Electrophysiological Models of Neural Processing Mark E
Focus Article Electrophysiological models of neural processing Mark E. Nelson∗
The brain is an amazing information processing system that allows organisms to adaptively monitor and control complex dynamic interactions with their environment across multiple spatial and temporal scales. Mathematical modeling and computer simulation techniques have become essential tools in understanding diverse aspects of neural processing ranging from sub-millisecond temporal coding in the sound localization circuity of barn owls to long-term memory storage and retrieval in humans that can span decades. The processing capabilities of individual neurons lie at the core of these models, with the emphasis shifting upward and downward across different levels of biological organization depending on the nature of the questions being addressed. This review provides an introduction to the techniques for constructing biophysically based models of individual neurons and local networks. Topics include Hodgkin-Huxley-type models of macroscopic membrane currents, Markov models of individual ion- channel currents, compartmental models of neuronal morphology, and network models involving synaptic interactions among multiple neurons. 2010 JohnWiley& Sons, Inc. WIREs Syst Biol Med 2011 3 74–92 DOI: 10.1002/wsbm.95
INTRODUCTION of emerging neuroinformatics approaches that can potentially link such models with a wealth of empirical nderstanding the electrophysiological basis of data currently being compiled and organized into Uneural coding, communication, and information large neuroscientific databases.3–5 In contrast, highly processing is central to modern neuroscience research. abstracted models lack sufficient biological detail Mathematical modeling and computer simulation to establish meaningful links to these databases, have become an integral part of the neuroscientist’s whereas large systems-level models involving multiple toolbox for exploring these phenomena at a variety brain regions are generally too diverse in structure of levels of organization, from the biophysical basis and function for neuroinformatics approaches to be of current flow through individual ion channels, to productive. In the intermediated term over the next the modeling of aspects of cognitive function arising several years, biophysically detailed models of single from the distributed activity of large populations of neurons and local networks will likely provide the neurons. Neural models can be constructed at many most fruitful level of analysis for uncovering new levels of abstraction. Some types of scientific questions functional relationships, dynamical principles, and can be addressed using highly reduced models that information-processing strategies. treat neurons as simple threshold devices, whereas How do electrophysiological models fit into an other questions require detailed models of membrane informatics approach to neuroscience? This question biophysics and intracellular signaling networks. is perhaps best answered in the context of a bottom- This article will focus on the tools and techniques up view of the problem. Starting at the molecular for constructing biophysically detailed compartmental level, sequence-based informatics approaches are models of individual neurons and local networks.1,2 being used to reveal information about structural and Such models are well positioned to take advantage evolutionary relationships among ion channels and ∗Correspondence to: [email protected] receptor proteins. Molecular dynamics simulations Department of Molecular and Integrative Physiology and The can help establish links from the structural level to Beckman Institute for Advanced Science and Technology, University the functional properties of individual ion channel of Illinois at Urbana-Champaign, Urbana, IL, USA and receptor complexes. Electrophysiological models DOI: 10.1002/wsbm.95 come into play at the next level of organization where
74 2010 John Wiley & Sons, Inc. Volume 3, January/February 2011 WIREs Systems Biology and Medicine Electrophysiological models information-processing properties emerge from the (e.g., Ref 15) and the biophysical properties of ion dynamic interactions of multiple channel and receptor channels (e.g., Ref 16). A brief glossary is provided types at the single-neuron level and interactions of here as a convenient reference for some of the key multiple neurons at the network level. terminology and functional concepts. Neurons typically contain numerous types of ion channels and membrane receptors. Different types of Glossary neurons express different combinations of these pro- teins, with varying densities and varying spatial dis- action potential A transient electrical impulse tributions. There are, for example, dozens of different + that propagates along an axon and serves as the types of voltage-gated K channels, but an individual most common form of electrical signaling between neuron may only express a few of these, and the neurons. The duration is typically on the order of expression might be restricted to the soma or to partic- 6,7 + a millisecond, and the amplitude is on the order ular regions of the dendrites. Different types of K of 100 mV. Action potentials are also called nerve channels vary in their electrophysiological properties, impulses or spikes. such as activation and inactivation voltages, time con- stants, and conductances. This heterogeneity suggests that K+ channels may be differentially expressed and axon An output branch of a neuron that conducts distributed in order to shape the electrophysiological action potentials away from the site of initiation response properties of individual neurons for carrying and conveys electrical signals to other neurons out particular types of information-processing tasks. or effectors. The axon usually starts off as a The contributions of different ion channels to the single long branch but may terminate in a complex information-processing capabilities of the system can- branched arbor that distributes outputs to large not be deduced from the properties of individual ion numbers of target neurons. channels alone. Rather, functional properties at the single-neuron level must be evaluated in the presence compartmental model A single-neuron model that of an appropriate mix of channel types, densities, and divides the cell into multiple spatial compartments. distributions and in the context of physiologically rel- Each compartment can have different properties evant spatiotemporal patterns of input. For example, + (length, diameter, membrane voltage, ion channel certain types of K channels from the Kv3 gene family densities, etc.). The model produces a coupled are known to be activated only at rather depolarized set of differential equations that are solved using membrane potentials and tend to have fast activation numerical integration techniques. and inactivation time constants.8 Using biophysically detailed compartmental models, neuroscientists have been able to achieve a detailed understanding of how conductance A measure of the ease with which Kv3 channel properties contribute to temporal signal electric current flows through a material; the processing in the electric sense of weakly electric reciprocal of resistance; conductance units are fish9–11 and in the mammalian auditory system.12 siemens; conductance (siemens) is a measure of If appropriate databases were available and suitable current (amperes) divided by voltage (volts). neuroinformatics tools existed, one could imagine undertaking a variety of interesting comparative dendrite An input branch of a neuron that typically investigations regarding the functional role of Kv3 receives synaptic contacts from other neurons and channels in other species, other sensory systems, and conveys graded electrical potentials to other parts other neural information-processing contexts. of the neuron.
equilibrium potential The membrane potential BACKGROUND at which the effects of the electrical potential This article assumes a general familiarity with the difference and the concentration gradient across neurophysiological and biophysical mechanisms the membrane are balanced so as to produce associated with electrical signaling in neurons. Good no net ion flux. Also called the Nernst poten- introductory material in this area can be found in tial. Calculated from the Nernst equation: Eion = numerous undergraduate textbooks (e.g., Refs 13 (RT/zF) ln([ion]out/[ion]in), where R is the univer- and 14). Recent advanced texts are available that sal gas constant, T the temperature in Kelvin, z provide detailed, up-to-date coverage in areas such the ionic charge, F Faraday’s constant, and [ion]in as the cellular and molecular biology of nerve cells and [ion]out are ionic concentrations.
