The inward rectifier potassium current IKir promotes the intrinsic pacemaker activity of thalamocortical neurons

Yimy Amarillo1,3,*, Angela I. Tissone1,3,4, Germán Mato2,3 & and Marcela S. Nadal1,3,4

1. Departamento de Física Médica. Centro Atómico Bariloche and Instituto Balseiro. Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). Av. E. Bustillo 9500. (8400) S. C. Bariloche (Río Negro) Argentina.

2. Departamento de Física Médica. Centro Atómico Bariloche and Instituto Balseiro. ComisiónNacional de Energía Atómica (CNEA). Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). Av. E. Bustillo 9500. (8400) S. C. Bariloche (Río Negro) Argentina.

3. Gerencia de Área Investigación y Aplicaciones no Nucleares. Gerencia de Física. Departamento Sistemas Complejos y Altas energías. División Física Estadística e Interdisciplinaria. Centro Atómico Bariloche, S. C. Bariloche (Rio Negro) Argentina.

4. Universidad Nacional del Comahue. Centro Regional Universitario Bariloche. S. C. Bariloche (Rio Negro) Argentina.

Keywords: Kir Channels; Thalamocortical Neurons; Repetitive Burst Firing; Sub-threshold Conductances

Running Head: IKir promotes the intrinsic pacemaker activity of TC neurons

* Correspondence:

Yimy AmarilloConsejo Nacional de Investigaciones Científicas y TécnicasFísica Estadística e InterdisciplinariaCentro Atómico BarilocheAvenida Bustillo 9500San Carlos de Bariloche, Rio Negro, Argentinaamarillo@cab.cnea.gov.ar

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NEW AND NOTEWORTHY

Up-regulation of IKir in TC neurons induces repetitive burst firing at slow and delta (<4 Hz)frequency bands

IKir induces bistability of the membrane potential

The interaction between IKir and Ih generates sustained robust subthreshold oscillations

IKir increases the robustness of IT mediated oscillations

ABSTRACT

Slow repetitive burst firing by hyperpolarized thalamocortical (TC) neurons correlates with global slow rhythms (< 4 Hz), which are the frequencies of the physiological oscillations during N-REM sleep, or pathological ones during idiopathic epilepsy. The pacemaker activity of TC neurons depends on the expression of several subthreshold conductances, which are modulated in a behaviorally dependent manner. Here we show that up-regulation of the small and overlooked inward rectifier potassium current IKir induces repetitive burst firing at slow and delta frequency bands. We demonstrate this in mice TC neurons in brain slices by manipulating the Kir maximum conductance with dynamic clamp. We also performed a thorough theoretical analysis that explains how the unique properties of IKir enable this current to induce slow periodic bursting in TC neurons. We describe a new ionic mechanism based on the voltage and time-dependent interaction of IKir and Ih that bestows TC neurons with the ability to oscillate spontaneously at very low frequencies, even below 0.5 Hz. The bifurcation analysis of conductance based models of increasing complexity demonstrates that IKir induces bistability of the membrane potential at the same time that induces sustained oscillations in combination with Ih and increases the robustness of IT mediated oscillations.

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INTRODUCTION2

Generation and maintenance of the global brain 3 rhythms in the slow and delta bands (< 4 Hz) that 4 characterize N-REM sleep and idiopathic epilepsies 5 are associated with the intrinsic oscillatory behavior of 6 thalamocortical (TC) neurons in the thalamus (Dossi 7 et al. 1992; McCormick and Pape 1990b; Soltesz et al. 8 1991; Steriade and Contreras 1995). Depending on the 9 level of their resting membrane potential, these 10 neurons display two firing modes that correlate with 11 distinct behavioral states. During the waking, alert 12 states, TC neurons are depolarized and fire action13 potentials tonically at variable frequencies (Steriade et 14 al. 1996), which are synchronized over specific15 thalamocortical circuits (Buzsaki 2006). In contrast, a16 large-scale synchronization of sustained, low-17 frequency oscillations characterizes unconscious states 18 like slow-wave sleep (Steriade 1997) and absence 19 seizures (Steriade and Contreras 1995). The slow 20 rhythms (< 4 Hz) that characterize these physiological 21 and pathological states are correlated with repetitive22 burst firing by hyperpolarized thalamocortical neurons23 (Steriade and Contreras 1995). At subthreshold 24 membrane potentials, TC neurons fire repetitive bursts 25 of action potentials thanks to a unique set of ion 26 conductances, including the essential calcium current 27 IT that generates low threshold spikes (LTS) (Amarillo 28 et al. 2014). Electrophysiological studies on 29 genetically modified animals also link the mechanisms 30 that control the subthreshold excitability and the 31 repetitive burst firing of TC neurons with the 32 expression of sleep rhythms and epileptic oscillations.33 For example, there is a large decrease of delta 34 oscillations during N-REM sleep in mice that lack the 35 ion channel subunit (Cav3.1) responsible for the low 36 threshold calcium current IT (Lee et al. 2004), which is 37 essential for the intrinsic oscillations in TC neurons. 38 On the other hand, either overexpression of the IT39 current, or elimination of the hyperpolarization 40 activated cationic current Ih –another important 41 current that controls the excitability of TC neurons42 (Amarillo et al. 2015; Amarillo et al. 2014)–, results in 43 a phenotype of absence epilepsy characterized by 44 spike and wave discharges at about 3 Hz (Ernst et al. 45 2009; Ludwig et al. 2003).46

From a theoretical point of view, the minimal 47 requirements to generate membrane potential 48 oscillations are the combination of an amplifying 49 variable and a resonant variable in the presence of leak 50 conductances (Hutcheon and Yarom 2000; Izhikevich 51 2005). The biophysical properties of the T-type 52 calcium channels allow for spontaneous, subthreshold 53 oscillations, since IT behaves as both an amplifying54

current (through the activation gate mT) and a resonant55 current (through the inactivation gate hT) (Amarillo et 56 al. 2015). The bifurcation analysis of the minimal IT-57 leaks model of TC neurons uncovered the dynamic 58 interplay of these two variables in the generation and 59 maintenance of intrinsic oscillations in the delta band60 (Amarillo et al. 2015).61

Besides IT and the leaks, another four subthreshold 62 conductances modulate and control the propensity of 63 TC neurons to oscillate at low frequencies (Amarillo 64 et al. 2014). The interaction of the voltage and time 65 dependent properties of these seven ion channels 66 establishes the sequence of events that underlie the 67 generation and maintenance of repetitive burst firing 68 in TC neurons (Amarillo et al. 2014). We 69 demonstrated that Ih has a stabilizing role in the 70 oscillations, since it amplifies the range of membrane 71 voltage at which the oscillations can occur (Amarillo 72 et al. 2015). Our analysis also suggested that the 73 inward rectifier potassium current IKir (largely 74 mediated by Kir2.2 channels in TC neurons; Amarillo 75 et al. 2014), promotes the periodicity of IT- mediated 76 burst firing by potentiating inter-burst 77 hyperpolarizations (Amarillo et al. 2014). IKir behaves 78 as a hyperpolarization-activated outward current due 79 to the existence of a negative slope region in the 80 current/voltage relationship. In this negative slope 81 conductance region, displacements of the membrane 82 potential in the hyperpolarizing direction produce the 83 regenerative activation (unblock) of IKir, and thereby,84 regenerative hyperpolarization. Progressive 85 hyperpolarization removes inactivation of IT at the 86 same time that induces activation of Ih, which brings 87 the membrane potential back to the region of 88 activation of IT. Here we show that an increase 89 (introduced with dynamic clamp) in Kir maximum 90 conductance in TC neurons recorded from mouse 91 brain slices, elicits oscillations at delta and slow 92 frequencies in all the recorded neurons. This suggests93 that up-regulation of IKir in TC neurons could amplify 94 the natural resonance of TC neurons at burst firing 95 frequencies, thereby increasing the propensity of the96 thalamocortical system to synchronize at these slow 97 frequencies. This mechanism could take place 98 physiologically during N-REM sleep, or 99 pathologically, during epilepsy.100

