Magnetic stimulation generates axonal firing in pyramidal neuron models but not dendritic calcium spikes.

<p>Simulations of the effect of intracellular current injection and magnetic stimulation on somatic and dendritic membrane potential in two different models of L5 pyramidal neurons. Somatic membrane potential is shown in black and the dendritic membrane potential (600 µm along the apical dendrite) in red. The left column of the figure displays the response of the Larkum et al. model (2009) <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi.1002022-Larkum2" target="_blank">[38]</a>. The right column of the figure displays the response of the Schaefer et al. model (2003) <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi.1002022-Schaefer2" target="_blank">[27]</a>. A, The response of the models to a 2 ms somatic current injection. B, Response of the model to a 2 ms dendrite current injection (at 600 µm along the apical dendrite). C, Response of the models to a 400 µs threshold MS. D, Response of the models to 400 µs MSs five times stronger than the threshold stimulation in C.</p

Magnetic stimulation of a peripheral axon.

<p>A straight axon was located in a plane below the coil and the coil was shifted along the y-axis by one coil radius. Distance from the plane of the coil was 1 cm, coil radius was 2 cm, 30 loops to the coil. The underdamped pulse was used (R = 0.09 Ω; L = 13 µH; C = 200 µF; τ = 0.4 ms). Axon diameter was 100 µm, length 16 cm. Magnetic threshold of the axon was 36 v. Nernst potentials, conductance, membrane capacitance and resistivity of axoplasm were all equal to NEURON's default of the Hodgkin-Huxley model. <b>A</b>, View of the system showing the magnetic coil from above and beneath it the straight axon (marked with scale). The coil was moved normal to the axon (y-axis). Here it is shown shifted away from the axon so that the coil's center lies at a distance of one coil radius from the axon. <b>B</b>, Membrane potential as a function of time and location along the axon for a subthreshold stimulus (V = 30 V). <b>C</b>, Membrane potential as a function of time and location along the axon for a suprathreshold stimulus (V = 36 V).</p

Basic neuronal structures affect magnetic stimulation.

<p>The artificial neurons were located at the center of the matrices as demonstrated with a pyramidal cell in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi-1002022-g006" target="_blank">Figure 6</a>. Distance from the plane of the coil was 1 cm, coil radius was 2 cm, 30 loops to the coil. The underdamped pulse was used (R = 0.09 Ω; L = 13 µH; C = 200 µF; τ = 0.4 ms). The artificial neurons contained a soma simulated with the Hodgkin-Huxley model to generate neuronal excitability, while the dendrites contained passive parameters. The diameters of the soma and dendrites were 20 µm and 5 µm respectively, with the exception of dendrites in the bifurcation structure, whose diameter was set at 3.1498 µm. This was calculated according to the d^3/2 law developed by Rall <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi.1002022-Rall1" target="_blank">[60]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi.1002022-Rall2" target="_blank">[61]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi.1002022-Koch1" target="_blank">[62]</a>. <b>Ai</b>, The dendrite had a bend at one point along its length. θ is the angle between the second dendrite and the imaginary continuation of the first dendrite. <b>Aii</b>, The magnetic threshold as a function of θ for the bent dendrite. <b>Bi</b>, The primary dendrite bifurcated into two branches with equal diameter. Here, θ is defined as the angle between the second and the third dendrite. <b>Bii</b>, The magnetic threshold as a function of θ for the bifurcating dendrite. <b>Ci</b>, A cell with a change in dendrite diameter. <b>Cii</b>, The magnetic threshold as a function of the ratio of the diameters of the first to the second dendrite segment.</p

Magnetic threshold correlates with current threshold in realistic morphologies.

<p>Neurons were located at the center of the matrix as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi-1002022-g004" target="_blank">Figure 4</a>. Distance from the plane of the coil was 1 cm, coil radius was 2 cm, 30 loops to the coil. The underdamped pulse was used (R = 0.09 Ω; L = 13 µH; C = 200 µF; τ = 0.4 ms). Excitability was added to all cells using two different models of L5 pyramidal neurons - Larkum et al.'s model (2009) <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi.1002022-Larkum2" target="_blank">[38]</a> (green) and Schaefer et al.'s model (2003) <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi.1002022-Schaefer2" target="_blank">[27]</a> (black). The magnetic threshold correlated with current threshold in pyramidal cell with different current thresholds. Sodium channel activation and inactivation were shifted towards hyperpolarizing potentials to reduce current threshold.</p

The location of action potential initiation and the magnetic threshold depend on the location of the soma relative to the coil.

