How can animals defecate or urinate in fear if the nervous system seems to opposes it?

Question

How do animals defecate and/or urinate in fear even though their 'flight or fight' response seems to oppose it?

I have seen many cats and dogs urinate and defecate when they experience extreme fear. It also happens with most other animals in shock or fear including humans.

The 'why' is probably to deter predators which has been made excessively clear by my research but this is what is puzzles me despite my attempts.

From a book(made by doctors and teachers for medical exams ): Another website for medical exams

It is clear that sympathetic nervous system inhibits both these functions. My teachers said that adrenaline is complementary to the sympathetic nervous system and has the same effects on the body.

But the real life observations seem to contradict.

Also this question:
Fear induced defecation does not clear my question regrading the aforementioned contradiction.

Answer 1

It's overly simplistic to think of the sympathetic (SNS) and parasympathetic (PNS) as purely inversely activated; it's possible to activate both at once or suppress both at once. It's also overly simplistic to think of them as unitary responses; it's possible to activate parts of the SNS and not others, or parts of the PNS and not others.

Freezing, urination, and defecation in stressful situations are associated with corticotropin releasing factor (CRF, also known as CRH) in rodents. CRF is part of the response to stress and influences both parts of the autonomic nervous system. The effects of CRF are different in different parts of the digestive system (Taché & Bonaz, 2007; bold emphasis added by me):

In contrast to the inhibitory effects of CRF and urocortin 1 injected into the CSF on gastric and small-intestinal motor function, these peptides stimulate colonic transit and defecation and induce diarrhea through increased sacral parasympathetic outflow to the large intestine in female and male rats, mice, and gerbils

In humans, parasympathetic activation is also associated with some peoples' responses to trauma and PTSD, specifically the feeling of immobility or inability to act (Corrigan et al 2011).

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Corrigan, F. M., Fisher, J. J., & Nutt, D. J. (2011). Autonomic dysregulation and the window of tolerance model of the effects of complex emotional trauma. Journal of psychopharmacology, 25(1), 17-25.

Taché, Y., & Bonaz, B. (2007). Corticotropin-releasing factor receptors and stress-related alterations of gut motor function. The Journal of clinical investigation, 117(1), 33-40.