METHODS101

Computational modeling102

For this study we used Hodgkin and Huxley-like 103 equations similar to those used previously (Amarillo 104 et al. 2014). Bifurcation analysis and phase plane 105

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portrait analysis were performed using XPPAUT 106 (Ermentrout 2002), using absolute values of maximum 107 conductance or permeability (in nS or cm3/s, 108 respectively). Time course simulations were 109 performed using the NEURON environment (Hines 110 and Carnevale 1997), where per unit area values were 111 used (in S/cm2 or cm/s, respectively). The 112 interconversion between absolute and per unit area 113 values was carried out considering a total capacitance 114 of 0.2 nF for all model cells. In table I we report the 115 absolute values. Time course simulations were 116 performed assuming a temperature of 28° C and the 117 equations of time dependence (time constants) were 118 adjusted accordingly by using the reported Q10119 conversion factors (Table I).120

The voltage equations have the form:121

dV/dt = (Iinj i)/C (1)122

where Iinj is the magnitude of injected current, C is the 123 capacitance, and i is the sum of all ionic currents as 124 follows: i = IKir + IKleak + INaleak for the minimal IKir-125

Ii = IKir + Ih + IKleak + INaleak for the 126 minimal IKir-Ih- Ii = IKir + Ih + IT + IKleak127 + INaleak for the IKir-Ih-IT- Ii = IKir +128 Ih + IT + IA + INaP+ INa + IK + IL + IAHP + IKCa + ICAN +129 IKleak + INaleak for the more complete model that 130 includes subthreshold and suprathreshold operating 131 conductances.132

The current equations for all ionic currents, except the 133 calcium currents IT and IL, have the form:134

Ii = i mip(V,t) h iq(V,t) i) (2)135

where Ii is a given ionic current; i is the maximum 136 conductance for current i; mi and hi are the voltage and 137 time dependent activation and inactivation variables 138 for that current respectively; p and q are exponents; V139 is voltage and Ei is the reversal potential for that 140 current. Activation of currents IKir, INaP, IAHP and IKCa141 was assumed to be instantaneous therefore, for these 142 currents the time dependent mi

p(V,t) variable was 143 replaced by m p(V) in equation (2).144

Current equations for the calcium currents are:145

Ii = pi mip hT

q G(V, Cao,Cai) (3)146

where pi is the maximum permeability, mi and hi are 147 the activation and inactivation variables respectively 148 and G(V, Cao,Cai) is a non-linear function of potential 149 and calcium concentration; 150

G(V, Cao,Cai) = z2F2V/RT (Cai o exp ])151 (4) 152

where Cao and Cai are the extracellular and the 153 intracellular concentrations of Ca++ and z, F, R and T154 are the valence, the Faraday constant, the gas constant 155 and the absolute temperature, respectively.156

Equations for the activation and inactivation variables 157 have the form:158

dmi/dt = (m (V) – mi)/ mi(V) (5)159

dhi/dt = (h (V) – hi hi(V) (6)160

where m (V), h mi hi(V) are the steady state 161 and time constants of activation and inactivation for a 162 given current Ii. The expressions for these steady state 163 and time constants are given in table I, together with 164 the default parameter values (those used in the last 165 section of results, Fig 5C). Parameter values subjected 166 to analysis are specified in the main text of the results 167 section and corresponding figure legends.168

The calcium dependent potassium current IAHP and the 169 voltage and calcium dependent potassium current IKCa170 were gated by the variation of intracellular calcium 171 linked to activation of the high threshold calcium 172 current IL, whereas the calcium dependent cationic 173 current ICAN was gated by the variation of intracellular 174 calcium linked to activation of IT. Intracellular 175 calcium dynamics, in turn, was modeled as previously 176 (Amarillo et al 2014, McCormick and Huguenard 177 1992) using a simple diffusion model from a shell of 178 100 nm depth just beneath the plasma membrane with 179 the dimensions of the model cell (22700 μm2). The 180 equation that describes this diffusion is:181

dCai/dt = –(IT or IL)/depth/F/2 (0.0000001) + (Cai0 -182 Cai (7)183

where depth is the thickness of the shell, F is the 184 faraday constant, Cai0 is the initial calcium 185 concentration (50 nM) and is the diffusion rate (1 186 ms).187

Addition of the calcium dependent cationic current 188 ICAN was necessary to reproduce the decaying plateau 189 potentials that follow LTSs observed experimentally. 190 ICAN was adapted from (Zhu et al. 1999) with the 191 parameters listed in table I.192

Slice preparation and electrophysiology193

All experiments were carried out in accordance with 194 the NIH Guide for the Care and Use of Laboratory 195

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Animals and were approved by the Institutional 196 Committee for the Care and Use of Laboratory 197 Animals (CICUAL) of the National Atomic Energy 198 Commission from Argentina (CNEA). All procedures 199 for obtaining brain slices and for electrophysiological 200 recording have been previously described (Amarillo et 201 al. 2008; Amarillo et al. 2014).202

Briefly, brain slices were prepared from 2 to 4 weeks 203 old NIH Swiss mice. Following induction of deep 204 anesthesia with pentobarbital sodium (50-75 mg/kg, 205 ip), mice were decapitated and the brains removed into 206 an ice-cold oxygenated artificial cerebrospinal fluid 207 (ACSF) that contained (in mM) 126 NaCl, 2.5 KCl, 208 1.25 Na2PO4, 26 NaHCO3, 2 CaCl2, 2 MgCl2 and 10 209 dextrose. The brain was blocked at a coronal plane 210 and 350 μm-thick slices were cut using a manual 211 vibroslicer (WPI, Sarasota, FL. USA). Slices 212 including the ventrobasal thalamic nuclei were213 incubated at 30°C for 1 hour and thereafter maintained 214 at room temperature in oxygenated ACSF (95% CO2, 215 5% O2) until they were transferred to the recording 216 chamber continuously perfused with oxygenated 217 ACSF. 218

Neurons from ventrobasal thalamic nuclei (ventral 219 posterolateral and ventral posteromedial nucleus) were 220 visualized using a CCD camera (Panasonic, Newark, 221 NJ, USA) mounted on a Carl Zeiss Axioskop FSII 222 plus fixed-stage microscope (Carl Zeiss AG, 223 Oberkochen, Germany) equipped with IR-DIC optics. 224 Patch pipettes were made from borosilicate glass in a 225 Sutter P-1000 horizontal puller (Sutter Instrument 226 Company, Novato, CA. USA) with resistances 227 between 2 and 5 M and filled with an intracellular 228 solution containing (in mM) 119 CH3KO3S, 12 KCl, 229 1 MgCl2, 0.1 CaCl2, 1 EGTA, 10 HEPES, 0.4 Na-230 GTP, 2 Mg-ATP, pH 7.4. Neurons were recorded in 231 fast current clamp mode using an Axoclamp 700B 232 amplifier (Molecular Devises, Sunnyvale, CA. USA) 233 after stabilization of the resting membrane potential. 234 Recordings were low pass filtered at 10 KHz, pipette 235 capacitance was canceled and bridge balance 236 compensated. A sampling rate of 10 kHz was used; 237 data was acquired on a personal computer using the 238 PClamp10 software (Molecular Devises, Sunnyvale, 239 CA. USA) and stored for further analysis.240

