Article Figures & Data
Figures
- Figure 1.
Pharmacological block of Ca2+-gated K+ channels. a, A somatic action potential measured from a L5 pyramidal neuron before (black trace) and after (red trace) adding apamin (100 nm) to the bath solution. b, The relationship between apamin concentration and inhibition of the repolarizing velocity of the action potential plotted on a logarithmic concentration scale (n ≥ 4). The repolarizing velocity was normalized to the minimal value. The data were fitted with the Hill equation (solid line), giving an IC50 of 30.82 ± 5.89 nm. Error bars indicate normalized SEM. c, A somatic action potential measured from a L5 pyramidal neuron before (black trace) and after (red trace) adding iberiotoxin (30 nm) to the bath solution. d, A somatic action potential measured from an L5 pyramidal neuron before (black trace) and after (red trace) adding TEA (1 mm) to the bath solution. e, Comparison of the repolarizing velocity of the action potential in the different bath solutions: control solution (ACSF, n = 9), solution containing TEA (1 nm, n = 4); and solution containing iberiotoxin (30 nm, n = 4). The asterisk indicates a significant difference (p < 0.00005, one-tailed t test) between the control solution and the solution containing TEA. The double asterisk indicates a significant difference (p < 0.00005, one-tailed t test) between the control solution and the solution containing iberiotoxin. Error bars indicate SEM. f, The relationship between TEA concentration and inhibition of the repolarizing velocity of the action potential plotted on a logarithmic concentration scale (n = 4) as in b. The fit gave an IC50 of 0.58 ± 0.046 mm. Errors bars indicate normalized SEM.
- Figure 2.
Application of the parameter peeling procedure to recordings from the apical dendrite of L5 pyramidal neurons. a, Reconstruction of a L5 pyramidal neuron stained with biocytin illustrating electrode placement. b, Constraining passive membrane parameters and channel density gradients of voltage-gated K+, Na+, Ca2+, and Ih channels (left), and Ca2+-gated K+ channels (right). Hyperpolarizing and depolarizing membrane potential traces were recorded at the soma (black traces) and at 415 μm along the apical dendrite (red traces). Ca2+-gated K+ channels were blocked by apamin (200 nm) and TEA (1 mm). The dashed blue traces show the membrane potential traces simulated at the dendrite using the best parameter set obtained by the genetic algorithm. Table 1 (cell 1) shows the final parameter set obtained from this optimization. c, In a simulation, the somatic membrane potential of this cell was clamped to an experimental waveform series of four APs at 41 and 157 Hz that were recorded from cell 1 (black traces; Table 1). The membrane potential traces recorded at 415 μm along the dendrite are shown in red. The dashed blue traces represent the membrane potential traces simulated at the dendrite using the best parameter set obtained by the genetic algorithm.
- Figure 3.
Conductance and permeability gradients in the optimized model. a, Passive parameters obtained by the peeling procedure in five experiments. b, The average dendritic conductance density of GNa (pink line), GKf (blue line), GKs (red line), and GIh (gray line). c, The average dendritic permeability density of PMVA (solid purple line). The dotted purple lines are the 90% confidence limits of the average. Inset, The average dendritic permeability density of PHVA (solid purple line). The dotted orange lines are the 90% confidence limits of the average. The dendritic permeability gradient of CaMVA was calculated using the following equation: PMVA(x) = PMVA,soma + PMVA,dend(exp ((−(x − CaMVA,dist)/CaMVA,width)2)) d, The average dendritic permeability density of PHVA (solid orange line). The dotted orange lines are the 90% confidence limits of the average. e, The average dendritic conductance density of GSK (solid dark blue line). The dotted dark blue lines are the 90% confidence limits of the average. f, The average dendritic conductance density of GBK (solid green line). The dotted green lines are the 90% confidence limits of the average.
- Figure 4.
Initiation of a local dendritic regenerative spike. a, Reconstruction of an L5 pyramidal neuron stained with biocytin illustrating electrode placement (left). Right, Simulated somatic and dendritic voltage responses to 50 ms current injection (0.6–1 nA) through the dendritic pipette at 600 μm, as illustrated on the left. A local dendritic spike was evoked by a 1 nA current. The optimized model used for this simulation was cell 5 (Table 1). b, The activation of the eight ion channels inserted into the model during the dendritic regenerative potential (black trace) simulated in a. c, The simulated threshold for dendritic potential (●) and somatic AP (○) is plotted as a function of the distance from soma. The solid lines are an exponential fit. To obtain a simulated somatic AP, an artificial axon was added to the model; the dendritic Na+ conductance density was divided by 10 to simulate the passive forward propagation toward the soma.
