PSICS - the Parallel Stochastic Ion Channel Simulator
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Model processing

In order to understand some of the model specification options it is necessary to know a little about how PSICS operates on a model to perform stochastic calculations. This section summarizes the key stages in the process. The numerical methods section provides more details on the calculation stage.

The input model specification is processed sequentially through the following stages, beginning with reading the XML files, and ending with writing the desired output to text and binary files for independent visualization and analysis.

Specification reading and dereferencing

The first step is to read the input model and find all the components it needs. In this context, the term "dereferencing" refers to the process of finding an actual component (such as an ion channel model) given its name. This is really only of interest to the user if it doesn't work - either by not finding a component at all or by finding the wrong one.

Each component that is referred to from elsewhere must have an "id" attribute (as in '<KSChannel id="Kdr" ...'). In general, the id should be the same as the file name for the component. When a component is referred to from elsewhere the id (not the file name) is used to identify it. By default, PSICS always looks in the current folder for components, and by adding "ModelFolder" elements to the main run file you can tell it to look in other folders too.

Model components that are referred to by their id do not have to be "top level" components defined in separate files: subcomponents of models can also have their own ids and be referred to from elsewhere. This is used, for example, in setting up channel distributions where you can define a few standard distribution rules, and then, for each channel population, say which rule applies. It is also used for parameter sweeps where a series of runs are performed for different values of a selected parameter. The quantity to be varied is specified by he id if the component it belongs to followed by the name of the parameter.

Structure discretization

Also sometimes known as "remeshing" this is the process of creating a set of discrete elements ("compartments") to represent the continuous structure of the cell. In the calculation, each compartment will be treated as being isopotential. In general, discretization and compartmentalization are not the same thing. Discretization implies picking a finite set of points at which to represent a continuous function, but the behavior of a system can be computed in wide variety of ways using these points. For example, quantities can be assumed to vary linearly in the space between points, or according to higher order polynomials. For neurons, it is conventional to take the lowest order approximation and treat the potential as flat around each point making a sudden transition to a new value half way to each neighbor. The main reason for this is convenience: channel gating equations are often complicated and it is a lot easier to assume a fixed potential than to compute the effective gate transition rates by integrating over the vicinity of a point. Also, if the equations are sufficiently complicated that the integrals would have to be evaluated numerically in any case, then there is unlikely to be any efficiency gain by using a coarser discretization and a more complicated interpolation rule. However, for neurons this is probably not the case: efficiency gains might well be made by using fewer points and computing channel updates by using expressions incorporating not just the potential of the nearest point but also those of its neighbors. Indeed, for continuous channels, higher orders may well be worth considering too.

For stochastic channels the situation is rather different because the population approximation assumes that channels within the same group (on the same compartment) are interchangeable. This means they must experience exactly the same membrane potential and so enforces the isopotential compartment picture for the channel equations. This still does not require the potential to be piecewise flat but it does reduce the benefits of more complex differencing schemes. At present PSICS sticks with the conventional use of the zeroth order approximation for the membrane potential on the discretized structure.

In this context, the discretization process involves selecting points at which to cut the dendrites or axon such that the sections between points can be treated as being isopotential in the voltage propagation and channel transition equations. Typically, these points are unrelated to the points used to specify the structure: there may be several cuts between a pair of points in the original structure, or there may be several original points, including branch points, within one element of the final discretization. In general, therefore, the elements used in the calculation are not cylindrical and may have rather complicated boundaries. The discretization process also calculates the volume, surface area and axial resistance to neighbors for each element. It does this by traversing the structure calculating the integral of 1 / sqrt(r) dx where x is measured along the structure and cutting the dendrite into sections such that they all have approximately the same value for the integral. The actual value is set in the input file in the "StructureDiscretization" block by the "baseElementSize" attribute. This size is specified in microns and the end result is that the structure is split into sections where (for simple cylinders) l2/3d1/3 = baseElementSize, where l is the length and d the diameter.

