We can identify many kinds of patterns of how neurons connect such as lateral inhibition, negative/positive feedback, convergence, divergence, and facilitation. But have any of these circuits been used to make claims about how a cognitive or psychological phenomena is computed? Is that even possible, given its complexity? In which context is neural circuitry useful?
Contrary to one of the other answers, I will have to respectfully disagree. Neural circuity is both the pinnacle and future of cognitive neuroscience. We already know the large areas of the brain are associated with specific cognitive processes, for example the NAc shell is associated with desire to seek out motivational objects such as food, but this could just as easily facilitate gaming, dancing or sex. The problem comes with investigating these behaviours in humans. We can easily test circuits on animals, but humans are far more difficult as we can only use scanning equipment like EEG and fMRI. While scanning equipment has improved the spatial or temporal resolution makes studying these circuits in humans difficult. However animal research supplements this and can show us circuits that are involved in particular cognitions, while the patterns of neural firing tell indicate particular activity. For instance we know the visual pathways in great detail how light engages neural activity in the retina, and how this information is transmitted to the occipital cortex to create a representation spread across the neural activity of the occipital cortex. In fact we understand this so well now that cognitive neuroscientists can take an EEG recording of activity while you sleep and interpret the neural activity into an image under specific circumstances. This fascinating method is called neural decoding, and it was developed out from our understanding of neural patterns of activity and mapping of neural networks developed over decades. Of course the same neural activity can be used to map networks and model neural activity in computational models which mimic cognitive processes.
Of course this is easier for vision, hearing and somatic perception as these are straightforward cognitive processes which are generating virtual representations of the world we experience. What is far more difficult to assess are the memories and experiences that are recalled and how these influence decision making relative to the environment you find yourself in. However this is all being investigated, research starts with large neural structures like the NAc shell and then moves down to more specific areas followed by the circuits connecting it to other parts of the brain relative to particular cognitions. An example may help, motivation like hunger drives you to eat this is used by both information coming from your environment and internal changes. For instance once food has been consumed our brains assess the value of the food, did we like it? was the food more or less pleasurable than last time? did it make us happy? This activates the pACC, PAG, OCC, lOFC, rACC. The areas mentioned largely deal with attention, affective assessment (feelings), and pleasure. The activation of these areas helps an individual to make future decisions about consuming food. In rats the order of area activation is well understood, in humans there is still work to be done. But overall the fundamental mechanisms are the same, and can be modelled computationally. Understanding how the fundamental structures and there activity in context for both people and animals is essential in understanding disorders ranging from schizophrenia to obesity, and helps other researchers to develop assessments and interventions, such as drugs or government legislation. Or for companies to taylor their products to their consumers needs. Crucially, to the question at least, this information can be used to predict behaviour, in fact that basically the role of any computational model.
Fundamentally our hard wiring is relatively similar, it is this that we investigate, and it is these hard wired mechanisms that have the greatest impact on our behaviour and cognition. Cognitive scientists use the activation patterns of neutrons as a means of mapping these cognitive processes in neural circuitry. Finally I will say that overall the brain isn't really so complicated, its quiet simple really if you consider it from an evolutionary perspective. We and all other animals have developed by means of natural selection to survive long enough to pass on our genes, both our behaviour and neural development demonstrate this. The extra complications in behaviour are merely complexities that enhance gene survival, until they are lost or supersede within the same selection process.
Check out [http://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.1002328] The Contribution of Network Organization and Integration to the Development of Cognitive Control by Scott Marek,Kai Hwang,William Foran,Michael N. Hallquist,Beatriz Luna
Just Published on PLOS Biology: December 29, 2015 DOI: 10.1371/journal.pbio.1002328
Cognitive control, which continues to mature throughout adolescence, is supported by the ability for well-defined organized brain networks to flexibly integrate information. However, the development of intrinsic brain network organization and its relationship to observed improvements in cognitive control are not well understood.
It's a fascinating read about how brain function may "top out" at a certain point, but the increasing integration of networks contribute to increased cognitive control.