Volume 3, January/February 2011 2010 John Wiley & Sons, Inc. 75 Focus Article www.wiley.com/wires/sysbio gating The process by which ion channels open and soma The cell body of the neuron; contains the close so as to regulate the flow of ions. Voltage- nucleus and much of the metabolic machinery of gated ion channels change their gating state the cell. based on the local electrical potential difference across the cell membrane. Ligand-gated channels reversal potential The membrane potential at change their gating state based on the binding which no net current flows through an open ion of signaling molecules (neurotransmitters). Gating channel or activated synapse. If the channel or usually involves a change in the conformation of synapse is permeable to a single ionic species, then the channel protein. the reversal potential is equal to the equilibrium potential for that ion. Otherwise, the reversal Hodgkin–Huxley model A model of the ionic potential reflects a weighted sum of the equilib- basis of the action potential in squid giant axon rium potentials for all permeant ionic species. developed by Hodgkin and Huxley and published in 1952. More generally, this term can refer to any voltage clamp A technique for recording the ionic model that uses a Hodgkin–Huxley (HH)-type currents across the cell membrane during con- mathematical formalism to describe macroscopic trolled changes in the membrane potential. Fast ionic currents in nerve cells based on voltage- feedback circuitry is used to maintain the mem- dependent gating properties. brane potential at the desired ‘command’ level. macroscopic conductance The electrical conduc- TECHNICAL DETAILS AND tance arising from a population of single-channel METHODOLOGY conductances; often quoted in terms of con- ductance per unit area of membrane; typically HH Models in the range of millisiemens (mS) per square The core mathematical framework for modern centimeter. biophysically based neural modeling was developed half a century ago by Sir Alan Hodgkin and Sir Markov model A formalism for modeling stochas- Andrew Huxley. They carried out an elegant series tic processes which treats the behavior of a system of electrophysiological experiments on the squid giant as a series of transitions between distinct states. In axon in the late 1940s and early 1950s. The squid the context of ion channels, these distinct states giant axon is notable for its extraordinarily large represent different conformational states. Some diameter (∼0.5 mm). Most axons in the squid nervous conformations will correspond to closed states, system and in other nervous systems are typically at some to open states, and some to inactivated least 100 times thinner. The large size of the squid states. giant axon is a specialization for rapid conduction of membrane potential The voltage across the cell action potentials that trigger the contraction of the membrane. If the potential outside of the cell squid’s mantle when escaping from a predator. In is used as a reference (0 mV), then a typical addition to being beneficial for the squid, the large membrane potential for a neuron at rest would diameter of the giant axon was beneficial for Hodgkin be about −70 mV. The choice of reference point, and Huxley because it permitted manipulations that however, is a matter of convention. In some cases, were not technically feasible in smaller axons that the inside of the cell is used as a reference, in had been used in biophysical studies up to that which case the resting membrane potential would point. In a well-designed series of experiments, be 0 mV and the external potential would be Hodgkin and Huxley systematically demonstrated +70 mV. how the macroscopic ionic currents in the squid giant axon could be understood in terms of changes in + + Monte Carlo method A numerical method for Na and K conductances in the axon membrane. simulating the behavior of a stochastic system Based on a series of voltage-clamp experiments, by using random numbers to generate possible they developed a detailed mathematical model of outcomes based on a model of the underlying the voltage-dependent and time-dependent properties + + probability distribution. of the Na and K conductances. The empirical work led to the development of a coupled set single-channel conductance The electrical con- of differential equations describing the ionic basis ductance of a single-ion channel in the open state; of the action potential,17 which became known typically between 1 and 150 pS (picosiemens). as the HH model. The real predictive power of
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vin FIGURE 1| Electrical equivalent circuit for a short segment of squid giant axon. The capacitor represents the capacitance of the cell membrane; the two variable I I I I I resistors represent voltage-dependent Na+ and K+ C Na K L ext conductances, the fixed resistor represents a voltage-independent leakage conductance, and the three batteries represent reversal potentials for the corresponding conductances. The pathway labeled ‘stim’ represents an externally applied current, such as might GNa GK GL Stim be introduced via an intracellular electrode. The sign CM conventions for the various currents are indicated by the directions of the corresponding arrows. Note that the ENa EK EL arrow for the external stimulus current Iext is directed from outside to inside (i.e., inward stimulus current is positive), whereas arrows for the ionic currents INa, IK, and IL are directed from inside to outside (i.e., outward ionic currents are positive). (Adapted with permission vout from Ref 17. Copyright 1952 John Wiley & Sons, Inc). the model became evident when Hodgkin and into three distinct components: a sodium current INa, Huxley demonstrated that numerical integration of a potassium current IK, and a small leakage current IL these differential equations (using a hand-cranked that is primarily carried by chloride ions. The behavior mechanical calculator!) could accurately reproduce of an electrical circuit of the type shown in Figure 1 all the key biophysical properties of the action can be described by a differential equation of the potential. For this outstanding achievement, Hodgkin general form: and Huxley were awarded the 1963 Nobel Prize in Physiology and Medicine (shared with Sir John Eccles dVm C + I = I (1) for his work on the biophysical basis of synaptic m dt ion ext transmission). where Iext is an externally applied current, such Electrical Equivalent Circuits as might be introduced through an intracellular In biophysically based neural modeling, the electrical electrode. Equation (1) is the fundamental equation properties of a neuron are represented in terms relating the change in membrane potential to the of an electrical equivalent circuit. Capacitors are currents flowing across the membrane. used to model the charge storage capacity of the cell membrane, resistors are used to model Macroscopic Ionic Currents the various types of ion channels embedded in The individual ionic currents INa, IK,andIL shown membrane, and batteries are used to represent the in Figure 1 represent the macroscopic currents electrochemical potentials established by differing flowing through a large population of individual intra- and extracellular ion concentrations. In their ion channels. In HH-style models, the macroscopic seminal paper on the biophysical basis of the action current is assumed to be related to the membrane 17 potential, Hodgkin and Huxley modeled a segment voltage through an Ohm’s law relationship of the of squid giant axon using an equivalent circuit form V = IR. In many cases, it is more convenient similar to that shown in Figure 1. In the equivalent to express this relationship in terms of conductance circuit, the current across the membrane has two rather than resistance, in which case Ohm’s law major components, one associated with the membrane becomes I = GV, where the conductance G is the capacitance and the other one associated with the flow inverse of resistance, G = 1/R. In applying this of ions through resistive membrane channels. The relationship to ion channels, the equilibrium potential capacitive current Ic is defined by the rate of change Ek for each ion type also needs to be taken into of charge q at the membrane surface: Ic = dq/dt.The account. This is the potential at which the net ionic charge q(t) is related to the instantaneous membrane current flowing across the membrane would be zero. voltage Vm(t) and membrane capacitance Cm by the The equilibrium potentials are represented by the relationship q = CmVm. Thus, the capacitive current batteries in Figure 1. The current is proportional to can be rewritten as Ic = CmdVm/dt. In the HH model the conductance times the difference between the of the squid axon, the ionic current Iion is subdivided membrane potential Vm and the equilibrium potential
