# Library models (Brian 1 –> 2 conversion)¶

## Neuron models¶

The neuron models in Brian 1’s `brian.library.IF` package are nothing more than shorthands for equations. The following table shows how the models from Brian 1 can be converted to explicit equations (and reset statements in the case of the adaptive exponential integrate-and-fire model) for use in Brian 2. The examples include a “current” `I` (depending on the model not necessarily in units of Ampère) and could e.g. be used to plot the f-I curve of the neuron.

### Perfect integrator¶

Brian 1

Brian 2

```eqs = (perfect_IF(tau=10*ms) +
Current('I : volt'))
group = NeuronGroup(N, eqs,
threshold='v > -50*mV',
reset='v = -70*mV')
```
```tau = 10*ms
eqs = '''dvm/dt = I/tau : volt
I : volt'''
group = NeuronGroup(N, eqs,
threshold='v > -50*mV',
reset='v = -70*mV')
```

### Leaky integrate-and-fire neuron¶

Brian 1

Brian 2

```eqs = (leaky_IF(tau=10*ms, El=-70*mV) +
Current('I : volt'))
group = ... # see above
```
```tau = 10*ms; El = -70*mV
eqs = '''dvm/dt = ((El - vm) + I)/tau : volt
I : volt'''
group = ... # see above
```

### Exponential integrate-and-fire neuron¶

Brian 1

Brian 2

```eqs = (exp_IF(C=1*nF, gL=30*nS, EL=-70*mV,
VT=-50*mV, DeltaT=2*mV) +
Current('I : amp'))
group = ... # see above
```
```C = 1*nF; gL = 30*nS; EL = -70*mV; VT = -50*mV; DeltaT = 2*mV
eqs = '''dvm/dt = (gL*(EL-vm)+gL*DeltaT*exp((vm-VT)/DeltaT) + I)/C : volt
I : amp'''
group = ... # see above
```

Brian 1

Brian 2

```eqs = (quadratic_IF(C=1*nF, a=5*nS/mV,
EL=-70*mV, VT=-50*mV) +
Current('I : amp'))
group = ... # see above
```
```C = 1*nF; a=5*nS/mV; EL=-70*mV; VT = -50*mV
eqs = '''dvm/dt = (a*(vm-EL)*(vm-VT) + I)/C : volt
I : amp'''
group = ... # see above
```

### Izhikevich neuron¶

Brian 1

Brian 2

```eqs = (Izhikevich(a=0.02/ms, b=0.2/ms) +
Current('I : volt/second'))
group = ... # see above
```
```a = 0.02/ms; b = 0.2/ms
eqs = '''dvm/dt = (0.04/ms/mV)*vm**2+(5/ms)*vm+140*mV/ms-w + I : volt
dw/dt = a*(b*vm-w) : volt/second
I : volt/second'''
group = ... # see above
```

### Adaptive exponential integrate-and-fire neuron (“Brette-Gerstner model”)¶

Brian 1

Brian 2

```# AdEx, aEIF, and Brette_Gerstner all refer to the same model
eqs = (aEIF(C=1*nF, gL=30*nS, EL=-70*mV,
VT=-50*mV, DeltaT=2*mV, tauw=150*ms, a=4*nS) +
Current('I:amp'))
group = NeuronGroup(N, eqs,
threshold='v > -20*mV',
```
```C = 1*nF; gL = 30*nS; EL = -70*mV; VT = -50*mV; DeltaT = 2*mV; tauw = 150*ms; a = 4*nS
eqs = '''dvm/dt = (gL*(EL-vm)+gL*DeltaT*exp((vm-VT)/DeltaT) -w + I)/C : volt
dw/dt=(a*(vm-EL)-w)/tauw : amp
I : amp'''
group = NeuronGroup(N, eqs,
threshold='vm > -20*mV',
reset='vm=-70*mV; w += 0.08*nA')
```

## Ionic currents¶

Brian 1’s functions for ionic currents, provided in `brian.library.ionic_currents` correspond to the following equations (note that the currents follow the convention to use a shifted membrane potential, i.e. the membrane potential at rest is 0mV):

Brian 1

Brian 2

```from brian.library.ionic_currents import *
defaultclock.dt = 0.01*ms
eqs_leak = leak_current(gl=60*nS, El=10.6*mV, current_name='I_leak')

eqs_K = K_current_HH(gmax=7.2*uS, EK=-12*mV, current_name='I_K')

eqs_Na = Na_current_HH(gmax=24*uS, ENa=115*mV, current_name='I_Na')

eqs = (MembraneEquation(C=200*pF) +
eqs_leak + eqs_K + eqs+Na +
Current('I_inj : amp'))
```
```defaultclock.dt = 0.01*ms
gl = 60*nS; El = 10.6*mV
eqs_leak = Equations('I_leak = gl*(El - vm) : amp')
g_K = 7.2*uS; EK = -12*mV
eqs_K = Equations('''I_K = g_K*n**4*(EK-vm) : amp
dn/dt = alphan*(1-n)-betan*n : 1
alphan = .01*(10*mV-vm)/(exp(1-.1*vm/mV)-1)/mV/ms : Hz
betan = .125*exp(-.0125*vm/mV)/ms : Hz''')
g_Na = 24*uS; ENa = 115*mV
eqs_Na = Equations('''I_Na = g_Na*m**3*h*(ENa-vm) : amp
dm/dt=alpham*(1-m)-betam*m : 1
dh/dt=alphah*(1-h)-betah*h : 1
alpham=.1*(25*mV-vm)/(exp(2.5-.1*vm/mV)-1)/mV/ms : Hz
betam=4*exp(-.0556*vm/mV)/ms : Hz
alphah=.07*exp(-.05*vm/mV)/ms : Hz
betah=1./(1+exp(3.-.1*vm/mV))/ms : Hz''')
C = 200*pF
eqs = Equations('''dvm/dt = (I_leak + I_K + I_Na + I_inj)/C : volt
I_inj : amp''') + eqs_leak + eqs_K + eqs_Na
```

