Fig 1.
(A) TMS induced electric fields in the cerebral cortex. (B) Coupling between induced electric fields and the states of cortical neurons (the focus of this work). (C) Circuit dynamics of cell populations in M1 following stimulation. The cortical output is determined via the firing behavior of L5 pyramidal cells projecting to the spinal cord. (D) Implanted epidural electrode recording of DI-waves descending the spinal cord following TMS, which are modulated by stimulation conditions. (E) Spinal cord microcircuit of alpha motor neurons (MNs) with global inhibition from Renshaw cells (RCs).
MNs send action potentials through peripheral nerves to activate motor units at the muscles. (F) Traces of motor evoked potentials (MEPs) at increasing stimulation intensities, measured by skin mounted electrodes. Motor units of the muscles are excited by action potentials from spinal motor neurons, causing contraction and a quantifiable MEP readout.
Fig 2.
Modeling flowchart of TMS coupling.
Electric fields enter as the input to a presynaptic populations, coupling directly to axons. The axonal delay kernel simulation quantifies the spatiotemporal output of the presynaptic populations as a density function r(z, t). This density function then enters the (synapto-) dendritic kernel simulation as an input to drive dynamics on the postsynaptic dendrites. Dendritic current entering the soma of the postsynaptic cells exits the model as the output measure, inheriting sensitivity to direction and dosing implicitly from the electric field input.
Fig 3.
Action potential backpropagation after application of parameterized electric field.
(A) Electric field parameters in relation to cell morphology: Polar angle , electric field intensity gradient
, azimuthal angle
, and electric field intensity
that scales the normalized waveform (Mono-/Biphasic, etc.). (B) Single frame at t = 0.525 ms of axonal delay simulation from S1 Video. Electric field parameters are
,
,
,
V/m, biphasic pulse. The color axis is the membrane potential of each compartment. One axon terminal is activated initially in this simulation at the lightning bolt and the depolarization backpropagates to the remaining axonal compartments as depicted by the gray arrows. The membrane potential time series below is shown for the axon terminal at the bottom right, which receives an action potential around 0.75 ms.
Fig 4.
Example Axonal delay histogram calculation.
(A) Action potential backpropagation for electric field parameters ,
,
,
, and biphasic pulse (Cell activation threshold is 140 V/m). Color axis indicates the arrival time of an action potential at the associated axon compartment in ms. Activation in this example occurs only at the terminal marked by the lightning bolt, with gray arrows depicting the backpropagation through the axonal arbor to other axon terminals. Boxed in red is an example z-bin between z = 0
m and z = 100
m. (B) Two-dimensional histogram of action potential arrival times and locations for each axon terminal in this cell. Boxed in red is a chosen histogram bin (with same z-values as in (A)) with depth between 0 and 100
m and time between 0.5 and 0.6 ms. (C) Membrane potential timeseries for the three axon terminals whose action potentials arrive within the time-depth-bin boxed in (B) (i.e., their position lies within the depth bin and their potentials cross V = 0 mV within the time bin). The boundaries of the bin in the time axis are also boxed in red.
Table 1.
Cell types and relative frequency per layer. Relative frequency weighting factors recovered from cell morphology and electric type resolved survey of rat somatosensory cortex via the Blue Brain Project online microcircuit explorer [24,27].
Fig 5.
Averaging of axonal delay simulations.
(A) Visual representation of averaging across azimuthal angle. (B) Visual representation of averaging across unique cell morphologies. (C) Average L2/3 PC axonal delay kernel for electric field parameters ,
, and
V/m. Overlaid at right is a L2/3 PC morphology with soma aligned to the vertical axis and a horizontal dotted line at the soma depth (zสน = 0). (D) Average L2/3 population axonal delay kernel calculated by correlation of the average cell kernel (C) with the cell density distribution [28] (left) and scaled by L2/3 PC/L5 PC cell frequency ratio. Overlaid at right are the M1 cortical layer boundaries.
Fig 6.
Synaptic inputs and model output of a single example synapto-dendritic delay simulation.
