# Models and neuron groups¶

## Model equations¶

The core of every simulation is a `NeuronGroup`

, a group of neurons that share
the same equations defining their properties. The minimum `NeuronGroup`

specification contains the number of neurons and the model description in the
form of equations:

```
G = NeuronGroup(10, 'dv/dt = -v/(10*ms) : volt')
```

This defines a group of 10 leaky integrators. The model description can be
directly given as a (possibly multi-line) string as above, or as an
`Equations`

object. For more details on the form of equations, see
Equations. Brian needs the model to be given in the form of differential
equations, but you might see the integrated form of synapses in some textbooks
and papers. See Converting from integrated form to ODEs for details on how
to convert between these representations.

Note that model descriptions can make reference to physical units, but also to scalar variables declared outside of the model description itself:

```
tau = 10*ms
G = NeuronGroup(10, 'dv/dt = -v/tau : volt')
```

If a variable should be taken as a *parameter* of the neurons, i.e. if it
should be possible to vary its value across neurons, it has to be declared
as part of the model description:

```
G = NeuronGroup(10, '''dv/dt = -v/tau : volt
tau : second''')
```

To make complex model descriptions more readable, named subexpressions can be used:

```
G = NeuronGroup(10, '''dv/dt = I_leak / Cm : volt
I_leak = g_L*(E_L - v) : amp''')
```

For a list of some standard model equations, see Neural models (Brian 1 –> 2 conversion).

## Noise¶

In addition to ordinary differential equations, Brian allows you to introduce random noise by specifying a stochastic differential equation. Brian uses the physicists’ notation used in the Langevin equation, representing the “noise” as a term \(\xi(t)\), rather than the mathematicians’ stochastic differential \(\mathrm{d}W_t\). The following is an example of the Ornstein-Uhlenbeck process that is often used to model a leaky integrate-and-fire neuron with a stochastic current:

```
G = NeuronGroup(10, 'dv/dt = -v/tau + sigma*sqrt(2/tau)*xi : volt')
```

You can start by thinking of `xi`

as just a Gaussian random variable
with mean 0 and standard deviation 1. However, it scales in an
unusual way with time and this gives it units of `1/sqrt(second)`

.
You don’t necessarily need to understand why this is, but it is
possible to get a reasonably simple intuition for it by thinking
about numerical integration: see below.

## Threshold and reset¶

To emit spikes, neurons need a *threshold*. Threshold and reset are given
as strings in the `NeuronGroup`

constructor:

```
tau = 10*ms
G = NeuronGroup(10, 'dv/dt = -v/tau : volt', threshold='v > -50*mV',
reset='v = -70*mV')
```

Whenever the threshold condition is fulfilled, the reset statements will be executed. Again, both threshold and reset can refer to physical units, external variables and parameters, in the same way as model descriptions:

```
v_r = -70*mV # reset potential
G = NeuronGroup(10, '''dv/dt = -v/tau : volt
v_th : volt # neuron-specific threshold''',
threshold='v > v_th', reset='v = v_r')
```

You can also create non-spike events. See Custom events for more details.

## Refractoriness¶

To make a neuron non-excitable for a certain time period after a spike, the refractory keyword can be used:

```
G = NeuronGroup(10, 'dv/dt = -v/tau : volt', threshold='v > -50*mV',
reset='v = -70*mV', refractory=5*ms)
```

This will not allow any threshold crossing for a neuron for 5ms after a spike. The refractory keyword allows for more flexible refractoriness specifications, see Refractoriness for details.

## State variables¶

Differential equations and parameters in model descriptions are stored as
*state variables* of the `NeuronGroup`

. In addition to these variables, Brian
also defines two variables automatically:

`i`

- The index of a neuron.
`N`

- The total number of neurons.

All state variables can be accessed and set as an attribute of the group. To get the values without physical units (e.g. for analysing data with external tools), use an underscore after the name:

```
>>> G = NeuronGroup(10, '''dv/dt = -v/tau : volt
... tau : second''', name='neurons')
>>> G.v = -70*mV
>>> G.v
<neurons.v: array([-70., -70., -70., -70., -70., -70., -70., -70., -70., -70.]) * mvolt>
>>> G.v_ # values without units
<neurons.v_: array([-0.07, -0.07, -0.07, -0.07, -0.07, -0.07, -0.07, -0.07, -0.07, -0.07])>
```

The value of state variables can also be set using string expressions that can refer to units and external variables, other state variables or mathematical functions:

```
>>> G.tau = '5*ms + (1.0*i/N)*5*ms'
>>> G.tau
<neurons.tau: array([ 5. , 5.5, 6. , 6.5, 7. , 7.5, 8. , 8.5, 9. , 9.5]) * msecond>
```

You can also set the value only if a condition holds, for example:

```
>>> G.v['tau>7.25*ms'] = -60*mV
>>> G.v
<neurons.v: array([-70., -70., -70., -70., -70., -60., -60., -60., -60., -60.]) * mvolt>
```

## Subgroups¶

It is often useful to refer to a subset of neurons, this can be achieved using Python’s slicing syntax:

```
G = NeuronGroup(10, '''dv/dt = -v/tau : volt
tau : second''',
threshold='v > -50*mV',
reset='v = -70*mV')
# Create subgroups
G1 = G[:5]
G2 = G[5:]
# This will set the values in the main group, subgroups are just "views"
G1.tau = 10*ms
G2.tau = 20*ms
```

Here `G1`

refers to the first 5 neurons in G, and `G2`

to the second 5
neurons. In general `G[i:j]`

refers to the neurons with indices from `i`

to `j-1`

, as in general in Python.

