Figure 1.
Modeling the descending motor system.
(A) Major fiber tracts of the descending motor system, redrawn from [105]. Axons of the pyramidal tract neurons (red) descend from the motor cortex to monosynaptically innervate motor neurons in the spinal cord. (B) Schematic representation of the dendritic arbors of a typical pyramidal tract neuron (PTN). The apical dendrites project widely throughout the superficial layers of cortex and thus are ideally placed to detect surface wave patterns in the neural activity (top). (C) Simulated cortical wave pattern. (D) The descending motor model. Cortical wave patterns are generated by a sheet of spatially-coupled phase oscillators (circles, 1–8). These wave patterns are spatially filtered by the dendritic trees of the pyramidal tract neurons to produce an amplitude-modulated oscillatory current at the soma. Spikes initiated by the PTN are transmitted to a randomly selected pool of motor neurons (MN) in the spine. Each MN integrates the incoming spikes to produce a muscle drive spike train. Net muscle drive is quantified by simulated Electromyogram (EMG). The cortical wave model is adapted from [17]. The MN and EMG models are adapted from [41].
Figure 2.
(A) Profiles of the spatial coupling kernel along its major and minor axes of orientation. (B) Contours of the spatial coupling kernel. (C) Exemplar oscillator pattern with this coupling kernel. Shading indicates phase. Black lines indicate the orientation of the kernel axes. (D) Frequency spectrum of the simulated local field potential (LFP) superimposed on the distribution of the autonomous oscillator frequencies (Osc). The latter is normally distributed with M = 20 Hz and SD = 4 Hz. (E) Time course of the simulated local field potential. (F) Time course of MEG signal recorded over human motor cortex during a precision grip task.
Figure 3.
Gabor filtering by excitatory and inhibitory receptor densities.
(A) Density profiles for excitatory (blue) and inhibitory (red) receptor populations which combine to form a Gabor filter (black). In this case, the excitatory density was nominated as Gaussian. (B) Spatial frequency response of the Gabor filter. Peaks correspond to waves of length 300 µm. (C) Excitatory (blue) and inhibitory (red) receptor samples taken from the density distributions in panel A. The combined receptor field (blue+red) represents the dendritic field of the neuron. (D) Spatial frequency response of the combined receptor field. Peaks correspond to vertically oriented waves of length 300 µm. (E) The combined receptor field superimposed on its preferred wave pattern. The wave pattern propagates from left to right at 6 mm/sec to simulate 20 Hz oscillations in the cortical field. (F) Time course of the net excitatory (blue shading) and inhibitory (red shading) conductances in response the preferred wave pattern. Faint lines show individual post-synaptic conductances for n = 40 randomly selected receptors (not to scale). Each receptor fires 20 spikes/sec on average. Heavy black line shows the dendritic current induced by the net changes in conductance. The amplitude of the dendritic current is modulated as the wave propagates across the receptor field. (G) The combined receptor field superimposed on the orthogonal wave pattern which propagates from top to bottom at 6 mm/sec. (H) Time course of the dendritic response to the orthogonal wave pattern. In this case the wave pattern does not modulate the dendritic current even though the individual receptors still fire at 20 spikes/sec on average.
Figure 4.
(A) Spatial profiles of the dendritic filter. (B) Spatial contours of the dendritic filter. (C) Preferred cortical oscillation pattern for this dendritic filter. (D) Orthogonal oscillation pattern. (E) Time course of the neural response to the preferred cortical pattern. Bottom trace (red) is the dendritic current. Top trace (black) is the somatic membrane potential. Light gray traces show the responses of four other PTNs located at random positions on the same cortical pattern. (F) Time course of neural response to the orthogonal pattern. Panels E and F have the same scales.
Figure 5.
Response properties of the PTN somatic compartment.
(A) Spike trains produced by the model in response to 20 Hz sinusoidal injection currents of amplitude 0.50 nA, 0.51 nA, 1.00 nA and 1.50 nA respectively. Bottom trace (red) shows the time course of the injection current. (B) Steady-state firing response of the model to 20 Hz sinusoidal injection current. The plateaus in the response curve are due to entrainment of the membrane potential to the oscillatory input. The main plateaus occur at 20 Hz and 40 Hz. Smaller plateaus also occur at 10 Hz, 13.25 Hz, 30 Hz and 33.25 Hz. (C) Steady-state firing response of the same model to constant injection currents. The parameters of the model were tuned so that this curve closely matched the physiological properties of pyramidal neurons [62]–[64]. Specifically, a mean slope of 42 Hz/nA and sudden onset of 10 Hz firing as the injection current approaches 0.5 nA.
Figure 6.
(A) Tuning curve of the PTN dendritic compartment. The amplitude of the dendritic response current (vertical axis) is modulated by the orientation of the cortical wave pattern (horizontal axis). Heavy black line indicates the mean amplitude of the dendritic response for any given wave orientation. Shaded region indicates the 90% confidence interval. The large variation is due to local defects in the wave pattern. (B) The likelihood of the soma responding at each of the dominant firing rates. (C) Net firing rates of a population of neurons in response to wave orientation.
Figure 7.
Variability of inter-spike intervals in the PTN model.
(A) Exemplar dendritic current (red) and resulting somatic spike train (black) exhibiting irregular inter-spike intervals. The coefficient of variation (CV = 0.76) and irregularity (IR = 0.35) measures were both computed over a 30 second window. (B) Coefficient of variation of the inter-spike intervals versus firing rate. (C) Irregularity metric for the same data. Box plot (yellow) reproduces the observed irregularity of PTN inter-spike intervals in primary motor cortex [69] where the whiskers indicate the extrema.
Figure 8.
Asymmetric dendritic kernels induce phase shifts in the PTN spike trains.
Profiles of the dendritic kernels are shown on the left. Spike trains produced by the PTN model are shown on the right. The thick gray line is the simulated LFP of the cortical pattern which is the same in all cases. (A) Case of a Gabor filter with zero phase shift. (B) Case of +90 degree phase shift. (C) Case of +180 degree phase shift. (D) Case of −90 degree phase shift. Light gray spike traces in B–D reproduce the case of zero phase shift for ease of comparison.
Figure 9.
The effect of wave orientation on the output of the descending motor system.
Each column presents the responses of the descending motor system for pyramidal neurons with a given dendritic orientation ( and
) relative to the cortical pattern. (A) Orientation of the dendritic kernels. The cortical pattern is the same in all cases. (B) Firing rate distribution of the pyramidal tract neurons. (C) Firing rate distribution of the motor neurons. (D) Time course of the simulated EMG. (E) Magnitude squared coherence between LFP and EMG. Light gray lines represent individual trials (n = 100). Black line shows the trial average. In red, average MEG-EMG coherence in 16 subjects while they perform a precision grip task at different force levels (2.0 N, 1.65 N, 0.95 N, 0.0 N). Dashed horizontal line indicates the 95% confidence level for the coherence distribution in each frequency bin. Peaks above that line are statistically significant at p = 0.05.
Table 1.
Parameters of the dendritic conductance model.
Table 2.
Parameters of the PTN soma model.
Table 3.
Parameters of the MN model.