Behaviour of neuron when membrane potential is maintained at threshold potential or more
Before I start, I should point out that in order to change the membrane potential (known as the Vm) of a neuron you really need an internal electrode - so that you can pump current in to it, so increasing the potential above the resting membrane potential (RMP). The RMP is simply the Vm of a neuron at rest, without any external meddling by pesky scientists (or synaptic inputs).

When you depolarise a neuron - in other words, make it less negatively charged, it will fire an action potential once it crosses the spike threshold. This value is different for different neurons, but -50mV is usually not enough to trigger an action potential - normally it's more in the range between -30 and -40. But let's say it does, as it doesn't change the thrust of the question.

The major part of what happens during an action potential is that sodium channels open, allowing the spike to fire. Once closed, these channels have a 'refractory period' - essentially a time out, during which they are less likely to open again. So even though your neuron is still above spike threshold, it can't fire because the channels can't open. (For the aficionados, you can actually think of this as the spike threshold becoming temporarily more positive - going up to, say, -20mV).

So your neuron doesn't fire again for, say, 10ms, until the sodium channels reactivate. Then it does. So you get a train of regularly spaced action potentials at a certain frequency. The more you depolarise the neuron, the faster this frequency becomes until a point where the channels just become totally inactivated and the neuron stops firing. If you hold the cell at this voltage for more than a few seconds, in general it will die.