The connectivity network in the brain forms the basis for its functionality. Past studies have focused on the use of random (Erdos-Renyi, ER) networks or simple distant-dependent connectivity rules to model biological neural networks (BNNs). However, experimental data shows that specific network motifs are either significantly overrepresented or underrepresented in BNNs as compared to random networks. The functional significance of such non-random motifs in BNNs is poorly understood. Here, we specifically ask the question of how the presence of 3-neuron motifs in the excitatory and inhibitory populations affects the population firing rates and dimensionality of the network. To test this, we set up simulations of excitatory (E)-inhibitory (I) networks. We found that 3-neuron motifs introduce heterogeneity in the degree of neurons and lead to the formation of densely or sparsely connected sub-networks.. Consistent with previous results, we found that overrepresentation of convergent and chain motifs in the E-population increases synchrony and oscillations in the network. By contrast, overrepresentation of convergent and chain motifs in the I-population decreases the network's oscillations and synchrony (dimensionality). Thus, the presence of convergent motifs in the I population can counter the oscillations caused due to 3-neuron motifs in the E-population. Given that cortical activity is usually asynchronous and irregular, we predict that similar to E-neurons, I-neurons should also show the prevalence of certain classes of 3-neuron motifs.