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Modelling Neuronal Excitation: The Hodgkin-Huxley Model Thierry Mondeel July 13, 2012 Contents 1 A short introduction to the biology of neurons 4 1.1 Neurons 5 1.2 The cell membrane and ion channels 6 1.2.1 The electrochemical gradient: Why does charge move? 7 1.2.2 The key ions: potassium and sodium . . . . . . . . . . 8 1.3 The action potential 10 1.3.1 The underlying currents . . . . . . . . . . . . . . . . . 10 2 Preliminary work: Membranes, channels and gradients 2.1 Single-compartment models 2.2 Capacitance and resistance 2.3 The membrane potential 2.4 The Nernst potential 2.4.1 The sodium and potassium equilibrium potentials 2.4.2 Ion pumps . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 The Goldman equation . . . . . . . . . . . . . . . 2.5 The form of Iion 2.6 The Hodgkin-Huxley gate model 2.6.1 Persistent conductance . . . . . . . . . . . . . . . . 2.6.2 Transient conductance . . . . . . . . . . . . . . . . . . . . . . . . . . 12 12 13 13 14 14 14 15 15 16 17 18 3 The Hodgkin-Huxley model 20 3.1 The work of Hodgkin and Huxley in perspective 20 3.2 Combining the preliminary work 21 3.2.1 The sodium and potassium conductances . . . . . . . 22 3.2.2 Summary of the equations . . . . . . . . . . . . . . . . 23 3.3 The dynamics 24 4 The 4.1 4.2 4.3 4.4 Fitzhugh-Nagumo model History and perspective The fast-slow simplification of the HH model The fast-slow phase plane The FitzHugh-Nagumo equations 1 26 26 27 28 29 0.0 2 5 Summary 5.1 Review 5.2 Futher studies 34 34 35 6 Sources 37 Preface This is a bachelor thesis about mathematical models that describe the development and propagation of action potentials. At the core of this subject is a model developed in the 1950s and named after the two scientists behind it: Alan Lloyd Hodgkin and Andrew Huxley. The content and usefulness of this model and others derived from it will hopefully become clear to the reader of this thesis. I have been especially drawn to this subject because of its inherent interdisciplinary nature, combining mathematics, physics and biology to create qualitative and quantitative models of one of the most fundamental processes in higher organisms: the spreading of electrical signals across neuronal axons and dendrites. Because of this interdisciplinary nature studying the models and their derivation will require many concepts from biology and physics to motivate the resulting mathematics. In fact, it might be said that most of the effort will be in motivating the mathematics instead of in the mathematics itself. I have tried to make these non-mathematical ideas clear wherever they are used. Therefore the following should be readable without any upfront knowledge of cell physiology, electricity or other knowledge not generally known to students of mathematics. My hope is that this thesis is as interesting to read as it was to work on, both for my thesis advisor(s) and any other reader who might someday, for whatever reason, decide to read it. Finally, I would like to thank my thesis advisor dr. Robert Planqué for tasking me with finding my own topic instead of handing me (a much less interesting) one. The process of figuring out which subjects I’m most excited about has shifted my ideas about what topics to study after my bachelor’s degree more than anything else. 3 Chapter 1 A short introduction to the biology of neurons The work of Hodgkin and Huxley is widely recognized as one of the most outstanding achievements in modern science; indeed they were awarded the Nobel prize for Physiology and Medicine in 1961. The bulk of this achievement is concentrated in 1952, the year they published four papers (a,b,c and d respectively). Of these the last is especially notable and its content will be the main focal point of this thesis. To start to make sense of this work we will have to start with the basic concepts of cell biology. We will be exclusively concerned with neurons in what follows, however it is wise to start at the most fundamental level of this discussion: the cell. What is a cell? A cell is the smallest structural unit of an organism that is capable of independent functioning. Or put differently: a cell is the basic unit of life. The human body is composed of trillions of cells, each a microscopic compartment, and these cells are virtually identical to the cells in other animals: elephants, mice etc. The word cell means “small chamber”, which is quite apt. The usual range for cell diameters in the human body is 10 − 20 µm, which is indeed quite small. What makes cells compartments is their cell membrane. This barrier serves several purposes but most importantly it separates the intracellular fluid from the extracellular fluid and can be very selective about what comes out or goes into the cell. As figure 1.1 points out, the cell contains many more fascinating components. These are however not very relevant for the rest of our story so we will not go into further detail about their function. The major chemical substances in the extracellular fluid are sodium and chloride ions, whereas the intracellular fluid contains high concentrations of potassium ions and ionized molecules, particularly proteins that are usually unable to cross the membrane. 4 1.1 Neurons 5 Figure 1.1: Structures found in most human cells. From: Vander’s Human Physiology. 1.1 Neurons Neurons are a specialized kind of cell. Specifically, they are specialized to transmit electrical signals to other neurons. There are three functional parts of the neuron that we have to understand: the cell body, dendrites and the axon. The cell body is the command center of the neuron. All inputs are gathered here and processed. Many extensions can sprout from the cell body; these are generally called processes of which there are two kinds: dendrites and axons. Dendrites gather the input from other neurons while the axon (each neuron has only one) sends signals to other neurons. An electrical signal that carries a command from the brain to, for instance, the hand travels along a sequence of neurons that can be visualized as in figure 1.2. When the signal arrives at the dendrites on the left side of the neuron, the stimuli given the dendrites are integrated at the cell body, which can then generate a nerve impulse. The nerve impulse travels along the axon to the branches of axons on the right side of the neuron. The impulse then ”jumps” to another set of dendrites and the process just described is repeated. This jumping of the electrical signal is also interesting, a lot more goes on there than just jumping. Neurons that are connected to each other are not actually touching; there is a space between them. We call this space the synaptic cleft, and the whole junction between the two neurons the synapse. At these synapses a release and uptake of ions takes place, that 1.2 The cell membrane and ion channels 6 Figure 1.2: A schematic drawing of a sequence of neurons. From: Vander’s Human Physiology 8th ed. results through a process that will be discussed below, in the propagation of the electrical signal. 