So we took the two extreme cases to start with, one the case of t equals 0, where v is 0. And one case where I inject the long, infinite long current paths and then I get the steady state case when t equal infinity. V equals IR. But let's see at other locations because we want to really see the whole development of voltage. And I want to take a very specific case where t equals RC. So, you can see that the units here I multiply by r the units of v, voltage and so this should be unitless. This means that R multiplied by C should be in units of time. And I take a very particular case. Where t, the time of the injection is exactly R multiplied by C. By the way, we'll mention this a lot. This RC value is many times called tau, or sometimes tau membrane, tau m. So, tau symbolizes the value of RC. It's in units of time. This is in units of seconds, yes? Units of second. So let's see what happens to voltage, to the voltage when you inject t exactly equal RC. So what do I really talk about? I'm looking at the case where I inject the current. I, I get that we saw before. Voltage, V. [COUGH] This is T. And, I'm looking at a particular time here. T equal RC, or tau membrane. Okay. So inject the current this direct-, the this, this long. And I'm looking for the value of V, the value of V here, after one time constant. So what would be the V is t, equal tau. Okay. We can see that when t equal RC this is becoming RC dived by RC becomes minus 1. So we see here one minus e to the power of minus one. So here, we see, the following. V, at t equal tau, is equal IR. Multiply by one minus e to the power minus 1. And this is this part, is 0.63. So this means. That after t equal tau, I get the voltage gets into 63% of the maximal voltage that it reaches. So here we get, at this location, 0.63, 63% of, the infinity. Which is my IR. This is the infinity. So that's something interesting, I, I, wait for a particular period of time. Tau m, I inject the current for this time, and I know from the equation that it will, the voltage will develop. To the value of 63% of the maximum that it can get, the maximum here, again this V infinity is IR. This marks my point. So that's when the voltage is being built and you can see that if I would inject another time constant. And another time constant, and another time constant, the voltage would grow and grow and grow as described here. It will grow and grow and grow exponentially, like 1 minus exponent. And eventually it will get, if I wait enough time, I will get close asymptotically, I will asymptotically close to the maximal value, which is IR. Okay, you can see, you can use the same equation exactly the same equation, you can use. And ask yourself, how does the voltage decays? Decays when I complete the current. So, I stop the current here, and I ask myself how will the voltage look. How will the attenuation look at the end of the voltage. I have already told you that it decays and goes back to its zero value. And one can show from the same equation that the attenuation of voltage. By the end of the injection, is also exponential, but in this case not 1 minus exponent, but just pure exponent. So let's say that we started with some value here, let's call it V zero here. Oh, sorry, V A, let's call it V star. Which will be the voltage when I stop the current and I look at the voltage simulation here. I can tell you that this curve, this attenuation is equal, so, so, sorry. That voltage. That the voltage after I stop the current will look like the initial voltage here. Attenuated like the same exponent, into the power of tau m. That means that if I wait one tau here, I would get attenuation to 63% from the start. I start at let's say with 10 I will attenuate it to, by 63%, I attenuate it by 63%, and I would be left with 37%. So I will get here after one time constant, or get 0.37 of V star. So, both the growth of voltage during current is 1 minus exponent. But governed by the same exponent here, which we call tau m, both the growth of the voltage and the attenuation of the voltage. After I stop the injection are governed by the same exponent, by the same tau, but the growth is 1 minus exponent. And the attenuation is just the initial, initial value multiplied by exponent. And so both the growth and the decay are mirror-image of one each other. So if I take this point and put it here and flip it, the tenuation is exactly like the build-up. It governed, it is governed by single-exponent, by tau m. And this is called. [SOUND] The membrane time constant. The membrane time constant. It's a very important parameter which we should discuss in a second. The membrane time constant. You really should remember this. It's a very, very important parameter for understanding nerve cells. Tau m, the membrane time constant. So, just to complete this part, okay, we started with an RC circuit that describes. The RC circuit that describes for a first approximation a spherical, isopotential, passive structure. And I'm using the word passive now for the first time. Saying that every parameter that I'm using, both R, C current, everything is linear and fixed. It doesn't change during the injection. It's fixed values. This is in ohms, this is in ferrets. This is in ampere, and this is fixed. So it's a passive system. And this RC circuit is a good representation. As an initial representation of a cell. And this, this would of been the case. Then you can see that when you inject current I you have during the injection. The voltage develops like 1 minus exponent. And after you, you finish injecting, you stop injecting the current, the voltage decays like exponent. And the, the controlling parameter, the important parameter that controls the timescale. How fast, how fast the voltage develops and how fast the voltage attenuates. After the injection is governed by this time constant, that's why it's so important, remembering time constant. For example, if the membrane time constant is long. This means it will take long time for the voltage to attenunate, long time. If the time constant is short, it will be much shorter to attenuate, or to build up. So the time constant controls how fast the membrane voltage responds to current. I inject a very fast current step. And it takes time for the voltage to respond. And after I stop the current, it takes time for the voltage to go back to its initiation value, which is 0. It takes it time, it has a memory. There is a memory for the membrane, if you want to call it this way, a memory for the previous current. It takes time to get rid for the, for the capacitor to discharge, to get rid of the current that I injected. It takes time, and this time is governed by tau membrane. It's a very important parameter, because it tells you something about the memory, the electrical memory of the cell. Show time constant means that the cell gets rid, develops fast voltage and gets rid of the voltage. and slow time constant means that it takes very long to get rid of this voltage. Another important parameter is R. So you can see that this, this, this equation is governed both by R. Which appears both here and here, and also by C, which when multiplied by R is tau. This R, this R, which is sometimes called R input also. The input resistance of the cell Input resistance, input resistance is another important parameter. So this R, or R input, is an important parameter. Because as you saw before it determines the maximum value of the voltage you can reach. So i multiplied by R is the maximum voltage that your voltage can reach. And so if R is very big, if the resistance of the membrane is very big you get very high, very large voltage. If R is very small, you will get IR, which is small, you will get less voltage. So this R is important parameter to tell you. How much voltage will I get? How much depolarization will I get when I have a given I? And RC tells you how fast the voltage develops. And how much, how fast the voltage attenuates following the current injection. So both our input and the time constant are the two critical parameters. Both the R input and the time constant are the critical parameters for understanding passive, RC circuit. Which is at first passive, linear approximation, for biological membranes. And in particular, neuronal membrane.