HH_cond_exp

models.Neurons.HH_cond_exp(
    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,
)

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 * m3 * 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 = ""
)