The Biological Neuron

Transcription

1 Biological Signal Processing Richard B. Wells Chapter 4 The Biological Neuron. The Diversity of Voltage-Gated Channels Although the iportance of the original Hodgkin-Huxley odel can hardly be overstated, it is nonetheless a odel of the giant axon of the squid. With the benefit of hindsight, today it does not see very surprising that the whole neuron is a uch ore coplicated object. Experiental research carried out in the 960s eventually showed the two types of voltage-gated ion channels (VGCs) described by Hodgkin and Huxley were not the only types in existence. It was discovered that potassiu channels cae in any flavors. Sodiu channels, too, were found to be of ultiple types. Voltage-gated calciu channels were likewise discovered and rounded out the collection of ajor contributors to voltage-gated ionotropic current flows in the neuron cell. Even voltage-gated chloride channels have been docuented [JENT]. The axon has a relatively siple task, naely the propagation of an all-or-nothing action potential. The principal signal processing tasks are carried out by the soa and dendrites of the neuron. The fast, transient Na + channel and the slower, persistent K + channel first described by Hodgkin and Huxley are adequate for the forer, but inadequate to describe the latter. What is ipressive, however, is the range over which the Hodgkin-Huxley odeling schea has been extended to take into account the whole neuron. Early work in extending Hodgkin-Huxley to the whole neuron appeared in 97 in a series of papers by Connor and Stevens, leading to the Connor-Stevens odel of the gastropod soa [CONN]. Extension of the basic H-H schea is accoplished by identifying and characterizing VGC channel types and adding the in parallel to the other VGCs in the basic H-H circuit odel. As Connor and Stevens rearked in a later paper, The strategy in this study has been to ake justifiable changes in an existing syste, the Hodgkin-Huxley equations, rather than pursue a copeting analysis... We have chosen this foral and, we hope, ore general approach for the following three reasons. First, we are unable at this tie to justify a novel and adequate kinetic description of current carried by potassiu ion in the repetitive-firing preparation and hence have tried to reain as close as possible to an existing schee. Second, despite specific shortcoings of the Hodgkin-Huxley equations in fitting all of the available voltage clap data fro axons... they reain the coonly used forulation in which to cast voltage-clap data and for the basis of the standard conceptual fraework for interpreting coplex electrical events in excitable cells. Third, the considerable literature on the repetitive characteristics of the equations should serve as a useful basis for criticis of this or any other analysis of repetitive activity in excitable ebranes [CONN]. Tie and experience has given testiony to the soundness of the Connor-Stevens strategy and 66

2 thus far has vindicated their treatent of H-H as a odeling schea. Today we know of what has been called a "veritable zoo" of voltage-gated ion channels. Let us look at the denizens of this zoo.. K + Channels K + channels are found in all living cells and are a core feature of cellular life. In higher anials they are classified in three ajor groups identified by the specific "architectures" of ebrane-spanning proteins [HILL4: 35]. These are naed the TM, 4TM, and 6TM topologies, where the nuerical designation identifies the nuber of transebrane regions in the protein. The principal voltage-gated K + channels belong to the 6TM group. There are currently 4 recognized sub-failies of K + channels, of which 6 belong to the 6TM faily. Genetic analysis indicates that all 6 sub-failies trace back to a coon ancestral protein ore than.4 billion years ago. Nine of the sixteen sub-failies are called "delayed rectifiers" and the original K + channel in the Hodgkin-Huxley odel belongs to this group. At least two sub-failies in the 6TM group, the "slo" and "SK" sub-failies, are odulated by concentration levels of cytoplasic free Ca +. Soe of these (the "BK" group of the slo sub-faily) are also voltage-dependent. Their single-channel conductances, g p, range fro as low as 4 ps to as high as 50 ps [HILL4: 44]. The seven sub-failies in the TM group are called "inward rectifiers." These have the interesting property that they stop conducting under ebrane depolarization and increase conduction under hyperpolarization. This is the precise opposite of behavior we saw for the K + VGC in the squid axon. The ebrane voltage at which they open and begin conducting is a function of the extra-cellular K + concentration, generally increasing (getting closer to 0 volts) as [K + ] o increases. They are also characterized by at least two tie constants, one very fast (less than s) and the other ranging fro illiseconds to 0.5 seconds, depending on the particular channel protein. Because glial cells are thought to regulate the levels of [K + ] o here is one speculative echanis by which it ight be possible for glia to participate in biological signal processing. Soe (the Kir6 sub-faily, also called K ATP ) open and hyperpolarize the cell when the level of ATP (adenosine triphosphate, a olecule responsible for powering the etabolis of a cell) falls low. One could say this channel "shuts down" neurons that are low on energy. Soe K + channels, belonging to the Kir8 sub-faily, are regulated by etabotropic signaling processes and can be opened in response to horone or neuroodulator signaling. Exploration of the 4TM faily is not yet very far along. It has currently not been divided into sub-failies, although future research is expected by any to reveal a rich sub-faily tree siilar 67