1.2 The cell membrane and ion channels Like all other cells, neurons are filled with a large number and variety of ions and molecules and collectively we call this medium the cytosol. Most of these molecules carry positive or negative charges often resulting in an excess concentration of negative charge inside a neuron. The cell membrane consists of a double layer of fat molecules, appropriately called the lipid bilayer, of about 3 to 4 nm thick which is essentially impermeable to charged molecules. This impermeability causes the cell membrane to act as a capacitor (it is able to store charge) by separating the charge lying along its interior and exterior. Many ion-conducting channels or pores are embedded in the cell membrane allowing ions to move in and out of the cell; see figure 1.4. These ions carry charge and therefore ion channels may induce current flows into or out of the cell. Ion channels can be selective, i.e. allowing only a single ion species to pass through them and the amount of ions that flows through them can be dependent upon several different things. For instance the membrane potential (voltage dependence), the shape of the membrane and the binding of certain ions or 1.2 The cell membrane and ion channels 7 Figure 1.3: The charges carried by molecules and ions are separated by the cell membrane from the charges in the extracellular fluid. From: Vander’s Human Physiology, 8th ed. molecules like neurotransmitters. By convention we define the potential of the extracellular fluid to be zero. When the neuron is at rest the excess internal negative charge of the neuron causes the potential inside the cell membrane to be negative with respect to the outside. We call this inner relative to outside potential the membrane potential; it is an equilibrium point at which the flow of ions into the cell matches the flow of ions out of the cell. The potential can change if the balance of ion flow is disturbed by the opening or closing of ion channels. Normally, neuronal membrane potentials vary over a range from about −90 to +50 mV. 1.2.1 The electrochemical gradient: Why does charge move? What we have not yet addressed is why charges move to the membrane and through ion channels. The reason for this is two-fold: electric forces and diffusion. These two processes are responsible for driving ions through channels and together are called the electrochemical gradient. Voltage differences between the exterior and interior of the cell produce forces on ions, negative membrane potentials attract positive ions into the neuron and repel negative ions and vice versa. Also, ions diffuse through channels because the ion concentrations differ inside and outside the neuron. Diffusion is basically a random walk followed by ions and molecules because of their thermal energy. Because of mass action, substances tend to diffuse from regions of high concentration to regions of low concentration. This can 1.2 The cell membrane and ion channels 8 Figure 1.4: (left) A schematic diagram of the lipid bilayer that forms the cell membrane with two ion channels embedded within it. From: Dayan & Abbott. (right) Measuring the membrane potential with a voltmeter, inside with respect to outside potential. From: Vander’s Human Physiology 8th ed. be visualized as in figure 1.5. Figure 1.5: If there are more ions in compartment C1 than in compartment C2 the random walks of the ions will cause a net flux from C1 to C2. From: Vander’s Human Physiology 8th ed. 1.2.2 The key ions: potassium and sodium One of the key things that Hodgkin and Huxley were able to show is that the electrical properties of neurons can be successfully described by considering only two ion species, specifically: sodium (Na+ ) and potassium (K + ). The concentrations of these two ionic species are summarized in figure 1.6. The concentration of Na+ is higher outside the cell than inside, so these ions are driven into the neuron by diffusion. Also, since the inside of the cell 1.2 The cell membrane and ion channels Potassium Sodium Intracellular 397 50 9 Extracellular 20 437 Figure 1.6: Extracellular and Intracellular Ion Concentrations in mM/L for the squid axon. Figure 1.7: The membrane potential is closer to the resting potential with increasing distance from the depolarization site. From: Vander’s Human Physiology, 8th ed. is negatively charged relative to the outside, the sodium ions are pulled into the cell by electrical forces. On the other hand, K + is more concentrated inside the neuron than outside, so it tends to diffuse out of the cell. However, it is also pulled into the cell as a result of the negative membrane potential. What is most relevant about this for us is that these ion flows generate changes in the membrane potential. These changes in the membrane potential from its resting level produce electric signals, which basically come in two flavours: graded potentials and action potentials. Graded potentials are changes in membrane potential that are confined to a relatively small region of the cell membrane and die out within a few millimetres of their origin. They are usually produced by a change in the cell’s environment acting on the membrane, and they are called “graded potentials because the magnitude of the potential change can vary. Whenever a graded potential occurs, charge flows between the place of origin of the potential and adjacent regions of the plasma membrane, which are still at the resting potential; see figure 1.7. Because the electric signal decreases with distance, graded potentials can function as signals only over very short distances. So how then do neurons transmit signals over larger distances? 1.3 The action potential 10 1.3 The action potential An action potential is a specific pattern of change in the membrane potential very different from a graded potential. Action potentials are rapid, all-encompassing large alterations to the membrane potential during which time the potential may depolarize 100 mV, from −70 to +30 mV, and then repolarize to its resting membrane potential. Action potentials also have so-called refractory periods, which we will return to later. Nerve and muscle cells as well as some other kinds of cells, have cell membranes capable of producing action potentials. These membranes are called excitable membranes, and their ability to generate action potentials is known as excitability. Whereas all cells are capable of conducting graded potentials, only excitable membranes can conduct action potentials. The propagation of action potentials is the mechanism used by the nervous system to communicate over long distances making it of fundamental importance to the functioning of the organism. 