Fig 1.
(A) The architecture of the reduced model consists of a single excitatory N1 and a single inhibitory N2 neural population that are reciprocally connected, constituting a minimal version of a system capable of generating oscillations. The excitatory and inhibitory projections are indicated by the blue arrow and the red circle, respectively. The neuronal population N1 receives a constant synaptic input H1. On the other hand, the neuronal population N2 receives an external current H2 constituting the electrical stimulation pattern (DBS signal). (B) Phase diagram of the intrinsic dynamics of the reduced model. The parameters G1 and G2 correspond to the synaptic efficacies of the efferent projections of the neural populations N1 and N2, respectively. The thick blue line corresponds to the Hopf bifurcation above which the system shows an oscillatory activity (light blue area). The red dot indicates the values of the synaptic efficacies used in the simulations (see Table 1).
Table 1.
Parameters for the reduced model corresponding to a state of stable oscillatory activity in the β-band.
Fig 2.
(A) The architecture of the model is composed of five populations of neurons. Three populations (C, Th, STN) are constituted by excitatory neurons, while the other two (St, GPi) are sets of inhibitory neurons. The excitatory and inhibitory projections are indicated by blue arrows and red circles, respectively. The model includes the hyperdirect (C-STN-GPi) and the direct (C-St-GPi) pathways which operate as competing feedback loops, HL and DL, via the efferent GPi-Th-C projections to close the BG-thalamocortical motor loop. (B) The neurons (light blue spheres) of each population are arranged on a ring, so that the model includes one-dimensional spatial structure of the BG nuclei through the angular coordinate θ. The effective volume of the tissue activated (angle highlighted in blue) in the STN and the effective volume of neural tissue contributing to the LFP signal in the C (angle highlighted in red) are defined by the function F in Eq 11. The probability of connection between two neurons belonging to two connected BG nuclei (green bell-shaped curve in the GPi), depends on the angular distance between those neurons and it is given by the function h in Eq 8. Neurons in the same population are not interconnected.
Table 2.
Values of the coupling parameters for the detailed network model.
Table 3.
Threshold values of the semi-linear transfer function for the detailed network model.
Fig 3.
Conditions for pole-zero cancellation.
(A) For clinically relevant DBS parameters (fDBS ≳ 100 Hz, δ ∼ 60-100 μs), the pole-zero cancellation produces the total suppression of frequencies above the β-band, specifically, in the γ-band from 30 to 40 Hz (thick red curve). (B) Stimulation amplitude as a function of the stimulation frequency fDBS for different pulse widths δ. This result shows that the required amplitude
decreases with increasing fDBS and δ, suggesting that the pole-zero cancellation mechanism depends only on the power delivered by the stimulation.
Fig 4.
Response of the reduced model in the case of periodic stimulation patterns.
The graphs (A-D) show the β-Power, root mean square value of the activity and power spectra computed from the synaptic current I1 corresponding to the neural population N1, when a periodic stimulation pattern configured with a pulse width of δ = 0.5 ms was applied to the population N2. (A) β-Power and the root mean square value of the activity as a function of the stimulation amplitude for a constant clinically relevant stimulation frequency of fDBS = 130 Hz. (B) β-Power and the root mean square value of the activity as a function of the stimulation frequency fDBS, keeping the stimulation amplitude unchanged at
. The frequency for the period doubling bifurcation fd ∼30 Hz and the frequency for activity total suppression fs ∼220 Hz, are indicated with solid vertical lines in order to highlight the transition points between the three states of the system: (1) Null activity (fDBS > fs), (2) β-Power suppression (fd < fDBS < fs) and (3) Pathological β-Power (fDBS < fd). In the graphs (A) and (B) the β-Power has been normalized with respect to that obtained for the system in the oscillatory state without stimulation (
). (C) Power spectrum obtained for the stimulation frequency configured at fDBS = 28 Hz < fd. In this case, the power spectral component in the β-band 10-20 Hz (dashed vertical lines), is produced by the period doubling induced by the stimulation frequency. The inset of the graphs show the resulting I1 signal which has a fundamental period given by the natural frequency of the system
. (D) Power spectrum obtained for the stimulation frequency configured at fd < fDBS = 50 Hz < fs. In this case, the stimulation pattern produces the suppression of the oscillations in the β-band without nullifying the activity A1. Thus, the period of the I1 signal (1/fDBS) is determined by the stimulation pattern (see the inset) and the resulting power spectral components correspond to the fundamental frequency of the stimulation fDBS and its harmonics.
Fig 5.
Frequency for activity total suppression (fs) and frequency for period doubling (fd), obtained from the reduced model.
(A) Effect on the of increasing the pulse width δ from 0.5 ms to 3.5 ms. The dashed vertical lines indicate the frequency for activity total suppression fs determined from the
curve. The solid vertical line indicates the frequency for period doubling fd determined from the β-Power curve (Fig 6A). (B) Transition frequencies fs (blue crosses) and fd (solid red circles) as functions of the
product. The fs and fd values were determined from the
plot (Fig 5A) and β-Power plot (Fig 6A), respectively. The analytical curve of fs values (solid blue line) is given by the Eq 16.
