hill_tononi

hill_tononi - Neuron model after Hill & Tononi (2005)

Description

This model neuron implements a slightly modified version of the neuron model described in 1. The most important properties are:

• Integrate-and-fire with threshold adaptive threshold.

• Repolarizing potassium current instead of hard reset.

• AMPA, NMDA, GABA_A, and GABA_B conductance-based synapses with beta-function (difference of exponentials) time course.

• Voltage-dependent NMDA with instantaneous or two-stage unblocking 1, 2.

• Intrinsic currents I_h, I_T, I_Na(p), and I_KNa.

• Synaptic “minis” are not implemented.

Documentation and examples can be found on the NEST Simulator repository (https://github.com/nest/nest-simulator/) at the following paths: - docs/model_details/HillTononiModels.ipynb - pynest/examples/intrinsic_currents_spiking.py - pynest/examples/intrinsic_currents_subthreshold.py

References

1(1,2)

Hill S, Tononi G (2005). Modeling sleep and wakefulness in the thalamocortical system. Journal of Neurophysiology. 93:1671-1698. DOI: https://doi.org/10.1152/jn.00915.2004

2

Vargas-Caballero M, Robinson HPC (2003). A slow fraction of Mg2+ unblock of NMDA receptors limits their contribution to spike generation in cortical pyramidal neurons. Journal of Neurophysiology 89:2778-2783. DOI: https://doi.org/10.1152/jn.01038.2002

Parameters

Name	Physical unit	Default value	Description
E_Na	mV	30.0mV
E_K	mV	-90.0mV
g_NaL	nS	0.2nS
g_KL	nS	1.0nS	1.0 - 1.85
Tau_m	ms	16.0ms	membrane time constant applying to all currents but repolarizing K-current (see [1, p 1677])
Theta_eq	mV	-51.0mV	equilibrium value
Tau_theta	ms	2.0ms	time constant
Tau_spike	ms	1.75ms	membrane time constant applying to repolarizing K-current
t_spike	ms	2.0ms	duration of re-polarizing potassium current
AMPA_g_peak	nS	0.1nS	Parameters for synapse of type AMPA, GABA_A, GABA_B and NMDApeak conductance
AMPA_E_rev	mV	0.0mV	reversal potential
AMPA_Tau_1	ms	0.5ms	rise time
AMPA_Tau_2	ms	2.4ms	decay time, Tau_1 < Tau_2
NMDA_g_peak	nS	0.075nS	peak conductance
NMDA_Tau_1	ms	4.0ms	rise time
NMDA_Tau_2	ms	40.0ms	decay time, Tau_1 < Tau_2
NMDA_E_rev	mV	0.0mV	reversal potential
NMDA_Vact	mV	-58.0mV	inactive for V << Vact, inflection of sigmoid
NMDA_Sact	mV	2.5mV	scale of inactivation
GABA_A_g_peak	nS	0.33nS	peak conductance
GABA_A_Tau_1	ms	1.0ms	rise time
GABA_A_Tau_2	ms	7.0ms	decay time, Tau_1 < Tau_2
GABA_A_E_rev	mV	-70.0mV	reversal potential
GABA_B_g_peak	nS	0.0132nS	peak conductance
GABA_B_Tau_1	ms	60.0ms	rise time
GABA_B_Tau_2	ms	200.0ms	decay time, Tau_1 < Tau_2
GABA_B_E_rev	mV	-90.0mV	reversal potential for intrinsic current
NaP_g_peak	nS	1.0nS	parameters for intrinsic currentspeak conductance for intrinsic current
NaP_E_rev	mV	30.0mV	reversal potential for intrinsic current
KNa_g_peak	nS	1.0nS	peak conductance for intrinsic current
KNa_E_rev	mV	-90.0mV	reversal potential for intrinsic current
T_g_peak	nS	1.0nS	peak conductance for intrinsic current
T_E_rev	mV	0.0mV	reversal potential for intrinsic current
h_g_peak	nS	1.0nS	peak conductance for intrinsic current
h_E_rev	mV	-40.0mV	reversal potential for intrinsic current
KNa_D_EQ	pA	0.001pA
I_e	pA	0pA	constant external input current

