Nodus 3.1. Manual. with Nodus 3.2 Appendix. Neuron and network simulation software for Macintosh computers

Transcription

1 Nodus 3.1 Manual with Nodus 3.2 Appendix Neuron and network simulation software for Macintosh computers Copyright Erik De Schutter, 1995

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3 Copyright This manual and the Nodus software described in it are copyrighted with all rights reserved. This manual or the Nodus software may not be copied without written consent of Erik De Schutter (E.D.S.), except in the normal use of the software or to make a backup copy. Nodus 3 was compiled with MPW Fortran II, Copyright Absoft Corporation Macintosh is a registered trademark of Apple Computer, Inc. Limited Warranty E.D.S. will replace the media on which this software is distributed at no charge if you report any physical defect within 90 days of purchase and return the item to be replaced. E.D.S. makes no warranty or representation, either express or implied, with respect to this software, its quality, performance, merchantability, or fitness for a particular purpose. As a result this software is sold as is, and you, the purchaser, are assuming the entire risk as to its quality and performance. In no event will E.D.S. be liable for direct, indirect, special, incidental, or consequential damages resulting from any defect in the software or its documentation. The warranty and remedies set forth above are exclusive and in lieu of all others oral or written, express or implied. This manual describes Nodus version E.D.S. does not guarantee that later versions of Nodus are accurately described by this manual. In the Appendix changes made in Nodus are described. Information and Service For further information or to report any problem or difficulty with the Nodus 3 software, please write to: Dr. E. De Schutter Born Bunge Foundation University of Antwerp (UIA) Universiteitsplein 1 B2610 Antwerp Belgium Fax 323/ , telex 33646, erikds@reks.uia.ac.be Reference Papers describing Nodus have been published. Please refer to thes papers when publishing results of modeling with Nodus. E. De Schutter: Computer software for development and simulation of compartmental models of neurons. Computers in Biology and Medicine 19: (1989). E. De Schutter: A consumer guide to neuronal modeling software. Trends in Neurosciences 15: (1992). E. De Schutter: Nodus, a user friendly neuron simulator for Macintosh computers. in Neural Systems: Analysis and Modeling, F.H. Eeckman editor, Kluwer Academic, Norwell MA (1993). The author would appreciate receiving reprints of any paper referring to work with Nodus.

4 TABLE OF CONTENTS I. Introduction 5 II. Installing Nodus III. Modeling with Nodus 9 Introduction 9 The Mathematics of Compartmental Modeling 11 Passive Membrane Models 13 Nodus Implementation of Compartmental Models 15 Excitable Membrane Models 20 Simulation of Synapses and Connections 22 Integration Methods 24 IV. Nodus Reference 27 Nodus Menus 27 Nodus Files 28 Making New Models 36 Making New Simulations 41 Selecting Simulation Parameters 47 V. Nodus Menu Commands 55 Apple Menu 55 File Menu 55 Edit Menu 62 Simulation Menu 64 Network Menu 80 Neuron Menu 83 Conductance Menu 99 VI. Using the Examples 103 Demo Files 103 Realistic Models 112 VII. Appendix 115 Maxima for Memory Storage (Nodus 3.2) 115 Shift and Option Key Menu Modifications 116 Bugs and Problems 117 Import Formats 118 References 120 Nodus 3.2 Update 125 Nodus ftp site 136 VIII. Index 137

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6 I-6 Introduction I. INTRODUCTION There is no Nodus 3.2 manual (yet). This Nodus 3.1 manual has been updated partially to document changes made in Nodus 3.2. This chapter and the next one have been rewritten. However, most of the differences between Nodus 3.2 and Nodus 3.1 are listed in the 'Nodus 3.2 Update' section of the Appendix. A complete rewrite of the manual will be done when additional changes have been made to Nodus. Nodus is a powerful, easy to use application for simulation of neuron and small network models. Compartmental models of neurons with voltage dependent ionic conductances described by Hodgkin-Huxley like equations and with pre- and postsynaptic sites can be created. Nodus runs simulations of these models. Experiments can be performed by injecting different kinds of currents, by simulating voltage clamps or by blocking ionic currents. Simulations can produce a wide variety of color graphics and text outputs. This manual contains all the information necessary to use Nodus 3. It assumes that you have read Macintosh, the owner's guide and are familiar with menus, scrolling, editing text and using the mouse. Read first II. Installing Nodus 3.2, which describes how to install Nodus 3.2 on your hard disk. An introduction to the theoretical aspects of modeling is presented in III. Modeling with Nodus. IV. Nodus Reference describes in depth the Nodus user interface and gives practical instructions about how to use Nodus. V. Nodus Menu Commands has a complete description of each menu command. VI. Using the Examples describes all the example files on the master disk and gives step by step instructions for using them. The Appendix contains some useful lists and references to the modeling literature. Users with no modeling experience should first read chapter III and consult the modeling literature. Refer to chapter VI to try out Nodus with the example files and discover the crucial steps in using Nodus. Every user (including experienced Nodus 2 users) should read chapter IV. The manual uses text attributes to help the user in relating information to the Nodus user interface. All menu commands and dialog window button names and text box titles are printed bold. All Nodus file types are printed italic when they are mentioned for the first time in a paragraph. Warnings are underlined. Titles of manual chapters and file names are quoted. If you want more information on Nodus 3 or if you have any comments or suggestions please contact the author. User feedback helps me to adapt future versions of Nodus better to the needs of neurobiologists. Please do not distribute copies of this program; it contains copyrighted software. If you have obtained a free copy you can still send your name and address to the author, you will be added to the Nodus mailing list and receive information on updates, etc.

