Fig 1.
(A) Diagram of model network with three functional groups (a, b, c). In the stimulation protocol, the BBCI records spikes from a single neuron in group a and delivers a large stimulus to group b after a delay d†. (B) Top: external rates νa(t) (solid black), νb(t) (dotted red) and νc(t) (dashed blue) delivered to each neural group. Middle: sample spike raster plot from a simulation subject to rates plotted above with sparse, homogeneous connectivity J(0) (N = 60). Bottom: same as middle but with spike-triggered stimulation turned “on”. Spike from neuron number 1, circled in red, trigger the stimulation of all of population b (neurons 20 to 40) after a delay of d† = 10 ms. (C) Top: snapshot of connectivity matrix J(t) for network in B after spiking simulation of 5000 seconds (grayscale: white = Jmin, black = Jmax). Bottom: example time traces of individual synapses from the same simulation.
Fig 2.
(A) Mean network correlations for synaptic equilibria under normal activity C(u) (thick green line) and under spike-triggered stimulation C†(u) (dashed red line). Mean correlations for normal network activity with synaptic equilibrium obtained from stimulation () is shown in dashed blue. External correlations
also shown (thin black line). (B) Evolution of synaptic weight averages
over time, computed with analytical estimates, color-coded as in C. External rates as in Fig 1B. Initiated at equilibrium (see Methods), spike triggered stimulation (†) is switched “on” for the indicated period. Inset shows evolutions of 10 randomly chosen synapses from group a to group b, from corresponding spiking network simulation. (C) Top: plots of mean equilibrium matrices
in normal activity and under spike-triggered stimulation (left/right resp.). Bottom: snapshot of full simulated matrix J(t) once equilibria are reached (N = 60). (D) Plot of relative synaptic changes between equilibria:
with α, β ∈ {a, b, c}.
Fig 3.
(A) ICMS with baseline connections. (B) ICMS after spike-triggered stimulation. Top left panels: Connectivity matrix J. Top right panels: Raster plot of entire network as in Fig 1. Bottom black bar shows a 50 ms stimulation of all neurons in group a at 100 Hz. Bottom: Filtered responses of group projections. Columns designate neural groups in MC being stimulated (see e.g. A for stimulation of group a) and rows designate the output EMG Mα evoked from each group, respectively. Circle indicates the biggest change induced by spike-triggered conditioning.
Fig 4.
(A) Experimentally obtained cross-correlograms of MC neurons in macaque monkey during a tracking task (blue) and Gaussian fit (red). Top, Middle: for neuron pairs recorded by the same electrode, respectively (using spike-sorting). Top shows the thinnest correlation peak (σ = 9.8 ms) and Middle shows the widest (σ = 89.3 ms). Bottom: For two neurons recorded by distinct electrodes. (B) Illustration of Gaussian-shaped external cross-correlations with width
(thin black line) and resulting network cross-correlation C(u) with width σ (thick blue line). (C) Relative differences
for all group-averaged synapses, as a function of stimulation delay d†, for the model network fitted to two extremal values of correlation width. Top: σ = 10 ms. Bottom: σ = 90 ms. (D) Relative differences for averaged synaptic strengths from group a to group b (
) as a function of stimulation delay d† and fitted correlation peak width σ. Black dotted line corresponds to best fit plotted in E. (E) Superposition of relative difference for
and normalized mean torque change from spike-triggered conditioning experiments on macaque monkeys. Experimental data from Figure 4 of [10]; error bars show the standard error of the mean. Best fit between model and experimental curve is for σ ≃ 17 ms (see black dotted line in D).
Fig 5.
