# 3.3. Built-in neuron types¶

ANNarchy provides standard spiking neuron models, similar to the ones defined in PyNN (http://neuralensemble.org/docs/PyNN/reference/neuronmodels.html).

## 3.3.1. LeakyIntegrator¶

ANNarchy.models.LeakyIntegrator = <class 'ANNarchy.models.Neurons.LeakyIntegrator'>[source]

Leaky-integrator rate-coded neuron, optionally noisy.

This simple rate-coded neuron defines an internal variable $$v(t)$$ which integrates the inputs $$I(t)$$ with a time constant :math: au and a baseline B. An additive noise $$N(t)$$ can be optionally defined:

$\tau \cdot \frac{dv(t)}{dt} + v(t) = I(t) + B + N(t)$

The transfer function is the positive (or rectified linear ReLU) function with a thresholf $$T$$:

$r(t) = (v(t) - T)^+$

By default, the input I(t) to this neuron is “sum(exc) - sum(inh)”, but this can be changed by setting the sum argument:

neuron = LeakyIntegrator(sum="sum('exc')")


By default, there is no additive noise, but the noise argument can be passed with a specific distribution:

neuron = LeakyIntegrator(noise="Normal(0.0, 1.0)")


Parameters:

• tau = 10.0 : Time constant in ms of the neuron.
• B = 0.0 : Baseline value for v.
• T = 0.0 : Threshold for the positive transfer function.

Variables:

• v : internal variable (init = 0.0):

tau * dv/dt + v = sum(exc) - sum(inh) + B + N

• r : firing rate (init = 0.0):

r = pos(v - T)


The ODE is solved using the exponential Euler method.

Equivalent code:

LeakyIntegrator = Neuron(
parameters="""
tau = 10.0 : population
B = 0.0
T = 0.0 : population
""",
equations="""
tau * dv/dt + v = sum(exc) - sum(inh) + B : exponential
r = pos(v - T)
"""
)


## 3.3.2. Izhikevich¶

ANNarchy.models.Izhikevich = <class 'ANNarchy.models.Neurons.Izhikevich'>[source]

Izhikevich neuron.

\begin{align}\begin{aligned}\frac{dv}{dt} = 0.04 * v^2 + 5.0 * v + 140.0 - u + I \\\frac{du}{dt} = a * (b * v - u) \end{aligned}\end{align}

By default, the conductance is “g_exc - g_inh”, but this can be changed by setting the conductance argument:

neuron = Izhikevich(conductance='g_ampa * (1 + g_nmda) - g_gaba')


The synapses are instantaneous, i.e the corresponding conductance is increased from the synaptic efficiency w at the time step when a spike is received.

Parameters:

• a = 0.2 : Speed of the recovery variable
• b = 0.2: Scaling of the recovery variable
• c = -65.0 : Reset potential.
• d = 2.0 : Increment of the recovery variable after a spike.
• v_thresh = 30.0 : Spike threshold (mV).
• i_offset = 0.0 : external current (nA).
• noise = 0.0 : Amplitude of the normal additive noise.
• tau_refrac = 0.0 : Duration of refractory period (ms).

Variables:

• I : input current (user-defined conductance/current + external current + normal noise):

I = conductance + i_offset + noise * Normal(0.0, 1.0)

• v : membrane potential in mV (init = c):

dv/dt = 0.04 * v^2 + 5.0 * v + 140.0 - u + I

• u : recovery variable (init= b * c):

du/dt = a * (b * v - u)


Spike emission:

v > v_thresh


Reset:

v = c
u += d


The ODEs are solved using the explicit Euler method.

Equivalent code:

Izhikevich = Neuron(
parameters = """
noise = 0.0
a = 0.2
b = 0.2
c = -65.0
d = 2.0
v_thresh = 30.0
i_offset = 0.0
""",
equations = """
I = g_exc - g_inh + noise * Normal(0.0, 1.0) + i_offset
dv/dt = 0.04 * v^2 + 5.0 * v + 140.0 - u + I : init = -65.0
du/dt = a * (b*v - u) : init= -13.0
""",
spike = "v > v_thresh",
reset = "v = c; u += d",
refractory = 0.0
)


## 3.3.3. IF_curr_exp¶

ANNarchy.models.IF_curr_exp = <class 'ANNarchy.models.Neurons.IF_curr_exp'>[source]

IF_curr_exp neuron.

