# What exactly is the time constant of an ion channel gating?

Let's take a channel that does not get inactivated, but is only in open and closed states. I understand what m (activation variable) represents. It's the fraction of the gates open.

But what does $\tau$ (tau) , the time constant represent? Does it mean the time taken to open the gate? If tau is higher then it mean more time taken to open the gate?

I'm slightly confused.

• It totally depends on what the model is. Can you please add the model equation? May 11, 2016 at 6:58
• Let's say can you describe the time constant for the activation gate of a sodium channel? May 11, 2016 at 9:27
• I need more information. Can you put your model equation in the question? The time constant can mean anything. May 12, 2016 at 7:58

Although the membrane time constant is very important in neuroscience, the time constant described in the original question is about ion channel gating rather than the rate of change of membrane voltage.

The "tau" when referring to ion channel gating is indeed referring to how long it takes to open/close.

When the membrane voltage changes, there is a new equilibrium value for the probability of the channel to be open versus closed (note: there can be other states for channels too, but for this explanation I will stick to open/closed). The rate at which this equilibrium is approached from the previous open probability is described by an exponential function, with "tau" as the time constant of that approach. A small value of tau means that the new equilibrium is reached very quickly. This site has a good explanation of the terminology and gating process, as do many others.

Time constants are involved in any system described by an exponential (these are called "linear time invariant" processes), whether decay or increase, and it is equal to the time it takes to progress approximately 63.2% of the way to the equilibrium. The cool thing about time constants (and linear time invariant processes in general) is that it doesn't matter what your starting point is or how far into the process you are, the time constant stays constant and every time "tau" passes you are now 63.2% closer to equilibrium than you were at time t-tau.

According to Wikipedia

In physics and engineering, the time constant, usually denoted by the Greek letter τ (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.

In Neuroscience:

The time constant, $\tau$, characterizes how rapidly current flow changes the membrane potential. $\tau$ is calculated as:

$$\tau = r_mc_m$$

where r$_m$ and c$_m$ are the resistance and capacitance, respectively, of the plasma membrane.

• The resistance across the membrane is a function of the number of open ion channels and the capacitance is a function of the properties of the lipid bilayer

Note: the time constant is a "passive property" of membranes because they are intrinsic properties of all biological membranes. [Source].

My guess is that you are referring to some equation reference from Hodgkin-Huxley Model for the Generation of Action Potentials, but without further information about your specific equation, I cannot comment about Tau's specific role in your equation.

Example Application:

This constant can be applied to one of a few equations describing changes in membrane voltage. For example, a fall in membrane voltage is modeled as:

$$V(t) = V_{max}e^{-t/\tau}$$

where voltage is in millivolts, time is in seconds, and $\tau$ is in seconds.

• You can read more about the mathematics behind this and passive membrane properties in general here and here.

Again, From Wikipedia:

The larger a time constant is, the slower the rise or fall of the potential of a neuron. A long time constant can result in temporal summation, or the algebraic summation of repeated potentials. A short time constant rather produces a coincidence detector through spatial summation.

• Hey, nice answer, but this wasn't the question: the question was about time constants in ion channel gating which is better thought of with equations describing binding or reaction rates, not the membrane time constant, it is a completely separate concept. Apr 3, 2017 at 22:30
• @BryanKrause Hmm so it is. Thanks for pointing this out! I wasn't quite sure what the OP was referring to when I first read it. Because of my uncertainty, I tried answering first broadly and then regarding the only tau I knew in this general area of biology. Your answer appears to be the correct answer in this case, though (and thus I gave you a +1). Apr 10, 2017 at 13:11

I Understand that you are referring to the 2-state model of ion channels. In this case, the transition from one steady-state to another is modeled as an exponential decay. $$\tau * dm/dt = (m_\inf - m)$$ The so called "fast" channels have $$\tau$$ of a few milliseconds, "slow" ion channels have time constants of hundreds of milliseconds or more.

Of course, the 2-state model is a simplified mathematical model of ion channels, but for many phenomenological purposes, it is good enough.