For convenience, you can also use a single index, i.e. `G[i]`

is equivalent
to `G[i:i+1]`

. In some situations, it can be easier to provide a list of
indices instead of a slice, Brian therefore also allows for this syntax. Note
that this is restricted to cases that are strictly equivalent with slicing
syntax, e.g. you can write `G[[3, 4, 5]]`

instead of `G[3:6]`

, but you
*cannot* write `G[[3, 5, 7]]`

or `G[[5, 4, 3]]`

.

Subgroups can be used in most places where regular groups are used, e.g. their
state variables or spiking activity can be recorded using monitors, they can be
connected via `Synapses`

, etc. In such situations, indices (e.g. the indices of
the neurons to record from in a `StateMonitor`

) are relative to the subgroup,
not to the main group

The following topics are not essential for beginners.

## Storing state variables¶

Sometimes it can be convenient to access multiple state variables at once, e.g.
to set initial values from a dictionary of values or to store all the values of
a group on disk. This can be done with the
`get_states()`

and
`set_states()`

methods:

```
>>> group = NeuronGroup(5, '''dv/dt = -v/tau : 1
... tau : second''', name='neurons')
>>> initial_values = {'v': [0, 1, 2, 3, 4],
... 'tau': [10, 20, 10, 20, 10]*ms}
>>> group.set_states(initial_values)
>>> group.v[:]
array([ 0., 1., 2., 3., 4.])
>>> group.tau[:]
array([ 10., 20., 10., 20., 10.]) * msecond
>>> states = group.get_states()
>>> states['v']
array([ 0., 1., 2., 3., 4.])
```

The data (without physical units) can also be exported/imported to/from
Pandas data frames (needs an installation of `pandas`

):

```
>>> df = group.get_states(units=False, format='pandas')
>>> df
N dt i t tau v
0 5 0.0001 0 0.0 0.01 0.0
1 5 0.0001 1 0.0 0.02 1.0
2 5 0.0001 2 0.0 0.01 2.0
3 5 0.0001 3 0.0 0.02 3.0
4 5 0.0001 4 0.0 0.01 4.0
>>> df['tau']
0 0.01
1 0.02
2 0.01
3 0.02
4 0.01
Name: tau, dtype: float64
>>> df['tau'] *= 2
>>> group.set_states(df[['tau']], units=False, format='pandas')
>>> group.tau
<neurons.tau: array([ 20., 40., 20., 40., 20.]) * msecond>
```

## Linked variables¶

A `NeuronGroup`

can define parameters that are not stored in this group, but are
instead a reference to a state variable in another group. For this, a group
defines a parameter as `linked`

and then uses `linked_var()`

to
specify the linking. This can for example be useful to model shared noise
between cells:

```
inp = NeuronGroup(1, 'dnoise/dt = -noise/tau + tau**-0.5*xi : 1')
neurons = NeuronGroup(100, '''noise : 1 (linked)
dv/dt = (-v + noise_strength*noise)/tau : volt''')
neurons.noise = linked_var(inp, 'noise')
```

If the two groups have the same size, the linking will be done in a 1-to-1 fashion. If the source group has the size one (as in the above example) or if the source parameter is a shared variable, then the linking will be done as 1-to-all. In all other cases, you have to specify the indices to use for the linking explicitly:

```
# two inputs with different phases
inp = NeuronGroup(2, '''phase : 1
dx/dt = 1*mV/ms*sin(2*pi*100*Hz*t-phase) : volt''')
inp.phase = [0, pi/2]
neurons = NeuronGroup(100, '''inp : volt (linked)
dv/dt = (-v + inp) / tau : volt''')
# Half of the cells get the first input, other half gets the second
neurons.inp = linked_var(inp, 'x', index=repeat([0, 1], 50))
```

## Time scaling of noise¶

Suppose we just had the differential equation

\(dx/dt=\xi\)

To solve this numerically, we could compute

\(x(t+\mathrm{d}t)=x(t)+\xi_1\)

where \(\xi_1\) is a normally distributed random number with mean 0 and standard deviation 1. However, what happens if we change the time step? Suppose we used a value of \(\mathrm{d}t/2\) instead of \(\mathrm{d}t\). Now, we compute

\(x(t+\mathrm{d}t)=x(t+\mathrm{d}t/2)+\xi_1=x(t)+\xi_2+\xi_1\)

The mean value of \(x(t+\mathrm{d}t)\) is 0 in both cases, but the standard deviations are different. The first method \(x(t+\mathrm{d}t)=x(t)+\xi_1\) gives \(x(t+\mathrm{d}t)\) a standard deviation of 1, whereas the second method \(x(t+\mathrm{d}t)=x(t+\mathrm{d}/2)+\xi_1=x(t)+\xi_2+\xi_1\) gives \(x(t)\) a variance of 1+1=2 and therefore a standard deviation of \(\sqrt{2}\).

In order to solve this
problem, we use the rule
\(x(t+\mathrm{d}t)=x(t)+\sqrt{\mathrm{d}t}\xi_1\), which makes
the mean and standard deviation of the value at time \(t\)
independent of \(\mathrm{d}t\).
For this to make sense dimensionally, \(\xi\) must have
units of `1/sqrt(second)`

.

For further details, refer to a textbook on stochastic differential equations.