Figure 1.
Framework of the basal ganglia-corticothalamic network.
Neural populations include excitatory pyramidal neurons (EPN);
inhibitory interneurons (IIN);
thalamic reticular nucleus (TRN);
specific relay nuclei (SRN);
striatal D1 neurons;
striatal D2 neurons;
substantia nigra pars reticulata (SNr);
globus pallidus external (GPe) segment;
subthalamic nucleus (STN). Note that we do not model the globus pallidus internal (GPi) segment independently but consider SNr and GPi as a single structure in this work. Red lines with arrow heads denote the excitatory projections mediated by glutamate receptors. Blue solid and dashed lines with round heads represent the inhibitory projections mediated by GABAA and GABAB receptors, respectively. Compared with the traditional model of corticothalamic system on absence seizures, the basal ganglia are also included in our biophysical model.
Table 1.
Default parameter values.
Figure 2.
Absence seizure activities induced by the slow kinetics of
receptors in TRN. A, B: Bifurcation diagrams of
as a function of the TRN-SRN inhibitory coupling strength
(A) and the delay parameter
(B), respectively. Four different dynamical states can be observed from the time series of
, which are: the saturation state (I), the SWD oscillation state (II), the simple oscillation state (III) and the low firing state (IV). C: Typical time series of
correspond to the above four dynamical states. Here we set
and chose
(I),
(II),
(III),
(IV), respectively. The colors in bifurcation diagrams (A) and (B) correspond to the typical time series plotted in (C). D, E: The state analysis (D) and frequency analysis (E) in the (
) panel. Different colors in (D) represent different dynamical state regions, corresponding to those dynamical states given in (A), (B) and (C). The yellow asterisk regions surrounded by black dashed lines in (D) and (E) represent the SWD oscillation regions falling into the 2–4 Hz frequency range. The other symbols in (D) and (E) are linked to parameter values used for different typical time series in (C): I (white filled circle), II (red filled square), III (yellow filled triangle), and IV (purple filled star). For all simulations, we set
.
Figure 3.
Control of absence seizures by the isolated SNr-TRN pathway.
A, B: The state analysis (A) and frequency analysis (B) in the () panel. Here
is the inhibitory coupling strength of the TRN-SRN pathway, whereas
is the excitatory coupling strength of the STN-SNr pathway. Different colors in (A) represent different dynamical state regions: the saturation region (I), the SWD oscillation region (II), the simple oscillation region (III) and the low firing region (IV). The suppression of SWDs appears to the right of the white dashed line in (A), where the white down arrow indicates that the SWD oscillation can be inhibited by decreasing
. The yellow asterisk region surrounded by black dashed lines in (B) denotes the typical 2–4 Hz SWD oscillation region. C, D: Bifurcation diagrams of
as a function of
for different
. The strengths of the inhibitory projections from the TRN to SRN are set as
(C) and
(D), respectively. Different colors in (C) and (D) represent different dynamical state regions, corresponding to those in phase diagram (A). E: The mean firing rates (MFRs) of several key neural populations as a function of
, with
. Here four neural populations are considered: SNr (blue dot), excitatory pyramidal neurons (green asterisk), SRN (black circle) and TRN (red square). Two black dashed lines in (E) represent the occurring positions of the low and high triggering mean firing rates (TMFRs), respectively. F: The low (red filled circle) and high (green filled square) TMFRs as a function of
. For all simulations, the SNr-SRN pathway is artificially blocked (i.e.,
).
Figure 4.
Control of absence seizures by the isolated SNr-SRN pathway.
A, B: The state analysis (A) and frequency analysis (B) in the () panel. Here
is the inhibitory coupling strength of the TRN-SRN pathway, whereas
is the excitatory coupling strength of the STN-SNr pathway. Similar to previous results, four different dynamical state regions are observed: the saturation region (I), the SWD oscillation region (II), the simple oscillation region (III) and the low firing region (IV), which correspond to those defined in Fig. 2 (D). The region between two white dashed lines in (A) represents the suppression region of SWDs, where the white up arrow indicates that the SWD oscillation can be inhibited by increasing
. The yellow asterisk region surrounded by black dashed lines in (B) denotes the typical 2–4 Hz SWD oscillation region. C: Two different types of SWD suppressions caused by linearly increasing
. Top (C1): The value of
as a function of time. Middle (C2): Corresponding
trace for
. Bottom (C3): Corresponding
trace for
. D: The MFRs of several key neural populations as a function of
, with
. Here four neural populations are considered: SNr (blue dot), excitatory pyramidal neurons (green asterisk), SRN (black circle) and TRN (red square). The black dashed line in (D) represents the occurring position of TMFR. E: The TMFR as a function of
. Note that the SNr-TRN pathway is artificially blocked (i.e.,
) for all simulations.
