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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).

ANNarchy.models.Neurons.LeakyIntegrator #

Bases: Neuron

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 \(\tau\) 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 threshold \(T\):

\[r(t) = (v(t) - T)^+\]

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)
    '''
)
Source code in ANNarchy/models/Neurons.py 
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class LeakyIntegrator(Neuron):
    r"""
    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 $\tau$ 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 threshold $T$:

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

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

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

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

    ```python
    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:

    ```python
    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)
        '''
    )
    ```
    """

    # For reporting
    _instantiated = []

    def __init__(self, tau=10.0, B=0.0, T=0.0, sum='sum(exc) - sum(inh)', noise=None):
        # Create the arguments
        parameters = """
            tau = %(tau)s : population
            B = %(B)s
            T = %(T)s : population
        """ % {'tau': tau, 'B': B, 'T': T}

        # Equations for the variables
        if not noise:
            noise_def = ''
        else:
            noise_def = '+ ' + noise

        equations="""
            tau * dv/dt + v = %(sum)s + B %(noise)s : exponential
            r = pos(v - T)
        """ % { 'sum' : sum, 'noise': noise_def}

        Neuron.__init__(self, 
            parameters=parameters, equations=equations,
            name="Leaky-Integrator", 
            description="Leaky-Integrator with positive transfer function and additive noise.")

        # For reporting
        self._instantiated.append(True)

ANNarchy.models.Neurons.Izhikevich #

Bases: Neuron

Izhikevich neuron as proposed in:

Izhikevich, E.M. (2003). Simple Model of Spiking Neurons, IEEE Transaction on Neural Networks, 14:6. http://dx.doi.org/10.1109/TNN.2003.820440

The equations are:

\[\frac{dv}{dt} = 0.04 * v^2 + 5.0 * v + 140.0 - u + I\]
\[\frac{du}{dt} = a * (b * v - u)\]

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 = 0.02 : Speed of the recovery variable
  • b = 0.2: Scaling of the recovery variable
  • c = -65.0 : Reset potential.
  • d = 8.0 : 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 = 0.02
            b = 0.2
            c = -65.0
            d = 8.0
            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
    )

The default parameters are for a regular spiking (RS) neuron derived from the above mentioned article.

Source code in ANNarchy/models/Neurons.py 
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class Izhikevich(Neuron):
    '''
    Izhikevich neuron as proposed in:

    > Izhikevich, E.M. (2003). *Simple Model of Spiking Neurons, IEEE Transaction on Neural Networks*, 14:6. <http://dx.doi.org/10.1109/TNN.2003.820440>

    The equations are:

    $$\\frac{dv}{dt} = 0.04 * v^2 + 5.0 * v + 140.0 - u + I$$

    $$\\frac{du}{dt} = a * (b * v - u)$$

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

    ```python
    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 = 0.02 : Speed of the recovery variable
    * b = 0.2: Scaling of the recovery variable
    * c = -65.0 : Reset potential.
    * d = 8.0 : 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:

    ```python

        Izhikevich = Neuron(
            parameters = """
                noise = 0.0
                a = 0.02
                b = 0.2
                c = -65.0
                d = 8.0
                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
        )
    ```

    The default parameters are for a regular spiking (RS) neuron derived from the above mentioned article.
    '''

    # For reporting
    _instantiated = []

    def __init__(self, 
        a=0.02, 
        b=0.2, 
        c=-65.0, 
        d=8.0, 
        v_thresh=30.0, 
        i_offset=0.0, 
        noise=0.0, 
        tau_refrac=0.0, 
        conductance="g_exc - g_inh"):

        # Extract which targets are defined in the conductance
        #import re
        #targets = re.findall(r'g_([\w]+)', conductance)

        # Create the arguments
        parameters = """
            noise = %(noise)s
            a = %(a)s
            b = %(b)s
            c = %(c)s
            d = %(d)s
            v_thresh = %(v_thresh)s
            i_offset = %(i_offset)s
            tau_refrac = %(tau_refrac)s
        """ % {'a': a, 'b':b, 'c':c, 'd':d, 'v_thresh':v_thresh, 'i_offset':i_offset, 'noise':noise, 'tau_refrac':tau_refrac}

