Genesis 2:7 states: "Jehovah God proceeded to form the man out of dust from the ground and to blow into his nostrils the breath of life, and the man came to be a living soul." In 1 Corinthians 15:45 it states: "It is even so written: 'The first man Adam became a living soul.' The last Adam became a life-giving spirit." 1 Peter 3:20 it states: "In Noah's days...a few people, that is eight souls were carried safely through the water." These Scriptures to name just a few clearly indicate that our bodies are souls as opposed to having a soul within us.
The brain is astoundingly complex. A lot of people can't help but see a soul in all of it. But most likely? No. No souls. Think about it rationally and critically. We aren't driving around this experience in our body vehicles.