Asynchronous State of Cortical Circuits: Dynamics of Densely Connected Networks

1. The asynchronous state of cortical circuits (Dynamics of densely connected networks of model neurons and of cortical circuits) Alfonso Renart (Rutgers) Jaime de la Rocha (NYU, Rutgers) Peter Bartho (Rutgers) LiadHollender (Rutgers) NéstorParga (UA Madrid) Alex Reyes (NYU) Kenneth Harris (Rutgers)) Science,in press. Online publication 28 Jan 2010 Marseille, Jan 2010

2. Neural correlations Spikingactivityiscorrelated Thesecorrelationscouldberelatedtoinformationprocessing, or theycouldlimittheefficiencyfor (e.g.) sensorydiscrimination … How are correlationsgenerated in cortical circuits? Common inputs In principle, itis plausible thatshared inputs play a role in generatingcorrelations

3. Whatistherelationship betweencommon input and correlations ? Do correlationsreallylimittheefficiency of computations?

4. Sparselyconnectednetworks In analyticalstudiestheeffect of common inputs isneglected (Amit, Brunel, …): onlysparsenetworksare considered, wheretheconnectionprobabilitydecreases as 1/N. Correlations are zerobyconstruction. However in thecorrespondingsimulationstheconnectionprobabilityisnottakensmall (e.g., 0.25). Tostudytheeffect of correlationswehaveconsidereddenselyconnectednetworks

5. Denselyconnectednetworkswithstronglycoupledneurons Connectivity Synapticefficacies In stronglycouplednetworksonly√N excitatoty neurons are needed to produce firing

6. Giventhatconnectivityisdense and neurons are stronglycoupleditisdifficulttounderstandhowanasynchronousstatecan bestable. Ourmainresultisthat in denselyconnectednetworkswithstrongcouplingsspikingcorrelations are smallbecause of a dynamicalcancellationbetweenthecorrelations of thecurrentcomponents.

7. Effect of shared inputs Let’sfirstneglect input correlations : total currentcorrelations spiking output correlations Both excitatory (E) and inhibitory (I) shared inputs cause positive correlations of moderate magnitude in the synaptic input and spiking activity of the postsynaptic pair

8. Effect of input spikingcorrelations One input population(E) Very weak input correlations give rise to strongly correlated synaptic currents and output spikes Input raster currents V’s simulation of a feed-forward network of LIF neurons

9. Correlated inputs Two input populations(E and I): Cancellation of currentcorrelations E-E and I-I firing correlations contribute positively to c, while E-I firing correlations contribute negatively large fluctuations in the excitatory and inhibitory currents occur simultaneously and cancel, leading to a significant reduction in the correlation of the total synaptic currents c and output spikes. Input raster currents V’s Correlations between E and I inputs tend to decorrelate the synaptic currents to post-synapticneurons

10. Can thedecorrelationoccur fromthedynamics of a recurrentnetwork? Westudiedthisproblemusing: • Binarynetwork: • analyticalsolution: self-consistentequationsforbothrates and correlations • numericalsimulations • LIF network: simulations • Experimental data: auditorycortex of urethane-anesthetizedrats

11. Binaryneurons: populations and connectivity Three neural populations: X, E, I E: network of excitatoryneurons I: network of inhibitoryneurons Bothreceiveexcitatoryprojectionsfroman externalpopulationX p:connectionprobability Feed-forward connections

12. Somedefinitions Connectivity Probthatthestate of thenetworkis : are O(1) and Averageactivity of cell i: State of neuron i: Afferentcurrenttocell i: Mean current (ss):

13. more definitions Populationaveraged mean current: Instantaneousspikingcovariance: Populationaveragedfiringrate: Populationaveragedspikingcovariance: Populationaveragedcurrentcovariance: Thequantities: are O(1)

14. Wewonderwhetherthisnetwork has anasynchronousstate Asynchronousstate:

15. Balance of theaveragefiringrates Thiswasnoticedforsparsenetworksbyvan Vreeswijk & Sompolinsky (1998). Italsoholdsfordense networks: Becauseeachneuronreceives ∼ O(N) synaptic inputs, but only ∼O(√N) are enough to make it fire, the net magnitude of the total excitation and inhibition felt by the neurons is very large compared tothefiringthreshold. Tohavefiniteratestheremustbe a cancellation: Thesolution of theseequations: asymptotically, the population averaged firing rate of each population is proportional to the population averaged rate of the external neurons

16. Pairwisecorrelations in the dense network A similar argument leads to equations for the population-averaged instantaneous pair-wise correlations in the steady state: anasynchronousstate is the leading-order population-averaged temporal variance of the activity of cells in population α Theserelationsgiverisetosomeinterestingproperties: Tracking of fluctuations in theasynchronousstate Balance of thecurrentcorrelations

17. Tracking of fluctuations in theasynchronousstate consider the difference between the normalized instantaneous activities of the excitatory and inhibitory populations and the instantaneous activity of theexternalpopulation the degree to which the activity in the recurrent network tracks the instantaneous activity in the external population can be measured by its variance at equilibrium, However, replacingthecorrelationsoneseesthat at thisorderthisvarianceiszero: thestandarddeviationis Thesameis true for theinstantaneousfiring rate in the three populations track each other. Tracking is perfect as N → ∞

18. Balance of thecurrentcorrelations TRACKING of the instantaneous population activities is equivalent to a precise cancellation of the different components of the (zero-lag) population-averaged current correlation c Thetotalcurrentcorrelation can bedecomposed as Presynaptic indexes Fromspikingcorrelations Fromshared inputs In theasynchronousstatetheseterms are O(1). However, substitutingthesolutionforther’sonefindsthat: Thecorrelations of thecurrentcomponentsare O(1), butthecorrelations of thetotal currents are small

19. Thepresence of shared inputs doesnotimplythatcorrelations are large, thedynamics of thenetwork produces a cancellationbetweenthecontributionsofcommon inputs and input correlationsthatleavesuswith a small total currentcorrelation Fromspikingcorrelations Fromshared inputs

20. Comparisonwithsparsenetworks Summary In sparsenetworks each component of the current correlation decreases with the network size in an asynchronous state. In a sparsely connected network the asynchronous state is a static feature of the network architecture, whereas in a densely connected network it is a purely dynamical phenomenon.

21. Anasynchronousstate in dense binarynetworks:simulations O(1): amplificationof weakfiringcorrelations O(1/√N): small total current correlations O(1/N): asynchronousstate

22. Tracking: simulations of binarynetworks Tracking becomes more accuratewithincreasingnetworksize

23. Cross-correlograms of thecurrentcomponents Distribution of thespikingcorrelations (spikecountcorrelationcoefficient)

24. Gating variables: populationindex: E, I, X LIF neurons neuronindex Belowthethreshold Abovethresholdneurons produces a spike. Thishappens at times Immediatelyafter V iskept in a resetvalueduring a refractory time t_ref. Synapticcurrents:

25. Active decorrelation in networks of spikingneurons tracking of instantaneous population-averaged activities (z-scores) (p = 0.2) reversal of excitation reversal of inhibition rest

26. Experimental data 7200 s Spontaneous alternations between brain states under urethane anesthesia

27. Experimental data

28. Conclusions Thesynchronyexplosionisnaturallyavoided in recurrentcircuits. Stablepropagation of ratesispossible In a dense network both the average firing rate (‘signal’ ) and the temporal fluctuations (‘noise’) are propagated with the same accuracy Anatomy (common inputs) isnotenoughto determine correlations: Spatialcorrelations are small, notbecause of sparseconnectividybutbecause of dynamiccancellation of ofcurrentcorrellations