Fig 6.
Period doubling transition in the reduced model.
(A) β-Power as a function of the stimulation frequency (fDBS) using the pulse width δ as a parameter. The β-Power has been normalized with respect to that obtained for the system in the oscillatory state without stimulation (). The solid vertical line indicates the stimulation frequency at which the period doubling bifurcation occurs (fd). (B) Synaptic current I1 when fDBS = 30 Hz + ϵ (top), fDBS = 30 Hz − ϵ (bottom), where ϵ ∼ 0.6 Hz.
Fig 7.
β-Power as a function of the stimulation frequency (fDBS) for different pulse shapes of the stimulation pattern.
The β-Power has been normalized with respect to that obtained for the system in the oscillatory state without stimulation (). This result suggest that the β-Power is invariant to the change in the shape of the pulses, and only depends on the area of the pulse constituting the stimulation pattern.
Fig 8.
β-Power and the root mean square value of the activity as functions of the coefficient of variation vf, in the case of the reduced model.
The β-Power and the were computed from the synaptic current I1 corresponding to the neural population N1, for non-periodic stimulation patterns configured with
and δ = 0.5 ms applied to the population N2. The β-Power has been normalized with respect to that obtained for the system in the oscillatory state without stimulation (
). The non-periodic stimulation patterns were generated using random instantaneous frequencies with a mean value of 〈fDBS〉 ∼ 130 Hz. Each point in the graph corresponds to the mean value and the standard deviation of 10 realizations for each vf value (see section “Materials and methods”).
Fig 9.
Response of the network model in the case of periodic stimulation patterns.
The graphs (A-B) show the β-Power and root mean square value of the activity computed from the cortical LFP signal (Eq 9), when a periodic stimulation pattern configured with a pulse width of δ = 0.5 ms was applied to the STN nuclei. β-Power and the root mean square value of the activity as a function of the stimulation frequency fDBS, keeping the stimulation amplitude unchanged at . The β-Power has been normalized with respect to that obtained for the system in the oscillatory state without stimulation (
). Besides, the
correspond to the root mean square value of the activity averaged over all the cortical neurons. The frequency for the period doubling bifurcation fd ∼ 25 Hz and the frequency for activity total suppression fs ∼ 130 Hz, are indicated with solid vertical lines in order to highlight the transition points between the three states of the system: (1) Null activity (fDBS > fs), (2) β-Power suppression (fd < fDBS < fs) and (3) Pathological β-Power (fDBS < fd).
Fig 10.
Response of the network model in the case of periodic stimulation patterns.
The graphs (A-B) show the β-Power and root mean square value of the activity computed from the cortical LFP signal (Eq 9), when a periodic stimulation pattern configured with an amplitude of was applied to the STN nuclei. (A) β-Power as a function of the stimulation frequency fDBS using the pulse width δ as a parameter. The β-Power has been normalized with respect to that obtained for the system in the oscillatory state without stimulation (
). The solid vertical line indicates the stimulation frequency at which the period doubling bifurcation occurs (fd). (B) Effect on the
of increasing the pulse width δ from 0.5 ms to 2 ms. The dashed vertical lines indicate the frequency for activity total suppression fs determined from the
curve. The solid vertical line indicates the frequency for period doubling fd determined from the β-Power curve (graph (A)). The
correspond to the root mean square value of the activity averaged over all the cortical neurons.
Fig 11.
β-Power and the root mean square value of the activity as functions of the coefficient of variation vf, in the case of the network model.
The β-Power and the were computed from the cortical LFP signal (Eq 9), when non-periodic stimulation patterns configured with
and δ = 0.5 ms were applied to the STN nuclei. The β-Power has been normalized with respect to that obtained for the system in the oscillatory state without stimulation (
). The
correspond to the root mean square value of the activity averaged over all the cortical neurons. The non-periodic stimulation patterns were generated using random instantaneous frequencies with a mean value of 〈fDBS〉 = 130 Hz. Each point in the graph correspond to the mean value and the standard deviation of 10 realizations for each vf value (see section “Materials and methods”).
Fig 12.
Spike rasters together with the spike histograms for STN neurons corresponding to the network model.
(A) Pathological oscillatory state without stimulation (). (B) Pathological oscillatory state under a periodic stimulation pattern with fDBS = 130 Hz. (C) Pathological oscillatory state under a non-periodic stimulation pattern with 〈fDBS〉 = 130 Hz and vf = 90%. (D) Pathological oscillatory state under a non-periodic stimulation pattern with 〈fDBS〉 = 130 Hz and vf = 70%. In the plots (B), (C) and (D), the stimulation patterns were applied on the STN nuclei and configured with
and δ = 0.5 ms.