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8 II-8 Setting up Nodus II. INSTALLING NODUS 3.2 The required hardware is a Macintosh computer with a or microprocessor and a numeric coprocessor (FPU) or a microprocessor having at least 4 megabytes (Mb) of memory and running System 6 or 7. You will get optimal performance using System 7. Nodus 3.2 requires at least 2.0 Mb free RAM to run, it has been preset to use 2.2 Mb. The size of the application heap memory mainly determines how may windows can be shown on the screen simultaneously. Simulation plot windows with color graphics can take up a lot of memory (up to 0.1 Mb). If available memory gets low, Nodus will warn the user and suggest to close some windows. Nodus 3.2 comes with a Nodus_Preferences file. Nodus 3 cannot run without this file. Nodus cannot make a default preferences file so be sure to have a backup! Note that the preferences file has been personalized for each registered user, do not mix files from different registered users. The Nodus Preferences file should be placed together with the Nodus 3.2 application file (i.e. both on the desktop or both in the same folder) or it can be placed in Preferences folder (System 7) in the System Folder. Nodus 3.2 is available in two versions: the standard version (Nodus 3.2) which runs on any Mac with 68020/30/40 CPU and a FPU and the Quadra version (Nodus 3.2Q) which runs only on Macs with a CPU (it is considerably faster than 3.2). Nodus does not run on Macs without a floating point unit (FPU) or with a disabled FPU (like the 68LC040). However, if you want to test Nodus, the Nodus ftp-site has a software patch SoftFPU that can replace the FPU (this is very slow though, so you want to upgrade your Mac if you are going to use Nodus extensively). You are allowed to use either one or both versions of Nodus. Both versions are completely file compatible with each other and use the same Nodus_Preferences file. Both versions are also 32-bit clean and compatible with the cache on the Quadras (using the cache results in faster performance).

9 Setting up Nodus II-9

10 III-10 Modeling III. MODELING WITH NODUS Introduction 9 The Mathematics of Compartmental Modeling 1 1 Linear cable theory...11 Compartmental models...11 Equivalent circuit...11 Membrane surface...13 Passive Membrane Models 1 3 Morphology...13 Cable parameters...14 Nodus Implementation of Compartmental Models 1 5 Reduced models...15 Weight factors...16 Node and branch connections Tree format model...17 An example of a model reduction...18 Excitable Membrane Models 2 0 Hodgkin Huxley equation...20 Equivalent circuit...21 Simulation of Synapses and Connections 2 2 Presynaptic transmitter release...22 Postsynaptic conductance...23 Connections between neurons...24 Integration Methods 2 4 Hybrid Euler method...25 Fehlberg method Experiments...26 Introduction This chapter describes how Nodus models neurons and small networks. It introduces the underlying mathematics and has some practical suggestions about making compartmental models. Modeling methods particular to Nodus are defined. Finally it specifies the different integration methods that are available. This chapter does not pretends to be a complete cookbook about modeling. Different approaches to modeling are possible. The big categories are single cell versus (small) network simulations and passive membrane versus excitable membrane models. All these models describe the electrical status of neurons. There is now a lot of interest in adding (limited) simulations of chemical events to these models, for example the movement and concentration of calcium ions. Which type of model is selected depends on the available experimental data and the particular interests of the modeler. An important issue in creating a model is the specification of the model parameters. An ideal model would have all its parameters based on hard experimental data. Usually only subsets of data are available, so that several parameters need to be estimated. Different model categories contrast in which model parameters are emphasized.

11 Modeling III-11 In passive membrane models detailed morphology is usually the key issue, while the abstraction is made that for the studied phenomena voltage dependent ionic currents are not critical. Excitable membrane models emphasize the ionic currents, with often a very simple morphology of the neuron (one compartmental models). In most network models connections between the neurons are more important than their morphology or their membrane properties. Nodus is suitable for simulations of all these model categories, but in always a realistic, biological approach is emphasized. All parameters have to be specified by the user, Nodus does not implement random variation of parameters. In the near future new modeling options will be added, including ion concentrations. The consensus in the modeling community about what are the characteristics of a good model evolves. The aspirant modeler should consult the recent literature, which has become quite extensive. See the Appendix for a list of good introductory papers and books. Figure III/1: compartmental models (right) can preserve the detailed morphology of the original neuron (left).

