This paper describes a carbon nanotube synapse circuit that exhibits Spike-Timing Dependant Plasticity (STDP). These synapses are found in cortical (e.g. pyramidal) neurons. Experiments with the synapse in a neuron circuit demonstrate changes in synaptic potential with pre- and post-spiking timing variations. The circuit design is biomimetic and changes in control voltages representing neurotransmitter concentration lead to changes in synaptic strength. The experiments are demonstrated with SPICE simulations using carbon nanotube transistor models.