So, before I start, some general remarks, this stuff, the resting membrane potential and the action potential, these are counter intuitive and usually students find it difficult to understand the concepts just by reading. So, to aid students, many many groups, professors, scientist have made programs which work on the Window, Mac or Linux platforms and which show the resting membrane potential like what happens to the potential when you change the ion, ionic concentrations of potassium, change the ionic concentration of sodium so on and so forth. These through a lot of experience over last 30-40 years by, in neuroscience programs, I have shown that students tend to pick up these things better and have a better intuitive kind of understanding if they actually see it. Now, it is not possible to show some of these experiments because you know, the action potential experiments use squids and those are difficult to get, you need seawater and so on and so forth.
So, these equations, you know, which govern the resting membrane potential and the action potentials happily enough, they provide a complete description of what we need to know to figure these potentials and these phenomena out. So, that is the basis of this demo. We will be showing some of these potentials using a software called MetaNeuron. A MetaNeuron was created by professors Newman and Newman from the University of Minnesota and it seems to be one of the more easier programs to learn these phenomena.
So, the paper which I refer to you is MetaNeuron, a free neuron simulation program for teaching cellular neurophysiology, in June which is journal of Undergraduate Neuroscience Education, from 2013 and it has been used in many programs satisfactorily and that is why we chose it. The other thing is it is free and it is a standalone program and can be used without restriction. And it is used to conduct neurophysiology experiments in silico as opposed in vivo. It works on Windows, Mac and Linux, Linux you have to use program called WINE which allows you to run this program.
So, the different neuronal parameters that is sodium, potassium concentrations, their equilibrium potentials and conductances can be easily be modified. Also you can inject current, either a single or a double current pulse into the neuron to see its effects on the potentials. And the responses are displayed graphically and can be measured with a cursor. Also families of traces can be easily generated.
Suppose you vary the extracellular sodium between 10 mill molar to 100 millimolar and you increase it in jumps or steps of 10, we can generate a family of traces. So, we can observe what happens at different concentrations. And finally, it can be viewed, these family of curves, traces can be viewed in rotatable 3D plots which helps in understanding it further.
So, I strongly advise you to download this program, as I said it is free, install it on your system, also download the paper and there are a lot of students exercises and if you go through the students exercises, you have a very good understanding of what exactly happens and how the phenomena of the membrane potential and action potential work.
So, a brief intro to the operation of the program. These simulations run automatically when the program is open and you select a lesson from the “Lesson” pull-down menu or from the function keys. So, the lessons deal with the membrane potentials, resting membrane potential, the action potential, voltage clamps, synaptic potentials, so on, the length constant and the time constant.
So, the parameter values can be changed in two ways. You can either type a new value in the parameter value box and by clicking on it and hit enter or you can click on the grey button to the right of the parameter value box and drag the mouse. If you drag it in the right side, it increases, if you drag it to the left side, it decreases. And finally, all the parameters can be reset to the default values by selecting “Restore All to Default” (control D) in the “File” pull-down menu. And this is for your one shows how you can control the parameter values.
Then the graphs of parameter values: some of the parameters are plotted in a graph on the lower part of the screen, the traces are colour coded and corresponding trace labels are shown to the right of the graph. Also the sweep duration, that the total time displayed on the X-axis is controlled by “Sweep duration” parameter just above the graph. And as mentioned earlier, the family of traces can be generated in a graph by selecting the check box to the right of the parameter value box.
A 3D display: A family of traces can be displayed in 3 dimensions by clicking the 3D graph above the plots. And finally, if you want to measure the trace with the cursor, the X and Y values of any point on the graph can be determined by moving the mouse over the graph and clicking.
So, the resting membrane potential, how do you measure the resting membrane potential? So, you need an intercellular electrode and on the left you see the intercellular electrode outside the cell and you do not have any potential because it is outside, so it is 0. It is 0. On the right, the intercellular electrode has been inserted inside the cell and we are recording from inside
the cell to the ground and soon as you insert the electrode, the potential goes to minus 60 millivolts and that is the resting membrane potential.
If you take it out, it will again come back to 0. Now, we are going to go through the demo to see how the resting membrane potential arises.
So, basically, why potassium and the sodium channels contribute to the rest generation of the resting membrane potential. The neuron is model by passive conductances so, of potassium and sodium and the intercellular and extracellular concentrations of potassium and sodium can be varied. So, these conductances remember our voltage-independent and the neuron does not generate action potentials in this model.