Dynamic clamp241

Dynamic clamp was implemented using a dedicated 242 DAQ card PCIe-6351 (National Instruments, Austin, 243 TX, USA) that was commanded by the open source 244 software StdpC under the Windows 7 OS (Kemenes et 245 al. 2011). The analog output of the Multiclamp 700B 246 amplifier was connected to analog inputs on both the 247 acquisition card Digidata 1440a (Molecular Devices, 248

Sunnyvale, CA. USA) and the dynamic clamp card 249 PCIe-6351. This signal was used to track the 250 membrane potential with the acquisition software 251 PClamp10 as well as the voltage input for 252 implementing the dynamic clamp loop. Analog 253 outputs of the two interface cards were connected to a 254 custom-made real time voltage summing device255 (patent pending, CONICET, Argentina), whose output 256 was connected, in turn, to the analog input of the 257 amplifier. This configuration allows us to 258 simultaneously perform current clamp protocols with 259 the PClamp10 software and dynamic clamp 260 simulations with the StdpC software, in the 261 Multiclamp 700B amplifier.262

RESULTS263

The unique biophysical properties of IKir induce 264 bistability of the membrane potential265

The inward rectification profiles of potassium 266 channels of the Kir2.X family show a distinctive 267 region of negative slope in the current/voltage 268 relationship (Dhamoon et al. 2004). This property of 269 Kir channels is produced by the voltage dependent 270 block of the channel pore by intracellular Mg2+ and 271 polyamines (reviewed in Anumonwo and Lopatin 272 2010). We have previously identified Kir2.2 as the 273 main ion channel subunit responsible for IKir in 274 thalamocortical neurons (Amarillo et al. 2014).275 Currents produced by Kir2.2 channels in heterologous 276 systems show the strongest voltage dependent inward277 rectification compared to those produced by Kir 2.1 278 and Kir 2.3 channels (Dhamoon et al. 2004).279 Accordingly, thalamocortical neurons display a fast 280 inward rectification that is not present on TC cells 281 from Kir2.2 KO mice or when Kir channels are 282 blocked by low concentrations of barium (10-50 μM)283 (Amarillo et al. 2014).284

In TC neurons from mice, the shape of the total 285 subthreshold steady state I/V curve is determined by 286 the combined contribution of seven conductances 287 (Amarillo et al. 2014). We started by analyzing a 288 model that includes IKir (modeled based on data from 289 thalamocortical neurons from mice –Amarillo et al. 290 2014–) and the potassium and sodium leak 291 conductances. The steady state I/V relationship for the 292 leaks alone (Fig 1A, dashed trace) is linear; whereas 293 the steady state I/V relationship for IKir shows strong 294 inward rectification and a negative slope conductance 295 region between 85mV and 60mV (Fig. 1A, dotted296 trace). For certain combination of conductance values 297 of Kir and leaks (see figure legend), the I/V curve of 298 this IKir-leaks minimal model is non-monotonic (Fig. 299 1A, solid black trace), and the steady state I/V curve 300 crosses the cero current axis at three values of 301

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membrane potential (Fig. 1A, dots). For a one 302 dimensional system, this I/V curve gives rise to a303 saddle-node bifurcation (S-N) (Izhikevich 2005) with 304 two stable equilibrium points (Fig. 1A, a and c) 305 separated by one unstable equilibrium point (Fig. 1A,306 b). The consequence of this dynamical property is the 307 expression of membrane potential bistability. Fig 1B 308 shows the simulated evolution in time of the 309 membrane potential for the IKir-leaks minimal model 310 for six different initial Vm values, around the unstable 311 equilibrium point. For the three initial values that are 312 above the unstable point ( 74.6 mV), the membrane 313 potential evolves towards the depolarized stable 314 equilibrium ( 57.7 mV), whereas for the other three 315 values, the membrane potential stabilizes at the 316 hyperpolarized stable equilibrium ( 87.2 mV). This 317 bistability is further evidenced when shown vis a vis318 the bifurcation diagram of the IKir-leaks model at 319 injected current = 0 (Fig. 1B, right).320

The physiological modulation of the potassium leak 321 current by neurotransmitters is considered the 322 mechanism that controls the resting membrane323 potential of thalamocortical neurons (McCormick and 324 Prince 1987; 1988). For this reason, we analyzed the 325 effect of varying the potassium leak maximum 326 conductance ( Kleak) on the dynamical behavior of the 327 IKir-leaks model. Similarly to using current injection as 328 the bifurcation parameter, changing Kleak displays two 329 saddle node bifurcation points and a region of 330 instability in the bifurcation diagram (Fig. 1C).331

The interaction of amplifying IKir and resonant Ih332 generates sustained subthreshold oscillations333

The hyperpolarization activated cationic current (Ih)334 plays an important role in establishing the resting 335 membrane potential and modulating the oscillations 336 mediated by the low threshold calcium current IT in 337 thalamocortical neurons (Amarillo et al. 2015;338 McCormick and Pape 1990b; Soltesz et al. 1991). Ih339 adds robustness to the repetitive low threshold spikes 340 (LTSs) by potentiating the initial phase of 341 depolarization. In addition, the resonant activation of 342 Ih allows transient, negative excursions of the 343 membrane potential between LTSs; without this 344 recovering mechanism, the membrane potential would 345 stabilize at the more negative total steady state 346 conductance level (Amarillo et al. 2015).347

The interaction of a resonant current such as Ih with an 348 amplifying current such as IKir can theoretically 349 support periodic subthreshold oscillations (Izhikevich 350 2005). In striatal cholinergic interneurons, these two 351 hyperpolarization-activated currents underlie the 352 cyclic transient hyperpolarizations that generate the 353 inter-bursts intervals in these neurons (Wilson 2005).354 In agreement with these results, we have previously 355 shown that the IKir-Ih-leaks minimal model exhibits356 sustained oscillations (Amarillo et al. 2014). Fig. 2A 357 shows the time course of the changes in the membrane358 potential produced by this two dimensional model 359 (dV/dt and dmh/dt) for a maximum conductance of Kir 360 of 41 nS, and physiological values of maximum 361

Figure 1. The Kir current induces bistability of the membrane potential

(A) Simulated steady state I/V relationships of the IKir-leaks model cell with parameters adjusted to generate bistability inthe absence of current injection. Kir = 15.9 nS, KLeak = Naleak = 0.68 nS, ENa = 0 mV and EK = –100 mV. The combined IKir-leaks I/V curve (solid trace) crosses the zero current level at three different voltage values (a, b, c dots). (B) Simulatedtime course of the membrane potential for the model in (A) for six different initial Vm values (from negative to positive:

80, 75, 74.7, 74.5, 74 and 70 mV). Projected on the right is the bifurcation diagram using I as the bifurcationparameter. The V/I curve crosses zero current at the stable points a and c, and at the unstable point b (the same Vm valuesas in A). The dashed segment indicates instability and the solid segments indicate stability. S-N indicates the saddle-node bifurcation points. (C) Bifurcation diagram of the IKir-leaks model in (A) using Kleak as the bifurcation parameter.

