Chaos!

Single Neuron Simulation

We can observe chaos in the activity of a single brain cell, or neuron. The simulator below allows you to change the electrical input being delivered to a virtual neuron. Stimulating a cell with a single electrical pulse can cause the neuron to respond by firing an electrical output spike. These electrical spikes, or action potentials, are a primary means that neurons communicate with each other. We call the sequence of electrical spikes from the neuron a spike train.

In this simulator, we are activating the neuron with a train of electrical pulses for the input, and we can adjust the parameters of the stimulus, including the height (voltage) of the pulses, the duration of each pulse, and the time interval between pulses. Play with the 3 sliders around the lower panel to adjust these stimulus parameters. You can see a diagram of the stimulus pulses in the lower left.

The top panel shows the activity of the simulated neuron as a function of time. For many combinations of the three stimulus parameters, the neuron responds with a simple, periodic spike train.

For some combinations of the input stimulus parameters, you can observe period-2 behavior, which corresponds to alternating big and small spikes in the output sequence of the neuron in the top panel. Play with the 3 input parameters until you find a combination that produces a result like this.

The lower right panel shows the behavior of the neuron in phase space, and it is very similar to the orbits seen in the graphical iteration of the Logistic or Quadratic functions. When the neuron is firing regularly with a period of 1, there is a single loop in the phase plot at the bottom right.

When the neuron fires a period-2 spike train, that is big spikes alternating with small spikes, the phase plot at the bottom right has two loops. Find a combination of stimulus parameters that generates a spike train like this. Try to figure out which loop in the phase plot corresponds to which spike in the spike train.

For certain combinations of the stimulus parameters, the neuron will behave erratically. In these areas, where the spike train is aperiodic or chaotic, the phase plot doesn't follow a single loop, or a double loop, but rather it traces out many different loops, which, over time, can fill up the phase plot.

Try to find combinations of the stimulus parameters that cause the neuron to respond chaotically. You can see three examples of chaotic behavior by pressing the 'chaos' buttons at the bottom of the applet. Play with the simulator until you can find at least one other set of input parameters that produce chaotic behavior.

Questions:

Set the input parameters to I = 0.335, T = 1.78 and w = 0.219.
Does the neuron show periodic or aperiodic behavior? [ _____ ]
If you said periodic, what period is it? (i.e. how many spikes occur in the train before the pattern repeats? [ ]



Set the input parameters to I = 0.469, T = 2.17 and w = 0.944.
Does the neuron show periodic or aperiodic behavior? [ _____ ]
If you said periodic, what period is it? (i.e. how many spikes occur in the train before the pattern repeats? [ ]


The behavior of a single neuron can be chaotic given the right input stimulus. Imagine how much more complicated is a small circuit of several neurons exciting and inhibiting each other, let alone an entire brain full of billions of neurons!