So, just to remind you, suppose you just have one ion, the resting equilibrium potential for that particular ion is given by the Nernst equation, Rt by Gf, natural log of extracellular concentration divided by intercellular concentration. So, that can be at the usual temperature use, all those coefficients can be reduced to 58, then use log to the base 10 and then it is an extracellular ion concentration, the square brackets indicate molar concentrations versus this intercellular.
Now, this is just for a single ion base sodium to be potassium. The resting membrane potential neuron, however, is determined by an interplay of the other ions. And that is an extension of the Nernst equation and is calculated from the Goldman-Hodgkin-Katz equation, the GHK constant field equation if you remember and here is a simplified version of the GHK equation where we just considering the main ions contributing to the resting membrane potential which is potassium and sodium. And P stands for permeability, how much the ion can go through the membrane, cell membrane.
And one note, as I said P, P k and P Na are the electrode membrane permeabilities of potassium and sodium. And usually chloride is omitted because it is freely permeable at its equilibrium potential and it is close to the potassium equilibrium potential and the potassium equilibrium potential is a one that contributes majorly to the resting membrane potential. So, ignore chloride and also we ignore active membrane conductances which cause action potentials, we are just looking at the passive membrane conductances.
So, some of the different manipulations you can do in the program which I will come to in a minute is vary the concentrations of potassium and sodium, both inside and outside the cell. And you can check it to see what effect this has on the equilibrium potential of each ion. Then also the resting membrane potential itself starting with the default parameters, we can vary the membrane permeability, how much of potassium or sodium is allowed to go through the membrane and see what effects this has on the resting membrane potential.
We can also look at the membrane conductance and membrane potential, we start with the default parameter values and we can plot the value of the membrane potential as a function of extracellular potassium over a range. For example, 0.2 to 100 millimolar and we can re-graph the data with the membrane potential plotted as a function of log of potassium outside and you can try and figure out why the second plot has the shape it does.
So, now let us go to the demo. So, this is MetaNeuron, and the resting membrane potential demo. So, just briefly, there are 6 lessons, the resting membrane potential, membrane time constant tau, the membrane length constant lambda, then the action potential, then the action voltage clamp and finally synaptic potentials and current. So, we will start with the resting membrane potential. So, the left, you see sodium concentrations outside the cell, so this is 120, concentration inside the cell is 16.48 and given these two, from the Nernst equation, we get a sodium equilibration potential of 50.
So, below are the plots and the membrane potential is yellow, right here. The sodium equilibration potential is 50, right on top over here and the potassium, equilibrium potential, given the values above is minus 77. So, potassium is on next to the sodium and again the concentrations inside and the concentrations outside, so outside is 3, inside is some about 63.7 and next to it is the relative membrane permeabilities, the sodium permeability is 1 and the potassium permeability is kept at 65
And with all these values you get the membrane potential at minus 65, that is over here. So, we will start and the sweep, whereas the sweep is not really important in this particular demo, but the sweep over here is 20 milliseconds.
So, we will start by varying the sodium concentration. So, I am changing the sodium concentration to a 100 and press enter and you see it comes down. So, outside remember is seawater, so NaCl. So, remember if sodium is not present outside, you will not have action potentials, so you see how the membrane potential goes down when we decrease the sodium concentration, so I make it 10
So, not only the sodium here with a major decrease in extracellular sodium, the sodium potential goes down, the sodium equilibration potential and so does the membrane potential. It is gone, it is really close to the potassium equilibration potential.
And what happens when I make it 0, it is gone. So, here since there is no sodium, there is no sodium outside, sodium inside does not matter. Sodium outside is 0, the equilibration potential, that is resting membrane potential is equal to the potassium equilibration potential because that is the only ion in play right now. So, we go back and restore all the values to the default and we get back the standard sodium potential, the equilibration potential, and the potassium potential. So, let us do the opposite. What happens when it mess with the or change the potassium concentration? So, the potassium concentration inside is what matter, so it is about 64.
So, I am going to make it 40. So, not only does it come up, but the equilibration potential also changes, it goes up and I will make it 0. So, when we make it 0, basically there is no ionic force due to the potassium and the membrane potential goes way up. There is only sodium acting over here. So, coming back to let us restore to default so, what happens when we change the permeability? Remember, at rest, at the resting membrane potential, there is no sodium influx and into the cell, whatever little there is, it is reverse by the sodium potassium ATPs. And it is very low compared to the potassium conductance.
