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"""
terub_gpe - Terman Rubin neuron model
#####################################
Description
+++++++++++
terub_gpe is an implementation of a spiking neuron using the Terman Rubin model
based on the Hodgkin-Huxley formalism.
(1) **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.
(2) **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
"""
neuron terub_gpe:
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 GPe Neuron
inline g_tau_n_0 ms = 0.05 ms
inline g_tau_n_1 ms = 0.27 ms
inline g_theta_n_tau mV = -40.0 mV
inline g_sigma_n_tau mV = -12.0 mV
inline g_tau_h_0 ms = 0.05 ms
inline g_tau_h_1 ms = 0.27 ms
inline g_theta_h_tau mV = -40.0 mV
inline g_sigma_h_tau mV = -12.0 mV
inline g_tau_r ms = 30.0 ms
# steady state values for gating variables
inline g_theta_a mV = -57.0 mV
inline g_sigma_a mV = 2.0 mV
inline g_theta_h mV = -58.0 mV
inline g_sigma_h mV = -12.0 mV
inline g_theta_m mV = -37.0 mV
inline g_sigma_m mV = 10.0 mV
inline g_theta_n mV = -50.0 mV
inline g_sigma_n mV = 14.0 mV
inline g_theta_r mV = -70.0 mV
inline g_sigma_r mV = -2.0 mV
inline g_theta_s mV = -35.0 mV
inline g_sigma_s mV = 2.0 mV
# time evolvement of gating variables
inline g_phi_h real = 0.05
inline g_phi_n real = 0.1 #Report: 0.1, Terman Rubin 2002: 0.05
inline g_phi_r real = 1.0
# Calcium concentration and afterhyperpolarization current
inline g_epsilon 1/ms = 0.0001 /ms
inline g_k_Ca real = 15.0 #Report:15, Terman Rubin 2002: 20.0
inline g_k1 real = 30.0
inline I_ex_mod real = -convolve(g_ex, spikeExc) * V_m
inline I_in_mod real = convolve(g_in, spikeInh) * (V_m-E_gg)
inline tau_n real = g_tau_n_0 + g_tau_n_1 / (1. + exp(-(V_m-g_theta_n_tau)/g_sigma_n_tau))
inline tau_h real = g_tau_h_0 + g_tau_h_1 / (1. + exp(-(V_m-g_theta_h_tau)/g_sigma_h_tau))
inline tau_r real = g_tau_r
inline a_inf real = 1. / (1. + exp(-(V_m-g_theta_a)/g_sigma_a))
inline h_inf real = 1. / (1. + exp(-(V_m-g_theta_h)/g_sigma_h))
inline m_inf real = 1. / (1. + exp(-(V_m-g_theta_m)/g_sigma_m))
inline n_inf real = 1. / (1. + exp(-(V_m-g_theta_n)/g_sigma_n))
inline r_inf real = 1. / (1. + exp(-(V_m-g_theta_r)/g_sigma_r))
inline s_inf real = 1. / (1. + exp(-(V_m-g_theta_s)/g_sigma_s))
inline I_Na real = g_Na * m_inf * m_inf * m_inf * gate_h * (V_m - E_Na)
inline I_K real = g_K * gate_n * gate_n * gate_n * gate_n * (V_m - E_K )
inline I_L real = g_L * (V_m - E_L )
inline I_T real = g_T * a_inf* a_inf * a_inf * gate_r * (V_m - E_Ca)
inline I_Ca real = g_Ca * s_inf * s_inf * (V_m - E_Ca)
inline I_ahp real = g_ahp * (Ca_con / (Ca_con + g_k1)) * (V_m - E_K )
# 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)
# V dot -- synaptic input are currents, inhib current is negative
V_m' = ( -(I_Na + I_K + I_L + I_T + I_Ca + I_ahp) * pA + I_e + I_stim + I_ex_mod * pA + I_in_mod * pA) / C_m
# channel dynamics
gate_h' = g_phi_h *((h_inf-gate_h) / tau_h) / ms # h-variable
gate_n' = g_phi_n *((n_inf-gate_n) / tau_n) / ms # n-variable
gate_r' = g_phi_r *((r_inf-gate_r) / tau_r) / ms # r-variable
# Calcium concentration
Ca_con' = g_epsilon*(-I_Ca - I_T - g_k_Ca * Ca_con)
end
parameters:
E_L mV = -55 mV # Resting membrane potential.
g_L nS = 0.1 nS # Leak conductance.
C_m pF = 1.0 pF # Capacity of the membrane.
E_Na mV = 55 mV # Sodium reversal potential.
g_Na nS = 120 nS # Sodium peak conductance.
E_K mV = -80.0 mV# Potassium reversal potential.
g_K nS = 30.0 nS # Potassium peak conductance.
E_Ca mV = 120 mV # Calcium reversal potential.
g_Ca nS = 0.15 nS # Calcium peak conductance.
g_T nS = 0.5 nS # T-type Calcium channel peak conductance.
g_ahp nS = 30 nS # afterpolarization current peak conductance.
tau_syn_ex ms = 1.0 ms # Rise time of the excitatory synaptic alpha function.
tau_syn_in ms = 12.5 ms # Rise time of the inhibitory synaptic alpha function.
E_gg mV = -100 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 nS <- inhibitory spike
spikeExc nS <- 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