  We investigate spontaneous critical dynamics of excitatory and inhibitory
(EI) sparsely connected populations of spiking leaky integrate-and-fire neurons
with conductance-based synapses. We use a bottom-up approach to derive a single
neuron gain function and a linear Poisson neuron approximation which we use to
study mean-field dynamics of the EI population and its bifurcations. In the low
firing rate regime, the quiescent state loses stability due to saddle-node or
Hopf bifurcations. In particular, at the Bogdanov-Takens (BT) bifurcation point
which is the intersection of the Hopf bifurcation and the saddle-node
bifurcation lines of the 2D dynamical system, the network shows avalanche
dynamics with power-law avalanche size and duration distributions. This matches
the characteristics of low firing spontaneous activity in the cortex. By
linearizing gain functions and excitatory and inhibitory nullclines, we can
approximate the location of the BT bifurcation point. This point in the control
parameter phase space corresponds to the internal balance of excitation and
inhibition and a slight excess of external excitatory input to the excitatory
population. Due to the tight balance of average excitation and inhibition
currents, the firing of the individual cells is fluctuation-driven. Around the
BT point, the spiking of neurons is a Poisson process and the population
average membrane potential of neurons is approximately at the middle of the
operating interval $[V_{Rest}, V_{th}]$. Moreover, the EI network is close to
both oscillatory and active-inactive phase transition regimes.
