traub_cond_multisyn

traub_cond_multisyn - Traub model according to Borgers 2017

Description

Reduced Traub-Miles Model of a Pyramidal Neuron in Rat Hippocampus 1. parameters got from reference 2 chapter 5.

AMPA, NMDA, GABA_A, and GABA_B conductance-based synapses with beta-function (difference of two exponentials) time course corresponding to “hill_tononi” model.

References

1
    1. Traub and R. Miles, Neuronal Networks of the Hippocampus,Cam- bridge University Press, Cambridge, UK, 1991.

2

Borgers, C., 2017. An introduction to modeling neuronal dynamics (Vol. 66). Cham: Springer.

See also

hh_cond_exp_traub

Parameters

Name

Physical unit

Default value

Description

t_ref

ms

2.0ms

Refractory period 2.0

g_Na

nS

10000.0nS

Sodium peak conductance

g_K

nS

8000.0nS

Potassium peak conductance

g_L

nS

10nS

Leak conductance

C_m

pF

100.0pF

Membrane Capacitance

E_Na

mV

50.0mV

Sodium reversal potential

E_K

mV

-100.0mV

Potassium reversal potentia

E_L

mV

-67.0mV

Leak reversal Potential (aka resting potential)

V_Tr

mV

-20.0mV

Spike Threshold

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

tau_AMPA_1

ms

0.5ms

rise time

tau_AMPA_2

ms

2.4ms

decay time, Tau_1 < Tau_2

NMDA_g_peak

nS

0.075nS

peak conductance

tau_NMDA_1

ms

4.0ms

rise time

tau_NMDA_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

tau_GABAA_1

ms

1.0ms

rise time

tau_GABAA_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

tau_GABAB_1

ms

60.0ms

rise time

tau_GABAB_2

ms

200.0ms

decay time, Tau_1 < Tau_2

GABA_B_E_rev

mV

-90.0mV

reversal potential for intrinsic current

I_e

pA

0pA

constant external input current

State variables

Name

Physical unit

Default value

Description

r

integer

0

number of steps in the current refractory phase

V_m

mV

-70.0mV

Membrane potential

Act_m

real

alpha_m_init / (alpha_m_init + beta_m_init)

Activation variable m for Na

Inact_h

real

alpha_h_init / (alpha_h_init + beta_h_init)

Inactivation variable h for Na

Act_n

real

alpha_n_init / (alpha_n_init + beta_n_init)

Activation variable n for K

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

\[\frac{ dV_{m} } { dt }= \frac 1 { C_{m} } \left( { (-(I_{Na} + I_{K} + I_{L}) + I_{e} + I_{stim} + I_{syn}) } \right)\]
\[\frac{ dAct_{n} } { dt }= \frac 1 { \mathrm{ms} } \left( { (\alpha_{n} \cdot (1 - Act_{n}) - \beta_{n} \cdot Act_{n}) } \right)\]
\[\frac{ dAct_{m} } { dt }= \frac 1 { \mathrm{ms} } \left( { (\alpha_{m} \cdot (1 - Act_{m}) - \beta_{m} \cdot Act_{m}) } \right)\]
\[\frac{ dInact_{h} } { dt }= \frac 1 { \mathrm{ms} } \left( { (\alpha_{h} \cdot (1 - Inact_{h}) - \beta_{h} \cdot Inact_{h}) } \right)\]

Source code

neuron traub_cond_multisyn:
  state:
    r integer = 0 # number of steps in the current refractory phase
    V_m mV = -70.0mV # Membrane potential
    Act_m real = alpha_m_init / (alpha_m_init + beta_m_init) # Activation variable m for Na
    Inact_h real = alpha_h_init / (alpha_h_init + beta_h_init) # Inactivation variable h for Na
    Act_n real = alpha_n_init / (alpha_n_init + beta_n_init) # Activation variable n for K
    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:
recordable    inline I_syn_ampa pA = -convolve(g_AMPA,AMPA) * (V_m - AMPA_E_rev)
recordable    inline I_syn_nmda pA = -convolve(g_NMDA,NMDA) * (V_m - NMDA_E_rev) / (1 + exp((NMDA_Vact - V_m) / NMDA_Sact))
recordable    inline I_syn_gaba_a pA = -convolve(g_GABAA,GABA_A) * (V_m - GABA_A_E_rev)
recordable    inline I_syn_gaba_b pA = -convolve(g_GABAB,GABA_B) * (V_m - GABA_B_E_rev)
recordable    inline I_syn pA = I_syn_ampa + I_syn_nmda + I_syn_gaba_a + I_syn_gaba_b
    inline I_Na pA = g_Na * Act_m * Act_m * Act_m * Inact_h * (V_m - E_Na)
    inline I_K pA = g_K * Act_n * Act_n * Act_n * Act_n * (V_m - E_K)
    inline I_L pA = g_L * (V_m - E_L)
    V_m'=(-(I_Na + I_K + I_L) + I_e + I_stim + I_syn) / C_m
    # Act_n
    inline alpha_n real = 0.032 * (V_m / mV + 52.0) / (1.0 - exp(-(V_m / mV + 52.0) / 5.0))
    inline beta_n real = 0.5 * exp(-(V_m / mV + 57.0) / 40.0)
    Act_n'=(alpha_n * (1 - Act_n) - beta_n * Act_n) / ms # n-variable
    # Act_m