- Figure 5.
Conductance and permeability activation during a low-frequency series of backpropagating action potentials. In a simulation, the somatic membrane potential was clamped to a waveform derived from a series of four APs with a firing rate of 39 Hz (black lines) recorded from cell 5 (Table 1). The changes in the membrane potential, the conductance of the six voltage-gated ion channels (Na+, Kf, Ks, Ih, SK, and BK), and the permeability of the two voltage-gated Ca2+ channels (CaHVA and CaMVA) included in the model are shown at the soma and at 200, 400, 600, and 800 μm along the apical dendrite. The traces are color coded according to the color of the simulated pipettes shown on the left.
- Figure 6.
Conductance and permeability activation during a high-frequency series of backpropagating action potentials. In a simulation the somatic membrane potential was clamped to a waveform consisting of a series of four APs with a firing rate of 147 Hz (black traces) recorded from cell 5 (Table 1). The changes in the membrane potential, the conductance of the six voltage-gated ion channels (Na, Kf, Ks, Ih, SK, and BK), and the permeability of the two voltage-gated Ca2+ channels (CaHVA and CaMVA) included in the model are shown at the soma and at 200, 400, 600, and 800 μm along the apical dendrite. The traces are color coded according to the simulated pipettes shown on the left.
- Figure 7.
Intracellular Ca2+ concentration during low- and high-frequency stimulation. In a simulation, the somatic membrane potential was clamped to a waveform comprising a series of four APs with a firing rate of 39 Hz (left) and 147 Hz (right) (black traces) recorded from cell 5 (Table 1). The parameter Cai in the optimized model was used to track the intracellular Ca2+ concentration changes that are shown at the soma and at 200, 400, 600, and 800 μm along the apical dendrite. The units of the intracellular Ca2+ concentration are millimolar due to the absence of a Ca2+ buffer in the model. The traces are color coded according to the simulated pipettes shown on the left.
- Figure 8.
Reproducing BAC firing. a, Reconstruction of an L5 pyramidal neuron stained with biocytin, illustrating electrode placement. b, EPSP-like current of 1.6 nA (rising τ = 2 ms, declining τ = 10 ms) was injected through the gray pipette (800 μm, bottom) into a dendrite of one of the optimized models (Table 1, cell 5). The simulated voltage response showed the shape of an EPSP at the soma (black trace) and at the apical dendrite (400 μm, blue trace; 600 μm, red trace). c, Injection of a square current of 0.5 nA through the somatic pipette (black, bottom). The action potential generated at the soma backpropagated along the apical dendrite. d, The combination of the two stimuli used in b and c with a time interval of 5 ms generated a BAC firing at the distal apical dendrite (red trace). e, EPSP-like current injection of 3 nA (rising τ = 2 ms, declining τ = 10 ms) through the gray pipette (800 μm, bottom) was sufficient to generated a Ca2+ spike at the distal apical dendrite (red trace).
- Figure 9.
The timing of BAC firing. Top, EPSP-like current of 0.6 nA (rising τ = 1 ms, declining τ = 5 ms) was injected at the distal dendrite (800 μm) into a dendrite of one of the optimized models (Table 1, cell 5). The simulated voltage response showed the shape of an EPSP at the soma (black trace) and at the apical dendrite (600 μm, red trace). Middle, Injection of a square current of 0.5 nA through the somatic pipette (black, bottom). The action potential generated at the soma backpropagated along the apical dendrite. Bottom, Combining the two stimuli used with a different time interval (a, −10 ms; b, 7 ms; c, 10 ms). A Ca2+ spike was simulated using a time interval of 7 ms. Δt was calculated as the time between the start of the somatic injection and the start of the dendritic injection.
- Figure 10.
Conductance and permeability activation during BAC firing. a, Reconstruction of an L5 pyramidal neuron stained with biocytin illustrating electrode placement. b, The activation of seven ion channels inserted into the model during the BAC firing simulated in Figure 5d at 600 μm from the soma. Left, Activation of GNa (pink trace) and GKf (blue trace) conductances. Middle, Activation of PHVA (orange trace) and PMVA (purple trace) permeabilities. Right, Activation of GKs (red trace), GBK (green trace), and GSK (dark blue trace) conductances. c, The activation of seven ion channels inserted into the model during the BAC firing simulated in Figure 5d at 470 μm from the soma. Left, Activation of GNa (pink trace) and GKf (blue trace) conductances. Middle, Activation of PHVA (orange trace) and PMVA (purple trace) permeabilities. Right, Activation of GKs (red trace), GBK (green trace), and GSK (dark blue trace) conductances.
