Table 1.
The modeled C. elegans ion channels (second column), grouped on the basis of ion selectivity (first column) are classified according to vertebrate homologues (third column) and corresponding nematode genes (fourth column). The fifth and the sixth columns report the gene expression in AWCON and RMD. The seventh column reports the references to experimental data used to model each channel type and the relative organism. The symbols denote the specific patch clamp recording conditions: • on dissected worm cultured myocytes [25], body wall muscles [15] or pharynx [36], *, ∘ and ⋄ respectively on Xenopus oocytes, HEK cells, and CHO cells expressing the desired genes. In the text, the genes encoding for the modeled channels and the corresponding channel proteins are reported in italic and capital letters, respectively (e.g egl-19 genes encode for EGL-19 channels). Further, we denote the current associated to a specific class of channels with the channel protein name, omitting the dash between the gene family name and the number (e.g. EGL19 are the currents carried by EGL-19 channels). When two or more genes encode for a channel type, we indicate the channels and the currents with the gene family name in capital letters (e.g. kcnl-1/4, KCNL).
Fig 1.
Simulated currents of the voltage-gated potassium channels SHL-1, KVS-1, SHK-1, IRK-1/3, KQT-3, EGL-36 and EGL-2.
Potassium currents present two different timescales for activation: some of them have fast activation (panels A-D, ), others have slow activation (panels E-G, ∼100 ms
). Stimulation protocol (sketched in panel H) consists in 10 mV voltage steps, from -120 mV to 40 mV. Each step lasts 600 ms for fast activating currents (panels A-D) and 6 s for slow activating ones (panels E-G). The holding potential is assumed equal to potassium reversal potential, i.e. Vh = −80 mV. The currents are expressed in pA (panels A-D) or nA (panels E-G), depending on the experimental data used as reference. The simulated currents in panels A-G are given by Eqs A5, A9, A14, A30, A21, A27, and A24 in S1 File, respectively, in which the conductances are assumed as follows:
,
,
,
,
,
, and
. A) SHL1 current. Experimental data from [25]. Activation time constant is
at V = −10 mV (Eq A2 in S1 File, S1 Fig). Current inactivation is described by the sum of two exponential functions, with time constants
and
at V > −10 mV for fast and slow components (Eq A4 in S1 File, S1 Fig). B) KVS1 current. Experimental data from [31]. KVS1 current activates significantly at positive potentials, with a time constant
at 0 mV (Eq A8 in S1 File, S1 Fig). The current inactivates exponentially with a time constant that is a decreasing function of voltage (
at 0 mV, S1 Fig). C) SHK1 current. Experimental data for steady-state activation and inactivation variables from [25], and from [29] for time constants. SHK1 current shows fast activation for depolarizing potentials (
at 0 mV, Eq A11 in S1 File), followed by exponential inactivation with a voltage independent time constant
(Eqs A12 and A13 in S1 File, S2 Fig). D) IRK current. Experimental data from [33–35]. The activation kinetics is described by a Boltzmann function (Eq A28 in S1 File) that is almost null at positive potentials and saturates at very negative potentials (V < −100 mV) (S2 Fig). The activation time constant voltage dependence is described by an asymmetric bell-shaped curve (Eq A29 in S1 File) with a peak of 8 ms at ∼-40 mV, and decreases to values below 4 ms for positive voltages (S2 Fig). E) KQT3 current. Experimental data from [26]. KQT3 current is characterized by the sum of two exponential functions with fast (
at V ∼ −20 mV, Eq A16 in S1 File) and slow (
at V ∼ −20 mV, Eq A17 in S1 File) activation components that account respectively for 30% and 70% of the total current (see S1 Fig). Inactivation is described by Eq A18 in S1 File [73]. F) EGL36 current. Experimental data from [30]. The current activates at high voltages V > 10 mV (S2 Fig) with three almost equally-weighted components at fast (13 ms), medium (63 ms) and slow (355 ms) timescales. Steady-state activation variable (Eq A25 in S1 File, S2 Fig) is shifted towards lower potentials (V0.5 = 6.9 mV) compared to SHL1, SHK1 and KVS1. G) EGL2 current. Experimental data from [28]. EGL2 current is slow activating (
) and non-inactivating. The activation function is described by a Boltzmann function with slow rise (ka = 14.9 mV) between -40 mV and 40 mV (Eq A22 in S1 File, S2 Fig). Activation time constant is weakly dependent on voltage, becoming almost constant for the considered voltages (Eq A23 in S1 File, S2 Fig). H) Simulation protocol. 10 mV voltage steps, range -120 mV to 40 mV, step duration 600 ms or 6 s.
