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.