# Functions

All equations, expressions and statements in Brian can make use of mathematical functions. However, functions have to be prepared for use with Brian for two reasons: 1) Brian is strict about checking the consistency of units, therefore every function has to specify how it deals with units; 2) functions need to be implemented differently for different code generation targets.

Brian provides a number of default functions that are already prepared for use with numpy and C++ and also provides a mechanism for preparing new functions for use (see below).

## Default functions

The following functions (stored in the `DEFAULT_FUNCTIONS` dictionary) are ready for use:

- Random numbers: `rand` (random numbers drawn from a uniform distribution between 0 and 1), `randn` (random numbers drawn from the standard normal distribution, i.e. with mean 0 and standard deviation 1), and `poisson` (discrete random numbers from a Poisson distribution with rate parameter \(\lambda\))

- Elementary functions: `sqrt`, `exp`, `log`, `log10`, `abs`, `sign`

- Trigonometric functions: `sin`, `cos`, `tan`, `sinh`, `cosh`, `tanh`, `arcsin`, `arccos`, `arctan`

- Functions for improved numerical accuracy: `expm1` (calculates `exp(x) - 1`, more accurate for `x` close to 0), `log1p` (calculates `log(1 + x)`, more accurate for `x` close to 0), and `exprel` (calculates `(exp(x) - 1)/x`, more accurate for `x` close to 0, and returning 1.0 instead of `NaN` for `x == 0`

- General utility functions: `clip`, `floor`, `ceil`

Brian also provides a special purpose function `int`, which can be used to convert an expression or variable into an integer value. This is especially useful for boolean values (which will be converted into 0 or 1), for example to have a conditional evaluation as part of an equation or statement which sometimes allows to circumvent the lack of an `if` statement. For example, the following reset statement resets the variable `v` to either `v_r1` or `v_r2`, depending on the value of `w`: `'v = v_r1 * int(w <= 0.5) + v_r2 * int(w > 0.5)'`

Finally, the function `timestep` is a function that takes a time and the length of a time step as an input and returns an integer corresponding to the respective time step. The advantage of using this function over a simple division is that it slightly shifts the time before dividing to avoid floating point issues. This function is used as part of the Refractoriness mechanism.

## User-provided functions

### Python code generation

If a function is only used in contexts that use Python code generation, preparing a function for use with Brian only means specifying its units. The simplest way to do this is to use the `check_units()` decorator:

    @check_units(x1=meter, y1=meter, x2=meter, y2=meter, result=meter)
    def distance(x1, y1, x2, y2):
        return sqrt((x1 - x2)**2 + (y1 - y2)**2)

Another option is to wrap the function in a `Function` object:

    def distance(x1, y1, x2, y2):
        return sqrt((x1 - x2)**2 + (y1 - y2)**2)
    # wrap the distance function
    distance = Function(distance, arg_units=[meter, meter, meter, meter],
                        return_unit=meter)

The use of Brian’s unit system has the benefit of checking the consistency of units for every operation but at the expense of performance. Consider the following function, for example:

    @check_units(I=amp, result=Hz)
    def piecewise_linear(I):
        return clip((I-1*nA) * 50*Hz/nA, 0*Hz, 100*Hz)

When Brian runs a simulation, the state variables are stored and passed around without units for performance reasons. If the above function is used, however, Brian adds units to its input argument so that the operations inside the function do not fail with dimension mismatches. Accordingly, units are removed from the return value so that the function output can be used with the rest of the code. For better performance, Brian can alter the namespace of the function when it is executed as part of the simulation and remove all the units, then pass values without units to the function. In the above example, this means making the symbol `nA` refer to `1e-9` and `Hz` to `1`. To use this mechanism, add the decorator `implementation()` with the `discard_units` keyword:

    @implementation('numpy', discard_units=True)
    @check_units(I=amp, result=Hz)
    def piecewise_linear(I):
        return clip((I-1*nA) * 50*Hz/nA, 0*Hz, 100*Hz)

Note that the use of the function outside of simulation runs is not affected, i.e. using `piecewise_linear` still requires a current in Ampere and returns a rate in Hertz. The `discard_units` mechanism does not work in all cases, e.g. it does not work if the function refers to units as `brian2.nA` instead of `nA`, if it uses imports inside the function (e.g. `from brian2 import nA`), etc. The `discard_units` can also be switched on for all functions without having to use the `implementation()` decorator by setting the codegen.runtime.numpy.discard_units preference.

