Bizarreness is a cognitive feature common to REM sleep dreams, which can be easily measured. Because bizarreness is highly specific to dreaming, we propose that it is most likely brought about by changes in neuronal activity that are specific to REM sleep. At the level of the dream plot, bizarreness can be defined as either discontinuity or incongruity. In addition, the dreamer's thoughts about the plot may be logically deficient. We propose that dream bizarreness is the cognitive concomitant of two kinds of changes in neuronal dynamics during REM sleep. One is the disinhibition of forebrain networks caused by the withdrawal of the modulatory influences of norepinephrine (NE) and serotonin (5HT) in REM sleep, secondary to cessation of firing of locus coeruleus and dorsal raphe neurons. This aminergic demodulation can be mathematically modeled as a shift toward increased error at the outputs from neural networks, and these errors might be represented cognitively as incongruities and/or discontinuities. We also consider the possibility that discontinuities are the cognitive concomitant of sudden bifurcations or "jumps" in the responses of forebrain neuronal networks. These bifurcations are caused by phasic discharge of pontogeniculooccipital (PGO) neurons during REM sleep, providing a source of cholinergic modulation to the forebrain which could evoke unpredictable network responses. When phasic PGO activity stops, the resultant activity in the brain may be wholly unrelated to patterns of activity dominant before such phasic stimulation began. Mathematically such sudden shifts from one pattern of activity to a second, unrelated one is called a bifurcation. We propose that the neuronal bifurcations brought about by PGO activity might be represented cognitively as bizarre discontinuities of dream plot. We regard these proposals as preliminary attempts to model the relationship between dream cognition and REM sleep neurophysiology. This neurophysiological model of dream bizarreness may also prove useful in understanding the contributions of REM sleep to the developmental and experiential plasticity of the cerebral cortex.