State variables

Name	Physical unit	Default value	Description
r_potassium	integer	0
g_spike	boolean	false
V_m	mV	(g_NaL * E_Na + g_KL * E_K) / (g_NaL + g_KL)	membrane potential
Theta	mV	Theta_eq	Threshold
IKNa_D	nS	0.0nS
IT_m	nS	0.0nS
IT_h	nS	0.0nS
Ih_m	nS	0.0nS
g_AMPA	real	0
g_NMDA	real	0
g_GABAA	real	0
g_GABAB	real	0
g_AMPA$	real	AMPAInitialValue
g_NMDA$	real	NMDAInitialValue
g_GABAA$	real	GABA_AInitialValue
g_GABAB$	real	GABA_BInitialValue

Equations

\[\frac{ dV_{m} } { dt }= \frac 1 { \mathrm{nF} } \left( { (\frac 1 { \Tau_{m} } \left( { (I_{Na} + I_{K} + I_{syn} + I_{NaP} + I_{KNa} + I_{T} + I_{h} + I_{e} + I_{stim}) } \right) + \frac{ I_{spike} \cdot \mathrm{pA} } { (\mathrm{ms} \cdot \mathrm{mV}) }) \cdot \mathrm{s} } \right)\]

\[\frac{ d\Theta } { dt }= \frac{ -(\Theta - \Theta_{eq}) } { \Tau_{\theta} }\]

\[\frac{ dIKNa_{D} } { dt }= \frac 1 { \mathrm{ms} } \left( { (D_{influx,peak} \cdot D_{influx} \cdot \mathrm{nS} - \frac 1 { \tau_{D} } \left( { (IKNa_{D} - \frac{ KNa_{D,EQ} } { \mathrm{mV} }) } \right) ) } \right)\]

\[\frac{ dIT_{m} } { dt }= \frac 1 { \mathrm{ms} } \left( { \frac 1 { \tau_{m,T} } \left( { (m_{\infty,T} \cdot \mathrm{nS} - IT_{m}) } \right) } \right)\]

\[\frac{ dIT_{h} } { dt }= \frac 1 { \mathrm{ms} } \left( { \frac 1 { \tau_{h,T} } \left( { (h_{\infty,T} \cdot \mathrm{nS} - IT_{h}) } \right) } \right)\]

\[\frac{ dIh_{m} } { dt }= \frac 1 { \mathrm{ms} } \left( { \frac 1 { \tau_{m,h} } \left( { (m_{\infty,h} \cdot \mathrm{nS} - Ih_{m}) } \right) } \right)\]

Source code

neuron hill_tononi:
  state:
    r_potassium integer = 0
    g_spike boolean = false
    V_m mV = (g_NaL * E_Na + g_KL * E_K) / (g_NaL + g_KL) # membrane potential
    Theta mV = Theta_eq # Threshold
    IKNa_D,IT_m,IT_h,Ih_m nS = 0.0nS
    g_AMPA real = 0
    g_NMDA real = 0
    g_GABAA real = 0
    g_GABAB real = 0
    g_AMPA$ real = AMPAInitialValue
    g_NMDA$ real = NMDAInitialValue
    g_GABAA$ real = GABA_AInitialValue
    g_GABAB$ real = GABA_BInitialValue
  end
  equations:
    inline I_syn_ampa pA = -convolve(g_AMPA,AMPA) * (V_m - AMPA_E_rev)
    inline I_syn_nmda pA = -convolve(g_NMDA,NMDA) * (V_m - NMDA_E_rev) / (1 + exp((NMDA_Vact - V_m) / NMDA_Sact))
    inline I_syn_gaba_a pA = -convolve(g_GABAA,GABA_A) * (V_m - GABA_A_E_rev)
    inline I_syn_gaba_b pA = -convolve(g_GABAB,GABA_B) * (V_m - GABA_B_E_rev)
    inline I_syn pA = I_syn_ampa + I_syn_nmda + I_syn_gaba_a + I_syn_gaba_b
    inline I_Na pA = -g_NaL * (V_m - E_Na)
    inline I_K pA = -g_KL * (V_m - E_K)
    # I_Na(p), m_inf^3 according to Compte et al, J Neurophysiol 2003 89:2707
    inline INaP_thresh mV = -55.7mV
    inline INaP_slope mV = 7.7mV
    inline m_inf_NaP real = 1.0 / (1.0 + exp(-(V_m - INaP_thresh) / INaP_slope))
    # Persistent Na current; member only to allow recording

    # Persistent Na current; member only to allow recording
recordable    inline I_NaP pA = -NaP_g_peak * m_inf_NaP ** 3 * (V_m - NaP_E_rev)
    inline d_half real = 0.25
    inline m_inf_KNa real = 1.0 / (1.0 + (d_half / (IKNa_D / nS)) ** 3.5)
    # Depol act. K current; member only to allow recording

    # Depol act. K current; member only to allow recording
recordable    inline I_KNa pA = -KNa_g_peak * m_inf_KNa * (V_m - KNa_E_rev)
    # Low-thresh Ca current; member only to allow recording
recordable    inline I_T pA = -T_g_peak * IT_m * IT_m * IT_h * (V_m - T_E_rev)
recordable    inline I_h pA = -h_g_peak * Ih_m * (V_m - h_E_rev)
    # The spike current is only activate immediately after a spike.