Fig 2.
Simulated currents of voltage-gated calcium channels EGL1-9, UNC-2, and CCA-1.
Calcium currents can be classified according to their activation voltage level: EGL19 and UNC2 currents activate at high membrane potential (V > ∼ − 40 mV), while CCA1 currents start to activate at low voltages (V ∼ −70 mV) [15, 20, 36, 37, 75]. Stimulation protocol (sketched in panel D) consists in 10 mV voltage steps ranging from -80 mV to 40 mV. Each step lasts 200 ms, and is applied from a holding potential Vh = −80 mV. The reversal potential for calcium is VCa = 60 mV. EGL19, UNC2, and CCA1 currents are given by Eqs B5, B10, and B14 in S1 File, respectively, in which ,
, and
(conductance values are chosen to match the currents of reference experimental data). The currents are expressed in nA, depending on the experimental data used as reference. A) EGL19 current. Experimental data for steady-state activation and inactivation variables from [15] and for activation and inactivation time constants from [36]. EGL19 current activates rapidly (
at 0 mV) at high voltage (V > −30 mV, see S3 Fig). The voltage dependence of the activation time constant is described by the sum of two Gaussian functions with shifted centers (Eq B2 in S1 File and S3 Fig), as in [76]. The steady-state inactivation function has a U-shape, with a minimum of about 0.5 at 0 mV (Eq B3 in S1 File and S3 Fig). The inactivation time constant voltage dependence is described by the sum of two sigmoids (Eq B4 in S1 File and S3 Fig), as in [76]. B) UNC2 current. Experimental data for steady-state activation and inactivation curves from [37], and for activation and inactivation time constants from [38] and [39], respectively. UNC2 current starts to activate at voltages slightly lower than in the case of EGL19 (V0.5 ∼ −10 mV and ∼6 mV for UNC2 and EGL19, respectively), with a steady-state activation function steeper than EGL19 (ka ∼ 4 mV and ∼8 mV for UNC2 and EGL19, respectively) (Eq B6 in S1 File and S3 Fig). The current shows fast activation with time constant voltage dependence described by a bell-shaped curve (Eq B7 in S1 File and S3 Fig) with a maximum value
at around -10 mV. Inactivation time constant is described by two sigmoids with ∼90 ms
(Eq B9 in S1 File, and S3 Fig). C) CCA1 current. Experimental data from [40]. CCA1 current exhibits a fast activation with a time constant
for V > −40 mV, followed by an inactivation with a time constant
for V > −30 mV. The current activates at more negative potentials than UNC2 and EGL19 currents (V0.5 ∼ −60 mV) (Eq B11 in S1 File and S3 Fig), and the steady-state activation and inactivation curves overlap between -70 mV and -30 mV giving rise to a sustained inward current, named window current (S3 Fig). D) Stimulation protocol. 10 mV voltage steps, range -80 mV to 40 mV, step duration 200 ms.
Fig 3.
Simulated currents of calcium-regulated potassium channels SLO-1, SLO-2 and KCNL.
Stimulation protocol consists in 10 mV voltage steps ranging from -80 mV to -40 mV. Each step lasts 600 ms, and is applied from a holding potential Vh = −80 mV. For both SLO1 and SLO2 we set the conductance to 15 nS (Eq 1). For KCNL current the conductance is . The modified set of parameters, i.e. fine-tuned to match neuron data (see S1 Table), has been used for CaV channels dynamics. A-B) SLO1 (A) and SLO2 (B) coupled with UNC-2 Ca2+ channels. SLO1 and SLO2 simulated currents show fast activation (
, and
, Eq 22). Current activation is followed by inactivation shaped by the inactivation variable of the coupled UNC2 current (Eq 23). C-D) SLO1 (C) and SLO2 (D) coupled with EGL-19 Ca2+ channels. Analogously to panels A-B the currents show fast activation, while partial inactivation is strictly related to the limited inactivation of EGL19 calcium current (Eq 23, Fig 2A and S3 Fig). E) KCNL current. Experimental data from [67, 68]. KCNL channels exhibit slow activation (
) and inactivation. The steady-state activation function depends only upon intracellular calcium concentration (Eq C6 in S1 File and S5 Fig). F) Simulated intracellular calcium concentration.
during voltage clamp simulation is calculated by solving Eq 28, where ICa is given by IEGL19 + IUNC2 + ICCA1 with
.
Table 2.
Ionic conductances for AWCON and RMD neurons.
We use VK = −80 mV, VCa = 60 mV and VNa = 30 mV as the reversal potentials for potassium, calcium and sodium currents, respectively. In AWCON the reversal potential for leakage current (Eq C10 in S1 File) is VL = −90 mV; in RMD it is VL = −80 mV.