<p><b>A</b>, A soma with a long straight axon containing sections of myelin and nodes of Ranviar was located in a plane below the coil. Soma, axon and node diameters were 20 µm, 1 µm and 0.75 µm, respectively. Soma, axon and nodes lengths were 20 µm, 100 µm and 1 µm, respectively. The view from above shows that the artificial neuron was shifted along the y-axis by one coil radius and along the x-axis by Δx. <b>B</b>, The magnetic threshold decreased with Δx until it reached a minimum at the location corresponding to the maximal gradient of the electric field. The action potential was initiated at the axon for small shifts (Δx<0.05 cm, green dots), and at the soma for larger shifts (black dots). The lowest magnetic threshold was achieved when the soma was located at the largest gradient of the induced electric field.</p

Magnetic threshold does not change with different angles of a bent axon.

<p>Neuron was located at the center of the matrix as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi-1002022-g004" target="_blank">Figure 4</a>. Distance from the plane of the coil was 1 cm, coil radius was 2 cm, 30 loops to the coil. The underdamped pulse was used (R = 0.09 Ω; L = 13 µH; C = 200 µF; τ = 0.4 ms). Excitability was added to all cells using a model of neocortical pyramidal neurons <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi.1002022-Schaefer2" target="_blank">[27]</a>. <b>A</b>, The pyramidal neuron with the bent axon. θ is the angle between the axon and its imaginary continuation. <b>B</b>, The magnetic threshold as a function of θ.</p

Neuronal activity modulates magnetic threshold.

<p>The effect of a change in membrane potential on the MS of the pyramidal cell model used in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi-1002022-g008" target="_blank">figure 8</a>. Location of the artificial cell relative to the center of the coil and the parameters for calculating the induced electric field as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi-1002022-g005" target="_blank">figure 5</a>. Cell parameters as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi-1002022-g007" target="_blank">figure 7</a>. <b>A</b>, Magnetic threshold as a function of a synaptic input Δt before stimulus onset. <b>B</b>, The membrane potential in the soma as a function of time. An excitatory postsynaptic potential was evoked Δt milliseconds before MS onset. <b>C</b>, Magnetic threshold as a function of an AP at Δt before stimulus onset. <b>D</b>, Membrane potential in the soma as a function of time. An AP was evoked Δt milliseconds before MS onset.</p

Magnetic threshold changes with pulse duration in realistic morphologies.

<p>Neurons were located at the center of the matrix as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi-1002022-g004" target="_blank">Figure 4</a>. Distance from the plane of the coil was 1 cm, coil radius was 2 cm, 30 loops to the coil. The underdamped pulse was used (R = 0.09 Ω; L = 13 µH; C = 200 µF; τ = 0.4 ms). Excitability was added to all cells using two different models of L5 pyramidal neurons - Larkum et al.'s model (2009) <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi.1002022-Larkum2" target="_blank">[38]</a> (green) and Schaefer et al.'s model (2003) <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi.1002022-Schaefer2" target="_blank">[27]</a> (black). <b>A</b>, Magnetic threshold as a function of pulse duration (strength-duration curve). Pulse duration modified by changing the capacity from 50 µF to 700 µF. <b>B</b>, The device energy as a function of pulse duration.</p

Dendrites act as current sinks modifying magnetic threshold.

<p><b>A</b>, Magnetic threshold as a function of soma diameter for a soma with one dendrite (black) and a soma with 11 dendrites (light blue). The lines is a curve fit of V_th = a+b/d² where V_th is the magnetic threshold, a and b are proportionality constants and d is the fiber diameter. <b>B</b>, Magnetic threshold as a function of the number of dendrites. The number of dendrites connected to the soma was increased and the magnetic threshold plotted for every cell. A soma with 6 dendrites is shown in the insert as an example. <b>C</b>, Magnetic threshold as a function of current threshold for cells with different numbers of dendrites. <b>D</b>, Magnetic threshold as a function of input resistance for cells with different numbers of dendrites. Location of the artificial cell with respect to the center of the coil and the parameters used for calculating the induced electric field as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002022#pcbi-1002022-g004" target="_blank">figure 4</a>. Soma and dendrite diameters were 20 µm and 1 µm, respectively.</p

Factors affecting magnetic stimulation of peripheral axons.

<p>A straight axon was located in a plane below the coil and the coil was shifted along the y-axis by one coil radius. Distance from the plane of the coil was 1 cm, coil radius was 2 cm, 30 loops to the coil. The underdamped pulse was used (R = 0.09 Ω; L = 13 µH; C = 200 µF; τ = 0.4 ms). Nernst potentials, conductances, membrane capacitance and resistivity of axoplasm were all equal to NEURON's default of the Hodgkin-Huxley model. <b>A</b>, Magnetic threshold as a function of fiber diameter shown on a log-log scale. The line is a curve fit of V_th = a+b/d² where V_th is the magnetic threshold, a and b are proportionality constants and d is the fiber diameter <b>B</b>, Magnetic threshold as a function of shifting the coil along the y-axis. Distance from the axon was measured with respect to the center of the coil. Axon diameter was 100 µm.</p