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conductances of h and the leaks ( h = 5 nS; Kleak = 2.3362 nS; Naleak = 0.7 nS). At rest with cero current363 injection the membrane potential stabilizes at 364

82.66mV. The injection of a current of 60 pA 365 induces robust periodic oscillations (black trace) that 366 are not observed with injection of either 40 pA (black 367 trace, damped oscillations) or 80 pA (gray traces). The 368 phase plane portrait of these three conditions (Fig. 2B) 369

shows that: 1) the trajectory of the dampened 370 oscillations (upper gray trajectory and V-nullcline) 371 obtained with injection of 40 pA coalesces to a stable 372 focus (s); 2) the orbit of the oscillations obtained with 373 60 pA (middle black trajectory and V-nullcline) enters374 a stable limit cycle around an unstable focus (u); and 375 3) the membrane potential stabilizes at a depolarized 376 stable focus (s) when the current injection is 80 pA 377

Figure 2. The interaction of IKir and Ih induces robust sustained subthreshold oscillations

(A) Time course of the membrane potential produced by the IKir-Ih-leaks model cell for different current injection magnitudes (indicated on the left of each trace). Simulations were initiated at the resting membrane potential ( 82.66mV; I = 0) and the injected current was applied during the time indicated by the bar above the upper trace. Kir = 41 nS, h= 5 nS, Kleak = 2.27 nS and Naleak = 0.68 nS. Gray traces indicate evolution towards a stable equilibrium and the black trace indicate sustained oscillations (B) Phase plane portrait for the three current levels shown in (A), starting at rest (I =0): 40pA, 60pA and 80 pA (Gray and black have the same meaning as in A). Note that for the three conditions the nullclines intersect in a single point: stable foci (s) for 40 and 80 pA and an unstable focus (u) for 60 pA. (C) Bifurcation diagram as the current injection changes showing the two supercritical Hopf bifurcation points (sup. Hopf) and the amplitude of the limit cycle (dots). (D) Bifurcation analysis using Kleak as the bifurcation parameter. At the saddle-node on invariant circle bifurcation (S-NIC) point, the stable limit cycle appear/disappear as Kleak increases/decreases due to the presence of a saddle node bifurcation that shrinks/expands a small heteroclinic trajectory (Izhikevich 2005). The inset shows the frequency of the limit cycle as function of Kleak Kleak – Kleakb when Kleakdecreases (from right to left) toward the bifurcation value Kleakb. The separation between slow and delta bands (at 1 Hz) is indicated by the shadowed area.

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(lower gray trajectory and V-nullcline). In the three 378 cases, the voltage and Ih activation nullclines intersect 379 in a single focus, which is either stable or unstable.380 This indicates that the transition from either of the two 381 stable foci (s) to the unstable focus (u) occurs via a 382 Hopf bifurcation, as shown in Fig. 2C. The bifurcation 383 diagram shows the two Hopf bifurcation points and 384 the amplitude of the limit cycle as the current injection 385 changes (the dots indicate the max/min of the limit 386 cycle). At the bifurcation points, the gradually 387 increasing/decreasing amplitude of the limit cycle 388 indicates that both transitions occur via supercritical 389

Hopf bifurcations. 390

As mentioned in the previous section, physiological 391 changes in Kleak induce changes in the resting 392 membrane potential of thalamocortical neurons. To 393 examine the range of values of Kleak that induces 394 oscillations, we performed the bifurcation analysis 395 using Kleak as the bifurcation parameter (Fig. 2D). The 396 system enters a stable limit cycle via a supercritical 397 Hopf as Kleak is decreased. Between this bifurcation 398 point ( Kleak 1.1 nS) and a Kleak value of 0.4 nS, the 399 membrane potential is unstable. At this last point, the 400

Figure 3. Effect of IKir on the dynamics of a model of thalamocortical neurons that includes Ih and IT

(A) Bifurcation diagram of the IT-Ih-leaks model using current injection as the bifurcation parameter, in the presence of IKir (9 nS, black diagram) or in the absence of IKir (grey diagram). Identical parameter values were used for the other conductances: h = 5 nS, Kleak = 2.27 nS, Naleak = 0.68 nS and pT = 14 cm3/s. sub. Hopf and sup. Hopf labels indicate subcritical and supercritical Hopf bifurcation points respectively. Dots are max/min of stable (filled dots) or unstable (empty dots) limit cycle. (B) Bifurcation diagram of the IT-Ih-leaks model using the maximum permeability of IT as the bifurcation parameter, in the presence of IKir (9 nS, black diagram) or in the absence of IKir (grey diagram). (C) Bifurcation diagram of the IT-Ih-leaks model using Kleak as the bifurcation parameter, in the presence of IKir (9 nS, black diagram) or in the absence of IKir (grey diagram). (D) Bifurcation diagram of the IT-Ih-leaks model using Kleak as the bifurcation parameter, in the presence of IKir (41 nS). The inset shows the time course of membrane potential oscillations at a Kleakvalue that is very close to the bifurcation point ( Kleak = 1.05973610281 nS). Conventions and parameter values for the other conductances in (B, C and D) are as in (A).

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limit cycle disappear via a saddle-node on invariant 401 circle bifurcation (S-NIC). This last bifurcation allows 402 the system to undergo oscillations at low frequencies403 (Izhikevich 2005), providing a possible mechanism to 404 generate physiological or pathological slow rhythms405 (< 1 Hz) (Fig. 2D: inset).406

Effect of IKir on the dynamics of a reduced model of 407 thalamocortical neurons408

The low threshold calcium current IT is absolutely 409 required for burst firing in thalamocortical neurons, as 410 evidenced by both pharmacological (Dreyfus et al. 411 2010) and genetic ablation (Kim et al. 2001) studies. 412 The interaction between the amplifying activation 413 variable mT and the resonant inactivation variable hT414 of IT creates an oscillatory unit that can support 415 sustained oscillations (repetitive LTSs, Amarillo et al. 416 2014; Hutcheon and Yarom 2000). The other 417 subthreshold conductances expressed by418 thalamocortical neurons modulate the generation, 419 repetitiveness, repolarization, frequency and voltage 420 range of the LTSs produced by IT (Amarillo et al. 421 2015; Amarillo et al. 2014).422

In thalamocortical neurons, the oscillatory system 423 formed by IKir-Ih described in the previous section co-424 exists with the oscillatory system formed by the gating 425 variables of IT. We sought to examine how the 426 interaction between the different modeled components 427 of these two oscillatory systems impact on the 428 dynamics of the membrane potential. We have 429 previously shown that the depolarizing drive 430 contributed by Ih in the IT-Ih-leaks model not only adds 431 to the regenerative activation of IT during the 432 ascending phase of the LTS, giving rise to larger and 433 faster oscillations, but it also permits the occurrence of 434 LTSs at more hyperpolarized potentials (Amarillo et 435 al. 2015). Including IKir (9 nS; Fig 3A, black diagram)436 in that model does not change the dynamical437 properties of the system, which transits between 438 stability and instability via a subcritical Hopf 439 bifurcation (sub. Hopf) at hyperpolarized potentials,440 and via a supercritical Hopf bifurcation (sup. Hopf) at 441 depolarized levels, either in the presence or absence of 442 IKir (Fig 3A, black or grey diagram, respectively). Yet,443 there are quantitative changes after addition of IKir to444 the IT-Ih-leaks model: First, there is a positive shift in 445 the range of current injections such that, in the 446 presence of IKir, no current injection is required to 447 produce maximum amplitude oscillations (see 448 oscillations produced at cero current value i.e., 449 resting state in Fig. 3A, black diagram). These results 450 indicate that IKir bestows the system with the ability to 451 oscillate spontaneously at lower levels of IT (see 452 discussion). And second, the minimum value of 453 permeability of IT (pT) required to generate sustained 454