Figure 5.
Control of absence seizures by the combination effects of the SNr-TRN and SNr-SRN pathways.
A, B: The state analysis (A) and frequency analysis (B) in the () panel. Here
is the scale factor, and
is the excitatory coupling strength of the STN-SNr pathway. Unlike previous results, only three dynamical state regions are observed in the phase diagram (A): the SWD oscillation region (II), the simple oscillation region (III) and the low firing region (IV), corresponding to the dynamical states defined in Fig. 2(D). For relatively weaker scale factor
, both increase and decrease in the activation level of SNr can inhibit the SWDs (double arrow, bidirectional suppression); whereas for sufficiently strong
, only reducing the activation level of SNr suppresses the SWDs (single arrow, unidirectional suppression). In (A), the white dashed line represents the boundary of suppression region, and the red dashed line stands for the demarcation between the bidirectional and unidirectional suppression regions. The yellow asterisk region surrounded by black dashed lines in (B) denotes the SWD oscillation region that falls into the 2–4 Hz frequency range. C: The MFRs of several key neural populations as a function of
, with the scale factor
. Here four neural populations are considered: SNr (blue dot), excitatory pyramidal neurons (green asterisk), SRN (black circle) and TRN (red square). Two black dashed lines in (C) represent the occurring positions of low and high triggering mean firing rates (TMFRs), respectively. D: The low (red filled circle) and high (green filled square) TMFRs as a function of
. For all simulations, the coupling strength of the TRN-SRN pathway is set as
.
Figure 6.
Effect of inhibitory coupling strength
on the control of absence seizures by the SNr-TRN and SNr-SRN pathways. Tow-dimensional state analysis (A) and corresponding frequency analysis (B) in the (
) panel for different values of
. Here only three dynamical state regions are observed in (A): the SWD oscillation region (II), the simple oscillation region (III) and the low firing region (IV). In (A1)–(A4), the regions marked by red diamonds denote the whole suppression regions of SWDs, the white dashed lines represent the boundaries of suppression regions, and the red dashed lines stand for the demarcations between the bidirectional (double arrow) and unidirectional (single arrow) suppression regions. In (B1)–(B4), the yellow asterisk regions surrounded by black dashed lines denote the typical 2–4 Hz SWD oscillation regions. From left to right, the strengths of inhibitory projections from the TRN to SRN are:
(A1, B1),
(A2, B2),
(A3, B3), and
(A4, B4), respectively. For better showing and comparing the bidirectional suppression regions among different subfigures, here we mainly consider the scale factor for all values of
within the same and small interval from 0 to 1. As an additional comparison, both the state analysis and frequency analysis for
in a relatively larger
interval is given in Fig. S2.
Figure 7.
An example illustrating that the bidirectional control of absence seizures by the basal ganglia is also available for the corticothalamic loop transmission delay induced SWDs.
A, B: Tow-dimensional state analysis (A) and corresponding frequency analysis (B) in the () panel. Here
is the scale factor, and
represents the excitatory coupling strength of the STN-SNr pathway. Four different dynamical states are observed in (A): the saturation state (I), the SWD oscillation state (II), the simple oscillation state (III) and the low firing state (IV), which also correspond to previous figures. In (A), the region between two red dashed lines denotes the main bidirectional suppression region of SWDs, where the double arrow represents both increasing and decreasing the excitatory coupling strength
can inhibit the SWDs. In (B), the yellow asterisk region surrounded by black dashed lines denotes the typical 2–4 Hz SWD oscillation region. (C) Three typical time series of
for different values of
, with
. Here we choose
(top),
(middle) and
(bottom), respectively. In all simulations, we set
,
,
,
,
and
.