3 to those of the other two failies. The 4TM channels that have been studied have not shown any strong voltage dependency and their presently known role sees to be that of "leakage" channels that help deterine the ebrane resting potential. Soe are known to be closed by etabotropic signaling processes, and these would appear to constitute another echanis by which the activity of a neuron can be odulated. The various K + channels play significant roles in odifying how a basic H-H-like circuit responds to different stiuli. For exaple, McCorick and Huguenard identified four distinct types of K + channels, one of which is a leakage channel, present in single thalaocortical relay neurons in the dorsal lateral geniculate nucleus (LGN) of rodents and cats [McCO]. McCorick has described soe of the effects these channels have on cell signaling properties in [McCO]. The list of known distinct types of K + channels is long and ipressive. We will not do an in-depth review of the specific channel types in this book. Extensive reviews are provided by Rudy [RUDY] and by Stor [STOR].. Na + Channels Na + VGCs are uch less diverse than the K + channels and belong to a different genetic "faily tree." They descend fro a 4TM protein "architecture." Na + channels are classified into 9 different channel types. Interestingly, the Na + channels appear to have branched off fro one line of Ca + channel proteins about 800 illion years ago [HILL4: 7]. They are classified according to aino acid sequences and bear the rather siple naes Na v. through Na v.9. The closer nuerically the last digit is to another channel, the ore siilar are their aino acid sequences. Functionally, Na + channels are divided into two groups, typically called Na (fast) and Na (slow) but also called Na t (for "transient") and Na p (for "persistent"). The forer is the class described by the H-H odel (that is, it is an "inactivating" channel). In the case of the latter, it is often not altogether clear if the channel is non-inactivating or if it is erely an inactivating channel with a very large inactivation tie constant. Functionally, this latter fine point sees to ake no known iportant difference insofar as biological signal processing is concerned since if it is "slowly inactivating" rather than non-inactivating, the neuron s signaling response sees the sae in either case. That is one reason for calling the "persistent" rather than inactivating vs. noninactivating. Transient ("fast") Na + channels typically have an activation threshold around about 50 V. They are the VGCs responsible for generation of the action potential. Persistent channels The LGN is part of the thalaus, which is the ain "switchboard" for sensory inforation en route to the neocortex fro the peripheral nervous syste. 68

4 typically have an activation threshold of around 65 V, and thus soe nuber of the ay be open already at the neuron s resting potential [JoWu: 08]. If these really are non-inactivating (as opposed to extreely slowly inactivating), they would constitute a for of "leakage" channel for the neuron at rest if the neuron s resting potential is V r > 65 V. Nonetheless, they are still VGCs. McCorick and Huguenard [McCO] described their persistent Na channel using an expression of the for I Nap = g ax (V E Na ). Unlike the H-H odel, was given as a direct function of the ebrane voltage and not fro a differential equation describing a rate process. Thus their does not have an interpretation in ters of gating kinetics. Rather, it is obtained based on experiental findings by French et al. [FREN]. French and his colleagues were able to deonstrate the existence of a voltage-activated and persistent Na + channel in rat hippocapus. The French odel is a curve-fit to easured data, and they were able to fit their results to a function of the for G Na = + exp g ax [( V V ) k] 50. (4.) An equation of the for of (4.) is called a Boltzann equation, naed after the Boltzann distribution function in classical statistical echanics. Here V 50 is the ebrane voltage at which 50% of the peak value of conductance occurs, V is the ebrane voltage, and k is a constant. They reported the best fit to their experiental data for V 50 = 50 V and k = 9 V. Their g ax averaged 7.8 ±. ns for the whole neuron and 4.4 ±.6 ns for dissociated cell bodies (neurons that had their dendrites reoved, leaving only the soa and soe dendrite "stubs"). This pair of findings led the to conclude the ajority of Na p channels were located on or near the soa. By estiating the cell s surface area fro easureents of its capacitance and assuing.0 µf/c, they estiated the specific g ax at.7 ± 0.6 ps/µ. 3 This copares to a easured specific axiu conductance for the transient Na + channels of 3 ±.6 ps/µ. Thus either the g p for the persistent channel is soe sixty ties saller than that of the transient channel pore (which sees unlikely), or the pore density is soe sixty ties less, or soe cobination of lower axiu conductance and lower density characterizes the persistent channel. The nature of the experiental technique used by French et al. did not perit the to characterize the gating kinetics of the persistent Na + channel because these kinetics were asked by the large-aplitude action potentials the neurons produced in response to stiulus. All that A negative current in their odel denotes current flow into the cytoplas. This has becoe a ore or less standard convention aong physiologists today. 3 A "specific" g ax is defined as g ax per unit area. 69

5 can be confidently concluded fro their experient is that the activation gate of the persistent channel opens no ore slowly than the inactivation gate of the transient channel closes. When g ax in (4.) is set to we obtain the variable used by McCorick and Huguenard. It still ranges fro 0 to, and can still be interpreted in ters of the fraction of available channels that are open. It does not, however, have an associated tie constant. This does not ean the physiological channel has no tie constant; it erely eans we do not know what it is. It is not inappropriate to ention here that soe channels not belonging to the Na + faily also conduct Na + currents along with K + currents. These are usually vaguely referred to as "cation channels." In addition, soe of the hyperpolarized K + slow inward-rectifier channels also conduct soe aount of Na +. As rearked upon in chapter, the "leakage circuit" part of a H-H-like circuit odel can hide soe rather interesting physiology..3 Ca + Channels Calciu ions are one of the ost potent etabotropic cheical "essengers" known. Free Ca + ions in the cytoplas are responsible for a vast range of effects, including long-lasting changes currently thought to be the biological basis for learning and eory. Free Ca + levels within the neuron are kept very low, on the order of 50 to 00 nm, by the action of various internal cell structures that capture and "warehouse" calciu, and by the action of calciu pups. In ost neurons a structure called the endoplasic reticulu (ER) serves as the priary "warehouse" for stored calciu. A typical [Ca + ] o concentration is on the order of about.0 M and so a Nernst potential for Ca + at 90 kelvin would range fro +4 to + 3 V. There are two ajor divisions of Ca + channels, called the high-voltage activated (HVA) and low-voltage activated (LVA) Ca channels. The HVA group is further divided into two branches, called the L branch and the "second branch." The L branch of the HVA group contains four subfailies of channels (called Ca v. through.4). These channels are slow, persistent ion channels. Their activation threshold is about 30 V. Although called "persistent" channels, they do have an inactivation range (about 60 to 0 V) but their inactivation tie constants are greater than 500 s [HILL4: 7]. They deactivate rapidly in the range fro 80 to 50 V and have a single channel conductance g p of about 5 ps. The second HVA branch contains three sub-failies (called Ca v. through.3, and also often called the P/Q, N, and R channels). These are transient channels with an activation threshold of about 0 V. Their inactivation range is fro 0 to 30 V with inactivation tie constants in the range fro 50 to 80 s. They deactivate slowly in the range fro 80 to 50 V and have a single channel conductance g p of about 3 ps [HILL4: 7]. Both branches of HVA channels 70