### Other code generation targets

To make a function available for other code generation targets (e.g. C++), implementations for these targets have to be added. This can be achieved using the `implementation()` decorator. The form of the code (e.g. a simple string or a dictionary of strings) necessary is target-dependent, for C++ both options are allowed, a simple string will be interpreted as filling the `'support_code'` block. Note that `'cpp'` is used to provide C++ implementations. An implementation for the C++ target could look like this:

    @implementation('cpp', '''
         double piecewise_linear(double I) {
            if (I < 1e-9)
                return 0;
            if (I > 3e-9)
                return 100;
            return (I/1e-9 - 1) * 50;
         }
         ''')
    @check_units(I=amp, result=Hz)
    def piecewise_linear(I):
        return clip((I-1*nA) * 50*Hz/nA, 0*Hz, 100*Hz)

Alternatively, `FunctionImplementation` objects can be added to the `Function` object.

The same sort of approach as for C++ works for Cython using the `'cython'` target. The example above would look like this:

    @implementation('cython', '''
        cdef double piecewise_linear(double I):
            if I<1e-9:
                return 0.0
            elif I>3e-9:
                return 100.0
            return (I/1e-9-1)*50
        ''')
    @check_units(I=amp, result=Hz)
    def piecewise_linear(I):
        return clip((I-1*nA) * 50*Hz/nA, 0*Hz, 100*Hz)

### Dependencies between functions

The code generation mechanism for user-defined functions only adds the source code for a function when it is necessary. If a user-defined function refers to another function in its source code, it therefore has to explicitly state this dependency so that the code of the dependency is added as well:

    @implementation('cpp','''
        double rectified_linear(double x)
        {
            return clip(x, 0, INFINITY);
        }''',
        dependencies={'clip': DEFAULT_FUNCTIONS['clip']}
        )
    @check_units(x=1, result=1)
    def rectified_linear(x):
        return np.clip(x, 0, np.inf)

Note

The dependency mechanism is unnecessary for the `numpy` code generation target, since functions are defined as actual Python functions and not as code given in a string.

### Additional compiler arguments

If the code for a function needs additional compiler options to work, e.g. to link to an external library, these options can be provided as keyword arguments to the `@implementation` decorator. E.g. to link C++ code to the `foo` library which is stored in the directory `/usr/local/foo`, use:

    @implementation('cpp', '...',
     libraries=['foo'], library_dirs=['/usr/local/foo'])

These arguments can also be used to refer to external source files, see below. Equivalent arguments can also be set as global Preferences in which case they apply to all code and not only to code referring to the respective function. Note that in C++ standalone mode, all files are compiled together, and therefore the additional compiler arguments provided to functions are always combined with the preferences into a common set of settings that is applied to all code.

The list of currently supported additional arguments (for further explications, see the respective Preferences and the Python documentation of the `distutils.core.Extension` class):

keyword | C++ standalone | Cython
--- | --- | ---
`headers` | ✓ | ❌
`sources` | ✓ | ✓
`define_macros` | ✓ | ❌
`libraries` | ✓ | ✓
`include_dirs` | ✓ | ✓
`library_dirs` | ✓ | ✓
`runtime_library_dirs` | ✓ | ✓

### Arrays vs. scalar values in user-provided functions

Equations, expressions and abstract code statements are always implicitly referring to all the neurons in a `NeuronGroup`, all the synapses in a `Synapses` object, etc. Therefore, function calls also apply to more than a single value. The way in which this is handled differs between code generation targets that support vectorized expressions (e.g. the `numpy` target) and targets that don’t (e.g. the `cpp_standalone` mode). If the code generation target supports vectorized expressions, it will receive an array of values. For example, in the `piecewise_linear` example above, the argument `I` will be an array of values and the function returns an array of values. For code generation without support for vectorized expressions, all code will be executed in a loop (over neurons, over synapses, …), the function will therefore be called several times with a single value each time.

In both cases, the function will only receive the “relevant” values, meaning that if for example a function is evaluated as part of a reset statement, it will only receive values for the neurons that just spiked.

### Functions with context-dependent return values

When using the `numpy` target, functions have to return an array of values (e.g. one value for each neuron). In some cases, the number of values to return cannot be deduced from the function’s arguments. Most importantly, this is the case for random numbers: a call to `rand()` has to return one value for each neuron if it is part of a neuron’s equations, but only one value for each neuron that spiked during the time step if it is part of the reset statement. Such function are said to “auto vectorise”, which means that their implementation receives an additional array argument `_vectorisation_idx`; the length of this array determines the number of values the function should return. This argument is also provided to functions for other code generation targets, but in these cases it is a single value (e.g. the index of the neuron), and is currently ignored. To enable this property on a user-defined function, you’ll currently have to manually create a `Function` object:

    def exponential_rand(l, _vectorisation_idx):
        '''Generate a number from an exponential distribution using inverse
           transform sampling'''
        uniform = np.random.rand(len(_vectorisation_idx))
        return -(1/l)*np.log(1 - uniform)

    exponential_rand = Function(exponential_rand, arg_units=[1], return_unit=1,
                                stateless=False, auto_vectorise=True)