12 III-12 Modeling The Mathematics of Compartmental Modeling Nodus uses compartmental models to simulate the electrophysiology of neurons. The compartmental approach is easy to use and makes accurate modeling of both anatomy and electrophysiology possible. Linear cable theory The linear cable theory makes exact mathematical formulations of neurophysiological events possible, but its practical use for realistic modeling is limited. Linear cable theory describes the electrophysiological events in a neuron by a single differential equation and for every electrical event specific solutions of this equation have to be obtained. The complexity of these solutions is very sensitive to the complexity of the model so that only simple models (e.g. ball and stick neuron model, 3/2 power branching, etc.) can be handled in practice. Linear cable models are often inadequate to simulate the behavior of real neurons with local interactions among dendrites, inhomogeneous distribution of synapses and ion channels, etc. It is also difficult to incorporate non linear behavior (like voltage dependent ionic currents) in linear cable models. Compartmental models Compartmental modeling is derived from linear cable theory. The basic assumptions are that the neuron is a system of connected membrane cylinders in which the intracellular current flow is essentially parallel to the cylinder axis, and that the resistance of the extracellular medium is negligible. Instead of describing the whole neuron by one large and complex equation, the neuron is divided into many small parts called compartments and the electrophysiology of each separate compartment is described by a single equation. This equation is simple because the compartments are kept small enough to be considered isopotential (i.e. the membrane voltage is constant over the whole compartment). For passive compartments (i.e. a compartment without voltage dependent ionic currents) the equation is always the same, for excitable compartments terms describing the ionic conductances have to be included. The first step in compartmental modeling is to divide the neuron in compartments, for mathematical reasons only cylinders and spheres are used. It is evident that accurate modeling of morphological details is possible, though it will not always be necessary to make the model as complex as in the example in Fig. III/1. All the compartments in the example, except the soma, are cylindrical with a length L and a diameter D: D L Figure III/2: a cylindrical compartment with length L and diameter D and bilateral connections. Equivalent circuit The compartment can be reduced to an electronic circuit which controls the membrane potential (Fig. III/3). The equivalent electronic circuit consists of a part describing the current flowing through the cell membrane, with a membrane capacitance (CM) and a potential source (E) coupled to a membrane resistance (RM), and a part describing the current flow to other compartments over a cytoplasmic resistance (RI). The values of CM, RM and RI depend on the size of the compartment and on neuron specific equivalent cable parameters (C m, R m and R i ). CM = C m π DL Eq. 1

13 Modeling III-13 RM = RI = 4R i L R m π DL Eq. 2 πd 2 Eq. 3 RI E CM RM Figure III/3: the equivalent circuit for an isopotential compartment. The potential source E in a passive compartment is the difference between the membrane potential and the resting membrane potential, in an excitable compartment it is made up of several ionic conductances coupled in parallel (see Fig. III/7 and Eq. 28). The change in membrane potential ( E) in a compartment #n with a synaptic current I s and an injected current I e can be described by a differential equation: E n E k E n = E n E r + k RI nk + I s + I e t RM n CM n CM n CM n CM n Eq. 4 The first term in equation 4 describes the current flowing through the cell membrane when the membrane potential is not equal to the resting membrane potential (E r ), the second term is the current flow to other compartments (#k) and the last two terms are the synaptic and injected currents. Each compartment in the model can be described by this simple equation: only the values for E, RM, CM, RI and k change. Mathematically compartmental modeling is simple because the same equation is always repeated. The complexity of the neuron model is contained in the size specific parameters RM, CM and RI, in the number of compartments and the connections between them. Nodus contains a loop which calculates equation 4 for each compartment (the parameters are stored in arrays), this loop is entered at least once during each integration step. From the size specific parameters RM, CM and RI two important values can be derived: the time constant τ m and the space constant λ for the compartment. The time constant τ m for the simple capacitive circuit describing a compartment is given by: τ m = RM i CM i or τ m = R m C m Eq. 5 The space constant λ for a cylinder is the length of a cable with the same diameter D that has RM equal to RI:

14 III-14 Modeling λ = R m D n 4R i Eq. 6 The electrotonic length l n of a compartment is its length relative to the space constant λ: l n = L n λ Eq. 7 The electrotonic length l n of a compartment is an important measure in determining the electrical accuracy of the model, if it is too large the compartment cannot be considered isopotential. Another important electrophysiological parameter for neurons is the input resistance R N (Eq. 8). It is measured by injecting a steady current I in the cell and it depends both on the local morphology (different R N in the soma compared to the dendrites) and the cable parameters. R N = E I Eq. 8 Membrane surface Membrane surfaces for the spheres and (hollow) cylinders (Eq. 1,2). are computed by standard geometric equations. For spherical compartments the holes in the membrane surface caused by connections to dendritic or axonal cylinders are subtracted from the computed surface. For cylindrical end compartments (at the tip of a dendrite or axon) the surface of the closed end is added to the surface of the cylinder. Inaccuracies in measurements of the size of compartments (due to shrinkage or membrane folding, see next section) usually underestimate the membrane surface. In Nodus corrections can be made for underestimation of membrane surface at 2 levels. For correction of known measurement errors one can Scale Sizes of all compartments (Neuron menu); lengths and diameters can be scaled separately. The changes in compartment lengths and diameters will affect the values for RM, CM and RI. An alternative solution, appropriate to compensate for membrane folding, is the use of a global scaling factor SF which changes for all compartments CM and RM, while leaving RI and τ m unchanged: RM ' = RM CM ' = SFCM SF RI ' = RI Eq. 9 Passive Membrane Models Passive membrane models do not contain voltage dependent ionic currents. The neuronal membrane is considered linear (i.e. the input resistance is constant) over the voltage range that is simulated. They are the most popular type of model, partially because most of the needed parameters are easy to obtain. Morphology Theoretically the best passive membrane model would be one that is a detailed morphological equivalent of the simulated neuron, with all the cable parameters measured in the same cell. In a lot of preparations it is difficult or impossible to get all these data out of one cell, therefore an acceptable approach is to combine accurate morphology of one neuron (obtained from camera lucida measurements on a neuron labeled by intracellular injection of HRP, Lucifer Yellow, etc.) with average physiological data from several neurons.

