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Neural circuits exhibit complex activity patterns, both spontaneously and evoked by external stimuli. Information encoding and learning in neural circuits depend on how well time-varying stimuli can control spontaneous network activity. We show that in firing-rate networks in the balanced state, external control of recurrent dynamics, i.e., the suppression of internally-generated chaotic variability, strongly depends on correlations in the input. A distinctive feature of balanced networks is that, because common external input is dynamically canceled by recurrent feedback, it is far more difficult to suppress chaos with common input into each neuron than through independent input. To study this phenomenon, we develop a non-stationary dynamic mean-field theory for driven networks. The theory explains how the activity statistics and the largest Lyapunov exponent depend on the frequency and amplitude of the input, recurrent coupling strength, and network size, for both common and independent input. We further show that uncorrelated inputs facilitate learning in balanced networks.

This article is part of the Physical Review Research collection titled Physics of Neuroscience. Recurrent networks are widely used as models of biological neural circuits and in artificial intelligence applications. Mean-field theory has been used to uncover key properties of recurrent network models such as the onset of chaos and their largest Lyapunov exponents, but quantities such as attractor dimension and Kolmogorov-Sinai entropy have thus far remained elusive. We calculate the complete Lyapunov spectrum of recurrent neural networks and show that chaos in these networks is extensive with a size-invariant Lyapunov spectrum and attractor dimensions much smaller than the number of phase space dimensions. The attractor dimension and entropy rate increase with coupling strength near the onset of chaos but decrease far from the onset, reflecting a reduction in the number of unstable directions. We analytically approximate the full Lyapunov spectrum using random matrix theory near the onset of chaos for strong coupling and discrete-time dynamics. We show that a generalized time-reversal symmetry of the network dynamics induces a point symmetry of the Lyapunov spectrum reminiscent of the symplectic structure of chaotic Hamiltonian systems. Temporally fluctuating input can drastically reduce both the entropy rate and the attractor dimension. We lay out a comprehensive set of controls for the accuracy and convergence of Lyapunov exponents. For trained recurrent networks, we find that Lyapunov spectrum analysis quantifies error propagation and stability achieved by different learning algorithms. Our methods apply to systems of arbitrary connectivity and highlight the potential of Lyapunov spectrum analysis as a diagnostic for machine learning applications of recurrent networks.

Neuronal activity in the central nervous system varies strongly in time and across neuronal populations. It is a longstanding proposal that such fluctuations generically arise from chaotic network dynamics. Various theoretical studies predict that the rich dynamics of rate models operating in the chaotic regime can subserve circuit computation and learning. Neurons in the brain, however, communicate via spikes and it is a theoretical challenge to obtain similar rate fluctuations in networks of spiking neuron models. A recent study investigated spiking balanced networks of leaky integrate and fire (LIF) neurons and compared their dynamics to a matched rate network with identical topology, where single unit input-output functions were chosen from isolated LIF neurons receiving Gaussian white noise input. A mathematical analogy between the chaotic instability in networks of rate units and the spiking network dynamics was proposed. Here we revisit the behavior of the spiking LIF networks and these matched rate networks. We find expected hallmarks of a chaotic instability in the rate network: For supercritical coupling strength near the transition point, the autocorrelation time diverges. For subcritical coupling strengths, we observe critical slowing down in response to small external perturbations. In the spiking network, we found in contrast that the timescale of the autocorrelations is insensitive to the coupling strength and that rate deviations resulting from small input perturbations rapidly decay. The decay speed even accelerates for increasing coupling strength. In conclusion, our reanalysis demonstrates fundamental differences between the behavior of pulse-coupled spiking LIF networks and rate networks with matched topology and input-output function. In particular there is no indication of a corresponding chaotic instability in the spiking network.