# Numerical integration¶

By default, Brian chooses an integration method automatically, trying to solve the equations exactly first (for linear equations) and then resorting to numerical algorithms. It will also take care of integrating stochastic differential equations appropriately.

Note that in some cases, the automatic choice of integration method will not be appropriate, because of a choice of parameters that couldn’t be determined in advance. In this case, typically you will get nan (not a number) values in the results, or large oscillations. In this case, Brian will generate a warning to let you know, but will not raise an error.

## Method choice¶

You will get an INFO message telling you which integration method Brian decided to use, together with information about how much time it took to apply the integration method to your equations. If other methods have been tried but were not applicable, you will also see the time it took to try out those other methods. In some cases, checking other methods (in particular the 'exact' method which attempts to solve the equations analytically) can take a considerable amount of time – to avoid wasting this time, you can always chose the integration method manually (see below). You can also suppress the message by raising the log level or by explicitly suppressing 'method_choice' log messages – for details, see Logging.

If you prefer to chose an integration algorithm yourself, you can do so using the method keyword for NeuronGroup, Synapses, or SpatialNeuron. The complete list of available methods is the following:

• 'exact': exact integration for linear equations (alternative name: 'linear')
• 'exponential_euler': exponential Euler integration for conditionally linear equations
• 'euler': forward Euler integration (for additive stochastic differential equations using the Euler-Maruyama method)
• 'rk2': second order Runge-Kutta method (midpoint method)
• 'rk4': classical Runge-Kutta method (RK4)
• 'heun': stochastic Heun method for solving Stratonovich stochastic differential equations with non-diagonal multiplicative noise.
• 'milstein': derivative-free Milstein method for solving stochastic differential equations with diagonal multiplicative noise

Note

The 'independent' integration method (exact integration for a system of independent equations, where all the equations can be analytically solved independently) should no longer be used and might be removed in future versions of Brian.

Note

The following methods are still considered experimental

• 'gsl': default integrator when choosing to integrate equations with the GNU Scientific Library ODE solver: the rkf45 method. Uses an adaptable time step by default.
• 'gsl_rkf45': Runge-Kutta-Fehlberg method. A good general-purpose integrator according to the GSL documentation. Uses an adaptable time step by default.
• 'gsl_rk2': Second order Runge-Kutta method using GSL. Uses an adaptable time step by default.
• 'gsl_rk4': Fourth order Runge-Kutta method using GSL. Uses an adaptable time step by default.
• 'gsl_rkck': Runge-Kutta Cash-Karp method using GSL. Uses an adaptable time step by default.
• 'gsl_rk8pd': Runge-Kutta Prince-Dormand method using GSL. Uses an adaptable time step by default.

The following topics are not essential for beginners.

## Technical notes¶

Each class defines its own list of algorithms it tries to apply, NeuronGroup and Synapses will use the first suitable method out of the methods 'exact', 'euler' and 'heun' while SpatialNeuron objects will use 'exact', 'exponential_euler', 'rk2' or 'heun'.

You can also define your own numerical integrators, see State update for details.

## GSL stateupdaters¶

The stateupdaters preceded with the gsl tag use ODE solvers defined in the GNU Scientific Library. The benefit of using these integrators over the ones written by Brian internally, is that they are implemented with an adaptable timestep. Integrating with an adaptable timestep comes with two advantages:

• These methods check whether the estimated error of the solutions returned fall within a certain error bound. For the non-gsl integrators there is currently no such check.
• Systems no longer need to be simulated with just one time step. That is, a bigger timestep can be chosen and the integrator will reduce the timestep when increased accuracy is required. This is particularly useful for systems where both slow and fast time constants coexist, as is the case with for example (networks of neurons with) Hodgkin-Huxley equations. Note that Brian’s timestep still determines the resolution for monitors, spike timing, spike propagation etc. Hence, in a network, the simulation error will therefore still be on the order of dt. The benefit is that short time constants occurring in equations no longer dictate the network time step.

In addition to a choice between different integration methods, there are a few more options that can be specified when using GSL. These options can be specified by sending a dictionary as the method_options key upon initialization of the object using the integrator (NeuronGroup, Synapses or SpatialNeuron). The available method options are:

• 'adaptable_timestep': whether or not to let GSL reduce the timestep to achieve the accuracy defined with the 'absolute_error' and 'absolute_error_per_variable' options described below. If this is set to False, the timestep is determined by Brian (i.e. the dt of the respective clock is used, see Scheduling). If the resulted estimated error exceeds the set error bounds, the simulation is aborted. When using cython or weave this is reported with an IntegrationError. Defaults to True.
• 'absolute_error': each of the methods has a way of estimating the error that is the result of using numerical integration. You can specify the maximum size of this error to be allowed for any of the to-be-integrated variables in base units with this keyword. Note that giving very small values makes the simulation slow and might result in unsuccessful integration. In the case of using the 'absolute_error_per_variable' option, this is the error for variables that were not specified individually. Defaults to 1e-6.
• 'absolute_error_per_variable': specify the absolute error per variable in its own units. Variables for which the error is not specified use the error set with the 'absolute_error' option. Defaults to None.
• 'max_steps': The maximal number of steps that the integrator will take within a single “Brian timestep” in order to reach the given error criterion. Can be set to 0 to not set any limits. Note that without limits, it can take a very long time until the integrator figures out that it cannot reach the desired error level. This will manifest as a simulation that appears to be stuck. Defaults to 100.
• 'use_last_timestep': with the 'adaptable_timestep' option set to True, GSL tries different time steps to find a solution that satisfies the set error bounds. It is likely that for Brian’s next time step the GSL time step will be somewhat similar per neuron (e.g. active neurons will have a shorter GSL time step than inactive neurons). With this option set to True, the time step GSL found to satisfy the set error bounds is saved per neuron and given to GSL again in Brian’s next time step. This also means that the final time steps are saved in Brian’s memory and can thus be recorded with the StateMonitor: it can be accessed under '_last_timestep'. Note that some extra memory is required to keep track of the last time steps. Defaults to True.
• 'save_failed_steps': if 'adaptable_timestep' is set to True, each time GSL tries a time step and it results in an estimated error that exceeds the set bounds, one is added to the '_failed_steps' variable. For purposes of investigating what happens within GSL during an integration step, we offer the option of saving this variable. Defaults to False.
• 'save_step_count': the same goes for the total number of GSL steps taken in a single Brian time step: this is optionally saved in the '_step_count' variable. Defaults to False.

Note that at the moment recording '_last_timestep', '_failed_steps', or '_step_count' requires a call to run() (e.g. with 0 ms) to trigger the code generation process, before the call to StateMonitor.

More information on the GSL ODE solver itself can be found in its documentation.