The main theory dealing with the relationship between thought/cognition and metaphor is George Lakoff’s The Neural Theory of Metaphor (2009). Before this, however, he wrote a book in 1979 with Mark Johnson detailing metaphors and their types titled Metaphors We Live By. In it, they explore conceptual metaphors and the processes that underlie them. They describe conceptual metaphors as one conceptual domain being described in terms of another.
This is slightly different, but may overlap, with literal metaphors — the kind one usually thinks of when thinking about metaphors. Conceptual metaphors encompass a whole domain. For example, argumentation is often talked about with words regarding battle and war (“attack” or “shot down”). Lakoff’s theory revolves around this concept, taking it one step further and saying these domains are embedded within neural processes.
The basics of his theory posit that ideas and concepts are physically encoded and computed by the brain. This is nothing new, as any input from life experience runs through a neural circuitry, but what he proposes is that metaphors themselves have a circuitry. Akin to a critical period for language acquisition, there’s a critical period for metaphor development. At birth, a baby is born with around 100 million neurons that average 2,500 connections, but as the child grows, the number of connections range from 5,000 to 15,000. Connections that are seldom used will be eliminated. Like molding a statue, continuous input of one concept will strengthen certain connections, and lack of input will weaken until they’re shaved away.
An example Lakoff gives in a lecture is the conceptual metaphor “Affection is Heat”. In a society in which the parents cuddle their baby often, cooing and accompanied with loving words, the baby will learn to associate expressions of affection with physical warmth. The neurons processing information of temperature will simultaneously fire with the cells processing language/behavior, and the number of synapses will increase between the two, making the connection integrated and more efficient in firing action potentials (to send electrical signals). This, then, manifests in language like “he is a warm person”, where warm is a metaphor for loving, welcoming, open, affectionate, etc.