a. Soma and dendrite channels:

We used seven types of ionic channels to simulate the active properties of the somatodendritic membrane: fast sodium (Na⁺), calcium (Ca⁺⁺), and five potassium currents: delayed rectifier (DR), small persistent muscarinic (M), A-type transient (A), short-duration [Ca]- and voltage-dependent (C) and long duration [Ca]-dependent (AHP). Conductance variables were described with Hodgkin-Huxley type formalism. The kinetics of all these channels has been obtained from Warman et al. (1994) with the following modifications:

Equilibrium potentials were +45 and -85 mV for Na⁺ and K⁺, respectively.

The activation time constant for the Na⁺ channel was still halved due to the faster rising slope of in vivo APs.

For the Ca⁺⁺ channel a 35ºC was used in the Nernst equation, E_Ca = -13.275Ln([Ca,1]_in/[Ca]_o), where [Ca]_o is 1.2 mM and [Ca,1]_in refers to pool number 1 (see the description for the calcium dynamics).

For the C-type K⁺ channel, alpha_c = -0.0077(V_m+V_shift+103)/{exp[(V_m+V_shift+103)/-12]-1}.

Here we reproduce the description of all channels in the standard Hodgkin-Huxley notation:

Na channel (Na_Wr):

I_Na = g_Nam³h(V_m-E_Na)

E_Na = 45 mV

alpha_m = -3.48(V_m-11.0)/{exp[(V_m-11.0)/-12.94]-1}

beta_m = 0.12(V_m-5.9)/exp[(V_m-5.9)/4.47]-1}

alpha_h = 3/exp[(V_m+80)/10]

beta_h = 12/{exp[(V_m-77)/-27]+1}
 

Ca channel (Ca_W):

I_Ca = g_Cas²r(V_m-E_Ca)

E_Ca = -13.3Ln([Ca,1]_in/1200)

alpha_s = -0.16(V_m+26.0)/{exp[(V_m+26)/-4.5]-1}

beta_s = 0.04(V_m+12)/exp[(V_m+12)/10]-1}

alpha_r = 2/exp[(V_m+94)/10]

beta_r = 8/{exp[(V_m-68)/-27]+1}
 

K_C channel (K_C_Wtn):

I_C = g_Cc²d(V_m-E_K)

E_K = -85 mV

alpha_c = -0.0077(V_m+V_shift+103)/{exp[(V_m+ V_shift +103)/-12]-1}

beta_c = 1.7/exp[(V_m+V_shift+237)/30]

V_shift = 40Log([Ca,1]_in)-105

tau_c = 1.1 ms

alpha_d = 1/exp[(V_m+79)/10]

beta_d = 4/{exp[(V_m-82)/-27]+1}
 

K_AHP channel (K_AHP_Wtn):

I_AHP = g_AHPq(V_m-E_K)

E_K = -85 mV

alpha_q = 0.0048/exp[(10Log([Ca,2]_in-35)/-2]

beta_q = 0.012/exp[(10Log([Ca,2]_in+100)/5]

tau_q = 48 ms
 

K_M channel (K_K_W):

I_M =g_Mu²(V_m-E_K)

E_K = -85 mV

alpha_u = 0.016/exp[(V_m+52.7)/-23]

beta_u = 0.016/exp[(V_m+52.7)/18.8]
 

K_Ap channel, proximal K_A type channel (K_A_3):

I_Ap =g_Apa⁴b(V_m-E_K)

E_K = -85 mV

a_inf = 1/{exp[(-V_m-5)/10]+1}

tau_a = 0.15ms

b_inf = 1/{exp[(V_m+56)/8]+1}

tau_b = 5ms if V_m<-30 mV and 5+0.26(V_m+30) if V_m>30 mV
 

K_Ad channel, distal K_A type channel (K_A_4):

I_Ad =g_Ada⁴b(V_m-E_K)

E_K = -85 mV

a_inf = 1/{exp[(-V_m-15)/8]+1}

tau_a = 0.15ms

b_inf = 1/{exp[(V_m+56)/8]+1}

tau_b = 5ms if V_m<-30 mV and 5+0.26(V_m+30) if V_m>30 mV
 

K_DR channel (K_DR_W):

I_DR =g_DRn⁴(V_m-E_K)

E_K = -85mV

alpha_n = (-0.018V_m)/[exp(V_m/-25)-1]

beta_n = 0.0036(V_m-10)/{exp[(V_m-10)/12]-1}
 
 

b. Axon channels:
 

In the axon, Na⁺ channels were identical as for the somatodendritic membrane, while the DR-type K⁺ channel was obtained from Traub et al. (1994):

 

K_DRA channel (K_DRA_T):

I_DRA =g_DRAn_ax⁴(V_m-E_K)

E_K = -85mV

alpha_n(ax)= -0.03(V_m+47.8)/{exp[(V_m+47.8)/-5]-1}

beta_n(ax) = 0.45exp[(V_m+53.0)/-40]
 
 

c. Ca dynamics:
 

As described by Warman et al. (1994) and Borg-Graham (1998), the [Ca]_i was simulated with two different Ca pools with different time constants, tau₁ = 0.9 ms for the calculation of E_Ca and modulating the C-type K⁺ current and tau₂ = 1 s for the AHP-type K⁺ current, as follows:

where tau_i is the i-th calcium pool removal time constant, f_i is the fraction of Ca influx affecting the i-th pool (f₁=0.7, f₂=0.024), w is the thickness of the diffusion shell (1 micron), A is the compartment area, z is the valence of the calcium ion and F is Faraday's constant.

 
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