6 are found in presynaptic terinals, and they are responsible for triggering neurotransitter exocytosis in response to action potentials. Their activation thresholds account for the value of Ω stated in chapter 3 for the odel of synaptic LGC conductance. HVA channels have also been found outside the presynaptic terinal, although their role here is not certain (owing to their high activation threshold). The LVA channel branch contains three sub-failies, denoted Ca v 3. through 3.3. These are also often called T-channels. As the nae iplies, their activation threshold is low, about 70 V. They are transient channels with an inactivation range fro 00 to 60 V. Thus, for the typical neuron at its resting potential these channels are inactivated. Hyperpolarization rapidly deactivates the, and then their low activation threshold causes the to re-open briefly. One interesting consequence of this is a phenoenon called post-inhibitory rebound. In soe neurons, inhibitory synaptic signals are followed, after cessation of inhibition, by the generation of a single action potential spike. This phenoenon is thought to be due to the T-channels reactivating after deep hyperpolarization. A typical value for g p in these channels is about 8 ps [HILL4: 7]. Genetic analysis suggests both branches of Ca + channels ste fro a coon ancestral protein, probably about.8 billion years ago. In turn, this coon ancestor is thought to have descended fro an earlier 6TM ancestor ore than.4 billion years ago. It is possible that this coon ancestor ight also have been the ancestor of the 6TM line of K + VGCs [HILL4: 7].. Extending and Augenting the Hodgkin-Huxley Model The presence of so any different kinds of VGCs, each with differing gating kinetics and voltage dependencies, akes it necessary to extend the basic Hodgkin-Huxley odel. Furtherore, the presence of Ca + channels poses an additional consideration. While the battery potentials E Na and E K are not changed significantly by their respective ion flows (because the ion concentrations in the cytoplas and extracellular region do not change by a large enough aount to significantly affect the Nernst potential), the situation is quite different in the case of calciu. The noral concentration of cytoplasic free Ca + for the cell at rest is quite low, and so the influx of Ca + via calciu VGCs is sufficient to register an effect on E Ca. Physiologists usually account for this effect by replacing the relatively siple expressions for ion current flow we have been using with the Goldan-Hodgkin-Katz current equation (which will be presented later). Furtherore, the calciu-dependent faily of K + channels also has a conductance that explicitly depends of the concentration of cytoplasic free calciu. In order to properly odel both effects, the cytoplasic concentration of free Ca +, which we will denote 7

7 using the sybol [Ca] f, ust be explicitly accounted for. Thus, [Ca] f ust be ade an additional state variable in the odel of the neuron. This is called augenting the Hodgkin-Huxley odel.. Extending the Ion Channel Model We will begin with the ethod for extending a VGC odel for one particular ion, say K + or Na +, to account for the variety of ebrane-spanning proteins that conduct the overall ion current. It has previously been noted the total conductance of a channel is the su of the conductances of the individual open pores. If these pores have different gating kinetics and different voltage dependencies, we ust divide the total channel conductance, G(t), into a su of conductances for the different channel types. For exaple, suppose we have two distinct types of K + channel proteins. Since each would have the sae E K potential, we would write the total channel conductance as the su of two ters, i.e. G(t) = G (t) + G (t). This is illustrated by Figure 4. below. The circuit at the left in this figure is called a Thévenin equivalent of the circuit on the right. This equivalence ethod can be extended to include as any different species of ion channels for a particular ion as necessary. For exaple, McCorick s and Huguenard s thalaocortical relay neuron odel [McCO], [HUGU] contains two types of Na channels, four types of K channels, two Ca channels, a sodiu leakage channel, and a potassiu leakage channel for a total of ten specific channels.. Calciu Channels Levels of free calciu in the cytoplas are always extreely low. Typical concentrations of free Ca + in the cytoplas are in the range of 50 to 00 nm. ( M = ole per liter). This is six orders of agnitude less than the concentration level of free Na + in the cytoplas. This low level of free Ca + is caused and aintained by physiocheical processes in the cell for transporting and Figure 4.: Extending an ion channel into two coponents. 7

