For a more complete summary of how NeuronC works, refer to the modeling paper: Smith, R.G. (1990) "NeuronC: A language based on C for dynamically simulating neural networks of the visual system."

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Another requirement of NeuronC is a video card with enough resolution to display high-res graphics on the screen. NeuronC can be run without graphics to produce a list of recordings but it is useful to display this information in graphical form.

Some excellent computer choices for running NeuronC include:

1) Notebook computer with Pentium 3000 MHz and: 1 - 4GB RAM Memory 100-1000 GB disk 1280x900 LCD display DVD-RW drive USB flash drive Linux operating system (download from ftp://carroll.aset.psu.edu/pub/linux/distributions) 2) Workstation/Server with: 8-12 core Opteron CPU (e.g. Opteron 6328, 3.2 GHz) 4-16 GB RAM memory 256 MB video card 200-1000 GB hard disk 1280x1024 monitor DVD-RW drive USB flash drive Linux operating system (download from ftp://carroll.aset.psu.edu/pub/linux/distributions) 3) Mac Mini, Mac Pro, or iMac with: Intel i5 or i7, 2.5 - 2.7 GHz, dual-core 1-4 GB RAM memory Radeon 6630M video card 200-500 GB hard disk 1280x1024 monitor OSX 10.7 - 10.9 4) PC with Xeon/Opteron with: 4-8 GB RAM memory 256 MB video card 200-2000 GB hard disk 1280x1024 monitor DVD-RW drive USB flash drive 5) Any of the above computers running Linux or Mac OSX. For Windows (or Mac OSX), install VirtualBox with a Linux virtual disk image (available at ftp://retina.anatomy.upenn.edu/pub/rob/linux.vdi.142.zip, ~7GB). 6) Array of 2-50 Linux computers running MOSIX, a parallel-job system (www.mosix.org) Opteron CPUS, 3.2 GHz Any Linux distribution Local area network (LAN), GB network interfaces.

The directory "nc" contains all the files needed to run NeuronC. The "nc" directory contains the source codes for "nc", "stim" and "plotmod", along with some examples of neural circuit programs. The "pl" directory contains the source codes for "vid" and "hpflt" and "ibmflt" which are the graphics drivers for the video screen and plotter, respectively. The "plot" directory contains the Ampex plot package which is the source code for the graphics language that "vid" and "hpflt" use. Normally the plot subroutines are compiled into a library called "libP.a" which is used when linking nc and plotmod.

1) Uncompress and "un-tar" the distribution. This will make a directory called "nc" and several subdirectories:

tar xvozf nc.tgz2) Go to the "nc" directory and type "make":

cd nc makeThis makes the programs "nc", "stim", "plotmod" and "vid". 3) After these programs are compiled by "make", they are linked (given a shortcut to) the nc/bin folder. To allow them to run from any other folder, you can include this nc/bin folder in your path. Then you must edit your .profile (bash) or .cshrc (tcsh) shell startup files to include ~/nc/bin in your program execution path. If you are using BASH (standard on many Linux and Mac systems): You can also do the same thing by creating a text file ~/.bashrc or ~/.profile (i.e. in your home folder) and adding

PATH="~/nc/bin:$PATH"You can check this by entering "vid" again -- it should pop up immediately.

If you are using the "tcsh" shell (recommended on Linux and Mac systems), you can add this to your ~/.cshrc file:

setenv PATH "~/nc/bin $PATH"

Alternately, on a Mac, you can add to your path by going to /etc/paths.d and adding a file that contains /Users/xxx/nc/bin (where xxx is your login name or name of your home folder).

Another way to allow these programs to be run from any other folder is to link these programs to a folder that is already in your path. For example, to install "vid", the graphics driver for "nc", you can go to the "pl" directory and:

mkdir ~/bin cd ../pl ln vid ~/bin // to use a ~/bin sub-folder in your home folder or ln vid /usr/bin // if you have root permission4) Go to the "nc/src" subdirectory and install "stim","nc", and "plotmod":

cd ../src ln nc stim plotmod ~/bin // to use a ~/bin sub-folder in your home folder or ln nc stim plotmod /usr/binIf you don't have a "bin" subdirectory in your home directory, you can "ln" the binaries anywhere else you like.