Fig 4.
AWCON response to current and voltage stimuli.
A) Voltage clamp simulation of the AWCON neuron. Experimental data from [59]. To test the model we apply to the in silico neuron the same voltage clamp stimulation protocol of experimental whole-cell recordings [59]. The protocol consists in voltage steps ranging between -110 mV and +110 mV with 20 mV increments. The holding potential is Vh = −70 mV, and the step duration is 100 ms. B) AWCON steady-state I-V relation. Comparison between experimental (red, data from [59]) and simulated (black) steady-state I-V curves. The simulated I-V curve is computed by averaging the currents in the last 5 ms of each voltage step, as in [59]. C) Current clamp simulation of the AWCON neuron. No published current clamp experimental data were found for AWCON. The in silico neuron shows active behavior when the injected current is above 6 pA (S6 Fig). The voltage response is characterized by an upstroke with a duration of ∼30 ms for Iext > 15 pA, followed by a plateau phase. The plateau height increases linearly with the stimulus amplitude (S6 Fig). A sensitivity analysis performed by varying the calcium removal rate τCa (Eq 28 and S6 Fig) do not show significant changes in the results. Current clamp stimulation protocol consists in 6 current steps ranging from -4 to 20 pA, with a duration of 500 ms. The holding current is Ih = 0 pA. The cell capacitance is 3.1 pF [59], that corresponds to 1.3μF/cm2 when scaled on the entire cell surface (Scell = 238.16μm2,from Neuromorpho.org). This value is in agreement with specific membrane capacitance reported for ASER [57], and with the one calculated for AIY by considering its total capacitance (∼0.7 pF [78]) and surface (65.89 μm2, from Neuromorpho.org).
Fig 5.
In silico analysis of AWCON voltage response to current injection.
The stimulation protocol consists in one single 15 pA current step with a duration of 500 ms. The holding current is Ih = 0pA. A) WT Voltage response to 15 pA current injection. B-F) Normalized conductance for modeled currents. In these panels we report the time evolution of the normalized conductance of each modeled current for the WT current clamp simulation shown in panel A. The normalized conductance is defined as the product of the activation and inactivation variables of the considered ion current (e.g. for SHL1 ). Panels B and C show
for voltage-gated K+ currents, panel D for voltage-gated Ca2+ currents, and panels E and F for calcium-dependent K+ currents. The
values for CCA1, UNC2 and EGL19 currents are multiplied by -1 to reproduce the sign of the associated currents.
Fig 6.
In silico AWCON knockouts voltage response to current injection.
The stimulation protocol consists in a single 15 pA current step with a duration of 500 ms. The holding current is Ih = 0pA. Each knockout is obtained by suppressing the contribution of the selected current, leaving unchanged the other conductances. In panels A-F we report the response of NCA (A), voltage-gated potassium channels (B and C), voltage-gated calcium channels (D), and calcium-regulated potassium channels (E and F) knockouts.
Fig 7.
In silico RMD neuron response to voltage and current stimuli.
A) Experimental current clamp. Experimental data from [13]. B) Simulated current clamp for the RMD neuron. The stimulation protocol, sketched in the middle column, consists in 4 current stimuli from -2 pA to 10 pA, followed by a negative step of 15 pA. The first step duration is 50 ms, while the duration of the negative step is 20 ms. The interval between the first and the second step is 50 ms. The holding current is Ih = 0pA. RMD shows a threshold-like response, characterized by large excursions of membrane voltage for Iext > 6 pA, followed by a sustained plateau phase. A sensitivity analysis performed by varying the calcium removal rate τCa (S7 Fig) does not show significant changes in the results. C) Simulated voltage clamp for RMD neurons. The stimulation protocol (sketched on the right) consists in a series of voltage steps from -120 mV to +60 mV with 15 mV increments. The holding potential is Vh = −70 mV and the step duration is 100 ms. D) RMD steady-state I-V relation. I-V curve is computed by averaging the current in the last 5 ms of the voltage step. The stimulation protocol consists in a series of 90 voltage steps between -120 mV and 60 mV. The step duration is 1200 ms and the holding potential is Vh = −70mV. The cell capacitance is set to 1.2 μF to match membrane time constant as observed from experimental current clamp data [13], i.e. to fit the rise time of the membrane potential in response to current stimulation.
Fig 8.
In silico analysis of RMD voltage response to current injection.