1.3.1 The underlying currents The magnitude of the resting membrane potential depends upon the concentration gradients and membrane permeabilities of different ions, particularly sodium and potassium. This is true for the action potential as well! It results from a transient change in membrane ion permeabilities, which allows certain ions to move down their electrochemical gradients. In the resting state, the open channels in the cell membrane are predominantly permeable to potassium ions. Very few sodium ion channels are open, which results in a membrane potential close to 70mV. During an action potential, however, the membrane permeabilities for sodium and potassium ions are strongly altered. See figure 1.8. Figure 1.8: (left) The excursion of the membrane potential that is the action potential. (right) The ionic currents of sodium and potassium that underlie the action potential. 1.3 The action potential 11 The depolarizing phase (upward movement towards zero) of the action potential is due to the opening of voltage-gated sodium channels, which increases the membrane permeability to sodium ions. Therefore, more positive charge enters the cell in the form of sodium ions than leaves in the form of potassium ions, and the membrane depolarizes. It even overshoots, becoming positive on the inside and negative on the outside of the membrane. In this phase (the top of figure 1.8), the membrane potential approaches but does not reach the sodium equilibrium potential. After reaching this peak the potential very quickly repolarizes back to the resting potential. Two processes are at work during this phase of the action potential. First, the sodium channels that opened during the depolarization phase are inactivated near the peak of the action potential, which effectively causes them to close. Secondly, voltage-gated potassium channels, opening more slowly than sodium channels, now open in response to the depolarization. Thus potassium flows out of the cell, contributing to the inside of the cell becoming more negatively charged relative to the outside of the cell. Chapter 2 Preliminary work: Membranes, channels and gradients This chapter will introduce the biological terms mentioned in the previous chapter in a more exact and mathematical way. It will introduce the relations and formulas we will need to derive the Hodgkin-Huxley model (from now on we will refer to this model as the HH model). Remember that our goal is to arrive at a mathematical model based on sound biology that has the desired behaviour, i.e. produces action potentials. To that end we will have to consider several subjects: the membrane potential, the Nernst potential, the ionic conductance and the opening and closing of ion channels, and try to come up with mathematical descriptions for these processes. 2.1 Single-compartment models Models that describe the membrane potential of a neuron by a single variable V are called single-compartment models. Only such models will be considered in this thesis. However, it is interesting to note why one would be interested in more complex or so called multi-compartment models. Membrane potentials measured at different places within a neuron can take different values. For example, the potentials in the cell body, dendrite, and axon can all be different. This causes ions to flow within the cell, which will tend to equalize these differences. The cytosol (intracellular fluid) provides resistance to such flow. This resistance is highest for long and narrow stretches of a dendrite or axon. Therefore a single-compartment model, i.e. models where we assume the membrane potential to be the same in the entire neuron, is realistic where there are few of such long and thin stretches of dendrite or axon or where the spatial effects are considered to be irrelevant. As stated before, we will consider only single-compartment models in this thesis. 12 2.3 The membrane potential 13 2.2 Capacitance and resistance The product of the membrane capacitance and the membrane resistance is a quantity with units of time called the membrane time constant, τm = Rm Cm . Because Cm and Rm have inverse dependences on the membrane surface area, the membrane time constant is independent of area. The membrane time constant sets the basic time scale for changes in the membrane potential and typically falls in the neighbourhood of 50 ms. 2.3 The membrane potential Capacitance is defined as charge held divided by the voltage needed to hold that charge Q Cm = . V This capacitance Cm is assumed to be constant, i.e. not dependent on time. Therefore by taking the time derivative we can derive the following formula for the flux of charge across the membrane dQ dV = Cm . dt dt (2.1) This formula is fundamental for all work to be done in this subject, it determines the membrane potential for a single-compartment model. What this equation really says is that the rate of change of the membrane potential is proportional to the rate at which the cell builds up charge. With it we can derive another fundamental result by considering what processes contribute to the charge build-up. Charge can only enter or leave the cell through ion-channels or through an applied current by an experimenter. Thus we have dQ = −Iion (V, t) + Iapp dt dV Cm = −Iion (V, t) + Iapp . dt (2.2) (2.3) Here the signs on the right are dictated by convention; positive charge flowing into the cell is defined as positive. When there is no applied current we have the following important relationship dV Cm + Iion (V, t) = 0. (2.4) dt This equation forms the basis for modelling the membrane potential. Really, all that remains for the derivation of the HH model is to be more specific about Iion , which is what we will do next. 2.4 The Nernst potential 14 2.4 The Nernst potential The electrical potential generated across the membrane at electrochemical equilibrium, the equilibrium potential, can be predicted by a simple formula called the Nernst equation. The Nernst equation gives a formula for one ion species that relates the numerical values of the concentration gradient to the electrical gradient that balances it. For an ion X with given outside concentration Xo and inside concentration Xi the Nernst potential VX is given by VX = RT [X]o ). ln( zF [X]i (2.5) Here R is the universal gas constant, T is the temperature in Kelvin, F is Faraday’s constant and Z is the valence of the ion in question. Since three of these parameters are environmental parameters one often comes across different formulas where these constants have been represented by their value in the problem that is being discussed. For instance at 27 degrees Celsius RT F = 25.8. 