If we plot the number of surviving cells in a structure over time (assuming no replacement), the shape of that curve should imply something about the underlying process responsible for cell death. For example, if surrounding conditions mean that each cell has a constant probability of dying within a given interval, then an exponential decay curve should be seen. If the cell has a characteristic lifetime (with say a gaussian distribution about that mean lifetime), then (I guess) we might expect something like a logistic S-shaped curve.

In Parkinson's disease, the onset of motor symptoms is related to the number of dopaminergic neurons in the substantia nigra pars compacta, a small structure of just several hundred thousand neurons. The decline in cell numbers due to the disease process is indeed usually modelled as something like an exponential decay.

But there is also a normal age-related decline in substantia nigra neuron numbers. This decline is usually modelled as a linear decrease from birth through to old age, at something like 0.5% to 1% per year (Rudow et. al., 2008. Morphometry of the human substantia nigra in ageing and Parkinson’s disease. Acta Neuropathologica, 115(4), 461-470 http://dacemirror.sci-hub.tw/journal-article/fc4e7854ee0cef0146d60756536b2a88/rudow2008.pdf )

Note that this percentage decline per year is not a percentage of the surviving number (which would yield an exponential curve), but of the original total. In effect, it is perhaps best to think of it as a decline by a constant absolute number of neurons per year, yielding a straight line survival curve.

Exponential decline is memoryless (conditions simply conspire such that the probability of any cell's death is constant in a given interval, uninfluenced by the history of that cell or any other). The death of a cell with a characteristic lifespan, meanwhile, does have memory, as the probability of a given cell dying increases as that cell ages and deteriorates. But the death of a constant number of cells in a given interval somehow implies a sort of memory across cells, such that the number dying in a given interval is related to the initial number, and not to the current surviving population level.

My question is what general physiological processes are consistent with such a linear decline? i.e. is a linear decline a common pattern for neurons or any other sort of cell death, and in general, what sort of process governs it?

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    $\begingroup$ Note that linear functions are good approximations to many underlying nonlinear processes. Simply because a process is described with a linear function does not mean the process itself is linear. $\endgroup$
    – Bryan Krause
    Oct 12 '18 at 15:14
  • $\begingroup$ Yes that’s certainly true and is perhaps what I’m getting at. i.e. could a linear model here actually be consistent with some underlying physiological process, or is it simply an approximate statistical fit to noisy data from some other more physiologically viable underlying non-linear process? $\endgroup$ Oct 13 '18 at 0:51

Progressive cell death along a spatial dimension would be consistent with this. For example, imagine a cylinder of tissue in which cells at one face die, inducing adjacent cells to die. This would yield linear decay in viability to 0.

  • $\begingroup$ Yes, a spatial process would be viable: nice approach (not for this particular situation I suspect, but it would work as a general concept). $\endgroup$ Oct 17 '18 at 22:43
  • $\begingroup$ I suspect that the comment by @BryanKrause is more likely to be correct but I've awarded the bounty to this answer, as it does provide a general and feasible physiological mechanism as requested, although it is likely not viable for the particular case I have in mind. $\endgroup$ Oct 23 '18 at 0:20

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