        # Equations for the variables
        equations="""
            I = %(conductance)s + noise * Normal(0.0, 1.0) + i_offset
            dv/dt = 0.04 * v^2 + 5.0 * v + 140.0 - u + I : init = %(c)s
            du/dt = a * (b*v - u) : init= %(u)s
        """ % { 'conductance' : conductance, 'c':c , 'u': b*c}

        spike = """
            v > v_thresh
        """
        reset = """
            v = c
            u += d
        """
        Neuron.__init__(self, 
            parameters=parameters, equations=equations, spike=spike, reset=reset, refractory='tau_refrac',
            name="Izhikevich", description="Quadratic integrate-and-fire spiking neuron with adaptation.")

        # For reporting
        self._instantiated.append(True)

ANNarchy.models.Neurons.IF_curr_exp #

Bases: Neuron

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
)
Source code in ANNarchy/models/Neurons.py 
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class IF_curr_exp(Neuron):
    '''
    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:

    ```python

    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
    )
    ```

    '''
    # For reporting
    _instantiated = []

    def __init__(self, v_rest=-65.0, cm=1.0, tau_m=20.0, tau_refrac=0.0, 
                tau_syn_E=5.0, tau_syn_I=5.0, v_thresh=-50.0, v_reset=-65.0, i_offset=0.0):

        # Create the arguments
        parameters = """
            v_rest = %(v_rest)s
            cm  = %(cm)s
            tau_m  = %(tau_m)s
            tau_refrac = %(tau_refrac)s
            tau_syn_E = %(tau_syn_E)s
            tau_syn_I = %(tau_syn_I)s
            v_thresh = %(v_thresh)s
            v_reset = %(v_reset)s
            i_offset = %(i_offset)s
        """ % {'v_rest':v_rest, 'cm':cm, 'tau_m':tau_m, 'tau_refrac':tau_refrac, 
                'tau_syn_E':tau_syn_E, 'tau_syn_I':tau_syn_I, 
                'v_thresh':v_thresh, 'v_reset':v_reset, 'i_offset':i_offset}

        # Equations for the variables
        equations="""    
            cm * dv/dt = cm/tau_m*(v_rest -v)   + g_exc - g_inh + i_offset : exponential, init=%(v_reset)s
            tau_syn_E * dg_exc/dt = - g_exc : exponential
            tau_syn_I * dg_inh/dt = - g_inh : exponential
        """ % {'v_reset':v_reset}

        spike = """
            v > v_thresh
        """
        reset = """
            v = v_reset
        """
        Neuron.__init__(self, parameters=parameters, equations=equations, 
            spike=spike, reset=reset, refractory='tau_refrac',
            name="Integrate-and-Fire", 
            description="Leaky integrate-and-fire model with fixed threshold and decaying-exponential post-synaptic current.")

        # For reporting
        self._instantiated.append(True)

ANNarchy.models.Neurons.IF_cond_exp #

Bases: Neuron

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
)
Source code in ANNarchy/models/Neurons.py 
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class IF_cond_exp(Neuron):
    '''
    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:

    ```python

    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
    )
    ```

    '''
    # For reporting
    _instantiated = []

    def __init__(self, v_rest=-65.0, cm=1.0, tau_m=20.0, tau_refrac=0.0, 
        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):

        # Create the arguments
        parameters = """
            v_rest = %(v_rest)s
            cm  = %(cm)s
            tau_m  = %(tau_m)s
            tau_refrac = %(tau_refrac)s
            tau_syn_E = %(tau_syn_E)s
            tau_syn_I = %(tau_syn_I)s
            v_thresh = %(v_thresh)s
            v_reset = %(v_reset)s
            i_offset = %(i_offset)s
            e_rev_E = %(e_rev_E)s 
            e_rev_I = %(e_rev_I)s
        """ % {'v_rest':v_rest, 'cm':cm, 'tau_m':tau_m, 'tau_refrac':tau_refrac, 
                'tau_syn_E':tau_syn_E, 'tau_syn_I':tau_syn_I, 'v_thresh':v_thresh, 
                'v_reset':v_reset, 'i_offset':i_offset, 'e_rev_E': e_rev_E, 'e_rev_I': e_rev_I}