So, if I changed it to 10, notice the change in the membrane potential. If I change it to a 100, so it is much more than the potassium, again it goes up. So, playing this, these different values, playing with the sodium concentration, the potassium concentration and the relative membrane permeabilities, you get an idea of the behaviour of the resting membrane potential. When you change the concentrations, when you change the permeabilities, etcetera. Now, this is a little contour intuitive because the equation is not a linear equation and we find that students find it much easier to understand the RMP when they change these values. So, I leave the rest of the manipulations to you and as I said this is just a demo and we will not be asking questions on the demo. But if you want to get a better understanding of how exactly the resting membrane potential works, I strongly advise you, suggest you to install the this program, MetaNeuron on your system, and also read the paper and it also has a manual for students. So, you systematically go through the different exercises and that way you know you can work at a phase at home and get an idea of the RMP.
So again, the software we are using is by Doctor Newman and Newman, it is called MetaNeuron. It is from the University of Minnesota. It is a free neuron simulation program for teaching cellular neurophysiology and was published in a journal of Undergraduate Neuroscience Education, June 2013 and it can be used without restriction. And it is used to conduct neurophysiology experiments in silico. It works on Windows, Mac and Linux. It is a Windows program, but on Linux you use WINE to run it.
So, as mentioned earlier, the neuronal parameters that is a sodium, potassium concentrations, their equilibrium potentials and conductances can be easily modified. A virtual simulator stimulator injects single or double current pulses into the neuron. Responses are displayed graphically and can be measured with a cursor. And finally, family of traces can be easily generated and viewed in rotatable 3D plots.
So, here we look at the time constant. So, now, the time constant is the time, suppose consider an axon, so at one end you inject current. So, ignore all the active conductances, just think of it is an electrical phenomenon. So, how long does it take to decay when passively if you inject a particular voltage here, how long does it go the axon and decays? We are not talking about axon potential, we are just talking of just the decay of little bit of current which is injected at one point in the axon.
So, that is what this demon would show you and here you see a current pulse being put in here, it is a little small, this red line which denotes this stimulus. And you have the yellow line which actually shows the membrane potentials. So, when you put in the current, you have a change in the membrane potential, it goes towards depolarisation and then it decays. Now, the purple line is the threshold for it to fire an action potential. We are not modeling the those action potential in this demo. We are just seeing what is the time course of this decay of the potential when you put, inject it at one point in the axon.
And this is important, before we get into the demo, little bit of background and also to remind you, the time constant affects many neuronal phenomenon. First of all it affects the time course of the neuron responses, I mean how long does it take to get back to 0. Then it majorly affects the propagation of action potentials in axon. And most importantly, it affects also the summation of synaptic potentials.
For example, if you have one synaptic potential, it does not reach the threshold, but just before it decays, if you have another synaptic potential, again it does not reach threshold, but just before decays if you add a third one, then it hits threshold and then you have an action potential. Now, this depends on tau, you know, how long it takes to decay.
And tau is very simply the product of the membrane resistance R subscript m and the membrane capacitance and a current source. And the membrane capacitance generally is assumed to be 1 microfarad per square centimetre. Membrane resistance, it depends on the axon, the dendrite, the neuron or whatever looking at. And in this demo, the value of the membrane resistance, as well as the time course and the amplitude of the current source, can be varied to get a sense of how it rises and then decays.
So, as mentioned, the membrane resistance and capacitance together determine tau and is described by the simple equation, tau equals R m into C m. Again, in this demo the neuron does not generate axon potentials because we are just looking at this passive property of the axon, not active conductances which give rise to the action potential.
So, here we see how the membrane resistance of x the time constant by varying R m, and the changes in the rise and fall of neuron responses evoke by injection of a square wave of current. Using the cursor we can measure the membrane potential at different times during the rising and falling phase of the response and empirically determine tau. And then we can compare this to the theoretical value of R m and C m and see how they match.
The default value of the stimulus current is a 1-millisecond depolarizing pulse. This is a very good approximation of the current generated by a fast excitatory synapse. When I say fast excitatory synapse, it assumes there are fast inhibitory synapses and also slow excitatory synapses and slow inhibitory synapses. But here we are looking at a fast excitatory synapse.
Using the default parameters we see the effect of membrane time constant on the temporal summation of axon, synaptic potentials.
The threshold potential indicates the voltage at which action potentials would be initiated. And remember, axon potentials all are nothing, once it reaches threshold, it fires an axon potential. These potentials, the synaptic potentials the graded. So, again, you have the centre play of a an analog system by a greater response and an all or nothing kind of a digital response. So, each neuron is a combination of an analog as well as a digital system.