(A) Relative differences (i.e. before versus after spike-triggered stimulation) for mean synaptic strengths as a function of stimulation delay d† and correlation peak width σ (as in Fig 4D). Subplots show outcome for all combinations of pre- and post-synaptic groups a, b, c, for 2.5 ≤ d† ≤ 12.5 (ms) and 1 ≤ σ ≤ 50 (ms). Star indicates parameter choice for panel B. (B) Bar plots of mean synaptic strength relative difference
for all combinations of pre- and post-synaptic groups, with d† = 5 ms and σ = 17 ms. Top: normal network (values marked by a star in panel A). Middle: network where b-to-c synapses are inactivated (Jcb ≡ 0). Bottom: network where a-to-c synapses are inactivated (Jca ≡ 0). Red crosses indicate the artificially inactivated synapses and red arrows show the most drastic change due to inactivation, in comparison to the normal network (top).
Fig 6.
(A) Network with group b subdivided into two subgroups: stimulated neurons (b†, red) and unstimulated neurons (b∘, white). (B) Top: plots of equilibrium synaptic strengths between group a and both subgroups of b in normal activity and under spike-triggered stimulation (left/right resp.). Proportion of stimulated neurons is set to ρ = 0.5. Bottom: Plot of relative synaptic changes between equilibria:
. Control group c not shown. (C) Relative synaptic changes as a function of proportion of neurons in group b receiving stimulation ρ. Plotted are subgroup averaged synaptic changes for synapses from a to b† and b∘ and from a to entire group b. (d† = 30 ms)
Fig 7.
(A) STDP function W(Δt, Jij). Dashed black line indicates the W+ and W− functions. Coloured lines indicate the full, weight-dependent rule W for different synaptic weight values Jij shown in the color bar. (B) Top: Periodic external rates delivered to each neural group (as in Fig 1). Middle: sample spike raster plot from a simulation subject to rates plotted above with sparse, homogeneous connectivity J(0) (N = 60). (C) Simulated synaptic evolution J(t) with Jmin = 0, Jmax = 0.1 and η = 10−8 under baseline conditions (stimulation “off”) as in B. Left: initial connectivity matrix with connectivity probability p = 0.3 and Jij(0) = Jmax/4 (grayscale: white = Jmin, black = Jmax). Middle: 10 evolving weights J(t) randomly selected from each of 9 possible pre- and post-synaptic group combinations. Color indicates whether a synapse connects neurons within the same group (orange) or across different groups (purple). Right: synaptic matrix J(t) at the end of the simulation, once group-averaged weights have reached steady state.
Table 1.
Spiking network simulation parameters.
Fig 8.
Correlation functions averaged over all pairs of connected cells in respective group combinations: external rate correlations (solid black); estimates for network correlations from full spiking network simulations (orange line with shaded green area showing standard deviation of 5 seconds batched estimates) and analytical estimates of order 4 (dashed green).
(A) For network under normal activity. (B) For network under spike-triggered stimulation (†). Boxes in the middle show zooms from C(†)(u) from A and B.
Fig 9.
(A) Plot of (with α, β ∈ {a, b, c}) where J(0) is sparse and homogeneous (see C) and
is as in Fig 8. Colors represent the pair of pre- and post-synaptic groups αβ as in the upper left key. (B) Plot of
as
evolves over time indicated by the color scheme. Curved arrows illustrate the iteration scheme
and culminate at the equilibrium
. (C) Middle: evolution of synaptic weights over time, color-coded as in A. Thick solid lines show analytical predictions of order 4, thick dotted lines show averages from simulations and thin lines show examples of individual synapses Jij(t). On either sides: plots of
and J(t) used for simulations for t = 0 (left) and t = 80 hours (right). (D) Scatter plot of
and corresponding averages of simulated J(t) over all time points plotted in C. For both C and D, N = 60, p = 0.3, Jmin = 0, Jmax = 0.1, η = 10−8 and Jij(0) = Jmax/4.
Fig 10.
(A) Cross-correlation functions for external commands (thin black line) and for network activity Cαα(u) (thick blue line) for an average pair of neurons in the same group (α ∈ {a, b, c}). Parameters are
ms and σ = 55.76 ms. See also Fig 4B. (B) Numerically derived linear relationship
. Blue dots are obtained from setting
and deriving network synaptic equilibria. Red line shows linear the regression. Dotted black line shows the identity.