Yes they are called senescent cells and removing them seems to have an anti-ageing effect.
All our cells (not just nerve cells) communicate chemically with each other to a certain degree.
One of the theories of ageing is that these senescent cells are able to cause disease and ill health without becoming cancerous, probably by releasing certain chemical agents which act to cause dysfunction in otherwise normal cells.
They seem to ratchet up the activity of the immune system in particular which can cause chronic inflammation - another process that seems to increase with age and also acts to cause damage to healthy cells, potentially causing a feedback loop which creates more senescent cells.
Under normal circumstances such cells are supposed to commit suicide by a process known as apoptosis (programmed cell death). However as we age these processes seem to get less efficient (possibly due to DNA damage) and we slowly build up more and more of them.
So to summarise they act lack bad leaders which signal other cells to misbehave and once they are removed the other cells go back to functioning normally.
I believe combining this sort of therapy to remove senescent cells and supplementing stem cells would create a very potent means of rolling back the chronological clock.
the process you are looking for for is called autophagy. It is where your immune system, in the form of phagocytes, search out and consume inefficient/ dying/ damaged cells and harvest their nutrients for reuse in other fresh cells.
Vampire therapy where an aged individual is leached off a youngling, is one way of replacing senescent proteins and cells, but a better way is through intermittent fasting either by way of time restricted eating windows of 8-10 hrs per day, or using something like Longo's fasting mimicking diet (www.prolonfmd.com) which is all the good of fasting without the downside like sarcopenia (muscle loss) which can be irreversible in some elderly, leading to their demise. (Thankfully this can be mitigated with a daily dose of 1-1.5g branched chain amino acids, and the use of exogenous ketone bodies). Heat shock proteins are also a way to induce autophagy, although not as powerful.
The discussion of autophagy raises a key point about a couple of interesting approaches to anti ageing, namely how long would you run rapalogs for given they inhibit autophagy to certain degree, and allow possible free radical production by cells containing damaged mitochondria leaking from disrupted electron chain pathways? The likely answer may be that we cycle this stuff, and have runs of treatments with an underlying persistent effort to detoxify, and unburden our mitochondria, then initiate mitochondrial biogenesis.