Is there a biological equivalent to the bias term used in artificial neural networks?
Let's call "response function" the function that gives a neuron's firing rate (or probability of firing) given its input. Then, different kinds of neurons have different response functions. All of them, as far as I'm aware, are nonlinear, and can usually be experimentally measured. 

There are neurons that will fire a little, even for small inputs (so they're basically rectified, nonlinear units) and neurons that need a specific, non-zero threshold of input to be crossed before they fire at all (somewhat analogous to having negative bias). These used to be called Type I and Type II neurons respectively, although I've rarely heard this definition recently. There are also neurons that always fire unless inhibited from doing so ("tonic activity"): you can see this as implementing a positive bias. 

About the second part of your question, the "modification mechanisms": there are many factors that influence a neuron's probability of firing which are independent both of the input and of the input weight. You can see these as changes in the bias, either temporary or long-term. The main causes of this are the neuron's recent firing history (usually neurons are less likely to fire if they've just fired, unless they like to burst) and other phenomena usually called 'intrinsic plasticity'.

I hope this gives you a bit more of an idea.