Assuming the original poster (now departed) was considering apathy as a distinct phenomena from depression, then current evidence points to the current neurological basis of apathy being connectivity in the pre-motor brain systems according to "Individual Differences in Premotor Brain Systems Underlie Behavioral Apathy" by Bonelle et al. (2016):
The results demonstrate that behavioral apathy is associated with
increased effort sensitivity as well as greater recruitment of neural
systems involved in action anticipation: supplementary motor area
(SMA) and cingulate motor zones. In addition, decreased structural and
functional connectivity between anterior cingulate cortex (ACC) and
SMA were associated with increased behavioral apathy. These findings
reveal that effort sensitivity and translation of intentions into
actions might make a critical contribution to behavioral apathy. We
propose a mechanism whereby inefficient communication between ACC and
SMA might lead to increased physiological cost—and greater effort
sensitivity—for action initiation in more apathetic people.
Note that the authors of the paper controlled for depression using questionnaires:
Self-reports of apathy traits were obtained using a modified, extended
version of the original Lille Apathy Rating Scale (LARS-e), available
online in Bonnelle et al. (2014) (see Supplementary Material for more
details). The LARS-e uses subscales that allow assessment of apathy
traits along several domains reflecting the distinct component of
apathy (behavioral, cognitive, and emotional). We used the “Action
Initiation” (AI) subscale of the LARS-e, which measures every-day
productivity and initiative and is an index of behavioral apathy
(Sockeel et al. 2006). This subscale was previously found specifically
to relate to the willingness to engage in an effort response in order
to obtain a reward on our paradigm (Bonnelle et al. 2014). In
addition, to control for a potential confound of depression and
anhedonia, we also used the Depression, Anxiety, Stress Scales (DASS)
(Lovibond and Lovibond 1995), a questionnaire developed in nonclinical
populations to measure depression, and the Snaith–Hamilton Pleasure
Scale (Snaith et al. 1995), which assesses anhedonia.
The authors assessed patients according to their willingness to complete a task while stakes (a combination of expected reward and expected effort) were manipulated. They found a high correlation between sensitivity to effort and behavioural apathy as assessed via the questionnaires in the previous quote. They then examined connectivity via fMRI of the patients completing the task and determined certain areas were more connected in patients with lower behavioural apathy. Specifically, the pre-motor brain systems were better connected in non-apathetic individuals.
It's unclear what this means cognitively, but the authors hypothesize, based on other activation, that this: "might be due to higher 'subjective experience' of effort cost in individuals who are more apathetic".