    # The spike current is only activate immediately after a spike.
    inline I_spike mV = (g_spike)?-(V_m - E_K) / Tau_spike:0
    V_m'=((I_Na + I_K + I_syn + I_NaP + I_KNa + I_T + I_h + I_e + I_stim) / Tau_m + I_spike * pA / (ms * mV)) * s / nF
    ############
    # Intrinsic currents
    ############
    # I_T
    inline m_inf_T real = 1.0 / (1.0 + exp(-(V_m / mV + 59.0) / 6.2))
    inline h_inf_T real = 1.0 / (1.0 + exp((V_m / mV + 83.0) / 4))
    inline tau_m_h real = 1.0 / (exp(-14.59 - 0.086 * V_m / mV) + exp(-1.87 + 0.0701 * V_m / mV))
    # I_KNa

    # I_KNa
    inline D_influx_peak real = 0.025
    inline tau_D real = 1250.0 # yes, 1.25 s
    inline D_thresh mV = -10.0
    inline D_slope mV = 5.0
    inline D_influx real = 1.0 / (1.0 + exp(-(V_m - D_thresh) / D_slope))
    Theta'=-(Theta - Theta_eq) / Tau_theta
    # equation modified from y[](1-D_eq) to (y[]-D_eq), since we'd not
    # be converging to equilibrium otherwise
    IKNa_D'=(D_influx_peak * D_influx * nS - (IKNa_D - KNa_D_EQ / mV) / tau_D) / ms
    inline tau_m_T real = 0.22 / (exp(-(V_m / mV + 132.0) / 16.7) + exp((V_m / mV + 16.8) / 18.2)) + 0.13
    inline tau_h_T real = 8.2 + (56.6 + 0.27 * exp((V_m / mV + 115.2) / 5.0)) / (1.0 + exp((V_m / mV + 86.0) / 3.2))
    inline I_h_Vthreshold real = -75.0
    inline m_inf_h real = 1.0 / (1.0 + exp((V_m / mV - I_h_Vthreshold) / 5.5))
    IT_m'=(m_inf_T * nS - IT_m) / tau_m_T / ms
    IT_h'=(h_inf_T * nS - IT_h) / tau_h_T / ms
    Ih_m'=(m_inf_h * nS - Ih_m) / tau_m_h / ms
    ############
    # Synapses
    ############
    kernel g_AMPA' = g_AMPA$ - g_AMPA / AMPA_Tau_2, g_AMPA$' = -g_AMPA$ / AMPA_Tau_1
    kernel g_NMDA' = g_NMDA$ - g_NMDA / NMDA_Tau_2, g_NMDA$' = -g_NMDA$ / NMDA_Tau_1
    kernel g_GABAA' = g_GABAA$ - g_GABAA / GABA_A_Tau_2, g_GABAA$' = -g_GABAA$ / GABA_A_Tau_1
    kernel g_GABAB' = g_GABAB$ - g_GABAB / GABA_B_Tau_2, g_GABAB$' = -g_GABAB$ / GABA_B_Tau_1
  end

  parameters:
    E_Na mV = 30.0mV
    E_K mV = -90.0mV
    g_NaL nS = 0.2nS
    g_KL nS = 1.0nS # 1.0 - 1.85
    Tau_m ms = 16.0ms # membrane time constant applying to all currents but repolarizing K-current (see [1, p 1677])
    Theta_eq mV = -51.0mV # equilibrium value
    Tau_theta ms = 2.0ms # time constant
    Tau_spike ms = 1.75ms # membrane time constant applying to repolarizing K-current
    t_spike ms = 2.0ms # duration of re-polarizing potassium current
    # Parameters for synapse of type AMPA, GABA_A, GABA_B and NMDA