For some of the old Mac compilers, the -O and -O3 optimizations caused retsim to crash. You may need to remove the optimizations defined by CFLAGS at the top of nc/src/makefile:

Change this (lines 1-2, nc/src/makefile):

CFLAGS = -O3 # CFLAGS = # for Mac OSXto this:

# CFLAGS = -O3 CFLAGS = # for Mac OSXWhen compiling retsim (see "Retsim: a retinal simulator" in ncman6.html), you will also have to modify the makefile for retsim's dynamic linking. The Mac OSX uses a slightly different way to link dynamic libraries. From the nc/models/retsim directory, you need to edit the "makefile" which is responsible for compiling and linking retsim. You only need to uncomment the line (i.e. remove the "#"):

Change this (line 1, nc/src/retsim/makefile):

CFLAGS = -Ofastto this:

# CFLAGS = -OfastChange this (line 6, nc/src/retsim/makefile):

# CFLAGS = # for Mac OSXto this:

CFLAGS = # for Mac OSXChange this (line 47, nc/src/retsim/makefile)

# OSFLAGS = -DMACOSXto this:

OSFLAGS = -DMACOSX

For example, to edit "makefile": cd nc/doc make cradd cradd.exe < makefile > m && mv m makefile notepad makefile

If you would like to run NeuronC under DOS you can compile it with the Delorie gcc compiler (www.delorie.com). A version of NeuronC that runs under DOS is available (with limited graphics).

The rationale behind using MOSIX to run multiple Neuron-C jobs in parallel is that computational modeling is extremely CPU-intensive but not very I/O-intensive. Models may take several minutes to several hours, or even days or weeks. Usually several models are run, each with a different set of parameters. The model jobs are started on a MOSIX server, either interactively or from batch processing scripts. After a few minutes the server detects the CPU load and migrates each job to equalize the load on all the CPUs on the MOSIX network. Although the jobs run on different machines, their I/O is automatically taken from the main server, completely transparently to the user. To monitor the progress of the jobs, a monitor command ("mosmon") is included that shows the status of the CPUs, their job load, CPU speed, and memory available. When jobs finish on a CPU, other jobs are migrated from other CPUs to it automatically. This makes the most efficient use of CPUs to run the most jobs as fast as possible.

MOSIX is helpful to run multiple jobs in parallel on multiple computers. However, MOSIX is not necessary when you have just one computer with multiple CPU cores, as Linux runing the Symmetric MultiProcessing (SMP) kernel can run your jobs in parallel. As long as the number of jobs is not much more than 2 per CPU core, and the machine has enough RAM to hold all the jobs in RAM without swapping, your Neuron-C/retsim jobs will run fast.

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- Biological modeling
- Numerical modeling
- Compartments
- Cable equation in time
- Synapses
- Reversal potentials, Nernst and GHK equation
- GHK current equation
- Photoreceptor responses
- Space constant, compartment size, and spacial accuracy
- Time steps and numerical integration
- Stability
- Three methods of numeric integration
- Difference equations for implicit method
- Solution methods: Gauss-Seidel iteration with successive over-relaxation
- Derivation of first and second-order implicit calculation of alpha and beta rate constants for Hodgkin-Huxley channels
- Derivation of second-order implicit calculation of calcium diffusion in shells and to calcium buffer.
- Time constant, time steps, and accuracy
- Drawbacks of compartmental modeling
- Parallel computation for compartmental modeling

One way to think of computational modeling is "quantitative imagination"; the computational approach to neuroscience is merely an addition to conceptual modeling. The value of the computational approach is to add a quantitative dimension to our intuition about the function of a neural circuit.

The relation between a biological system and the computational model of it depends on analogy in exactly the same way that a conceptual model does. With this type of analogy, we assert that the model represents the biology in a direct quantitative way, but this assertion depends on our intuitive knowledge and experience about the biological system. Seen in this context, a language for neural simulation provides a concise description of a neural circuit's biological form/function for us, but a set of difference equations for a computer.

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Several numerical method conventions for modeling neural circuitry exist in the literature. Rall's (1959) cable model allowed currents and voltages in neurons with linear passive membrane properties and branching dendritic trees to be described reasonably accurately. More recently, cable theory has been extended into the time domain (Jack et al. 1975), but still typically requires linear membrane conductances. Ideally we would like a way to model any neural element (cable, synapse, voltage-sensitive channel) in this type of continuous model.