LTSs decreases from 12.8 x 10–9 cm3/s to 10.7 x 10–9455 cm3/s. When we performed the bifurcation analysis 456 using the maximum permeability of IT as the 457 bifurcation parameter (for I = 0), both in the presence 458 and absence of IKir, we were able to visualize the 459 effect of IKir on the range of permissive pT values (i.e. 460 values of pT that allow sustained oscillations) (Fig. 461 3B). Without current injection, pT is permissive for 462 values between 12 x 10-9 cm3/s and 30 x 10-9 cm3/s 463 when using a Kir of 9 nS (Fig. 3B, black diagram);464 whereas in the absence of IKir, these permissive pT465 values are between 14 x 10-9 cm3/s to 22 x 10-9 cm3/s466 (Fig. 3B, gray diagram). The amplitude of the 467 membrane potential oscillations is notably larger in 468 the presence of IKir.469

When using Kleak as the bifurcation parameter without 470 current injection, the system has two different 471 dynamical behaviors depending on the magnitude of 472

Kir: At low levels of Kir (9 nS), the transitions 473 between rest and the limit cycle occur via Hopf 474 bifurcations (Fig 3C, colored diagram), similar to the 475 behavior of the system in the absence of IKir (Fig 3C,476 gray diagram). Addition of low magnitudes of IKir477 again produces a quantitative effect by increasing the 478 range of permissive values of Kleak. On the other 479 hand, for high values of Kir (41 nS, Fig 3D), the limit 480 cycle disappear via a S-NIC (saddle-node on invariant 481 circle) bifurcation as Kleak decreases. As mentioned 482 above, this S-NIC bifurcation endows the system with 483 the ability to oscillate at very slow frequencies. 484 Curiously, for values of Kleak infinitesimally close to485 the bifurcation point, the time course changes from the 486 typical triangle shaped oscillations to oscillations with 487 prolonged plateaus after the LTSs (Fig 3D, inset), as 488 seen in recordings from rodent thalamocortical 489 neurons in vitro (see below).490

Increasing the maximum conductance of IKir induces 491 sustained repetitive burst firing in TC neurons from 492 mice493

Recorded membrane potential of TC neurons from 494 mice is stable at about 67 mV. In recordings 495 performed in brain slices from rodents, it is very rare 496 to observe spontaneous repetitive bursting without 497 manipulating the extracellular concentration of 498 divalent ions and/or without stimulating the cortico-499 thalamic/reticulo-thalamic afferents (Jacobsen et al. 500 2001; Leresche et al. 1991; Warren et al. 1994).501 Moreover, in typical experiments in rodent brain 502 slices, even sustained injection of current in the 503 depolarizing or hyperpolarizing direction does not 504 induce oscillations at subthreshold levels. Larger 505 magnitudes of depolarizing current (suprathreshold 506 depolarization) elicit either tonic firing or single burst 507 firing depending on the membrane potential prior to 508

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the onset of the stimulus (holding potential). Either 509 mode of firing is dependent on the level of 510 inactivation of the low threshold calcium current IT,511 which reflects the history of the membrane potential.512 At depolarized holding potentials, T channels are 513 inactivated and the neuron responds with a train of 514

action potentials at a frequency that correlates with the 515 magnitude of the injected current. At holding 516 potentials negative to about -65 mV, T channels are 517 increasingly de-inactivated and the neuron responds to 518 a depolarizing input with a low threshold calcium 519 spike (LTS), crowned by a burst of Na+/K+-mediated 520

Figure 4. Artificial increase of Kir induces sustained repetitive burst firing in TC neurons from mice.

(A) Current clamp recordings of a TC neuron from the ventro basal complex of mouse thalamus, where different amounts of maximum conductance of Kir, ranging from 10 to 30 nS, were artificially introduced with dynamic-clamp (the arrow and vertical dashed line indicate the onset of the dynamic clamp). The inset shows a magnified view of one of the bursts of action potentials riding on a low threshold spike. (B) Current clamp recordings of a different TC neuron in response to increasing magnitudes of injected hyperpolarizing current before (left traces) and after introduction of 50 nS of Kir with dynamic-clamp (right traces). (C) Current clamp recordings of another TC neuron with a protocol of square pulses in 20 pA increments from –100 pA, under control conditions (upper left) and after bath application of 10 μM barium (upper right). Note the disappearance of the fast inward rectification (measured at an early time point –vertical dashed line) after application of Ba++. The lower left traces show the rescue of the fast inward rectification after introduction of 5 nS of Kirwith dynamic clamp. An excess of d Kir (15 nS) further increases the fast inward rectification in this cell (lower right). (D) Current/Voltage curves for the same TC neuron recorded in (C) under these different conditions, measured at the time indicated by the vertical dashed lines in (C). Control (open circles), 10 μM barium (full circles), and 10 μM barium + either 5 nS (triangles) or 15nS d Kir (squares).

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action potentials (see recordings of TC neurons from 521 rodents elsewhere; for example, Amarillo et al. 2014;522 Kim et al. 2001; Llinas and Jahnsen 1982; Meuth et al. 523 2006; Zhu et al. 1999). In order to test the predicted 524 pro-oscillatory effect of IKir, we recorded 19 TC 525 neurons in the ventrobasal nuclei of the thalamus from 526 mice and manipulated the maximum conductance of 527 Kir with dynamic-clamp (d Kir, see methods). In all 528 tested cells, sole introduction of artificial IKir induces 529

repetitive burst firing (examples from three different 530 cells are shown in Fig 4A, Fig4B and Fig. 5A). The 531 shape of the burst cycle changes depending on the 532 magnitude of artificial Kir introduced by dynamic 533 clamp (Fig. 4A, the arrow marks the onset of the 534 dynamic clamp). For the minimal d Kir that induces 535 oscillations, the cycle consists of an initial 536 hyperpolarization followed by a slow recovering 537 depolarization that brings the membrane potential to 538

Figure 5. Mechanism of dIKir induced oscillations in TC neurons.

(A) Membrane potential time course of repetitive bursts in a TC neuron (upper trace) induced by introducing 10 nS d Kir. Aligned below is the time course of the current injected by the dynamic-clamp system (lower trace). Regenerative activation of IKir (dashed vertical line) initiates the down stroke of a transient hyperpolarization (B)Membrane potential (upper trace) and IKir (lower trace) time courses during repetitive bursts in a complete model of TC neuron using a large Kir value and a low Kleak value (34 nS and 0.3 nS, respectively) in the absence of ICAN. See parameters for the other conductances in table I. (C) Simulations in the complete TC neuron model that includes the cationic calcium activated current ICAN (2.5 nS) using Kir = 30.6 nS and a more physiological Kleak value (2.5 nS). All parameters for these simulations are listed in table I. The upper trace shows the time course of the membrane potential followed below by the time course of IKir and ICAN. The lower traces show the time course of the modeled activation ofIh (mh, dotted trace) and the activation (mT

2, dashed trace) and inactivation (hT, solid trace) variables of IT. The vertical dashed line is placed at an onset of regenerative activation of IKir, which coincides with an onset of activation of Ih and a large deinactivation of IT. Horizontal dashed lines indicate the zero current level.