8 storing calciu and for returning Ca + to the extracellular region [HILL4: ]. The iediate consequence of this is that it is rather eaningless to try to assign a Nernst potential for calciu channels. Even if the ebrane potential V becoes positive, either due to action potential generation or because of voltage clap easureents, the cytoplas contains so few free Ca + ions relative to the extracellular region that no outward flow of calciu current results. Physically, a Nernst potential is the representation of a condition of therodynaic equilibriu, but calciu channel current flow is an inherently non-equilibriu therodynaics phenoenon. For this reason, and for the practical reason that attepts to describe Ca + currents in the sae way Na + and K + are described do not work, odels of calciu VGCs are based on another theoretical approach, called constant-field theory, originally developed in 939 by Nevill Mott for describing electron conduction in a copper/copper oxide rectifier. D.E. Goldan introduced the ethod to biology in 943 [GOLD], and the theory was further polished up and developed by Hodgkin and Katz in 949 [HODG7]. The result is the Goldan-Hodgkin-Katz (GHK) current equation. The GHK current equation 4 odels the current density (current per unit area) flowing through a pereable ebrane as a function of ebrane voltage and ion concentrations. We will only consider Ca + currents here, but the GHK current equation applies equally well to other kinds of ion flux. Flux density is defined as the nuber of particles flowing per unit area of cross section per unit tie for a group of particles oving in an organized anner. (The integral of flux density taken across the entire surface area is called the particle flux). When the nuber of particles is expressed in oles ( ol = Avogadro s nuber of particles), the flux density is called the olar flux density (which is usually expressed in units of ol/c s). The pereability constant P of the ebrane (often just called its "pereability") is defined to be the ratio of the olar flux density to the difference in ion concentrations on the two sides of the ebrane. When the concentrations are expressed in units of oles per c 3 (oles per illiliter) and the olar flux density is expressed as before, the pereability constant has the units of c/s. Forally, P is erely a atheatical quantity eant to convey in soe sense how "easy" or "hard" it is for an ion to penetrate through the ebrane. At our present state of knowledge we cannot derive P fro ore fundaental considerations, and its quantity is experientally deterined. In aking this deterination it is not unusual for the easured quantity to actually be the product of this atheatical pereability constant and the surface area of the cell, p = PA. P is then deterined by dividing out the surface area. 4 There is also a GHK voltage equation. It is often used by theoretical biologists, but we will have little need of it in this text. 73

9 Ared with these ideas, the GHK current equation is derived fro the Nernst-Planck equation for electrodiffusion, which describes ion otion [HILL4: 3-39]. The final result can be written 5 as th + + [ Ca ] exp( V V ) [ Ca ] zf i th o ighk = s P V (4.) V exp ( V V ) th where V th = kt/ze is called the theral equivalent voltage, V is the ebrane potential (relative to the extracellular region), F = coulob/ole is Faraday s constant, z = is the valence of Ca +, and i GHK is the current density. s is a scale factor that depends on the units used to express the concentrations and the pereability. When P is in c/s and the concentrations are expressed in oles per illiliter, s = and i GHK has units of aperes/c. If the concentrations are expressed in nm (0-9 oles/liter) and s is set to 000, then i GHK is in na/c. To obtain the Ca + channel currents for the whole cell, we factor in the gating dynaics of the voltage-dependent calciu pores and either ultiply i GHK by the surface area A (in c ) or, equivalently, replace P by p = PA in (4.). This gives us I Ca ( V ) = j Ca h q Ca th + + [ Ca ] exp( V V ) [ Ca ] zf i th o s pax V (4.3) V exp ( V V ) where Ca and h Ca are the activation and inactivation gate factors. A typical epirical value for L-type Ca + and T-type Ca + channels is j =. p ax is called the axiu pereability factor and has units of c 3 /s. If the cell s internal echaniss for clearing out free Ca + (called "calciu buffering") are efficient, then q = 0 and the channel is non-inactivating 6. As an exaple of the activation gate kinetics, McCorick and Huguenard used the following expressions for the highvoltage L-type channel in thalaocortical relay neurons: th Ca α = α + β.6 α = (4.4) + exp β = exp ( 0.07 ( 5.0)) V 0.0 ( V.3) [( V.3) 5.36] with V expressed in V. As always, these three quantities are diensionless. 5 Biologists and cheists are accustoed to seeing the GHK current equation in a different for fro this. To obtain this expression we ake use of the identity R/F = k/e, where k is the Boltzann constant. 6 It is known that for calciu channels h is a function of [Ca + ] i. If soething interferes with the calciu buffering echaniss, allowing a large rise in free cytoplasic calciu, the channel is inactivating. 74

10 Biologists usually do not express the calciu channel in ters of a conductance, although by noting that (4.3) is in the for of soething ties the ebrane voltage, we are perfectly free to regard this expression as a conductance, i.e., G GHK + ( V, Ca ) = j Ca h q Ca s p ax zf V th + + [ Ca ] exp( V V ) [ Ca ] i exp( V V ) th th o. (4.5) Doing this is equivalent to saying the calciu battery potential is E Ca = 0. This G GHK is, however, a rather exotic conductance. First we note that it is calciu dependent as well as voltage dependent. The extracellular calciu concentration can be regarded as constant (typically with a value of about M), but the free cytoplasic calciu concentration is not. Therefore, in siulations this expression ust be augented by a odel coponent describing the concentration of free [Ca + ] i. Second, note that for V = 0, the denoinator of (4.5) goes to zero, eaning G GHK is infinite (like a "short circuit"). However, (4.3) reains finite because li 0 V V exp th V th ( V V ) and so I Ca reains finite and well-defined. For V < 0, G GHK is positive (because the extracellular calciu concentration is so uch larger than the free cytoplasic calciu concentration). But for V > 0 there is a range of ebrane voltages for which G GHK is negative. Electrical engineers are used to dealing with negative resistances and negative conductances they are a coonplace occurrence in feedback circuits containing active gain eleents such as transistors but the occurrence of negative conductance in a biological odel severely clouds the issue of aking any sort of interpretation of the physical significance of G GHK. Certainly we lose the cofortable ental picture of channel conductance as the siple conductance of a pore in a ebrane. What one should bear in ind is that constant-field theory is at best an approxiate theory and akes a nuber of siplifying assuptions that do not stand up under close physiological scrutiny. As Bertil Hill has rearked, The GHK theory is a superb tool for reporting results, but is less useful as a guide to the physical structure of channels. Two quite different concepts of solubility-diffusion theory the partition coefficient and the obility are blended into one pereability paraeter. The channel is assued to be hoogeneous. The ions are assued not to interact either physically or electrostatically. But these assuptions are wrong, so the predictions cannot be right in detail. Indeed, the deviations fro GHK theory... have stiulated the ajor advances since 970 [HILL4: 449]. If we bear this in ind and understand that G GHK is a atheatical creature, we can go ahead and use (4.5), understanding we are not to read into it things that are siply not there. 75