The stimulation protocol consists in a single 10 pA current step with a duration of 50 ms. The holding current is Ih = 0 pA. A) WT Voltage response to 10 pA current injection. B-F) Normalized conductance for modeled currents. In these panels we report the time evolution of the normalized conductance of each modeled current for WT current clamp simulation shown in panel A. The normalized conductance is defined as the product of the activation and inactivation variables of the considered ion current. Panels B and C show for voltage-gated K+ currents, panel D for voltage-gated Ca2+ currents, and panels E and F for calcium-dependent K+ currents. The
values for CCA1, UNC2 and EGL19 currents are multiplied by -1 to reproduce the sign of the associated currents.
Fig 9.
In silico RMD knockouts voltage response to current injection.
The stimulation protocol consists in a single 10 pA current step with a duration of 50 ms. The holding current is Ih = 0 pA. Each knockout is obtained by suppressing the contribution of the desired current, leaving unchanged the other conductances. In panels A-F we report the response of NCA (A), voltage-gated potassium channels (B and C), voltage-gated calcium channels (D), and calcium-regulated potassium channels (E and F) knockouts.
Fig 10.
Analysis of the bistable behavior of RMD neuron.
A) Steady-state I-V curves at varying . Steady-state I-V curves are obtained from the currents computed during a voltage clamp stimulation with steps ranging between -90 mV and -10 mV, with 1 mV increments. The value of
is varied between 0.5 nS and 5 nS with 0.5 nS increments. B) Calcium channels knockouts steady-state I-V curves. Voltage clamp simulation follows the same protocol of panel A. Knockout I-V curves are computed by removing the contribution of one of the CaV from the total current, leaving unchanged the other conductances. C) Bifurcation diagram with
as bifurcation parameter. The black empty triangle represents the saddle-node (or limit point LP-H) bifurcation point [V,
] = [-59.5 mV, 1.14 nS]. At
the system exhibits three fixed points: two stable (Vr and Vs) and one unstable (Vu). D)-E) Steady-state I-V curves for different values of
. The passive conductance is varied between 0.04 nS and 0.9 nS to obtain the steady-state I-V curve in correspondence of the different regimes highlighted also by the bifurcation diagram analysis (panel F). The applied voltage stimuli ranges from -90 mV to -30 mV with 0.8 mV increments, in panel D, and from -35 mV to +5 mV with 0.8 mV increments, in panel E. Step duration 1300 ms, holding potential Vh = −70 mV. F) Bifurcation diagram with
as bifurcation parameter. For 0.25 nS
two stable solutions are separated by an unstable one. These correspond to Vr, Vs and Vs in panel C. At lower values of
, and by decreasing the parameter, two unstable solutions
and
arise from a LP bifurcation (LP-H), with
. By decreasing the control parameter,
gains stability through a subcritical Hopf bifurcation (HB-H), while at lower values of the control parameter, Vs loses stability through a second subcritical Hopf bifurcation (HB-L). Finally, by further lowering the parameter,
and Vs collide in a second fold bifurcation (LP-L) and
remains the only stable solution at even lower values of
. A second bistable regime may occur in between the two Hopf bifurcations.
Fig 11.
Bifurcation diagram in the plane .
Fold (LP) and Hopf (HB) bifurcation curves are highlighted in red and blue respectively, while black and green curves denote fold and torus bifurcations of cycles (LP cycle and TR, respectively), i.e., periodic solutions of the model. A) Two-dimensional bifurcation diagram at low values. At low values of
we recover the one-dimensional bifurcation diagram described in Fig 10, and within the red area enclosed by HB-L and HB-H the system shows bistabilty, with Vs and
as stable non-periodic states (nodes). At increasing values of
two unstable cycles originate from a fold bifurcation of cycles (LP cycle) one of which collides with
in the HB-H bifurcation at increasing values of
. In the very proximity of the fold bifurcation of cycles, the larger periodic solution gains stability via a torus bifurcation (TR). In between TR and HB-H bifurcations, the system shows a different kind of bistability, characterized by the coexistence of
and a stable periodic solution (gray area). At higher values of
, on the right of HB-H, only the oscillatory solution survives (blue area), which eventually disappears through the HB-L bifurcation. White areas in the diagram denote the presence of a unique stable steady-state,
and Vs at low and high values of
, respectively. B) Two-dimensional bifurcation diagram at high
values. Below the LP bifurcation at low voltage (LP-L, red dashed line), there is a unique stable state Vs, while after the bifurcation curve a second stable state Vr appears, together with an unstable state Vu. Further increasing
, the stable state Vs coalesces with Vu in a second LP bifurcation at high voltage (LP-H, red continuous line), making the system collapse in the unique stable state Vr. The bistability regime is guaranteed within the red area.