15 Modeling III-15 Often some of the cable parameters have to be found by trial and error: several values are used in the model till experimental measurements can be replicated during simulation. Sometimes one is confronted with the problem of multiple possible model parameter values; where one cannot determine from the available experimental data which of these parameter values is the real one. This problem has been reported for a limited number of neuron models only. One can argue that it does not make much sense to use a detailed morphological model if average physiological data are used. Usually there is a great variance in morphological details of branching (from second or third order on) between functionally equivalent cells, both in vertebrates and in invertebrates and the branching pattern in some neurons may change over time. The accuracy of morphology may be illusory. Because the measurements are done on fixed and dehydrated material, the neuron has shrinked and the amount of shrinkage can be different for diverse parts of the cell. The shrinkage can be estimated by measuring some easy to recognize structures before (for example on photographs of a Lucifer Yellow labeled cell) and after preparation but the results will be approximate. Computation time is related to the size and complexity of a model and simulations of large, detailed models may take a lot of time on microcomputers. It makes sense to reduce the size of the model to increase calculation speed, but there is no golden rule on where the best balance between accuracy and computation time is situated. A good approach is to design first a detailed model, then reduce it to about 30 to 150 actual calculated compartments (see the next section on how to do this) and compare the reduced model and the detailed model for important physiological parameters as input resistance, time constants and attenuation of membrane potentials. Increase the complexity of the reduced model if necessary, or reduce it further if possible. Compartments should be neither too large nor too small. Compartments are supposed to be isopotential, this puts an upper limit on their size. It would seem that very small compartments give a greater accuracy, but this is not true in computer simulations because extremely small (or large) numbers give larger round off errors and may even cause numeric overflow (i.e. the number is too large to be represented in the numeric format used by the computer). Try to keep connected compartments to a similar size, large differences in size between connected compartments may also give larger calculation errors. A good measure for the electrical size of a compartment is its electrotonic length (see previous section) which should be between 0.200λ and 0.015λ. Cable parameters The delicate spot in most passive membrane models are the equivalent cable parameters; in a lot of cases their values cannot be measured experimentally. Most authors take for specific membrane capacitance the universal value of 1 µf/cm2, though one can find measured values of 0.3 to 5 µf/cm2. The discrepancies in these measurements are probably due to underestimation of the membrane surface. Specific membrane resistance varies between different species and between different neurons in an animal. Membrane resistance has to be defined for passive compartments, in excitable compartments it is the inverse of the sum of all voltage dependent ionic conductances. One should try to find an exact value for R m. Two approaches can be used: either R m is calculated from time constant data (Eq. 5) or R m is found by trial and error from input resistance measurements in the model. No matter which method is used, one should always use the other one as a final check on the accuracy of the parameter value. To calculate the time constant accurate measurements of the passive response to a hyperpolarizing current step or the relaxation phase after a depolarizing pulse should be made. The choice of depolarization versus hyperpolarization depends on the nonlinear behavior of the cell and R N : stay within a linear range of membrane potential.

16 III-16 Modeling The potential response is usually determined by several time constants, these can be found by the exponential peeling method. Some time constants may depend on activation or inactivation time constants of ionic currents instead of the shape and cable parameters of the cell. R m can be determined from the slowest time constant τ m of the neuron (Eq. 5). Another method is to find R m with simulations. The input resistance at the soma should be measured in the original preparation in the linear range of membrane potential. If the electrode leak was small this value is mainly determined by the size and shape of the cell (which are supposed to be replicated accurately in the model) and the membrane resistance. R N for the neuron model is measured in simulations with different values for R m till a good approximation is found. If R N can also be measured at others sites (axon, dendrites) one should do so. This method is superior to using exponential peeling data if R m is not constant in the cell. Some authors found that to get an accurate model, R m had to be much larger in the dendrites than in axon and soma (reflecting differences in local ionic conductances). Nodus supports variable membrane resistance in a neuron model. Cytoplasmic resistance can be measured if the axon or a similar structure can be penetrated at two different sites to measure the attenuation of an injected current pulse between these two positions. Often the neuron is too small to be sticked at two sites and even if the experiment is possible the actual measurements may be made inaccurate by bridge imbalances, etc. Most authors therefore use values obtained in other neurons of the same animal or of the same phylogenetic class. Simulations should be done for a range of cytoplasmic resistances if no measured values are available. Nodus allows free mixing of passive and excitable compartments in a neuron model. Any compartment can also have postsynaptic currents and/or presynaptic transmitter release. Nodus Implementation of Compartmental Models Nodus uses an experiment look-a-like approach to modeling neurons and emphasizes the importance of morphology. This is implemented by placing the nodes of the equivalent circuits at the center of the compartments (Fig. III/4) and by using alternative methods to lump branches together. 1 2 Figure III/4: equivalent circuit for a node connection between 2 cylindrical compartments. Reduced models The computation time for a simulation depends on several factors: the integration method, precision of the method, size of the model and total number of conductances. One can simplify the model of a neuron to keep the number of compartments (and the computation time) reasonable. The classic way to simplify dendritic trees is to lump branches together into an equivalent cylinder, i.e. to use one large dendrite instead of several small ones. An extreme example of this method is the ball and stick model used in linear cable models: one spherical soma compartment and a large cylindrical compartment equivalent to the axon, the dendrites, etc.