2.4.1 The sodium and potassium equilibrium potentials Since we will be relying on sodium and potassium ions to do most of the work in what follows, it might be instructive to use the just derived Nernst equation to calculate the sodium and potassium equilibrium potentials. Taking the extracellular and intracellular concentrations for sodium mentioned in table 1.6, we find: 25.8 437 ln( ) = 56 mV. 1 50 Similarly for potassium 25.8 397 ln( ) = −77 mV. 1 20 2.4.2 Ion pumps Now, the attentive reader will have noticed that the experimentally measured resting membrane potential is not equal to either equilibrium potentials. The resting membrane potential is close but not equal to the potassium equilibrium potential, because a small number of sodium channels are open in the resting state, and some sodium ions continually move into the cell, cancelling the effect of an equivalent number of potassium ions simultaneously moving out. Thus, there is net movement through ion channels of sodium into the cell and potassium out. If the resting membrane potential is to be maintained this cannot be all that is going on, and indeed it is not. 2.5 The form of Iion 15 Active-transport mechanisms in the plasma membrane utilize energy derived from cellular metabolism to pump the sodium back out of the cell and the potassium back in. Actually, the pumping of these ions is linked because they are both transported by the same pump, appropriately named the Na,K-ATPase pump. ATPase because it gets its energy from the molecule ATP. 2.4.3 The Goldman equation As a short side note, it is interesting to consider that we have used the Nernst potential in the story above. However, the Nernst potential only considers a membrane permeable to one ion species, which in reality is of course not the case. So what happens if multiple ionic species are able to diffuse across the membrane? In this case a more complex formula is available: the Goldman equation. In the Goldman equation the equilibrium potential is not determined by equation 2.5, but takes a value between the equilibrium potentials of the individual ion species that it conducts. Suppose wish to know the membrane potential for a membrane permeable to sodium, potassium and chloride. The Goldman equation would look like this RT PK [K]o + PNa [Na]o + PCl [Cl]i Vrest = ln , F PK [K]i + PNa [Na]i + PCl [Cl]o where the Pion stand for the permeabilities of the different ion species. The Goldman equation is “Nernst-like” but has a term for each permeable ion and is thus an extended version of the Nernst equation that takes into account the relative permeabilities of each of the ions involved. The relationship between the two equations becomes obvious in the situation where the membrane is permeable only to one ion, say, K + . In this case, the Goldman expression collapses back to the Nernst equation. We will however, for the HH model, not use this description of the equilibrium potential, but use the simpler Nernst potential instead. 2.5 The form of Iion As discussed in the previous section on the Nernst potential, the current carried by a certain type of ion channel with equilibrium potential Vion , vanishes when the membrane potential satisfies V = Vion . For many types of channels the current changes approximately linearly when the membrane potential deviates from this equilibrium value. The difference V − Vion is called the driving force for that ion and the membrane current per unit area due this ion species is written as: Iion = gion (V − Vion ), (2.6) 2.6 The Hodgkin-Huxley gate model 16 Figure 2.1: Movements of sodium and potassium ions across the cell membrane of a resting neuron. The diffusion movements (black arrows) are exactly balanced by the active transport (red arrows) of the ions in the opposite direction. From: Vander’s Human Physiology, 8th ed. where the term gion is the conductance per unit area of the membrane due to the ion channels for this ion species. We call the relationship in 2.6 a Linear I-V curve. It is important to realize that we have made an assumption here, the linearity of the I-V curve. For some neurons this is not a realistic assumption; their ion currents are better described by other relationships. 2.6 The Hodgkin-Huxley gate model Now, our final point of interest will be how to describe conductances of specific ions across the membrane. Again, we are mainly interested in sodium and potassium and these appear to have two very specific types of conductance. Potassium has a persistent conductance while sodium has a transient conductance; see figure 2.2. 2.6.1 Persistent conductance The opening of the persistent conductance gate (we now know) involves a number of conformational changes (this was unknown at the time of Hodgkin and Huxley). For example, the K + channel consists of four identical subunits, and it appears that all four must undergo a structural change for the channel to open. When k identical but independent events are needed 2.6 The Hodgkin-Huxley gate model 17 Figure 2.2: (A) An example of a persistent conductance gate. The gate is opened and closed by a kind of sensor that responds to the membrane potential. (B) An example of a transient conductance gate, the activation gate is coupled to a voltage sensor that acts like the gate in A. A second gate, denoted by the ball, can block that channel once it is open. The top figure shows the channel in a deactivated state. The middle panel shows an activated ion conducting channel, and the bottom panel shows the channel in its inactivated state. for a channel to open, and one of these events happens with probability n, we can write the conductance as gion = nk . (2.7) If large numbers of channels are present, and if they act independently of each other (which experimental evidence suggests they do to a good approximation), then, the fraction of channels open at any given time is approximately equal to the probability that any one channel is in an open state. This is an application of the law of large numbers. Suppose a percentage n of subunit gates is in the open state at a given moment in time. Thus the probability of one specific subunit gate to be open is also n and the probability of it being closed is 1 − n. We can describe the transition of each subunit gate by a simple scheme in which the gating transition from closed to open occurs at a voltage-dependent rate αn (V ), and the reverse transition open to closed occurs at a voltagedependent rate βn (V ). The probability that a subunit gate opens over a 2.6 The Hodgkin-Huxley gate model 18 short