I've read two of Dr. David Perlmutter's books; Grain Brain & Brain Maker, and he delves heavily into the gut-brain connection.

The Empowering Neurologist - David Perlmutter, MD, and Dr. Michael Ruscio 5-20-2018 [ Perlmutter's bio ]. He describes patients where probiotics improved their situation, sometimes drastically. He talks about the gut-brain connection - through the vagus nerve. He mentions how the vagus nerve is considered by some to be the second brain.

Think Twice: How the Gut's "Second Brain" Influences Mood and Well-Being - The emerging and surprising view of how the enteric nervous system in our bellies goes far beyond just processing the food we eat - 2-12-2010

How Does the Vagus Nerve Convey Gut Instincts to the Brain? 5-23-2014

They say that mitochondria was commonly believed to once have been bacteria that were engulfed by eukaryotic cells.

Perlmutter also explains the relationship between the gut bacteria and the immune system. Also, there is a passing of a mother's microbiome to the newborn child during vaginal delivery, and a distinct lack of gut flora if they are born by caesarean section. And he sees advantages when newborn are breastfeed for at least a year instead of formula fed. That does not mean you are SOL if born by c-section and not breast fed - probiotics will restore what you may be lacking.

Mood, Gut Bacteria, and the Immune System 4-6-2011

In Brain Maker, he lists 5 Core Species:

Lactobaccilus plantarum
Lactobaccilus acidophilus
Lactobaccilus brevis
Bifidobacterium lactis (also called B. animalis)
Bifidobacterium longum

I just started taking Garden of Life - RAW Probiotics Ultimate Care and have started on some of the prebiotics (feeds gut bacteria) he lists; jicama (tastes like water chestnuts), dandelion greens, kimchee, sauerkraut, raw leeks, raw garlic, and some others. They all pass the taste test (I was brought up on kimchee).

I believe Perlmutter said the gut is the largest producer of immune hormones in the human body, more than the brain produces.

Stress & the gut-brain axis: Regulation by the microbiome 10-2016

5.4. Immune signaling

The immune system plays an important intermediary role in the dynamic equilibrium that exists between the brain and the gut (Bengmark, 2013). The HPA axis, ANS and ENS all directly interact with the immune system (Bateman et al., 1989; Genton and Kudsk, 2003; Hori et al., 1995; Leonard, 2006; Nance and Sanders, 2007), and the gut itself is an important immune organ that provides a vital defensive barrier between externally-derived pathogens and the internal biological environment. Gut-associated lymphoid tissues form the largest immune organ of the human body, comprising more than 70% of the total immune system (Vighi et al., 2008).