.D. & de l Roch, J. Stochstic trnsitions into silence cuse noise correltions in corticl circuits. Proc. Ntl. Acd. Sci. USA, (5). 3. Ecker, A.S. et l. Stte dependence of noise correltions in mcque primry visul cortex. Neuron 8, (4). 3. Hnsen, B.J., Chelru, M.I. & Drgoi, V. Correlted vriility in lminr corticl circuits. Neuron 76, 59 6 (). 33. Smith, M.A., Ji, X., Zndvkili, A. & Kohn, A. Lminr dependence of neuronl correltions in visul cortex. J. Neurophysiol. 9, (3). 34. Lund, J.S., Angelucci, A. & Bressloff, P.C. Antomicl sustrtes for functionl columns in mcque monkey primry visul cortex. Cere. Cortex 3, 5 4 (3). 35. Wimmer, K. et l. The dynmics of sensory integrtion in hierrchicl network explins choice proilities in MT. Nt. Commun. 6, 677 (5). 36. Smith, M.A. & Kohn, A. Sptil nd temporl scles of neuronl correltion in primry visul cortex. J. Neurosci. 8, (8). 37. Rosenum, R. & Doiron, B. Blnced networks of spiking neurons with sptilly dependent recurrent connections. Phys. Rev. X 4, 39 (4). 38. Yu, B., Kohn, A. & Smith, M.A. Estimting shred firing rte fluctutions in neurl popultions. Soc. Neurosci. str (). 39. Lin, I.C., Okun, M., Crndini, M. & Hrris, K.D. The nture of shred corticl vriility. Neuron 87, (5). 4. Schölvinck, M.L., Sleem, A.B., Benucci, A., Hrris, K.D. & Crndini, M. Corticl stte determines glol vriility nd correltions in visul cortex. J. Neurosci. 35, 7 78 (5). 4. Yu, B.M. et l. Gussin-process fctor nlysis for low-dimensionl single-tril nlysis of neurl popultion ctivity. J. Neurophysiol., (9). 4. Litwin-Kumr, A. & Doiron, B. Slow dynmics nd high vriility in lnced corticl networks with clustered connections. Nt. Neurosci. 5, (). 43. Chelru, M.I. & Drgoi, V. Negtive correltions in visul corticl networks. Cere. Cortex 6, (6). 44. Bosking, W.H., Zhng, Y., Schofield, B. & Fitzptrick, D. Orienttion selectivity nd the rrngement of horizontl connections in tree shrew strite cortex. J. Neurosci. 7, 7 (997). 45. Pfeffer, C.K., Xue, M., He, M., Hung, Z.J. & Scnzini, M. Inhiition of inhiition in visul cortex: the logic of connections etween moleculrly distinct interneurons. Nt. Neurosci. 6, (3). 46. Kriener, B., Helis, M., Rotter, S., Diesmnn, M. & Einevoll, G.T. How pttern formtion in ring networks of excittory nd inhiitory spiking neurons depends on the input current regime. Front. Comput. Neurosci. 7, 87 (4). 47. Rikhye, R.V. & Sur, M. Sptil correltions in nturl scenes modulte response reliility in mouse visul cortex. J. Neurosci. 35, (5). 48. Tn, A.Y., Chen, Y., Scholl, B., Seidemnn, E. & Priee, N.J. Sensory stimultion shifts visul cortex from synchronous to synchronous sttes. Nture 59, 6 9 (4). dvnce online puliction nture NEUROSCIENCE

9 6 Nture Americ, Inc., prt of Springer Nture. All rights reserved. ONLINE METHODS Description of computtionl model. We modeled squre of cortex with N neurons, N e of which re excittory nd N i inhiitory. The memrne potentil of neuron j from the excittory ( = e) or inhiitory ( = i) popultion oeyed exponentil integrte-nd-fire (EIF) dynmics, dvj Cm IL Vj f Vj I j t dt = ( ) + ( ) + ( ) Ech time tht V j exceeds threshold t Vth, the neuron spikes nd the memrne potentil is held for refrctory period τ ref, then reset to fixed vlue V re. The lek current is given y IL( V) = gl( V EL) nd spike-generting current is defined y f ( V) = gl T exp[( V VT )/ T ] For excittory neurons, τ m = C m /g L = 5 ms, E L = 6 mv, V T = 5 mv, V th = mv, T = mv, V re = 65 mv nd τ ref =.5 ms. Inhiitory neurons were the sme except τ m = ms, T =.5 mv nd τ ref =.5 ms. Synptic input currents were defined y Cm I j ( t) = Fj ( t) + R j ( t) where F (t) is the feedforwrd input nd R (t) the recurrent input to neuron j in popultion = e, i. The feedforwrd input ws modeled differently for different figures, s descried elow. The recurrent input ws defined y N J jk Rj ( t t tn, )= h k N = e,i k= n ( ), k where t n is the nth spike time of neuron k in popultion = e, i. The N scling of synptic weights is defining feture of the lnced network formlism nd cptures the lnce etween excittory nd inhiitory currents s well s intrinsiclly generted temporl vriility for lrge N (ref. 7). Ech term J jk represents the synptic weight from presynptic neuron k in popultion to postsynptic neuron j in popultion. For ll simultions, we modeled synptic kinetics using h ( t) = exp( t / t)/ t for t > where τ e = 6 ms nd τ i = 5 ms. All networks were dense in the sense tht connection proilities re O() (ref. 3). For the model in Figure, there were N =, neurons, hlf of which were excittory nd hlf inhiitory. For ech (presynptic) neuron, we rndomly nd uniformly chose,5 excittory nd,5 inhiitory postsynptic neurons in the network. Postsynptic