        # Equations for the variables
        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=%(v_reset)s
            tau_syn_E * dg_exc/dt = - g_exc : exponential
            tau_syn_I * dg_inh/dt = - g_inh : exponential
        """% {'v_reset':v_reset}

        spike = """
            v > v_thresh
        """
        reset = """
            v = v_reset
        """
        Neuron.__init__(self, parameters=parameters, equations=equations, 
            spike=spike, reset=reset, refractory='tau_refrac',
            name="Integrate-and-Fire", 
            description="Leaky integrate-and-fire model with fixed threshold and decaying-exponential post-synaptic conductances.")

        # For reporting
        self._instantiated.append(True)

ANNarchy.models.Neurons.IF_curr_alpha #

Bases: Neuron

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
)
Source code in ANNarchy/models/Neurons.py 
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class IF_curr_alpha(Neuron):
    '''
    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:

    ```python

    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
    )
    ```

    '''
    # For reporting
    _instantiated = []

    def __init__(self, v_rest=-65.0, cm=1.0, tau_m=20.0, tau_refrac=0.0, 
        tau_syn_E=5.0, tau_syn_I=5.0, v_thresh=-50.0, v_reset=-65.0, i_offset=0.0):

        # Create the arguments
        parameters = """
            v_rest = %(v_rest)s
            cm  = %(cm)s
            tau_m  = %(tau_m)s
            tau_refrac = %(tau_refrac)s
            tau_syn_E = %(tau_syn_E)s
            tau_syn_I = %(tau_syn_I)s
            v_thresh = %(v_thresh)s
            v_reset = %(v_reset)s
            i_offset = %(i_offset)s
        """ % {'v_rest':v_rest, 'cm':cm, 'tau_m':tau_m, 'tau_refrac':tau_refrac, 
                'tau_syn_E':tau_syn_E, 'tau_syn_I':tau_syn_I, 'v_thresh':v_thresh, 'v_reset':v_reset, 'i_offset':i_offset}

        # Equations for the variables
        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=%(v_reset)s
            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
        """  % {'v_reset':v_reset}

        spike = """
            v > v_thresh
        """
        reset = """
            v = v_reset
        """
        Neuron.__init__(self, parameters=parameters, equations=equations, 
            spike=spike, reset=reset, refractory='tau_refrac',
            name="Integrate-and-Fire", 
            description="Leaky integrate-and-fire model with fixed threshold and alpha post-synaptic currents.")

        # For reporting
        self._instantiated.append(True)

ANNarchy.models.Neurons.IF_cond_alpha #

Bases: Neuron

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
)
Source code in ANNarchy/models/Neurons.py 
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class IF_cond_alpha(Neuron):
    '''
    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:

    ```python

    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
    )
    ```
    '''
    # For reporting
    _instantiated = []

    def __init__(self, v_rest=-65.0, cm=1.0, tau_m=20.0, tau_refrac=0.0, 
        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):

        # Create the arguments
        parameters = """
            v_rest = %(v_rest)s
            cm  = %(cm)s
            tau_m  = %(tau_m)s
            tau_refrac = %(tau_refrac)s
            tau_syn_E = %(tau_syn_E)s
            tau_syn_I = %(tau_syn_I)s
            v_thresh = %(v_thresh)s
            v_reset = %(v_reset)s
            i_offset = %(i_offset)s
            e_rev_E = %(e_rev_E)s 
            e_rev_I = %(e_rev_I)s
        """ % {'v_rest':v_rest, 'cm':cm, 'tau_m':tau_m, 'tau_refrac':tau_refrac, 
                'tau_syn_E':tau_syn_E, 'tau_syn_I':tau_syn_I, 'v_thresh':v_thresh, 
                'v_reset':v_reset, 'i_offset':i_offset, 'e_rev_E': e_rev_E, 'e_rev_I': e_rev_I}

        # Equations for the variables
        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=%(v_reset)s
            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
        """ % {'v_reset': v_reset}

        spike = """
            v > v_thresh
        """
        reset = """
            v = v_reset
        """
        Neuron.__init__(self, parameters=parameters, equations=equations, 
            spike=spike, reset=reset, refractory='tau_refrac',
            name="Integrate-and-Fire", 
            description="Leaky integrate-and-fire model with fixed threshold and alpha post-synaptic conductances.")

