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So, here, in this session, this is going to be a little complicated, I am warning you right now. But if you go through it slowly and if you take breaks and try and understand each slide, you will have a fairly good knowledge of it. So, here we are going to consider the properties of axons. , so far we have considered action potentials, resting membrane potentials, and how axon potentials are formed and how they spread, and so forth. But what about the passive properties of the axon? The neuronal-biophysics, the electrotonic properties of axon dendrites. Why should we model them? We model them because using modeling, we can have a pretty good idea and a realistic idea of how it works. And to do that, we need to know the different properties which are involved, which cause electrical transmission in axons.
So, some background. So, as we saw in the earlier lecture on the microscopic anatomy of the central nervous system, neurons have elaborate dendritic trees, like regular trees, arising from their cell bodies and single axons with their own terminal branching patterns. So, you have your dendritic trees on top and then you have your axons going down transmitting information and they also have their own trees where they transmit information. There are five basic functions. One is the neuron generates intrinsic activity. Second, they receive synaptic inputs from other neurons, axons, cells. So, then they integrate all this, they combine the synaptic activity with intrinsic membrane activity, so it is integrating stuff. Then they encode output patterns in the form of graded action potentials or graded electrotonic potentials, sub-threshold potentials. And finally, they distribute the synaptic inputs.
So, here you see inputs coming in and they get integrated in the somatic area. Then you have the intrinsic activity of the cell and then it is encoded and then you have output in the form of an action potential or graded electrotonic potentials.
So, a fundamental goal of neuroscience is to develop quantitative descriptions of all these functional operations. One, we can test the experiment-driven hypothesis, and two, we can do realistic computational models of neurons. When I say realistic, this is compared and contrasted with a very simple form of computational depiction where you have a neuron that integrates everything coming inside and produces a spike. So, these are called spike and integrate neurons, these were earlier ways of modeling neurons. But now we put in details of the dendritic trees, different conductances, the actual shape of the branches and their diameter so that computationally it requires more resources, but it is more realistic. For example, you have the rat brain on the right and you have electrodes going into different parts of the brain and it shows different kinds of activity of neurons or neuronal cells. So, here you have regular firing, here you have burst firing, then you have intermittent bursts happening in the cerebellum, which we have not talked about much. And in the mid-brain, the thalamus, you have burst modes or fast spikes happening. So, all these different kinds of neuronal activity can all be modeled.
The first task is to understand how the activity spreads. So, to do this for a single process such as an axon is difficult, to begin with. Now, when you have branching dendritic trees, it becomes very complicated and challenging. And then you have interactions between two of these trees, it becomes even more complex, and this is one of the main frontiers of neuro-science how to figure out all the details of what happens in single cells, in cortical columns and neuron.
So, here you have four possible excitatory inputs that come to the pyramidal cell. This is a cell in layer three of the cortex and you have cortical afferents that are inputs coming from other cortical cells. You have spiny stellate layers cells - these are cells in layer four which give inputs to layer three. Then you have other inputs coming from layer five. And then you have inputs coming from the thalamus - typical, this is a typical sensory pyramidal cell in layer three. And the cells as mentioned can have very different dendritic trees. This is the dendritic tree of a Purkinje cell which is a pyramidal cell in the cerebella cortex. And this in the cerebral cortex, a pyramidal cell. Very very different, like a banyan tree versus a pine tree.
So, to model this, we borrowed an electrical theory from the engineers and this is called cable theory and compartment models. So, in the 19th century, cables were laid between Europe, England, and America to transmit telegraphic signals. Now, a cable is like a co-axial cable. So, you have a central conductor and then there is insulation and then it is surrounded by sea-water. So, the theory of cables can be easily applied to neurons. Very similar maths are involved. And on the right is a famous cabling ship, the Great Eastern. And that laid the first Atlantic sea cable for a telephoning between England and America.
Just consider this tree. So, it is mathematically very difficult t apply cable theory to complex branching dendrites. But in the 1960s, a great computational neuro-physiologist, Wilfrid Rall from the United States, he solved this problem by developing computational compartmental models. And these models have provided a theoretical basis for dendritic function. So, combined with mathematical models for the generation of synaptic potentials and action potentials, they can provide a complete theoretical description of neuronal activity. And the more data your plug-in, the more realistic it and faithfully follows what we record from the brain.
So, many open-source packages allow even a beginning student like you to explore functional properties and construct realistic neuronal models. There is a whole branch of neuroscience, computational neuroscience, where scientists only do this. And for this, you just need a computer and some theoretical background and you can start doing it at home. Of course, the more you model the more computation you need. But for starters, a laptop is more than enough to do this. So, the program, there are many programs, genesis, neurons, etc which do this.