- Figure 11.
Conductance and permeability activation during a complex Ca2+ spike. a, Reconstruction of a filled L5 pyramidal neuron stained with biocytin, illustrating electrode placement. b, The activation of seven ion channels inserted into the model during the dendritic Ca2+ spike simulated in Figure 5e at 600 μm from the soma. Left, Activation of GNa (pink trace) and GKf (blue trace) conductances. Middle, Activation of PHVA (orange trace) and PMVA (purple trace) permeabilities. Right, Activation of GKs (red trace), GBK (green trace), and GSK (dark blue trace) conductances. c, The activation of seven ion channels inserted into the model during the dendritic Ca2+ spike simulated in Figure 5e at 470 μm from the soma. Left, Activation of GNa (pink trace) and GKf (blue trace) conductances. Middle, Activation of PHVA (orange trace) and PMVA (purple trace) permeabilities. Right, Activation of GKs (red trace), GBK (green trace), and GSK (dark blue trace) conductances.
- Figure 12.
Modulation of the dendritic spike propagation. a, Reconstruction of a filled L5 pyramidal neuron stained with biocytin, illustrating electrode placement. b, A dendritic spike changed to BAC firing. b1, An EPSP-like current of 2 nA (rising τ = 2 ms, declining τ = 10 ms) was injected through the gray pipette (800 μm) into a dendrite of one of the optimized models (Table 1, cell 5). The simulated voltage response showed the shape of an EPSP at the soma (black trace) and at the apical dendrite (300 μm, blue trace; 600 μm, red trace). b2, A small depolarizing current at the proximal dendrite (300 μm, blue electrode) was injected (0.2 nA, 50 ms, onset 30 ms before the EPSP-like current). The dendritic spike converted to a Ca2+ spike at the distal dendrite (red trace). c, A dendritic AP changed to a dendritic spike. c1, An EPSP-like current injection of 3 nA (rising τ = 2 ms, declining τ = 10 ms) through the gray pipette (800 μm, bottom) generated a dendritic potential at the distal apical dendrite (red trace) that propagated to the soma (black trace). c2, Hyperpolarizing current (−0.4 nA, 50 ms, onset 30 ms before the EPSP-like current) was injected into the proximal dendrite (300 μm, blue electrode). The hyperpolarizing current caused the dendritic AP to convert to a dendritic spike (red trace).
- Figure 13.
Simulated NMDA spikes within and between dendrite branches. a, Reconstruction of a layer 5 pyramidal neuron stained with biocytin, illustrating electrode placement. b, A NMDA-like spike was evoked through two distal pipettes (green and orange) after inserting NMDA and AMPA synapses (Larkum et al., 2009) into one of the optimized models (Table 1, cell 5). Left, Integration within branches. First, the distal pipettes (green and orange) were activated separately (black traces) and then simultaneously (red traces). The EPSPs were simulated 150 μm below the orange pipette. The blue trace represents the summation of the two individual activations. The summation is shown for weak, medium, and strong stimuli. The distance between the two pipettes at the same branch was 26 μm. Right, Integration between branches. First, the distal pipettes (gray and orange) were activated separately (black traces) and then simultaneously (red traces). The EPSPs were simulated 150 μm below the orange pipette. The summation is shown for weak, medium, and strong stimuli. c, Expected versus actual simulated EPSP intervals are plotted for a range of stimulus intensities for summation within the branch (red line) and between branches (blue line).
Tables
- Table 1.