interval of time is proportional to the probability of finding the gate closed, 1−n, multiplied by the opening rate αn (V ). Likewise, the probability that a subunit gate closes during a short time interval is proportional to the probability of finding the gate open, n, multiplied by the closing rate βn (V ). The rate at which the open probability for a subunit gate changes is given by the difference of these two terms dn = αn (ν)(1 − n) − βn (ν)n. dt (2.8) Now, we can mold this equation into another often used form by dividing by αn + βn , resulting in τn (ν) dn = n∞ − n, dt where (2.9) τn (V ) = 1 αn (V ) + βn (V ) (2.10) n∞ (V ) = αn (V ) . αn (V ) + βn (V ) (2.11) and So what does this equation actually say? It says that for a fixed voltage V , n approaches the value n∞ (V ) exponentially with time constant τn (V ). The key elements in the equation for n are the opening and closing rate functions αn (V ) and βn (V ). These are obtained by fitting experimental data. The data upon which these fits are based are typically obtained using a technique called voltage clamping; indeed this is what Hodgkin and Huxley used in their experiments. 2.6.2 Transient conductance Some channels only open transiently when the membrane potential is depolarized because they are gated by two processes (subunits) with opposite voltage dependences; see figure 2.2. One of the gates behaves exactly like the gate of the persistent conductance. The probability that it is open is written as mk where m is an activation variable like n, and k is an integer that again signifies the number of subunits in the channel (or the number of events that need to happen for it to open). The probability that the ball (see figure 2.2) does not block the channel is denoted by h and called the inactivation variable. Now, for the transient conducting channel to be conducting both gates must be open, and if the two gates act independently, the probability of this occurring is mk h. Thus we can write the transient conductance of a certain ion species as gion = mk h. (2.12) 2.6 The Hodgkin-Huxley gate model 19 More generally, if there is not one process blocking the channel but several, we could describe this with the following formula for the current flow due to a certain ion species, sometimes referred to as the HodgkinHuxley gating equation I = gmα hβ (V − Vion ). (2.13) Chapter 3 The Hodgkin-Huxley model Now that we have found some mathematical descriptions for the various parts of and processes within neurons we are ready to consider the actual model derived by Hodgkin and Huxley. In this thesis we are concerned with the space-clamped dynamics of the system; that is, we consider the spatially homogeneous dynamics of the membrane. With a real axon this state can be obtained experimentally by having a wire down the middle of the axon maintained at a fixed potential difference to the outside. The spatial propagation of action potentials along the nerve axon through the concept of travelling waves, is a very interesting and crucial subject to study, but is not included in this thesis. For the interested reader we refer to Keener & Sneyd - Mathematical Physiology volume I, chapter 6 or to Murray’s Mathematical Biology Volume II, chapter 1. Here, we will derive the Hodgkin-Huxley model and the reduced analytically tractable FitzHugh-Nagumo model (See FitzHugh 1961, Nagumo et al, 1962) which captures the key phenomena of the HH model. 3.1 The work of Hodgkin and Huxley in perspective Now that we have some basic notions under our belt we are in a better position to understand the importance of the work of Hodgkin and Huxley. In 1952, Hodgkin and Huxley wrote five papers (Hodgkin & Huxley a, b, c, d and e) that described their experiments that aimed at determining the laws that govern the movement of ions in a nerve cell during an action potential. The first paper examined the function of the neuron membrane under normal conditions and explained the basic experimental set-up used in each of their subsequent studies. The second paper examined the effects of changes in sodium concentration on the action potential as well as the 20 3.2 Combining the preliminary work 21 Figure 3.1: The inventors of the model, (left) Hodgkin and (right) Huxley. partitioning of the ionic current into sodium and potassium currents. The third paper examined the effects of potential changes on the action potential. The fourth paper outlined how the inactivation process reduces sodium permeability. Hodgkin and Huxley’s accomplishment is really two-fold. Using the already existing (but at the time, relatively novel) technique of voltageclamping they were able to gather large amounts of experimental data from the giant squid axon. This axon was used because of its size, making it practical to do experiments on. Secondly they then in their final paper supplied a mathematical model consisting of four non-linear ordinary differential equations, which they showed captured the gathered data quite well, meaning that for certain parameter values action potential like excursions of the variables can be observed. The influence of the work of Hodgkin and Huxley can easily be seen when opening any textbook on computational neuroscience or mathematical physiology since this subject will always be treated. However, when opening a biologically oriented textbook this is somewhat harder. Since the mathematics of the HH model is considered to complicated, there is often little mention of Hodgkin and Huxley even though the results of their work are visible in any chapter on action potentials, ion channels and membranes. The final paper put together all the insights from the earlier papers and revealed the mathematical model for which they received the 1962 Nobel prize for Physiology and medicine. We will now focus on its derivation. 