1 We investigated the types and distribution of voltage-gated K⁺ channels in the soma and apical dendrite of layer 5 (L5) neocortical pyramidal neurones, of young rats (postnatal days 13–15), in acute brain slices. 2 A slow inactivating outward K⁺ current and a fast inactivating outward K⁺ current were detected in nucleated patches. The slow K⁺ current was completely blocked by tetraethylammonium (TEA) with an IC₅₀ of 5 ± 1 mm (mean ±s.e.m.) and was partially blocked by 4-aminopyridine (4-AP). The fast K⁺ current was blocked by 4-AP with an IC₅₀ of 4.2 ± 0.5 mM, but was not blocked by TEA. 3 The activation kinetics of the slow K⁺ current were described by a second order Hodgkin-Huxley model. The slow K⁺ current displayed bi-exponential inactivation. A fourth order Hodgkin-Huxley model for activation and first order for inactivation described the kinetics of the fast K⁺ current. 4 In somatic cell-attached recordings, three classes of single K⁺ channels could be differentiated based on their unitary conductance and inactivation kinetics, a fast inactivating channel having a conductance of 13 ± 1 pS, a slow inactivating channel having a conductance of 9.5 ± 0.5 pS, and a very slowly inactivating channel having a conductance of 16 ± 1 pS. 5 The inactivation time constants of the slow and of the very slow K⁺ channel corresponded to the two inactivation time constants of the slow K⁺ current observed in nucleated patches. This suggested that two distinct K⁺ channels mediated the slow K⁺ current in nucleated patches. 6 The three subtypes of K⁺ channels that were observed in somatic recordings were present along the apical dendrite. The amplitude of ensemble K⁺ currents in cell-attached patches decreased along the apical dendrite as the distance from the soma increased, with a slope of −0.9 ± 0.3 pA per 100 μm. 7 The results suggest that the decrease of the voltage-gated K⁺ channel density from the soma along the apical dendrite of L5 pyramidal neurones helps to define a distal, low threshold region for the initiation of dendritic regenerative potentials.

Transcranial magnetic stimulation (TMS) is a stimulation method in which a magnetic coil generates a magnetic field in an area of interest in the brain. This magnetic field induces an electric field that modulates neuronal activity. The spatial distribution of the induced electric field is determined by the geometry and location of the coil relative to the brain. Although TMS has been used for several decades, the biophysical basis underlying the stimulation of neurons in the central nervous system (CNS) is still unknown. To address this problem we developed a numerical scheme enabling us to combine realistic magnetic stimulation (MS) with compartmental modeling of neurons with arbitrary morphology. The induced electric field for each location in space was combined with standard compartmental modeling software to calculate the membrane current generated by the electromagnetic field for each segment of the neuron. In agreement with previous studies, the simulations suggested that peripheral axons were excited by the spatial gradients of the induced electric field. In both peripheral and central neurons, MS amplitude required for action potential generation was inversely proportional to the square of the diameter of the stimulated compartment. Due to the importance of the fiber's diameter, magnetic stimulation of CNS neurons depolarized the soma followed by initiation of an action potential in the initial segment of the axon. Passive dendrites affect this process primarily as current sinks, not sources. The simulations predict that neurons with low current threshold are more susceptible to magnetic stimulation. Moreover, they suggest that MS does not directly trigger dendritic regenerative mechanisms. These insights into the mechanism of MS may be relevant for the design of multi-intensity TMS protocols, may facilitate the construction of magnetic stimulators, and may aid the interpretation of results of TMS of the CNS. Transcranial magnetic stimulation (TMS) is a widely applied tool for probing cognitive function in humans and is one of the best tools for clinical treatments and interfering with cognitive tasks. Surprisingly, while TMS has been commercially available for decades, the cellular mechanisms underlying magnetic stimulation remain unclear. Here we investigate these mechanisms using compartmental modeling. We generated a numerical scheme allowing simulation of the physiological response to magnetic stimulation of neurons with arbitrary morphologies and active properties. Computational experiments using this scheme suggested that TMS affects neurons in the central nervous system (CNS) primarily by somatic stimulation. Since magnetic stimulation appears to cause somatic depolarization, its effects are highly correlated with the neuron's current threshold. Our simulations therefore predict that subpopulations of CNS neurons with different firing thresholds will respond differently to magnetic stimulation. For example, low-intensity TMS may be used to stimulate low-threshold cortical inhibitory interneurons. At higher intensities we predict that both inhibitory and excitatory neurons are activated. These predictions may be tested at the cellular level and may impact cognitive experiments in humans. Furthermore, our simulations may be used to design TMS coils, devices and protocols.