So, you just have to learn one properly. So, I suggest neurons. I use Genesis but I suggest Neuron because there is a lot of documentation available on the internet and Neuron is from Yale, it is a simulation environment for modeling individual neurons and networks. And it provides tools for building, managing and using models in a numerically sound and computationally efficient manner. ModelDB is an archive of published models. So, you do not have to have to do things from scratch, you can, for example, get a model of a pyramidal cell or a network of neurons from the somatosensory cortex, plug it into your neuron, and then start playing with it, modifying it, as you wish. So, you do not have to do it from scratch. So, ModelDB is the database where all the neuron models are there. So, Neuron is free, it is an open-source program. You have to download it, go through the tutorials, and then use realistic published data from ModelDB and take off.
So, now we talk about actually how do you model and what are the assumptions involved. Now, regardless of which modeling software you use, you need to have these facts under your belt. So, first point - segments are cylinders, we treat them as cylinders. So, an axon segment is a cylinder, so the electrotonic potential or the local potential, which spreads, that is called an electrotonic potential as opposed to resting membrane or action potential. That electrotonic potential is due to a change in the resting membrane potential. Also, it is ohmic, and in the steady-state membrane, capacitance is ignored. What we are interested in is the delta or the change in the resting membrane potential.
So, we have to make some simplifying assumptions. So, this local spread of current is also called electrotonic current. So, there are two pathways it can take. One is, it can go through the axon and the other is it can go out of the axon via the membrane. So, when it goes through the axon, it encounters resistance and this is internal resistance or ri. And when it goes through the membrane out, its membrane resistance is rm. And here, d is the diameter.
So, with more assumptions, the axial current is inversely proportional to the diameter. So, because the resistance is in parallel sum to decrease the overall resistance, so axial current is inversely proportional to the cross-sectional area. This is straight-forward geometry.
The other thing is axial resistance that is the resistance through the longitudinal axis of the axon, it is assumed to be uniform through its process. Hence, the total cross-sectional axial resistance of a segment is represented by a single resistance. So, ri =Ri /A which is a specific internal resistance divided by the cross-sectional area. So, keep in mind that the units for internal resistance are different from the specific internal resistance. So, the units for resistance is in ohms per centimeter of axial length. While ri is a specific internal resistance, it is ohm centimeter. It is the resistance of a patch. So, A is the cross-sectional area. So, ri is ohms per centimeter while Ri is a specific internal resistance ohm centimeter. In voltage-clamps, the only current is via rm. So, that is going through the membrane. So, this permits the isolation and analysis of different ionic membranes conductances as shown in the Hodgkin–Huxley experiments earlier.
So, membrane current is inversely proportional to the membrane surface area. So, for a unit length of the cylinder, that is the axial cylinder, the membrane current im and the membrane resistance rm they are assumed to be uniform over the entire surface. So, thus, by the same rule of summing parallel resistances, the membrane current is inversely proportional to the membrane area of the segment so that a thicker process has lower overall membrane resistance. Thus, rm = Rm /c where Rm is the membrane resistance of a unit length of the cylinder again, ohm centimeter and Rm is a specific membrane resistance and c = 2πr, the circumference.
So, for a segment, the entire membrane resistance is assumed as concentrated at one point, that is no axial current flow within a segment but only between segments. Membrane current passes through the ionic channels in the membrane as shown earlier, the sodium channels, the potassium channels, the calcium channels, etc. So, variations in the channel density and type in different processes and branches can be easily incorporated into the compartment models. So, because when we model, we handle segment by segment, we can change, depending on if it is an initial segment, whether a more sodium channels or in the interneuron areas where less sodium channels. We can handle all these within our segment models.
So, the final assumptions are the external mediums, the extracellular fluid fuel, it is assumed to have zero resistance, because of its large volume. And also the resistance of the parts either along a process to the ground or straight to the ground is regarded as negligible. And the potential outside the membrane is assumed to be everywhere equivalent to the ground. We also assume that the ionic concentrations, the driving force, the emf across the membrane are constant.
So, cables, a cable can have different boundary conditions. So, it can be infinite boundaries or it can be a particular length, it can be closed or open or partial. So, for long-distance communication, like a cable in the sea, it is considered of infinite length and these kinds of assumptions carry over to the application of cable theory to long axons. However, if you look at the dendrites, they are very short. So, this imposes boundary conditions as solutions of cable equations which have important effects on the electrotonic spread. So, suppose it is highly branched, boundary conditions are very difficult to deal with analytically like you see a neuron over here. It is very very difficult to handle this on a computer. But if you use compartment models and divide this into, first use cable models, divide into different cables of different diameters and each of them is a particular compartment with different resistance and capacitances. Then it is tractable, this problem.
