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''')
>>> G.v = -70*mV
>>> G.v
<neurongroup.v: array([-70., -70., -70., -70., -70., -70., -70., -70., -70., -70.]) * mvolt>
>>> G.v_  # values without units
<neurongroup.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
<neurongroup.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
<neurongroup.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. 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.


Shared variables

Sometimes it can also be useful to introduce shared variables or subexpressions, i.e. variables that have a common value for all neurons. In contrast to external variables (such as Cm above), such variables can change during a run, e.g. by using run_regularly(). This can be for example used for an external stimulus that changes in the course of a run:

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

Note that there are several restrictions around the use of shared variables: they cannot be written to in contexts where statements apply only to a subset of neurons (e.g. reset statements, see below). If a code block mixes statements writing to shared and vector variables, then the shared statements have to come first.

By default, subexpressions are re-evaluated whenever they are used, i.e. using a subexpression is completely equivalent to substituting it. Sometimes it is useful to instead only evaluate a subexpression once and then use this value for the rest of the time step. This can be achieved by using the (constant over dt) flag. This flag is mandatory for subexpressions that refer to stateful functions like rand() which notably allows them to be recorded with a StateMonitor – otherwise the monitor would record a different instance of the random number than the one that was used in the equations.

For shared variables, setting by string expressions can only refer to shared values:

>>> G.shared_input = '(4.0/N)*mV'
>>> G.shared_input
<neurongroup.shared_input: 0.4 * mvolt>

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''')
>>> 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
<neurongroup.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.