Yes, the distance between a chemical synapse and the axon hillock is important for whether an action potential is generated because changes in V m in the dendrites depend on diffusion, which varies with distance.
Chemical synapses are junctions through which two neurons can send signals to each other and to non-neuronal cells (muscle cell, glandular cell, sensory cell).
And, the axon hillock is an important step for generating an action potential because the axon hillock contains a very high concentration of voltage-gated Na+ channels that become activated once a critical membrane potential is reached, the threshold potential.
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