  Dynamic excitatory-inhibitory (E-I) balance is a paradigmatic mechanism
invoked to explain the irregular low firing activity observed in the cortex.
However, we will show that the E-I balance can be at the origin of other
regimes observable in the brain. The analysis is performed by combining
simulations of sparse E-I networks composed of N spiking neurons with
analytical investigations of low dimensional neural mass models. The
bifurcation diagrams, derived for the neural mass model, allow to classify the
asynchronous and coherent behaviours emerging any finite in-degree K. In the
limit N >> K >> 1 both supra and sub-threshold balanced asynchronous regimes
can be observed. Due to structural heterogeneity the asynchronous states are
characterized by the splitting of the neurons in three groups: silent,
fluctuation and mean driven. The coherent rhythms are characterized by regular
or irregular temporal fluctuations joined to spatial coherence similar to
coherent fluctuations observed in the cortex over multiple spatial scales.
Collective Oscillations (COs) can emerge due to two different mechanisms. A
first mechanism similar to the pyramidal-interneuron gamma (PING) one. The
second mechanism is intimately related to the presence of current fluctuations,
which sustain COs characterized by an essentially simultaneous bursting of the
two populations. We observe period-doubling cascades involving the PING-like
COs finally leading to the appearance of coherent chaos. For sufficiently
strong current fluctuations we report a novel mechanism of frequency locking
among collective rhythms promoted by these intrinsic fluctuations. Our analysis
suggest that despite PING-like or fluctuation driven COS are observable for any
finite in-degree K, in the limit N >> K >> 1 these solutions result in two
coexisting balanced regimes: an asynchronous and a fully synchronized one.