And just to reiterate, the neuron model used in this lesson does not include active membrane conductances and it will not generate action potentials when the membrane potential exceeds thresholds. Just bear that in mind.
So, when a constant current is injected into a neuron, the membrane potential depolarizes with an exponential time course (assuming there are no active membrane conductances). Similarly, when the stimulus is turned off, it returns back to the rmp, the resting membrane potential in an exponential, with an exponential time course. So, the rate at which the membrane potential increases or decreases is described by the constant tau. And tau is defined as the time it takes for the membrane to increase or decrease to approximately 63 percent of its final value, 1-1/e.
So, there are exercises, again for this lesson, I wish I encourage you to go through. So, some of these are, one is using the default parameter values, determine the tau by measuring the time it takes for the amplitude of the membrane potential to fall to 63 percent way back to baseline value, after the current source is turned off.
You can also generate a family of curves showing the effect of varying the membrane resistance and here you use the range function of MetaNeuron. So, I will show this to you in the demo. So, you essentially you check the membrane resistance range box and choose range values of 0.5, 20 and 2 (begin value, end value and increment) and you get a family of responses like so.
So, here a 150 microcurrent is injected for 1 millisecond and it rapidly decays over time. Each of these is a family of curves and this is an exponential time course. And at the end of the pulse the decay begins. The time constant of the decay decreases as the membrane resistance is reduced from 20 to 0.5k.
And here is what I mentioned about temporal summation. You have 3 stimuli, 3 current injections over here. The first one, the second one, and the third one. So, with the long time membrane constant, membrane time constant, here we are talking of 10k ohms, they summate and reach the threshold potential for firing an action potential. So, this is the 10k ohms and when it reaches here you get an action potential.
As I said that is not a model but just assume that happens. But if you have a short time constant, they do not summate enough to reach the threshold. So, you get a sense of how the membrane resistance effects the firing of action potentials.
I open MetaNeuron and I have chosen a lesson 2 which is a membrane time constant. Briefly, on the left you have the membrane resistance and here we have kept with default is 10 kilo ohms into square centimetre and membrane capacitance is constant at 1 microfarad per centimetre square. and the stimulus, we can put in the delay, we can put in the width, we can change the amplitude, we can change the period and we can change the number of stimuli.
And below is the 3D graph function and I will get to that just in a second. So, briefly, the red line right below the plot, the trace is the current injection, is a stimulus and it is in microamps. The yellow trace is the membrane potential, how it changes and how it decays and purple is the threshold. When it hits threshold, just in your mind think an action potential fires.
So, we will start simply by changing the membrane resistance from 10 to 2 and you see it comes back to normal much much earlier. Now, over here, let us change the number of stimuli to 3 I hit enter. So, you have a, they sum, this is a little higher, then this because it is coming in before it is completely decayed and this is little higher than this comes back, but it does not and it come back to normal. But it does not hit the purple line. So, it does not hit the threshold, so there are no action potentials.
Now if we change it back to 10, there you go. So, the you have nothing else has changed, we have just changed the membrane resistance and you see the tau changes and it takes much longer to decay and by the third stimulus current injection, it reaches threshold. So, why does this happen? Intuitively you can think of some of the current when it is just 2, we will change it back to 2, it leaks out of the membrane, in the sides. It leaks out.
And if it is the resistance is low, more of it leaks out. So, it does not reach threshold, but if it is high, if it goes back to 10, then it is forced to go through the axon on the current. And that is why it is, the reaches threshold. So, now what we can do is we can actually we can change the delay.
Let me go back to the default values. The default value is, the membrane resistance is 10k ohm centimetre square and one thing I did not mention is the way to change values without actually entering number into the box. So, next to it is this button, so you can hit the button and move it and you can see the tau increasing or you can make it much shorter and it goes. So, that is an easier way to change things rather than entering values each time.
So, let us go back to default and likewise with the number of stimuli, so you have 1, you have 2, 3, so you can keep doing this way, you do not have to keep entering values and so you see, that when it is 10, by about 3 reaches a threshold and you can easily see that over here. And you can also change this sweep duration if you want to make it much shorter, more tangible or easier to see. So, you can make it much shorter or you can make it much longer. So, let us go back to defaults and we can have range.
So, we can have range of membrane resistances. So, what you do is, you press click this box over here and then this is activated, this the range panel. So, here you have a begin value of 0.5 and end value of 20 and it increments by 1 and the sweep duration we saw that you can
make it shorter, this is making it longer or you can make it much shorter so you can see it in detail.