    # Act_m
    inline alpha_m real = 0.32 * (V_m / mV + 54.0) / (1.0 - exp(-(V_m / mV + 54.0) / 4.0))
    inline beta_m real = 0.28 * (V_m / mV + 27.0) / (exp((V_m / mV + 27.0) / 5.0) - 1.0)
    Act_m'=(alpha_m * (1 - Act_m) - beta_m * Act_m) / ms # m-variable
    # Inact_h'

    # Inact_h'
    inline alpha_h real = 0.128 * exp(-(V_m / mV + 50.0) / 18.0)
    inline beta_h real = 4.0 / (1.0 + exp(-(V_m / mV + 27.0) / 5.0))
    Inact_h'=(alpha_h * (1 - Inact_h) - beta_h * Inact_h) / ms # h-variable
    ############
    # Synapses
    ############

    kernel g_AMPA' = g_AMPA$ - g_AMPA / tau_AMPA_2, g_AMPA$' = -g_AMPA$ / tau_AMPA_1
    kernel g_NMDA' = g_NMDA$ - g_NMDA / tau_NMDA_2, g_NMDA$' = -g_NMDA$ / tau_NMDA_1
    kernel g_GABAA' = g_GABAA$ - g_GABAA / tau_GABAA_2, g_GABAA$' = -g_GABAA$ / tau_GABAA_1
    kernel g_GABAB' = g_GABAB$ - g_GABAB / tau_GABAB_2, g_GABAB$' = -g_GABAB$ / tau_GABAB_1
  end

  parameters:
    t_ref ms = 2.0ms # Refractory period 2.0
    g_Na nS = 10000.0nS # Sodium peak conductance
    g_K nS = 8000.0nS # Potassium peak conductance
    g_L nS = 10nS # Leak conductance
    C_m pF = 100.0pF # Membrane Capacitance
    E_Na mV = 50.0mV # Sodium reversal potential
    E_K mV = -100.0mV # Potassium reversal potentia
    E_L mV = -67.0mV # Leak reversal Potential (aka resting potential)
    V_Tr mV = -20.0mV # Spike Threshold
    # 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
    tau_AMPA_1 ms = 0.5ms # rise time
    tau_AMPA_2 ms = 2.4ms # decay time, Tau_1 < Tau_2
    NMDA_g_peak nS = 0.075nS # peak conductance
    tau_NMDA_1 ms = 4.0ms # rise time
    tau_NMDA_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
    tau_GABAA_1 ms = 1.0ms # rise time
    tau_GABAA_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
    tau_GABAB_1 ms = 60.0ms # rise time
    tau_GABAB_2 ms = 200.0ms # decay time, Tau_1 < Tau_2
    GABA_B_E_rev mV = -90.0mV # reversal potential for intrinsic current
    # constant external input current

    # constant external input current
    I_e pA = 0pA
  end
  internals:
    AMPAInitialValue real = compute_synapse_constant(tau_AMPA_1,tau_AMPA_2,AMPA_g_peak)
    NMDAInitialValue real = compute_synapse_constant(tau_NMDA_1,tau_NMDA_2,NMDA_g_peak)
    GABA_AInitialValue real = compute_synapse_constant(tau_GABAA_1,tau_GABAA_2,GABA_A_g_peak)
    GABA_BInitialValue real = compute_synapse_constant(tau_GABAB_1,tau_GABAB_2,GABA_B_g_peak)
    RefractoryCounts integer = steps(t_ref) # refractory time in steps
    alpha_n_init real = 0.032 * (V_m / mV + 52.0) / (1.0 - exp(-(V_m / mV + 52.0) / 5.0))
    beta_n_init real = 0.5 * exp(-(V_m / mV + 57.0) / 40.0)
    alpha_m_init real = 0.32 * (V_m / mV + 54.0) / (1.0 - exp(-(V_m / mV + 54.0) / 4.0))
    beta_m_init real = 0.28 * (V_m / mV + 27.0) / (exp((V_m / mV + 27.0) / 5.0) - 1.0)
    alpha_h_init real = 0.128 * exp(-(V_m / mV + 50.0) / 18.0)
    beta_h_init real = 4.0 / (1.0 + exp(-(V_m / mV + 27.0) / 5.0))
  end
  input:
    AMPA nS <-spike
    NMDA nS <-spike
    GABA_A nS <-spike
    GABA_B nS <-spike
    I_stim pA <-current
  end

  output: spike

  update:
    U_old mV = V_m
    integrate_odes()
    # sending spikes:
    if r > 0: # is refractory?
      r -= 1
    elif V_m > V_Tr and U_old > V_Tr:
      r = RefractoryCounts
      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