Activity in the brain propagates as waves of firing neurons, namely avalanches. These waves’ sizes and duration distributions have been experimentally shown to display a power law profile and to have long-range correlations *in vivo* and *in vitro*. We study an avalanching model of the primary visual cortex and find a minimum value for the time that the network takes to process input information by varying the excitatory postsynaptic potential parameter (EPSP). We determine this point analytically by mean field approximation, which matches the expected experimental value of EPSP. Surprisingly, this point is located inside a Griffiths phase where avalanches are power law distributed. These power law distributed avalanches are associated with a critical or quasi-critical state of the system. We also characterize these critical avalanches nontrivial scaling, order parameter and temporal correlations through their power spectrum.