Spiking neural networks (SNN) are dynamic models of biological neurons, that communicates with event-based signals called spikes. SNN that reproduce observed properties of biological senses like vision are developed to better understand how such systems function, and to learn how more efficient sensor systems can be engineered. A branching parameter describes the average probability for spikes to propagate between two different neuron populations. The adaptation of branching parameters towards critical values is known to be important for maximizing the sensitivity and dynamic range of SNN. In this thesis, a recently proposed SNN model for visual feature learning and pattern recognition known as the E-I Net model is studied and extended with a critical branching mechanism. The resulting modified E-I Net model is studied with numerical experiments and two different types of sensory queues. The experiments show that the modified E-I Net model demonstrates critical branching and power-law scaling behavior, as expected from SNN near criticality, but the power-laws are broken and the stimuli reconstruction error is higher compared to the error of the original E-I Net model. Thus, on the basis of these experiments, it is not clear how to properly extend the E-I Net model properly with a critical branching mechanism. The E-I Net model has a particular structure where the inhibitory neurons (I) are tuned to decorrelate the excitatory neurons (E) so that the visual features learned matches the angular and frequency distributions of feature detectors in visual cortex V1 and different stimuli are represented by sparse subsets of the neurons. The broken power-laws correspond to different scaling behavior at low and high spike rates, which may be related to the efficacy of inhibition in the model.