Nope. Every cell is an autonomous circadian oscillator. Not just every neuron, but every cell. Red blood cells don’t have a nucleus, but they do have a metabolic clock.

You and I are clusters of about 36 trillion cells that are all able to individually keep time. They synchronize and orchestrate their timekeeping, and entrain to the external environment, and there is a cluster of cells, the suprachiasmatic nucleus, that is the master orchestrator of circadian time, sitting right behind the eyes as light is the main signal it uses to tell external time.

We know its function can get dampened with age.

However, please note this is not how we perceive time, in the sense of being able to count 60 seconds and have it match the wall clock. That’s a subsystem. But the internal temporal order of your body is cell by cell, not dictated from one spot in your body.