3.2 Combining the preliminary work We saw in the previous chapter that the membrane potential for a singlecompartment model satisfies Cm dV + Iion (V, t) = 0, dt 3.2 Combining the preliminary work 22 or, including an applied current Cm dV = −Iion (V, t) + Iapp . dt Figure 3.2: The equivalent circuit for the HH model. This equation can also be arrived at by considering the equivalent circuit of the HH model; see figure 3.2. The cell membrane can be modelled as a capacitor in parallel with an ionic current resulting in the formulas above. We also saw the concept of linear I-V curves in the previous chapter. There are numerous ions present in the fluids surrounding a cell, however the HH model assumes they can be reduced to two ions and one extra variable in which we lump together all others. This results in dV = −gN a (V − VN a ) − gK (V − VK ) − gL (V − VL ). (3.1) dt The previous chapter also covered how to model the conductances in terms of activating and inactivating mechanisms. This last step will result in the actual HH model. Cm 3.2.1 The sodium and potassium conductances It is important to step back for a moment and revel at the magnitude of what Hodgkin and Huxley did in the following step of the model’s derivation. In the previous chapter we deduced equations for the conductance based on the assumption of subunits that all have to undergo the same conformational change. However, for Hodgkin and Huxley it was the other way around. This interpretation was not available at the time; indeed, they came up with it. To their measurements of sodium and potassium conductances they fitted functions m, n and h, and in a stroke of genius gave them the interpretation we now use. 3.2 Combining the preliminary work 23 The specific forms for the conductances they came up with are as follows. For potassium they found gK = ḡK n4 (3.2) where g¯K is a constant. This formula is to be interpreted as a channel consisting of four identical subunits. And for sodium they found gN a = ḡN a m3 h (3.3) where gN ¯ a is a constant. This formula is to be interpreted as again consisting of four subunits, three of which are activating and one of which is inactivating (think of the ball-like structure in figure 2.2). 3.2.2 Summary of the equations Combining the results in this chapter we arrive at the full HodgkinHuxley model of 4 non-linear ordinary differential equations: Cm dV = −ḡN a m3 h(V − VN a ) − ḡK n4 (V − VK ) − gL (V − VL ), dt (3.4) where m, h and n are the gating variables that satisfy: dm = m∞ − m dt dh τh (ν) = h∞ − h dt dn = n∞ − n τn (ν) dt τm (ν) Figure 3.3: (left) The steady-state functions and (right) the time constants of the HH model. Note: τm is much smaller than the other two. From: Keener & Sneyd, 2nd ed. Remember that the parameters τm , m∞ etc. are defined in terms of αm , βm etc. and thus have to be estimated from measurements. For the HH 3.3 The dynamics 24 Figure 3.4: (left) The action potential and (right) the gating variables during the action potential, both on the same time scale. From: Keener & Sneyd, 2nd edition. model these functions are given in figure 3.3 and figure 3.4. The interested reader can find the actual expressions and values of the parameters in, for instance, Keener & Sneyd - Mathematical Physiology Volume I, 2nd ed. (p. 206). 3.3 The dynamics In this section we will explain some features of the dynamics the HH model without any proof since these dynamics are not easily investigated. Because of the complexity of the system, various simpler mathematical models, which capture the key features of the full system, have been proposed, the best known and most useful one of which is the FitzHugh-Nagumo model (FitzHugh 1961, Nagumo et al. 1962), which we will derive in the next chapter. If Iapp = 0, the model is linearly stable but excitable. That is, if the perturbation from the steady state is large enough there is a large excursion of the variables in their phase space before returning to the steady state. If Iapp 6= 0 there is a range of values where regular repetitive firing occurs, that is, the system exhibits limit cycles. These two points remind us of course of the threshold potential and action potential we discussed in the first chapter. Both types of phenomena have been observed experimentally and this indicates that the HH model is very effective at describing what actually happens during the action potential. The behaviour of periodic orbits as Iapp is increased can be nicely summarized in a bifurcation diagram. For each value of Iapp we plot the value of V at the resting state, and the maximum and minimum values of V over the associated periodic orbit, see figure 3.5. The point here is just to notice that the HH model does indeed produce solutions with the desired behaviour and that oscillations are possible with increased applied current. Investigation of 3.3 The dynamics 25 Figure 3.5: Bifurcations of the HH model with the applied current Iapp as the bifurcation parameter. HB denotes a Hopf bifurcation, SNP denotes a saddle-node of periodics bifurcation, osc max and osc min denote, respectively, the maximum and minimum of an oscillation, and ss denotes a steady state. Solid lines denote stable branches, dashed or dotted lines denote unstable branches From: Keener & Sneyd, 2nd ed. the bifurcations would require more time than we have here and is therefore excluded. In the following chapter we will focus on explaining the dynamics of simplified systems that will also still produce action potential behaviour even though the biological context may have been lost. These simplified systems allow us to more easily investigate the dynamics and we can therefore draw conclusions from their analysis. Chapter 4 The Fitzhugh-Nagumo model 4.1 History and perspective In 1961, Fitzhugh sought to reduce the Hodgkin-Huxley model to a twovariable model for which phase plane analysis is applicable. His general observation was that the gating variables n and h have slow kinetics relative to m. Using this he came up with a two-dimensional system that has the same qualitative features as the HH model. This model is not only named after Fitzhugh but also after Nagumo, who in 1962 built the equivalent circuit for the above model (see figure 4.1). In the original papers of FitzHugh, this model was called the Bonhoeffer-van der Pol oscillator because it contains the Van Der Pol oscillator as a special case. Figure 4.1: Circuit diagram of the tunnel-diode nerve model of Nagumo et al. (1962) 26 4.2 The fast-slow simplification of the HH model 27 4.2 The fast-slow simplification of the HH model In the previous chapter we did not include a very detailed description of the dynamics of the HH model. The dynamics would be much easier to investigate if the model could somehow be reduced to two dimensions. The crucial observation that allows for this follows from the graphs of the gating variables of the HH model and their time constants. For ease of reference these have been included again (see figure 4.2). Figure 4.2: (left) The gating variables and (right) their associated time constants. From: Keener & Sneyd 2nd edition. To go from four to two dimensions we will have to drop some information, that is unavoidable, but we can try to make reasonable assumptions. The clearest of which is that the time constant of m, τm , is small or rather fast compared to the other two constants. Remember that τm signifies the speed with which m converges on its steady state value m∞ . Our first assumption could therefore be that that m is instantaneously at this value, i.e. m = m∞ . This is called a quasi steady-state approximation. The second reduction is a bit more obscure. Looking at the graph of the gating variables a symmetry between the graphs of h and n can be seen. Specifically the graph of 1 − h and n seem very similar. This can be used to eliminate one of the variables. For instance by setting h = 1 − n. Combining our two assumptions we come upon the simplified Hodgkin-Huxley model (without an applied current) that contains one fast variable V and one slow variable n dV = −ḡN a m3∞ (V )(1 − n)(V − VN a ) − ḡK n4 (V − VK ) − gL (V − VL ) dt (4.1) dn τn (ν) = n∞ − n. (4.2) dt Cm 4.3 The fast-slow phase plane 28 This is often referred to as the fast-slow model and its phase plane as the fast-slow phase plane. 