<p> Supplementary material: single neuron models</p><p>A.V. Chizhov1, E.Yu. Smirnova1, K.Kh. Kim2, A.V. Zaitsev2</p><p>1A.F.Ioffe Physical-Technical Institute of the Russian Academy of Sciences, Saint-Petersburg 2Sechenov Institute of Evolutionary Physiology and Biochemistry of the Russian Academy of Sciences, Saint-Petersburg</p><p>The individual models, taken from literature or constructed, are listed below. All voltage values are given in millivolts; time values are in milliseconds. The input is specified by two values of u and s, the voltage-independent input current and conductance, respectively, thus the external ext voltage-dependent current is I  u  s(V Vus ) with Vus  65 mV . </p><p>Model 1. Leaky integrate-and-fire (LIF) model with fixed threshold.</p><p> dV C  G (V V )  I ext  2 G (t) dt L L V L and the threshold criterion T if V V then V VL where VL is the rest and reset potential, C is the membrane capacity, GL is the membrane conductance, ξ(t) is the white Gaussian noise with zero mean and unit dispersion; and σV is the amplitude of noise in terms of the steady-state, sub-threshold voltage dispersion. The parameters T are as follows: GL = 10 nS, τm = C/GL = 10 ms, and σV = 2 mV, V -VL=10 mV. </p><p>Suppl. Fig.1. f-I-curves for Model 1.</p><p>Model 2. One-compartment Hodgkin-Huxley-like model with slow inactivating sodium channels (Migliore et el. 1999).</p><p> dV 3 ext C  GL (V VL )  g Na m h i (V VNa )  gK n (V VK )  I , dt dm 2 a   m  m ,   , m  , m dt  m a  b  a  b</p><p>1  0.4(V  30) 0.124(V  30) a  , b  , if  h  0.02 then  h  0.02 , (exp((V  30) /7.2) 1) (exp((V  30) / 7.2) 1) dh 2 1   h  h ,   , h  , h dt  h a  b 1 exp(V  50) / 4)  0.03(V  45) 0.01(V  45) a  , b  , if  h  0.5 then  h  0.5 , (exp((V  45) /1.5) 1) (exp((V  45) /1.5) 1) di 3000b 1 0.5exp((V  58) / 2)   i  i ,   , i  , i dt  i 1 a 1 exp(V  58) / 2) a  exp(0.45(V  60)) , b  exp(0.09(V  60)) , if  i  10 then  i  10 , dn 50b 1   n  n ,   , n  , n dt  n 1 a  1 a a  exp(0.11(V 13)) , b  exp(0.08(V 13)) , if  n  2 then  n  2 .</p><p>C  27 pF , GL  1.2nS , gNa  874nS , gK  273nS , </p><p>VL  65.4mV , VNa  55mV , VK  72mV .</p><p>Suppl. Fig.2. Model 2: spike shape and phase-space plot in response to current step stimulation with u = 9.1 pA, starting at t=0.</p><p>Model 3. One-compartment reduction of the model (Borg-Graham 1999) with 4-state Markov approximation of sodium channels, where the effects of calcium channels were approximated by the AHP-channel. Described by Chizhov and Graham (2007): dV C = I  I  I  I  I  I  I  G (V V )  I ext , (1) dt Na DR A M H L AHP L L Different types of ionic currents were inserted, including the leak current, IL , the fast sodium current, INa , the voltage-dependent potassium currents responsible for spike repolarization, IDR and I A , the voltage-dependent potassium current that contributes to spike frequency adaptation, IM , the voltage-dependent cation current, IH , and calcium-dependent potassium current that also contributes to spike frequency adaptation, I AHP . The approximating formulas for the currents INa , IDR , I A , IM and IH were adapted from [Graham (1999)]; the approximation for I AHP was taken from [Kopel (2000)]. 