# Equations¶

## Equation strings¶

Equations are used both in NeuronGroup and Synapses to:

• define state variables
• define continuous-updates on these variables, through differential equations

Equations are defined by multiline strings.

An Equation is a set of single lines in a string:
1. dx/dt = f : unit (differential equation)
2. x = f : unit (subexpression)
3. x : unit (parameter)

The equations may be defined on multiple lines (no explicit line continuation with \ is necessary). Comments using # may also be included. Subunits are not allowed, i.e., one must write volt, not mV. This is to make it clear that the values are internally always saved in the basic units, so no confusion can arise when getting the values out of a NeuronGroup and discarding the units. Compound units are of course allowed as well (e.g. farad/meter**2).

Aliases are no longer available in Brian 2. Some special variables are defined: t, dt (time) and xi (white noise). Variable names starting with an underscore and a couple of other names that have special meanings under certain circumstances (e.g. names ending in _pre or _post) are forbidden.

For stochastic equations with several xi values it is now necessary to make clear whether they correspond to the same or different noise instantiations. To make this distinction, an arbitrary suffix can be used, e.g. using xi_1 several times refers to the same variable, xi_2 (or xi_inh, xi_alpha, etc.) refers to another. An error will be raised if you use more than one plain xi. Note that noise is always independent across neurons, you can only work around this restriction by defining your noise variable as a scalar parameter and update it using a user-defined function (e.g. a CodeRunner).

### Flags¶

A new syntax is the possibility of flags. A flag is a keyword in brackets, which qualifies the equations. There are several keywords:

event-driven
this is only used in Synapses, and means that the differential equation should be updated only at the times of events. This implies that the equation is taken out of the continuous state update, and instead a event-based state update statement is generated and inserted into event codes (pre and post). This can only qualify differential equations of synapses. Currently, only one-dimensional linear equations can be handled (see below).
unless refractory
this means the variable is not updated during the refractory period. This can only qualify differential equations of neuron groups.
constant
this means the parameter will not be changed during a run. This allows optimizations in state updaters. This can only qualify parameters.
scalar
this means that a parameter or subexpression isn’t neuron-/synapse-specific but rather a single value for the whole NeuronGroup or Synapses. A scalar subexpression can only refer to other scalar variables.

Different flags may be specified as follows:

dx/dt = f : unit (flag1,flag2)


### Event-driven equations¶

Equations defined as event-driven are completely ignored in the state update. They are only defined as variables that can be externally accessed. There are additional constraints:

• An event-driven variable cannot be used by any other equation that is not also event-driven.
• An event-driven equation cannot depend on a differential equation that is not event-driven (directly, or indirectly through subexpressions). It can depend on a constant parameter. An open question is whether we should also allow it to depend on a parameter not defined as constant (I would say no).

Currently, automatic event-driven updates are only possible for one-dimensional linear equations, but it could be extended.

### Equation objects¶

The model definitions for NeuronGroup and Synapses can be simple strings or Equations objects. Such objects can be combined using the add operator:

eqs = Equations('dx/dt = (y-x)/tau : volt')
eqs += Equations('dy/dt = -y/tau: volt')


In contrast to Brian 1, Equations objects do not save the surrounding namespace (which led to a lot of complications when combining equations), they are mostly convenience wrappers around strings. They do allow for the specification of values in the strings, but do this by simple string replacement, e.g. you can do:

eqs = Equations('dx/dt = x/tau : volt', tau=10*ms)


but this is exactly equivalent to:

eqs = Equations('dx/dt = x/(10*ms) : volt')


In contrast to Brian 1, specifying the value of a variable using a keyword argument does not mean you have to specify the values for all external variables by keywords. [Question: Useful to have the same kind of classes for Thresholds and Resets (Expression and Statements) just for convenience?]

The Equations object does some basic syntax checking and will raise an error if two equations defining the same variable are combined. It does not however do unit checking, checking for unknown identifiers or incorrect flags – all this will be done during the instantiation of a NeuronGroup or Synapses object.

## External variables and functions¶

Equations defining neuronal or synaptic equations can contain references to external parameters or functions. During the initialisation of a NeuronGroup or a Synapses object, this namespace can be provided as an argument. This is a group-specific namespace that will only be used for names in the context of the respective group. Note that units and a set of standard functions are always provided and should not be given explicitly. This namespace does not necessarily need to be exhaustive at the time of the creation of the NeuronGroup/Synapses, entries can be added (or modified) at a later stage via the namespace attribute (e.g. G.namespace['tau'] = 10*ms).

At the point of the call to the Network.run() namespace, any group-specific namespace will be augmented by the “run namespace”. This namespace can be either given explicitly as an argument to the run method or it will be taken from the locals and globals surrounding the call. A warning will be emitted if a name is defined in more than one namespace.

To summarize: an external identifier will be looked up in the context of an object such as NeuronGroup or Synapses. It will follow the following resolution hierarchy:

1. Default unit and function names.
2. Names defined in the explicit group-specific namespace.
3. Names in the run namespace which is either explicitly given or the implicit namespace surrounding the run call.

Note that if you completely specify your namespaces at the Group level, you should probably pass an empty dictionary as the namespace argument to the run call – this will completely switch off the “implicit namespace” mechanism.

The following three examples show the different ways of providing external variable values, all having the same effect in this case:

# Explicit argument to the NeuronGroup
G = NeuronGroup(1, 'dv/dt = -v / tau : 1', namespace={'tau': 10*ms})
net = Network(G)
net.run(10*ms)

# Explicit argument to the run function
G = NeuronGroup(1, 'dv/dt = -v / tau : 1')
net = Network(G)
net.run(10*ms, namespace={'tau': 10*ms})

# Implicit namespace from the context
G = NeuronGroup(1, 'dv/dt = -v / tau : 1')
net = Network(G)
tau = 10*ms
net.run(10*ms)


External variables are free to change between runs (but not during one run), the value at the time of the run() call is used in the simulation.

## Examples¶

### Equation objects¶

Concatenating equations

>>> membrane_eqs = Equations('dv/dt = -(v + I)/ tau : volt')
>>> eqs1 = membrane_eqs + Equations('''I = sin(2*pi*freq*t) : volt
...                                    freq : Hz''')
>>> eqs2 = membrane_eqs + Equations('''I : volt''')
>>> print eqs1
I = sin(2*pi*freq*t)  : V
dv/dt = -(v + I)/ tau  : V
freq : Hz
>>> print eqs2
dv/dt = -(v + I)/ tau  : V
I : V


Substituting variable names

>>> general_equation = 'dg/dt = -g / tau : siemens'
>>> eqs_exc = Equations(general_equation, g='g_e', tau='tau_e')
>>> eqs_inh = Equations(general_equation, g='g_i', tau='tau_i')
>>> print eqs_exc
dg_e/dt = -g_e / tau_e  : S
>>> print eqs_inh
dg_i/dt = -g_i / tau_i  : S


Inserting values

>>> eqs = Equations('dv/dt = mu/tau + sigma/tau**.5*xi : volt',
mu = -65*mV, sigma=3*mV, tau=10*ms)
>>> print eqs
dv/dt = (-0.065 * volt)/(10.0 * msecond) + (3.0 * mvolt)/(10.0 * msecond)**.5*xi  : V