Classical Reductionism has its limits in explaining the properties of complex systems. Most neuroscientists tend to view consciousness as an emergent property of organisms that possess a sufficiently complex nervous system.
Essentially, an emergent property of a system is one that cannot be found in any of the constituents parts.
Some very loose and simple examples:
Combine the colors yellow and red and you get something with totally novel properties that neither color possess -- pink. Of course, those who like shades of the color pink do not say, "I like red and green combined"; they say ,"I like Pink."
Neither Oxygen or Hydrogen possess the properties of water, but the combination of hydrogen and oxygen yields a system with emergent properties that cannot be found in the constituents.
The notion of a soul appears on the scene because it seems like a logical way to reconcile the apparent conflict when we try to use a classical reductionist approach. In the view of most scientists(and many philosophers), consciousness is simply what you get when you put a system together containing these constituent elements -- a novel property that is manifest in the physical world and one that cannot be equated with the components themselves.
Like many other complex systems, consciousness is a symmetrical property in that it can induce feedback onto the component parts. For instance, the emergent property itself can interact holistically with the physical system. In the case of consciousness, this means a feedback that affects the components and neurons and causes a chain reaction that can alter the physical state of the organisms -- limb movement, memory etc etc...(analogy: self-induction of a magnetic coil). Hence, we have the notion of 'mind over matter', when in fact it would be more aptly described as 'a complex system with emergent properties undergoing feedback.'