11 0 L-current vs Mebrane Voltage 0.5 Effective L-channel Conductance L-current (na) Ica j Conductance (us) G j V j V j Mebrane Voltage (V) Mebrane Voltage (V) Figure 4.: I Ca and G GHK steady-state curves with [Ca + ] i held constant for an L-type Ca + channel. Although there is no particular physiological value to doing so, it is instructive to plot (4.3) and (4.5) as functions of ebrane voltage with [Ca + ] i held constant and with =. This is done in Figure 4. using p ax = c 3 /s, [Ca + ] i = 50 nm, [Ca + ] o = M, T = 97 kelvin, j =, and q = 0. The McCorick-Huguenard expressions (4.4) were used to calculate. (4.3) was scaled to give I Ca in na, and V and V th are in V. G GHK is in µs. Our first observation is that I Ca is negative (denoting current flowing into the cytoplas) for all values of V. The peak occurs just slightly above V = 0, and the half-axiu current first occurs at around V = 5 V. This illustrates why the L-type Ca + channel is called a highvoltage-activated (HVA) channel. G GHK is positive for all V < 0. It has a singularity at V = 0 and is negative throughout the range of V > 0 shown in the plot. These plots are instructive for the purpose of gaining a qualitative "picture" of the properties of the calciu channel, although the artificial constraint of keeping the calciu concentrations constant prevents us fro aking serious quantitative judgents of how the channel will behave in siulation. Their usefulness is akin to the steady-state activation variable and tie constant plots in chapter 3 for the Hodgkin- Huxley odel. Note, too, the singularity at V = 0 is a feature of G GHK that is independent of calciu concentrations for any physiological nuerical values for these concentrations. LVA calciu channels are also usually odeled using the GHK current equation ethod, although there are exceptions to this in the odeling literature. The two principal differences between LVA curves and HVA curves are these: () the activation voltage for the LVA channel shifts draatically to the left, fro about 5 V to around 40 V; () the LVA channel is a rapidly inactivating channel (q = ) and requires Hodgkin-Huxley-like expressions for the voltage-dependent inactivation factor h. This is often in the for of a Boltzann equation (V 50 on the order of about 80 V, the Boltzann equation k factor on the order of about 4 V). It is a peculiarity of the LVA T-type channel that its tie constant function is often biphasic, i.e, the curve fit equation for V < V 50 and that for V > V 50 are often different. An exaple of this is 76

12 provided by [HUGU]..3 The Linvill Modeling Schea Models of cytoplasic Ca + concentration in neurons are usually fairly siple, direct, and to the point. Nonetheless, there is soething to be said in favor of having a general and versatile odeling schea for particle accuulation, transport, cheical reaction, and storage in the cell. A general schea is an aid for transforing qualitative odels into quantitative ones and for developing relationships descriptive of ore coplex physiological processes. After all, if the Hodgkin-Huxley schea is useful in part because it provides a guide for dealing with the coplexities of VGC and LGC signaling, is it not also likely that a siilar schea ight prove useful for dealing with cellular biocheical signaling processes? In this text we resurrect and adapt to our purposes a odeling schea originally developed in 958 by John G. Linvill [LINV: 7-48] and extended to application in neuroscience by his forer student, Wells, in 007. The Linvill odel was developed to represent carrier transport, storage, and generation-recobination phenoena in seiconductors in ters of carrier densities and current flows. The odel had never previously been applied to biological signaling processing odels. After all, the neuron is not a transistor or a diode. Nonetheless, with only a few inor adaptations of the Linvill odel, we can apply its as a schea for representing ion flux and concentration in the cell. This section introduces the basic odeling eleents and their atheatical description. The next section applies it to the relatively siple task of odeling cytoplasic free Ca + concentration. The Linvill odeling schea allows us to represent the odel as a network of eleents, each of which is characterized by a specific eleent law. Figure 4.3 illustrates the five basic Linvill network eleents. 7 Ion or olecule concentration is denoted by the sybol ν. Ion or olecule flux is denoted by the sybol φ. Flux is positive in the direction denoted by the arrows. Figure 4.3: The five basic network eleents of the Linvill odeling schea. 7 The notations and variables used here are odified fro Linvill s original for. Linvill was concerned with odeling charge densities and currents, whereas we have broader and ore diverse needs. 77

