Figure 1.
(A) Illustrating a relay neuron. Ensemble activity of all the distal synapses (stars) is modulating input . The proximal synapses (diamonds) form the driving input
. The output is the axonal voltage
. (B) A block diagram of a relay neuron showing two inputs and output
.
Table 1.
Details of function in (1).
Table 2.
Parameter's values in (1).
Figure 2.
(A) Illustrates the equilibrium point , the steady state orbit
and the orbit tube,
, for
given by (3) and
. The orbit tube is shown for
. (B) Illustrates
, the threshold voltage
and threshold current
. Note that these parameters are defined by the undriven system (9). (C) Illustrates the critical hypersurface
, a successful response trajectory, an unsuccessful response trajectory, and the refractory zone,
for the undriven system (9). The time it takes for the solution to leave
after generating a successful response is called the refractory period,
. Note that refractory zone depends on
and therefore
also depends on
. Additionally, note that the region shaded in the darker grey is also in the refractory zone, because if
is in this region then
such that
Therefore, a successful response cannot be generated if
is in this region by definition. (D) Dependence of
on
. Note that
is approximately a straight line with slope
, i.e
. (E) Illustrates
vs
and
.
Figure 3.
Illustrates the critical hypersurface , which defines the threshold for a successful response.(9) generates a successful response for any initial condition that is to the right of the hypersurface i.e.
. Whereas, any initial condition to the left of the hypersurface results in unsuccessful response.
Figure 4.
Illustrates and
. When an
pulse arrives, the solution jumps from
to
. Now, whether the neuron generates a successful response or not is governed by the local dynamics. Therefore, we linearize (4) about
to analyze the behaviour of
for
. If a successful response is generated,
such that
else if an unsuccessful response is generated
such that
.
Figure 5.
Plots the theoretical and numerically computed reliability as a function of , with
. The dotted lines are the lower and upper bounds on reliability from the (48) and (47), respectively. The solid line is
calculated by running simulations of (1), and the error bars indicate
.
Figure 6.
Plots the theoretical and numerically computed reliability as a function of , with
. B. Plots the theoretical and numerically computed reliability as a function of
with
,
. The dotted lines are the lower and upper bounds on reliability from the (48) and (47), respectively. The solid line is
calculated by running simulations of (4), and the error bars indicate
.
Figure 7.
Dependence of and
on model parameters.
A. Plots as a function of
B.
(see (35) versus
and
. Note that
depends largely upon
, whereas its dependence upon
is minimal.
changes the maximum value of
but does not effect it much in the high frequency range.
Figure 8.
A. Voltage profile of the 3rd order model in the bursting mode () B. zoomed in view of a burst C.
vs
for the
order model.
In this Figure, we illustrate the results from a 3rd order model of a thalamic neuron. A. Plots the voltage profile obtained from the model in response to pulses in . Note that each pulse in
either generates a burst of spikes or does not spike at all. B. Zoomed in view of a burst. C. Plots the theoretical and numerically computed reliability as a function of
, with
,
,
. The dotted lines are the lower and upper bounds on reliability from the (48) and (47), respectively. The solid line is plots
calculated by running simulations of (4), and the error bars indicate
. We estimated
as the minimum height of a
pulse that makes the neuron generate a successful response.
Figure 9.
A. Voltage profile of the 3rd order model in the tonic mode () B. zoomed in view of a spike C.
vs
for the
order model.
In this Figure we illustrate the results from a 3rd order model of a thalamic neuron. A. Plots the voltage profile obtained from the model in response to pulses in . Note that each pulse in
either generates a successful spike or generates unsuccessful spike. B. Zoomed in view of a successful spike. C. Plots theoretical and numerically computed reliability versus
, with
,
,
,
,
,
. The dotted line is plotting the lower and upper bounds on reliability from the (48) and (47), respectively. Note that here
, therefore
. The solid line plots
calculated by running simulations of (4), and the error bars indicate
. We estimated
as the minimum height of a
pulse that makes the neuron spike.
Table 3.
Parameters and functions for (50).
Figure 10.
Thalamocortical loop in motor signal processing.
(A) Simplified view of basal ganglia thalamo-cortical motor signal processing. Sensorimotor cortex generates the driving input and projects to the motor thalamus. The thalamus relay of cortical input is modulated by the basal ganglia (BG). (B) Relay reliability curves computed from our analysis as a function of and
from (49). (C) Simulations of
(basal ganglia output) from the computational study [15] for the Healthy, PD and PD with high frequency deep brain stimulation (HFDBS) cases. As we can see in the healthy case, the amplitude of the BG output,
, is smaller compared to the PD BG output, resulting in a higher relay reliability. HFDBS increases the frequency,
, of the BG output, resulting in a higher relay reliability. (D) Intuition of how reliability changes in the three cases. In PD,
is larger, therefore, the diameter of the orbit tube is larger compared to the orbit tube for healthy. This results in more time spent in the unsuccessful response region
, which leads to poor reliability. In contrast, in PD case with HFDBS applied,
is larger and the gains
decrease, which generates a smaller orbit tube. In this case, the state spends more time in the successful response region
of the orbit tube, resulting in high reliability.