A signature feature of cortical circuits is the irregularity of neuronal firing, which manifests itself in the high temporal variability of spiking and the broad distribution of rates. Theoretical works have shown that this feature emerges dynamically in network models if coupling between cells is strong, i.e. if the mean number of synapses per neuron K is large and synaptic efficacy is of order 1/\sqrt{K}. However, the degree to which these models capture the mechanisms underlying neuronal firing in cortical circuits is not fully understood. Results have been derived using neuron models with current-based synapses, i.e. neglecting the dependence of synaptic current on the membrane potential, and an understanding of how irregular firing emerges in models with conductance-based synapses is still lacking. Moreover, at odds with the nonlinear responses to multiple stimuli observed in cortex, network models with strongly coupled cells respond linearly to inputs. In this talk, I will discuss the emergence of irregular firing and nonlinear response in networks of leaky integrate-and-fire neurons. First, I will show that, when synapses are conductance-based, irregular firing emerges if synaptic efficacy is of order 1/\log(K) and, unlike in current-based models, persists even under the large heterogeneity of connections which has been reported experimentally. I will then describe an analysis of neural responses as a function of coupling strength and show that, while a linear input-output relation is ubiquitous at strong coupling, nonlinear responses are prominent at moderate coupling. I will conclude by discussing experimental evidence of moderate coupling and loose balance in the mouse cortex.

Neural circuits exhibit complex activity patterns, both spontaneously and in response to external stimuli. Information encoding and learning in neural circuits depend on the ability of time-varying stimuli to control spontaneous network activity. In particular, variability arising from the sensitivity to initial conditions of recurrent cortical circuits can limit the information conveyed about the sensory input. Spiking and firing rate network models can exhibit such sensitivity to initial conditions that are reflected in their dynamic entropy rate and attractor dimensionality computed from their full Lyapunov spectrum. I will show how chaos in both spiking and rate networks depends on biophysical properties of neurons and the statistics of time-varying stimuli. In spiking networks, increasing the input rate or coupling strength aids in controlling the driven target circuit, which is reflected in both a reduced trial-to-trial variability and a decreased dynamic entropy rate. With sufficiently strong input, a transition towards complete network state control occurs. Surprisingly, this transition does not coincide with the transition from chaos to stability but occurs at even larger values of external input strength. Controllability of spiking activity is facilitated when neurons in the target circuit have a sharp spike onset, thus a high speed by which neurons launch into the action potential. I will also discuss chaos and controllability in firing-rate networks in the balanced state. For these, external control of recurrent dynamics strongly depends on correlations in the input. This phenomenon was studied with a non-stationary dynamic mean-field theory that determines how the activity statistics and the largest Lyapunov exponent depend on frequency and amplitude of the input, recurrent coupling strength, and network size. This shows that uncorrelated inputs facilitate learning in balanced networks. The results highlight the potential of Lyapunov spectrum analysis as a diagnostic for machine learning applications of recurrent networks. They are also relevant in light of recent advances in optogenetics that allow for time-dependent stimulation of a select population of neurons.