Brief Answer:
Yes there are fungal infections

Detailed Answer:
Hello and welcome to HealthcareMagic.

Yes there are fungal infections of the brain. Several fungi can cause central nervous system infections like Candida, Aspergillus, Cryptococcus, Histoplasma etc to name a few. Treatment is attempted through anti-fungal agents like amfotericin B, protocols vary depending upon the responsible fungus. However the prognosis is not that good. That is also because brain involvement happens generally in the setting of generalized infection, mainly in patients with low immunity like in those with HIV.

I am not sure what you mean by "what fungal infection would be most closely related to this condition"? Do you mean Alzheimer's? (please explain if otherwise). There are some theories that a preceding fungal infection may be at the origin of the abnormal amyloid plaques formation in the brain. Remember though they these theories are not proven, only theories made by small studies. Such theories have been raised about other types of infections as well. There is not a specific fungus which has been identified, several of them have.
Even if we were to assume that it was the case it is a question of a past infection, which has happened years ago and triggered the formation of those plaques. It is not an active infection anymore, there is nothing to treat, it has triggered a process which is maintained by the patients own body not the infection anymore.

I remain at your disposal for other questions.

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