## Synapses¶

Brian 1’s synaptic models, provided in `brian.library.synpases` can be converted to the equivalent Brian 2 equations as follows:

### Current-based synapses¶

Brian 1

Brian 2

```syn_eqs = exp_current('s', tau=5*ms, current_name='I_syn')
eqs = (MembraneEquation(C=1*nF) + Current('Im = gl*(El-vm) : amp') +
syn_eqs)
group = NeuronGroup(N, eqs, threshold='vm>-50*mV', reset='vm=-70*mV')
syn = Synapses(source, group, pre='s += 1*nA')
# ... connect synapses, etc.
```
```tau = 5*ms
syn_eqs = Equations('dI_syn/dt = -I_syn/tau : amp')
eqs = (Equations('dvm/dt = (gl*(El - vm) + I_syn)/C : volt') +
syn_eqs)
group = NeuronGroup(N, eqs, threshold='vm>-50*mV', reset='vm=-70*mV')
syn = Synapses(source, group, on_pre='I_syn += 1*nA')
# ... connect synapses, etc.
```
```syn_eqs = alpha_current('s', tau=2.5*ms, current_name='I_syn')
eqs = ... # remaining code as above
```
```tau = 2.5*ms
syn_eqs = Equations('''dI_syn/dt = (s - I_syn)/tau : amp
ds/dt = -s/tau : amp''')
group = NeuronGroup(N, eqs, threshold='vm>-50*mV', reset='vm=-70*mV')
syn = Synapses(source, group, on_pre='s += 1*nA')
# ... connect synapses, etc.
```
```syn_eqs = biexp_current('s', tau1=2.5*ms, tau2=10*ms, current_name='I_syn')
eqs = ... # remaining code as above
```
```tau1 = 2.5*ms; tau2 = 10*ms; invpeak = (tau2 / tau1) ** (tau1 / (tau2 - tau1))
syn_eqs = Equations('''dI_syn/dt = (invpeak*s - I_syn)/tau1 : amp
ds/dt = -s/tau2 : amp''')
eqs = ... # remaining code as above
```

### Conductance-based synapses¶

Brian 1

Brian 2

```syn_eqs = exp_conductance('s', tau=5*ms, E=0*mV, conductance_name='g_syn')
eqs = (MembraneEquation(C=1*nF) + Current('Im = gl*(El-vm) : amp') +
syn_eqs)
group = NeuronGroup(N, eqs, threshold='vm>-50*mV', reset='vm=-70*mV')
syn = Synapses(source, group, pre='s += 10*nS')
# ... connect synapses, etc.
```
```tau = 5*ms; E = 0*mV
syn_eqs = Equations('dg_syn/dt = -g_syn/tau : siemens')
eqs = (Equations('dvm/dt = (gl*(El - vm) + g_syn*(E - vm))/C : volt') +
syn_eqs)
group = NeuronGroup(N, eqs, threshold='vm>-50*mV', reset='vm=-70*mV')
syn = Synapses(source, group, on_pre='g_syn += 10*nS')
# ... connect synapses, etc.
```
```syn_eqs = alpha_conductance('s', tau=2.5*ms, E=0*mV, conductance_name='g_syn')
eqs = ... # remaining code as above
```
```tau = 2.5*ms; E = 0*mV
syn_eqs = Equations('''dg_syn/dt = (s - g_syn)/tau : siemens
ds/dt = -s/tau : siemens''')
group = NeuronGroup(N, eqs, threshold='vm>-50*mV', reset='vm=-70*mV')
syn = Synapses(source, group, on_pre='s += 10*nS')
# ... connect synapses, etc.
```
```syn_eqs = biexp_conductance('s', tau1=2.5*ms, tau2=10*ms, E=0*mV,
conductance_name='g_syn')
eqs = ... # remaining code as above
```
```tau1 = 2.5*ms; tau2 = 10*ms; E = 0*mV
invpeak = (tau2 / tau1) ** (tau1 / (tau2 - tau1))
syn_eqs = Equations('''dg_syn/dt = (invpeak*s - g_syn)/tau1 : siemens
ds/dt = -s/tau2 : siemens''')
eqs = ... # remaining code as above
```