terub_stn

terub_stn - Terman Rubin neuron model

Description

terub_stn is an implementation of a spiking neuron using the Terman Rubin model based on the Hodgkin-Huxley formalism.

Post-syaptic currents: Incoming spike events induce a post-synaptic change of current modelled by an alpha function. The alpha function is normalised such that an event of weight 1.0 results in a peak current of 1 pA.
Spike Detection: Spike detection is done by a combined threshold-and-local-maximum search: if there is a local maximum above a certain threshold of the membrane potential, it is considered a spike.

References

1: Terman, D. and Rubin, J.E. and Yew, A.C. and Wilson, C.J. Activity Patterns in a Model for the Subthalamopallidal Network of the Basal Ganglia. The Journal of Neuroscience, 22(7), 2963-2976 (2002)
2: Rubin, J.E. and Terman, D. High Frequency Stimulation of the Subthalamic Nucleus Eliminates Pathological Thalamic Rhythmicity in a Computational Model Journal of Computational Neuroscience, 16, 211-235 (2004)

Parameters

Name	Physical unit	Default value	Description
E_L	mV	-60mV	Resting membrane potential
g_L	nS	2.25nS	Leak conductance
C_m	pF	1pF	Capacity of the membrane
E_Na	mV	55mV	Sodium reversal potential
g_Na	nS	37.5nS	Sodium peak conductance
E_K	mV	-80mV	Potassium reversal potential
g_K	nS	45nS	Potassium peak conductance
E_Ca	mV	140mV	Calcium reversal potential
g_Ca	nS	0.5nS	Calcium peak conductance
g_T	nS	0.5nS	T-type Calcium channel peak conductance
g_ahp	nS	9nS	Afterpolarization current peak conductance
tau_syn_exc	ms	1ms	Rise time of the excitatory synaptic alpha function
tau_syn_inh	ms	0.08ms	Rise time of the inhibitory synaptic alpha function
E_gs	mV	-85mV	Reversal potential for inhibitory input (from GPe)
t_ref	ms	2ms	Refractory time
I_e	pA	0pA	constant external input current

State variables

Name	Physical unit	Default value	Description
r	integer	0	counts number of tick during the refractory period
V_m	mV	E_L	Membrane potential
gate_h	real	0.0	gating variable h
gate_n	real	0.0	gating variable n
gate_r	real	0.0	gating variable r
Ca_con	real	0.0	gating variable r

Equations

\[\frac{ dV_{m} } { dt }= \frac 1 { C_{m} } \left( { (-(I_{Na} + I_{K} + I_{L} + I_{T} + I_{Ca} + I_{ahp}) + I_{e} + I_{stim} + I_{exc,mod} + I_{inh,mod}) } \right)\]

\[\frac{ dgate_{h} } { dt }= \phi_{h} \cdot (\frac{ (h_{\infty} - gate_{h}) } { \tau_{h} })\]

\[\frac{ dgate_{n} } { dt }= \phi_{n} \cdot (\frac{ (n_{\infty} - gate_{n}) } { \tau_{n} })\]

\[\frac{ dgate_{r} } { dt }= \phi_{r} \cdot (\frac{ (r_{\infty} - gate_{r}) } { \tau_{r} })\]

\[\frac{ dCa_{con} } { dt }= \epsilon \cdot (\frac{ (-I_{Ca} - I_{T}) } { \mathrm{pA} } - k_{Ca} \cdot Ca_{con})\]

Source codeneuron terub_stn:
  state:
    r integer = 0 # counts number of tick during the refractory period
    V_m mV = E_L # Membrane potential
    gate_h real = 0.0 # gating variable h
    gate_n real = 0.0 # gating variable n
    gate_r real = 0.0 # gating variable r
    Ca_con real = 0.0 # gating variable r
  end
  equations:
    #time constants for slow gating variables
    inline tau_n_0 ms = 1.0ms
    inline tau_n_1 ms = 100.0ms
    inline theta_n_tau mV = -80.0mV
    inline sigma_n_tau mV = -26.0mV
    inline tau_h_0 ms = 1.0ms
    inline tau_h_1 ms = 500.0ms
    inline theta_h_tau mV = -57.0mV
    inline sigma_h_tau mV = -3.0mV
    inline tau_r_0 ms = 7.1ms # Guo 7.1 Terman02 40.0
    inline tau_r_1 ms = 17.5ms
    inline theta_r_tau mV = 68.0mV
    inline sigma_r_tau mV = -2.2mV
    #steady state values for gating variables
    inline theta_a mV = -63.0mV
    inline sigma_a mV = 7.8mV
    inline theta_h mV = -39.0mV
    inline sigma_h mV = -3.1mV
    inline theta_m mV = -30.0mV
    inline sigma_m mV = 15.0mV
    inline theta_n mV = -32.0mV
    inline sigma_n mV = 8.0mV
    inline theta_r mV = -67.0mV
    inline sigma_r mV = -2.0mV
    inline theta_s mV = -39.0mV
    inline sigma_s mV = 8.0mV
    inline theta_b real = 0.25 # Guo 0.25 Terman02 0.4
    inline sigma_b real = 0.07 # Guo 0.07 Terman02 -0.1
    #time evolvement of gating variables