So, how do you do it? How do you construct a compartment model of passive electrical properties of a nerve cell either Neuron or Genesis? So, there are discreet steps. The first step is, you identify which part of the axon you want to model, which is the segment, what is in there, stuff. Based on this, you abstract this into an electrical equivalent circuit. So, you have cm, which is the membrane capacitance. You have internal resistance, ri and then you have, of course, the resting membrane potential, here it is Er. And then you also have r m. So, we abstract this whole thing into this and it works pretty well, surprisingly.
So, suppose you want to do a study state spread. So, if it is a steady-state spread, you can ignore the capacitance and the electromotive force because it is a steady-state spread. And if you use the voltage clamp, then even it can be further reduced to only the membrane resistance because everything else is held constant. And instead of rm, you call it g because you call it conductance which is the inverse of resistance. And finally, the equivalent circuit parameters can be scaled to the size of each segment. One segment may be big so you can scale that and the next segment may be small so we can adjust to the size. So, thank you and in the next session, we will get into details of neuronal biophysics.
So, we were talking about the passive spread or electrotonic spread of potentials across dendrites, across axons. And this depends on something called the characteristic length or Lambda. So, consider the spread of electrotonic potential under steady-state conditions. In standard cable theory, this equation defines it where V is voltage, then you have the, a term for membrane resistance, you have a term for internal resistance and this is d square v by dx square is its spread over the length of the axon, the spread of the voltage length of the axon. So, the steady-state solution for this equation in an infinite cable for positive values of x gives V = V0e-x/ λ And here, this is very straight forward to derive from cable theory. And here, this term Lambda is defined, it is defined as the √rm/ri. So, the resistance of the membrane, neuronal membrane, is divided by the internal resistance of the neuron. And V0 is the value of v at X=0. So, when x = 1, the ratio of V to V0e-1/ λ or 1/e or 0.37. So, it reaches 0.37 of its original value, where x=1.
So, therefore, Lambda is a critical parameter defining the length over which the electrotonic potential decays through a value of 0.37 from the original value at the site of input. So, this is referred to as lambda, it is also referred to as characteristic length, it is referred to as space length or it is referred to as the length constant of the cable.
So, the higher the specific membrane resistance, capital Rm leads to a higher value of rm for the segment, which is the resistance of a patch of membrane. So, therefore the value of Lambda is bigger and so the electrotonic spread, potential spreads more. So thus, specific membrane resistance capital Rm is an important variable in determining the spread of activity, this passage electrical activity in a neuron.
So, length and space constant and membrane resistance. So consider, you know, an axon, 3 axons with 3 different values of lambda, this is Lambda of the first, this curve, this is the second, this curve and this is the third, this curve. So, the potential profiles for all these different values are on top over here and the dotted lines represent the location of lambda on each of these processes, the physical distance is shown in red. So in the first case, because of the properties, it is only so much, in the second case it is much more and in the third case, it is even more.
So Rm, some numbers Rm can vary in values from less than 1 k ohm centimeter square to more than a 100 thousand ohm square centimeter in different neurons and different parts of a neuron. So note, that lambda varies with the √Rm. So, a 100 fold difference in Rm translates to only a tenfold difference in lambda. Conversely, a higher value of specific internal resistance Ri, so the higher the internal resistance, ri, for that segment the smaller the lambda and the less the spread of the potential through the segment. Because the resistance prevents the electric current from spreading. So, the value of Ri is approximately in the range of 50 to 100-ohm centimeter muscle cells and squid axon. In mammalian axons, you know, in mammalian neurons, it is much higher, about 200-ohm centimeter.
So, this kind of limited range suggests that Ri is less important than Rm in controlling the passive of current spread in the neuron i the internal resistance of an axon is not so important as the membrane resistance of the axon, as far as the passive spread of electronic spread of electrical activity in the neuron. Furthermore, it is a square root relationship. So, it further reduces the sensitivity of lambda to Ri. However, there are some caveats. So, in the cytoplasm, you have membranous and filamentous organelles, tubules, various things mitochondria, endoplasmic reticulum, which we looked at in the microscopic anatomy of the central nervous system. All these may change the effective Ri and also the relative significance of Ri and Rm depends on the length of the given processes and how branched it is and so forth.
