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I think both Dickinson's and Descartes' interpretations of the brain have credibility to them, but I don't think they should be two, standalone interpretations of the brain, i.e. perhaps neither is entirely correct without the other. 

I feel that this is often the case in psychology, neurobiology, and behavior - one example is the diathesis-stress model used to predict an individual's disposition towards a certain psychological disorder.  A diathesis is the individual's genetic predisposition and the stress can be a combination of various environmental factors.  In this sense, perhaps arguments such as the age-old "nature vs nurture" debate are moot, as both nature and nurture have significant bearing on studies in animal and human behavior. 

Similarly, Dickinson and Descartes could both be right, together.  While the brain's neural networks, chemistry, and specific structures are key in determining many things (e.g. personality, as in the case of Phineas Gage or psychological and behavioral patterns, especially those affected by pharmaceuticals), I agree with Descartes' opinion that our thoughts are neither physical nor physical manifestations of strictly biological processes in the brain.