The main idea of cable modeling is that a neural dendrite acts like an electric cable with an inside conductor (cytoplasm of neuron) covered by an insulator (membrane). Two ideas form the basis for this analogy:

1) Current flow through a patch of passive membrane is proportional to the voltage difference between the inside and outside of the membrane.

im = (Vout-Vin) / rm (1) where: im = current through membrane rm = resistance of membrane2) Voltage along the inside of the dendrite (the conductor) is reduced by axial current flow through the dendritic cytoplasm.

d(Vin)/dx = ri * I (2) where: ri = axial resistance I = axial current x = axial distance along dendritedifferentiating with respect to x:

d2(Vin)/dx2 = ri * dI/dx (3)but the change in axial current within one small patch of dendrite is precisely equal to the membrane current:

dI/dx = im (4)combining (1) (3) and (4):

d2Vin/dx2 = (Vin-Vout) * ri/rm (5)The solution of this differential equation defines an electric circuit with exponential decay of voltage along the length of a neural dendrite. Voltage and current signals along a passive dendritic tree can be calculated easily using exponential functions (Rall, 1959).

For an infinite cable:

V = I * ri * lambda * e^-x/lambda (6) where: V = voltage along cable at distance x from end I = current injected into cable at end ri = axial resistance of cable, ohm/cm lambda = space constant, sqrt (rm/ri), rm = membrane resistance of cable, ohm*cm

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Each compartment consists of a volume of cytoplasm small enough that its internal voltage is nearly the same everywhere; this condition is called "isopotential". The compartments are connected together by resistors which represent the dendritic axial resistance. The reduction of a neural circuit to a series of connected compartments is justified by analogy because a series of connected isopotential compartments acts very much like a cable if each compartment is small compared with the length of the cable. In addition to the appropriate amount of membrane, each compartment may have other properties such as multiple connections to neighboring compartments and membrane channels with special properties.

<--------------dx------------> <------------dx-------------> compartment 1 -----ri------ compartment 2 ------ri------- compartment 3 V1 < ---------- I1 ---------> V2 <-------- I2 ----------> V3 | Im | Gnd

Derivation of cable equation for compartments:

dV3/dx = V3 - V2 = I2 * ri (7) dV2/dx = V2 - V1 = I1 * ri (8) d2V/dx2 = (V3 - V2) - (V2 - V1) (9)By Ohm's Law, the voltage across a resistor is equal to the electric current multiplied by the resistance:

d2V/dx2 = I2 * ri - I1 * ri (10) = ri * (I2 - I1) (11)by Kirchoff's Law, the current flow into a node equals the current flow out of the node:

I2 - I1 = Im (12)But the membrane current equals the compartment voltage divided by the membrane resistance:

Im = V2 / rm (13)Combining (11), (12), and (13),

d2V/dx2 = V * ri/rm (14)Note that (14) is identical to (5).

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im = (Vout-Vin) / rm + dVin/dt * cm (15)where:

im = current through membrane rm = resistance of membrane cm = capacitance of membrane t = timeIncluding axial voltage drop (as in equation 3):

d2Vin/dx2 = (Vin-Vout) * ri/rm + dVin/dt * cm (16)where:

ri = axial resistance rm = membrane resistance cm = membrane capacitance t = time Vin = voltage inside cellThis is the standard equation that neurophysiologists use to describe voltage in a cable in space and time.

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To simulate the mechanism of neurotransmitter release, NeuronC defines a static "transfer function" which defines how much neurotransmitter is released as a function of presynaptic voltage. The neurotransmitter action is modified by a time function which delays and low-pass filter the neurotransmitter action. The time-filtered neurotransmitter level defines the conductance of the postsynaptic channel.

This synaptic model is very simplified because it leaves out details such as: presynaptic calcium channel activity, calcium activated vesicle release, neurotransmitter binding kinetics, etc. These mechanisms are available in NeuronC by selecting special values for the synapse (calcium sensitivity, postsynaptic channel type, etc.). However, the simplified NeuronC synaptic parameters are easy to understand and can simulate a wide variety of synaptic types. Each parameter is "separable"; that is changing its value does not affect other synaptic function parameters. In addition, synaptic parameters can be modified during a runtime to allow a wide variety of neural plasticity functions to be simulated.

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G = I / V where: I = current G = conductance of open channel V = reversal potential minus membrane voltageIf the ionic concentrations in the internal and external medium are known, it is possible to calculate the voltage at which no current flows for an ion. This is known as the ion's Nernst potential and is very similar to the definition of "reversal potential". It is a function of the internal and external ion concentrations and temperature:

V = RT / zF * ln (co / ci) where: V = Nernst potential T = temperature (^To^TK) co = external concentration ci = internal concentration z = valency of ionThe Nernst potential is a useful approximation of the driving force for an ion through a membrane channel because it defines the voltage at which no net current flows through the channel. Knowing the Nernst potential for the major ions is a great help in understanding electric current flow through a channel.