Leaky integrate-and-fire model with fixed threshold and decaying-exponential post-synaptic current. (Separate synaptic currents for excitatory and inhibitory synapses).

Parameters:

• v_rest = -65.0 : Resting membrane potential (mV)
• cm = 1.0 : Capacity of the membrane (nF)
• tau_m = 20.0 : Membrane time constant (ms)
• tau_refrac = 0.0 : Duration of refractory period (ms)
• tau_syn_E = 5.0 : Decay time of excitatory synaptic current (ms)
• tau_syn_I = 5.0 : Decay time of inhibitory synaptic current (ms)
• i_offset = 0.0 : Offset current (nA)
• v_reset = -65.0 : Reset potential after a spike (mV)
• v_thresh = -50.0 : Spike threshold (mV)

Variables:

• v : membrane potential in mV (init=-65.0):

cm * dv/dt = cm/tau_m*(v_rest -v)   + g_exc - g_inh + i_offset

• g_exc : excitatory current (init = 0.0):

tau_syn_E * dg_exc/dt = - g_exc

• g_inh : inhibitory current (init = 0.0):

tau_syn_I * dg_inh/dt = - g_inh


Spike emission:

v > v_thresh


Reset:

v = v_reset


The ODEs are solved using the exponential Euler method.

Equivalent code:

IF_curr_exp = Neuron(
parameters = """
v_rest = -65.0
cm  = 1.0
tau_m  = 20.0
tau_syn_E = 5.0
tau_syn_I = 5.0
v_thresh = -50.0
v_reset = -65.0
i_offset = 0.0
""",
equations = """
cm * dv/dt = cm/tau_m*(v_rest -v)   + g_exc - g_inh + i_offset : exponential, init=-65.0
tau_syn_E * dg_exc/dt = - g_exc : exponential
tau_syn_I * dg_inh/dt = - g_inh : exponential
""",
spike = "v > v_thresh",
reset = "v = v_reset",
refractory = 0.0
)


## 3.3.4. IF_cond_exp¶

ANNarchy.models.IF_cond_exp = <class 'ANNarchy.models.Neurons.IF_cond_exp'>[source]

IF_cond_exp neuron.

Leaky integrate-and-fire model with fixed threshold and decaying-exponential post-synaptic conductance.

Parameters:

• v_rest = -65.0 : Resting membrane potential (mV)
• cm = 1.0 : Capacity of the membrane (nF)
• tau_m = 20.0 : Membrane time constant (ms)
• tau_refrac = 0.0 : Duration of refractory period (ms)
• tau_syn_E = 5.0 : Decay time of excitatory synaptic current (ms)
• tau_syn_I = 5.0 : Decay time of inhibitory synaptic current (ms)
• e_rev_E = 0.0 : Reversal potential for excitatory input (mV)
• e_rev_I = -70.0 : Reversal potential for inhibitory input (mv)
• i_offset = 0.0 : Offset current (nA)
• v_reset = -65.0 : Reset potential after a spike (mV)
• v_thresh = -50.0 : Spike threshold (mV)

Variables:

• v : membrane potential in mV (init=-65.0):

cm * dv/dt = cm/tau_m*(v_rest -v)  + g_exc * (e_rev_E - v) + g_inh * (e_rev_I - v) + i_offset

• g_exc : excitatory current (init = 0.0):

tau_syn_E * dg_exc/dt = - g_exc

• g_inh : inhibitory current (init = 0.0):

tau_syn_I * dg_inh/dt = - g_inh


Spike emission:

v > v_thresh


Reset:

v = v_reset


The ODEs are solved using the exponential Euler method.

