Dual mechanism theory
The Dual Mechanism theory also known as the "words and rules" theory was summarised in Steven Pinker's book of the same name.
In more explicit neurological terms that is that Regulars are generated in the Left interior frontal lobe which supports rules, and irregulars are stored in the Temporal lobe which houses the lexicon.
This theory is backed up by evidence from aphasics with damage in one of the two areas listed above having difficulty with either the regulars or irregulars depending on the area damaged.
Crossmodal priming data (Marslen-Wilson, 1993) also stands up to the dual mechanism theory as facilitation occurs with the regular verbs but not the irregular ones.
Marslen-Wilson & Tyler (1998), however, showed that both regulars and irregulars pattern with repetition priming.
Albright, A, and Hayes, B P. 2003. Rules vs. analogy in English past tenses: a computational/experimental study. Cognition 90:119-161
Marslen-Wilson, W. D. 1993. Issues of process and representation in lexical access. In G. T. Altmann & R. ShiUcock (Eds.), Cognitive models of speech processing: The second Sperlonga meeting (pp. 187-210). Hillsdale, NJ: Erlbaum.
Marslen-Wilson, W. D., & Tyler, L. K. 1998. Rules, representations, and the English past tense. Trends in Cognitive Sciences, 2(11), 428-435.
Pinker, S. 1999. Words and Rules: The Ingredients of Language. Basic Books