Using electrocorticogram (ECoG) analysis, we compared age-related dynamics of general neuronal activity and convulsive epileptiform responsiveness induced by intracortical microinjections of 4-aminopyridine (4-AP) in control Wistar rats and those subjected to prenatal hypoxia (Hx; E14; 7% O2, 3 h). The studies were carried out in three age periods roughly corresponding to childhood (P20–27), adolescence (P30–45), and adulthood (P90–120). It was found that in the process of postnatal development of the control rats, the peak of the ECoG power spectrum density (PSD) of the theta rhythm during wakefulness shifted from the low to the higher frequency, while in the Hx rats this shift had the opposite direction. Moreover, the Hx rats had different frequency characteristics of the ECoG PSD and longer episodes of spike-and-wave discharges caused by 4-AP injections compared to the controls. The total ECoG PSD of slow-wave sleep (1–5 Hz) was also dramatically decreased in the process of development of the Hx rats. Such alterations in PSD could be explained by the changes in balance of the excitation and inhibition processes in the cortical networks. Analyzing protein levels of neurotransmitter transporters in the brain structures of the Hx rats, we found that the content of the glutamate transporter EAAT1 was higher in the parietal cortex in all age groups of Hx rats while in the hippocampus it decreased during postnatal development compared to controls. Furthermore, the content of the vesicular acetylcholine transporter in the parietal cortex, and of the inhibitory GABA transporter 1 in the hippocampus, was also affected by prenatal Hx. These data suggest that prenatal Hx results in a shift in the excitatory and inhibitory balance in the rat cortex towards excitation, making the rat’s brain more vulnerable to the effects of proconvulsant drugs and predisposing animals to epileptogenesis during postnatal life.