Equivalent code:

IF_cond_exp = Neuron(
parameters = """
v_rest = -65.0
cm  = 1.0
tau_m  = 20.0
tau_syn_E = 5.0
tau_syn_I = 5.0
e_rev_E = 0.0
e_rev_I = -70.0
v_thresh = -50.0
v_reset = -65.0
i_offset = 0.0
""",
equations = """
cm * dv/dt = cm/tau_m*(v_rest -v)   + g_exc * (e_rev_E - v) + g_inh * (e_rev_I - v) + i_offset : exponential, init=-65.0
tau_syn_E * dg_exc/dt = - g_exc : exponential
tau_syn_I * dg_inh/dt = - g_inh : exponential
""",
spike = "v > v_thresh",
reset = "v = v_reset",
refractory = 0.0
)


## 3.3.5. IF_curr_alpha¶

ANNarchy.models.IF_curr_alpha = <class 'ANNarchy.models.Neurons.IF_curr_alpha'>[source]

IF_curr_alpha neuron.

Leaky integrate-and-fire model with fixed threshold and alpha post-synaptic currents. (Separate synaptic currents for excitatory and inhibitory synapses).

The alpha currents are calculated through a system of two linears ODEs. After a spike is received at t_spike, it peaks at t_spike + tau_syn_X, with a maximum equal to the synaptic efficiency.

Parameters:

• v_rest = -65.0 : Resting membrane potential (mV)
• cm = 1.0 : Capacity of the membrane (nF)
• tau_m = 20.0 : Membrane time constant (ms)
• tau_refrac = 0.0 : Duration of refractory period (ms)
• tau_syn_E = 5.0 : Rise time of excitatory synaptic current (ms)
• tau_syn_I = 5.0 : Rise time of inhibitory synaptic current (ms)
• i_offset = 0.0 : Offset current (nA)
• v_reset = -65.0 : Reset potential after a spike (mV)
• v_thresh = -50.0 : Spike threshold (mV)

Variables:

• v : membrane potential in mV (init=-65.0):

cm * dv/dt = cm/tau_m*(v_rest -v) + alpha_exc - alpha_inh + i_offset

• g_exc : excitatory current (init = 0.0):

tau_syn_E * dg_exc/dt = - g_exc

• alpha_exc : alpha function of excitatory current (init = 0.0):

tau_syn_E * dalpha_exc/dt = exp((tau_syn_E - dt/2.0)/tau_syn_E) * g_exc - alpha_exc

• g_inh : inhibitory current (init = 0.0):

tau_syn_I * dg_inh/dt = - g_inh

• alpha_inh : alpha function of inhibitory current (init = 0.0):

tau_syn_I * dalpha_inh/dt = exp((tau_syn_I - dt/2.0)/tau_syn_I) * g_inh - alpha_inh


Spike emission:

v > v_thresh


Reset:

v = v_reset


The ODEs are solved using the exponential Euler method.

Equivalent code:

IF_curr_alpha = Neuron(
parameters = """
v_rest = -65.0
cm  = 1.0
tau_m  = 20.0
tau_syn_E = 5.0
tau_syn_I = 5.0
v_thresh = -50.0
v_reset = -65.0
i_offset = 0.0
""",
equations = """
gmax_exc = exp((tau_syn_E - dt/2.0)/tau_syn_E)
gmax_inh = exp((tau_syn_I - dt/2.0)/tau_syn_I)
cm * dv/dt = cm/tau_m*(v_rest -v)   + alpha_exc - alpha_inh + i_offset : exponential, init=-65.0
tau_syn_E * dg_exc/dt = - g_exc : exponential
tau_syn_I * dg_inh/dt = - g_inh : exponential
tau_syn_E * dalpha_exc/dt = gmax_exc * g_exc - alpha_exc  : exponential
tau_syn_I * dalpha_inh/dt = gmax_inh * g_inh - alpha_inh  : exponential
""",
spike = "v > v_thresh",
reset = "v = v_reset",
refractory = 0.0
)


## 3.3.6. IF_cond_alpha¶

ANNarchy.models.IF_cond_alpha = <class 'ANNarchy.models.Neurons.IF_cond_alpha'>[source]

IF_cond_exp neuron.