Results of fitting the model to the data using a linear gradient for the MVA Ca2+ channel
Parameter Unit Lower boundary Upper boundary Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Average SD CV Distance μm 415 580 417 445 413 454 72 Ra Ωcm 90 230 106 90 70 91 120 96 19 0.20 Rm Ωcm2 3000 42,000 27,673 17,141 12,569 8374 25,812 18,314 8322 0.45 Cm μF/cm2 0.5 1.8 0.79 0.59 0.81 0.60 0.60 0.68 0.11 0.16 Epassive mV −50 −30 −49.1 −47.6 −49.9 −46.8 −47.8 −48.3 1.3 0.03 GIh,dend pS/μm2 90 210 92 114 92 159 118 115 28 0.24 Ih,X1/2 μm 350 650 385 498 407 452 352 419 57 0.14 Ih,slope 1/μm −0.2 −0.01 −0.011 −0.016 −0.019 −0.021 −0.014 −0.016 0.004 0.26 GIh,soma pS/μm2 1 3.5 1.31 3.49 1.09 2.77 2.51 2.24 1.01 0.45 GKs,dend pS/μm2 3 15 4.79 3.65 14.46 3.93 3.79 6.13 4.68 0.76 Ks,slope 1/μm −0.1 −0.02 −0.071 −0.060 −0.085 −0.081 −0.092 −0.078 0.012 0.16 GKs,soma pS/μm2 60 290 163 224 251 258 206 220 38 0.17 GKf,dend pS/μm2 5 30 24 31 35 21 28 28 6 0.20 Kf.slope 1/μm −0.1 −0.005 −0.020 −0.008 −0.030 −0.020 −0.012 −0.018 0.009 0.48 GKf,soma pS/μm2 70 320 248 298 248 347 332 294 46 0.16 GNa,soma pS/μm2 150 600 222 507 187 150 352 284 146 0.52 GNa,dend pS/μm2 65 150 148 66 57 79 56 81 38 0.47 Nadist μm 400 700 669 695 584 489 481 583 99 0.17 Nashift,act mV −11 −9 −10.2 −10.9 −9.3 −9.0 −11.0 −10.1 0.9 0.09 Nashift,inact mV −11 −9 −11.0 −9.9 −10.8 −10.0 −9.6 −10.2 0.6 0.06 PHVA,soma μm/s 0 500 11.9 31.0 151.1 139.4 0.9 66.9 72.5 1.08 PHVA,dend μm/s 0 500 0.06 1.65 2.48 1.17 1.56 1.39 0.88 0.64 CaHVA,dist μm 0 990 145 44 101 13 10 63 59 0.93 CaHVA,shift,act mV −13 10 −10.1 −11.8 −8.6 −9.3 −4.5 −8.8 2.7 0.31 CaHVA,shift,inact mV −13 10 1.9 7.0 3.5 −1.1 −7.1 0.8 5.3 6.40 PMVA,soma μm/s 0 200 0.05 108.3 81.5 140.8 31.5 72.4 56.9 0.79 PMVA,dend μm/s 0 200 11.3 10.1 30.3 10.1 4.9 13.3 9.8 0.73 CaMVA,dist μm 0 990 214 602 609 88 925 487 337 0.69 CaMVA,shift,act mV −10 10 −9.8 −9.8 −2.9 −9.9 −9.7 −8.4 3.1 0.36 CaMVA,shift,inact mV −15 10 0.6 −5.0 4.9 −11.2 −2.1 −2.6 6.1 2.37 GSK,soma pS/μm2 0 4 1.85 3.33 3.62 3.25 3.18 3.05 0.69 0.23 GSK,dend pS/μm2 0 4 0.36 1.23 1.10 0.06 0.52 0.65 0.50 0.76 SKdist μm 0 900 401 580 677 367 239 453 175 0.39 GBK,soma pS/μm2 0 4 3.54 1.29 0.93 3.23 0.64 1.93 1.36 0.70 GBK,dend pS/μm2 0 4 0.53 3.51 3.87 0.13 1.23 1.85 1.73 0.93 BKdist μm 0 900 16 104 269 4 28 84 111 1.31 Parameter values obtained by constraining the model of the apical dendrite of L5 pyramidal neurons using five experimental datasets. The distance of the dendritic pipette from the soma is noted in the top row. The average parameter value (n = 5), the SD, and the CV are presented for each parameter. The passive parameters are the Rm, the axial resistance (Ra), the membrane capacitance (Cm), and the passive reversal potential (Epassive). The dendritic conductance gradient of Ih was calculated using GIh(x) = GIh,soma + GIh,dend/(1 + exp(Ih,shope(x−Ih,x1/2))) (where x stands for the distance from the soma along the apical dendrite). The dendritic conductance gradient of Ks was calculated using the exponential equation GKs(x) = GKs,dend + FKs,soma(exp(Ks,slope · x)). This formula was also used to calculate the dendritic conductance gradient of Kf. The dendritic conductance gradient of Na+ was calculated using GNa(x) = GNa,soma + x(GNa,dend − GNa,soma)/Nadist. This formula was also used to calculate the dendritic permeability gradient of CaHVA and CaMVA, and the dendritic conductance gradient of SK and BK.
