# Refractoriness

Brian allows you to model the absolute refractory period of a neuron in a flexible way. The definition of refractoriness consists of two components: the amount of time after a spike that a neuron is considered to be refractory, and what changes in the neuron during the refractoriness.

## Defining the refractory period

The refractory period is specified by the `refractory` keyword in the `NeuronGroup` initializer. In the simplest case, this is simply a fixed time, valid for all neurons:

    G = NeuronGroup(N, model='...', threshold='...', reset='...',
                    refractory=2*ms)

Alternatively, it can be a string expression that evaluates to a time. This expression will be evaluated after every spike and allows for a varying refractory period. For example, the following will set the refractory period to a random duration between 1ms and 3ms after every spike:

    G = NeuronGroup(N, model='...', threshold='...', reset='...',
                    refractory='(1 + 2*rand())*ms')

In general, modelling a refractory period that varies across neurons involves declaring a state variable that stores the refractory period per neuron as a model parameter. The refractory expression can then refer to this parameter:

    G = NeuronGroup(N, model='''...
                                ref : second''', threshold='...',
                    reset='...', refractory='ref')
    # Set the refractory period for each cell
    G.ref = ...

This state variable can also be a dynamic variable itself. For example, it can serve as an adaptation mechanism by increasing it after every spike and letting it relax back to a steady-state value between spikes:

    refractory_0 = 2*ms
    tau_refractory = 50*ms
    G = NeuronGroup(N, model='''...
                                dref/dt = (refractory_0 - ref) / tau_refractory : second''',
                    threshold='...', refractory='ref',
                    reset='''...
                             ref += 1*ms''')
    G.ref = refractory_0

In some cases, the condition for leaving the refractory period is not easily expressed as a certain time span. For example, in a Hodgkin-Huxley type model the threshold is only used for counting spikes and the refractoriness is used to prevent the count of multiple spikes for a single threshold crossing (the threshold condition would evaluate to `True` for several time points). When a neuron should leave the refractory period is not easily expressed as a time span but more naturally as a condition that the neuron should remain refractory for as long as it stays above the threshold. This can be achieved by using a string expression for the `refractory` keyword that evaluates to a boolean condition:

    G = NeuronGroup(N, model='...', threshold='v > -20*mV',
                    refractory='v >= -20*mV')

The `refractory` keyword should be read as “stay refractory as long as the condition remains true”. In fact, specifying a time span for the refractoriness will be automatically transformed into a logical expression using the current time `t` and the time of the last spike `lastspike`. Specifying `refractory=2*ms` is basically equivalent to specifying `refractory='(t - lastspike) <= 2*ms'`. However, this expression can give inconsistent results for the common case that the refractory period is a multiple of the simulation timestep. Due to floating point impreciseness, the actual value of `t - lastspike` can be slightly above or below a multiple of the simulation time step; comparing it directly to the refractory period can therefore lead to an end of the refractory one time step sooner or later. To avoid this issue, the actual code used for the above example is equivalent to `refractory='timestep(t - lastspike, dt) <= timestep(2*ms, dt)'`. The `timestep` function is provided by Brian and takes care of converting a time into a time step in a safe way.

Added in version 2.1.3: The `timestep` function is now used to avoid floating point issues in the refractoriness calculation. To restore the previous behaviour, set the legacy.refractory_timing preference to `True`.

## Defining model behaviour during refractoriness

The refractoriness definition as described above only has a single effect by itself: threshold crossings during the refractory period are ignored. In the following model, the variable `v` continues to update during the refractory period but it does not elicit a spike if it crosses the threshold:

    G = NeuronGroup(N, 'dv/dt = -v / tau : 1',
                    threshold='v > 1', reset='v=0',
                    refractory=2*ms)

There is also a second implementation of refractoriness that is supported by Brian, one or several state variables can be clamped during the refractory period. To model this kind of behaviour, variables that should stop being updated during refractoriness can be marked with the `(unless refractory)` flag:

    G = NeuronGroup(N, '''dv/dt = -(v + w)/ tau_v : 1 (unless refractory)
                          dw/dt = -w / tau_w : 1''',
                    threshold='v > 1', reset='v=0; w+=0.1', refractory=2*ms)

In the above model, the `v` variable is clamped at 0 for 2ms after a spike but the adaptation variable `w` continues to update during this time. In addition, a variable of a neuron that is in its refractory period is read-only: incoming synapses or other code will have no effect on the value of `v` until it leaves its refractory period.

The following topics are not essential for beginners.

## Arbitrary refractoriness

In fact, arbitrary behaviours can be defined using Brian’s refractoriness mechanism.

A `NeuronGroup` with refractoriness automatically defines two variables:

`not_refractory`

  A boolean variable stating whether a neuron is allowed to spike.

`lastspike`

  The time of the last spike of the neuron.

The variable `not_refractory` is updated at every time step by checking the refractoriness condition – for a refractoriness defined by a time period, this means comparing `lastspike` to the current time `t`. The `not_refractory` variable is then used to implement the refractoriness behaviour. Specifically, the `threshold` condition is replaced by `threshold and not_refractory` and differential equations that are marked as `(unless refractory)` are multiplied by `int(not_refractory)` (so that they have the value 0 when the neuron is refractory).

This `not_refractory` variable is also available to the user to define more sophisticated refractoriness behaviour. For example, the following code updates the `w` variable with a different time constant during refractoriness:

    G = NeuronGroup(N, '''dv/dt = -(v + w)/ tau_v : 1 (unless refractory)
                          dw/dt = (-w / tau_active)*int(not_refractory) + (-w / tau_ref)*(1 - int(not_refractory)) : 1''',
                    threshold='v > 1', reset='v=0; w+=0.1', refractory=2*ms)