No, staring at a flame will not slow brain activity.
First of all, let's define brain activity, and see what 'slow' would mean in this context.
I had a lot of trouble finding a source that's neither completely technical nor completely nonsensical. In the end I opted for a combination of the two. Here is a semi-nonsensical one from which I am quoting the good parts:
Neurons communicate with each other by electrical changes. We can actually see these electrical changes in the form of brain waves as shown in an EEG
(electroencephalogram). Brain waves are measured in cycles per second
(Hertz; Hz is the short form). We also talk about the "frequency" of
brain wave activity. The lower the number of Hz, the slower the brain activity or the
slower the frequency of the activity.
Therefore, neurons discharge at a certain rate, and we can measure this discharge rate (notably, a lot of neurons have to discharge simultaneously for us to pick it up). This happens pretty quickly, and is expressed in how many times per second a group of neurons discharges. When measured by EEG, it comes out as a wavy line on the screen; hence the concept of brain waves.
On to slow and fast brain waves:
Researchers in the 1930's and 40's identified several different types
of brain waves. Traditionally, these fall into 4 types:
Delta waves (below 4 hz) occur during sleep
Theta waves (4-7 hz) are associated with sleep, deep relaxation (like hypnotic relaxation), and visualization
Alpha waves (8-12 hz) occur when we are relaxed and calm
Beta waves (13-38 hz) occur when we are actively thinking, problem-solving, etc.
Since these original studies, other types of brainwaves have been
identified.
The Sensory motor rhythm (or SMR; around 14 hz) was originally discovered to prevent seizure activity in cats.
Gamma brain waves (40-100 hz) are involved in higher mental activity and consolidation of information.
So, neurons discharge at different rates, and these rates are associated to different brain states. The ones of interest to us right now are the alpha and the gamma waves, because they are typically connected to activity in the visual system.
Relative to each other, alpha waves are slow (they happen when neurons discharge about 10 times per second), and gamma waves are fast (they happen when neurons discharge about 40-70 times per second, though even faster ones have been described).
So, to redefine your question a bit, when staring at a flame, one could say that we are in a slow brain state if we can find evidence of alpha waves, and a fast brain state if we can find evidence of gamma waves. The question is, which state are we going to find?
The answer is: gamma. Fast waves.
From a Nature paper:
In visual areas, attended stimuli induce enhanced responses and an
improved synchronization of rhythmic neuronal activity in the gamma
frequency band (40–70 Hz)
From the European Journal of Neuroscience:
During voluntary orienting of attention, we found
alpha-synchronization to dominate over desynchronization, to be
topographically specific for each of eight attention positions, and to
occur over areas processing unattended space in a retinotopically
organized pattern. This indicates that alpha-synchronization is an
important component of selective attention, serving active suppression
of unattended positions during visual spatial orienting.
Put together, these two quotes mean the following: an attended visual stimulus is connected to increased fast/gamma activity, while pulling attention away from a visual stimulus is connected to increased slow/alpha activity. Pulling attention away is typically done my making people stare at a spot where nothing is happening.
So, if you look at a fire, your visual brain will be displaying fast activity. If you look away from the fire at a place where nothing much is happening, your visual brain will go into a slower state. Looking at a fire does not slow brain activity.