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Ek. The total ionic current Iion is the algebraic sum seem more natural to speak of gates as being open of the individual contributions from all participating or closed, a great deal of confusion can be avoided channel types found in the cell membrane: by consistently using the terminology permissive and nonpermissive for gates while reserving the terms open = = − Iion Ik Gk(Vm Ek)(2)and closed for channels. k k The voltage-dependence of ionic conductances is incorporated into the HH model by assuming that which expands to the following expression for the the probability for an individual gate to be in the HH model of the squid axon: permissive or nonpermissive state depends on the value of the membrane voltage. If we consider gates Iion = GNa(Vm − ENa) + GK(Vm − EK) of a particular type i, we can define a probability p , ranging between 0 and 1, which represents + GL(Vm − EL)(3)i the probability of an individual gate being in the Note that individual ionic currents can be positive or permissive state. If we consider a large number of negative depending on whether the membrane voltage channels, rather than an individual channel, we can is above or below the equilibrium potential. This also interpret pi as the fraction of gates in that raises the question of sign conventions. Is a positive population that are in the permissive state. At some ionic current flowing into or out of the cell? The most point in time t,letpi(t) represent the fraction of gates − commonly used sign convention in neural modeling that are in the permissive state. Consequently, 1 pi is that ionic current flowing out of the cell is positive (t) must be in the nonpermissive state. and ionic current flowing into the cell is negative (see α (V) Section on ‘Sign Conventions’ for more details). fraction in −−→i fraction in In general, the conductances are not constant nonpermissive ←−− permissive − values, but can depend on other factors like state, 1 pi (t) βi(V) state, pi (t) the membrane voltage or the intracellular calcium concentration. In order to explain their experimental The rate at which gates transition from the data, Hodgkin and Huxley postulated that GNa nonpermissive state to the permissive state is denoted and GK were voltage-dependent quantities, whereas by a variable αi(V), which has units of 1/s. Note the leakage current GL was taken to be constant. that this ‘rate constant’ is not really constant, but Thus, the resistor symbols in Figure 1 are shown as depends on membrane voltage V. Similarly, there is a variable resistors for GNa and GK, and as a fixed second rate constant, βi(V), describing the transition resistor for GL. Today, we know that the voltage- rate from the permissive to the nonpermissive state. dependence of GNa and GK can be related to the Transitions between permissive and nonpermissive biophysical properties of the individual ion channels states in the HH model are assumed to obey first-order that contribute to the macroscopic conductances. kinetics: Although Hodgkin and Huxley did not know about the properties of individual membrane channels when dpi = αi(V)(1 − pi) − βi(V)pi (4) they developed their model, it will be convenient for dt us to describe the voltage-dependent aspects of their model in those terms. where αi(V)andβi(V) are voltage-dependent. If the membrane voltage Vm is clamped at some fixed value Gates V, then the fraction of gates in the permissive state will = The macroscopic conductances of the HH model can eventually reach a steady-state value (i.e., dpi/dt 0) →∞ be considered to arise from the combined effects of a as t given by large number of microscopic ion channels embedded in the membrane. Each individual ion channel can be = αi(V) pi,t→∞ + (5) thought of as containing one or more physical gates αi(V) βi(V) that regulate the flow of ions through the channel. An individual gate can be in one of two states, permissive The time course for approaching this equilibrium or nonpermissive.Whenall of the gates for a particular value is described by a simple exponential with time channel are in the permissive state, ions can pass constant τi(V) given by: through the channel and the channel is open. If any of 1 the gates are in the nonpermissive state, ions cannot τ = i(V) + (6) flow and the channel is closed. Although it might αi(V) βi(V)
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When an individual channel is open, it con- specify the behavior of the membrane potential Vm in tributes some small, fixed value to the total con- the HH model of the squid giant axon. ductance and zero otherwise. The macroscopic con- ductance for a large population of channels is thus Sign Conventions proportional to the number of channels in the open Note that the appearance of Iion on the left-hand side state, which is, in turn, proportional to the proba- of Eq. (1) and Iext on the right indicates that they have bility that the associated gates are in their permissive opposite sign conventions. As the equation is written, state. Thus, the macroscopic conductance Gk due to a positive external current Iext will tend to depolarize channels of type k, with constituent gates of type i, the cell (i.e., make Vm more positive) while a positive is proportional to the product of the individual gate ionic current Iion will tend to hyperpolarize the cell probabilities pi: (i.e., make Vm more negative). This sign convention for ionic currents is sometimes referred to as the =¯ Gk gk pi (7) neurophysiological or physiologists’ convention. This i convention is conveniently summarized by the phrase ‘inward negative’, meaning that an inward flow of ¯ where gk is a normalization constant that determines positive ions into the cell is considered a negative the maximum possible conductance when all the current. This convention perhaps arose from the fact channels are open (i.e., all gates are in the permissive that when one studies an ionic current in a voltage- state). clamp experiment, rather than measuring the ionic We have presented Eqs. (4)–(7) using a current directly, one actually measures the clamp generalized notation that can be applied to a wide current that is necessary to counterbalance it. Thus, variety of conductances beyond those found in the an inward flow of positive ions is observed as a squid axon. To conform to the standard notation negative-going clamp current, hence explaining the of the HH model, the probability variable pi in Eqs. ‘inward negative’ convention. Some neural simulation (4)–(7) is replaced by a variable that represents the gate software packages, such as GENESIS, use the opposite type. For example, Hodgkin and Huxley modeled the sign convention (inward positive), because that allows sodium conductance using three gates of a type labeled all currents to be treated consistently. In the figures ‘m’ and one gate of type ‘h’. Applying Eq. (7) to the shown in this article, membrane currents are plotted sodium channel using both the generalized notation using the neurophysiological convention (inward and the standard notation yields: negative). =¯ 3 =¯ 3 GNa gNapmph gNam h (8) Voltage Conventions Although we are on the topic of conventions,thereare Similarly, the potassium conductance is modeled with two more issues that should be discussed here. The four identical ‘n’ gates: first concerns the value of the membrane potential V . Recall that potentials are relative; only potential G =¯g p4 =¯g n4 (9) m K K n Na differences can be measured directly. Thus, when defining the intracellular potential V , one is free Summarizing the ionic currents in the HH model in m to choose a convention that defines the resting standard notation, we have intracellular potential to be zero (the convention used by Hodgkin and Huxley), or one could choose a I =¯g m3h(V − E ) +¯g n4(V − E ) ion Na m Na K m K convention that defines the extracellular potential to + − gL(Vm EL) (10) be zero, in which case the resting intracellular potential dm would be around −70 mV. In either case, the potential = α (V)(1 − m) − β (V)m (11) dt m m difference across the membrane is the same, it is simply dh a matter of how ‘zero’ is defined. Most simulation = α (V)(1 − h) − β (V)h (12) dt h h software packages allow the user to select a voltage dn reference convention they like. = α (V)(1 − n) − β (V)n (13) dt n n The second convention we need to discuss concerns the sign of the membrane potential. The To completely specify the model, the one task modern convention is that depolarization makes the that remains is to specify how the six rate constants membrane potential Vm more positive. However, in Eqs. (11)–(13) depend on the membrane voltage. Hodgkin and Huxley17 used the opposite sign Then Eqs. (10)–(13), together with Eq. (1), completely convention (depolarization negative) in their article.