    # Parameters for synapse of type AMPA, GABA_A, GABA_B and NMDA
    AMPA_g_peak nS = 0.1nS # peak conductance
    AMPA_E_rev mV = 0.0mV # reversal potential
    AMPA_Tau_1 ms = 0.5ms # rise time
    AMPA_Tau_2 ms = 2.4ms # decay time, Tau_1 < Tau_2
    NMDA_g_peak nS = 0.075nS # peak conductance
    NMDA_Tau_1 ms = 4.0ms # rise time
    NMDA_Tau_2 ms = 40.0ms # decay time, Tau_1 < Tau_2
    NMDA_E_rev mV = 0.0mV # reversal potential
    NMDA_Vact mV = -58.0mV # inactive for V << Vact, inflection of sigmoid
    NMDA_Sact mV = 2.5mV # scale of inactivation
    GABA_A_g_peak nS = 0.33nS # peak conductance
    GABA_A_Tau_1 ms = 1.0ms # rise time
    GABA_A_Tau_2 ms = 7.0ms # decay time, Tau_1 < Tau_2
    GABA_A_E_rev mV = -70.0mV # reversal potential
    GABA_B_g_peak nS = 0.0132nS # peak conductance
    GABA_B_Tau_1 ms = 60.0ms # rise time
    GABA_B_Tau_2 ms = 200.0ms # decay time, Tau_1 < Tau_2
    GABA_B_E_rev mV = -90.0mV # reversal potential for intrinsic current
    # parameters for intrinsic currents

    # parameters for intrinsic currents
    NaP_g_peak nS = 1.0nS # peak conductance for intrinsic current
    NaP_E_rev mV = 30.0mV # reversal potential for intrinsic current
    KNa_g_peak nS = 1.0nS # peak conductance for intrinsic current
    KNa_E_rev mV = -90.0mV # reversal potential for intrinsic current
    T_g_peak nS = 1.0nS # peak conductance for intrinsic current
    T_E_rev mV = 0.0mV # reversal potential for intrinsic current
    h_g_peak nS = 1.0nS # peak conductance for intrinsic current
    h_E_rev mV = -40.0mV # reversal potential for intrinsic current
    KNa_D_EQ pA = 0.001pA
    # constant external input current
    I_e pA = 0pA
  end
  internals:
    AMPAInitialValue real = compute_synapse_constant(AMPA_Tau_1,AMPA_Tau_2,AMPA_g_peak)
    NMDAInitialValue real = compute_synapse_constant(NMDA_Tau_1,NMDA_Tau_2,NMDA_g_peak)
    GABA_AInitialValue real = compute_synapse_constant(GABA_A_Tau_1,GABA_A_Tau_2,GABA_A_g_peak)
    GABA_BInitialValue real = compute_synapse_constant(GABA_B_Tau_1,GABA_B_Tau_2,GABA_B_g_peak)
    PotassiumRefractoryCounts integer = steps(t_spike)
  end
  input:
    AMPA nS <-spike
    NMDA nS <-spike
    GABA_A nS <-spike
    GABA_B nS <-spike
    I_stim pA <-current
  end

  output: spike

  update:
    integrate_odes()
    # Deactivate potassium current after spike time have expired
    if (r_potassium > 0) and (r_potassium - 1 == 0):
      g_spike = false # Deactivate potassium current.
    end
    r_potassium -= 1
    if (not g_spike) and V_m >= Theta:
      # Set V and Theta to the sodium reversal potential.
      V_m = E_Na
      Theta = E_Na
      # Activate fast potassium current. Drives the
      # membrane potential towards the potassium reversal
      # potential (activate only if duration is non-zero).
      if PotassiumRefractoryCounts > 0:
        g_spike = true
      else:
        g_spike = false
      end
      r_potassium = PotassiumRefractoryCounts
      emit_spike()
    end
  end

function compute_synapse_constant(Tau_1 msTau_2 msg_peak real) real:
    # Factor used to account for the missing 1/((1/Tau_2)-(1/Tau_1)) term
    # in the ht_neuron_dynamics integration of the synapse terms.
    # See: Exact digital simulation of time-invariant linear systems
    # with applications to neuronal modeling, Rotter and Diesmann,
    # section 3.1.2.
    exact_integration_adjustment real = ((1 / Tau_2) - (1 / Tau_1)) * ms
    t_peak real = (Tau_2 * Tau_1) * ln(Tau_2 / Tau_1) / (Tau_2 - Tau_1) / ms
    normalisation_factor real = 1 / (exp(-t_peak / Tau_1) - exp(-t_peak / Tau_2))
    return g_peak * normalisation_factor * exact_integration_adjustment
end

end

Characterisation