So, the space constant also depends on the diameter of the process besides the resistance which we considered before. So, thus from the relationships of rm and Rm, ri and Ri, discussed in the previous slides you can, λ =Rm / Ri and that reduces to the square root of the specific membrane resistance, specific internal resistance, and the d/4 is the, a term for the cross-sectional area. So, neuronal processes vary very widely in diameter. The thinnest processes are the distal branches of dendrites and the necks of dendritic spines. These have diameters of greater or equal to 0.1 mu. Note again, that the relationship of Lambda is to the square root. So, a tenfold difference in diameter increases lambda by only 3 times. (
So here, we have the relationship of lambda to diameter. So again, you have 3 different axons with 3 different diameters 1 mu, 4 mu, and 16 mu. And you see the potential profile, so lambda in the first one is over here, in the second one is here and the third one is here because you know, it spreads much further because its cross-sectional area is much more. So the 3, to double lambda, the diameter must be quadrupled.
So, the real length of a giant axon is several centimeters. So, to relate this real length to the characteristic length or lambda, we define electrotonic length or L of a cylindrical neurite as its physical length divided by its space constant. So thus, if x is 30 millimeters, then L would be, that is the electrotonic length 30/4.5 = 7. So, the electrotonic potential decays to a small percentage of the original value by 3 characteristic lengths. So, this has implications when stimulating a nerve. So, you can stimulate a nerve and for whatever reason, experimental conditions, you know, it goes down, it can move by 3 characteristic lengths and then you have a brand new zone to stimulate and the previous area does not affect the new stimulatory site.
But these were axons that we discussed so far. If you look at dendrites, they have lengths that are very much shorter in size than axonal lengths and characteristic lengths. So, therefore in dendrites, what is important is the branching patterns, you know, which modulate the extent of potential spread. So, action potentials overcome the attenuation of passively spreading potentials that occur over axon length. But this occurs to long axons, not too short axons and their collateral.
So, putting all this together, we have combined analog and digital signaling in neurons. So, you have the pyramidal cells over here on the right, couple of them, and you have their extensive dendritic branches and you would have from the base, these cells axons leading onto other cells- one over here, one over here and there is one behind actually 3 of them over there. So, you have excitatory synaptic sites on the soma and they get inputs from different cells, different networks, and, depending on the amount of excitation, the amount of inhibition, they modify the analog signals. The analog signals reach a threshold. Once it reaches a threshold, you have action potentials that are kind of digital, all or nothing, and coded by frequency. So, these mechanisms are seen in, have been found in pyramidal cells. So again, so you, think of analog computer over here, analog computation occurring in the dendritic branches and this modulates action potentials coming out from the digital neurons. So, a single cell is a combined analog and a digital supercomputer.
So fast signals, so how do electrotonic properties affect the spread of fast signals? So, many neural signals change rapidly. So, in mammals, you have fast action potentials that last from 1 to 5 milliseconds. And fast synaptic potentials that last from 5 to 30 milliseconds. So, rapid signal spread depends not only on all these factors discussed so far but also on the membrane capacitance because when there is a rapid change, capacitative effects come into play. And of course, to remind you, the capacitance is due to the lipid part or the lipid moiety of the cell membrane. And classically, we put the value of specific membrane capacitance, Cm as 1 microfarad per centimeter square.
So, continuing with electrotonic properties and the spread of fast signals. So, here you have the equivalent circuit of the renewal process. The membrane capacitance is in parallel with the ohmic components of the membrane conductance the electromotive force and the driving potential. And neglecting the resting membrane potential, what we do is, we inject current into a cell body. So, the time course of the spread of current is described by two currents one is the capacitative current, the part which discharges and recharges the membrane, and the resistive current plus of course the input current which you give during the pulse. So, this would describe it, the capacitance. Then you have a resting membrane potential that changes. Then you have a membrane potential and then you have the resistance and that is equal to the current pulse.
So if we rearrange this, and we substitute RC = τ which is the time constant of the membrane, we get this by rearranging, we get this. And the solution for this equation for a response to a step current change in potential is given by this equation where T = τ/t. When the pulse is terminated, the decay of the initial potential to rest is given by this equation. So, it comes back to baseline and again you see e and e-t.
So, what happens? So you have and "on" transient and then you have an "off" transient. This is how it looks. You have a step current coming in here and it takes some time for the membrane capacitance to discharge, recharge, to reach that thing. And when you switch it off, this is a square pulse. It again decays back to normal. So, the on and off transient shown in the time required for the voltage change across the membrane to reach 1/e of its final value which is 0.37 So, this is similar to the way the length constant defines the spread of voltage over distance. So, the equivalent circuit of a single isolated compartment responds to an injected step current by charging and discharging along a time course determined by the time constant tau. And we, for this figure, V is a steady-state voltage. Im is the injected current applied to the membrane, just an electrical square pulse on top of the axon. Ic is a current through the capacitance. Ii is the current through the ionic leak conductance. And τ is the membrane time constant.
