IF_curr_alpha

models.Neurons.IF_curr_alpha(
    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,
)

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
)