You have learned that when sodium rushes through membrane channels, the neuron tends to get depolarized. You have also learned that when potassium rushes through its membrane channels, it tends to repolarize the neuron.
In fact, potassium can actually hyperpolarize the neuron (like you see at the end of the action potential). And other ions, like chloride ions, can also cause hyperpolarization.
When a synapse occurs, the effect it will have on the postsynaptic neuron depends on the neurotransmitter receptors, or ligand-gated channels, that the postsynaptic neuron has. If the ligand-gated channels allow sodium ions to pass through, the effect of the synapse will be a depolarization-- in other words, it will be excitatory for the postsynaptic neuron, and possibly even lead to an action potential.
Another possibility is that the ligand-gated channels on the postsynaptic neuron may allow chloride or potassium ions through. This would cause a hyperpolarization of the postsynaptic neuron. This would cause a hyperpolarization postsynaptically, and would be inhibitory (and prevent an action potential).
How do you know what effect the neurotransmitter will have?
Each neuron puts one type of ligand-gated channel on its membrane for each neurotransmitter that it receives. Skeletal muscle is a postsynaptic cell that only ever receives acetylcholine from neurons in humans. It only has one type of ligand-gated channel that it expresses on its sarcolemma-- the acetylcholine receptor in muscle. Every time it is activated, it allows sodium ions (and some potassium ions) to cross... the sodium ions cause enough of a depolarization that the sarcolemma starts an action potential.
Some neurons receive more than one neurotransmitter. An example of this possibility is shown in this diagram. One postsynaptic neuron is shown here. It is receiving information via synapses from many other neurons, but only two of these are indicated on the drawing. One of the two neurons releases some particular neurotransmitter that interacts with postsynaptic receptors and causes a depolarization. Depolarizations tend to bring neurons closer to the point where sodium voltage-gated channels can open, and cause an action potential.
At the same time, there's another neuron shown that is synapsing on this postsynaptic neuron. This neuron must have a different neurotransmitter that it releases-- one that causes a hyperpolarization after contacting its receptor. This is an inhibitory effect on the postsynaptic neuron.
Think about this, now... if only the top presynaptic neuron is active, the postsynaptic neuron may fire an action potential. If only the bottom (inhibitory) presynaptic neuron is active, the postsynaptic neuron will not fire an action potential.
What if both presynaptic axons active at the same time? They might just cancel each other out (a depolarization plus a hyperpolarization equals no change in membrane potential at all), if they are the same strength.
Whatever change in the membrane potential that occurs postsynaptically then joins up together in the cell body (integration of information by the cell body) and reaches the axon hillock. If the membrane potential that reaches the axon hillock is enough to open voltage-gated sodium channels, then and action potential will begin on the axon and run all the way down it.
Therefore, this change in membrane potential, due to the synapse, is the "some electrical event happens" that we talked about before... it has to happen to begin the entire action potential. I have some more information about this in a PowerPoint presentation that I wrote on this topic... take a look! Please note that the size of the p.s.p. indicated in the presentation is what you would record at the axon hillock-- but at the actual postsynaptic membrane, the size could be a lot larger.
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