Fig 1.
Reshaping of synaptic connectivity by PMCS.
A: During PMCS, periodic stimulus trains (red) are delivered to multiple neuronal subpopulations (1,2,3) with phase shifts (Δα1,Δα2). D: Schematic of PMCS delivery using a multisite DBS electrode where phase-shifted stimulus trains are delivered to three contacts (1,2,3). Neurons are mostly affected by stimulus trains that are delivered to nearby stimulation contacts, neuron i by stimuli delivered to contact 1 and neuron j by stimuli delivered to contact 2. B: For illustration, in this figure we assume that neurons respond with a single spike (black vertical bar) of high fidelity to a stimulus (red). During stimulation, the spike train of the postsynaptic neuron, j, and that of the presynaptic neuron, i, become time shifted by Δα1/f, where f is the stimulation frequency. Due to STDP, time lags, Δt, between postsynaptic spikes and delayed presynaptic spike arrivals (gray) lead to weight updates, W, according to the STDP function (C, see Eq 10). Negative weight updates result from pairings of presynaptic spike arrivals with earlier postsynaptic spikes (Δt− < 0) and positive updates from pairings with later postsynaptic spikes (Δt+ > 0). E: The sign of the overall weight update per inter-stimulus interval (ISI), , determines whether the synapse between neurons i and j is up-regulated (thick arrow) or down-regulated (thin arrow). F: If the neurons are bidirectionally coupled, PMCS may either induce effective unidirectional coupling (a,b), bidrectional coupling (d), or lead to decoupling (c). Blue dashed lines in panels E and F separate parameter regions with
from regions with
. Dark blue dashed lines separate corresponding regions for
.
Fig 2.
Statistics of LIF neurons’ spiking responses to periodic delivery of single-pulse (top) and burst stimuli (bottom).
A: Raster plot of spiking activity of a network of 103 LIF neurons during periodic single-pulse stimulation (red bars) of 333 neurons. B: Mean number of spikes per ISI and neuron as function of the dimensionless stimulation amplitude, Astim, and pulse width, de (Methods). Gray contour lines correspond to 0.75 and 1.25 spikes per ISI and neuron (from bottom to top). C: Cumulative distribution function of the timing of the first spike of LIF neurons after stimulus delivery (Methods) for the stimulus parameters marked orange, black, and brown, respectively, in panel B. The inset zooms into the first 5 ms. The stimulation current, Istim, corresponding to de = 10 is shown in panel D. E: Same as A but for burst stimuli with five pulses per burst and an intraburst frequency of 120 Hz. F: Mean number of spikes per ISI and neuron as function of the number of stimulus pulses per burst and the stimulation amplitude. G: Cumulative distribution function of the timing of the kth spike per ISI
for the parameter set marked white in panel F. Color code: F1(t) (black), F2(t) (blue), F3(t) (green), F4(t) (brown), and F5(t) (red). H: Corresponding stimulation current, Istim. Parameters: f = 5 Hz (all); Astim = 0.4, and de = 10 (A,C,D) and de = 1 (C, orange) and de = 20 (C, brown); and Astim = 0.8, de = 1, and 120 Hz intraburst frequency (E, F, G, H).
Fig 3.
Effect of PMCS with (M = 2) on the mean weight of synapses with postsynaptic neuron in subpopulation 2 and presynaptic neuron in subpopulation 1 as function of the phase lag, Δα, between stimulus trains delivered to the two subpopulations.
Panels show results for single-pulse (A-C) and burst stimuli with three (A’-C’) and five (A”-C”) pulses per burst. Results for high (120 Hz) and low (60 Hz) intraburst frequencies are shown in red and blue, respectively. Curves show theoretical approximations (Eq 19) and markers
obtained from simulations and averaged over five network realizations and initial conditions (Methods). Parameters: de = 1 and Astim = 0.4 (A,B,C), Astim = 0.8 (A’-C’, A”-C”). Note that different intervals are plotted on the y-axes.
Fig 4.
PMCS-induced network structure depends on phase lags and stimulus type.
Panels A-I show results for the PMCS pattern illustrated in panel A. Red rectangles mark stimuli. Individual stimuli were either charge-balanced single-pulse (D-F) or burst stimuli (G-I). B: Phase shifts, Δϕxy, between stimulus trains delivered to the postsynaptic subpopulation y and presynaptic subpopulation x. C: limt→∞〈wx → y(t)〉 obtained from Eq 1 for single-pulse stimuli (1P) and burst stimuli with three pulses (3P) (see panel I for color code). Panels D-F show a raster plot of simulated spiking activity of the LIF network model 1000 sec after stimulation onset (D), and snapshots of the network’s synaptic weight matrix taken 20 sec (E) and 1000 sec (F) after stimulation onset. After 1000 sec, acute effects of stimulation have fully developed. The scale bar refers to a 100 ms time interval. The block
of the synaptic weight matrix is marked red in panel F. G-I: Same as D-F but for burst stimuli with three pulses per burst and an intraburst frequency of 120 Hz. A’-I’: Same as A-I but for the PMCS pattern illustrated in panel A’. Parameters: f = 5 Hz; Astim = 0.4 (single-pulse stimuli) and 0.8 (burst stimuli), de = 1. Δα1/2 = 0.5 (A-I), and Δα1 = 0.1 and Δα2 = 0.3 (A’-I’).
