Desynchronizing electrical and sensory coordinated reset neuromodulation

Abstract

Coordinated reset (CR) stimulation is a desynchronizing stimulation technique based on timely coordinated phase resets of sub-populations of a synchronized neuronal ensemble. It has initially been computationally developed for electrical deep brain stimulation (DBS), to enable an effective desynchronization and unlearning of pathological synchrony and connectivity (anti-kindling). Here we computationally show for ensembles of spiking and bursting model neurons interacting via excitatory and inhibitory adaptive synapses that a phase reset of neuronal populations as well as a desynchronization and an anti-kindling can robustly be achieved by direct electrical stimulation or indirect (synaptically-mediated) excitatory and inhibitory stimulation. Our findings are relevant for DBS as well as for sensory stimulation in neurological disorders characterized by pathological neuronal synchrony. Based on the obtained results, we may expect that the local effects in the vicinity of a depth electrode (realized by direct stimulation of the neurons' somata or stimulation of axon terminals) and the non-local CR effects (realized by stimulation of excitatory or inhibitory efferent fibers) of deep brain CR neuromodulation may be similar or even identical. Furthermore, our results indicate that an effective desynchronization and anti-kindling can even be achieved by non-invasive, sensory CR neuromodulation. We discuss the concept of sensory CR neuromodulation in the context of neurological disorders.

Keywords: anti-kindling; coordinated reset neuromodulation; electrical stimulation; neuronal synchronization; sensory stimulation; spike timing-dependent plasticity.

Figure 1

Desynchronized and synchronized collective dynamics of the HH ensemble (1), (2). (A) Time courses of the membrane potentials V_j of three selected neurons of a desynchronized neuronal ensemble and the post-synaptic potential s_i(t) (multiplied by a factor of 100) (black curve) triggered by the blue action potentials. (B), (C) Raster plots of the neuronal firing (left plots) and the corresponding normalized histograms of the spiking frequencies (number of spikes per second averaged over 100 s, right plots) for (B) the desynchronized regime, and (C) the synchronized regime. The corresponding DC-balanced LFPs (multiplied by a factor of 1000) are depicted by red curves at the bottom of the raster plots. Parameters I_i are randomly and uniformly distributed in the interval [10.55, 11.45], coupling c_ij = 0 in plots (A) and (B), and c_ij = 0.5 in plot (C).

Figure 2

Desynchronized and synchronized collective dynamics of the FHR ensemble (5), (2). (A) Time courses of the membrane potentials V_j of three selected neurons of the desynchronized neuronal ensemble and the post-synaptic potential s_i(t) (multiplied by a factor of 2) (black curve) triggered by the blue action potentials. (B), (C) Raster plots of the neuronal firing (left plots) and the corresponding normalized histograms of the bursting frequencies (number of bursts per second averaged over 500 s, right plots) for (B) desynchronized regime, and (C) synchronized regime. The corresponding DC-balanced LFPs (multiplied by a factor of 400) are shown by red curves at the bottom of the raster plots. Parameters I_i are randomly and uniformly distributed in the interval [0.347, 0.353], coupling c_ij = 0 in plots (A) and (B), and c_ij = 0.2 in plot (C).

Figure 3

Stimulation setup for electrical and sensory CR stimulation. (A) Schematic localization of four stimulation sites within the neuronal population and the corresponding spatial profiles (7) of current decay in the neuronal tissue with the distance from the stimulation site. (B) Stimulation signals of electrical CR stimulation composed of short pulse trains of charge-balanced pulses (8). (C) Stimulation signals of sensory CR stimulation composed of post-synaptic potentials modeled by α-functions (9).

Figure 4

Left plot: normalized plasticity function (12) vs. the difference of the spike/burst timing of post- and pre-synaptic neuron Δt = t_post −t_pre. Right plot: The average “net” STPD effect Δ c(ε) for uniformly distributed Δt ∈ [−ε, ε] (see text for details). Parameters τ = 1, β₁ = 1, β₂ = 5.3, γ₁ = 5, and γ₂ = 4.

Figure 5

Plasticity-induced multistability of synchronized and desynchronized states in the FHR ensemble (5) with STDP (12). (A) Time courses of the mean synaptic weights C(t) = N⁻² ∑_i,j sgn(M_ij)c_ij(t) for different initial coupling matrices [c_ij(0)] whose elements are Gaussian distributed around the mean value c₀ (indicated in the legend) with standard deviation 0.005. (B) Time courses of the DC-balanced LFP observed in the stable weakly coupled and desynchronized regime (green curve) and strongly coupled and synchronized regime (red curve). (C) The corresponding coupling matrices, where the excitatory synaptic weights are suppressed and the inhibitory connections are potentiated in desynchronized regime (left plot) and the opposite situation in the synchronized regime (right plot). Parameters N = 200, δ = 0.005, β₁ = 1, β₂ = 5.3, γ₁ = 5, γ₂ = 4, τ = 350, and the other parameters as in Figure 2.

Figure 6

Plasticity-induced multistability of synchronized and desynchronized states in the HH ensemble (1) with STDP (12). (A) Time courses of the mean synaptic weights C(t) (see Figure 5 for definition) for different initial coupling matrices [c_ij(0)] whose elements are Gaussian distributed around the mean value c₀ (indicated in the legend) with standard deviation 0.01. (B) Time courses of the DC-balanced LFP of four stable synchronized and desynchronized regimes observed for the initial coupling matrices with mean value c₀ indicated in the legend. (C) The corresponding coupling matrices developed in the neuronal ensemble due to STDP for the above four stable states. The initial mean values c₀ are indicated in the plots. Parameters N = 200, δ = 0.002, β₁ = 1, β₂ = 16, γ₁ = 1/0.12, γ₂ = 1/0.15, τ = 14, and the other parameters as in Figure 1.