(,
,
,
V/m, 0.01 ms step size simulated for 50 ms). (A) Inhibitory axonal delay kernel (L2/3 BC population). (B) Excitatory axonal delay kernel (L2/3 PC population). Overlaid over both kernels is the postsynaptic L5 PC with soma at -1584
, an example dendritic compartment marked by a green cross, and the bin containing that compartment boxed in red. (C) Zoomed-in view of L5 PC morphology near the green cross. (D) Partial integral of the inhibitory axonal delay kernel across the red boxed, 50
bin in (A). (E) Partial integral of the excitatory axonal delay kernel across the red boxed, 50
bin in (B). (F) Double exponential conductance kernels describing AMPA, NMDA, GABAa, GABAb synapses. (G) Synaptic conductance of each synapse type at green cross-marked compartment, computed by the weighted convolution of (D) and (E) with the kernels in (F). (H) Simulated trans-membrane current of each conductance-based synapse at green cross-marked compartment. Positive valued current flows into the cell. (I) Dendritic current flowing into the L5 PC soma from the dendrites due to synaptic inputs.
Table 2.
Double Exponential Mechanism Constants. Synaptic time constants, reversal potential, and peak conductance for the 2 excitatory (AMPA/NMDA) and 2 inhibitory (GABAa/GABAb) synapse types used.
Fig 7.
Averaging algorithm of synapto-dendritic delay simulation across cortical depth, spanning the space occupied by L5 PCs.
(left) Cell density function of L5 PCs in M1 [28], overlaid with 25 equidistant samples in orange and the mean soma depth. For a given set of electric field and synapse distribution parameters, each synapto-dendritic delay simulation is computed and averaged for 25 different depths (right), weighted by the cell density function for L5 PCs. The weight function is normalized such that the sum of weights across the 25 samples is 1.
Fig 8.
Directional and intensity sensitivity of axonal delay kernel for the average L2/3 PC.
Kernels are shown in a grid with electric field intensity increasing to the right and polar angle
increasing to the bottom (electric field orientation w.r.t somatodendritic axis overlaid). Each kernel is plotted with respect to time and the depth with respect to cell soma. The color scale is spike density.
Fig 9.
Directional and intensity sensitivity of axonal delay kernel for the L2/3 PC population (Fig 8 after correlation with the L2/3 cell density function from Fig 5D and scaling by L2/3 PC:L5 PC cell count ratio 5.04).
Kernels are shown in a grid with electric field intensity increasing to the right and polar angle
increasing to the bottom (electric field orientation w.r.t somatodendritic axis overlaid). Each kernel is plotted with respect to time and the cortical depth with respect to CSF boundary. The color scale is spike density.
Fig 10.
Directional sensitivity of axonal delay kernel for L2/3 population with z-axis dependence averaged out.
Half polar plots for six electric field intensity values included. 2D kernels as in Fig 9 are averaged across z to produce a time-dependent spike density for each angle. The radial axis of each half polar plot is time, the polar axis is the angle,
, the electric field vector makes with the somatodendritic axis, and the color scale is partial spike density.
Fig 11.
Mechanisms of axon terminal recruitment saturation and spike output synchrony.
(A) Axonal delay simulations for a L2/3 PC at increasing electric field intensities (where ,
,
). The color axis quantifies the time delay after TMS at which an action potential reaches each axonal compartment. Lightning bolts indicate points of action potential generation. (B) Membrane potential at the boxed axon terminal compartment (the first terminal to be activated) as a function of time and electric field intensity. 0 mV is marked by a dashed line to show the shift of action potential generation time toward t = 0 as intensity increases. The biphasic TMS coil current waveform (normalized) is plotted beneath.
Fig 12.
Directional and intensity sensitivity of dendritic delay kernel.
The average dendritic current to L5 due to TMS activation of L2/3 PC and BC populations is plotted for six electric field intensities . The radial axis of each half polar plot is time, the polar axis is the angle,
, the electric field vector makes with the somatodendritic axis, and the color scale is dendritic current in nanoamperes entering the L5 soma. Top right and bottom right are crosscuts showing average current when
for
V/m and 400 V/m respectively.
Fig 13.
Sensitivities of the dendritic current with respect to the input parameters over time.
(A) Average dendritic current its standard deviation; the grey shaded area shows the probability density. Depicted in blue are the average dendritic currents for cases when fex is fixed to 0.5 and 0.8 respectively. (B) Highest normalized Sobol indices over time reflecting their relative contribution to the total variance. (C) Global average derivatives of the parameters over time.
Fig 14.
Time averaged Sobol indices of the dendritic current.
(i.e., average of Fig 13B.).