Fig 5.
PMCS-induced network structure depends on phase lags between stimulus trains.
Predicted block structure of the synaptic weight matrix after long PMCS employing single-pulse stimuli (A, B) and PMCS employing burst stimuli consisting of three pulses (C, D). A: limx→y〈wx→y(t)〉, obtained from Eq 1, (color map) is compared to simulation results for the LIF network model after 1000 sec of stimulation (filled circles). B: Predicted mean synaptic weight limt→∞〈w(t)〉 obtained from Eq 4. C,D: Same as A,B but for burst stimuli with three pulses per burst and an intraburst frequency of 120 Hz. Parameters: Astim = 0.4, f = 10 Hz, de = 1 (A,B), and Astim = 0.8, f = 5 Hz, de = 1, and an intraburst frequency of 120 Hz (C,D).
Fig 6.
Mean synaptic weight during single-pulse PMCS depends on pulse duration and stimulation frequency.
Theoretical predictions (color maps) obtained from Eq 4 after 20 sec and 1000 sec of PMCS are compared to simulation results for the LIF network model (filled circles). w0 = 0.38 is the mean synaptic weight prior to stimulation. Labeled columns show results for different stimulation frequencies: f = 10.0 Hz (A, D), f = 5.0 Hz (B, E), and f = 2.5 Hz (C, F). Rows show results for short pulses (de = 1, top) and long pulses (de = 20, bottom) (Methods). Simulated spike trains, snapshots of the connectivity matrix, and theoretical predictions of the block structure of the synaptic weight matrix are shown below for the parameter sets marked by red crosses (see red figure labels). Parameters: Astim = 0.4.
Fig 7.
Mean synaptic weight during PMCS with burst stimuli depends on intraburst frequency and number of pulses per burst.
Panels A,C and B,D show results for PMCS with burst stimuli with high (120 Hz) and low (60 Hz) intraburst frequencies, respectively. Rows show results for burst stimuli with three (top, 3P) and five (bottom, 5P) pulses, respectively. Columns show the mean synaptic weight 20 sec (left), 100 sec (middle), and 1000 sec (right) after onset of PMCS. Color maps show theoretical predictions, obtained from Eq 4, and filled circles resulted from simulations of the LIF network model. Simulated spike trains, snapshots of the synaptic weight matrix, and corresponding theoretical predictions of the block structure of the synaptic weight matrix, obtained from Eq 1, are shown below for the parameter sets marked by colored crosses (see corresponding figure labels). Parameters: f = 5 Hz, Astim = 0.8, de = 1.
Fig 8.
Decoupling effects of three PMCS patterns compared to decoupling by CRS RVS.
A: Raster plot of spiking activity of LIF neurons after 1000 sec of periodic stimulation (Δα1/2 = 0) using burst stimuli of three pulses per burst and an intraburst frequency of 60 Hz. Furthermore, snapshots of the synaptic weight matrix of the LIF network (sim) and the corresponding theoretical approximation obtained from Eq 1 (theory) are shown. Successful stimulation-induced decoupling led to synaptic weights that were close to zero (white). B, C: The mean synaptic weight after 20 sec and 1000 sec of periodic stimulation as function of the stimulation frequency, f, for single-pulse stimuli (1P) and burst stimuli with two (2P), three (3P), and four (4P) pulses per burst (see legend). Panels B and C show results for an intraburst frequency of 60 Hz (B) and 120 Hz (C), respectively. Markers show simulation results (averaged over five network realization) and curves theoretical approximations (Eq 4). Vertical lines mark the frequency of the synchronous rhythm prior to stimulation (red dashed line) and the set of stimulation parameters corresponding to the raster plot shown in panel A (colored dotted line), respectively. A’-C’ and A”-C”: Same as A-C but for PMCS with Δα1/2 = 0.5 (A’-C’) and Δα1/2 = 0.33 (A”-C”), respectively. (a’’’-c’’’): Similar plots for CRS RVS. Parameters: de = 1 (all panels). Astim = 0.8 for PMCS with burst and CR RVS with single-pulse and burst stimuli. Astim = 0.4 for PMCS with single-pulse stimuli (B, C, B’, C’, and B”, C”).
Fig 9.
Periodic stimulation may either up-regulate or down-regulate all synapses depending on the employed type of stimuli.