It would be convenient if the reversal potential for a channel could be calculated directly from the Nernst potential of the channel's major ion. However, the driving force for electric current through a channel is not exactly the same as the Nernst potential for the channel's major ion, for several reasons. Every channel allows other ions to pass too. The relative permeability of "other" ions can range from 0.0005 for K through a Ca channel to 0.2 for Ca through an NMDA channel. The effect of having several ions pass through a channel is that its reversal potential is a function of all the ion concentrations and their respective permeabilities. That function is the GHK voltage equation:

( Nao * pna + Ko * pk ) V = RT / zF * ln -------------------- ( Nai * pna + Ki * pk ) where: V = GHK potential (reversal potential for the channel) T = temperature (deg K) Nao = Na external concentration Nai = Na external concentration pNa = Na permeability through channel Ko = K external concentration Ki = K external concentration pK = K permeability through channel z = valency of ions ( must be same valencies, if different use eqn below)To accurately compute the current through a channel, we need to compute the reversal potential from the GHK voltage equation and for this we need to know ion concentrations and relative permeabilities.

In NeuronC, each channel type has a data structure that contains the channel's conductance (and its temperature dependency) and the channel's permeability to several different ions so that an accurate reversal potential can be computed.

A more detailed version of the GHK voltage equation is required when both monovalent and divalent ions are permeable through a channel:

( Nao * pNa + Ko * pK + 4 * p'Ca * Cao ) V = RT / F * ln ----------------------------------------- + Vs ( Nai * pNa + Ki * pK + 4 * p'Ca * Cai * ef) where: V = GHK potential (reversal potential for the channel) Vs = surface potential due to external Cao (set by dcavoff, dcaspvrev) F = Faraday constant 96,485 C/mol R = Gas constant 8.314 J/K/mol T = temperature (deg K) Nao = Na external concentration Nai = Na external concentration pNa = Na permeability through channel Ko = K external concentration Ki = K external concentration pK = K permeability through channel Cao = Ca external concentration Cai = Ca external concentration pCa = Ca permeability through channel ef = exp((V-Vs) F / RT ) p'Ca = modified Ca permeability = pCa / (1 + ef)This more detailed GHK voltage equation is an "implicit" function, i.e. V is on both left and right sides of the equation. The reason for the extra terms for Ca is that Ca++ has a valence of 2 so the normal GHK equation for RT/zf is RT/2F. The extra terms have the effect of adding the extra 2. [ This equation is derived from the GHK current equations for mono- and di-valent ions assuming "independence" between ion motion in a "constant field theory" by Lewis (1979), J. Physiol. 286: 417-445. Though more complete than the simple GHK equation above, ions in some combinations do not have independence so even a more complex equation would be necessary at some combinations of ion concentrations to completely specify ion flux. ]

To solve this equation, a starting value for V is given and "ef" computed, and the equation solved for a new estimate of V. The process continues for 2-5 iterations until the difference between the new and old estimates for V is small ( 0.1 mV ).

The surface potential Vs is a function of the concentration of divalent ions on the external surface of the membrane (for our purposes, Cao). External calcium shifts both gating properties and reversal potentials, though to different amounts. The shift in the gating properties is about 18 mV negative for each factor of 10 increase in Cao. Therefore an increase in Cao makes the membrane voltage more negative (for gating channels) and causes Na channels to be less activated. The effect on the reversal potential is positive but less, and is set by the "dcaspvrev" variable (default = 0.18), which multiplies dcavoff (.018) for a value of 3.2 mV positive shift / factor of 10 increase in cao.

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df = exp (-z * v * F*F/R/T); i = p / [iono] * z * z * ([ioni] - io * df) / (1 - df); where: i = current due to ion flux through selective channel p = permeability of ion through selective channel z = valence of ion (Na,K ->1 Cl -> -1; Ca -> 2) v = membrane voltage F = Faraday constant 96,485 C/mol R = Gas constant 8.314 J/K/mol T = temperature, degrees K iono = external concentration ioni = internal concentrationThis equation can readily be extended to multiple ions with the assumption that their flux through the channel is independent. The nonlinear nature of this flux equation means that the current through most membrane ion channels is also nonlinear, and must be calculated from the voltage during each time step. This is particularly difficult for a simulator such as "nc" that uses an iterative method to solve the voltage and current flows in compartments.