17 Modeling III-17 Alternative methods are used in Nodus to decrease the number of compartments with a constant apparent morphological detail: weight factors and branch connections. These techniques are conceptually simpler and make switching between reduced and complex models easy. Nodus can make reduced models automatically. While these methods can make models more manageable, one should realize that there is always a loss in accuracy. Inexperienced users should not use weight factors or branch connections; one make nice models in Nodus without ever using them. Nodus defaults always to (standard) node connections and a neutral weight factor (of one). Weight factors Weight factors are a way to lump branches together without changing their size. Instead of lumping n branches together into one huge branch, an average branch is made and connected n times to the parent compartment (Eq ; Fig. III/6); n is the weight factor. All compartments in the averaged branch are calculated only once by Nodus; there can be no difference in membrane voltage between the n branches. Mathematically this is similar to lumping branches together, the advantage is that a normal", morphological size of the branch is used. Node and branch connections Branch connections are used to increase the electrical accuracy for some connections in the model, without increasing the number of compartments. In a classic compartmental model all compartments are connected by node connections: the end of one cylinder is connected to the end of the next cylinder, etc. (Fig III/4) If several compartments connect together at one node (for example where a branch splits), each compartment is connected to all other compartments. RI node, 1 2 = 2 R i n L 1 πd L 2 2 πd 2 Eq. 10 RI node, 2 1 = nri node, 1 2 Eq Figure III/5: equivalent circuit for a branch connection between 2 compartments.

18 III-18 Modeling If more than 2 compartments are connected by nodes to the same parent compartment, there is electrotonically no difference between a binary branch (the parent splits into 2 similar sized branches, typical for vertebrate neurons) or a continuation of the parent branch (Nodus assumes this for the first connection) and a smaller side-branch (invertebrate neurons). If the number of compartments in a parent branch is reduced, the accuracy of the model is decreased because several side-branches, which do not originate from the same point on the parent branch, will have to be connected to the same node. Branch connections connect to the center of the cylindrical compartment of the parent branch instead of to the end (Fig. III/5). If several compartments make branch connections to the same compartment they are not connected to each other. Current flow has to pass through the parent branch first, which is more accurate if the compartments are equivalent to branches which do not connect to the same point on the parent branch. RI branch, 1 2 = 2 R i n L 2 2 πd 1 Eq. 12 RI branch, 2 1 = nri branch, 1 2 Eq. 13 Branch connections allow more accurate modeling of complex branching from a lumped parent branch. For each parent compartment connections can be made at two sites (as a node and as a branch), with a distance of halve the length of the compartment between them. To use branch connections the tree format model option has to be on. Branch connections are not necessary to use compartmental models; node connections are sufficient. Use branch connections only if needed and if you understand the concept. They are particularly useful in modeling invertebrate neurons were a lot of thin dendrites originate close to each other from a thick neurite. They can also be used to model dendritic spines. They are less useful in neurons were dendrites show binary branching (as in vertebrate neurons). Tree format model The tree format option for neuron models helps in automatically creating and checking connections between compartments. A tree format neuron model has to be to defined in a centrifugal way. First the soma is defined, then branches originating from the soma, etc. In other words: if each division increases the order of the more distal sections of the branch, then the low order compartments should be defined before the high order compartments. The user is free to define either all compartments of a main branch first and then the compartments of its side-branches, or to define the compartments of a complete side-branch before the more distal parts of the main branch are defined. If the tree format model is selected Nodus enforces some simple rules for the neuron model: - compartments can be connected by only one connection. - a spherical compartment cannot be connected to another spherical compartment. - all connections and their weight factors are defined at the parent compartment (the compartment proximal to the soma, having the lowest order of branching). Connections can be changed or deleted only in the parent compartment definition window. - connecting compartment #n to #k automatically connects #k to #n. - if compartment #k and #l are connected to the cylindrical compartment #n by node connections, then #k and #l are also connected to each other by a node connection ( cross connection ). Node connections to spherical compartments are not interconnected. - compartments connected as a branch can have only one parent compartment. - reverse weight factors (i.e. child to parent) are always equal to one.

