In the book "Thinking, Fast and Slow", the author claims that there are two systems of thinking. System 1 is quick, instinctive and emotionally driven while System 2 is more logical and deliberate.
Firstly, has there ever been any physiological proof, in terms of brain activation areas or other signs, that these two systems exist. If they do exist, have they ever been capture in a cognitive model, regardless of biological plausibility?
I would disagree with @Krysta: the distinction between system 1 and system 2 processes goes far beyond that of cognition with/without emotion: it's a complex debates that's been going on since the 70s (in it's current form, it's echoed in a wider debate going back centuries). For the sake of brevity, let me resort to bullet points.
For psychologists, Evans (2008) is the go to reference on this matter. Obviously, Kahneman is the grandfather of this topic, and his book is a great introduction for the layman, you'll find a much more rigorous discussion here.
There's a lot of variation in terminology here - while "dual process theory" is the commonly accepted generic term for this idea, you'll also see it as "type, or system 1/2 processes", "intuition/logic", "associative/controlled processes", etc. Again, see Evans (2008).
A nice simplification of the distinction would be to say there's two kinds of processes, System 2 processes, which requires Working Memory (i.e. maintaining, updating, and manipulating a representation of information over a period of time), and System 1 processes, which don't. System 1, when taken this way, isn't a single "subsystem", its a diverse set of processes (this idea is nicely laid out in Stanovich, 2009, who uses the term "The Set of Autonomous Systems (TASS")), amongst other sources) including evolved, domain specific modules for solving specific problems (i.e. recognising faces), the emotion subsystem, and rules and associations which have become automatic through experience, and implicit learning (stereotypes, for example).
To finally get to your question, yes, this does translate into specific neural systems.
Working Memory (and thus system 2 processes) is pretty clearly localised to the prefrontal cortex (see Braver et al, 1997, for example).
System 1, not being a single "system", obviously can't be localized to a single brain region. However, many of it's component processes can be: emotion to the limbic system, face recognition to the fusiform face area, etc. (I don't know much about localization of implicit learning, but I would guess the hippocampus is involved).
I've just realised I haven't answered your second question, about cognitive models. I'll come back to this at a later date. EDIT I've just noticed you're working with Chris Eliasmith. I can think of few people better qualified to answer your second question than he is.
Neuroanatomically speaking I think Kahneman may be talking about the "hot" and "cold" executive function pathways (this wikipedia article is fairly informative on the subject). Hot executive function is thought to involve affect or reward processing, and the ventromedial prefrontal cortex is generally thought to be crucial for this. Cold executive function is thought to be invoked more by abstract tasks with little reward involvement, and is associated with the dorsolateral prefrontal cortex.
This extract from Prencipe et al. (2011, cited below) has more to say about the cognitive modeling of this theory--the article is from a developmental standpoint, so it speaks specifically to children and adolescents.
Rather than considering executive function (EF) in hot and cool situations in a dichotomous fashion, this characterization recognizes the interplay between relatively hot aspects of EF and relatively cool aspects of EF (Cunningham & Zelazo, 2007). According to this model, information is processed hierarchically, with relatively quick evaluative reactions followed by the generation of approach–avoidance-oriented responses. These relatively simple responses may suffice, or, if necessary, further processing and reprocessing of information may ensue that allows reflection on context and future consequences, and that supports the top-down control of behavior. The extent to which a reflective response is likely to be generated depends on a number of factors, including time, motivation, and neural and cognitive maturity. Given that neural development also proceeds in a generally hierarchical fashion, with areas of the brain associated with more complex processing developing later than areas of the brain associated with more automatic processing (e.g., Bunge & Zelazo, 2006), children and adolescents may be expected to be less reflective than adults. Moreover, difficulty in generating reflective responses is likely to be most evident in emotionally charged situations that have meaningful personal consequences; in these situations, emerging cognitive control must be implemented in the context of potential interference from relatively automatic responses to potential rewards and punishments.
I haven't read "Thinking Fast & Slow" but I would in general be pretty skeptical of claims that there are two "systems" for thinking; it is a bit more accurate in my view to say that these are two capabilities that can be brought to bear on a situation, and most situations will involve some combination of both. In addition, I have never heard any evidence that there are processing-speed differences between hot and cold.
Prencipe, Angela; Kesek, Amanda; Cohen, Julia; Lamm, Connie; Lewis, Marc D.; Zelazo, Philip David (2011). "Development of hot and cool executive function during the transition to adolescence". Journal of Experimental Child Psychology 108 (3): 621–637. doi:10.1016/j.jecp.2010.09.008. ISSN 0022-0965.
Regarding the first question:
It is well documented that emotions guide/modulate "higher" cognitive processes. A more specific theory of how this could work is, for instance, Antonio Damasio's somatic marker hypothesis. Additionally, I'm not so sure emotion and "cold" cognition can really be categorically seperated, seeing as they share a lot of the same brain structures/processes: "Brain structures at the heart of the neural circuitry for emotion (e.g., the amygdala) impact cognitive processing from early attention allocation (Holland & Gallagher 1999) through perceptual processing to memory (for a recent review, see Phelps 2006). Similarly, brain structures involved in the neural circuitry for cognition, such as DMPFC and VLPFC, have an intrinsic role in the experience of emotion". So (although his work is obviously great and meaningful), his mistake seems to be the categorical distinction.
The previous answers have elaborated quite well in terms of whether an actual physiological division of the two types/speeds of processing actually exist. In regards to the "model" aspect of my question, the only model that I know intimately, Spaun. Spaun does have these two types/speeds of processing mapped to two different circuits, if we are focusing on Eoin's definition of working-memory vs. non-working-memory.
In Spaun, most (if not all) of the tasks using working memory must be routed through the basal-ganglia-thalamus loop and tasks that don't require working memory are encoded in the neuron weights.
For example, in the counting task, Spaun is shown some commands that give it a range for counting. The command recognition is done quickly once via neuron weights adapted from a trained deep-belief network. The actual counting is done via the much slower and iterative basal-ganglia-thalamus loop.