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In the figures in this article, we use the modern 1 convention that depolarization is positive. At a conceptual level, the choice of conventions 0.5 for currents and voltages is inconsequential; however, (normalized)
at the implementation level it matters a great deal, K 0 because inconsistencies will cause the model to G behave incorrectly. The most important thing in 150 choosing conventions is to ensure that the choices are internally consistent. One must pay careful attention 100 to these issues when implementing a simulation using (mV)
c 50 equations from a published model, because it may be V necessary to convert the empirical results reported 0 using one set of conventions into a form that is 2 0 2 4 6 8 10 consistent with one’s own model conventions. Time (msec)
FIGURE 2| Simulated voltage-clamp data illustrating Rate Constants + How did Hodgkin and Huxley go about determining voltage-dependent properties of the K conductance in squid giant axon. The command voltage (mV) is shown in the lower panel, and the voltage-dependence of the rate constants α and β Vc the K+ current is shown in the upper panel. Simulation parameters are that appear in Eqs. (11)–(13)? How did they determine from the HH model.17 that the potassium conductance should be modeled with four n gates, but that the sodium conductance = required three m gates and one h gate? In order to Initially, with Vm 0, the state variable n has a steady- answer these questions, we need to look in more state value (i.e., when dn/dt = 0) given by Eq. (5): detail at the type of data that can be obtained from α voltage-clamp experiments. = n(0) n∞(0) + (15) Figure 2 shows simulated voltage-clamp data, αn(0) βn(0) similar to those obtained by Hodgkin and Huxley in When V is clamped to a new level V , the gating their studies of squid giant axon. In these experiments, m c variable n will eventually reach a new steady-state Hodgkin and Huxley used voltage-clamp circuitry to value given by step the membrane potential from the resting level (0 mV) to a steady depolarized level. The figure αn(Vc) shows the time course of the change in normalized n∞(Vc) = (16) α (V ) + β (V ) K+ conductance for several different voltage steps. n c n c Three qualitative effects are apparent in the data. The solution to Eq. (14) that satisfies these boundary First, the steady-state conductance level increases conditions is a simple exponential of the form: with increasing membrane depolarization. Second, −t/τn(Vc) the onset of the conductance change becomes faster n(t) = n∞(Vc) − (n∞(Vc) − n0(0))e (17) with increasing depolarization. Third, there is a slight temporal delay between the start of the voltage step Given Eq. (17), which describes the time course of n and the change in conductance. in response to a step change in command voltage, one In the simulated voltage-clamp experiments could try fitting curves of this form to the conductance illustrated in Figure 2, the membrane potential starts data shown in Figure 2 by finding the values of n∞(Vc), = in the resting state (Vm 0, using the HH voltage n∞(0), and τn(Vc) that give the best fit to the data for convention) and is then instantaneously stepped to a each value of Vc. Figure 3 illustrates this process, new clamp voltage Vc. What is the time course of using some simulated conductance data generated by the state variable n, which controls gating of the K+ the HH model. Recall that n takes on values between channel, under these circumstances? Recall that the 0 and 1, so in order to fit the conductance data, differential equation governing the state variable n is n must be multiplied by a normalization constant given by: g¯K that has units of conductance. For simplicity, the normalized conductance GK/g¯K is plotted. The dotted line in Figure 3 shows the best-fit results for a dn simple exponential curve of the form given in Eq. (17). = αn(V)(1 − n) − βn(V)n (14) dt Although this simple form does a reasonable job of
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1.0 0.3 0.2 0.8 0.1
(normalized) 0 Na
0.6 G 0.1 n n 2 n3 n4 0.10 0.08 150 0.4 0.06 (normalized) K 0.04 100 G 0.02
0.2 0.00 (mV) c 50
0 0.2 0.4 0.6 0.8 1.0 V 0 0.0 2 0 2 4 6 8 10 0 2 4 6 8 10 time (msec) time (msec) FIGURE 4| Simulated voltage-clamp data illustrating activation and FIGURE 3| Best-fit curves of the form G = g¯ nj (j = 1–4) for k K inactivation properties of the Na+ conductance in squid giant axon. The simulated conductance versus time data. The inset shows an command voltage is shown in the lower panel, and the Na+ current enlargement of the first millisecond of the response. The initial Vc is shown in the upper panel. Simulation parameters are from the HH inflection in the curve cannot be well-fit by a simple exponential (dotted model.17 line) which rises linearly from zero. Successively higher powers of j = = (j 2: dot–dashed; j 3: dashed line) result in a better fit to the potassium channels do not. To model this process, initial inflection. In this case, = (solid line) gives the best fit. j 4 Hodgkin and Huxley postulated that the sodium capturing the general time course of the conductance channels had two types of gates, an activation gate, change, it fails to reproduce the sigmoidal shape which they labeled m, and an inactivation gate, and the temporal delay in onset. This discrepancy which they labeled h. Again, boundary conditions is most apparent near the onset of the conductance dictated that m and h must follow a time course change, shown in the inset of Figure 3. Hodgkin given by and Huxley realized that a better fit could be − = − − t/τm(Vc) obtained if they considered the conductance to be m(t) m∞(Vc) (m∞(Vc) m∞(0))e (19) proportional to a higher power of n. Figure 3 shows −t/τ (V ) h(t) = h∞(V ) − (h∞(V ) − h∞(0))e h c (20) the results of fitting the conductance data using a c c =¯ j form GK gkn with powers of j ranging from 1 to 4. Using this sort of fitting procedure, Hodgkin Hodgkin and Huxley made some further simplifica- and Huxley determined that a reasonable fit to the K+ tions by observing that the sodium conductance in the conductance data could be obtained using an exponent resting state is small compared with the value obtained of j = 4. Thus, they arrived at a description for during a large depolarization; hence, they were able the K+ conductance under voltage-clamp conditions to neglect m∞(0) in their fitting procedure. Likewise, given by: steady-state inactivation is nearly complete for large depolarizations, so h∞(Vc) could also be eliminated 4 from the fitting procedure. With these simplifications, GK =¯gKn =¯gK [n∞(Vc) − (n∞(Vc) Hodgkin and Huxley were able to fit the remaining −t/τ 4 parameters from the voltage-clamp data. The sodium −n∞(0))e n (18) conductance GNa was thus modeled by an expression 3 of the form GNa =¯gNam h. Activation and Inactivation Gates The strategy that Hodgkin and Huxley used for Parameterizing the Rate Constants modeling the sodium conductance is similar to that By fitting voltage-clamp data as discussed above, described above for the potassium conductance, steady-state conductance values and time constants except that the sodium conductance shows a can be empirically determined as a function of more complex behavior. In response to a step command voltage for each of the gating variables change in clamp voltage, the sodium conductance associated with a particular channel. Using Eqs. (5) exhibits a transient response (Figure 4), whereas the and (6), the steady-state conductance values and time potassium conductance exhibits a sustained response constants can be transformed into expressions for (Figure 2). Sodium channels inactivate whereas the the forward and backward rate constants α and β.
