ExplicitStateUpdater class

(Shortest import: from brian2 import ExplicitStateUpdater)

class brian2.stateupdaters.explicit.ExplicitStateUpdater(description, stochastic=None, custom_check=None)[source]

Bases: brian2.stateupdaters.base.StateUpdateMethod

An object that can be used for defining state updaters via a simple description (see below). Resulting instances can be passed to the method argument of the NeuronGroup constructor. As other state updater functions the ExplicitStateUpdater objects are callable, returning abstract code when called with an Equations object.

A description of an explicit state updater consists of a (multi-line) string, containing assignments to variables and a final “x_new = ...”, stating the integration result for a single timestep. The assignments can be used to define an arbitrary number of intermediate results and can refer to f(x, t) (the function being integrated, as a function of x, the previous value of the state variable and t, the time) and dt, the size of the timestep.

For example, to define a Runge-Kutta 4 integrator (already provided as rk4), use:

k1 = dt*f(x,t)
k2 = dt*f(x+k1/2,t+dt/2)
k3 = dt*f(x+k2/2,t+dt/2)
k4 = dt*f(x+k3,t+dt)
x_new = x+(k1+2*k2+2*k3+k4)/6

Note that for stochastic equations, the function f only corresponds to the non-stochastic part of the equation. The additional function g corresponds to the stochastic part that has to be multiplied with the stochastic variable xi (a standard normal random variable – if the algorithm needs a random variable with a different variance/mean you have to multiply/add it accordingly). Equations with more than one stochastic variable do not have to be treated differently, the part referring to g is repeated for all stochastic variables automatically.

Stochastic integrators can also make reference to dW (a normal distributed random number with variance dt) and g(x, t), the stochastic part of an equation. A stochastic state updater could therefore use a description like:

x_new = x + dt*f(x,t) + g(x, t) * dW

For simplicity, the same syntax is used for state updaters that only support additive noise, even though g(x, t) does not depend on x or t in that case.

There a some restrictions on the complexity of the expressions (but most can be worked around by using intermediate results as in the above Runge- Kutta example): Every statement can only contain the functions f and g once; The expressions have to be linear in the functions, e.g. you can use dt*f(x, t) but not f(x, t)**2.

Parameters:

description : str

A state updater description (see above).

stochastic : {None, ‘additive’, ‘multiplicative’}

What kind of stochastic equations this state updater supports: None means no support of stochastic equations, 'additive' means only equations with additive noise and 'multiplicative' means supporting arbitrary stochastic equations.

Raises

ValueError
If the parsing of the description failed.

See also

euler, rk2, rk4, milstein

Notes

Since clocks are updated after the state update, the time t used in the state update step is still at its previous value. Enumerating the states and discrete times, x_new = x + dt*f(x, t) is therefore understood as \(x_{i+1} = x_i + dt f(x_i, t_i)\), yielding the correct forward Euler integration. If the integrator has to refer to the time at the end of the timestep, simply use t + dt instead of t.

Attributes

Methods

__call__(eqs[, variables]) Apply a state updater description to model equations.
can_integrate(equations, variables)
replace_func(x, t, expr, temp_vars, eq_symbols) Used to replace a single occurance of f(x, t) or g(x, t): expr is the non-stochastic (in the case of f) or stochastic part (g) of the expression defining the right-hand-side of the differential equation describing var().

Details

DESCRIPTION = {[Group:({~{"x_new"} W:(abcd...,abcd...) Suppress:("=") Re:('.*')})]... Group:({Suppress:("x_new") Suppress:("=") Re:('.*')})}

A complete state updater description

EXPRESSION = Re:('.*')

A single expression

OUTPUT = Group:({Suppress:("x_new") Suppress:("=") Re:('.*')})

The last line of a state updater description

STATEMENT = Group:({~{"x_new"} W:(abcd...,abcd...) Suppress:("=") Re:('.*')})

An assignment statement

TEMP_VAR = {~{"x_new"} W:(abcd...,abcd...)}

Legal names for temporary variables

DESCRIPTION()

Requires all given C{ParseExpression}s to be found in the given order. Expressions may be separated by whitespace. May be constructed using the C{‘+’} operator.

EXPRESSION()

Token for matching strings that match a given regular expression. Defined with string specifying the regular expression in a form recognized by the inbuilt Python re module.

OUTPUT()

Converter to return the matched tokens as a list - useful for returning tokens of C{L{ZeroOrMore}} and C{L{OneOrMore}} expressions.

STATEMENT()

Converter to return the matched tokens as a list - useful for returning tokens of C{L{ZeroOrMore}} and C{L{OneOrMore}} expressions.

TEMP_VAR()

Requires all given C{ParseExpression}s to be found in the given order. Expressions may be separated by whitespace. May be constructed using the C{‘+’} operator.

__call__(eqs, variables=None)[source]

Apply a state updater description to model equations.

Parameters:

eqs : Equations

The equations describing the model

variables: dict-like, optional :

The Variable objects for the model. Ignored by the explicit state updater.

Examples

>>> from brian2 import *
>>> eqs = Equations('dv/dt = -v / tau : volt')
>>> print(euler(eqs))
_v = -dt*v/tau + v
v = _v
>>> print(rk4(eqs))
__k_1_v = -dt*v/tau
__k_2_v = -dt*(0.5*__k_1_v + v)/tau
__k_3_v = -dt*(0.5*__k_2_v + v)/tau
__k_4_v = -dt*(__k_3_v + v)/tau
_v = 0.166666666666667*__k_1_v + 0.333333333333333*__k_2_v + 0.333333333333333*__k_3_v + 0.166666666666667*__k_4_v + v
v = _v
can_integrate(equations, variables)[source]
replace_func(x, t, expr, temp_vars, eq_symbols, stochastic_variable=None)[source]

Used to replace a single occurance of f(x, t) or g(x, t): expr is the non-stochastic (in the case of f) or stochastic part (g) of the expression defining the right-hand-side of the differential equation describing var(). It replaces the variable var() with the value given as x and t by the value given for t. Intermediate variables will be replaced with the appropriate replacements as well.

For example, in the rk2 integrator, the second step involves the calculation of f(k/2 + x, dt/2 + t). If var() is v and expr is -v / tau, this will result in -(_k_v/2 + v)/tau.

Note that this deals with only one state variable var(), given as an argument to the surrounding _generate_RHS function.