**3. Computational simulations**

514 Etiology and Pathophysiology of Parkinson's Disease

whenever there is a neural spiking. Therefore *gc* increases and inhibits the cell, if the cell is

The dopaminergic action on *gc* is modeled by the summation of alpha functions (Carvalho, 1994) representing the rise and the decrease of the dopaminergic level, in each of the *N*

ˆ exp[ / ]

4 4 () ()

= − −−

*D di i pd g tt tt t*

Here, the constant *tpd* stands for the peak time for the alpha function, and *ĝd4* is the conductance constant of the dopaminergic projection. For further details, see ref. (Madureira

Finally, we present the synaptic modeling (Carvalho, 1994). For the synaptic conductance

() ()

ˆ exp /

= − −−

*syn syni i <sup>p</sup> g g tt tt t*

where *ĝsyn* is the maximal conductance, which assumes different values for each particular synapse. In fact, each modeled synapse has a specific associated conductance, reflecting its influence: *ĝc-trn* and *ĝc-t* for synapses between the cortex and the TRN, and between the cortex and the thalamus, respectively; *ĝt-trn* and *ĝtrn-t* for synapses between the thalamus and the TRN, and vice versa; and *ĝe-t* for synapses between somatosensory projections and the

The synaptic conductance *gsyn* is also represented by a summation of alpha functions, for each of the *N* presynaptic spikes that occurred at times *ti* before *t*, for *1 ≤ I ≤ N.* We denote by *tp* the peak time for the alpha function, and it assumes the values *tpe* and *tpi* for excitatory

We used the ANSI C ® programming language to implement the model. The differential equations are integrated by the Euler's method. Ref. (Madureira et al., 2010) and Table 1

I*ahp* Hyperpolarizing potassium current (μA.cm-2) g*ahp* Conductance of I*ahp* (m.mhos.cm-2) θ*Ca* Threshold for calcium channel's opening ( 0 mV) θ*Na* Threshold for sodium channel's opening (1 mV)

activation (mM)

Intracellular calcium concentration threshold for I*ahp*

β*ahp* Variation rate of g*ahp* (100) τ*ahp* Time constant of g*ahp* (2 ms)

1

i 1

=

*N*

=

i

*N*

presynaptic spikes that occurred at times *ti*, before *t*, with 1 ≤ *i* ≤ *N*:

\*

excited beyond a threshold.

**2.3 Synaptic projections** 

*gsyn*, appearing in Equation (1), it follows that

and inhibitory synapses respectively.

Table 1. Glossary of parameters.

present glossaries with all necessary parameter values.

et al., 2010).

thalamus.

Θ*Ca*

Due to a mechanism of inhibitory feedback between thalamic and TRN neurons in the thalamocortical circuit, when a projected stimulus on the central thalamic area *Tx* is propagated for posterior cortical processing, its neighboring thalamic area *Ty* suffers inhibition from *TRN*. This property was highly explored in (Madureira et al., 2010), because our major concern was the attentional focus formation.

Here, we explore such inhibitory feedback to inspect how the activity degree in the TRN influences the thalamic excitatory state. Summarizing, our simulations illustrate how dopamine modulates the activation of TRN neurons and, consequently, that of the thalamic cells.

With relation to the dopaminergic action in the TRN, we assume a relationship between the level of mesothalamic dopamine released in the TRN and the nigral spiking frequency. Consequently, we simulate variations in the level of dopamine released in the TRN by varying the SN spiking pattern. We also address the dopamine receptor D4 degree of activity, through the term *ĝd4* in (3). Indeed, *ĝd4* model the weight of the connection between the SN and *TRN* neurons. Then, *ĝd4* tells us how much receptor D4, in the TRN, is affected by the dopamine release due to a nerve impulse from SN, or due to the action of exogenous factors as drugs that alter the dopamine level throughout synaptic clefts.

Overall, throughout these simulations our major concern is the dopaminergic effect on the thalamocortical dynamics. We do not intend to focus our exploration on the consequences of variations in external or cortical stimuli.