11

the LTS threshold. Immediately after the LTS, the 539 next cycle re-initiates with a similar sequence. In 540 contrast, for larger d Kir magnitudes, a plateau 541 potential of varying lengths follows the LTSs, 542 introducing a delay to the hyperpolarization that 543 initiates the next cycle (Fig. 4A, 24 and 30 nS). A544 similar response was obtained by varying the 545 magnitude of injected hyperpolarizing current for a546 large value of d Kir (50 nS, Fig 4B). The plateau 547 potential is only observed for current injections around 548 threshold level (–30 pA), and not for larger549 hyperpolarizations (–60 pA). A large 550 hyperpolarization (–90 pA) overpowers the amount of 551 artificial d Kir introduced by dynamic clamp and no 552 subthreshold oscillations are observed (Fig 4B, bottom 553 right).554

We have previously estimated that the maximum 555 conductance of native Kir in most TC neurons is 556 approximately 5 nS (Amarillo et al. 2014, yet there is 557 significant variability among some cells). This was 558 done by subtracting the IV curves in presence and 559 absence of barium, which –in low concentrations– is 560 considered a specific blocker of Kir 2.X channels 561 (Dhamoon et al. 2004; Schram et al. 2003). Here we562 used dynamic clamp to reinstate the endogenous levels 563 of Kir –and hence the fast inward rectification– in TC 564 cells treated with 10 μM barium. In agreement with 565 our previous estimations, the d Kir magnitude that 566 restores the fast inward rectification was 5 nS in two567 cells tested and 7 nS in another cell. Fig. 4C shows 568 that the characteristic fast rectification of TC neurons 569 (control, upper left) is lost in the presence of 10 μM 570 barium (upper right), and it is subsequently 571 reconstituted after re-introducing IKir with dynamic 572 clamp (lower traces): a value of d Kir of 5 nS is 573 sufficient to restore the native Kir in the cell shown 574 (lower left), whereas a larger amount (15 nS in this 575 cell) bestows TC neurons with a stronger inward 576 rectification (i.e. amplification of hyperpolarization; 577 lower right) and slow oscillations (repetitive bursts are 578 truncated by finalization of current injection steps in 579 the lower right panel of Fig 4C). As expected, a larger 580 d Kir value is required to induce oscillations in the 581 presence of low concentrations of barium (10 μM) 582 (data not shown). Fig 4D shows the I/V curves for the 583 same TC neuron recorded in Fig. 4C under the 584 different conditions described above: control, 10 μM 585 barium and 10 μM barium + 5 nS or 15nS d Kir.586

Role of IKir on the oscillatory behavior of TC neurons587

To further explore the mechanism by which increasing 588 IKir in rodent TC neurons induces repetitive burst 589 firing, we examined the membrane potential time 590 course of induced repetitive bursts (Fig. 5A, upper 591 trace), at the same time that we recorded the Kir 592

current injected by the dynamic clamp system (Fig. 593 5A, lower trace). As expected from previous 594 computational analysis (Amarillo et al. 2014), IKir595 decreases as the membrane potential recovers towards 596 the base of the LTS, reaching its minimal value during 597 the LTSs and the bursts of action potentials. After the 598 firing of the LTS, the magnitude of IKir began to 599 increase during the slow initial hyperpolarization that 600 follows the LTS (plateau potential) until an inflection601 point (vertical dashed line in Fig 5A), at which the 602 activation of IKir becomes regenerative and thus 603 increases abruptly to its maximum value. This is 604 mirrored in the membrane potential by a sudden 605 hyperpolarization, which removes the inactivation of 606 IT at the same time that induces the activation of Ih. To 607 closely examine the interaction among IKir, Ih and the 608 gating variables of IT during this process, we used a609 more complete model of TC neurons –which includes 610 all the subthreshold conductances as well as the 611 suprathreshold conductances that generate and control 612 Na+/K+ spiking– to reproduce the time course of 613 induced oscillations (Fig 5B and C, also see methods 614 and table I). In order to test the possibility that IKir615 could mediate the very slow oscillations and the 616 plateau potentials observed experimentally by a 617 mechanism involving a S-NIC bifurcation as in Figs 618 2D and 3D, we first try to reproduce the time course 619 of the recorded oscillations combining large Kir620 values with low Kleak values (Fig 5B, 34 nS and 0.3 621 nS, respectively). Indeed, we were able to generate 622 both oscillations at frequencies below 0.2 Hz and large 623 plateau potentials (Fig 5B). Yet, it was not possible to 624 reproduce the exact shape of the plateaus by 625 iteratively manipulating Kir and Kleak, We then 626 performed additional simulations including the 627 cationic calcium activated current ICAN, which is 628 known to mediate plateau potentials by producing an 629 after-depolarization that decays slowly due to its slow 630 deactivation (Hughes et al. 2002; Zhu et al. 1999). 631 Including ICAN in the model (Fig 5C, third trace) 632 accurately and robustly reproduces the experimental 633 traces for values of Kir and Kleak other than those 634 underlying the S-NIC bifurcation ( Kir = 30 nS and 635

Kleak = 2.5 nS in Fig 5C, and data not shown). Using 636 this set of parameters (see table 1), the modeled IKir637 (Fig 5C, second trace) reproduces the time course of 638 the dynamic clamp injected current shown in Fig. 5A 639 (second trace). The comparison with the time course 640 of the gating variables of IT (Fig. 5C, bottom trace)641 shows that the abrupt amplification of 642 hyperpolarization induced by IKir removes nearly 643 100% of the inactivation of IT (hT increases, solid 644 trace). This hyperpolarization also induces the delayed 645 activation of Ih (mh, dotted trace). After that, IKir646 decreases as the membrane potential recovers towards 647

12

the next LTS threshold, when the regenerative648 activation of IT takes place (mT

2, dashed trace).649

650

DISCUSSION651

All members of the inward rectifier potassium channel 652 family Kir2, except the Kir2.6 subunit, are expressed 653 in the central nervous system of rodents (Hibino et al. 654 2010; Karschin et al. 1996; Ryan et al. 2010). The 655 main attributed role of these channels in the brain is 656 the regulation of the cerebral blood flow by 657 controlling the microvascular tone and the 658 neurovascular coupling (Longden and Nelson 2015),659 this control is mediated by non-neuronal Kir2 660 channels. Few studies have provided direct evidence 661 for a role of Kir2.X channels on the intrinsic 662 physiology of neurons (Amarillo et al. 2014; Carr and 663 Surmeier 2007; Shen et al. 2007). In striatal 664 cholinergic interneurons, Kir2.X like currents seem to 665 play a pro-oscillatory function (Wilson 2005);666 whereas in neurons of the nociceptive pathway, in the 667 dorsal spinal horn of neonate rats, Kir2 channels 668 prevent the oscillations (Li et al. 2013). Here we used 669 conductance-based models to demonstrate the 670 theoretical basis for a pacemaker promoting function 671 of Kir2-like currents in thalamocortical neurons. To 672 verify this prediction, we recorded rodent 673 thalamocortical neurons in vitro and manipulated the 674 levels of Kir with dynamic clamp. 675

We showed that the singular negative slope 676 conductance region that characterizes the I/V curve of 677 Kir2 channels, generates bistability of the membrane 678 potential when combined with leak conductances (Fig 679 1). The linear, monotonic I/V relationship generated 680 by the background leak conductances is transformed 681 into a non-linear, non-monotonic I/V curve by effect 682 of the Kir negative slope conductance, introducing 683 instability in the membrane potential via saddle-node 684 bifurcations (Fig 1). Note that the range at which this 685 instability occurs coincides with the range of 686 membrane potential at which physiological slow 687 oscillations occur in thalamocortical neurons (see 688 below). The dynamical properties of this one-689 dimensional system (formed by dV/dt) can be 690 manipulated either by changing the injected current 691 (as typically performed during electrophysiological 692 experiments) or by changing the maximum potassium 693 leak conductance (as it occurs physiologically). Yet, 694 this is not sufficient to induce oscillations. Stable, 695 robust oscillations are produced by the interaction696 between IKir and Ih in an IKir-Ih-leaks model (Fig 2). 697 The I/V relationship of this minimal model increases 698 monotonically as the membrane potential changes 699 from negative to positive values, indicating that 700

transitions between rest and oscillations (and vice-701 versa) occur via Hopf bifurcations (Izhikevich 2005).702 Furthermore, these bifurcations are of the supercritical 703 type, in which the resting state losses stability with the 704 appearance of a stable limit cycle. 705

