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)

Author¶

Martin Ebert

Parameters¶

Name	Physical unit	Default value	Description
E_L	mV	-60mV	Resting membrane potential.
g_L	nS	2.25nS	Leak conductance.
C_m	pF	1.0pF	Capacity of the membrane.
E_Na	mV	55mV	Sodium reversal potential.
g_Na	nS	37.5nS	Sodium peak conductance.
E_K	mV	-80.0mV	Potassium reversal potential.
g_K	nS	45.0nS	Potassium peak conductance.
E_Ca	mV	120mV	Calcium reversal potential.
g_Ca	nS	140nS	Calcium peak conductance.
g_T	nS	0.5nS	T-type Calcium channel peak conductance.
g_ahp	nS	9nS	afterpolarization current peak conductance.
tau_syn_ex	ms	1.0ms	Rise time of the excitatory synaptic alpha function.
tau_syn_in	ms	0.08ms	Rise time of the inhibitory synaptic alpha function.
E_gs	mV	-85.0mV	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
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¶

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

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

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

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

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

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:
    #Parameters for Terman Rubin STN Neuron

    #time constants for slow gating variables
    inline tau_n_0 ms = 1.0 ms
    inline tau_n_1 ms = 100.0 ms
    inline theta_n_tau mV = -80.0 mV
    inline sigma_n_tau mV = -26.0 mV

    inline tau_h_0 ms = 1.0 ms
    inline tau_h_1 ms = 500.0 ms
    inline theta_h_tau mV = -57.0 mV
    inline sigma_h_tau mV = -3.0 mV

    inline tau_r_0 ms = 7.1 ms # Guo 7.1 Terman02 40.0
    inline tau_r_1 ms = 17.5 ms
    inline theta_r_tau mV = 68.0 mV
    inline sigma_r_tau mV = -2.2 mV

    #steady state values for gating variables
    inline theta_a mV = -63.0 mV
    inline sigma_a mV = 7.8 mV
    inline theta_h mV = -39.0 mV
    inline sigma_h mV = -3.1 mV
    inline theta_m mV = -30.0 mV
    inline sigma_m mV = 15.0 mV
    inline theta_n mV = -32.0 mV
    inline sigma_n mV = 8.0 mV
    inline theta_r mV = -67.0 mV
    inline sigma_r mV = -2.0 mV
    inline theta_s mV = -39.0 mV
    inline sigma_s mV = 8.0 mV

    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
    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
    inline epsilon 1/ms = 0.00005 / ms # 1/ms Guo 0.00005 Terman02 0.0000375
    inline k_Ca real = 22.5
    inline k1 real = 15.0

    inline I_ex_mod pA = -convolve(g_ex, spikeExc) * V_m
    inline I_in_mod pA = convolve(g_in, spikeInh) * (V_m - E_gs)

    inline tau_n ms = tau_n_0 + tau_n_1 / (1. + exp(-(V_m-theta_n_tau)/sigma_n_tau))
    inline tau_h ms = tau_h_0 + tau_h_1 / (1. + exp(-(V_m-theta_h_tau)/sigma_h_tau))
    inline tau_r ms = tau_r_0 + tau_r_1 / (1. + exp(-(V_m-theta_r_tau)/sigma_r_tau))

    inline a_inf real = 1. / (1. +exp(-(V_m-theta_a)/sigma_a))
    inline h_inf real = 1. / (1. + exp(-(V_m-theta_h)/sigma_h));
    inline m_inf real = 1. / (1. + exp(-(V_m-theta_m)/sigma_m))
    inline n_inf real = 1. / (1. + exp(-(V_m-theta_n)/sigma_n))
    inline r_inf real = 1. / (1. + exp(-(V_m-theta_r)/sigma_r))
    inline s_inf real = 1. / (1. + exp(-(V_m-theta_s)/sigma_s))
    inline b_inf real = 1. / (1. + exp((gate_r-theta_b)/sigma_b)) - 1. / (1. + 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_ex_mod + I_in_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
    Ca_con' = epsilon*( (-I_Ca  - I_T ) / pA - k_Ca * Ca_con)

    # synapses: alpha functions
    ## alpha function for the g_in
    kernel g_in = (e/tau_syn_in) * t * exp(-t/tau_syn_in)
    ## alpha function for the g_ex
    kernel g_ex = (e/tau_syn_ex) * t * exp(-t/tau_syn_ex)
  end

  parameters:
    E_L        mV = -60 mV  # Resting membrane potential.
    g_L        nS = 2.25 nS # Leak conductance.
    C_m        pF = 1.0 pF # Capacity of the membrane.
    E_Na       mV = 55 mV   # Sodium reversal potential.
    g_Na       nS = 37.5 nS # Sodium peak conductance.
    E_K        mV = -80.0 mV# Potassium reversal potential.
    g_K        nS = 45.0 nS # Potassium peak conductance.
    E_Ca       mV = 140 mV  # Calcium reversal potential.
    g_Ca       nS = 0.5 nS  # Calcium peak conductance.
    g_T        nS = 0.5 nS  # T-type Calcium channel peak conductance.
    g_ahp      nS = 9 nS    # afterpolarization current peak conductance.
    tau_syn_ex ms = 1.0 ms  # Rise time of the excitatory synaptic alpha function.
    tau_syn_in ms = 0.08 ms # Rise time of the inhibitory synaptic alpha function.
    E_gs       mV = -85.0 mV# reversal potential for inhibitory input (from GPe)
    t_ref      ms = 2 ms    # refractory time

    # constant external input current
    I_e pA = 0 pA
  end

  internals:
    refractory_counts integer = steps(t_ref)
  end

  input:
    spikeInh pA <- inhibitory spike
    spikeExc pA <- excitatory spike
    I_stim pA <- continuous
  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 > 0 mV and U_old > V_m:
      r = refractory_counts
      emit_spike()
    end

  end

end