#### **3.1 Asymmetrical architecture**

In this section, we describe a series of simulations performed using an artificial neural network that presents the architecture illustrated in the Figure 2. Since such network is the one used in (Madureira et al., 2010), we set it as our departure point.

Fig. 2. The asymmetrical network architecture (from (Madureira et al., 2010)).

Mesothalamic Dopaminergic Activity: Implications in Sleep Alterations in Parkinson's Disease 517

initiate with a low SN spike frequency, characterizing the mesothalamic dopamine hypoactivity, and raise the SN activity in successive steps. Table 3 presents our simulations results. Overall, we gather that as the mesothalamic dopaminergic activity decreases, the *TRN* neurons become more excited. Also, *Tx* becomes be more inhibited than *Ty*, enlarging thus the difference between the activations of *Tx* and *Ty*, as showed in Table 4. Therefore the mesothalamic dopamine hypoactivity forces the attention to focus on the stimulus *Y*, implying in attentional shifting difficulty and mental rigidity. On the other hand, the almost identical neural activity of *Tx* and *Ty*, enhanced by the mesothalamic dopamine hyperactivity, lead to a no-winner competition between stimuli *X* and *Y,* which may represent distraction or lack of attentional focus. These results are compatible with the ones

Spikes in 100ms

Tx Ty TRNx TRNy (100ms – 200ms) (400ms – 500ms)

11 0 25 14 2 14 14 28

14 0 23 9 4 14 16 25

12 0 21 10 7 13 13 21

18 0 17 0 10 15 5 10

18 0 0 0 18 19 0 0

In the next series of experiments, we keep up our focus on the tonic mode of spiking. And, for each imposed SN spike frequency, we examine the effects of changes in the receptor D4 activation. Since the different degrees of D4 activation can be associated to not modeled exogenous or endogenous factors, that are not modeled, this approach makes it possible to speculate plausible outcomes of the dopaminergic agonists (or antagonists) action at the synaptic cleft. The simulations results are summarized in Table 4**.** First, we may note that the results presented in the column relative to *ĝd4* = 1.0 agree with the previous set of experiments. It is more interesting, however, to observe that as the receptor D4 activity diminishes, the thalamic neurons become less active, thus reaching a completely inhibited state. Conversely, as the receptor D4 activity increases, thalamic neurons become more active and tend to spike at the same frequency. Finally, we highlight that, except for the baseline case where the interval between spikes in SN equals 10, when *ĝd4* assumes values lower than 1.0, the differences between *Tx* and *Ty* spiking frequencies disappear. So, the mesothalamic hypoactivity does not

provided by our previous model.

Table 3. SN spiking frequency and the thalamic tonic state.

**3.2.2 Receptor D4 activity and the attentional focus** 

impose the attention to focus on the stimulus *Y* anymore.

Interval between Spikes in SN (ms)

50

30

20

10

5

Here, we investigate if variations in the activity of receptor D4 in the TRN may influence the mode of spiking in neurons of the thalamic complex, along different SN spiking frequencies. We impose a drastic decrease in the nigral dopamine level, reflecting a disturbance in the mesothalamic system, and raise the dopaminergic level afterwards. Through the 500mssimulations, variations in the dopaminergic receptor activation are modeled by altering the parameter *ĝd4* after 250ms.

Table 2 describes all simulated variations in the SN spiking frequency, the imposed changes on receptor D4 activation, as well as the characteristic spike modes related to each situation. From these results, we gather that the bursting mode was elicited in two opposing situations: increase of D4 activation under mesothalamic hypoactivity (interval between spikes in SN equals 50 and above) and, decrease of D4 activation under mesothalamic hyperactivity (interval between spikes in SN equals 5).

In the first case, the mesothalamic hypoactivity turns the *TRNx* neuron so highly excited, that *Ty* becomes strongly inhibited, thus activating the calcium current. The posterior increase of the D4 activity, plausibly representing the effect of some dopaminergic agonist, was able to elicit LTSs. Consequently, the calcium concentration reached a threshold value that activated the hyperpolarizing current, promoting the oscillatory pattern in the thalamic cell *Ty*.