The voltage-clamp technique is applicable only to spherical cells. In nonspherical cells, such as neurons, the membrane potential is not clamped distal to the voltage-clamp electrode. This means that the current recorded by the voltage-clamp electrode is the sum of the local current and of axial currents from locations experiencing different membrane potentials. Furthermore, voltage-gated currents recorded from a nonspherical cell are, by definition, severely distorted due to the lack of space clamp. Justifications for voltage clamping in nonspherical cells are, first, that the lack of space clamp is not severe in neurons with short dendrites. Second, passive cable theory may be invoked to justify application of voltage clamp to branching neurons, suggesting that the potential decay is sufficiently shallow to allow spatial clamping of the neuron. Here, using numerical simulations, we show that the distortions of voltage-gated K⁺ and Ca²⁺ currents are substantial even in neurons with short dendrites. The simulations also demonstrate that passive cable theory cannot be used to justify voltage clamping of neurons due to significant shunting to the reversal potential of the voltage-gated conductance during channel activation. Some of the predictions made by the simulations were verified using somatic and dendritic voltage-clamp experiments in rat somatosensory cortex. Our results demonstrate that voltage-gated K⁺ and Ca²⁺ currents recorded from branching neurons are almost always severely distorted.

Scientific models are abstractions that aim to explain natural phenomena. A successful model shows how a complex phenomenon arises from relatively simple principles while preserving major physical or biological rules and predicting novel experiments. A model should not be a facsimile of reality; it is an aid for understanding it. Contrary to this basic premise, with the 21st century has come a surge in computational efforts to model biological processes in great detail. Here we discuss the oxymoronic, realistic modeling of single neurons. This rapidly advancing field is driven by the discovery that some neurons don't merely sum their inputs and fire if the sum exceeds some threshold. Thus researchers have asked what are the computational abilities of single neurons and attempted to give answers using realistic models. We briefly review the state of the art of compartmental modeling highlighting recent progress and intrinsic flaws. We then attempt to address two fundamental questions. Practically, can we realistically model single neurons? Philosophically, should we realistically model single neurons? We use layer 5 neocortical pyramidal neurons as a test case to examine these issues. We subject three publically available models of layer 5 pyramidal neurons to three simple computational challenges. Based on their performance and a partial survey of published models, we conclude that current compartmental models are ad hoc, unrealistic models functioning poorly once they are stretched beyond the specific problems for which they were designed. We then attempt to plot possible paths for generating realistic single neuron models.

Compartmental models with many nonlinearly and nonhomogeneous distributions of voltage-gated conductances are routinely used to investigate the physiology of complex neurons. However, the number of loosely constrained parameters makes manually constructing the desired model a daunting if not impossible task. Recently, progress has been made using automated parameter search methods, such as genetic algorithms (GAs). However, these methods have been applied to somatically recorded action potentials using relatively simple target functions. Using a genetic minimization algorithm and a reduced compartmental model based on a previously published model of layer 5 neocortical pyramidal neurons we compared the efficacy of five cost functions (based on the waveform of the membrane potential, the interspike interval, trajectory density, and their combinations) to constrain the model. When the model was constrained using somatic recordings only, a combined cost function was found to be the most effective. This combined cost function was then applied to investigate the contribution of dendritic and axonal recordings to the ability of the GA to constrain the model. The more recording locations from the dendrite and the axon that were added to the data set the better was the genetic minimization algorithm able to constrain the compartmental model. Based on these simulations we propose an experimental scheme that, in combination with a genetic minimization algorithm, may be used to constrain compartmental models of neurons.

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 Ca²⁺channels decreased linearly from the soma, and leveled at the distal apical dendrite. The density of the small-conductance Ca²⁺-activated channel decreased along the apical dendrite, whereas the density of the large-conductance Ca²⁺-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.