Leaky integrate-and-fire model with fixed threshold and alpha post-synaptic conductance.

Parameters:

• v_rest = -65.0 : Resting membrane potential (mV)
• cm = 1.0 : Capacity of the membrane (nF)
• tau_m = 20.0 : Membrane time constant (ms)
• tau_refrac = 0.0 : Duration of refractory period (ms)
• tau_syn_E = 5.0 : Rise time of excitatory synaptic current (ms)
• tau_syn_I = 5.0 : Rise time of inhibitory synaptic current (ms)
• e_rev_E = 0.0 : Reversal potential for excitatory input (mV)
• e_rev_I = -70.0 : Reversal potential for inhibitory input (mv)
• i_offset = 0.0 : Offset current (nA)
• v_reset = -65.0 : Reset potential after a spike (mV)
• v_thresh = -50.0 : Spike threshold (mV)

Variables:

• v : membrane potential in mV (init=-65.0):

cm * dv/dt = cm/tau_m*(v_rest -v)  + alpha_exc * (e_rev_E - v) + alpha_inh * (e_rev_I - v) + i_offset

• g_exc : excitatory conductance (init = 0.0):

tau_syn_E * dg_exc/dt = - g_exc

• alpha_exc : alpha function of excitatory conductance (init = 0.0):

tau_syn_E * dalpha_exc/dt = exp((tau_syn_E - dt/2.0)/tau_syn_E) * g_exc - alpha_exc

• g_inh : inhibitory conductance (init = 0.0):

tau_syn_I * dg_inh/dt = - g_inh

• alpha_inh : alpha function of inhibitory current (init = 0.0):

tau_syn_I * dalpha_inh/dt = exp((tau_syn_I - dt/2.0)/tau_syn_I) * g_inh - alpha_inh


Spike emission:

v > v_thresh


Reset:

v = v_reset


The ODEs are solved using the exponential Euler method.

Equivalent code:

IF_cond_alpha = Neuron(
parameters = """
v_rest = -65.0
cm  = 1.0
tau_m  = 20.0
tau_syn_E = 5.0
tau_syn_I = 5.0
e_rev_E = 0.0
e_rev_I = -70.0
v_thresh = -50.0
v_reset = -65.0
i_offset = 0.0
""",
equations = """
gmax_exc = exp((tau_syn_E - dt/2.0)/tau_syn_E)
gmax_inh = exp((tau_syn_I - dt/2.0)/tau_syn_I)
cm * dv/dt = cm/tau_m*(v_rest -v)   + alpha_exc * (e_rev_E - v) + alpha_inh * (e_rev_I - v) + i_offset : exponential, init=-65.0
tau_syn_E * dg_exc/dt = - g_exc : exponential
tau_syn_I * dg_inh/dt = - g_inh : exponential
tau_syn_E * dalpha_exc/dt = gmax_exc * g_exc - alpha_exc  : exponential
tau_syn_I * dalpha_inh/dt = gmax_inh * g_inh - alpha_inh  : exponential
""",
spike = "v > v_thresh",
reset = "v = v_reset",
refractory = 0.0
)


## 3.3.7. HH_cond_exp¶

ANNarchy.models.HH_cond_exp = <class 'ANNarchy.models.Neurons.HH_cond_exp'>[source]

HH_cond_exp neuron.

Single-compartment Hodgkin-Huxley-type neuron with transient sodium and delayed-rectifier potassium currents using the ion channel models from Traub.

Parameters:

• gbar_Na = 20.0 : Maximal conductance of the Sodium current.
• gbar_K = 6.0 : Maximal conductance of the Potassium current.
• gleak = 0.01 : Conductance of the leak current (nF)
• cm = 0.2 : Capacity of the membrane (nF)
• v_offset = -63.0 : Threshold for the rate constants (mV)
• e_rev_Na = 50.0 : Reversal potential for the Sodium current (mV)
• e_rev_K = -90.0 : Reversal potential for the Potassium current (mV)
• e_rev_leak = -65.0 : Reversal potential for the leak current (mV)
• e_rev_E = 0.0 : Reversal potential for excitatory input (mV)
• e_rev_I = -80.0 : Reversal potential for inhibitory input (mV)
• tau_syn_E = 0.2 : Decay time of excitatory synaptic current (ms)
• tau_syn_I = 2.0 : Decay time of inhibitory synaptic current (ms)
• i_offset = 0.0 : Offset current (nA)
• v_thresh = 0.0 : Threshold for spike emission

