The mathematics behind the neuron model simulation.

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The system to model a neuron used is very basic, but captures some interesting aspects of neuronal signaling. The assumptions are:

• The simulated neuron has a distinct threshold.
  • Below the threshold, the neuron behaves as a simple RC circuit. The R refers to the resistance of membrane channels, which allow ions (electrical current) to flow through the membrane. Each channel has a fixed resistance. More channels mean lower total R, because the individual channels are effectively in electrical "parallel". In other words, they share current, making it easier to flow. The C refers to membrane capacitance, which allows ions to "pile up" on the membrane.
  • Above the threshold, the neuron behavior depends on the kind of model used.
    • If a sigmoid output is used, then the RC circuit continues to accumulate charge.
    • If a spike output is used, then the RC circuit is instantly reset to resting potential
• Inputs are currents which flow into the neuron from external sources, for instance, synapes or receptors. Current applied to the neuron may charge the membrane capacitance or flow through open channels. The sum of the flow into C and R must add up to the applied current (conservation of electrical charge).

In order to implement the description given above, the behavior was written as a first order differential equation (below threshold) driven by a variety of inputs.