A: Raster plot of simulated spiking activity for PMCS with Δα1/2 = 0 and burst stimuli with low intraburst frequency. The scale bar indicates a time window of 100 ms. Stimuli are illustrated by red blocks. B: Snapshots of the simulated synaptic weight matrix after 1000 sec of stimulation (sim) and the corresponding theoretical prediction obtained from Eq 1 (theory). C: Schematic of stimulation-induced network structure. Circles represent subpopulations one, two, and three. Dashed arrows illustrate down-regulated synaptic connections. A’-C’: Same as A-C but for burst stimuli with high intraburst frequency. Thick arrows in panel C’ illustrate up-regulated synaptic connections. Parameters: Astim = 0.8, f = 5 Hz, and de = 1 (all panels). An intraburst frequency of 60 Hz and five pulses per burst were used in panels A-C, and an intraburst frequency of 120 Hz and five pulses per burst were used in panels A’-C’.
Fig 10.
PMCS can be employed to modulate intra- and inter-population synapses in opposite ways.
A: Raster plot of simulated spiking activity for PMCS with burst stimuli with high intraburst frequency. B: Snapshots of simulated synaptic weight matrix after 1000 sec of stimulation (sim) and the corresponding theoretical prediction obtained from Eq 1 (theory). C: Schematic of stimulation-induced network structure (see caption of Fig 9 for symbols). A’-C’: Same as A-C but for burst stimuli with low intraburst frequency. Parameters: Astim = 0.8, f = 5 Hz, and de = 1 (all panels). In panels A-C, we used an intraburst frequency of 120 Hz, three pulses per burst, and Δα1/2 = 0.33 (A-C), In panels A’-C’, we used 60 Hz, eight pulses per burst, and Δα1/2 = 0.31 (A’-C’).
Fig 11.
PMCS-induced tree and reverse-tree network structures.
A: Raster plot of the spiking activity in the LIF network model 1000 sec after onset of PMCS with single-pulse stimuli and small positive phase lags between stimuli delivered to subpopulation two and subpopulations one and three. B: A snapshot of the synaptic weight matrix after 1000 sec of stimulation (sim) and the corresponding theoretical prediction obtained from Eq 1 (theory). C: Schematic of stimulation-induced tree network structure (see caption of Fig 9 for symbols). A’-C’: Same as A-C but for single-pulse PMCS with small negative phase lags between stimuli delivered to subpopulation two and subpopulations one and three. Parameters: Astim = 0.4, f = 10 Hz, and de = 1 (all panels) and Δα1 = 0.7, and Δα2 = 0.3 (A-C), and Δα1 = 0.3, and Δα2 = 0.7 (A’-C’).
Fig 12.
PMCS-induced feed-forward and circular network structure.
A: Raster plot of simulated spiking activity in the LIF network model 1000 sec after onset of PMCS with single-pulse stimuli and small positive phase lags between subsequent subpopulations. B: Snapshots of simulated synaptic weight matrix after 1000 sec of stimulation (sim) and the corresponding theoretical prediction obtained from Eq 1 (theory). C: Schematic of stimulation-induced feed-forward structure (see caption of Fig 9 for symbols). A’-C’: Same as A-C but for slow PMCS with long single-pulse stimuli. The phase lags were adjusted to cause a circular network structure. Parameters: Astim = 0.4 and Δα1/2 = 0.3 (A-C), and Δα1/2 = 0.33 (A’-C’) and de = 1 and f = 10 Hz (A-C), and de = 20 and f = 2.5 Hz (A’-C’).
Fig 13.
PMCS-induced network structures in which either all-but-one or only one synaptic population was up-regulated while all others were down-regulated.
A: Raster plot of simulated spiking activity of LIF neurons 1000 sec after onset of PMCS with burst stimuli with high intraburst frequency and short phase lags between subsequent subpopulations. B: Snapshots of simulated synaptic weight matrix after 1000 sec of stimulation (sim) and the corresponding theoretical prediction obtained from Eq 1 (theory). C: Schematic of stimulation-induced network structure (see caption of Fig 9 for symbols). A’-C’: Same as A-C but for single-pulse PMCS. One phase lag was chosen to be significantly smaller than the others in order to up-regulate only one population of synapses. Parameters: de = 1 (all panels). In panels A-C, we used Astim = 0.8, an intraburst frequency of 120 Hz, five pulses per burst, f = 5 Hz, and Δα1/2 = 0.1. In panels A’-C’, we used single-pulse stimuli with Astim = 0.4, a stimulation frequency of f = 10 Hz, and the phase lags Δα1 = 0.1 and Δα2 = 0.5.
Fig 14.
Schematics of the derivation of the estimated mean rate of weight change.
During PMCS, the postsynaptic neuron and the presynaptic neuron (left) received stimuli at frequency f. Stimulus deliveries (red arrows) were phase shifted by a phase lag ϕ ∈ [0, 1]. Each stimulus could trigger several spikes (vertical bars). Presynaptic spikes traveled along the axon and arrived with a time delay ta at the synapse (blue). Backpropagating postsynaptic spikes traveled along the dendrites and arrived with a delay time td at the synapse. Interspike intervals ϵ are marked by horizontal arrows. Indices count interspike intervals during respective PMCS cycles. Dashes refer to interspike intervals from the previous PMCS cycle. Suffixes ‘post’ and ‘pre’ indicate interspike intervals of the postsynaptic or presynaptic neurons, respectively.