Variables:

• Voltage-dependent rate constants an, bn, am, bm, ah, bh:

an = 0.032 * (15.0 - v + v_offset) / (exp((15.0 - v + v_offset)/5.0) - 1.0)
am = 0.32  * (13.0 - v + v_offset) / (exp((13.0 - v + v_offset)/4.0) - 1.0)
ah = 0.128 * exp((17.0 - v + v_offset)/18.0)

bn = 0.5   * exp ((10.0 - v + v_offset)/40.0)
bm = 0.28  * (v - v_offset - 40.0) / (exp((v - v_offset - 40.0)/5.0) - 1.0)
bh = 4.0/(1.0 + exp (( 10.0 - v + v_offset )) )

• Activation variables n, m, h (h is initialized to 1.0, n and m to 0.0):

dn/dt = an * (1.0 - n) - bn * n
dm/dt = am * (1.0 - m) - bm * m
dh/dt = ah * (1.0 - h) - bh * h

• v : membrane potential in mV (init=-65.0):

cm * dv/dt = gleak*(e_rev_leak -v) + gbar_K * n**4 * (e_rev_K - v) + gbar_Na * m**3 * h * (e_rev_Na - v)

+ g_exc * (e_rev_E - v) + g_inh * (e_rev_I - v) + i_offset

• g_exc : excitatory conductance (init = 0.0):

tau_syn_E * dg_exc/dt = - g_exc

• g_inh : inhibitory conductance (init = 0.0):

tau_syn_I * dg_inh/dt = - g_inh


Spike emission:

v > v_thresh and v(t-1) < v_thresh (the spike is emitted only once when v crosses the threshold from below)


The ODEs for n, m, h and v are solved using the midpoint method, while the conductances g_exc and g_inh are solved using the exponential Euler method.

Equivalent code:

HH_cond_exp = Neuron(
parameters = """
gbar_Na = 20.0
gbar_K = 6.0
gleak = 0.01
cm = 0.2
v_offset = -63.0
e_rev_Na = 50.0
e_rev_K = -90.0
e_rev_leak = -65.0
e_rev_E = 0.0
e_rev_I = -80.0
tau_syn_E = 0.2
tau_syn_I = 2.0
i_offset = 0.0
v_thresh = 0.0
""",
equations = """
# Previous membrane potential
prev_v = v

# Voltage-dependent rate constants
an = 0.032 * (15.0 - v + v_offset) / (exp((15.0 - v + v_offset)/5.0) - 1.0)
am = 0.32  * (13.0 - v + v_offset) / (exp((13.0 - v + v_offset)/4.0) - 1.0)
ah = 0.128 * exp((17.0 - v + v_offset)/18.0)

bn = 0.5   * exp ((10.0 - v + v_offset)/40.0)
bm = 0.28  * (v - v_offset - 40.0) / (exp((v - v_offset - 40.0)/5.0) - 1.0)
bh = 4.0/(1.0 + exp (( 10.0 - v + v_offset )) )

# Activation variables
dn/dt = an * (1.0 - n) - bn * n : init = 0.0, exponential
dm/dt = am * (1.0 - m) - bm * m : init = 0.0, exponential
dh/dt = ah * (1.0 - h) - bh * h : init = 1.0, exponential

# Membrane equation
cm * dv/dt = gleak*(e_rev_leak -v) + gbar_K * n**4 * (e_rev_K - v) + gbar_Na * m**3 * h * (e_rev_Na - v)
+ g_exc * (e_rev_E - v) + g_inh * (e_rev_I - v) + i_offset: exponential, init=-65.0

# Exponentially-decaying conductances
tau_syn_E * dg_exc/dt = - g_exc : exponential
tau_syn_I * dg_inh/dt = - g_inh : exponential
""",
spike = "(v > v_thresh) and (prev_v <= v_thresh)",
reset = ""
)


## 3.3.8. EIF_cond_exp_isfa_ista¶

ANNarchy.models.EIF_cond_exp_isfa_ista = <class 'ANNarchy.models.Neurons.EIF_cond_exp_isfa_ista'>[source]

EIF_cond_exp neuron.