19 Modeling III-19 An example of a model reduction The goal of simplifying a model is to reduce the number of compartments to a minimum with respect for significant morphological details like the number of side-branches, the distance between their point of origin and their cross section and membrane surface. The use of weight factors and of branching connections to obtain this goal is demonstrated in Fig. III/6. Fig. III/6 shows a part of a neuron model: a branch with 5 side-branches, connected to a larger branch through compartment C. This model is a simple example: usually the side-branches themselves will also consist of several compartments. In the original model all connections are node connections and 13 compartments are necessary to describe the complete branch. The first step in making a reduced model is to lump the compartments of the parent branch together. The choice of which compartments are to be lumped depends on two factors: the diameter of the compartments should be comparable and the distance between the branches connected to these compartments should be either small or have no influence on the simulated electrophysiological events. In this example the choice is simple because some compartments have already a common diameter; 3 compartments can be fused to a long compartment D in and 4 small compartments to a compartment E (Fig. III/6B). If the diameters of the compartments selected for lumping are not identical, an average diameter has to be calculated (Eq ). A A F G C D E B B A F G n B b b b b b C D E n n C A F G 2 3 C D E B Figure III/6: Successive steps in reducing a branch of a compartmental model (A). First the main branch is lumped together and all node connections (n) are converted into branch connections (b) (B). Then the small branches are lumped together and connected with weight factors (C).

20 III-20 Modeling The connections between compartments C and D and between D and E remain node connections (n). The connections between the side-branches and the parent branch (F to D and G to E) become branch connections (b). The position on the parent branch is more accurate than it would be with a node connection at the end of the respective parent compartments, but the distance in origin between the 2 side-branches F and between the 3 side-branches G has disappeared. The effect of this reduction on the electrical accuracy of this model is limited because the branches F and G are small and far removed from the soma of the model (interposition of several compartments: D, C, A, ). The next step is to lump the side-branches together into one, average branch and to connect this branch n times to the parent branch; the 2 branches F and the 3 branches G are lumped together (Fig. III/6C). The size of the lumped compartment is not just the average of the lengths (L) and diameters (D) of the original compartments. The important electrical sizes are the membrane surface of the compartment (for CM and RM) and its length divided by cross section (for RI). Calculate first the surface (S) and length divided by cross section (C) of all the (cylindrical) compartments: S = πld Eq. 14 4L C = πd 2 Eq. 15 Calculate the averages of the surfaces (S a ) and cross sections (C a ) and use these to calculate length (L a ) and diameter (D a ) of the lumped compartment: L a = S a πd a Eq. 16 4S a D a = 3 π 2 C a Eq. 17 An important characteristic of branch connections is their asymmetry. From parent compartment D it looks like there are 2 child compartments F, but from F there is only one D. This gives a weight factor of 2 for D->F and a weight factor of 1 for F->D. For all connections only the weight factor from parent to child has to be supplied, the reverse weight factor is calculated and controlled by Nodus. The use of branch connections and weight factors has reduced the number of compartments in this part of the example from 13 to 5! Of course there is a decrease in morphological accuracy in the reduced model. In most simulations this will not matter. As long as the branch beginning at compartment C is completely passive and the membrane voltage is monitored in compartments connected to A or B, the effect of the reduction on for example R N will be small. The reduction can be acceptable for current injections (as for example in synapses) in all the side-branches F and/or G together, if membrane potential is monitored in compartments connected to A or B: the effect of the reduction will be greater, but in most cases not substantial. The reduction is not acceptable if one monitors membrane voltage in one of the reduced compartments or if one wants to study local interactions between side-branches F and G. The example shown here is reproduced in the neuron models Test-cell 5a, Test-cell 5b and Test-cell 5c on the Nodus Master disk. Using the examples shows numerical results for current injections in these models. It is important to check the accuracy of a reduced model. The best way to do this is to compare it with the original, complex model, but if this is not available one should compare with a model that has at least twice the number of compartments.

21 Modeling III-21 Compare the reduced and complex model by repeating simulations of the relevant experiments. Significant values are in most cases values that can be compared with biological measurements like the R N and τ m measured in the soma (in some models other structures like a large axon may be more important) or size and timing of action potentials and synaptic events. The acceptable relative error will vary, but in most cases errors of up to 5% will be no problem (one compares with biological events which show even greater variability). Another way to keep reduced models to an acceptable level of accuracy, is to start with the complete, complex model and then reduce it progressively till the error becomes too large. Excitable Membrane Models Compartments can have excitable membrane in Nodus. Any voltage dependent ionic conductance that can be described with Hodgkin-Huxley like equations can be included. The ionic conductances are defined apart from the neuron model definition. Hodgkin Huxley equations Hodgkin and Huxley based their equations on a model of the conducting channel that consists of activation and inactivation gates, each with a voltage dependent probability of being open (M and H). An important simplification in this model is that though several activation and inactivation gates can be connected in series in one channel, all act independently of each other and all obey the same voltage dependent equations. Transitions between the open and closed form of a gate follow the simple reaction: α M closed β M open Eq. 18 M closed = 1 M open Eq. 19 The rate factors α and β are voltage dependent. The equations for these rate factors are of the general form: α = b ( d + E ) c + e (d + E ) / f or α = a + be c + e (d +E )/f with a = d b Eq. 20 With a, b, c, d and f as terms which are specific for each conductance and E is the membrane potential. Singularities may be found in Hodgkin Huxley like equations when c=-1. If E is the opposite of d then the denominator becomes zero. Nodus obtains a value for α at this singular point by interpolation between values around the point, but the rate factor functions may still behave irregularly. Opening and closing of the gates can be described with the differential equation: M t = α( 1 M ) βm Eq. 21 From this equations follows the steady state open gate fraction M : M = α α +β Eq. 22 and the gate time constant τ M : 1 τ M = α +β Eq. 23

