This topic for me is old hat, since I read the report of Philippe Bottero, MD in 2000, Role of Rickettsiae and Chlamydiae in the Psychopathology of Chronic Fatigue Syndrome (CFS) Patients, Journal of Chronic Fatigue Syndrome. In it he mentions several groups with results:
- “59 psycho-somatic cases; 5 schizophrenia; 3 borderline; 10 children with agressivity, excitement; 1 autistic child; 1 delirium with relapses.”
- “Group two: 82.3% good and very good results; 2.5% fairly good; 15.2% failed.”
- In addition, “other psychopathological states (300): Sydney 98 CFS Conference, Australia.” reported similar results.
How was this obtained? Using long term antibiotics — a very well known way of altering the microbiome.
Jumping ahead two decades
There are multiple articles for neurological issues having a strong statistically significant association to the microbiome..
- Alzheimer’s Disease
- Chronic Fatigue Syndrome
- Mood Disorders
- Stress / Post Traumatic Stress Disorder
For other conditions, we find similar findings just looking at recent studies:
- Gut microbial clues to bipolar disorder: State-of-the-art review of current findings and future directions 
- The Gut Microbiome in Bipolar Disorder and Pharmacotherapy Management.
- Sleep and Microbiome in Psychiatric Diseases. 
- “Alterations of the gut microbiome in anorexia nervosa have been demonstrated, but their clinical relevance remains unclear. ” 
- “We have recently reported specific fecal metabolomic changes in acute and short-term weight restored patients with anorexia nervosa (AN). In this study we explored the association between those metabolomic changes and patients’ gut microbiome” 
The Microbiome is only a Part — but the easiest to modify
For almost all of the above conditions there are known DNA mutations associated (SNPs). We know that there is an interaction between the DNA and the microbiome — effectively the microbiome is an organ. Organs have compatibility issues when transplanted. A similar failure to take has been observed with Fecal Matter Transplant.
The simplest model to understand what is happening is this: the DNA favors bacteria that favors the DNA desired behavior (be it good or bad). Basic self-serving “microbiome forming” by the DNA. Over time, the DNA may alter its behavior due to environmental factors like stress (epigenetics) and in a few cases, RNA from a virus may be taken up into the DNA. In reality, it is estimated that up to 50% of your DNA originated from a virus [Src 2020]… ongoing incorporation of new virus fragments remains a possibility.
The question is how do you stop the undesirable behavior? You could attempt to alter the DNA — that is very experimental with considerable risk. You could use drugs that appear to inhibit the undesirable behavior — drugs tried at random on a population in a study with positive results. Often we do not know the mechanism of the drug — simply that they worked.
My more unconventional approach is to “strong arm” the bacteria. Yes, DNA is encouraging one pattern — but with food, diet, supplements it appears possible to alter the pattern. I have seen my own SPECT scan go from appearing to be early Alzheimer’s disease back to normal by this method.
Is 100% recovery expected — no, improvement is expected. Often very significant improvement. Is there actual evidence? FMT or Fecal Matter Transplant is the clearest demonstration… you change the gut bacteria and see if there are changes!
Clinical trials with FMT have been performed in patients with autism spectrum disorder and showed beneficial effects on neurological symptoms. For multiple sclerosis and Parkinson’s disease, several animal studies suggested a positive effect of FMT, supported by some human case reports. For epilepsy, Tourette syndrome, and diabetic neuropathy some studies suggested a beneficial effect of FMT, but evidence was restricted to case reports and limited numbers of animal studies. For stroke, Alzheimer’s disease and Guillain-Barré syndrome only studies with animal models were identified. These studies suggested a potential beneficial effect of healthy donor FMT. … Whether positive findings from animal studies can be confirmed in the treatment of human diseases awaits to be seen. Several trials with FMT as treatment for the above mentioned neurological disorders are planned or ongoing, as well as for amyotrophic lateral sclerosis.Fecal Microbiota Transplantation in Neurological Disorders 
My Garage Project
The above was the spark that launched me on a project that resulted in https://microbiomeprescription.com/. Using studies from the National Library of Medicine National Center for Biotechnology Information and 16s microbiome samples (from American Gut, uBiome, Biome Sight with “MICRO” as discount code , and Thryve Inside) I applied my Data Science, Artificial Intelligence, Software developer skills to generate suggestions on what may alter the microbiome in a positive way.