Final answer:
The statement is false because transmitter-gated ion channels do rely on the presence of neurotransmitters to open and influence the neuron's membrane potential, which is essential for the initiation of an action potential.
Step-by-step explanation:
The statement "Transmitter-gated ion channels are insensitive to membrane potential; in the absence of neurotransmitters, they cannot generate an action potential" is false. Transmitter-gated ion channels (also known as ligand-gated ion channels) are indeed sensitive to the presence of neurotransmitters; however, they are not directly influenced by the membrane potential. These channels open when a specific neurotransmitter binds to them, allowing ions to enter or leave the neuron. This changes the neuron's membrane potential, potentially triggering an action potential if the depolarization threshold is reached. Action potentials are primarily mediated by voltage-gated ion channels, which are sensitive to changes in membrane potential. A ligand-gated channel by itself cannot generate an action potential in the absence of neurotransmitters; it is the binding of neurotransmitters to these channels that influences the neuron's excitability.