traub_cond_multisyn¶
Name: traub_cond_multisyn - Traub model according to Borgers 2017.
Reduced Traub-Miles Model of a Pyramidal Neuron in Rat Hippocampus[1]. parameters got from reference [2] chapter 5.
- 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.
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] R. D. 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.
SeeAlso: 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 |
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 |
I_e |
pA |
0pA |
constant external input current |
State variables¶
Name |
Physical unit |
Default value |
Description |
|---|---|---|---|
V_m |
mV |
-70.0mV |
Membrane potential |
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)) |
|
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 |
nS |
0.0nS |
|
g_AMPA__d |
nS / ms |
0.0nS / ms |
|
g_NMDA |
nS |
0.0nS |
|
g_NMDA__d |
nS / ms |
0.0nS / ms |
|
g_GABAA |
nS |
0.0nS |
|
g_GABAA__d |
nS / ms |
0.0nS / ms |
|
g_GABAB |
nS |
0.0nS |
|
g_GABAB__d |
nS / ms |
0.0nS / ms |
Equations¶
Source code¶
neuron traub_cond_multisyn:
state:
r integer = 0 # number of steps in the current refractory phase
V_m mV = -70. mV # 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.) / (1. - exp(-(V_m / mV + 52.) / 5.))
inline beta_n real = 0.5 * exp(-(V_m / mV + 57.) / 40.)
Act_n' = ( alpha_n * ( 1 - Act_n ) - beta_n * Act_n ) / ms # n-variable
# Act_m
inline alpha_m real = 0.32 * (V_m / mV + 54.) / (1.0 - exp(-(V_m / mV + 54.) / 4.))
inline beta_m real = 0.28 * (V_m / mV + 27.) / (exp((V_m / mV + 27.) / 5.) - 1.)
Act_m' = ( alpha_m * ( 1 - Act_m ) - beta_m * Act_m ) / ms # m-variable
# 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.) / 5.))
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 = 2.0 ms # Refractory period 2.0
g_Na nS = 10000.0 nS # Sodium peak conductance
g_K nS = 8000.0 nS # Potassium peak conductance
g_L nS = 10 nS # Leak conductance
C_m pF = 100.0 pF # Membrane Capacitance
E_Na mV = 50. mV # Sodium reversal potential
E_K mV = -100. mV # Potassium reversal potentia
E_L mV = -67. mV # Leak reversal Potential (aka resting potential)
V_Tr mV = -20. mV # Spike Threshold
# Parameters for synapse of type AMPA, GABA_A, GABA_B and NMDA
AMPA_g_peak nS = 0.1 nS # peak conductance
AMPA_E_rev mV = 0.0 mV # reversal potential
tau_AMPA_1 ms = 0.5 ms # rise time
tau_AMPA_2 ms = 2.4 ms # decay time, Tau_1 < Tau_2
NMDA_g_peak nS = 0.075 nS # peak conductance
tau_NMDA_1 ms = 4.0 ms # rise time
tau_NMDA_2 ms = 40.0 ms # decay time, Tau_1 < Tau_2
NMDA_E_rev mV = 0.0 mV # reversal potential
NMDA_Vact mV = -58.0 mV # inactive for V << Vact, inflection of sigmoid
NMDA_Sact mV = 2.5 mV # scale of inactivation
GABA_A_g_peak nS = 0.33 nS # peak conductance
tau_GABAA_1 ms = 1.0 ms # rise time
tau_GABAA_2 ms = 7.0 ms # decay time, Tau_1 < Tau_2
GABA_A_E_rev mV = -70.0 mV # reversal potential
GABA_B_g_peak nS = 0.0132 nS # peak conductance
tau_GABAB_1 ms = 60.0 ms # rise time
tau_GABAB_2 ms = 200.0 ms # decay time, Tau_1 < Tau_2
GABA_B_E_rev mV = -90.0 mV # reversal potential for intrinsic current
# constant external input current
I_e pA = 0 pA
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.) / (1. - exp(-(V_m / mV + 52.) / 5.))
beta_n_init real = 0.5 * exp(-(V_m / mV + 57.) / 40.)
alpha_m_init real = 0.32 * (V_m / mV + 54.) / (1.0 - exp(-(V_m / mV + 54.) / 4.))
beta_m_init real = 0.28 * (V_m / mV + 27.) / (exp((V_m / mV + 27.) / 5.) - 1.)
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.) / 5.))
end
input:
AMPA nS <- spike
NMDA nS <- spike
GABA_A nS <- spike
GABA_B nS <- spike
I_stim pA <- continuous
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: # threshold && maximum
r = RefractoryCounts
emit_spike()
end
end
function compute_synapse_constant(Tau_1 ms, Tau_2 ms, g_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