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For example, for the potassium channel n gate we varies continuously in the model. Typically, voltage- have dependences are expressed as a continuous function of voltage, and the task for the modeler becomes n∞(V) αn(V) = (21) one of determining the parameter values that best τn(V) fit the data. As an illustration, the closed circles in Figure 5a and b represent empirical data on n∞(V ) 1 − n∞(V) c β (V) = (22) and τ (V ) obtained by Hodgkin and Huxley (Table n τ (V) n c n 1 of Hodgkin and Huxley17). The data points in Thus, there are two equivalent representations for the Figure 5c and d show the same data set transformed voltage-dependence of a channel. One representation into the α/β representation. Hodgkin and Huxley used the following functional forms to parameterize specifies the voltage-dependence of the rate constants, + which we will call the α/β representation. The other their K conductance results (shown as solid lines in representation specifies the voltage-dependence of Figure 5): the steady-state conductance and the time constant, 0.01 (10 − V) which we will call the n∞/τ representation. These α (V) = (23) n exp (10 − V/10) − 1 two representations are interchangeable, and one can easily convert between them using the algebraic −V β (V) = 0.125 exp (24) relationships in Eqs. (5) and (6) (for transforming from n 80 α/β to n∞/τ) and Eqs. (21) and (22) (for transforming from n∞/τ to α/β). In general, experimentalists If Eqs. (23) and (24) above are compared with Eqs. tend to use the n∞/τ representation because it (12) and (13) from the original article,17 you will maps more directly onto the results of voltage- note that the sign of the membrane voltage has been clamp experiments. Modelers, on the other hand, changed to correspond to the modern convention (see tend to express voltage-dependences using the α/β Section on ‘Voltage Conventions’ above). Hodgkin representation, because it maps more directly onto the and Huxley used similar functional forms to describe gating Eqs. (11)–(13) in the standard formulation of the voltage-dependence of the m and h gates of the the HH model. sodium channel: Voltage-clamp experiments yield estimates of − n∞/τ or α/β only at the discrete clamp voltages 0.1(25 V) αm(V) = (25) Vc used in the experiment. Numerical integration exp (25 − V/10) − 1 of the HH model, however, requires that n∞/τ or −V α/β values be specified over a continuous range of βm(V) = 4exp (26) membrane voltages, because the membrane potential 18
1.0 8.0 (a) (b) 0.8 6.0 0.6 4.0 τ n• n
0.4 (msec) τ 2.0 (Dimensionless) 0.2
0.0 0.0 FIGURE 5| Parametric fits to 40 0 40 80 120 40 0 40 80 120 voltage-dependence of the K+ conductance in the HH model. (A) Steady-state value ∞;(B) 1.2 1.2 n (c) (d) time constant τn (C) forward rate constant αn; and (D) backward rate constant β .Data ) ) n 1 1 − 0.8 − 0.8 points are from Table 1 of Hodgkin and Huxley.17 Solid lines in panels (C) and (D) are α β n n parametric fits to the rate data. The best-fit 0.4 0.4 curves correspond to Eqs. (23) and (24), Rate (msec Rate (msec respectively. Solid lines in panels (A) and (B) are the transformations of the α/β functions 0.0 0.0 40 0 40 80 120 40 0 40 80 120 into the n∞/τ representation using Eqs. (5) V (mV) V (mV) and (6).
82 2010 John Wiley & Sons, Inc. Volume 3, January/February 2011 WIREs Systems Biology and Medicine Electrophysiological models
−V membrane voltage V and intracellular calcium con- α (V) = 0.07 exp (27) h 20 centration χ.Traub18 modeled the slow potassium conductance as Gs =¯gsq, where q is a standard HH 1 β (V) = (28) gating variable with first-order kinetics: h exp (30 − V/10) + 1 dq = α (1 − q) − β q (30) In neural simulation software packages, the rate dt q q constants in HH-style models are often parameterized using a generic functional form: The voltage- and calcium-dependence were incorpo- rated into the rate equations as follows: A + BV α(V) = (29) + ( + / ) V 0.005(200 − χ) C H exp V D F α (χ, V) = exp (31) q 27 exp (200 − χ/20) − 1 In general, this functional form may require up to six βq = 0.002 (32) parameters (A, B, C, D, F,andH) to fully specify the rate equation. However, in many cases, adequate fits Conductances that depend on both membrane to the data can be obtained using far fewer parameters. voltage and calcium concentration are rarely as Fortunately, Eq. (29) is flexible enough that it can be well characterized experimentally as are ordinary transformed into simpler functional forms by setting voltage-dependent channels. In part this is due certain parameters to either 0 or 1. For example, to the technical challenges in trying to achieve a if the voltage-clamp data can be adequately fit by ‘calcium clamp’ to precisely quantify the calcium- an exponential function over the relevant range of = = = = dependence. Furthermore, voltage-clamp experiments voltages, then setting B 0, C 0, D 0, and H 1 on these conductances are more difficult to interpret in Eq. (29), results in a simple exponential form, = − because even though the membrane voltage is a(V) A exp( V/F), with just two free parameters held fixed by the clamp circuitry, the intracellular (A and F) to be fit to the data. Similarly, setting B = 0, = = calcium concentration is varying during the clamp. C 1, and H 1 gives a sigmoidal function with Consequently, modelers must often devise rate three free parameters (A, D,andF). equations for such channels based on more qualitative One other technical note is that certain function criteria than are used for regular voltage-dependent forms can become indeterminate at certain voltage channels. To simplify this task, it is common to take α values. For example, the expression for n(V)in one of the α/β rate equations as a constant [as was Eq. (23) evaluates to the indeterminate form 0/0 = done for βq in Eq. (32) above] and to put all of at V 10. The solution to this problem is to the voltage- and calcium-dependence into the other apply L’Hopital’sˆ rule, which states that if f(x)and rate equation. This reduces the number of unknown g(x) approach 0 as x approaches a,andf (x)/g(x) parameters in the model, and it simplifies searching approaches L as x approaches a, then the ratio the parameter space. f(x)/g(x) approaches L as well. Using this rule, it For understanding the effects on channel gating, can be shown that α (10) = 0.1. When implementing n the region of space in which the calcium concentration HH-style rate functions in computer code, care must must be known is a thin shell just inside the membrane be taken to handle such cases appropriately. surface. The calcium concentration in this region can be significantly different from the bulk concentration Calcium-Dependent Channels in the interior of the cell. Calcium enters this shell Certain types of ion channels are influenced by both + region primarily through the influx of Ca2 ions membrane voltage and intracellular calcium concen- through membrane calcium channels. Calcium leaves tration. Although calcium-dependence was not part of the shell region because of diffusion and buffering. the original HH model, it is straightforward to extend A simple model of intracellular calcium dynamics the HH framework to handle this case. Calcium- describes this process by a differential equation of the dependence is typically implemented by modifying form18: the α/β rate equations to include an additional state variable representing the intracellular calcium con- dχ 18 = AI − Bχ (33) centration. For example, Traub proposed a model dt Ca of intrinsic bursting in hippocampal neurons that included a slow calcium-dependent potassium con- where A is a constant related to the volume of the ductance Gs. This conductance was modeled using shell and the conversion of coulombs to moles of an HH-style rate equation that depends on both ions, while B is a rate constant representing the
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effects of diffusion and buffering. As a technical p12 p23 note, recall that ionic currents are typically defined as S1 S2 S3 p ‘inward negative’ (see Section on ‘Sign Conventions’ p21 32 above). Using this convention, the constant A in Eq. (33) will be a negative number, such that inward p p p p (negative) calcium current will cause a positive change 42 24 53 35 in calcium concentration χ. For a discussion of more advanced techniques for modeling calcium dynamics, p45 see Ref 19. S4 S5 p54 | Markov Models of Individual Channels FIGURE 6 A representative Markov model diagram. This particular model has five distinct states S1–S5 and 10 transition probabilities pij. The HH framework has been extremely successful In Markov models of ion channels, each state represents a putative for developing quantitative models of macroscopic conformational state of the ion channel complex. Some conformations currents observed in single neurons. However, a dif- will correspond to closed states, some to open states, and some to ferent approach must be used if one is interested inactivated states. in modeling the currents flowing through individual channels. At the microscopic level, gating of individ- ual ion channels is a stochastic process. Transitions the system in a new state j if it has recently been in between permissive and nonpermissive gating states state i. The time evolution of Pi(t) can be written as: take place by probabilistic transitions between differ- ent