Do fish indeed rot from the head? Some fishing-experienced Mudcatter ought to know the answer to this.

        As a third-generation commercial fisherman (of the female persuasion) I can report that the gills are the first part to get smelly. Dad told me, "Blood spoils, and the gills have a lot of blood in them." They also have plenty of surface area, and no tough skin or protective slime to delay infection.

        So before icing our catch, we cut out the gills, and after removing the guts, we cut and then scrape out the contents of the "blood line" - an organ which fills the top of the body cavity, against the spine. I believe it serves like our kidneys.
        For a large salmon, we may even take time to scrape out the blood from larger veins (squeezing it into the space where the bloodline was, gently, so as not to damage the skin of the belly walls). (If our catch is kept in chilled brine, it will bleed itself thoroughly without this step.)

        If the brain rots first, it doesn't matter to us, since bone isolates it from the muscle tissue which we plan to eat. Gut rotting speed depends largely on how the fish has been feeding.
        A coho salmon with a stomach full of krill or other small feed may secrete so much stomach acid that its belly wall starts to show signs of chemical burn (reddening and fragility) almost as soon as it arrives on deck. The only sure way to prevent "belly burn" is to clean the fish immediately (which may still be too late if you caught several at once). It happens even faster if the fish are warm (from sun-warmed deck, or relatively warm water). Your hands, if ungloved, may suffer the same fate ("coho burn") after cleaning a lot of actively feeding salmon (coho or spring/chinook especially).
        Yet a coho the same size, caught in the same waters at the same time, but with an empty stomach, will keep much longer round (i.e. uncleaned) before the warmth of decaying internal organs begins to damage the belly wall.

        Aren't you glad you asked?

        By the way, if you want to spot a fresh fish in the market, look at the eyes (both sides if possible). The eyes sink and grow dull as the fish warms or dehydrates. Storekeepers get mad when you poke your finger into the fish, to see if it springs back.