I have one particular area to offer as a potential example. Keep in mind that likely no formalization of psychological concepts has heretofore been comprehensive (and the utility of a comprehensive "model" may be questionable, but I'll leave this for my more philosophical colleagues to consider).
I am not a psychologist, but from what I can tell, both classical and instrumental conditioning are formalized in reinforcement learning. For instance, many but not all aspects of classical conditioning may be captured by models such as the Rescorla-Wagner rule. One of the most successful models of instrumental control has been the temporal-difference learning algorithm (Sutton & Barto 1998), which contributed significantly to our understanding of the computational function of mesencephalic dopaminergic neurons in several species (Schultz, Dayan, and Montague 1997; Schultz, 2015; Eschel et al 2015). The mathematical details will not be presented here since excellent treatments exist elsewhere (Sutton & Barto 1998; Dayan & Abbott, 2001). Briefly put, not only has the temporal difference learning rule mathematically formalized elements of psychological theories of behavioural control (mapping from behavioural to algorithmic level), but its critical element (the reward prediction error) has also been successfully mapped to neural substrates. (For many years, these neural substrates were independently thought to be involved in reward learning.)
A note on further implications
I know you didn't ask directly about this, but I can't help myself :P
Of particular interest in recent years relates to the fact that instrumental control can be further subdivided into model-free (MF; aka "habitual") and model-based (MB; "goal-directed") systems (Dolan & Dayan, 2013), and that humans ostensibly use a combination of the two during sequential planning tasks (Daw et al. 2011). Probing the balance of MB/MF control in healthy controls and neuropsychiatric patients has suggested that deficits in goal-directed control may, in fact, be a cross-cutting neuropsychiatric symptom dimension spanning eating disorders, obsessive-compulsive disorder, and addiction (Gillan et al. 2016).
1.Sutton & Barto. Reinforcement Learning: An Introduction (1998)
2. Schultz, W., Dayan, P., & Montague, P. R. (1997). A neural substrate of prediction and reward. Science (New York, N.Y.), 275(5306), 1593–1599.
3. Schultz, W. (2015). Neuronal Reward and Decision Signals: From Theories to Data. Physiological Reviews, 95(3), 853–951.
4. Eshel, N., Bukwich, M., Rao, V., Hemmelder, V., Tian, J., & Uchida, N. (2015). Arithmetic and local circuitry underlying dopamine prediction errors. Nature.
6. Dayan, P., & Abbott, L. F. (2001). Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. Computational and Mathematical Modeling of Neural Systems. Cambridge, Massachusetts: MIT Press.
6. Dolan, R. J., & Dayan, P. (2013). Goals and habits in the brain. Neuron, 80(2), 312–325.
7. Daw, N. D., Gershman, S. J., Seymour, B., Dayan, P., & Dolan, R. J. (2011). Model-based influences on humans’ choices and striatal prediction errors. Neuron, 69(6), 1204–1215.
8. Gillan, C. M., Kosinski, M., Whelan, R., Phelps, E. A., & Daw, N. D. (2016). Characterizing a psychiatric symptom dimension related to deficits in goal-directed control. eLife, 5, 1–24.
9. Voon, V. et al. (2014). Disorders of compulsivity: a common bias towards learning habits. Molecular Psychiatry, 20(October 2013), 1–8.
10. Voon, V. et al. (2015). Motivation and value influences in the relative balance of goal-directed and habitual behaviours in obsessive-compulsive disorder. Translational Psychiatry, 5(11), e670.