izhikevich.nestml

"""
izhikevich - Izhikevich neuron model
####################################

Description
+++++++++++

Implementation of the simple spiking neuron model introduced by Izhikevich [1]_. The dynamics are given by:

.. math::

   dv/dt &= 0.04 v^2 + 5 v + 140 - u + I\\
   du/dt &= a (b v - u)


.. math::

   &\text{if}\;\; v \geq V_{th}:\\
   &\;\;\;\; v \text{ is set to } c\\
   &\;\;\;\; u \text{ is incremented by } d\\
   & \, \\
   &v \text{ jumps on each spike arrival by the weight of the spike}

As published in [1]_, the numerics differs from the standard forward Euler technique in two ways:

1) the new value of :math:`u` is calculated based on the new value of :math:`v`, rather than the previous value
2) the variable :math:`v` is updated using a time step half the size of that used to update variable :math:`u`.

This model will instead be simulated using the numerical solver that is recommended by ODE-toolbox during code generation.


Authors
+++++++

Hanuschkin, Morrison, Kunkel


References
++++++++++

.. [1] Izhikevich, Simple Model of Spiking Neurons, IEEE Transactions on Neural Networks (2003) 14:1569-1572
"""
neuron izhikevich:

  initial_values:
    V_m mV = V_m_init         # Membrane potential
    U_m real = b * V_m_init   # Membrane potential recovery variable
  end

  equations:
    V_m' = ( 0.04 * V_m * V_m / mV + 5.0 * V_m + ( 140 - U_m ) * mV + ( (I_e + I_stim) * GOhm ) ) / ms
    U_m' = a*(b*V_m-U_m * mV) / (mV*ms)
  end

  parameters:
    a real = 0.02        # describes time scale of recovery variable
    b real = 0.2         # sensitivity of recovery variable
    c mV = -65 mV        # after-spike reset value of V_m
    d real = 8.0         # after-spike reset value of U_m
    V_m_init mV = -65 mV # initial membrane potential
    V_min mV = -inf * mV # Absolute lower value for the membrane potential.

    # constant external input current
    I_e pA = 0 pA
  end

  input:
    spikes mV <- spike
    I_stim pA <- current
  end

  output: spike

  update:
    integrate_odes()
    # Add synaptic current
    V_m += spikes

    # lower bound of membrane potential
    V_m = (V_m < V_min)? V_min : V_m

    # threshold crossing
    if V_m >= 30 mV:
      V_m = c
      U_m += d
      emit_spike()
    end

  end

end