Figure 7

Cross-trial diagrams of the distribution densities of the mean phase Φ_k(t) averaged across stimulation trials for (A)–(C) synchronized FHR neurons (5), (2), and (D)–(F) synchronized HH neurons (1), (2) without STDP. Note, for the phase resetting analysis, STDP is turned off to avoid any long-term effects of the stimulation and to guarantee that after the post-stimulus transients the network has re-established its natural, pre-stimulus dynamics. In plots (A) and (D) direct electrical stimulation was administered, whereas in plots (B) and (E) indirect (synaptically-mediated) excitatory stimulation and in plots (D) and (F) indirect (synaptically-mediated) inhibitory stimulation was used, see Section 2.4 for details. In color diagrams (left column) the values of the phase distribution densities are encoded in color ranging from 0 (blue) to (A) 2, (B) 3, (C) 4, and (D)–(F) 1 (red). The white curves depict the phase resetting index E(t) with the scale indicated on the right vertical axis. The plots in the right column illustrate the corresponding phase distribution densities which are shown for fixed times in the pre-stimulus interval (blue curves) and in the post-stimulus interval (red curves) as indicated in the legends. Stimulation strength (A) K = 2, (B) K = 0.2, (C) K = 0.8, (D) K = 50, (E) K = 0.4, and (F) K = 2. The other parameters as in Figure 1C for HH neurons and in Figure 2C for FHR neurons.

Figure 8

Stimulation-induced rewiring and desynchronization of the FHR ensemble (5), (2) with STDP (12) by CR stimulation. The time courses of the mean synaptic weights C(t) (see Figure 5 for definition) are shown for different stimulation intensities K as indicated in the legends. The stimulation time interval is indicated by the red bar in plot (A) and by vertical dashed lines for (A) direct electrical stimulation, (B) indirect (synaptically-mediated), e.g., sensory excitatory stimulation, and (C) indirect (synaptically-mediated), e.g., sensory inhibitory stimulation administered to a strongly coupled and synchronized regime as in Figure 5 for c₀ = 0.4. Other parameters as in Figures 2 and 5.

Figure 9

Stimulation-induced rewiring and desynchronization of the HH ensemble (1), (2) with STDP (12) by CR stimulation. The time courses of the mean synaptic weights C(t) (see Figure 5 for definition) are plotted for different stimulation intensities K as indicated in the legends. The stimulation time interval is indicated by the red bar in plot (A) and by vertical dashed lines for (A) direct electrical stimulation, (B) indirect (e.g., sensory) excitatory stimulation, and (C) indirect (e.g., sensory) inhibitory stimulation administered to a strongly coupled and synchronized regime as in Figure 6 for c₀ = 0.5. Other parameters as in Figures 1 and 6.

Figure 10

Stimulation-induced rewiring of the FHR ensemble (5), (2) with STDP (12) by CR stimulation versus the stimulation intensity K. Dashed and solid green curves depict the mean synaptic weights C_on and C_off registered during (A) direct electrical, (B) indirect (synaptically-mediated, e.g., sensory) excitatory, and (C) indirect (synaptically-mediated, e.g., sensory) inhibitory stimulation and at the end of the post-stimulation transient, respectively. The red solid curves show the order parameter 〈 R(t)〉 time averaged over the last 3 s of the post-stimulation transient, see the scale on the right vertical axis. Vertical dashed lines indicate the parameter interval K ∈ (K₁, K₂), where the corresponding CR stimulation is effective. Number of neurons N = 100, and the other parameters as in Figures 2, 5, and 8.

Figure 11

Stimulation-induced rewiring of the HH ensemble (1), (2) with STDP (12) by CR stimulation versus the stimulation intensity K. Dashed and solid green curves depict the mean synaptic weights C_on and C_off registered during (A) direct electrical, (B) indirect (e.g., sensory) excitatory, and (C) indirect (e.g., sensory) inhibitory stimulation and at the end of the post-stimulation transient, respectively. The red solid curves show the order parameter 〈 R(t)〉 time averaged over the last 2 s of the post-stimulation transient, see the scale on the right vertical axis. Vertical dashed lines indicate the parameter interval K ∈ (K₁, K₂), where the corresponding CR stimulation is effective. Number of neurons N = 200, and the other parameters as in Figures 1, 6, and 9.

Figure 12

Stimulation-induced rewiring and desynchronization of the FHR ensemble (5), (2) with STDP (12) and with 20% of randomly selected inhibitory connections by indirect (synaptically-mediated) mixed excitatory-inhibitory CR stimulation. (A) The time courses of the mean synaptic weights C(t) (see Figure 5 for definition) are shown for different fractions of the neuronal population of randomly selected neurons receiving an inhibitory stimulation as indicated in the legend. The rest of the neuronal ensemble receives an excitatory stimulation. The stimulation time interval is indicated by the red bar and by vertical dashed lines. (B) Time courses of the DC-balanced LFP observed in the stable strongly coupled and synchronized regime (red curve for the mean coupling C(t) ≈ 0.38) and weakly coupled and desynchronized regime (green curve for the mean coupling C(t) ≈ −0.07). (C) The corresponding coupling matrices, where the excitatory synaptic weights are potentiated and the inhibitory connections are suppressed in the synchronized regime (left plot) and the opposite situation in the desynchronized regime (right plot). Parameters M_ij of the spatial profile of coupling (2) are uniformly and randomly distributed in the interval [−0.2, 0.8], initial mean coupling c₀ = 0.48, stimulation strength K = 1, and the other parameters as in Figures 2 and 5.