2 The sodium current INa</p><p>I Na (t) = gNa x1(t)(V (t) VNa ), (2)</p><p> x1  x2  x3  x4 = 1, (3) 4 4 dxi =  Aj,i x j  xi  Ai, j , i = 1,2,3 (4) dt j=0, ji j=0, j i</p><p>1 1,3 1,4 A1,2 = 3 ms , A1,3 = f1 (V ), A1,4 = f1 (V ), 2,3 A2,1 = 0, A2,3 = f2 (V ), A2,4 = 0, 3,1 3,4 A3,1 = f1 (V ), A3,2 = 0, A3,4 = f2 (V ), 4,1 A4,1 = f1 (V ), A4,2 = 0, A4,3 = 0</p><p>1  V V i, j  f i, j (V ) =  i, j 1/exp 1/2  , 1  min  i, j    k  1 1  i, j  i, j  i, j  i, j i, j 1 V V1/2   f2 (V ) =  min  ( max  min )  exp i, j   ,  k       Note that there is an implicit coefficient of the exponential term of 1/ms in the equations i, j i, j for f1 (V ) and f2 (V ) . 1,3 1,3 1,3 min = 1/3 ms, V1/2 = 51 mV , k = 2 mV , 1,4 1,4 1,4 min = 1/3 ms, V1/2 = 57 mV , k = 2 mV , 2,3 2,3 2,3 2,3 min = 1 ms, V1/2 = 53 mV , k = 1 mV , max = 100 ms, 3,1 3,1 3,1 min = 1/3 ms, V1/2 = 42 mV , k = 1 mV , 3,4 3,4 3,4 3,4 min = 1 ms, V1/2 = 60 mV , k = 1 mV , max = 100 ms, 4,1 4,1 4,1 min = 1/3 ms, V1/2 = 51 mV , k = 1 mV ,</p><p>The voltage-dependent potassium current IDR</p><p>IDR (t) = gDR x(t) y(t) (V (t) VDR ), (5)</p><p> dx dy = x (V )  x (V ), = y (V )  y (V ) dt  x dt  y  x = 1/(a  b)  0.8 ms; x = a/(a  b), a = 0.17exp((V  5) 0.090) ms 1, b = 0.17exp((V  5) 0.022) ms 1,</p><p> y = 300 ms, y = 1/(1 exp((V  68) 0.038));</p><p>The voltage-dependent potassium current I A 4 3 I A(t) = g A x (t) y (t) (V (t) VA ), (6)</p><p> dx dy = x (V )  x (V ), = y (V )  y (V ) dt  x dt  y 3  x = 1/(ax  bx ) 1 ms; x = ax/(ax  bx ), 1 ax = 0.08exp((V  41) 0.089) ms , 1 bx = 0.08exp((V  41) 0.016) ms ,</p><p> y = 1/(ay  by )  2 ms; y = ay/(ay  by ), 1 ay = 0.04 exp((V  49) 0.11) ms , 1 by = 0.04 ms ;</p><p>The voltage-dependent potassium current IM</p><p>2 IM (t) = gM x (t) (V (t) VM ), (7)</p><p> dx = x (V )  x (V ), dt  x  x = 1/(a  b)  8 ms, x = a/(a  b), a = 0.003exp((V  45) 0.135) ms 1, b = 0.003exp((V  45) 0.090) ms 1;</p><p>The cation current IH</p><p>IH (t) = gH y(t) (V (t) VH ), (8)</p><p> dy = y (V )  y (V ), dt  y 1  y = 180 ms, y = 1/(1 exp((V  98) 0.075)) ms ;</p><p>The adaptation current I AHP</p><p>I AHP (t) = g AHP w(t) (V (t) VAHP ), (9)</p><p> dw = w (V )  w (V ), dt  w  w = 4005/(3.3exp((V  35)/20)  exp((V  35)/20)) ms,</p><p> w = 1/(1 exp((V  35)/10)); where the reversal potentials and maximum conductances are VDR = 70 mV , VA = 70 mV , VM = 80 mV ,</p><p>VH = 17 mV , VAHP = 70 mV ,</p><p> gNa = 1.2 S, gDR = 0.4 S, g A = 2.3 S,</p><p> gM = 0.4 S, gH = 0.003 S, g AHP = 0.32 S,</p><p>GL = 0.02 S (i.e.  m = 14.4 ms),</p><p>VL = 58.5 mV (i.e. Vrest = 65 mV ),</p><p>4 Suppl. Fig.3. Model 3: spike trains (top), spike shape (bottom left) and phase-space plot (bottom right) in response to current step stimulation, starting at t=0. Stimulus amplitude for the bottom plots is u = 317 pA.