neurons were smpled with replcement, so tht single presynptic neuron could mke multiple contcts with postsynptic neuron. The synptic weight of ech connection depended on the pre- nd postsynptic neuron types (excittory or inhiitory). Specificlly, J jk =( numer of contcts) where j ee =.5 mv, j ie = mv nd j ii = j ei = 5 mv. Note tht synptic weights were scled y N = 4 in eqution (4), so tht the ctul synptic weight of ech contct ws on the order of. mv. For Figure d, the feedforwrd input to ech neuron ws given y the sum of n input is nd smoothly vrying signl, Fj ( t) = Nm + sss( t) j Here, s(t) is shred source of smooth, unised Gussin noise defined y its uto-covrince function, ( ) cov s( t) s( (, t + t ))= exp t t s, (4) (5) τ s = 4 ms sets the correltion timescle nd σ s =. mv/ms scles the mgnitude of the fluctutions. The terms m e =.5 mv/ms nd m i =. mv/ms introduce sttic is to the input current. The model for Figure e h ws identicl, except tht two independent reliztions, s (t) nd s (t), of the shred input were generted. Hlf of the neurons received s (t) nd the other hlf received s (t). Firing rtes for Figure d were 7.6 Hz on verge for excittory neurons nd 3.8 Hz for inhiitory neurons. For Figure e h, verge firing rtes were 7.4 Hz for excittory nd 3.8 Hz for inhiitory neurons. To model the sptilly extended recurrent network in Figures 3 nd 4, we rrnged N e = 4, excittory nd N i =, inhiitory EIF model neurons on uniform grid covering two- dimensionl squre domin. The feedforwrd lyer ws modeled y popultion of N F = 5,65 excittory Poisson-spiking neurons on uniform grid covering squre tht is prllel to the recurrent network. Feedforwrd input to the recurrent lyer ws defined y J Fj N F F jk ( t e t tn F, ) = h k N k= n ( ) F, k where t n is the nth spike time of neuron k in the feedforwrd lyer. Ech spike trin in the feedforwrd lyer ws modeled s independent Poisson processes with rte r F = 5 Hz. To simplify clcultions, we mesured distnce in units of the side-length of the squre domin. In these units, the domin is represented s the unit squre, Γ = [, ] [, ]. Neurons were connected rndomly nd the proility tht two neurons were connected depended on their distnce mesured periodiclly on Γ. The precise lgorithm for generting connections is descried in Supplementry Note 5. This lgorithm ssures tht the expected numer of synptic contcts from presynptic neuron t coordintes y = (y, y ) in popultion to postsynptic neuron t x = (x, x ) in popultion is given y out K p ( x y)= N g ( x y ; ) g ( x y ; ) where g ( u;) is wrpped Gussin distriution 37. Out-degrees were Kee out out = Kei =,, Kie out = Kii out out =5, K ef =, nd out KeF =8. It follows tht the network-wide verge numer of synptic inputs to excittory nd inhiitory neurons in the recurrent network ws K = 3,75. Synptic weights were determined y eqution (5) where j ee = 4 mv, j ie = mv, j ei = j ii = 4 mv, nd j ef = j if = mv. Note gin tht these terms were divided y N 4 s indicted in eqution (4), so tht the ctul synptic weights were etween.8 mv nd.8 mv. Excittory nd inhiitory recurrent projection widths were α rec = α e = α i =.5 for Figure 3 nd α rec =.5 for Figure 4. Feedforwrd connection widths were α ffwd =. in oth figures. For the simultions in Figure 3, verge firing rtes were 3.9 Hz for excittory nd 6. Hz for inhiitory neurons. For the simultions in Figure 4, verge rtes were 4. Hz for excittory nd 6. Hz for inhiitory neurons. The first lyer (L4C) in the model in Figure 6 ws identicl to the model in Figure 3 except tht α ffwd =., α e =.5 nd α i =.3. The length units used in Figures 6 nd 7 were determined y interpreting the network domin, Γ, s mm y mm squre. Thus, in physicl dimensions, α ffwd = mm, α e =.5 mm nd α i =.3 mm. Averge firing rtes in the L4C lyer were 3.7 Hz for excittory nd 6. Hz for inhiitory neurons. Connectivity in the second lyer (L/3) in Figure 6 ws identicl to tht in the L4C lyer except tht α e =.5 (or.5 mm), α ffwd =.5 (or.5 mm). The spike times from eqution (6) for the L/3 lyer were given y the spike times of neurons in the L4C lyer, so tht N F = 5,. The mgnitude of feedforwrd connectivity ws lso modified y setting K ef =,46, K if = 3 nd j ef = j if = mv. A shred gin modultion ws implemented y ltering feedforwrd input currents ccording to [ ] Fj ( t) Fj ( t) + wl ( x) L( t) The shred gin modultion, L(t), is reliztion of unised Gussin noise defined y its uto-covrince function ( ) cov L( t) L( (, t + t ))= exp t t L with correltion timescle τ L = 4 ms. The dimensionless weight fctor w L (x) depended on the coordintes, x = (x, x ) Γ, of the neuron nd ws given y wl ( x, x)=. 5g ( x c; sl ) g x c; sl ( ) doi:.38/nn.4433 nture NEUROSCIENCE