conformational states of the ion channel complex. dP (t) n n i = P (t)p − P (t)p (34) Certain conformational states allow ions to move dt j ji i ij through the channel, while others do not. When j=1 j=1 monitored experimentally in single-channel patch- clamp recordings, for example, individual channels The first term on the right-hand side of this equation are observed to fluctuate randomly between open and represents the increase in probability of finding the closed states. system in state Si because of transitions entering this Markov models provide a framework for state from other states. The second term represents the describing the microscopic currents through individ- decrease in probability because of transitions out of 20 ual ion channels. The basic assumption underlying state Si into other states. If there is a large population the Markov model formalism is that the opening of identical channels, then Pi(t) can be interpreted as and closing of ion channels can be described as a the fraction of channels in state Si and the transition series of transitions between distinct conformational probabilities pij can be interpreted as rate constants. states. Certain states may correspond to the channel Thus, Markov models provide a convenient formalism being open, closed, inactivated, and so on. Tran- for linking the gating properties of individual channels sitions between different states occur according to to the behavior of macroscopic currents as described a set of transition probabilities. Figure 6 shows a by the HH model. generic Markov model consisting of 5 states Si and 10 Figure 7a shows a five-state Markov model that 4 + transition probabilities pij. Note that the number of corresponds to the n gating kinetics of the HH K transition probabilities will depend on the topology channel model. The Markov model has five distinct of the Markov model. For example, a fully connected states, n0–n4, where the subscript represents the five-state model, in which any state could transition number of HH gates in the permissive state. When the to any other state, would have 20 transition proba- channel is in state n1, for example, one of the gates is in bilities. Part of the task of designing a Markov model the permissive configuration and three of the gates are involves determining how many states are involved, nonpermissive. Ions can flow through the channel only which transitions are allowed, and which are forbid- when all gates are in the permissive state (state n4); all den. The forbidden transitions do not appear in the other states correspond to closed states. The transition diagram. The modeler’s task then becomes one of probabilities between states can be calculated from the determining values for the remaining allowed transi- forward (αn) and reverse (βn) rate constants of the HH tion probabilities. K+ channel model and the assumption that each gate The probability to find the system in state Si behaves independently. There are four possible ways at some time t is defined as Pi(t). The transition that the n0 state can transition to the n1 state, so probability pij is the conditional probability of finding the corresponding transition rate is 4αn.Thefullset
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Closed states Open (a) state α α α α 4 n 3 n 2 n n n0 n1 n2 n3 n4 β β β β n 2 n 3 n 4 n
60 (b)
30 (mV) c V 0 + FIGURE 7| A Markov model of the HH K n 4 (c) conductance. (a) The Markov model has four closed n3 − states n0 n3 and one open state n4.Thesubscript n2 corresponds to the number of n gates in the n1 permissive state. (b) Command voltage in a simulated n0 voltage-clamp experiment. (c) Monte Carlo simulation (d) of state transitions of the Markov model in response 1
to a step change in command voltage. (d) Normalized norm K
+ G conductance of the K channel. The channel is open 0 norm = whenever the system is in state , (GK 1) n4 100 50 0 50 100 otherwise the channel is closed (Gnorm = 0). K Time (msec) of transition probabilities that correspond to the HH that activation and inactivation processes are not model kinetics are shown in the figure. completely independent. Figure 8b shows a more The sequence of openings and closings of recent Markov model of Na+ channel gating21 that an individual channel can be simulated using provides a better description of the data. Monte Carlo techniques to randomly generate state transitions with the specified probabilities. Recall that the HH rate constants are voltage-dependent, so a Synaptic Models change in membrane voltage (Figure 7b) will result in Thus far, the techniques in this article have a shift of all the transition probabilities and hence a focused primarily on modeling voltage-dependent shift in the probability distribution of states. Figure 7c channels. Equally important from a functional shows a Monte Carlo simulation of the time history perspective are the ligand-gated channels that mediate of state transitions before and after a step change in chemical synaptic transmission. When an action clamp voltage. When the membrane is clamped to the potential arrives at the presynaptic terminal of a chemical synapse, neurotransmitter is released resting voltage (VC = 0), the system spends most of into the synaptic cleft. Neurotransmitter molecules its time in states n0 − n2, which are all closed states. When the membrane is clamped to a depolarized subsequently bind to ligand-gated receptors in the postsynaptic membrane, causing changes in voltage (VC = 60), the system spends most of its time ionic current flow across the membrane. In an in states n2 − n4. The channel is open whenever the equivalent electrical circuit model (Figure 1), ligand- model is in state n4, as reflected in the conductance record shown in Figure 7d. gated channels are represented by additional resistive Figure 8a shows an eight-state Markov model pathways across the membrane. that corresponds to the m3h gating kinetics of the For simulating synaptic activation in neural HH Na+ channel. The model is in the open state models, the details of synaptic release, diffusion, and receptor binding are often abstracted into a simpler only when all gates are permissive (state m3h1). Any state in which the inactivation gate is nonpermissive form that describes the postsynaptic conductance as a time-dependent function. The arrival of an action (h0) corresponds to an inactivated state of the channel. According to the HH model, the behavior potential at a synapse at time tspike gives rise to a transient change in a postsynaptic conductance that is of the inactivation gate (h) is independent of the 22 three activation gates (m). This is reflected in the often modeled using the alpha function : Markov model by the fact that transitions to an g e − − G (t) = peak (t − t )e (t tspike)/τsyn for t ≥ t inactivated state can potentially occur from any syn τsyn spike spike open or closed state. However, careful experimental studies of Na+ channel gating kinetics have revealed (35)
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(a) HH Na channel second-order differential equation23 or reorganization Inactivated states of the computation to require the storage of only two running sums per synapse, rather than a complete time 3α 2α α m m m history of activation.24 m0h0 m1h0 m2h0 m3h0 β 2β 3β Another technique for modeling synaptic con- m m m ductances utilizes a Markov model approach.25 The β α β α β α β α simplest form of such models involves only a single h h h h h h h h open state and a single closed state. Such two-state models can be represented by: α α α 3 m 2 m m α m0h1 m1h1 m2h1 m3h1 −→ β β β m 2 m 3 m C + T ←− O (36) β Closed states Open where C is a closed state, O an open state, T represents Inactivated states neurotransmitter, and α and β are forward and backward rate constants, respectively. Unlike the (b) Na channel (Patlak21) Hodgkin and Huxley model, the rate constants, α I2 I1 and β, are independent of membrane voltage. Let the fraction of receptors in the open state be represented by r, and let the neurotransmitter concentration be denoted by [T]. Then the first-order kinetic equation for this system is