        # For reporting
        self._instantiated.append(True)

ANNarchy.models.Neurons.HH_cond_exp #

Bases: Neuron

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 * n4 * (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 (the spike is emitted only once when v crosses the threshold from below):

v > v_thresh and v(t-1) < v_thresh

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 = ""
)
Source code in ANNarchy/models/Neurons.py 
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class HH_cond_exp(Neuron):
    '''
    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 (the spike is emitted only once when v crosses the threshold from below):

        v > v_thresh and v(t-1) < v_thresh

    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:

    ```python

    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 = ""
    )
    ```

    '''
    # For reporting
    _instantiated = []

    def __init__(self, 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):

        parameters = """
        gbar_Na    = %(gbar_Na)s   
        gbar_K     = %(gbar_K)s     
        gleak      = %(gleak)s      
        cm         = %(cm)s        
        v_offset   = %(v_offset)s 
        e_rev_Na   = %(e_rev_Na)s 
        e_rev_K    = %(e_rev_K)s   
        e_rev_leak = %(e_rev_leak)s   
        e_rev_E    = %(e_rev_E)s    
        e_rev_I    = %(e_rev_I)s   
        tau_syn_E  = %(tau_syn_E)s 
        tau_syn_I  = %(tau_syn_I)s 
        i_offset   = %(i_offset)s  
        v_thresh   = %(v_thresh)s  
        """ % {
        'gbar_Na'    : gbar_Na   ,
        'gbar_K'     : gbar_K     ,
        'gleak'      : gleak      ,
        'cm'         : cm        ,
        'v_offset'   : v_offset  ,
        'e_rev_Na'   : e_rev_Na  ,
        'e_rev_K'    : e_rev_K   ,
        'e_rev_leak' : e_rev_leak   ,
        'e_rev_E'    : e_rev_E    ,
        'e_rev_I'    : e_rev_I   ,
        'tau_syn_E'  : tau_syn_E ,
        'tau_syn_I'  : tau_syn_I ,
        'i_offset'   : i_offset  ,
        'v_thresh'   : v_thresh  
        }

        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=%(e_rev_leak)s

            # Exponentially-decaying conductances
            tau_syn_E * dg_exc/dt = - g_exc : exponential
            tau_syn_I * dg_inh/dt = - g_inh : exponential
        """ % {'e_rev_leak': e_rev_leak}

        spike = "(v > v_thresh) and (prev_v <= v_thresh)"

        reset = ""

        Neuron.__init__(self, parameters=parameters, equations=equations, 
            spike=spike, reset=reset,
            name="Hodgkin-Huxley", 
            description="Single-compartment Hodgkin-Huxley-type neuron with transient sodium and delayed-rectifier potassium currents.")

        # For reporting
        self._instantiated.append(True)

ANNarchy.models.Neurons.EIF_cond_exp_isfa_ista #

Bases: Neuron

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
)
Source code in ANNarchy/models/Neurons.py 
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class EIF_cond_exp_isfa_ista(Neuron):
    '''
    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:

    ```python

    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
    )
    ```

    '''

    # For reporting
    _instantiated = []

    def __init__(self, v_rest = -70.6, cm = 0.281, tau_m = 9.3667, 
        tau_refrac = 0.1, 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):