Exponential integrate-and-fire neuron with spike triggered and sub-threshold adaptation currents (isfa, ista reps.) according to:

Brette R and Gerstner W (2005) Adaptive Exponential Integrate-and-Fire Model as an Effective Description of Neuronal Activity. J Neurophysiol 94:3637-3642

Parameters:

• v_rest = -70.6 : Resting membrane potential (mV)
• cm = 0.281 : Capacity of the membrane (nF)
• tau_m = 9.3667 : Membrane time constant (ms)
• tau_refrac = 0.1 : Duration of refractory period (ms)
• tau_syn_E = 5.0 : Decay time of excitatory synaptic current (ms)
• tau_syn_I = 5.0 : Decay time of inhibitory synaptic current (ms)
• e_rev_E = 0.0 : Reversal potential for excitatory input (mV)
• e_rev_I = -80.0 : Reversal potential for inhibitory input (mv)
• tau_w = 144.0 : Time constant of the adaptation variable (ms)
• a = 4.0 : Scaling of the adaptation variable
• b = 0.0805 : Increment on the adaptation variable after a spike
• i_offset = 0.0 : Offset current (nA)
• delta_T = 2.0 : Speed of the exponential (mV)
• v_thresh = -50.4 : Spike threshold for the exponential (mV)
• v_reset = -70.6 : Reset potential after a spike (mV)
• v_spike = -40.0 : Spike threshold (mV)

Variables:

• I : input current (nA):

I = g_exc * (e_rev_E - v) + g_inh * (e_rev_I - v) + i_offset

• v : membrane potential in mV (init=-70.6):

tau_m * dv/dt = (v_rest - v +  delta_T * exp((v-v_thresh)/delta_T)) + tau_m/cm*(I - w)

• w : adaptation variable (init=0.0):

tau_w * dw/dt = a * (v - v_rest) / 1000.0 - w

• g_exc : excitatory current (init = 0.0):

tau_syn_E * dg_exc/dt = - g_exc

• g_inh : inhibitory current (init = 0.0):

tau_syn_I * dg_inh/dt = - g_inh


Spike emission:

v > v_thresh


Reset:

v = v_reset
u += b


The ODEs are solved using the explicit Euler method.

Equivalent code:

EIF_cond_exp_isfa_ista = Neuron(
parameters = """
v_rest = -70.6
cm = 0.281
tau_m = 9.3667
tau_syn_E = 5.0
tau_syn_I = 5.0
e_rev_E = 0.0
e_rev_I = -80.0
tau_w = 144.0
a = 4.0
b = 0.0805
i_offset = 0.0
delta_T = 2.0
v_thresh = -50.4
v_reset = -70.6
v_spike = -40.0
""",
equations = """
I = g_exc * (e_rev_E - v) + g_inh * (e_rev_I - v) + i_offset
tau_m * dv/dt = (v_rest - v +  delta_T * exp((v-v_thresh)/delta_T)) + tau_m/cm*(I - w) : init=-70.6
tau_w * dw/dt = a * (v - v_rest) / 1000.0 - w
tau_syn_E * dg_exc/dt = - g_exc : exponential
tau_syn_I * dg_inh/dt = - g_inh : exponential
""",
spike = "v > v_spike",
reset = """
v = v_reset
w += b
""",
refractory = 0.1
)


## 3.3.9. EIF_cond_alpha_isfa_ista¶

ANNarchy.models.EIF_cond_alpha_isfa_ista = <class 'ANNarchy.models.Neurons.EIF_cond_alpha_isfa_ista'>[source]

EIF_cond_alpha neuron.

