Distributed correlations in motor cortex suggest virtually unstable linearized dynamics

Despite the large amount of shared input between nearby neurons in cortical circuits, massively parallel spiking recordings of various in vivo networks exhibit pairwise covariances in ensembles of neuronal spike trains that are on average close to zero [1]. The low average has been well understood in terms of active decorrelation by inhibitory feedback [2,3] in networks that operate far away from the critical point, which marks the onset of avalanche-like activity [4]. Experiments, however, also show large variability of covariances across pairs of neurons. An explanation for their wide distribution in relation to the static (quenched) disorder of the connectivity in recurrent networks is so far elusive.Here we combine ideas from spin-glass theory [5] with a generating function representation for the joint probability distribution of the network activity [6] to derive a finite-size mean-field theory that reduces a disordered to a highly symmetric network with fluctuating auxiliary fields (Fig. 1). The theory relates the statistics of covariances to the statistics of connections, in particular the largest eigenvalue of the connectivity matrix, and explains the experimentally observed covariance distributions [7]. The analytical expressions expose that both, average and dispersion of the latter, diverge at a critical point which has been studied in terms of a transition from regular to chaotic dynamics [8,9,10]. This critical point does not arise from net excitation, but rather from disorder in networks with balanced excitation and inhibition. Applying these results to recordings from motor cortex suggests its operation close to this breakdown of linear stability. References:1. Ecker AS, Berens P, Keliris GA, Bethge M, Logothetis NK: Decorrelated Neuronal Firing in Cortical Microcircuits. Science 2010, 327:584-587.2. Renart A, De La Rocha J, Bartho P, Hollender L, Parga N, Reyes A, Harris KD: The asynchronous State in Cortical Circuits. Science 2010, 327:587-590.3. Tetzlaff T, Helias M, Einevoll G, Diesmann M: Decorrelation of neural-network activity by inhibitory feedback. PLOS Comput. Biol. 2010, 8(8):e1002596.4. Beggs JM, Plenz D: Neuronal avalanches in neocortical circuits. J. Neurosci. 2003, 23:11167-11177.5. Sompolinsky H, Zippelius A: Relaxational dynamics of the Edwards-Anderson model and the mean-field theory of spin-glasses. Phys. Rev. B 1982, 25:6860-6875.6. Chow C, Buice M: Path Integral Methods for Stochastic Differential Equations. J Math Neurosci. 2015, 5:8. 7. Dahmen D, Diesmann M, Helias M: Distributions of covariances as a window into the operational regime of neuronal networks. arXiv 2016, 1605.04153 [cond-mat.dis-nn].8. Sompolinsky H, Crisanti A, Sommers HJ: Chaos in Random Neural Networks, Phys. Rev. Lett. 1988, 61:259-262