Drones are used in an increasing number of applications including inspection, environment mapping, and search and rescue operations. During these missions, they might face complex environments with many obstacles, sharp corners, and deadlocks. Thus, an obstacle avoidance strategy that allows them to successfully navigate in such environments is needed. Different obstacle avoidance techniques have been developed. Most of them require complex sensors (like vision or a sensor array) and high computational power. In this study, we propose an alternative approach that uses two simple ultrasonic-based distance sensors and neural control with synaptic plasticity for adaptive obstacle avoidance. The neural control is based on a two-neuron recurrent network. Synaptic plasticity of the network is done by an online correlation-based learning rule with synaptic scaling. By doing so, we can effectively exploit changing neural dynamics in the network to generate different turning angles with short-term memory for a drone. As a result, the drone can fly around and adapt its turning angle for avoiding obstacles in different environments with a varying density of obstacles, narrow corners, and deadlocks. Consequently, it can successfully explore and navigate in the environments without collision. The neural controller was developed and evaluated using a physical simulation environment.