Exponential integrate-and-fire neuron with spike triggered and sub-threshold adaptation conductances (isfa, ista reps.) according to:

Brette R and Gerstner W (2005) Adaptive Exponential Integrate-and-Fire Model as an Effective Description of Neuronal Activity. J Neurophysiol 94:3637-3642

Parameters:

• v_rest = -70.6 : Resting membrane potential (mV)
• cm = 0.281 : Capacity of the membrane (nF)
• tau_m = 9.3667 : Membrane time constant (ms)
• tau_refrac = 0.1 : Duration of refractory period (ms)
• tau_syn_E = 5.0 : Decay time of excitatory synaptic current (ms)
• tau_syn_I = 5.0 : Decay time of inhibitory synaptic current (ms)
• e_rev_E = 0.0 : Reversal potential for excitatory input (mV)
• e_rev_I = -80.0 : Reversal potential for inhibitory input (mv)
• tau_w = 144.0 : Time constant of the adaptation variable (ms)
• a = 4.0 : Scaling of the adaptation variable
• b = 0.0805 : Increment on the adaptation variable after a spike
• i_offset = 0.0 : Offset current (nA)
• delta_T = 2.0 : Speed of the exponential (mV)
• v_thresh = -50.4 : Spike threshold for the exponential (mV)
• v_reset = -70.6 : Reset potential after a spike (mV)
• v_spike = -40.0 : Spike threshold (mV)

Variables:

• I : input current (nA):

I = g_exc * (e_rev_E - v) + g_inh * (e_rev_I - v) + i_offset

• v : membrane potential in mV (init=-70.6):

tau_m * dv/dt = (v_rest - v +  delta_T * exp((v-v_thresh)/delta_T)) + tau_m/cm*(I - w)

• w : adaptation variable (init=0.0):

tau_w * dw/dt = a * (v - v_rest) / 1000.0 - w

• g_exc : excitatory current (init = 0.0):

tau_syn_E * dg_exc/dt = - g_exc

• g_inh : inhibitory current (init = 0.0):

tau_syn_I * dg_inh/dt = - g_inh

• alpha_exc : alpha function of excitatory current (init = 0.0):

tau_syn_E * dalpha_exc/dt = exp((tau_syn_E - dt/2.0)/tau_syn_E) * g_exc - alpha_exc

• alpha_inh: alpha function of inhibitory current (init = 0.0):

tau_syn_I * dalpha_inh/dt = exp((tau_syn_I - dt/2.0)/tau_syn_I) * g_inh - alpha_inh


Spike emission:

v > v_spike


Reset:

v = v_reset
u += b


The ODEs are solved using the explicit Euler method.

Equivalent code:

EIF_cond_alpha_isfa_ista = Neuron(
parameters = """
v_rest = -70.6
cm = 0.281
tau_m = 9.3667
tau_syn_E = 5.0
tau_syn_I = 5.0
e_rev_E = 0.0
e_rev_I = -80.0
tau_w = 144.0
a = 4.0
b = 0.0805
i_offset = 0.0
delta_T = 2.0
v_thresh = -50.4
v_reset = -70.6
v_spike = -40.0
""",
equations = """
gmax_exc = exp((tau_syn_E - dt/2.0)/tau_syn_E)
gmax_inh = exp((tau_syn_I - dt/2.0)/tau_syn_I)
I = alpha_exc * (e_rev_E - v) + alpha_inh * (e_rev_I - v) + i_offset
tau_m * dv/dt = (v_rest - v +  delta_T * exp((v-v_thresh)/delta_T)) + tau_m/cm*(I - w) : init=-70.6
tau_w * dw/dt = a * (v - v_rest) / 1000.0 - w
tau_syn_E * dg_exc/dt = - g_exc : exponential
tau_syn_I * dg_inh/dt = - g_inh : exponential
tau_syn_E * dalpha_exc/dt = gmax_exc * g_exc - alpha_exc  : exponential
tau_syn_I * dalpha_inh/dt = gmax_inh * g_inh - alpha_inh  : exponential
""",
spike = "v > v_spike",
reset = """
v = v_reset
w += b
""",
refractory = 0.1
)