Wash your brushes and combs regularly in... what else? Soapy
water with a bit of shampoo added! And remember, don't let wood combs or
brushes soak in water, dry them as thoroughly and as quickly as possible
without using any sort of heat source.

Wood combs should be oiled or waxed regularly to keep the wood from drying out.  Bone/horn combs can benefit from a bit of lanolin or a light oil.