The reason for this choice of spacing is that it balances the charging rates of adjacent compartments. The charging rate depends on the capacitance (proportional to the area), the cross section which increases the conductance and the length which reduces it. Altogether, taking uniform intervals in the integral 1 / sqrt(r) dx gives uniform set of voltage differences between compartments under standard charging conditions for wide range of geometries. The more common discretization for neurons is to use the electrotonic length which matches charging against loss through membrane conductance under steady state conditions. But such conditions are not necessarily relevant for most neurons firstly because the membrane conductance is not a fixed quantity but is strongly voltage dependent and secondly because neurons are rarely in a steady state. Most charge flow goes into membrane capacitors, not out of the membrane leak, so a discretization based on charging rates depending on capacitance and conductance might be expected to be more suitable (in the sense of producing smaller errors) than one based on a notion of membrane resistance. In practice there is unlikely to be a lot of difference, but since PSICS does not have a concept of membrane resistance the charging rate based discretization is used.

A possible confusion arises between the discretized structure and the original structure specification. In the original specification, it is assumed that the section between adjacent points is always a uniform or truncated cone (in fact a frustum, or, in the code, a carrotoid). In the discretized specification, a center of gravity and typical radius can be associated with each element, but it is not necessarily the case that the surface area, volume or axial resistance can be computed by assuming a tapering cylinder between neighbors.

This can present a problem for visualizing the results since joining adjacent points in the solution mesh can be a poor approximation to the original structure. For this purpose a file is written to the results folder with name ending "-vgrid.txt" that contains element boundaries as viewed from the z-axis. For sub-slices of original segments, there are just four points to each boundary, but there may be many more when the element covers multiple points from the original structure.

Channel conversion

This only takes place for channels that are specified with multiple independent gating complexes. Such channels are converted to an equivalent single kinetic scheme. For example, for a HH style sodium channel with a conductance given by p3q for gating particles p and q, the multi-complex interpretation is that there are three p-type complexes and one q-type complex, each having two possible states, open and closed. The p-type complexes contribute four states to the final scheme: the channel can have 0, 1, 2, or 3 of the complexes in the open state. For each of these four states, the q-complex can be either open or closed. This then doubles the number of states to eight. The transition rates between the final states are simple multiples of the transitions for the original two-state complexes.

This step is provided mainly for the convenience of validating against legacy channel models. Given that the destination format is a kinetic scheme it is generally preferable (for computational efficiency as well as physiological relevance) to start new models with a single-scheme specification rather than the HH subset.

Transition rate tabulation

The model specification allows a variety of formats for expressing the rates of transitions between the states of a channel. The core calculation does not, however, use these expressions, but rather interpolates for the current membrane potential in a table of transition matrices. The range and voltage increment of the table can be specified in the main model file. Using a very small increment leads to a larger table and more memory use although this is rarely an issue. A large increment would reduce the accuracy of the interpolation. The default interval is 1.5mV.

For channels that are expressed as serial independent gating complexes, including those with multiple HH-style two-state complexes, two sets of transition rate tables are written: one for the single scheme model, and one for the original multi-scheme model. This is done for computational efficiency. The single scheme table is suitable for both stochastic and continuous calculations whereas the multi-scheme table is only suitable for continuous calculations (see the methods section for details). However, for continuous calculations it is generally more efficient to use the multi-scheme model (for example, four 2x2 matrix by vector product cost 16 multiplications whereas one 8x8 matrix by vector product costs 64 multiplications). Since this matrix operation is often the dominant cost the saving achieved by using the independent schemes where possible can be considerable. Note that this is not an argument against kinetic scheme models per se, but only against computing continuous multi-complex channel models using their single scheme equivalent. Since such schemes are functionally equivalent to the original multi-complex model they suffer all the same problems as such models in addition to being unnecessarily large. A better comparison might be between a model with four independent two state complexes and an independently fitted single four state scheme. Depending on the geometry they could both have four independently parameterized transitions and would each take 16 multiplications for a single continuous update step.