</p><p>5 Model 4. One-compartment Hodgkin-Huxley-like model (Migliore et el. 1999) with two sodium channel subtypes distinguished by a shift (14 mV) of the activation and inactivation characteristics for the second channel. dV 3 ~ ~ 3 ~ ~ ext C  GL (V VL )  g Na m h i (V VNa )  g Na m h i (V VNa )  g K n (V VK )  I , dt ~ dm ~ 2 a   m  m ,   , m  , m dt  m a  b  a  b  0.4(V  V  30) 0.124(V  V  30) a  , b  , (exp((V  V  30) / 7.2) 1) (exp((V  V  30) / 7.2) 1) if  h  0.02 then  h  0.02 , ~ dh ~ 2 1   h  h ,  h  , h  , h dt  a  b 1 exp(V  V  50) / 4)  0.03(V  V  45) 0.01(V  V  45) a  , b  , (exp((V  V  45) /1.5) 1) (exp((V  V  45) /1.5) 1) if  h  0.5 then  h  0.5, ~ di ~ 3000b 1 0.5exp((V  58) / 2)   i  i ,  i  , i  , i dt  1 a 1 exp(V  58) / 2) a  exp(0.45(V  60)) , b  exp(0.09(V  60)) , if  i  10 then  i  10 . ~ V  14mV , gNa  437nS , gNa  437nS . The rest parameters and equations are the same as in Model 2.</p><p>6 Suppl. Fig.4. Model 4: top, f-I curves; bottom, spike shape and phase-space plot in response to current step stimulation with u = 14.1 pA, starting at t=0.</p><p>Model 5. Model 3, but with two sodium channel subtypes distinguished by a shift (14 mV) of the transition characteristics for the second channel. Below, the second channel current, conductance etc. are marked by tilde. dV ~ C = I  I  I  I  I  I  I  I  G (V V )  I ext , dt Na Na DR A M H L AHP L L ~ ~ ~ I Na (t) = gNa x1(t)(V (t) VNa ), (2)</p><p> x1  x2  x3  x4 = 1, (3) 4 4 dxi =  Aj,i x j  xi  Ai, j , i = 1,2,3 (4) dt j=0, ji j=0, j i</p><p>1 1,3 1,4 A1,2 = 3 ms , A1,3 = f1 (V  V ), A1,4 = f1 (V  V ),</p><p>7 2,3 A2,1 = 0, A2,3 = f2 (V  V ), A2,4 = 0, 3,1 3,4 A3,1 = f1 (V  V ), A3,2 = 0, A3,4 = f2 (V  V ), 4,1 A4,1 = f1 (V  V ), A4,2 = 0, A4,3 = 0, ~ V  14mV , gNa  1S , gNa  1S . The rest parameters and equations are the same as in Model 2.</p><p>Suppl. Fig.5. Model 5: top, f-I curves; bottom, spike shape and phase-space plot in response to current step stimulation with u = 322 pA, starting at t=0.</p><p>Model 6. One-compartment, Hodgkin-Huxley-based but reduced to threshold model with slow inactivating sodium channels (Fernandez and White 2010). dV C  g m h3 (V V )  g (V V )  I ext dt Na  Na L L 1 m   1 exp((V  30) / 4) ,</p><p>8 dh h  h 1  , h  dt  h 1 exp((V  52) / 2) if V  15mV then V VL 2 2 2 C  1.5F / cm , gNa  6mS / cm , gL  0.03mS / cm , VNa  50mV , VL  65mV ,</p><p> h  200ms .</p><p>Suppl. Fig.6. Model 6: top, f-I curves; bottom, spike shape and phase-space plot in response to current step stimulation with u = 0.73 μA/cm2, starting at t=0.