4.3 The fast-slow phase plane Since we have now reduced the model to two variables, we can draw the phase plane for it by considering the nullclines of V and n. Setting dn dt = 0 we find that n = n∞ (V ) which is the monotonically increasing function that we already know. Similarly setting dV dt = 0 we find that V = ḡN a m3∞ (1 − n)VNa + ḡK n4 VK + gL VL , ḡN a m3∞ (1 − n) + ḡK n4 + gL which has a cubic shape. The nullclines have been plotted in figure 4.3 along with an action potential resulting from the dynamics. Notice that for the HH model parameters there is one intersection of the nullclines which is thus the dV only steady-state of this system. From the signs of dn dt and dt we can explain how this system responds to a perturbation. V is a fast variable and n is a slow variable and they are usually called the excitation and recovery variable, respectively. This makes sense since V causes the rise in potential, and n causes the return to the resting state. Because of the relative speed differences the trajectories in the phase plane are nearly horizontal except on the nullcline of V where the flow is vertical. This cubic curve is called the slow manifold. On this curve the trajectories move slowly in the direction determined by the sign of dn dt , but elsewhere the trajectories move quickly in a horizontal direction, since V is fast. From the sign of dV dt it follows that trajectories flow away from the middle branch of the slow manifold, meaning in between the local minimum and maximum of the function, and toward the left or right branches. Thus, the middle branch is called the unstable branch of the slow manifold. This functions as a threshold. When the perturbation from the resting state is so small that V does not cross the unstable manifold, then the trajectories move horizontally toward the left and return to the resting state, i.e. no action potential is generated. However, if the perturbation is so large that V crosses the unstable manifold, then the trajectory moves to the right until it reaches the right branch of the slow manifold. Here dn dt > 0, and so the solution moves up and along the branch (since it is stable) until the start of the middle branch is reached (the local maximum). There, the flow is nearly horizontal to the left and away from the middle branch, and so the solution moves over to the left branch of the slow manifold. Here dn dt < 0, so the solution moves down and along this branch until the resting state is reached again, completing the excursion we have come to know as the action potential. If we turn on the applied current in this fast-slow model what happens? As Iapp increases, the V nullcline shifts up, until the two nullclines intersect 4.4 The FitzHugh-Nagumo equations 29 Figure 4.3: The fast-slow phase plane of the Hodgkin-Huxley equations. Both nullclines and an action potential are drawn for Iapp = 0. From: Keener & Sneyd, 2nd ed. on the middle branch of the cubic. This branch is unstable so the trajectory will never approach the resting state again, continuously switching between the two stable branches for the same reasons we saw above. This behaviour is called a relaxation limit cycle or relaxation oscillation, see figure 4.4. Biologically, this represents a continuous stream of action potentials. 4.4 The FitzHugh-Nagumo equations What FitzHugh did was look at a simplified two-dimensional system that has the same qualitative characteristics as the fast-slow phase-plane. The system can be represented in several forms but is often written in the abstract form dv = f (v) − w + Iapp dt dw = (v − γw). dt (4.3) (4.4) Here f (v) is understood to be cubic, just like the V nullcline in the reduced system, and Iapp is the same parameter as in the HH model. One form for f (v) that is often used is f (v) = v(1 − v)(1 − α), with α < 1, although other choices are available. The nullcline for w is the linear equation w = γv which roughly looks the same as the n nullcline in the reduced HH system. Remember that increasing the Iapp parameter will shift the nullcline of the v upwards, just like we saw in the previous section. Also remember that the parameter signifies the time-scale difference between the two variables. It is custom to take small, thereby making the recovery variable w slow compared to the excitation variable v, as it should. One of the other ways 4.4 The FitzHugh-Nagumo equations 30 Figure 4.4: Fast-slow phase plane of the Hodgkin-Huxley equations, with Iapp = 50, showing the nullclines and an oscillation. From: Keener & Sneyd, 2nd ed. the system can thus be represented is by putting in the other equation, making dv dt very large and thus v very fast. dv = f (v) − w + Iapp dt dw = v − γw. dt (4.5) (4.6) The nullclines are assumed to have a single intersection point, which, without loss of generality, is placed to be at the origin and are drawn in figure 4.5. Now, the analysis of the dynamics of the FitzHugh-Nagumo system is analogous to the analysis of the reduced HH model, i.e. the two phase planes are basically the same. When the resting state (the intersection of the nullclines) is on the left branch V− and not too far from the minimum W∗ , the system is excitable. This is because, as in the reduced model, a sufficiently large perturbation from the resting state puts the system on a trajectory that moves away from the resting state before eventually returning to rest. Such a trajectory, again just as in the reduced model, goes to the right branch V+ , which it follows as it moves up. When the maximum W ∗ is reached, it moves to the left branch V− and then returns to rest, following the branch. See figure 4.6. As we increased Iapp the reduced HH system produced limit cycles; the FitzHugh-Nagumo system does too. The v nullcline moves up as Iapp increases. So, when Iapp takes values in some intermediate range (not too big because then it will lie on the right branch and be stable again), the steady state lies on the middle branch, V0 , and is unstable. Instead of returning to 4.4 The FitzHugh-Nagumo equations 31 Figure 4.5: A schematic diagram of the