Neurons in cortical slices emit spikes or bursts of spikes regularly in response to a suprathreshold current injection. This is in marked contrast to the behavior of cortical neurons in-vivo, whose response to electrical or sensory input displays a strong degree of irregularity. Correlation measurements show significant degree of synchrony in the temporal fluctuations of neuronal activities in cortex. We explore the hypothesis that these phenomena are the result of the synchronized chaos generated by the deterministic dynamics of local cortical networks. A model of a "hypercolumn" in the visual cortex is studied. It consists of two populations of neurons, one inhibitory and one excitatory. The dynamics of the neurons is based on a Hodgkin-Huxley type model with several cellular and synaptic conductances. The pattern of connectivity is correlated with the internal organization of hypercolumns in orientation columns. Numerical simulations of the model show that in an appropriate parameter range, the network settles in a synchronous chaotic state, characterized by a strong temporal variability of the neural activity which is correlated across the hypercolumn. Strong inhibitory feedback is essential for the stabilization of this state. Auto-correlation and cross-correlation functions of neuronal spike trains are computed, and analyzed. The relation between the results of the model and experiments in visual cortex is discussed.