## What is the time constant when modeling a biological neuron?

The first few paragraphs of the Wikipedia article on time constants are incomprehensible for noobs like me, so let’s try to look for a more intuitive understanding of what a time constant is.

**Intuitive understanding:**Let’s imagine the membrane potential of a neuron. There is some point where this potential is resting. In other words: everything being the same, a value of the potential, where it returns to after some time . This value is called the resting potential . Now, we apply an input current (some electrode). As a consequence, changes immediately. So, now the membrane potential has to adapt to the resting potential. It surely would be interesting to know how long it takes for the membrane potential to reach it, right? This measure of delay is called time constant (specifically, the time constant measures how long it takes for an electric circuit to relax to 63.2% of its final/asymptotic value, e.g. the resting potential).

**Formalized understanding:**When we look at a neuron, the time constant is given formally by:

with proportionality constant for the resistance of the membrane (, with A being the area of the neurons membrane) and being the proportionality constant for the capacity of the membrane, the membrane capacitance per unit area of the membrane. This is as we expected: the higher the resistance of the membrane, the longer it takes for the membrane to charge. Also, if the membrane has more capacitance, it takes longer as well.

Now, let’s look at a single-compartment model of a neuron. This basically means that the neuron has a capacitance current, a membrane current and an input current.

The function for is then given by¹:

with being the equilibrium potential, resting potential or sometimes also called leak potential (sometimes denoted by ¹). Here we can see that the higher the time constant, the longer it takes for the system to arrive at its resting potential.

**Sources:**

1) Miller, P. (2018)

*An Introductory Course in Computational Neuroscience*. 1st edition. Cambridge, Masachusetts: The MIT Press.