To speed the computation of currents and voltages in the compartments, we compute a local linear approximation of the GHK current equation. For each voltage, the simulator computes a linear "slope conductance" and a "slope reversal potential", which are utilized just like the ordinary channel conductance and the channel reversal potential:

vf = exp(-z * v * F/R/T); vfi = 1 / (1 - vf); cvfi += p * z*z/zm/zm * (ioni-iono*vf)/(1-vf)/ionm; gfrac += p * F/R/T * v *z*z*z/zm/zm * (ioni-iono)*vf*vfi*vfi/ionm; itot = v * cvfi where: cvfi = varible used to calculate slope and vrev gfrac = variable used to calculate slope itot = total current through channel p = permeability of ion through selective channel z = valence of ion v = membrane voltage zm = valence of main ion F = Faraday constant 96,485 C/mol R = Gas constant 8.314 J/K/mol T = temperature, degrees K iono = external concentration of ion ioni = internal concentration of ion ionm = external concentration of main ionFor each ion, we compute the "cvfi" and "gfrac" variables and sum them over all the ions. Then we then compute the "slope cond" and "slope vrev" from them:

scond = cvfi - gslope; svrev = v - v*cvfi/(cvfi-gslope);To increase simulation speed, the "scond" and "svrev" variables are computed for each voltage before runtime, and placed into tables indexed by voltage. At run time, the values are looked up and linearly interpolated.

For more on the derivation of this equation, see DeSchutter & Smolen (1998), Methods in Neuronal Modeling, p 218.

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NeuronC uses an automatic method for determining compartment size. It calculates each neural cable's space constant (from diameter, axial and membrane resistance) then calculates the compartment size as a fraction of the space constant (normally 1/10th). This causes the size of the compartments to be varied according to the properties of the cable that defines them.

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It is relatively easy to simulate almost any time-related change in properties of neural elements using a compartmental model, because by analogy these properties do not change for the purposes of the simulation during the course of one time step. Small jumps in channel properties, for example, are allowable over a sequence of time steps because if they're small enough, they faithfully represent the effect of continuous change.

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The implicit method of numerical integration is more difficult to calculate than the Euler method because the voltages and current flows for the end of the time step are not known until the voltage calculation for each compartment is complete. However this type of computation is equivalent to solving a system of N simultaneous equations in N unknowns (N=number of compartments), which is solvable by matrix techniques.

Although there are higher-order methods to calculate the continuous voltage during a time step, these methods usually require more computation per time step. Thus, there is a compromise between accuracy and speed, and higher-order numerical integration schemes are generally only useful if their time steps can be made proportionately longer. Empirically, the second-order Crank-Nicolson method is often faster than the Euler or implicit methods because larger time steps can be taken as a consequence of its better accuracy (see Moore and Ramon, 1974). However, fourth-order (in time) methods are often slower than second- order methods for simulations of neural circuits with commonly used timesteps (between 10 and 100 usec).

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Crank-Nicolson method: Voltage change in a compartment = dt/c * average of new and old currents v2n-v2 = k * ( (i1 + i3 + i1n + i3n) / 2 - ( i2 + i2n) / 2 ) v2n-v2 = k/2 * ((v1-v2)*s1 + (v3-v2)*s2 + (v1n-v2n)*s1 + (v3n-v2n)*s2 - (v2-vm + v2n-vm)*sm ) v2n-v2 = k/2 * ( v1*s1 + v3*s2 - v2 * (s1+s2+sm) + v1n*s1 + v3n*s2 + 2*vm*sm - v2n * (s1+s2+sm) ) v2n * (1 + k/2 * (s1+s2+sm)) = v2 + k/2 * ( v1*s1 + v3*s2 - v2 * (s1+s2+sm) + v1n*s1 + v3n*s2 + 2*vm*sm ) v2n = (v2 + k/2 * ( v1*s1 + v3*s2 - v2 * (s1+s2+sm) + v1n*s1 + v3n*s2 + 2*vm*sm)) / (1 + k/2 * (s1+s2+sm)) v2n = (v2 + k/2 * ( v1*s1 + v3*s2 - v2 * (s1+s2+sm) + v1n*s1 + v3n*s2 + 2*vm*sm)) * implicit factorwhere:

v2 = voltage in this compartment at start of time step v2n = voltage in this compartment at end of time step vm = membrane leakage reversal potential s1 = conductance to neighbor compartment 1 s2 = conductance to neighbor compartment 2 sm = membrane conductance in this compartment k = dt/c c = membrane capacitance of compartment dt = time step implicit factor = 1 / ( 1 + k/2 * (s1+s2+sm))