</p><p>Model 7. One-compartment model with Markov, 26-state approximation of sodium channels (Milescu et al. 2010), fitted to experimental data obtained at room temperature. The rest ionic channels are from Model 3.</p><p>The schematic of the model of Na-channels is given in Suppl. Fig.7A,B. </p><p>A</p><p>9 B</p><p>C</p><p>Suppl. Fig.7. Model 7: Na channel kinetic model (extracted from [Milescu et al. 2010]). Model has two gating modes, differing only in their slow inactivation properties. Channels reside mostly in mode I during slow spiking, with a fraction switching to mode II during fast spiking. Rates were</p><p> exponential functions of voltage ( k  k0 exp(k1 V ) ). The notations for the rates are given in A and B. C, spike shape and phase-space plot in response to current step stimulation with u = 690 pA, starting at t=0.</p><p>Numerical values for the rates and maximum conductance are given below: for m : k0= 5.254, k1= 0.01474; for m: k0= 0.3454, k1= -0.08526; for : k0= 85.07, k1=</p><p>0.005784; for : k0= 5.895, k1= -0.01043; for ’ : k0= 25.45, k1= 0.005784; for ’ : k0= 26.06, k1= -0.01043; for h: k0= 0.01068, k1= -0.04270; for h: k0= 0.05954, k1= 0.02803; for ho: k0=</p><p>0.004182, k1= -0.04270; for ho: k0= 1.805, k1= 0.02803; a= 0.71098; b= 8.1799; for s : k0= -5 5.987×10 , k1= -0.06819; for s : k0= 0.4260, k1= 0.01496; for s2 : k0= 0.01876, k1= 0.01534; for s2 : k0= 0.3355, k1= -0.02956; for c : k0= 0.03731, k1= 0.1111; for c : k0= 0.001808, k1=</p><p>0.01301; for o : k0= 1.059, k1= 0.08020; for o : k0= 0.05132, k1= -0.01791; g Na = 2.6 S . The model showed stationary spiking regime only in very limited range of stimulation parameters</p><p>10 and only without the adaptation channels, i.e. with gM  g AHP  0 . That is why, the f-I-curves have not been plotted for the model.</p><p>Model 8. One-compartment model with Markov, 12-state approximation of sodium channels (Carter et al. 2010), fitted to experimental data obtained at 37 oC. The rest ionic channels are from Model 3.</p><p>The schematic of the model of Na-channels is given in Suppl. Fig.7B. Numerical values for the rates and maximum conductance are given below: for m : k0= 250, k1= 1/24; for m: k0= 12, k1= -1/24; = 250; = 60; ’= ; ’=; h= 2; h= </p><p>0.01; ho= 0.05; ho= 8; a= 2.51; b= 5.32; g Na = 2.6 S .</p><p>Suppl. Fig.8. Model 8: top, f-I curves; bottom, spike shape and phase-space plot in response to current step stimulation with u = 175 pA, starting at t=0.</p><p>11 Model 9. Spatially distributed neuron, consisting of soma and axon, based on Hodgkin-Huxley- like approximations (Yu et al. 2008).