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That is normal. The circuit you are using is a pretty common design. Here is an example using MOSFETS for the transistors. We are not turning the transistors On and Off as in instantly. If you were to look at the base leads using a scope you would see a sawtooth type waveform. So unlike banging the bases with a square wave we see a different result. Nice effect and actually normal and what would be expected of the circuit.
Ron
The first circuit you posted is nothing like the final circuit you used.
You should read how the first circuit works to see why a re-arrangment will not work.
The LEDs in the first circuit produce a chasing pattern similar the running LEDs display in video shops.
In fact the effect is called: "Running Hole." All transistors will try to come on at the same time when the power is applied, but some will be faster due to their internal characteristics and some will get a different turn-on current due to the exact value of the 22u electrolytics. The last 22u will delay the voltage-rise to the base of the first transistor and make the circuit start reliably.
The 22u's between each of the transistors have a secondary purpose. They keep the next transistor OFF when the previous is ON. They do this by providing a negative voltage on the base.
This feature is not possible in the second circuit you used. That is why it will not work.
What FET's are you using?
If you are using 2N7000 the 1M resistors will need to be 47k and the 2u2 can be 10u or 22u
Have you done this:
The circuit relies on discharging the 2u2 via the 1M resistor and the voltage characteristic of the FET gate.
You will have to see how long it takes to charge the 2u2 by taking the 1M to positive (via a LED) then take it directly to 0v to see how long it takes to turn the LED off. Do this as a separate, individual, section. Reduce the 1M to 47k if the circuit does not work.
You should put a resistor in parallel with the LEDs. I suggest 10 kΩ or so.
At really low currents, the voltage across an LED will be nearly its forward voltage (Vf). So an LED rated at 2 V might be 1.6 V at less than 1 μA. Just for fun, at no current an LED with a Vf or 2 V will have a voltage across it of around 1.6 V in bright light.
If you are running at 5 V, and loosing 1.6 V across your LEDs, you do not have the 4 V needed to turn on the IRF630B.
The 10 kΩ resistor will only have 10 mV across it at 1 μA, and it will absorb any current generated by the LED in bright light. That means you will get the full 5 V to turn on the IRF630B. When the LED is powered up, there will only be about 0.2 mA though the 10 kΩ resistor, so you won't loose much power.