13 Particle (ion or olecule) otion in space is forally describable in ters of partial differential equations with boundary conditions. While such a description is atheatically rigorous and precise, it is also rather cubersoe and coputationally difficult to deal with in all but the siplest cases of physical geoetries. This issue is often overcoe in the various sciences through the introduction of the luped eleent approxiation. For exaple, electric and agnetic phenoena are rigorously described by Maxwell s equations, a set of partial differential equations. Circuit theory is the luped eleent approxiation of these equations, applicable when electroagnetic radiation (a consequence of Maxwell s equations) is not an iportant factor. (Neglect of radiation effects is called the quasi-static approxiation). Siilarly, the Linvill odel is a luped eleent approxiation to the partial differential equation description of particle transport, storage, and accuulation. The storage eleent ("storance") represents the accuulation of particles in a region due to influx fro soe other location or source. Its ν-φ relationship is based on the atheatical for of the divergence theore which, in nontechnical language, erely states, "What goes in ust coe out or else reain inside." The eleent law erely states that the net influx of particles into the storage eleent is proportional to the tie rate of change of concentration, dν C = C & ν = φ (4.6) dt Diensional analysis shows C has units of volue. C reflects the fact that a region of greater volue builds up particle concentration less rapidly than a region of saller volue for the sae aount of flux. We ay at once note the siilarity between (4.6) and the eleent law for a capacitor in circuit theory. φ is analogous to electric current and ν is analogous to voltage. There are, however, liits to this electric circuit analogy. The accuulation of ν in a storage eleent does not induce the accuulation of soe other particle elsewhere (no equivalent to charge on one plate of a capacitor inducing the opposite charge on the other plate), and there is nothing analogous to the "displaceent current" through a capacitor which is a consequence of the Maxwell equations. Likewise, there is no "Kirchhoff s voltage law" for this network. Particle flux is not required to flow in a closed circuit path (hence the odel is a "network" odel rather than a "circuit" odel). Particle conservation, however, is required for the first four network eleents because only the reactance eleent represents a cheical reaction, e.g. a + b c. The flux source eleent represents influx/efflux fro soe exterior source. Our ost coon use for this eleent will be to convert electric currents obtained fro the H-H odel to particle flux. The current I due to a flux φ of particles with valence z is siply I = zeφ when φ is 78

14 expressed in particles per second. It is ore convenient, however, to express flux in units such as oles/second. In this case we would write I = zfφ, where F is Faraday s constant. (If we had a current density aperes per unit area we would replace flux by flux density). Letting β = /zf we obtain the eleent law as φ = β I with I in aperes and β in oles/coulob. For z =, β is nuerically equal to oles/coulob; it is one-half this value if z =. Note that the sign of the valence is irrelevant to flux (since this is erely the sign of the electric charge carried). Therefore if one is odeling the flux of a negative-valence particle, the absolute value of z would be used. The transporter sybol is our generalization of an eleent Linvill called a "cobinance" [LINV: 5]. In his original odel this eleent was used to odel charge generation and recobination in seiconductors. That is a process by which bound charge is converted to free charge and vice versa. The analog to this within the cell (not including cheical reactions that produce new copounds) are the processes by which free particles are introduced or reoved by various transport processes, such as the transport process by which free Ca + is reoved fro the cytoplas and stored in the endoplasic reticulu [HILL4: 69-73]. In effect, these are puping processes in which the average flux is deterined by the aount of particle transport per cycle of the pup and the concentration of the particle being transported. Thus, the siplest for of an eleent law for the transporter is the relationship φ = ρ ν (4.7) where ρ is an epirically-deterined transport paraeter with units of volue/second. Note that this expression is unidirectional. The concentration variable in (4.7) is placed at the "boxed" end of the network eleent, and whatever the concentration ay be at the other terinal is irrelevant. The transporter does not represent a passive diffusion process. That dynaic is odeled by the fourth network eleent. The diffusion eleent ("diffusance") represents particle flux due to concentration differences. Flux is positive in the direction fro higher concentration to lower concentration. In its siplest for, the diffuser eleent law is erely ( ν ν ) φ = D (4.8) where D is an epirically-deterined paraeter with units of volue/second. If C, ρ, and D are represented by siple constants we have a linear odel of transport and storage. Making these variables functions of ν, and possibly other variables, produces a nonlinear odel. At our present state of knowledge of neural physiology no nonlinear odel of the 79

15 neuron s internal processes is in widespread use and, consequently, ost neuron odeling work has used a linear odel of the cytoplasic Ca + transport and storage processes. Using the first four eleents we can construct network odels of arbitrary coplexity to represent various transport and storage processes excepting those involving cheical reactions (for which we need to represent what happens to the reaction copounds afterward). Cheical reactions are odeled by the reactance eleent, which is discussed at the end of this chapter. If it happens to be the case where we do not care what happens later to these reaction products, a transporter eleent (called a reactor in this case) cobined with a storage eleent can suffice for representing the introduction or the reoval of free particles. The next section illustrates the application of this odeling schea to Ca + augentation of the basic H-H odel..4 Siple Models of Ca + Processes in the Neuron The siplest and ost coon odel of Ca + buffering in the neuron represents the gross influx of Ca + fro calciu channels and the reoval of free Ca + fro the cytoplas by a transport process. Figure 4.4 illustrates the odel network. The endoplasic reticulu is regarded as having infinite volue and so the tie rate of change of [Ca + ] at this node is zero. I Ca is obtained fro the electrical odel of the neuron and its nuerical value is negative or zero. Thus, the network of Figure 4.4 receives an influx of Ca + into node ν. Suing the effluxes fro this node, we obtain the dynaical equation dν C = ρ ν β I dt Ca () t. (4.9) We convert (4.9) to difference equation for by the sae ethod used previously. Figure 4.4: Siple odel of free Ca + buffering. I Ca is obtained fro the electrical odel of the neuron. Because its nuerical value is negative, the Ca network receives a Ca + influx. 80