</p><p>The cable equation is dV d 2V C  g  2  i (t,V )  I ext , dt L dz2 m 3 where im (t,V )  gL (V VL )  gNa m h i (V VNa )  gK n (V VK ) . The boundary conditions are</p><p>V  V (t,0) ext  soma  ri C  im (t,V (t,0))  I  S , z z0  t  V  0 z zL a Here   is the characteristic length, a is the radius of the fiber (0.5 μm), rL is the 2 gL rL intracellular resistivity (150 Ωcm), axon length L  50 m , somatic membrane area S soma  3105 cm2 . dm 1 a  m  m  m ,  m  , m  , dt (a  b)T a  b  0.182(V  43) 0.124(V  43) a  , b  ; (exp((V  43) / 7) 1) (exp((V  43) / 7) 1) dh 1 1  h  h  h ,  h  , h  , dt (a  b)T 1 exp(V  72) / 6.2)  0.024(V  50) 0.0091(V  75) a  , b  ; (exp((V  50) / 5) 1) (exp((V  75) /5) 1) dn 1 a  n  n  n ,  n  , n  , dt (a  b)T a  b  0.02(V  25) 0.002(V  25) a  , b  ; (exp((V  25) / 9) 1) (exp((V  25) / 9) 1) 2 2 2 2 C  0.75 F / cm , gL  33 S / cm , at soma gNa  80mS / cm , gK  32 mS / cm , at axon 2 2 gNa  800mS / cm , gK  160mS / cm , 2.3 VL  70mV , VNa  50mV , VK  90mV , the temperature factor T  (T  296) /10 with T=37OC.</p><p>12 Suppl. Fig.9. Model 9: spike shape and phase-space plot in response to current step stimulation with u = 1.7 μA/cm2, starting at t=0.</p><p>Model 10. Adaptive exponential integrate-and-fire model (Brette and Gerstner 2005):</p><p>T dV (V V ) / T ext C  GL (V VL )  GLT e  w  I , dt dw  w  a(V VL )  w , dt</p><p> if V  15 ms then V VL , w  w  w , T VL  65 mV , GL  26 nS , C  370 pF , V  50.3mV , T  2 mV ,  w  14.4 ms , a  4 nS , w  0.0805 nA</p><p>13 Suppl. Fig.10. Model 10: spike train (top), spike shape (bottom left) and phase-space plot (bottom right) in response to current step stimulation, starting at t=0. Stimulus amplitude is 4μA/cm2 for the top plot and u = 1.15 μA/cm2 for the bottom plots.</p><p>Model 11. Exponential integrate-and-fire model with spike-dependent parameters (Badel et al. 2008). </p><p>T dV (V V ) / T ext C  GL (V VL )  GLT e  I , dt</p><p>0 (t tk ) / 32.8 (t tk ) / 42.9 VL VL  49.8 e  42.7 e ,</p><p>0 (ttk ) / 20 GL  GL  16 e</p><p>T T (t tk ) /14 V V0 18.4 e</p><p> if V  15 ms then V VL , </p><p> tk - spike time moment; 0 0 T VL  57 mV , GL  18 nS , C  370 pF , V0  42 mV , T  1.51 mV .</p><p>14 Suppl. Fig.11. Model 11: spike train (top), spike shape (bottom left) and phase-space plot (bottom right) in response to current step stimulation, starting at t=0. Stimulus amplitude is 4μA/cm2 for the top plot and u = 1.15 μA/cm2 for the bottom plots.</p><p>15 Model 12. Model 3, but with a new semi-phenomenological model of sodium channels with 3 Markov states (closed (C), open (O), and inactivated (I)) and the dynamic, I-dependent threshold of C-to-O transition (see Sections 2.2.2 and 3.3).</p><p>Suppl. Fig.12. Model 12: spike shape and phase-space plot in response to current step stimulation with u = 320 pA, starting at t=0.</p><p>Model 13. LIF model with dynamic threshold. The model description and simulation results are given in Sections 2.2.3 and 3.4).</p><p>The models 1-12 were programmed in Delphi-7 environment. The source code and executive file are available from the webpage http://www.ioffe.ru/CompPhysLab/AntonV3.html.</p><p>16</p>