Believe it or not, but the biggest challenge to dualism does not come from neuroscience or physiology, and in fact is shared with materialism, it is the threat of epiphenomenlism. Whether mental is ideal or material it is clearly something successfully used in empirical reasoning. Neuroscience and physiology at this point are in the early preliminary stages of approaching what it is, how it comes to be, or even what the physical correlates of qualia, etc. are. Attempts to explain it away in functionalist terms largely failed. With that reductive materialism is unattractive, and non-reductive materialism and dualism are essentially equivalent for all empirical purposes. It really makes no difference if one thinks that mental is physical but there are no laws reducing the former to the latter (Davidson, Searle), or if one thinks that they are fundamentally different.
There is some sentimental aversion to letting mental causally affect physical, as in dualism, because of an air of mysticism, perhaps, but no one has a problem with letting physical affect mental. And if both are allowed to affect each other we might as well count them both as material, or ideal, or as idealizations of some unifying stuff, as Bergson suggested, or reduxes of some qualitatively different from both stuff, as Russell once held. The last position is called neutral monism. In 19th century many had doubts as to whether electromagnetic field was matter, some courts even ruled that stealing electricity is not stealing, because electricity is not material.
The above sentiment is not philosophically compelling, Burge, who is not even a dualist, dismisses it outright. A more serious issue with mental causation is a potentially inconsistent interference with physical causation which is subject to natural law. For materialism mental is a manifestation of physical, so mental causation reflects physical causation, but this does not clearly resolve the issue unless mental is also epiphenomenal, i.e. translatable into impersonal terms where mental reasons and impulses play no role. That is a problem because it seems to reduce man to machine, not to mention free will issues. Burge explicitly advocates an agnostic middle path, where we acknowledge phenomenological uses of the mental, but withhold judgement on its ontological nature and interaction with the physical until further investigation:
"Why should mental causes of physical effects interfere with the physical system if they do not consist in physical processes? Thinking that they must surely depends heavily on thinking of mental causes on a physical model... But whether the physical model of mental causation is appropriate is part of what is at issue".
This is less explicitly embraced by other non-reductivists, see e.g. 1950-2000 historical survey in Burge's Foundations of Mind (Ch.20).
Among those who dare step into the physical models most prominent, predictably, are not professional philosophers but philosophizing physicists. The strategy is known since Boussinesq pioneered it in 1870s. Find "causal gaps" in laws of nature and describe how mental provides a "guiding principle" to resolve the indeterminacy, while respecting the conservation laws, or emerges from one. Boussinesq made do with non-uniqueness in (non-Lipschitz) classical models, but since Heisenberg quantum mechanics is used instead. The idea is that "consciousness causes collapse", and the task is to find a mechanism that obviates Tegmark's objections to "quantum mind" wrapped into his metaphor that brain is "too warm, wet, and noisy". Penrose made an extravagant but empirically testable suggestion involving quantum gravity and microtubules. More recently a theory in this vein appears in Kauffman's Physics and Five Problems in the Philosophy of Mind, who relies in particular on a recent discovery of the role of quantum effects in photosynthesis, which seems like an exception to Tegmark's "too warm, wet, and noisy".