22 III-22 Modeling The ionic conductance itself can be described by one of the following equations: G (E,t ) = g Eq. 24 G (E,t ) = gm x Eq. 25 G (E,t ) = gm x H z Eq. 26 g-bar is the maximum conductance, which may be proportional to the membrane surface of the compartment (in the Nodus dialog windows and printouts it is marked as Gmax). Equation 24 describes a leak conductance, Eq. 25 and 26 are true voltage dependent ionic conductances with respectively x activation gates M, and x activation gates M and z inactivation gates H. Note that in Hodgkin-Huxley like equations inactivation is treated as another channel, but as inactivation increases the channel closes (go to zero) and vice versa. Equivalent circuit RI CM E Na G Na E K G K E L G L Figure III/7: equivalent circuit for a compartment with voltage dependent Na + and K + currents and a leak current. In compartments with excitable membrane the equivalent circuit is expanded, it includes for each voltage dependent current a variable conductance and a potential source (Fig. III/7). The circuit shown is for a compartment with the classic Hodgkin and Huxley currents. The ionic current I is determined by the conductance and its reversal potential E j : I k = G k ( E, t )( E E k ) Eq. 27 The first term in equation 4 is replaced by a term describing the voltage dependent ionic currents: E n t = j ( ) E n E j G j E n,t ( ) CM n + k E n E k RI nk CM n + I s CM n + I e CM n Eq. 28 For j ionic currents the voltage and time dependent conductance G j (E,t) (conductance is the inverse of resistance) is calculated and multiplied with the potential source which is determined by the difference between the membrane potential and the conductance specific reversal potential E j.

23 Modeling III-23 To model action potentials and other voltage dependent phenomena accurately, voltage clamp data for all ionic currents present in the neuron should be obtained to construct Hodgkin-Huxley equations. This represents a lot of work, and may not be experimentally feasible. Therefore a lot of authors use published equations. Try to measure the time constant, reversal potential and maximum conductance of the important ionic currents and adapt the standard equations, because these values vary between preparations. If these values cannot be measured directly with voltage clamps, find indirect measurements (refractory period, period of firing, etc.) and correct the values in the simulation by trial and error. Equations derived from voltage clamp experiments may not perform as expected in models where all the ionic currents were included. This is usually a consequence of the artificial conditions under which voltage clamp experiments were done (blockers, unusual ionic concentrations, etc.); in such cases G max may be changed. A small library of equations for voltage dependent ionic conductances is included on the Nodus Master disk. Nodus can plot any factor related to ionic conductances: α, β, M, H, τ M, τ H and G in factor versus membrane potential plots and M, H, τ M, τ N, G and I dynamically during simulations. These plots can help in developing new Hodgkin-Huxley like equations or in adapting existing equations to describe the conductances in the model. Simulation of Synapses and Connections Several functions to simulate synapses are available to the user. One can also implement special functions with user-defined processes (not available in Nodus 3.1). Presynaptic transmitter release In network models presynaptic activity fires synapses. Presynaptic transmitter release can be constant or variable. In both cases transmitter is released only when the presynaptic potential is higher than a threshold potential. With constant transmitter release the release is signaled to the postsynaptic cell once each time the threshold is crossed, where it arrives at a time t 0 which is needed to compute the postsynaptic conductance (use Eq. 33 or 34). With variable transmitter release the amount of transmitter is signaled continuously to the postsynaptic site (as long as the presynaptic potential is over the threshold); there is no t 0 available (use Eq. 32, 35 or 36). Three types of variable transmitter release T s are available: Linear to presynaptic potential: T s = b + f E E th ( ),T s b Eq. 29 The minimal transmitter release is the base amount b. The variable part is determined by the factor f and is computed relative to the threshold potential E th. Exponential to presynaptic potential: ( T s = b + fe E E th) /c,t s b Eq. 30 The minimal transmitter release is the base amount b. The variable part is determined by the factor f and the characteristic potential c; it is computed relative to the threshold potential E th. Process dependent T s = U( E,t, ),T s b Eq. 31 The transmitter release is determined by a user defined function U which is defined as a separate process.

24 III-24 Modeling Postsynaptic conductance Nodus has five types of postsynaptic conductances G s, which are grouped in 2 types. The type 1 synapses should be used if presynaptic transmitter release is constant or in single neuron models; type 2 synapses should be connected to variable transmitter presynaptic sites. Constant conductance (type 2) G s = gt s Eq. 32 g-bar is the maximum conductance (G max ), which may be proportional to the membrane surface of the compartment. The postsynaptic conductance is completely determined by the presynaptic variable transmitter release T s. Alpha function (type 1) G s = gt s t α e αt / τ Eq. 33 The postsynaptic conductance is determined by the alpha function, which is controlled by 2 factors: α and τ. The alpha function is computed relative to the start of the synaptic firing (t 0 = 0). Nodus uses a normalized version (i.e. 0<=G s <=1), it should be used with constant transmitter release. Factor α determines the steepness of the initial slope of the synaptic conductance, usually the value α=1 is used. The time to peak τ determines when the alpha function reaches its maximum conductance. Dual exponential function (type 1) ( ) G s = gt s e t / τ o e t / τ c Eq. 34 The postsynaptic conductance is determined by the dual exponential function, which is controlled by two time constants. The dual exponential function is computed relative to the start of the synaptic firing (t 0 = 0). Nodus uses a normalized version (i.e. 0<=G s <=1), it should be used with constant transmitter release. The open time constant τ ο determines how fast the conductance increases, while the close time constant τ c controls its decrease. Conductance dependent (type 2) G s = gt s G( E,t) Eq. 35 The postsynaptic conductance is voltage dependent and controlled by a Hodgkin-Huxley equation. Conductance dependent synapses should be used with variable transmitter release; the synapse can only be shut off by decreasing T s to zero. One can model NMDA-receptors with this type of postsynaptic conductance. Process dependent (type 2) G s = gt s U( E,t, ) Eq. 36 The postsynaptic conductance is determined by a user defined function U which is defined as a separate process. The postsynaptic current I s is given by: ( ) Eq. 37 I s =G s E E s E s is the synaptic reversal potential.