C C C C O dr 4 3 2 1 = α[T](1 − r) − βr (37) dt Open Closed states One simple way to model the neurotransmitter concentration is to assume that a constant amplitude FIGURE 8| Two different Markov models of the Na+ conductance. pulse of transmitter is released when the action (a) The Markov model corresponding to independent activation and inactivation gating of the HH model has eight distinct states. The potential arrives at the presynaptic terminal, in which 26 subscripts on the states represent the number of gates of each type in case Eq. (37) can be solved analytically for r(t). The the permissive state. The channel is open only when all three m gates synaptic current is then modeled by: and the h gate are in the permissive state (m3h1). (b) A model proposed 21 by Patlak which includes interactions between the activation and Isyn(t) =¯gsynr(t)(V − Esyn) (38) inactivation gates. This model provides a better description of actual voltage-clamp data from squid axon than does the HH model. In general, Markov models of this type can be much more sophisticated than the two-state model presented above. These more detailed models can have The peak of the conductance change occurs at time multiple states representing various open, closed, and = + t tspike τsyn, and the conductance value at this desensitized configurations. Such biophysically rich time is gpeak. The synaptic current Isyn associated with Markov models may be particularly useful when = − the synapse is modeled by Isyn(t) Gsyn(t)(V Esyn), using a neuroinformatics approach to investigate where Esyn is the reversal potential of the synapse. how receptor properties are altered by variations in When a synapse is activated by a sequence the molecular structure and subunit composition of of action potentials, the net change in conductance particular ligand-gated receptors. is often modeled as a linear summation of the contributions from each individual action potential. Metabotropic Receptors A straightforward implementation based on Eq. (35) Up to this point, we have been discussing ionotropic would require keeping a time history of spike receptors for which neurotransmitter binding causes activity and summating over all previous spike direct and immediate gating of an associated ion times. However, this approach is computationally channel. Metabotropic receptors, on the other inefficient and rarely used in large-scale simulations. hand, exert their influence indirectly by acting There are more efficient methods involving either through an intracellular second messenger system. the reformulation of the conductance change as a For metabotropic receptors, neurotransmitter binding
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leads to the activation of intracellular biochemical and decaying phases of a G-protein-activated GABAB pathways, which may ultimately link to the opening receptor current. or closing of second messenger gated ion channels. The cascade of reactions that take place in such systems can be modeled using a combination of Markov models for Multicompartment Models the components that have discrete states and standard A simple electrical equivalent circuit, such as that biochemical reaction kinetics for describing chemical shown in Figure 1, can be used to model a localized concentrations that vary continuously.26 For example, region of nerve cell membrane. In general, however, the binding of transmitter T to a metabotropic neurons have spatially extended axons and dendrites receptor R, leading to the formation of an activated with heterogeneous properties. Different regions of receptor state R∗, might be described by a two-state the cell will have different diameters and varying types Markov model: and densities of ion channels and receptors. Further- more, quantities such as the local membrane potential −−→ and the local intracellular calcium concentration can + ∗ R T ←−− R (39) vary significantly across the spatial extent of a neuron. Multicompartment models provide a means for han- Following receptor activation, there could be several dling the spatial complexity of neuron morphology intermediate biochemical reactions of the general and the heterogeneity of physical properties. Figure 9 form: illustrates the compartmental modeling approach for
α −−→ (a) dendrite A + B ←−− X + Y (40) β which can be modeled using standard reaction kinetics.27 In Eq. (40), α and β are forward and backward rate constants for the reaction. The chemical concentrations are governed by a rate (b) compartmentalization equation of the form:
d[A] =−α[A][B] + β[X][Y] (41) dt i - 1 ii + 1 and a set of relationships that reflect the stoichiometry of the reaction
d[A] d[B] d[X] d[Y] (c) equivalent circuit = =− =− (42) dt dt dt dt vi-1 vi vi+1
In a second messenger cascade, one of the reactants appearing on the left-hand side of one of the biochemical reactions would be the activated receptor R∗, and one of the products appearing on the right- hand side would be a second messenger Z that could serve as a ligand for a postsynaptic ion channel. The v v v gating of this second messenger gated channel could out out out then be described by a Markov model, such as the FIGURE 9| Compartmental approach for single-neuron modeling. following two-state model, The dendrites (a) are divided into distinct regions that are represented by cylindrical compartments (b). Each compartment can have different −−→ C + Z O (43) physical characteristics (membrane potential, length, diameter, channel ←−− types, channel densities, etc.). The physical properties are modeled by an electrical equivalent circuit (c). In the circuit model, neighboring or by a more complex multi-state model. For example, compartments are coupled by resistors representing the axial resistance Destexhe et al.26 found that a four-state Markov of the intracellular space. Branch points are handled in a similar manner model was needed to adequately fit both the rising (data not shown).
Volume 3, January/February 2011 2010 John Wiley & Sons, Inc. 87 Focus Article www.wiley.com/wires/sysbio a segment of dendritic membrane. The multicom- technique in which action potentials are represented as partment modeling approach divides the neuron into discrete temporal events. In this event-based approach, a number of smaller spatial compartments, each of an action potential generated by neuron i at time ti which can be modeled with an electrical equivalent is represented as a time-stamped event that is used circuit similar to Figure 1. The components of the to trigger synaptic input to a target neuron j after equivalent circuit and their numerical values can vary some time delay tij. Propagation along the axon from compartment to compartment, depending on is not modeled explicitly; rather it is implicit in the the particular types of conductances found in differ- axonal propagation delay tij. Recall that a single ent regions of the cell. Neighboring compartments axon typically makes synaptic contacts with multiple are coupled by axial currents that flow between com- target neurons. In general, the propagation delay tij partments in the intracellular space. The membrane can have different numerical values for each of the potential for compartment i, Vi, is related to the mem- possible postsynaptic targets. brane potentials in neighboring compartments, Vi−1 The second issue in network modeling involves and Vi+1,by specification of the connectivity between neurons. For small network models, this is often handled on a case- − − dVi (Vi−1 Vi) (Vi+1 Vi) by-case basis, whereas large network models usually Cm + Iion = + (44) dt ri−1,i ri+1,i require a rule-based approach. For example, a model of an invertebrate central pattern generator might where Cm and Iion are based on the equivalent circuit involve 10 neurons with an average of five synapses for compartment i.Thetermsri±1,i represent the per neuron, resulting in approximately 50 synaptic axial resistances between neighboring compartments, connections. Specification of the synaptic properties − and the terms (Vi±1,i Vi)/ri±1,i represent the axial (receptor type, reversal potential, peak conductance, currents. Similar relationships exist for branch points propagation delay, etc.) could easily be handled on where an axonal or dendritic segment splits into a synapse-by-synapse basis. In contrast, a network two or more subsegments. Using these techniques, model of a local region of mammalian visual cortex multicompartment models can describe arbitrarily might involve on the order of 10,000 neurons with an complex cell morphologies. Detailed advice on how to average of 100 synapses per neuron, resulting in one construct, parameterize, and test multicompartment million synaptic connections. In this case, a synapse- models can be found in Refs 28, 29 and 37. by-synapse specification would be unfeasible and a rule-based approach would be utilized. For example, a Network Models connection rule might specify that all neurons of type Previous sections have covered techniques for mod- A (e.g., inhibitory interneurons) make a particular eling single neurons, ion channels, and individual type of synaptic connection (e.g., GABAergic) with all synapses. Using these techniques, it is relatively neurons of type B (e.g., pyramidal cells) that lie within straightforward to create network models, in which a fixed radius. The rule might also specify how the the spike outputs from certain model neurons pro- peak conductance and axonal propagation delay vary vide synaptic inputs to other neurons in the network. with target distance. There are two main issues to consider in constructing network-level models. One involves choosing an appropriate mathematical representation for the prop- Software Tools agation of action potentials between neurons. The Fortunately, sophisticated software packages are other issue has to do with techniques for specifying available to facilitate the development, implemen- the synaptic connectivity within the network. tation, and dissemination of biophysically detailed In principle, the propagation action potentials neural models. Two of the most widely used tools are between neurons could be handled using HH con- GENESIS (GEneral NEural SImulation System)32,33 ductances and a multicompartmental description of and NEURON.34 Both of these modeling environ- the axon and its terminal arbor. This approach is ments are designed for constructing biophysically sometimes used when the scientific questions being detailed multicompartment models of single neurons, addressed pertain explicitly to mechanisms of action and they also provide modeling tools that span from potential propagation.30 However, it is computation- the molecular level to the network level. Both GEN- ally expensive to use a full multicompartment model to ESIS and NEURON provide high-level languages for describe every axon and terminal arbor in a large net- model specification, predefined sets of neural build- work. Because of the all-or-none nature of the action ing blocks, and graphical user interface elements for potential, it is often possible to use a more efficient simulation control and visualization. To construct a
