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In this paper, the design of a neuromorphic cortical neuron with synaptic and ion-channel variability is presented and and simulated using carbon nanotube circuit elements. Since the variability could be because of noise or chaos, in this paper, both possible sources of variability are considered by embedding either Gaussian noise or a chaotic signal into the synaptic circuit and the axon circuit and observing the results. The paper also presents a chaotic signal generator using carbon nanotube transistors that could be embedded in the electronic neural circuit. The circuit uses a chaotic piecewise linear one-dimensional map implemented by switched-current circuits. The design was simulated using carbon nanotube SPICE models. Spontaneous firing of neurons due to intrinsic variability was demonstrated.