So, consider a two-compartment model. So you have two compartments, there is one compartment here and there is one compartment here. This is compartment A and this compartment B. So in compartment A, you have its equivalent circuit, you have the capacitative term and you have the resistive and the EMF term. Likewise, with B. So, this very simple two-compartment model can be applied to the generation spread of any arbitrary transient signal. So, consider the simplest case. So, this is compartment A and you inject current, positive charge here. And this flows outward across the membrane, partially opposing the negative charge on the inside of the lipid membrane. This negative charge is responsive to the resting membrane potential. And thereby, it depolarizes the membrane capacitance at this site.
At the same time, the charge begins to flow as a current across the membrane through the resistance of the ionic membrane channels that are open in this site, across the membrane. And the proportion of charge divided between Cm and Ri determines the rate of charge of the membrane, which is the membrane constant. So, the electrotonic current which spreads from one compartment to another is also called the local current.
So, in unmyelinated axons, if you remember, some axons have myelin insulating them, some of them are unmyelinated. In an unmyelinated axon, the local current spreading through the internal resistance allows the propagation to the next compartment. So, each of these cable properties is relevant in specific ways. So, for brief signals such as an action potential, Cm the capacitance is critical in controlling the rate of change of membrane potential. For long processes such as axon, the internal resistance is important because it opposes the electrotonic current flow.
So continuing, the effect of rm, you know, membrane resistance, decreases with increased membrane areas because of parallel current paths. This is greater in thinner axons that have a shorter lambda. Specific resistance, Rm, can vary widely. So if it is high, then the current is forced to spread along the membrane, increasing lambda and spread. However, at the same time, tau is also increased, because if you remember, τ = RC. Thus slowing the response of the neighboring compartment to this change. So, consider what happens when you increase the diameter of an axon, it lowers, so here it is 1 mu, 4 mu, and 16 mu. So, it lowers the effective internal resistance, thereby it increases the space constant lambda but without an effect on the time constant tau. So, these are the effects, cable properties are very important, they have very distinct effects on a local current and electrotonic spread of current in the dendritic trees.
So, the question is how do you make it conduct faster? Because once you have large brains, I mean large brains compared to you know, invertebrates. So, we need to control large bodies and that requires communications over long distances. Even though it feels very fast for us, thought and stuff, the maximum speeds of axonal conduction are only about 100 to 120 meters per second in humans. And much much slower in lower forms. So, there is this problem in evolution how do you make axons conduct faster? So one way to increase, you know, axonal conduction is to increase the diameter. But there is a limit, you know, you cannot have, the squid giant axon is probably the axon which we know with the maximum diameter. You cannot increase axon's diameters indefinitely because larger diameters mean fewer axons within a given space and fewer axons mean less cognitive processing.
So, coming back to myelin. So one way is to increase the rate of conduction, which is to make the kinetics of the impulse mechanism faster. So, if you remember, we talked about myelin, so now we will talk about biophysics. So, an action potential in the membrane, mammalian nerves are very fast. So, with the wrapping of these cell membranes from the oligodendrocytes and the Schwann cells, more resistances are added in the series of the membrane resistances. More capacitances are in series of the membrane capacitance, remember capacitance is added as reciprocals. And this kind of insulates and forces, you know, faster conduction. We will get to it in just a bit. So, just to remind you, these myelin layers are provided by the Schwann cells in the peripheral nervous system and the oligodendrocytes in the central nervous system, they wrap their cell membranes on an axon. And because of this, they are the fastest, myelinated axons are fastest in the central nervous system.
So, myelinated axons are not myelinated across the entire length. If you remember, we have nodes, nodes of Ranvier, they interrupted. So here you have a cross-section of an axon with the myelin layers and the Schwann cell over there. And here, you have the longitudinal section and here you see the node in between. So, the density of sodium channels in the node is very high, you know, about 10,000. It is much lower in the internodal regions. An internode is approximately 300 mu to 2000 mu. This difference in density means that most of the action, potential action happens at the nodes, not so much in the internodes.
So, the impulse jumps from one node to another. And it is called saltatory conduction or leaping conduction. So, this makes it much faster. And myelinated axons, one way to think of it, resemble passive cables with active booster stations. And Hursh, in 1939, found an empirical law, that states that the rate of propagation of an impulse along a myelinated axon per second is 6 times the diameter of the axon in mu. So, the
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