16 Figure 4.5: Two-copartent odel of a Ca + network. Most calciu buffering odels incorporate a constraint that the iniu level of [Ca + ] i is not allowed to fall below soe iniu value [Ca + ] in. This is easily incorporated into the network of Figure 4.4 by adding a phenoenological "calciu leakage flux" in parallel with the calciu flux obtained fro the electrical odel. Setting the derivative in (4.9) to zero, we obtain for this leakage flux the nuerical value β I leak = ρ [Ca + ] in. The direction of the arrow for this flux source is the sae as that of the source shown in Figure 4.4. McCorick and Huguenard [McCO] argued that the literature on neuron physiology suggested HVA calciu channels, I Ca(L), and LVA T-current channels, I Ca(T), are probably located in different regions of the neuron. They used this arguent to justify aking their Ca + -dependent K + VGC eleent depend on the contribution to [Ca + ] fro the L-current only. In their odel they kept track of the individual contributions fro I Ca(L) and I Ca(T) while still letting their GHK current odel depend on the su of the two coponents. In effect, their odel is soething like a two-copartent odel of the calciu network, although not entirely rigorous in its forulation. A ore foral representation of two-copartent odeling is illustrated in Figure 4.5. The schea is easily extended for representing any nuber of calciu copartents. Suing the effluxes fro each node and rearranging ters we obtain a syste of two differential equations, dν dν dt = dt ( ρ + D) D C C D C ( ρ + D) ν β C + C ν 0 0 I β C I Ca Ca. (4.0) Using Euler s ethod, the corresponding difference equation representation is ν ν ( t + t) ( t + t) = t ( ρ + D) t D C C t t D C ( ρ + D) ν C ν ( t) () t t β C t β I C I Ca Ca () t (). t 8

17 Figure 4.6: Siplified odel of etabotropic-signal-induced calciu release. As a final exaple we will consider a etabotropic signaling process in which internal Ca + stores are released fro the endoplasic reticulu to becoe free Ca + in the cytoplas. Figure 4.6 illustrates the odel. This odel is intended to illustrate the general ideas conveyed by the qualitative odels of this process. It should be noted that our present state of knowledge of the quantitative details of this process is incoplete and so the odel presented here is to be regarded as conceptual but not an established accurate representation of this process. The odel contains two types of cheical concentrations, [Ca + ] i represented on the left by node variable Ca, and cytoplasic concentration of inositol triphosphate (IP 3 ) represented by node variable IP3 on the right. Ca represents the concentration of Ca + stored in the endoplasic reticulu (ER). Typical concentrations of stored Ca + is typically greater than 00 µm under noral physiological conditions and likely reaches illiolar levels. The ER s supply of Ca + is by no eans unliited, but for noral signal processing functions we can regard the volue C ER as effectively infinite so that concentration Ca ay be regarded as constant. We will also assue flux source β I Ca includes a "leakage flux" that aintains the iniu level of Ca at its resting concentration level (on the order of 50 to 00 nm). Transporter ρ is the sae as described previously. We will coe back to transporter ρ oentarily. In the absence of synaptic etabotropic signaling, IP 3 is norally bound in the cytoplasic ebrane wall. It is liberated by the action of a ebrane-spanning G-protein that acts as a receptor for etabotropic neurotransitters. A G-protein receptor does not itself open to produce a pore for ionotropic current influx into the cytoplas. Rather, it acts as a "olecular switch" to turn on the production of "second essenger" cheicals, IP 3 in this case. Flux source β IP3 odels the generation of free IP 3 by this echanis. The units of β IP3 are flux (oles per second). Free IP 3 oves to the ER and binds with Ca-release channels in the ebrane of the ER. As IP 3 again becoes bound by this process, it depletes the pool of free IP 3 and we can represent the 8

18 rate of this depletion using transporter r IP3. Thus, the concentration IP3 is described by 8 () t dip3 C = rip3 IP3( t) + β IP3. (4.) dt Transporter ρ (IP3) represents the influx of free Ca + due to the opening of the calciurelease channels in the ER. The kinetics of this process are coplicated [MORA], but we can pose a few likely-seeing approxiations. First, since the Ca + influx is zero in the absence of IP 3, the siplest plausible odel for this effect is to presue the influx is proportional to IP3(t). This assuption is analogous to that used in odeling calciu buffering in (4.9). Second, it is known that the opening probability, π o, of the ER s calciu-release channels is strongly affected by the concentration level of cytoplasic free calciu, Ca [BEZP], [FINC], [MORA]. Bezprozvanny et al. report a bell-shaped curve function for release probability vs. [Ca + ] i that reaches a peak of π o = at around Ca = 0. µm. The bell shape of the π o dependency shows up on a logarith plot of [Ca + ] i, i.e. the π o = 0.5 points on the curve occur at approxiately µm and 0.55 µm [BEZP]. Bezprozvanny et al. report the fitted dependency as s n n n n n ( Ca + k ) ( Ca + k ) n Ca k π = π (4.) o where π s is a scale factor chosen to ake π o equal to at Ca = 0. µm, n =.8, k = k = 0. µm. Figure 4.7 graphs (4.) as a function of calciu concentration Ca. Taking these factors into account, we would write the transporter eleent law as ( Ca ) ρax IP3( t) φ = Ca ρ = ρ. π o ; One noteworthy property of this syste is the following. For concentrations Ca less than about 0. µm, π o is an increasing function of Ca, and thus there is a positive-feedback effect taking place inducing a strong rise in free cytoplasic calciu. Above this level, π o is a decreasing function of Ca, and so the total rise in free calciu is self-inhibited by the kinetics of the release probability. This has been known to produce oscillations in the concentration levels of free [Ca + ] i in response to etabotropic signaling. This results in calciu-ediated odulation of ionotropic potassiu currents [HILL5]. The oscillations are spike-like and very slow, with period 8 There are other dynaics we are not representing accurately in this siplified odel. For exaple, IP 3 does not reain indefinitely bound to the calciu-release channels, and this odel depicts neither how long IP 3 reains bound (thus activating the channel) nor what happens to it later. An understanding of these additional dynaics is necessary for a coplete and accurate odel of this process [MORA]. The exaple given here is intended to erely illustrate an application of the Linvill odel. 83

