A quantitative description of dendritic conductances and its application to dendritic excitation in layer 5 pyramidal neurons

Abstract

Postsynaptic integration is a complex function of passive membrane properties and nonlinear activation of voltage-gated channels. Some cortical neurons express many voltage-gated channels, with each displaying heterogeneous dendritic conductance gradients. This complexity has hindered the construction of experimentally based mechanistic models of cortical neurons. Here we show that it is possible to overcome this obstacle. We recorded the membrane potential from the soma and apical dendrite of layer 5 (L5) pyramidal neurons of the rat somatosensory cortex. A combined experimental and numerical parameter peeling procedure was implemented to optimize a detailed ionic mechanism for the generation and propagation of dendritic spikes in neocortical L5 pyramidal neurons. In the optimized model, the density of voltage-gated Ca(2+) channels decreased linearly from the soma, and leveled at the distal apical dendrite. The density of the small-conductance Ca(2+)-activated channel decreased along the apical dendrite, whereas the density of the large-conductance Ca(2+)-gated K(+) channel was uniform throughout the apical dendrite. The model predicted an ionic mechanism for the generation of a dendritic spike, the interaction of this spike with the backpropagating action potential, the mechanism responsible for the ability of the proximal apical dendrite to control the coupling between the axon and the dendrite, and the generation of NMDA spikes in the distal apical tuft. Moreover, in addition to faithfully predicting many experimental results recorded from the apical dendrite of L5 pyramidal neurons, the model validates a new methodology for mechanistic modeling of neurons in the CNS.

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 I_h 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 G_Na (pink line), G_Kf (blue line), G_Ks (red line), and G_Ih (gray line). c, The average dendritic permeability density of P_MVA (solid purple line). The dotted purple lines are the 90% confidence limits of the average. Inset, The average dendritic permeability density of P_HVA (solid purple line). The dotted orange lines are the 90% confidence limits of the average. The dendritic permeability gradient of Ca_MVA was calculated using the following equation: P_MVA(x) = P_MVA,soma + P_MVA,dend(exp ((−(x − Ca_MVA,dist)/Ca_MVA,width)²)) d, The average dendritic permeability density of P_HVA (solid orange line). The dotted orange lines are the 90% confidence limits of the average. e, The average dendritic conductance density of G_SK (solid dark blue line). The dotted dark blue lines are the 90% confidence limits of the average. f, The average dendritic conductance density of G_BK (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⁺, K_f, K_s, I_h, SK, and BK), and the permeability of the two voltage-gated Ca²⁺ channels (Ca_HVA and Ca_MVA) 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, K_f, K_s, I_h, SK, and BK), and the permeability of the two voltage-gated Ca²⁺ channels (CaH_VA and Ca_MVA) 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 Ca_i 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 G_Na (pink trace) and G_Kf (blue trace) conductances. Middle, Activation of P_HVA (orange trace) and P_MVA (purple trace) permeabilities. Right, Activation of G_Ks (red trace), G_BK (green trace), and G_SK (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 G_Na (pink trace) and G_Kf (blue trace) conductances. Middle, Activation of P_HVA (orange trace) and P_MVA (purple trace) permeabilities. Right, Activation of G_Ks (red trace), G_BK (green trace), and G_SK (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 G_Na (pink trace) and G_Kf (blue trace) conductances. Middle, Activation of P_HVA (orange trace) and P_MVA (purple trace) permeabilities. Right, Activation of G_Ks (red trace), G_BK (green trace), and G_SK (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 G_Na (pink trace) and G_Kf (blue trace) conductances. Middle, Activation of P_HVA (orange trace) and P_MVA (purple trace) permeabilities. Right, Activation of G_Ks (red trace), G_BK (green trace), and G_SK (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).