Channel allocation

Channel number densities can be expressed in a variety of ways including expressions that depend on local properties of the cell such as path length to the soma or the dendritic radius. These expressions are evaluated across the structure and multiplied by element surface areas to get a real (as opposed to integer) value for the number of channels. For both stochastic and continuous calculations, however, there must be a whole number of channels of each type on each isopotential element. Rounding down the real channel numbers leaves a few extra channels that are randomly allocated according to the residual densities such that the total number of channels on the cell is within one of the expected number given the densities.

Note that this introduces a small random elements even into deterministic calculations. For such calculations, it can, however, be made arbitrarily small by scaling down the single channel conductance and scaling up the the density by the same factor.

It is worth noting that this variation only arises from allocating uniformly distributed point channels to discrete elements. It does not correspond to the distribution that would arise from randomly allocating channels across the structure according to a channel density distribution. This can be achieved by setting other attributes of the density distribution in the model specification.

Membrane potential balancing

If the channel density specification includes a DensityAdjustment block, then this applied next and may modify the channel densities. It provides the behavior that might physiologically be implemented by a homeostatic mechanism, increasing or decreasing certain channel densities in order to achieve a desired resting potential. Typically, the density adjustment block should specify two leak channels (one sodium and one potassium) and a target resting potential. The densities of these two channels are modified across the cell to achieve as nearly as possible densities that give no local net current across the membrane.

This is implemented as a non-local process since it is not always the case that the densities can be well balanced on a compartment by compartment basis. In particular, for large leak conductances, several compartments may have to share the same small number of channels, and in some cases the computed densities are negative wiche are then balanced against the nearest positive populatins of that channel. The details of the process are presented in the cell properties section.

Writing and reading the calculation-ready specification

At this stage the model has been converted into a set of discrete elements with coupling constants and channel numbers, and transition rate tables fore each channel type. This information is written to a plain text file with name ending ".ppp". For the fortran implementation, no further use is made of the XML model specification: it reads only the .ppp file which contains everything necessary to define the calculation.

In a parallel computing environment, where different realizations of the stochastic system are run on different nodes, each node requires only the .ppp file and the core executable. The format of this file is documented within the file itself using end-of-line comments. It would be possible to generate such a file from another system and to use only the native core of PSICS but it is generally inadvisable to edit it by hand because of the risk of producing a specification that is not self-consistent.

Numerical calculation

The next step sees the main java program handing over to computational core to actually perform the calculation. On most platforms this is done by spawning a separate system process to run the native executable of the core calculation. Platform specific executables are included within the standard jar file and are unpacked to a local folder before being executed. PSICS also supports loading the native code as a shared library, but route via a spawned process running a separate executable is generally more robust. The numerical algorithms implemented in the core calculation are described in the numerical methods section. The output of this calculation is a single calculation summary file (in xml) and a set of result files. Each run generates one block of results that is saved in both binary (.dat) and text (.txt) formats. Where a model defines multiple runs, as in a parameter sweep defined by the "RunSet" element, the file names are composed of the model name and the value of the parameter being changed.

Output consolidation and documentation

The final stage returns control to java. The calculation summary is converted to an html page containing links to the input files for the model and the output data files. If the model specification contains any plot definitions then the corresponding plots are generated and included in the html page.

As well as summarizing the model, the summary file also contains information on how long the calculation took. This timing is reported by the computational core and is for the numerical calculation itself. It excludes time spent on generating plots or formatting the output. In particular, for small models generating large volumes of output, the time taken to format the output into the text files can actually be as long or even longer than the calculation itself. If many such models are to be run it may be more efficient to suppress the generation of plain text files and instead extend analysis programs to work with the binary files written by the core. The time to write the binary file is included in the timing for the calculation, but is generally small compared with the calculation time itself.

A final step of the output consolidation process is to update the index file in the parent folder of the output to link to the new file, and to update the index blocks in sibling files in order to provide convenient navigation between models.

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