Frenchman's refusal to admit regret back in cell (6)
I believe the answer is:
neuron
'cell' is the definition.
(I know that neuron is a type of somatic cell)
'frenchman's refusal to admit regret back' is the wordplay.
'frenchman's refusal' becomes 'non' ('no' in French).
'to admit' is an insertion indicator.
'regret' becomes 'rue' (to rue something is to regret it).
'back' is a reversal indicator.
'non' placed around 'rue' is 'noruen'.
'noruen' backwards is 'NEURON'.
'in' acts as a link.
(Other definitions for neuron that I've seen before include "Part of one's nervous system" , "Nerve impulse cell" , "It transmits nerve impulses" , "Conductor" , "that makes for nervous transmission" .)