19 Open Probability π j Ca j [Ca] (M) Figure 4.7: Epirical curve fit to calciu release channel open probability. on the order of to 0 seconds per spike for the case where IP 3 is produced in response to etabotropic action by the neuropeptide GnRH (gonadotropin-releasing horone). Calciu spikes with peaks in the range of about.5 to.5 µm have been observed. Suing effluxes fro the Ca node and incorporating (4.) gives us the syste of first order differential equations dca dt ρ = dip3 dt 0 C r 0 IP3 Ca + ρ C C IP3 0 β C 0 0 C Ca I Ca β IP3. (4.3) Although the equations in (4.3) appear to be uncoupled, in fact they are not. IP3 couples into the first equation through ρ. An additional consideration not incorporated into this siplified odel would account for loss of free calciu through binding with the calciu-releasing channels depicted by ρ. The kinetics odel of Moraru et al. [MORA] assues two Ca + ions and four IP 3 olecules participate in each binding event at the receptor site for the calciu-releasing channels. This, however, could be taken into account in an approxiate fashion by the nuerical value assigned to ρ. More iportant is the absence in the siplified odel of a tie-dependent rate process description for transporters ρ and r IP3. In the siplest case we would have at least one additional differential equation, possibly siilar to a rate process odel such as is used in the Hodgkin-Huxley schea, capturing the opening- and closing-kinetics of the calciu releasing channels. Such a process would affect the tie-dependencies of both ρ and r IP3. Judging fro the findings reported in 84

20 [MORA], tie constants for such a process would be on the order of a few illiseconds. As a final note, the odel just presented is a very siplified representation of these dynaics. Much ore elegant treatents based on diffusion theory and partial differential equation odels have been forulated [DeSC]. A luped odel approxiation for this type of sophisticated odel would require the odel of Figure 4.6 to be turned into a ulti-copartent odel..5 Calciu-activated Potassiu Channels An iportant class of channels not yet discussed is the calciu-activated K + voltage-gated channel. There are at least two iportant signaling sub-classes of Ca + activated K + channels, which we will denote by currents I K(Ca) and I K(AHP). The latter is further subdivided into the categories "interediate channel" (IK) and "sall channel" (SK). The first type, correspondingly, is often called the "big channel" (BK). Epirical odels for the various denizens of the "zoo" of K + channels are often expressed in ters of the steady-state activation variable,, and a tie constant, τ. Assuing, as did Hodgkin and Huxley, that channel activation follows a first-order rate process, the activation variable is described by the difference equation ( t t) = ( ( t) ) ( t τ ) + exp. (4.4) is typically described either in ters of Hodgkin-Huxley rate paraeters, α and β, or else in ters of a Boltzann equation = + exp [( V V ) k] 50. (4.5) While the BK class of channels is both voltage- and calciu-dependent, the IK and SK classes are not strongly voltage dependent and therefore do not follow the for (4.5). Their priary effect is to produce a long-lasting hyperpolarization (the "after-hyperpolarization") following intense spiking activity by the neuron, but they also odulate the resting potential and excitability of the neuron in response to IP 3 -producing etabotropic signaling. Yaada et al. [YAMA] presented a BK odel derived fro the sypathetic ganglion B-type cell of the bullfrog. (McCorick and Huguenard also used this sae odel in [McCO]). The current equation for their odel is I K ( Ca) = g ax ( V EK ) (4.6) with the auxiliary equations 85

21 Activation Constants vs [Ca], V 0 Tie Constants vs [Ca], V 00 j 0.75 τ00 j 7.5 k j 0k j 0.5 τk j τ0k j V j V j Mebrane Voltage (V) Mebrane Voltage (V) 00 nm Ca um Ca 0 um Ca 00 nm Ca um Ca 0 um Ca Figure 4.8: Steady-state activation constants and tie constants for I K(ca) vs. calciu concentration. d = dt τ = τ τ = 50 + [ Ca ] i exp( V 4) + 0. exp( V 4) + [ Ca ] exp( V 4) 50 i. (4.7) In these expressions, [Ca + ] i is in M and V is in V. The tie constant is in s. Yaada et al. gave g ax as. µs. Figure 4.8 illustrates the dependency of the steady-state activation constant,, and tie constant, τ, on [Ca + ] i as a function of ebrane potential. At noral resting levels of Ca + (around 00 nm) the calciu-dependent K channel is effectively closed at the resting potential (around 65 to 60 V) and the channel tie constant during action potential generation is slow in coparison with its values for elevated Ca + levels. Increasing the cell s internal concentration of Ca + shifts the activation curve to ore negative ebrane potentials, and the axiu tie constant falls draatically. An I K(AHP) odel was also presented in [YAMA]. This after-hyperpolarization current (that is, the after-hyperpolarization that occurs following the generation of an action potential) is not a strong function of ebrane potential. Rather, it is a potassiu channel current that depends on the internal concentration of Ca +. Yaada et al. give the current equation as I K ax ) ( AHP) = g ( V EK (4.8) with a axiu channel conductance g ax of µs. The differential equation for is the sae as for the Ca-dependent BK current channel but the expressions for the calciu dependency are different and are given by 86