The thing you were reading about is known as the action potential. It is a mechanism that governs how information flows within a neuron.
It works like this: Neurons have an electrical potential, which is a voltage difference inside and outside the cell. They also have a default resting potential, and an activation potential. The neuron tends to move towards the resting potential if it is left alone, but incoming electric activations from dendrites can shift its electric potential.
If the neuron reaches a certain threshold in electric potential (the activation potential), the entire neuron and its connecting axons goes through a chain reaction of ionic exchange inside/outside the cell that results in a "wave of propagation" through the axon.
TL;DR: Once a neuron reaches a certain activation potential, it electrically discharges. But if the electric potential of the neuron doesn't reach that value then the neuron does not activate.
Does the human brain use a specific activation function?
IIRC neurons in different parts of the brain behave a bit differently, and the way this question is phrased sounds as if you are asking if there is a specific implementation of neuronal activation (as opposed to us modelling it).
But in general behave relatively similar to each other (Neurons communicate with each other via neurochemicals, information propagates inside a neuron via a mechanism known as the action potential...) But the details and the differences they cause could be significant.
There are various biological neuron models, but the Hodgkin-Huxley Model is the most notable.
Also note that a general description of neurons don't give you a general description of neuronal dynamics a la cognition (understanding a tree doesn't give you complete understanding of a forest)
But, the method of which information propagates inside a neuron is in general quite well understood as sodium / potassium ionic exchange.
It (activation potential) sounds a lot like ReLU...
It's only like ReLU in the sense that they require a threshold before anything happens. But ReLU can have variable output while neurons are all-or-nothing.
Also ReLU (and other activation functions in general) are differentiable with respect to input space. This is very important for backprop.
This is a ReLU function, with the X-axis being input value and Y-axis being output value.
And this is the action potential with the X-axis being time, and Y being output value.