    #time evolvement of gating variables
    inline phi_h real = 0.75
    inline phi_n real = 0.75
    inline phi_r real = 0.5 # Guo 0.5 Terman02 0.2
    # Calcium concentration and afterhyperpolarization current

    # Calcium concentration and afterhyperpolarization current
    inline epsilon 1/ms = 5e-05 / ms # 1/ms Guo 0.00005 Terman02 0.0000375
    inline k_Ca real = 22.5
    inline k1 real = 15.0
    inline I_exc_mod pA = -convolve(g_exc,exc_spikes) * V_m
    inline I_inh_mod pA = convolve(g_inh,inh_spikes) * (V_m - E_gs)
    inline tau_n ms = tau_n_0 + tau_n_1 / (1.0 + exp(-(V_m - theta_n_tau) / sigma_n_tau))
    inline tau_h ms = tau_h_0 + tau_h_1 / (1.0 + exp(-(V_m - theta_h_tau) / sigma_h_tau))
    inline tau_r ms = tau_r_0 + tau_r_1 / (1.0 + exp(-(V_m - theta_r_tau) / sigma_r_tau))
    inline a_inf real = 1.0 / (1.0 + exp(-(V_m - theta_a) / sigma_a))
    inline h_inf real = 1.0 / (1.0 + exp(-(V_m - theta_h) / sigma_h))
    inline m_inf real = 1.0 / (1.0 + exp(-(V_m - theta_m) / sigma_m))
    inline n_inf real = 1.0 / (1.0 + exp(-(V_m - theta_n) / sigma_n))
    inline r_inf real = 1.0 / (1.0 + exp(-(V_m - theta_r) / sigma_r))
    inline s_inf real = 1.0 / (1.0 + exp(-(V_m - theta_s) / sigma_s))
    inline b_inf real = 1.0 / (1.0 + exp((gate_r - theta_b) / sigma_b)) - 1.0 / (1.0 + exp(-theta_b / sigma_b))
    inline I_Na pA = g_Na * m_inf * m_inf * m_inf * gate_h * (V_m - E_Na)
    inline I_K pA = g_K * gate_n * gate_n * gate_n * gate_n * (V_m - E_K)
    inline I_L pA = g_L * (V_m - E_L)
    inline I_T pA = g_T * a_inf * a_inf * a_inf * b_inf * b_inf * (V_m - E_Ca)
    inline I_Ca pA = g_Ca * s_inf * s_inf * (V_m - E_Ca)
    inline I_ahp pA = g_ahp * (Ca_con / (Ca_con + k1)) * (V_m - E_K)
    # V dot -- synaptic input are currents, inhib current is negative
    V_m'=(-(I_Na + I_K + I_L + I_T + I_Ca + I_ahp) + I_e + I_stim + I_exc_mod + I_inh_mod) / C_m
    #channel dynamics
    gate_h'=phi_h * ((h_inf - gate_h) / tau_h) # h-variable
    gate_n'=phi_n * ((n_inf - gate_n) / tau_n) # n-variable
    gate_r'=phi_r * ((r_inf - gate_r) / tau_r) # r-variable
    #Calcium concentration

    #Calcium concentration
    Ca_con'=epsilon * ((-I_Ca - I_T) / pA - k_Ca * Ca_con)
    # synapses: alpha functions
    # alpha function for the g_inh
    kernel g_inh = (e / tau_syn_inh) * t * exp(-t / tau_syn_inh)
    # alpha function for the g_exc

    # alpha function for the g_exc
    kernel g_exc = (e / tau_syn_exc) * t * exp(-t / tau_syn_exc)
  end

  parameters:
    E_L mV = -60mV # Resting membrane potential
    g_L nS = 2.25nS # Leak conductance
    C_m pF = 1pF # Capacity of the membrane
    E_Na mV = 55mV # Sodium reversal potential
    g_Na nS = 37.5nS # Sodium peak conductance
    E_K mV = -80mV # Potassium reversal potential
    g_K nS = 45nS # Potassium peak conductance
    E_Ca mV = 140mV # Calcium reversal potential
    g_Ca nS = 0.5nS # Calcium peak conductance
    g_T nS = 0.5nS # T-type Calcium channel peak conductance
    g_ahp nS = 9nS # Afterpolarization current peak conductance
    tau_syn_exc ms = 1ms # Rise time of the excitatory synaptic alpha function
    tau_syn_inh ms = 0.08ms # Rise time of the inhibitory synaptic alpha function
    E_gs mV = -85mV # Reversal potential for inhibitory input (from GPe)
    t_ref ms = 2ms # Refractory time
    # constant external input current

    # constant external input current
    I_e pA = 0pA
  end
  internals:
    refractory_counts integer = steps(t_ref)
  end
  input:
    inh_spikes pA <-inhibitory spike
    exc_spikes pA <-excitatory spike
    I_stim pA <-current
  end

  output: spike

  update:
    U_old mV = V_m
    integrate_odes()
    # sending spikes: crossing 0 mV, pseudo-refractoriness and local maximum...
    if r > 0:
      r -= 1
    elif V_m > 0mV and U_old > V_m:
      r = refractory_counts
      emit_spike()
    end
  end

end