And finally, you can 3D graph it. So, well. You can see it, it is not very prominent but you can see the values increasing and this is the voltage and this is the time course, the time course over here and you see the membrane resistance values. Another way of looking at is, this is traces, individual traces with all the increments shown over here, but you can also look at the surface, not so prominent over here but you can, for the action potential stuff, this is the useful way to look at things.
So, thus basically, as far as this demo is concern, please, again I strongly encourage you to download this and it would be very useful if you look at the lecture in tantrum with the demo and play with these values, the you cannot change the capacitance, but the resistance values. If you look at the delay, the width, the amplitude, period, etcetera and get a good sense of how it actually work.
In my experience I found that this is a better way to understand tau rather than you know, dry dreary numbers, you where you keep calculating R m into C m.
So again, we are using MetaNeuron made by professors Newman and Newman in the University of Minnesota. It is a free standalone program that can be used without restriction and it works on Windows, Mac and Linux, if you are using Linux, you have to use WINE to get it going. And it is used to conduct neurophysiology experiments in silico.
So, as mentioned earlier, neuronal parameters, sodium, potassium concentrations, equilibrium potentials and conductances can be easily modified. A virtual stimulator injects single or double current pulses into the neuron and responses are displayed graphically and can be measured with a cursor. And families of traces can be generated and viewed in rotatable 3D plots.
So, this is similar conceptually to tau. But here we are looking at distance rather than time. So, consider this image. This is a part of the demo. You inject the red trace indicates stimulus, you inject stimulus in the central of the dendrite or the axon, it does not matter and then you see how it decays. Again, it is an exponential decay on both sides of the injected stimulus.
And we can change different parameters and get an idea of how lambda is dependent on these parameters like properties of the dendrite and axon, the stimulus intensity and then you can look at potential versus time.
So, just to remind you the membrane length constant lambda, so passive conductance of responses through dendrites and axons is an important principle of cellular neurophysiology. So, these principles were first formally enunciated by a researcher such as Wolfed Roll and if you want to model a neuron realistically, when I say realistically, I mean put in exactly all the physics of the dendrites, the denditictory, the axon, the cell body, etcetera. Then you need these concepts of the length constant lambda as well as tau.
Many people do not bother and they just use, and this is an older way of doing it where you consider a neuron, a single unit and it is called a spike and integrate. Integrate and spike, so you have many different impulses coming through the neuron, it integrates everything in a very very primitive fashion and it generates spikes.
But when you use lambda and tau, you can get a realistic idea of how the neuron actually works and this obviously is computationally much more intense and needs a lot of power compared to just the simple integrate and spike units. But this is realistic.
So, coming back from the integration, the length constant lambda, it show shows the effect of the spread of voltage down a neuron process and again we are looking at the distance, not time, time is for tau. So, for simplicity this process is assumed to be a dendrite, but it can also represent an axon.
So, the default parameters illustrate the steady state exponential decay of the membrane potential with distance what you saw earlier. So, you see the, you see a stimulus being put in here and then the decay over a distance. And here, the dendrite is depolarised for 50 milliseconds and the membrane potential plotted at t equal 50 milliseconds. Lambda or the length constant is the distance where the membrane potential falls to 37 percent, 1 by eth of its value.
And this demo models the dendrite as a cylindrical process of uniform diameter and infinite length having a passive leak conductance. Membrane capacitance is kept at one microfarad per centimetre square. So, lambda depends on the membrane resistance. It also depends on the internal, that is the cytoplasmic resistivity. So, you have the axon, the membrane and you have the impulse going in here. The membrane resistance is the resistance of the membrane. The internal resistance is the resistance along the dendrites longitudinal axis or the axon’s longitudinal axis. And it also of course, depends on the diameter, diameter has an effect on lambda which we will see shortly and the stimulus is applied to the dendrite at X equals 0. Again this is distance.
So, a MetaNeuron simulates passive conduction by solving the cable equations. This was developed by Lord Kelvin in the 1850’s because they were laying the cable between Europe-England and the United States and that is very similar to an axon. You have a cable, the cable has an outer conductor, it has insulation and then you have a central conductor which is pretty much how the axon also functions.
So, those equations were very useful and to model the cable properties of an axon or a dendrite. The third dimension, distance along the dendrite can be viewed using the 3D graph option and a range of distances is displayed by selecting the “Range” function for “Position”.
So, this is how it looks. Here you have the time, here you have the millivolts on the y axis and here you have the distance, how it decays and this is a surface contour plot. So, some of the exercises
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