FitzHugh-Nagumo phase plane. V− , V0 and V+ respectively denote the left, middle and right branch we saw in the reduced model. V− and V+ are stable and V0 is unstable. We denote the minimal value of w for which V(w) exists by W∗ , and the maximal value of w for which V+ (w) exists by W ∗ . From: Keener & Sneyd, 2nd ed. rest after one action potential, the trajectory alternates periodically between the left and right branches. As before with the HH model, the behaviour of periodic orbits as Iapp is increased can be nicely summarized in a bifurcation diagram. For each value of Iapp we plot the value of V at the resting state, and the maximum and minimum values of V over the periodic orbit. As Iapp increases, a branch of periodic orbits appears in a Hopf bifurcation at Iapp = 0.1 and disappears again in another Hopf bifurcation at Iapp = 1.24. Between these two points there is a branch of stable periodic orbits. The bifurcation diagram is drawn in figure 4.8. This figure shares a lot of its features with the bifurcation diagram of the HH model. It helps us understand the dynamics of the HH model just a little better by making it a lot easier to investigate them. We have lost the biological interpretation of the variables involved but those are still available when looking at the closely associated HH model. This system of equation that FitzHugh found has been a cornerstone model for many other subjects in this field. Its tractability allows easier mathematical analysis and thus makes it a very popular model. 4.4 The FitzHugh-Nagumo equations 32 Figure 4.6: Phase portrait for the FitzHugh-Nagumo system, with α = 0.1, γ = 0.5, = 0.01 and no applied current. For these parameter values the system has a stable resting state, and is excitable. From: Keener & Sneyd, 2nd ed. Figure 4.7: Phase portrait for the FitzHugh-Nagumo system, with α = 0.1, γ = 0.5, = 0.01 and Iapp = 0.5. For these parameter values, the resting state is unstable and there is a periodic orbit. From: Keener & Sneyd, 2nd ed. 4.4 The FitzHugh-Nagumo equations 33 Figure 4.8: Bifurcation diagram of the FitzHugh-Nagumo equations, with α = 0.1, γ = 0.5, = 0.01, with the applied current as the bifurcation parameter. The steady-state solution is labeled ss, while osc max and osc min denote, respectively, the maximum and minimum of V over an oscillation. HB denotes a Hopf bifurcation point. From: Keener & Sneyd, 2nd ed. From: Keener & Sneyd, 2nd ed. Chapter 5 Summary 5.1 Review In this thesis we looked at the problem of finding mathematical models for describing the biological phenomenon of action potential generation. After a brief introduction to the relevant biology we focused on mathematical formulations of the basic properties and processes of neurons. To this end we started out with the fundamental circuit equation Cm dV = −Iion (V, t) + Iapp . dt Then by assuming that the ionic current is composed of only sodium, potassium and a leakage current and that the I-V curves are linear, we stumbled upon Cm dV = −gN a (V − VNa ) − gK (V − VK ) − gL (V − VL ) + Iapp . dt After that we considered the specific forms of the ionic conductances which we modeled with a simple kinetic scheme and curve fitted the resulting Hodgkin-Huxley gating equation to measurements. All in all we derived the four-dimensional HH model dV dt dm τm (ν) dt dh τh (ν) dt dn τn (ν) dt Cm = −gN ¯ a m3 h(V − VN a ) − g¯K n4 (V − VK ) − gL (V − VL ) = m∞ − m = h∞ − h = n∞ − n, 34 5.2 Futher studies 35 where Iapp denotes the externally applied current. This model works quite well, and the desired behaviour (action potentials) is reproduced by this model very nicely. From this starting point, research went in two directions: one direction tried to make the model even more realistic, taking into account more ion species or other phenomena (like synapses etc.). The other direction is a simplification of the model, such that it is possible to see clearly how the spike is produced. The most important of these simplifications is the FizHugh-Nagumo model that we considered in the next chapter. We started out by making two assumptions, one about the time constants (m moves fast) and one about the apparent symmetry in h and n. This resulted in the fast-slow model dV = −gN ¯ a m3∞ (V )(1 − n)(V − VN a ) − g¯K n4 (V − VK ) − gL (V − VL ) dt (5.1) dn τn (ν) = n∞ − n. (5.2) dt Cm After the reduction of the HH model from 4 to 2 variables we reduced even further and found the FitzHugh-Nagumo equations which are a simpler representation of the dynamics of the fast-slow phase plain dv = f (v) − w + Iapp dt dw = (v − γw). dt This system in particular appeared to be a very useful model with the same qualitative features as the HH model and indeed is a standard model in that virtually every book on the subject of mathematical biology or biological oscillators treats it. 5.2 Futher studies What we did not get to in this thesis due to a lack of time and the fact that it is an entire topic in itself, is the propagation of these action potentials through travelling waves. For this we refer the reader to the standard texts by Keener & Sneyd and Murray. Or for a very readable introduction to travelling waves in general to the book by Edelstein-Keshet. Rest assured though, that it is indeed possible to show that travelling waves exist, at least for the FitzHugh-Nagumo system. There are numerous other subjects that are now ready to be explored. This is one of the reasons why the Hodgkin-Huxley model is so important; it opens doors to all sorts of other interesting problems. Below we name a few fun ones and the relevant literature. 5.2 Futher studies 36 Consider networks of spiking neurons. Such neurons can synchronize and exhibit collective behaviour that is not intrinsic to any individual neuron. For example partial synchrony in cortical networks is believed to generate various brain oscillations, such as the alpha and gamma EEG rhythms. For more, see the book by Izhikevich, Chapter 10. Other interesting subjects that are perhaps more biological than mathematical include: synaptic plasticity and learning, for this Gerstner’s book seems to be a good starting point. All in all we can conclude that the theory of neuron firing and propagation of nerve action potentials is one of the major successes of mathematical biology as a discipline. The contents of this thesis are merely the tip of the iceberg, but a very useful tip nonetheless. Chapter 6 Sources • Keener & Sneyd - Mathematical physiology, Volume I: Cellular Physiology • Vander - Human Physiology • Purves - Neuroscience • Cronin - Mathematical aspects of Hodgkin-Huxley neural theory • Abbott & Dayan - Theoretical neuroscience • Izhikevich - Dynamical systems in neuroscience • Gerstner - Spiking neuron models 37