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

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Neural correlations Spiking activity is correlated These correlations could be related to information processing, or they could limit the efficiency for (e.g.) sensory discrimination … In principle, it is plausible that shared inputs play a role in generating correlations Common inputs How are correlations generated in cortical circuits?

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What is the relationship between common input and correlations ? Do correlations really limit the efficiency of computations?

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In analytical studies the effect of common inputs is neglected (Amit, Brunel, …): only sparse networks are considered, where the connection probability decreases as 1/N. Correlations are zero by construction. However in the corresponding simulations the connection probability is not taken small (e.g., 0.25). Sparsely connected networks To study the effect of correlations we have considered densely connected networks

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Connectivity Synaptic efficacies Densely connected networks with strongly coupled neurons In strongly coupled networks only √N excitatoty neurons are needed to produce firing

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Given that connectivity is dense and neurons are strongly coupled it is difficult to understand how an asynchronous state can be stable. Our main result is that in densely connected networks with strong couplings spiking correlations are small because of a dynamical cancellation between the correlations of the current components.

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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 Effect of shared inputs Let’s first neglect input correlations : total current correlations spiking output correlations

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

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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. E-E and I-I firing correlations contribute positively to c, while E-I firing correlations contribute negatively Correlations between E and I inputs tend to decorrelate the synaptic currents to post- synaptic neurons Two input populations (E and I): Cancellation of current correlations Input raster currents V’s Correlated inputs

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Can the decorrelation occur from the dynamics of a recurrent network? Binary network: analytical solution: self-consistent equations for both rates and correlations numerical simulations LIF network: simulations Experimental data: auditory cortex of urethane-anesthetized rats We studied this problem using:

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Binary neurons: populations and connectivity Three neural populations: X, E, I Both receive excitatory projections from an external population X E: network of excitatory neurons I: network of inhibitory neurons Feed-forward connections p: connection probability

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and are O(1) Connectivity Some definitions Afferent current to cell i: State of neuron i: Prob that the state of the network is : Mean current (ss):Average activity of cell i:

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more definitions Instantaneous spiking covariance: Population averaged current covariance: Population averaged spiking covariance: The quantities: are O(1) Population averaged firing rate: Population averaged mean current:

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Asynchronous state: We wonder whether this network has an asynchronous state

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Because each neuron receives ∼ 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 to the firing threshold. This was noticed for sparse networks by van Vreeswijk & Sompolinsky (1998). It also holds for dense networks: Balance of the average firing rates To have finite rates there must be a cancellation: asymptotically, the population averaged firing rate of each population is proportional to the population averaged rate of the external neurons The solution of these equations :

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A similar argument leads to equations for the population-averaged instantaneous pair-wise correlations in the steady state: Pairwise correlations in the dense network is the leading-order population-averaged temporal variance of the activity of cells in population α These relations give rise to some interesting properties: Tracking of fluctuations in the asynchronous state Balance of the current correlations an asynchronous state

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Tracking of fluctuations in the asynchronous state consider the difference between the normalized instantaneous activities of the excitatory and inhibitory populations and the instantaneous activity of the external population 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, replacing the correlations one sees that at this order this variance is zero: the standard deviation is The same is true for the instantaneous firing rate in the three populations track each other. Tracking is perfect as N → ∞

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Balance of the current correlations 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 The total current correlation can be decomposed as Presynaptic indexes From spiking correlations From shared inputs In the asynchronous state these terms are O(1). However, substituting the solution for the r’s one finds that: The correlations of the current components are O(1), but the correlations of the total currents are small

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From spiking correlations From shared inputs The presence of shared inputs does not imply that correlations are large, the dynamics of the network produces a cancellation between the contribution sof common inputs and input correlations that leaves us with a small total current correlation

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Comparison with sparse networks In sparse networks 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. Summary

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An asynchronous state in dense binary networks:simulations O(1): amplification of weak firing correlations O(1/N): asynchronous state O(1/√N): small total current correlations

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Tracking: simulations of binary networks Tracking becomes more accurate with increasing network size

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Distribution of the spiking correlations Cross-correlograms of the current components (spike count correlation coefficient)

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LIF neurons population index: E, I, X neuron index Synaptic currents: Above threshold neurons produces a spike. This happens at times Immediately after V is kept in a reset value during a refractory time t_ref. Below the threshold Gating variables:

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

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

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Experimental data

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Conclusions Anatomy (common inputs) is not enough to determine correlations: Spatial correlations are small, not because of sparse connectividy but because of dynamic cancellation of of current correllations In a dense network both the average firing rate (‘signal’ ) and the temporal fluctuations (‘noise’) are propagated with the same accuracy The synchrony explosion is naturally avoided in recurrent circuits. Stable propagation of rates is possible

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