Implementations for other code generation targets can then be added using the `add_implementation` mechanism:

    cpp_code = '''
    double exponential_rand(double l, int _vectorisation_idx)
    {
        double uniform = rand(_vectorisation_idx);
        return -(1/l)*log(1 - uniform);
    }
    '''
    exponential_rand.implementations.add_implementation('cpp', cpp_code,
                                                        dependencies={'rand': DEFAULT_FUNCTIONS['rand'],
                                                                      'log': DEFAULT_FUNCTIONS['log']})

Note that by referring to the `rand` function, the new random number generator will automatically generate reproducible random numbers if the `seed()` function is use to set its seed. Restoring the random number state with `restore()` will have the expected effect as well.

### Additional namespace

Some functions need additional data to compute a result, e.g. a `TimedArray` needs access to the underlying array. For the `numpy` target, a function can simply use a reference to an object defined outside the function, there is no need to explicitly pass values in a namespace. For the other code language targets, values can be passed in the `namespace` argument of the `implementation()` decorator or the `add_implementation` method. The namespace values are then accessible in the function code under the given name, prefixed with `_namespace`. Note that this mechanism should only be used for numpy arrays or general objects (e.g. function references to call Python functions from Cython code). Scalar values should be directly included in the function code, by using a “dynamic implemention” (see `add_dynamic_implementation`).

See `TimedArray` and `BinomialFunction` for examples that use this mechanism.

### Data types

By default, functions are assumed to take any type of argument, and return a floating point value. If you want to put a restriction on the type of an argument, or specify that the return type should be something other than float, either declare it as a `Function` (and see its documentation on specifying types) or use the `declare_types()` decorator, e.g.:

    @check_units(a=1, b=1, result=1)
    @declare_types(a='integer', result='highest')
    def f(a, b):
        return a*b

This is potentially important if you have functions that return integer or boolean values, because Brian’s code generation optimisation step will make some potentially incorrect simplifications if it assumes that the return type is floating point.

### External source files

Code for functions can also be provided via external files in the target language. This can be especially useful for linking to existing code without having to include it a second time in the Python script. For C++-based code generation targets (i.e. the C++ standalone mode), the external code should be in a file that is provided as an argument to the `sources` keyword, together with a header file whose name is provided to `headers` (see the note for the codegen.cpp.headers preference about the necessary format). Since the main simulation code is compiled and executed in a different directory, you should also point the compiler towards the directory of the header file via the `include_dirs` keyword. For the same reason, use an absolute path for the source file. For example, the `piecewise_linear` function from above can be implemented with external files as follows:

    //file: piecewise_linear.h
    double piecewise_linear(double);

    //file: piecewise_linear.cpp
    double piecewise_linear(double I) {
        if (I < 1e-9)
            return 0;
        if (I > 3e-9)
            return 100;
        return (I/1e-9 - 1) * 50;
    }

    # Python script

    # Get the absolute directory of this Python script, the C++ files are
    # expected to be stored alongside of it
    import os
    current_dir = os.path.abspath(os.path.dirname(__file__))

    @implementation('cpp', '// all code in piecewise_linear.cpp',
                    sources=[os.path.join(current_dir,
                                          'piecewise_linear.cpp')],
                    headers=['"piecewise_linear.h"'],
                    include_dirs=[current_dir])
    @check_units(I=amp, result=Hz)
    def piecewise_linear(I):
        return clip((I-1*nA) * 50*Hz/nA, 0*Hz, 100*Hz)

For Cython, the process is very similar (see the Cython documentation for general information). The name of the header file does not need to be specified, it is expected to have the same name as the source file (except for the `.pxd` extension). The source and header files will be automatically copied to the cache directory where Cython files are compiled, they therefore have to be imported as top-level modules, regardless of whether the executed Python code is itself in a package or module.

A Cython equivalent of above’s C++ example can be written as:

    # file: piecewise_linear.pxd
    cdef double piecewise_linear(double)

    # file: piecewise_linear.pyx
    cdef double piecewise_linear(double I):
        if I<1e-9:
            return 0.0
        elif I>3e-9:
            return 100.0
        return (I/1e-9-1)*50

    # Python script

    # Get the absolute directory of this Python script, the Cython files
    # are expected to be stored alongside of it
    import os
    current_dir = os.path.abspath(os.path.dirname(__file__))

    @implementation('cython',
                    'from piecewise_linear cimport piecewise_linear',
                    sources=[os.path.join(current_dir,
                                          'piecewise_linear.pyx')])
    @check_units(I=amp, result=Hz)
    def piecewise_linear(I):
        return clip((I-1*nA) * 50*Hz/nA, 0*Hz, 100*Hz)