In the IKir-leaks model, bistability occurs via saddle 706 node bifurcations when we used either Kleak (Fig 1C)707 or current injection (Fig. 1A and B) as the bifurcation 708 parameter. When Ih is present, the model becomes a709 two-dimensional system formed by dV/dt and dmh/dt,710 and different dynamical behaviors are obtained in 711 response to either changes in Kleak or changes in 712 current injection at hyperpolarized potentials. At these 713 membrane potentials, changes in current injection in 714 either direction (increase or decrease) produce a 715 transition via a supercritical Hopf bifurcation (Fig 716 2C). As the magnitude of injected current decreases, 717 the frequency of the oscillation suddenly drops to 718 cero, imposing a lower limit of about 0.6 Hz (not 719 illustrated). This supercritical Hopf bifurcation is 720 equivalent to the supercritical Hopf bifurcation that 721 occurs with values of Kleak of 1.1 nS (Fig. 2D). In 722 contrast, for lower values of Kleak, the stable limit 723 cycle disappears via a saddle-node on invariant circle 724 bifurcation (S-NIC, Fig 2D). This admits oscillations 725 at very low frequencies (i.e. type I excitability) since726 the frequency decreases with the square root of Kleak 727 (inset in fig 2D; see also Izhikevich 2005). This 728 dynamical behavior is compatible with experimentally 729 recorded oscillations in thalamocortical neurons at 730 frequencies below 0.5 Hz (Figs 4 and 5; see also 731 Hughes et al. 2002).732

To unveil the role of IKir in a model that also includes 733 the oscillatory system provided by the activation and 734 inactivation of IT, we examined the bifurcation 735 diagrams of the IT-IKir-Ih-leaks model in the presence 736 (black diagrams) or absence (gray diagrams) of IKir.737 We analyzed the effect of using the following738 bifurcation parameters: current injection (Fig 3A), IT739 maximum permeability (pT, Fig 3B), or Kleak (Fig 3C). 740 In the three cases, slow oscillations have larger 741 amplitudes when IKir is present (black diagrams), and 742 they also take place over an extended range of the 743 bifurcation parameter. For example, the presence of 744

Kir (9 nS) diminishes the requirement of IT, since 745 strong oscillations can now be produced with pT as 746 low as 10.7 x 10–9 cm3/s (Fig 3B). Furthermore, these 747 oscillations occur under normal, physiological values 748 of Kleak (2.7 nS). This indicates that the interaction 749 between a large T current and/or a small leak 750 conductance is not the only possible mechanism for 751 intrinsic slow wave sleep oscillations in 752 thalamocortical neurons, as previously suggested753 (Crunelli et al. 2005). The dynamical behavior of the 754

13

the IT-IKir-Ih-leaks system in the presence of high Kir 755 conductance and very low leak conductance further 756 supports this notion. A high value of Kir (41nS) gives 757 rise to a S-NIC bifurcation (Fig 3D), which bestows 758 the system with the ability to oscillate at frequencies 759 below 0.5 Hz as the bifurcation point is approached at 760 very low values of Kleak ( 0.106 nS). These 761 conditions also reproduce the time course of the762 oscillations observed experimentally (compare inset in 763 Fig 3D with recording traces in Fig 4 and Fig 5). 764

Although the narrow conditions of high Kir and low 765 Kleak used to model the dynamical behavior in Fig 3D 766

reproduce the time course of the oscillations observed 767 experimentally –including the plateau potentials–,768 adding the cationic calcium dependent current ICAN is 769 required for the system to undergo slow oscillations 770 and plateau potentials on a larger parameter space (see 771 below). This setting would be more compatible with a772 highly variable, physiological scenario.773

The results obtained with the reduced computational 774 models of TC neurons predict that an increase in IKir775 induces robust slow oscillations of large amplitude in 776 thalamocortical neurons. To test this experimentally, 777 we recorded TC neurons from mouse brain slices and 778 manipulated the amount of Kir conductance by 779 dynamic clamp. In all TC neurons, the addition of 780 determined magnitudes of IKir induces subthreshold 781 oscillations at low frequencies, indicating that all the 782 cells possess the necessary and sufficient ionic 783 machinery that would interact with the added IKir to 784 produce this periodic behavior. Significantly, all TC 785 neurons have the adequate balance of IT, Ih and leak 786 currents, since no manipulation of these currents was 787 required to induce oscillations. This agrees with the 788 robustness of the mechanism evidenced by the 789 dynamical systems analysis (i.e. the large range of 790 permissive parameters in Figs 3B and C).791

In Fig 5 we provide a mechanistic explanation for the 792 oscillations induced by adding artificial IKir to TC 793 neurons from mice. Regenerative activation (unblock) 794 of IKir amplifies hyperpolarization (dashed vertical 795 lines in Figs 5A and C); this powerful 796 hyperpolarization strongly activates resonant Ih (dotted797 trace in lower graph of Fig 5C) and almost totally 798 removes inactivation of IT (hT approaches 1.0; solid799 trace in lower graph of Fig 5C); the slow activation of 800 Ih brings the membrane potential towards the point of 801 regenerative activation of highly available IT; 802 regenerative activation of IT produces the upstroke of 803 the LTS and the subsequent inactivation of this current 804 produces the repolarization of the membrane potential 805 after the burst of Na+/K+ mediated action potentials. 806 This repolarization brings the membrane potential to 807

the point where regenerative activation of IKir re-808 initiates the cycle. 809

Besides the inactivation of IT, several ionic 810 conductances participate in the repolarization of the 811 membrane potential during the down stroke of the 812 LTS (Amarillo et al. 2014; McCormick and 813 Huguenard 1992, Hughes et al. 2002; Zhu et al. 1999).814 In particular, the cationic calcium dependent current 815 ICAN is activated during the LTS producing a slowly 816 decaying afterdepolarization. The kinetics of this 817 depolarization depends on the slow time constant of 818 deactivation of ICAN (third trace in Fig 5C; Hughes et 819 al. 2002; Zhu et al. 1999). Without ICAN in our 820 simulations, it was possible to generate oscillations 821 that resemble the recorded behavior (very low 822 frequencies and plateaus) only if we use a very 823 specific set of Kir and Kleak parameters (Fig 5B). 824 Plateaus are generated in this case by the balance 825 between the weak hyperpolarizing drive provided by a 826 small IKleak and all the depolarizing conductances that 827 are active at the end of the LTS (i.e. INaleak, steady state 828 activated INaP and Ih and the window current 829 component of IT –Amarillo et al 2014–). In this 830 scenario, regenerative activation of IKir is triggered by 831 membrane potential fluctuations that reach a not-832 returning point (see Vm fluctuations following LTSs 833 in fig 5B). Including ICAN in the model reproduces –for 834 a more physiological range of IKleak magnitudes– the 835 slow frequency of the oscillations and the shape of the 836 time course of the membrane potential, especially the 837 decaying plateaus. The depolarizing drive of activated 838 ICAN counteracts IKleak-mediated hyperpolarization,839 forcing the membrane potential to follows its slow 840 deactivation kinetics. In this last scenario, regenerative 841 activation of IKir is reached smoothly at a point where 842 ICAN is sufficiently deactivated (vertical dashed lines 843 in figs 5A and 5C).844