Fully implicit (Backwards Euler) method: v2n-v2 = k * ( (i1n + i3n) - i2n) v2n-v2 = k * ((v1n-v2n)*s1 + (v3n-v2n)*s2 - (v2n-vm)*sm ) v2n-v2 = k * (v1n*s1 + v3n*s2 + vm*sm - v2n * (s1+s2+sm) ) v2n * (1 + (k * s1+s2+sm)) = v2 + k* (-v2 * (s1+s2+sm) + v1n*s1 + v3n*s2 + vm*sm ) v2n = (v2 + k * (v1n*s1 + v3n*s2 + vm*sm - v2 * s1+s2+sm)) / (1 + k * (s1+s2+sm)) v2n = (v2 + k * (v1n*s1 + v3n*s2 + vm*sm - v2 * s1+s2+sm)) * implicit factorwhere:

v2 = voltage in this compartment at start of time step v2n = voltage in this compartment at end of time step vm = membrane leakage reversal potential s1 = conductance to neighbor compartment 1 s2 = conductance to neighbor compartment 2 sm = membrane conductance in this compartment k = dt/c c = membrane capacitance of compartment dt = time step implicit factor = 1 / (1 + k * (s1+s2+sm))

When the model contains only neurons, each with a soma and a tree-structured dendritic and/or axonal tree, the implicit method of integration can be solved efficiently using Gaussian elimination. This method works if the equivalent connection matrix for the compartments is symmetrical, that is, the only connection back to the soma from each compartment is along just one path from the soma to the compartment. However, when there are loops in the model, for example, when one or more gap junctions connect between neurons, Gaussian elimination fails and another solution method must be used. Often forwards-Euler integration is a backup method. However, the "Gauss-Seidel" method works for non-symmetrical matrices and is the method used by Neuron-C.

Synapses are slow enough that propagation of a voltage through them is not necessary to take into account in the implicit mode. In Neuron-C the synaptic conductance is simply added onto the other compartmental conductances. The synaptic conductance is then computed on a slower time scale than the compartments so that it doesn't take much computation time.

The "Gauss-Seidel" method for solving N equations in N unknowns is an iterative method. For implicit methods, because the new voltages v1n and v3n are not known, the solution to all the compartments must be iterated until none of the compartments change their values beyond a criterion value. Then the new voltages v1n and v3n are close enough to being correct. In Neuron-C, the compartments are organized into a linked list. Each compartment points to the next one, and they are computed sequentially starting at the beginning of the linked list. Each compartment is calculated one at a time, so e.g. v1n voltage is calculated before v2n, which in turn is calculated before v3n. However, the calculation of v2n is improved because v1n is closer to the final calculation. Each iteration improves the solution to all the compartments. This iteration process allows voltage changes to propagate through a neuron's dendritic tree. The compartments are calculated in the opposite order every other iteration, to allow voltage changes to propagate in both directions. Typical numbers of iteration for compartments are between 10 and 1000 iterations, depending on how fast the compartment voltage is changing, on the compartment size and the resistive coupling between compartments, and on the value of "crit" which is the accuracy limit.

In addition, the calculation can be accelerated by simply multiplying the change from the previous iteration by an "over-relaxation" factor greater than 1, typically 1.1 to 1.6. This is called "successive over-relaxation" (SOR) and it can speed the iterative calculation to the compartments. The over-relaxation factor is different for each compartment. If it is too large, the calculation of the compartment's voltage may be "under-damped", i.e. the difference between the previous estimate and the present estimate will oscillate from positive and negative values. If the over-relaxation factor is too small (e.g. 1), the difference between the present and previous estimates is "over-damped" and will asymtote slowly towards zero. This takes many iterations and wastes computation time. The simulator finds the value for the over-relaxation factor that makes the compartment voltage change (difference present and previous estimates) have just one sign change. Each compartment has its own over-relaxation factor.

Moreover, the compartments that change their voltage from one iteration to the next less than a criterion value ("crit") don't need to be recalculated, so they are dropped from the calculation. This generates a small error in the neighboring compartments that may increase with iterations, so every 20 iterations all of the compartments are recomputed. This saves computation time because often most of the compartments in a large neuron don't need more than 10 iterations.

References: Joyner et al, 1978 Crank and Nicolson, 1947 Hines, 1984; 1989 Atkinson KE (1989), "An introduction to Numerical Analysis", Wiley & Sons.