        # Create the arguments
        parameters = """
            v_rest     = %(v_rest)s
            cm         = %(cm)s
            tau_m      = %(tau_m)s
            tau_refrac = %(tau_refrac)s
            tau_syn_E  = %(tau_syn_E)s
            tau_syn_I  = %(tau_syn_I)s
            e_rev_E    = %(e_rev_E)s
            e_rev_I    = %(e_rev_I)s
            tau_w      = %(tau_w)s
            a          = %(a)s
            b          = %(b)s
            i_offset   = %(i_offset)s
            delta_T    = %(delta_T)s
            v_thresh   = %(v_thresh)s
            v_reset    = %(v_reset)s
            v_spike    = %(v_spike)s
        """ % {
            'v_rest'     : v_rest    ,
            'cm'         : cm        ,
            'tau_m'      : tau_m     ,
            'tau_refrac' : tau_refrac,
            'tau_syn_E'  : tau_syn_E ,
            'tau_syn_I'  : tau_syn_I ,
            'e_rev_E'    : e_rev_E   ,
            'e_rev_I'    : e_rev_I   ,
            'tau_w'      : tau_w     ,
            'a'          : a         ,
            'b'          : b         ,
            'i_offset'   : i_offset  ,
            'delta_T'    : delta_T   ,
            'v_thresh'   : v_thresh  ,
            'v_reset'    : v_reset   ,
            'v_spike'    : v_spike   ,
        }
        # Equations for the variables
        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=%(v_reset)s

            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
        """ % {'v_reset': v_reset}

        spike = """
            v > v_spike
        """
        reset = """
            v = v_reset
            w += b
        """
        Neuron.__init__(self, parameters=parameters, equations=equations, 
            spike=spike, reset=reset, refractory='tau_refrac',
            name="Adaptive exponential Integrate-and-Fire", 
            description="Exponential integrate-and-fire neuron with spike triggered and sub-threshold adaptation currents (isfa, ista reps.).")

        # For reporting
        self._instantiated.append(True)

ANNarchy.models.Neurons.EIF_cond_alpha_isfa_ista #

Bases: Neuron

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
)
Source code in ANNarchy/models/Neurons.py 
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class EIF_cond_alpha_isfa_ista(Neuron):
    ''' 
    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:

    ```python

    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
    )
    ```

    '''
    # For reporting
    _instantiated = []

    def __init__(self, v_rest = -70.6, cm = 0.281, tau_m = 9.3667, 
        tau_refrac = 0.1, 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):

        # Create the arguments
        parameters = """
            v_rest     = %(v_rest)s
            cm         = %(cm)s
            tau_m      = %(tau_m)s
            tau_refrac = %(tau_refrac)s
            tau_syn_E  = %(tau_syn_E)s
            tau_syn_I  = %(tau_syn_I)s
            e_rev_E    = %(e_rev_E)s
            e_rev_I    = %(e_rev_I)s
            tau_w      = %(tau_w)s
            a          = %(a)s
            b          = %(b)s
            i_offset   = %(i_offset)s
            delta_T    = %(delta_T)s
            v_thresh   = %(v_thresh)s
            v_reset    = %(v_reset)s
            v_spike    = %(v_spike)s
        """ % {
            'v_rest'     : v_rest    ,
            'cm'         : cm        ,
            'tau_m'      : tau_m     ,
            'tau_refrac' : tau_refrac,
            'tau_syn_E'  : tau_syn_E ,
            'tau_syn_I'  : tau_syn_I ,
            'e_rev_E'    : e_rev_E   ,
            'e_rev_I'    : e_rev_I   ,
            'tau_w'      : tau_w     ,
            'a'          : a         ,
            'b'          : b         ,
            'i_offset'   : i_offset  ,
            'delta_T'    : delta_T   ,
            'v_thresh'   : v_thresh  ,
            'v_reset'    : v_reset   ,
            'v_spike'    : v_spike   ,
        }
        # Equations for the variables
        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=%(v_reset)s

            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
        """ % {'v_reset': v_reset}

        spike = """
            v > v_spike
        """
        reset = """
            v = v_reset
            w += b
        """
        Neuron.__init__(self, parameters=parameters, equations=equations, 
            spike=spike, reset=reset, refractory='tau_refrac',
            name="Adaptive exponential Integrate-and-Fire", 
            description="Exponential integrate-and-fire neuron with spike triggered and sub-threshold adaptation conductances (isfa, ista reps.).")

        # For reporting
        self._instantiated.append(True)