25 Modeling III-25 The electronic circuit for a passive compartment with one synaptic conductance is: RI CM E S G S E RM Figure III/8: equivalent circuit for a passive compartment with one synaptic conductance. Postsynaptic currents can be used in both network and single cell models in Nodus. In network models the presynaptic cell drives the synapse; in single cell models the user has to specify a synaptic firing time. To improve computation speed the user can specify a synaptic switch off time. This controls the switching off of alpha function or dual exponential function postsynaptic conductances. Without this switch off all synapses that have started firing would go on continuously, even when the actual conductance is infinitesimally small. The synapses will be switched off when the simulation time equals t 0 plus switch off time. Note that other types of synapses are always connected to variable presynaptic transmitter releases, they will be switched off when transmitter release goes to zero. Connections between neurons Connections between neurons in a network are defined in a straightforward manner. A presynaptic site is connected to a postsynaptic site and a delay time is specified. The delay time can be used to simulate axons; the presynaptic transmitter release will arrive at the postsynaptic site after the delay time has passed. A single presynaptic site can have connections with multiple postsynaptic sites (simulating multiple branchlets at the end of the axon). A single postsynaptic site can also have multiple presynaptic sites. One has to be careful about connecting the right type of transmitter release with the right synapse (type 1 or 2). Nodus does not check for this kind of error in connections and the result will be synapses that behave strangely (not firing at all, or firing infinitely). One should connect constant transmitter release to postsynaptic type 1 synapses (alpha functions or dual exponential functions) and variable transmitter release to type 2 synapses (constant, conductance or process dependent). Integration Methods Two integration methods are available: a fast hybrid Euler method for initial exploratory experiments and an accurate, slower fifth order Runge-Kutta method for confirmation of the final results, both with variable time steps. All simulation variables and integration steps are computed in double precision.

26 III-26 Modeling Both integration methods are sensitive to the stiffness of the differential equations, they will compute slowly if membrane voltage or concentration is changing very fast and/or if very small compartments are present in the model. The integration methods control the minor time step t. Nodus uses in simulations also a major time step, which is specified by the user. The major time step is used to control the switching on or off of experiments. The minor time step is always less than or equal to the major time step. Hybrid Euler method The hybrid method (Moore, J.W., and Ramon, F.: On numerical integration of the Hodgkin and Huxley equations for a membrane action potential. J. Theor. Biol., 45 (1974) ) is a fast forward Euler method. (Eq. 38). The accuracy of the computation is controlled by setting absolute maxima for changes in membrane potential and concentration in any compartment. The minor time step is adapted dynamically to keep the changes below the maxima. E = E 0 + t E 0 t Eq. 38 For excitable membrane models the hybrid method simplifies the differential equations describing an excitable compartment (Eq. 20, 25 and 28) by making the rate factors for (in)activation (α and β) voltage independent for each time step t. The time constant τ M (Eq. 23) and the steady state open gate fraction M (Eq. 22) are determined from α and β. The approximate solution for M becomes then a simple exponential function: = M + ( M 0 M ) e t /τ M M Eq. 39 To increase computation speed a table of the (in)activation factors M and N is calculated before the simulation starts. An information dialog window shows the progress of this computation. The maximum error shown in the dialog window is estimated by halve the difference between the calculated value and the value obtained from interpolation between the preceding and the next entry in the table. If this error becomes too large one cannot trust the results from the hybrid Euler method. During the simulation the actual (in)activation factors are determined by linear interpolation between the values in the table. The hybrid method may oscillate when an equilibrium is reached in the simulation (for example after a current injection), more so in small compartments than in large ones. The error of the hybrid method relative to the Fehlberg method is usually smaller than 1%, input resistances and period of firing are less accurate than for example maximum amplitude of an action potential. Fehlberg method The Fehlberg method (Forsythe, G.E., Malcolm, M.A., and Moler, C.B.: Computer methods for mathematical computations. Prentice-Hall, Englewood Cliffs (1977) pp ) is based on the Runge-Kutta integration method. In its standard formulation the Runge-Kutta method is a fourth-order integration method: E = E 0 + k 1 + 2k 2 + 2k 3 +k 4 6 Eq. 40 k 1 = t E 0 t 0