Here, we narrow this gap by developing aneffectivemethod fortraining acanonical model ofcortical neural circuits, the stabilized supralinear network (SSN), that in previous work had to beconstructed manually ortrainedwithundueconstraints.

Excitation-inhibition balance is ubiquitously observed in the cortex. Recent studies suggest an intriguing link between balance on fast timescales, tight balance, and efficient information coding with spikes. We further this connection by taking a principled approach to optimal balanced networks of excitatory (E) and inhibitory(I) neurons. By deriving E-I spiking neural networks from greedy spike-based optimizations of constrained minimax objectives, we show that tight balance arises from correcting for deviations from the minimax optimum. We predict specific neuron firing rates in the networks by solving the minimax problems, going beyond statistical theories of balanced networks. We design minimax objectives for reconstruction of an input signal, associative memory, and storage of manifold attractors, and derive from them E-I networks that perform the computation. Overall, we present a novel normative modeling approach for spiking E-I networks, going beyond the widely-used energy-minimizing networks that violate Dale's law. Our networks can be used to model cortical circuits and computations.

Weaknesses: Although I believe the math derivation of the novel minimax objective function is correct, I have two major concerns. My first concern is whether this minimax objective function provides some novel insight on network dynamics which cannot be captured by traditional framework that network dynamics is minimizing an "energy" function. My concern is resulted from that the minimax objective (Eq. It seems to me that the only difference between the minimax and minimized objective function is that the network state converges to the saddle point in the former case, while in later case the network state converges to a stable fixed point. I really hope the authors explain this and correct me if I understood something wrong.

The reviewers were originally divergent in their opinions of this paper, but came to some agreements in discussion. It was agreed that the paper provides an interesting contribution for neuroscience by extending the previous work of Seung et al. (1997) to more biologically realistic networks, but the actual theoretical insights beyond that original paper are not large. In the end, an "accept" decision was reached, but it was agreed that the authors should better clarify the strong links to the Seung paper and be more cautious in their claims of "detailed" or "tight" balance in cortical networks.

Excitation-inhibition balance is ubiquitously observed in the cortex. Recent studies suggest an intriguing link between balance on fast timescales, tight balance, and efficient information coding with spikes. We further this connection by taking a principled approach to optimal balanced networks of excitatory (E) and inhibitory(I) neurons. By deriving E-I spiking neural networks from greedy spike-based optimizations of constrained minimax objectives, we show that tight balance arises from correcting for deviations from the minimax optimum. We predict specific neuron firing rates in the networks by solving the minimax problems, going beyond statistical theories of balanced networks.

In many cases the crosscorrelations between the activities of cortical neurons are approximately symmetric about zero time delay. These have been taken as an indication of the presence of "functional connectivity" between the correlated neurons (Fetz, Toyama and Smith 1991, Abeles 1991). However, a quantitative comparison between the observed cross-correlations and those expected to exist between neurons that are part of a large assembly of interacting population has been lacking. Most of the theoretical studies of recurrent neural network models consider only time averaged firing rates, which are usually given as solutions of mean-field equations. They do not account for the fluctuations about these averages, the study of which requires going beyond the mean-field approximations. In this work we perform a theoretical study of the fluctuations in the neuronal activities and their correlations, in a large stochastic network of excitatory and inhibitory neurons. Depending on the model parameters, this system can exhibit coherent undamped oscillations. Here we focus on parameter regimes where the system is in a statistically stationary state, which is more appropriate for modeling non oscillatory neuronal activity in cortex. Our results for the magnitudes and the time-dependence of the correlation functions can provide a basis for comparison with physiological data on neuronal correlation functions.

In many cases the crosscorrelations between the activities of cortical neurons are approximately symmetric about zero time delay. These have been taken as an indication of the presence of "functional connectivity" between the correlated neurons (Fetz, Toyama and Smith 1991, Abeles 1991). However, a quantitative comparison between the observed cross-correlations and those expected to exist between neurons that are part of a large assembly of interacting population has been lacking. Most of the theoretical studies of recurrent neural network models consider only time averaged firing rates, which are usually given as solutions of mean-field equations. They do not account for the fluctuations about these averages, the study of which requires going beyond the mean-field approximations. In this work we perform a theoretical study of the fluctuations in the neuronal activities and their correlations, in a large stochastic network of excitatory and inhibitory neurons. Depending on the model parameters, this system can exhibit coherent undamped oscillations. Here we focus on parameter regimes where the system is in a statistically stationary state, which is more appropriate for modeling non oscillatory neuronal activity in cortex. Our results for the magnitudes and the time-dependence of the correlation functions can provide a basis for comparison with physiological data on neuronal correlation functions.

In many cases the crosscorrelations betweenthe activities of cortical neurons are approximately symmetric about zero time delay. These have been taken as an indication of the presence of "functional connectivity" between the correlated neurons (Fetz, Toyama and Smith 1991, Abeles 1991). However, a quantitative comparison between the observed cross-correlations and those expected to exist between neurons that are part of a large assembly of interacting population has been lacking. Most of the theoretical studies of recurrent neural network models consider only time averaged firing rates, which are usually given as solutions of mean-field equations. They do not account for the fluctuations about these averages, the study of which requires going beyond the mean-field approximations. In this work we perform a theoretical study of the fluctuations in the neuronal activities and their correlations, in a large stochastic network of excitatory and inhibitory neurons. Depending on the model parameters, this system can exhibit coherent undamped oscillations. Here we focus on parameter regimes where the system is in a statistically stationary state, which is more appropriate for modeling non oscillatory neuronal activity in cortex. Our results for the magnitudes and the time-dependence of the correlation functions can provide a basis for comparison with physiological data on neuronal correlation functions.