88 2010 John Wiley & Sons, Inc. Volume 3, January/February 2011 WIREs Systems Biology and Medicine Electrophysiological models specific neural model, the model specification language [Eq. (35)] that allows for different time constants is used to define and link appropriate sets of predefined for the rising and falling phases of the synaptic building blocks to create a functional model. The basic waveform.23 Figure 10 shows the response of the building blocks include such things as compartments model to a large synchronous synaptic activation over or cable segments for modeling neuron morphology, a large portion of the dendritic tree. This pattern of voltage-gated and ligand-gated conductances, compo- synaptic input represents activation of the Purkinje nents for intracellular diffusion and buffering of ions, cell by a climbing fiber input. The so-called ‘complex chemical and electrical synapses, and various forms spike’ response of a Purkinje cell to climbing fiber of synaptic plasticity. Other building blocks provide stimulation has been well studied experimentally. The the model with external inputs and outputs, includ- ability of the model to reproduce known membrane ing file I/O and graphical displays. Some building voltage and intracellular calcium characteristics of blocks provide models of electrophysiological instru- a complex spike was one of the benchmarks for mentation like stimulus generators and voltage-clamp tuning certain model parameters and for evaluating circuits, which allow users to closely model the exper- the underlying modeling assumptions. The simulation imental setups that are used in empirical studies. results summarized in Figure 10 represent only one Custom user-defined elements can be created if the of several studies carried out using the Purkinje cell required modeling component is not already part of model.31,35 After tuning the model to reproduce a the predefined set of building blocks. More infor- range of in vitro firing behaviors, the model was used mation on these modeling environments, including to make predictions about the in vivo firing patterns documentation, tutorials, users groups, and work- of Purkinje cells. The model has been particularly shop announcements, can be found on the Web by useful in elucidating the role of dendritic inhibition in following the links provided in the Web Resources shaping neural response properties. section. LIMITATIONS CURRENT APPLICATIONS There are several limitations to keep in mind when Hundreds of biophysically detailed neural models developing biophysically detailed neural models. have been developed using GENESIS, NEURON, and Perhaps one of the most important is that such similar modeling tools. The scientific issues addressed models are actually highly impoverished relative to in these models span a broad range of topics, including the true richness and complexity of the underlying intracellular signaling, dendritic processing, neural biology. Even though these models are described oscillations, central pattern generation, motor control, as ‘biophysically detailed’, many aspects of cell and sensory coding, feature extraction, learning, and membrane physiology have been stripped away in the memory. See the ‘Web Resources’ section for links to modeling process. The art of creating a good model research publications that have been generated using involves knowing which details are important and GENESIS and NEURON. An illustrative example which details can be safely disregarded. However, of this type of biophysically detailed modeling details that are unimportant in one functional context approach is provided by the cerebellar Purkinje cell may become pivotal in a different context. Thus, one model developed by De Schutter and Bower31,35 should avoid thinking of any particular model, such using GENESIS. The dendritic morphology shown as the Purkinje cell model described above, as a full in Figure 10 contains approximately 1600 distinct and complete description of the underlying biological compartments with lengths and diameters based system. on detailed anatomical reconstructions of an actual It is better to think of a neural model as Purkinje cell.36 The model includes 10 different types an extended hypothesis that is designed to address of voltage-dependent channels: two Na+ channels a restricted range of neurobiological function. As (fast and persistent), two Ca2+ channels (T-type an extended hypothesis, each model embodies a and P-type), three voltage-dependent K+ channels, large number of assumptions. Certain assumptions and two Ca2+-dependent K+ channels. The channel will be well supported by empirical data, while properties were modeled using HH equations, and the others will be largely speculative. For the purpose modeling parameters were constrained by empirical of hypothesis testing, it is important to keep track of voltage-clamp data where available. The channels all the underlying assumptions and the corresponding were distributed differentially over three zones of the empirical constraints on those assumptions. This is Purkinje cell. Synaptic inputs were modeled using one area where neuroinformatics tools can play a key a dual exponential version of the alpha function role in helping modelers establish and document links
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(a) (b) (c) Volt +40 +20 0
−20 −40 −60
(d) (e) (f)
[Ca2+] 16 8 5 3 2 20 mV 1 AB C 5 ms 0
FIGURE 10| Representations of the membrane potential and calcium concentration in a large compartmental model of a cerebellar Purkinje cell following synaptic activation. (a–c) Membrane potential 1.4, 4.0, and 10.0 ms after synaptic activation. (D and E) Intracellular Ca2+ concentration 1.4 and 4.0 ms after activation. (f) Membrane potential (red trace) and Ca2+ concentration (green trace) in the cell body following activation. The vertical white bars indicate the times at which the false color images in panels (a)–(e) were generated. (Adapted with permission from Ref 31. Copyright 1994 John Wiley & Sons, Inc.). between each assumption and the set of empirical OUTLOOK results that impact that particular assumption. In Based on research trends over the past decade, it is terms of hypothesis testing, an important limitation to clear that both neuroinformatics and electrophysio- keep in mind is that even if a neural model successfully logical modeling are becoming increasingly important reproduces certain empirical results, it does not imply tools for exploring the functional properties of neu- that all the underlying assumptions in the model are ral systems. Several ongoing research and develop- true. Likewise, if a model fails to agree with some ment efforts are leading toward a convergence and piece of empirical data, the fact that the ‘extended integration of neuroinformatics and modeling tools hypothesis’ is falsified does not directly indicate which that will greatly enhance the ability of neurosci- of the underlying assumptions might be responsible entists to make use of these powerful approaches. for the disagreement. Therefore, it is not particularly Much of this development effort is taking place in the context of the Human Brain Project.3–5 Several useful to simply label a neural model as ‘right’ or research groups are actively developing large electro- ‘wrong’. Instead, neural modeling should be viewed physiological databases, common data representations as an integral component of the scientific method, in to facilitate information sharing, software tools for which progress is made through multiple iterations electrophysiological data analysis and visualization, of experimental observation, hypothesis generation neuroinformatics tools for search and retrieval, and (model building), prediction (model simulation), and neuroinformatics-based extensions to neural modeling testing (comparison with empirical data). software packages.
90 2010 John Wiley & Sons, Inc. Volume 3, January/February 2011 WIREs Systems Biology and Medicine Electrophysiological models
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WEB RESOURCES CATACOMB: a simulation system for biologically based network models (http://www.compneuro.org/catacomb/) GENESIS: a general-purpose simulation system for single neuron and network models (http://www.genesis-sim.org/ GENESIS/). For a list of research publication using the GENESIS simulator, see http://www.genesis-sim.org/GENESIS/ pubs.html NEURON simulator: a general-purpose simulation system for single neuron and network models (http://www.neuron.yale. edu/). For a list of research publication using the NEURON simulator, see http://www.neuron.yale.edu/neuron/bib/usednrn. html NeuroML: a prototype markup language for describing neuroscience simulation models (http://www.neuroml.org/) SenseLab: prototype databases of cell properties, membrane properties, and neural models (http://senselab.med.yale.edu/ senselab/)
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