Abstract
We present a simple model for coherent, spatially correlated chaos in a recurrent neural network. Networks of randomly connected neurons exhibit chaotic fluctuations and have been studied as a model for capturing the temporal variability of cortical activity. The dynamics generated by such networks, however, are spatially uncorrelated and do not generate coherent fluctuations, which are commonly observed across spatial scales of the neocortex. In our model we introduce a structured component of connectivity, in addition to random connections, which effectively embeds a feedforward structure via unidirectional coupling between a pair of orthogonal modes. Local fluctuations driven by the random connectivity are summed by an output mode and drive coherent activity along an input mode. The orthogonality between input and output mode preserves chaotic fluctuations by preventing feedback loops. In the regime of weak structured connectivity we apply a perturbative approach to solve the dynamic mean-field equations, showing that in this regime coherent fluctuations are driven passively by the chaos of local residual fluctuations. When we introduce a row balance constraint on the random connectivity, stronger structured connectivity puts the network in a distinct dynamical regime of self-tuned coherent chaos. In this regime the coherent component of the dynamics self-adjusts intermittently to yield periods of slow, highly coherent chaos. The dynamics display longer time-scales and switching-like activity. We show how in this regime the dynamics depend qualitatively on the particular realization of the connectivity matrix: a complex leading eigenvalue can yield coherent oscillatory chaos while a real leading eigenvalue can yield chaos with broken symmetry. The level of coherence grows with increasing strength of structured connectivity until the dynamics are almost entirely constrained to a single spatial mode. We examine the effects of network-size scaling and show that these results are not finite-size effects. Finally, we show that in the regime of weak structured connectivity, coherent chaos emerges also for a generalized structured connectivity with multiple input-output modes.
Highlights
Firing-rate fluctuations and irregular spiking are ubiquitous in the neocortex [1, 2]. This temporal variability is often observed to be correlated across spatial scales ranging from local cortical circuits to the entire brain: in local cortical circuits both in membrane potential fluctuations [3] and on the level of spiking [4,5,6,7], in the coherency measured in brain-wide EEG signals [8, 9], and in the global signal observed across all voxels in fMRI measurements [10,11,12]
We can think of the row vector, νT, as an “output mode” performing a read-out of the network activity, and the column vector, ξ, as a corresponding “input mode” along which the output mode activity is fed back to the network (Fig 1A)
Coherent fluctuations are prevalent in cortical activity ranging in spatial scale from shared variability in membrane potential and spiking in local circuits to global signals measured across the scalp via EEG or across voxels via fMRI [3, 4, 9, 11]
Summary
Firing-rate fluctuations and irregular spiking are ubiquitous in the neocortex [1, 2]. This temporal variability is often observed to be correlated across spatial scales ranging from local cortical circuits to the entire brain: in local cortical circuits both in membrane potential fluctuations [3] and on the level of spiking [4,5,6,7], in the coherency measured in brain-wide EEG signals [8, 9], and in the global signal observed across all voxels in fMRI measurements [10,11,12]. A major challenge to theorists has been to produce network models which generate spatially coherent, temporally irregular fluctuations which can account for broad spatial correlations observed in experiments
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