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dm = alpham - (alpham+betam) * m (original HH rate equation) dmest = alpham - (alpham+betam) * mest (implicit version) mest = m + (dm+dmest) / 2 (second-order C-N version) mest = m + (alpham-(alpham+betam)*m + alpham-(alpham+betam)*mest) / 2 mest (1+alpham+betam) / 2) = alpham + m * (1-(alpham+betam)) / 2) mest = (alpham + m * (1 - (alpham+betam)/2)) / (1 + (alpham+betam)/2) ( CN ) mest = k1 * m + k2 where: k1 = (1 - (alpham+betam)/2) / (1 + (alpham+betam)/2) k2 = alpham / (1+(alpham+betam)/2)Fully implicit version:

mest = m + alpham-(alpham+betam)*mest (implicit version) mest (1+alpham+betam) = m + alpham mest = (alpham + m) / (1 + alpham + betam) mest = k1 * m + k2 where: k1 = 1 / (1 + alpham + betam) k2 = alpham / (1 + alpham + betam)Since the multipliers k1 and k2 are functions of only voltage, they are precomputed and placed in lookup tables, after the method of Hines (1984). Since the rate equations are second-order, they need not be re-computed for each iteration for a compartment's voltage. Instead, the implicit estimated value is used and this allows the calculation to run substantially faster.

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B + Ca.B = Bt ( free buffer + Ca-bound buffer = total buffer ) f > B + Ca <--------> Ca.B ( defines forward and reverse rates ) < r Where: Ca = free [Ca] in shell B = free buffer in shell Ca.B = Ca-bound buffer Bt = total buffer f = forward rate, /M/sec r = reverse rate, /sec [ Buffer equations here from Yamada et al. (1998) ] d[Ca]/dt = r * [Ca.B] - f [Ca] [B] ( change in Ca from Ca buffer ) ( change in Ca from spatial diffusion ) d[Ca]/dt = ( [Ca(left neighbor shell)] - [Ca(this shell)] ) * S + ( [Ca(right neighbor shell)] - [Ca(this shell)] ) * S dCa = dt * r * (Bt - B) - f * Ca * B + (Caj - Cai) * S + (Cak - Cai) * S dCaest = dt * r * (Bt - Best) - f * Caest * Best + (Cajest-Caest) * S + (Cakest-Caest) * S Where: dca = change in Ca at the beginning of the time step dcaest = change in Ca at the end of the time step i = shell being computed j = i - 1 (left neighbor shell) k = i + 1 (right neighbor shell) Best = new estimate of B Caest = new estimate of Ca in this shell Cajest = new estimate of left neighbor Ca Cakest = new estimate of right neighbor Ca S = shell diffusion multiplier

Caest = Ca + dCaest Caest = Ca + r*Bt - r*Best - f*Caest*Best + (Cajest+Cakest-2*Caest)*S Caest (1+f*Best+2*S) = Ca + r*(Bt-Best) + (Cajest+Cakest)*S Caest = Ca + r*(Bt-Best) + (Cajest+Cakest)*S / (1+f*Best+2*S) (1) Best = B + dBest = B + r * (Bt - Best) - f*Caest*Best Best (1+r+f*Caest) = B + r*Bt Best = (B + r*Bt) / (1+r+f*Caest) (2)

For Calcium: Caest = Ca + (dca + dcaest) / 2 Substituting: Caest = Cai + ( r * (Bt - B) - f * Ca * B + r * (Bt - Best) - f * Caest * Best + (Caj - Cai) * S + (Cak - Cai) * S + (Cajest - Caest) * S + (Cakest - Caest) * S ) / 2 Rearranging: Caest = Ca + ( 2*r*Bt - r*B - r*Best - f*Ca*B - f*Caest*Best + (Caj + Cak - 2 * Ca + Cajest + Cakest - 2 * Caiest) * S ) / 2 Caest = Ca * (1 - f*B/2 - S) + r*Bt - r*(B+Best)/2 - f*Caest*Best/2 + (Caj + Cak + Cajest + Cakest) * S / 2 - Caest * S Solving for Caest: Caest * (1 + f*Best/2 + S) = Ca * (1 - f*B/2 - S) + r*Bt - r*(B+Best)/2 + (Caj + Cak + Cajest + Cakest) * S / 2 Switch f*Best with f*B: (allowed by Taylor series expansion after Yamada et al, 1998) Caest * (1 + f*B/2 + S) = Ca * (1 - f*Best/2 - S) + r*Bt - r*(B+Best)/2 + (Caj + Cak + Cajest + Cakest) * S / 2 Caest * (1 + f*B/2 + S) = Ca * (1-S) + r*Bt - r*B/2 - Best*(Ca*f + r) / 2 + (Caj + Cak + Cajest + Cakest) * S / 2 Caest = [ Ca + r*(Bt-B/2) - Best*(Ca*f+r)/2 + (Caj+Cak+Cajest+Cakest-2*Ca) * S/2 ] / implf (3) Where: implf = (1 + f*B/2 + S)