It has been proposed that the bistability that underlies 845 intrinsic slow oscillations in TC neurons is generated 846 by the interaction of the window current component of 847 IT and the leak current (Crunelli et al. 2005; Hughes et 848 al. 2002; Hughes et al. 1999). According to this 849 model, the magnitudes of these two currents have to 850 be balanced in such a way that the I/V relationship of 851 the system becomes non-monotonic. When this852 happens, oscillations result from the alternating 853 destabilization of two stable equilibriums: one is 854 destabilized by deactivation of ICAN at depolarized 855 potentials, followed by the destabilization of the other 856 one by activation of Ih at hyperpolarized potentials 857 (Crunelli et al. 2005; Hughes et al. 1999). Here we 858 proposed an alternative scenario in which the 859 dynamical interaction between the voltage and time-860 dependent properties of IKir (fast amplifying current) 861

14

and Ih (slow resonant current) generates slow 862 oscillations even though the total steady state I/V 863 curve is monotonic. This slow system interacts with 864 the intrinsically oscillatory system formed by the 865 activation and inactivation gates of IT, increasing the 866 robustness of the oscillation.867

Delta and slow brain rhythms (< 4 Hz) have been 868 linked to changes in synaptic strength and structural 869 remodeling of dendritic spines, which are believed to 870 be the basis of memory consolidation (Wei et al. 2016;871 Yang et al. 2014). TC neurons seem to play a critical 872 role in the generation and maintenance of these 873 globally synchronized low-frequency oscillations,874 which are characteristic of unconscious states like 875 deep stages of N-REM sleep (Steriade 1997) and 876 absence seizures (Steriade and Contreras 1995). The 877 pacemaker activity of TC neurons could be controlled 878 by modulating their intrinsic excitability in a 879 behaviorally dependent manner. For this reason, great 880 attention has been placed on studying modulatory 881 mechanisms of IKleak, Ih and IT in these neurons 882 (Cheong et al. 2008; Leresche et al. 2004; McCormick 883 1992; McCormick and Pape 1990a; Yue and 884 Huguenard 2001; Zhu and Uhlrich 1998). Our results 885 suggest that the strong inward rectifier potassium 886 current IKir should be added to this list, given its pro-887 oscillatory role in TC neurons demonstrated here. 888 Several signaling cascades, including the PIP2 889 pathway via lipid kinases and the PKA and src kinase 890 pathways, are known to regulate the activity of Kir2.2 891 channels (reviewed in Hibino et al 2010). These and 892 other Kir2.2 undiscovered modulatory mechanisms 893 could probe important in the regulation of the 894 pacemaker propensity of TC neurons, and henceforth,895 in the regulation of the slow brain rhythms and their 896 physiological or pathological consequences.897

Finally, the overexpression of strong inward rectifier 898 potassium channels of the Kir2 family (see for 899 example Nitabach et al. 2002; Yoon et al. 2008) is a 900 commonly used technique for silencing a particular 901 population of neurons. Here we demonstrate that the 902 negative slope conductance region in the I/V curve of 903 these channels amplifies the hyperpolarization and 904 promotes oscillations when combined with resonant Ih.905 Therefore, caution must be taken when using this 906 technique, since overexpression of Kir channels in 907 cells that express Ih could result in a “paradoxical” 908 increase of excitability.909

910

DISCLOSURES911

The authors declare not conflicts of interest, financial 912 or otherwise.913

GRANTS914

This work was subsidized by Consejo Nacional de 915 Investigaciones Científicas y Técnicas-Argentina PIP 916 0256 and doctoral fellowship to AIT.917

918

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Ii m h mi hi p q i(nS)

Ei(mV)

IKir 1/(1+exp[(V+97.9)/9.7)] 1 0 30.6 –100IKleak 0 0 2.27 –100INaleak 0 0 0.68 0

IT1/(1+exp[(V+53 1/(1+exp[(V+75)/4])

+exp[(V+12.8)/18.2]))/1.44(exp[(V+461)/66.6])/3 for V

for V2 1 9.1*

Ih1/(1+exp[(V+82)/5.49]) 1 0 1.82 –43

IAa

m 1 = 1/(1+exp[–(V+60)/8.5])m 2 = 1/(1+exp[–(V+36)/20])

1/(1+exp[(V+78)/6]) (0.37+1/(exp[(V+35.8)/19.7]+exp[-(V+79.7)/12.7]))/1.67

hA1= (1/(exp[(V+46)/5]+exp[(–V+238)/37.5]))/1.67 for V<–63hA1= 0.735 for V>–63hA2=(1/(exp[(V+46)/5]+exp[(–V+238)/37.5]))/1.67 for V<–73hA2= 44.12 for V>–73

4 1 1248.5 –100

INaP 1/(1+exp[–(V+57.9)/6.4)] 1/(1+exp[(V+58.7)/14.2)] (1000+(10000/(1+exp[(V+60)/10])))/1.73 1 1 1.25 +45

INa

m/( m m)m=0.32(–V–37)/(exp[(–V–37)/4]–1)m=0.28(V+10)/(exp[(V+10)/5]–1)

h/( h h)h=0.128exp[(–V–33)/18]h=4/(1+exp[(–V–10)/5])

(1/( m m))/0.41m=0.32(–V–37)/(exp[(–V–37)/4]–1)m=0.28(V+10)/(exp[(V+10)/5]–1)

(1/( h h))/0.41h=0.128exp[(–V–33)/18]h=4/(1+exp[(–V–10)/5])

3 1 1135 +45

IK

m/( m m)m=0.032(–V–35)/(exp[(–V–35)/5]–1)m= 0.5exp[(–V–40)/40]

(1/( m m))/0.41m=0.032(–V–35)/(exp[(–V–35)/5]–1)m= 0.5exp[(–V–40)/40]

4 0 1135 –100

IL

1/(1+exp[(V+10)/-10]) (1/( m m))/1.32m=1.6/(1+exp[–0.072(V-5)])m= 0.02(V–1.31)/(exp[(V–1.31)/5.36]–1)

2 0 9.1*

IAHP

m/( m m)m= CaiL / 10000

for CaiL <0.0001m=0.01 for CaiL >0.0001m=0.01

1 0 1.14 –100

IKCab

m/( m m)m=2/37.95(exp[(V+50)/11–(V+53.5)/27])

for V<–10m=2exp[(–V–53.5)/27] for V>–10m=2exp[(–V–53.5)/27]– m for V<–10m=0 for V>–10

1 0 11.35 –100

ICAN

m/( m m)m=0.0004(CaiT /0.0001)8

m=0.0004

(1/( m m))/1.93 for mCAN

0.1 for mCAN <0.1m=0.0004(CaiT /0.0001)8

m=0.0004

2 0 2.27 –15

Table I. Ion channel kinetics and parameters

a. As in the original description by Huguenard and Prince 1991, IA is modeled as the sum of two currents. The current expression for IA is: IA =A(0.6 hA1mA14+0.4 hA2mA24) (V–EA). b. As for IAHP, IKCa is gated by changes in the internal concentration of Ca++ that depend on IL (CaiL, see

methods). Thus, the current equation of IKCa reflecting this Ca++ dependency is: IKCa = 0.004 KCa mKCa CaiL (V–EKCa) if 0.004 KCa < 0.0000001. Otherwise, IKCa = KCa mKCa (V-EKCa). * Permeability units for the calcium currents IT and IL are in 10-9 cm3/s.