What is dualist interactionism (DI)? In broad strokes, it's really no mystery (in fine strokes, of course, it's nothing but mystery!): there are minds, incorporeal entities, and they interact somehow with physical brains. Nowhere does Eccles specify the interaction in question; and he gladly admits that such specification is currently beyond reach. However, he's well aware of the standard objection:
The materialist critics argue that insuperable difficulties are encountered by the hypothesis that immaterial mental events such as thinking can act in any way on material structures such a neurons of the cerebral cortex Such a presumed action is alleged to be incompatible with the conservation laws of physics, in particular of the first law of thermodynamics (, 187).
How does Eccles address this problem? By turning to quantum physics, specifically to the work of Margenau :
Following Margenau, the hypothesis is that mind-brain interaction is analogous to a probability field of quantum mechanics, which has neither mass nor energy yet can cause effective action at microsites. More specifically it is proposed that the mental concentration involved in intentions or planned thinking can cause neural events by a process analogous to the probability fields of quantum mechanics (, 189).
Such a scheme isn't likely to make any philosophical headway -- for the simple reason that that which Eccles appeals to here (as physicists and philosophers of science can confirm) is as controversial as interactionism itself.