For Buffer: Remove spatial diffusion, solve for Best: Best = B + (dB + dBest) / 2 = B + (r*(Bt - Best) - f*Caest*Best + r*(Bt - B) - f*Ca*B) / 2 Solving for Best: Best (1 + (r+f*Caest)/2) = B + r*(Bt-B/2) - f*Ca*B/2 Best = (B + r*(Bt-B/2) - f*Ca*B/2) / (1 + (r+f*Caest)/2) (4)

Derivation of driving force for calcium current, when both internal and external calcium may change:

GHK current equation (for Ca): I = P * z2 * Vm * F^2 / RT * (Cai - Cao * exp (-vs)) ------------------------ 1 - exp(-vs) Defining for simulator (for Ca): Ica = conduct * Vm * (Cai - Cao * exp (-vs)) ------------------------ 1 - exp(-vs) where: conduct = channel conductance, defined by user vs = 2 * Vm * F/RT Simplifying: Ica = conduct * Vm * (Cai - Cao * exp (-vs)) * exp(vs) ------------------------ ------- 1 - exp(-vs) exp(vs) Ica = conduct * Vm * Cao * (1 - caratio * exp (vs)) ------------------------ 1 - exp(vs) where: caratio = Cai / Cao However, this doesn't give a correct Ica because the conductance already takes into account the default Cao. Normalizing the Cao by its default, this simplifies to: Ica = conduct * Vm * Cao/dcao (1 - caratio * exp (vs)) ----------------------- (1 - exp(vs)) This simplifies to: Ica = conduct * cadf where: cadf = driving force = Vm * Cao (1 - caratio * exp (vs)) ---- ------------------------ dcao (1 - exp(vs))

The idea is that we rely on "conduct" to represent the conductance defined by the user. When Cao or Cai change, this effectively modulates the conductance. In order to capture this effect, we define the driving force as a function of Vm and Cai and Cao. Then the driving force changes the current depending on changes in the calcium. In order to keep the driving force a voltage, we normalize by Cao / dcao, which then maintains the driving force correct for the default Cai and Cao.

Strategy for solving Ca diffusion equations:

The idea here is to write an "implicit" equation that can be solved for the Ca concentration at the end of the time step. This number is dependent on the neighboring Ca concentrations at the end of the time step. These are unknown, but are gradually updated by iterating over all the compartments. Calculate Caest and Best alternately so that buffer gets updated along with Ca.

Iterate equations 1 and 2 (or 3,4) over all the compartments. When change in Caest from one iteration to next is small ( < 1e-10) stop.

Typical number of iterations is 2-5 so computation is relatively fast.

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Accuracy of voltage change over time can be determined by successive simulation runs with different size time steps. For example, one could decrease the time step by a factor of 2 until the voltage response of the model does not change from a previous run more than a criterion of say, 0.1 percent. This assures an absolute accuracy of 0.1 percent in the voltage calculated.

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Second, compartmental modeling breaks up neural circuits into discrete spatial "chunks" which causes the voltage response to be inaccurate. If many tiny compartments are used to describe a neural cable segment, the inaccuracy is insignificant but the model runs slowly. If just a few larger compartments are used to describe the same cable segment, the model runs faster, but waveshapes and amplitudes of the voltage response are noticeably wrong.

The overall problem with compartmental modeling, then, is the generally slow computational speed. Both of the problems with inaccuracy that occur with compartmental modeling (space, time) can be solved by making the model run more calculations, therefore slower. Thus most ambitious compartmental modelers usually end up eyeing large main-frame computers as a "way out" of their modeling problems.

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Using a parallel library such as "PVM" or "MPI" and a parallel implementation of a simulator running on multiple CPUs, it is possible to run a simulation job much faster than it can run on just one CPU. However, often we need to run many simulation jobs, each with a different set of parameters. In this case, a truly parallel implementation may actually decrease the overall speed. If there are 20 simulation jobs, and each one runs on one CPU, it would take 20 CPUs the same (or similar) time to run 20 jobs in parallel, each running on one CPU, or using all the CPUs for a parallel implementation of each job. A parallel implementation usually runs a little slower than the expected 20-fold increase in speed, because of inefficiencies in the parallel computation -- during each time step, a CPU must communicate information about connections between compartments that are computed by a different CPU, so it must often wait a few microseconds to receive the information. Thus with a parallel job system we can run many jobs simultaneously very fast without a parallel implentation of the simulator. This is the rationale for using a parallel dynamic balancing job system such as MOSIX or the Linux SMP kernel.

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