Table 2. Spiking modes examined through an asymmetrical network.

Conversely, in the second case, the mesothalamic hyperactivity generated the strong inhibition in the TRN neuron, which enabled the calcium currents activation. Then, the imposed decrease of D4 activity was sufficient to diminish the TRN inhibition, thus allowing LTSs and, finally, the bursting mode of spiking.

#### **3.2 Symmetrical architecture**

Following the first series of experiments, we extend the network architecture to incorporate the symmetry between neighboring thalamic areas. The symmetrical architecture is represented in the Figure 1**.** 

#### **3.2.1 SN spiking frequency and the attentional focus**

We start exploring the extended architecture by addressing the dopaminergic action in neurons under the tonic mode of spiking. Therefore, as a first approximation, we apply the mathematical model developed in (Madureira et al., 2010). In this set of simulations, a weaker stimulus *X* is presented to the network before the a stronger one, *Y.* Again, we

Here, we investigate if variations in the activity of receptor D4 in the TRN may influence the mode of spiking in neurons of the thalamic complex, along different SN spiking frequencies. We impose a drastic decrease in the nigral dopamine level, reflecting a disturbance in the mesothalamic system, and raise the dopaminergic level afterwards. Through the 500mssimulations, variations in the dopaminergic receptor activation are modeled by altering the

Table 2 describes all simulated variations in the SN spiking frequency, the imposed changes on receptor D4 activation, as well as the characteristic spike modes related to each situation. From these results, we gather that the bursting mode was elicited in two opposing situations: increase of D4 activation under mesothalamic hypoactivity (interval between spikes in SN equals 50 and above) and, decrease of D4 activation under mesothalamic

In the first case, the mesothalamic hypoactivity turns the *TRNx* neuron so highly excited, that *Ty* becomes strongly inhibited, thus activating the calcium current. The posterior increase of the D4 activity, plausibly representing the effect of some dopaminergic agonist, was able to elicit LTSs. Consequently, the calcium concentration reached a threshold value that activated the hyperpolarizing current, promoting the oscillatory pattern in the thalamic

Dopamine Receptor D4 (*ĝd4*) Neuron Spike Mode

Tonic to Bursting

Tonic to Bursting

Bursting

Changes in the Activity of

10 1.0 to 1.2 - Tonic 5 1.0 to 0.8 TRN Tonic to

Conversely, in the second case, the mesothalamic hyperactivity generated the strong inhibition in the TRN neuron, which enabled the calcium currents activation. Then, the imposed decrease of D4 activity was sufficient to diminish the TRN inhibition, thus allowing

Following the first series of experiments, we extend the network architecture to incorporate the symmetry between neighboring thalamic areas. The symmetrical architecture is

We start exploring the extended architecture by addressing the dopaminergic action in neurons under the tonic mode of spiking. Therefore, as a first approximation, we apply the mathematical model developed in (Madureira et al., 2010). In this set of simulations, a weaker stimulus *X* is presented to the network before the a stronger one, *Y.* Again, we

100, 150 and 200 1.0 to 1.2 Ty

50 1.0 to 1.2 Ty

Table 2. Spiking modes examined through an asymmetrical network.

LTSs and, finally, the bursting mode of spiking.

**3.2.1 SN spiking frequency and the attentional focus**

**3.2 Symmetrical architecture** 

represented in the Figure 1**.** 

parameter *ĝd4* after 250ms.

cell *Ty*.

Interval between Spikes in SN (ms)

hyperactivity (interval between spikes in SN equals 5).

initiate with a low SN spike frequency, characterizing the mesothalamic dopamine hypoactivity, and raise the SN activity in successive steps. Table 3 presents our simulations results. Overall, we gather that as the mesothalamic dopaminergic activity decreases, the *TRN* neurons become more excited. Also, *Tx* becomes be more inhibited than *Ty*, enlarging thus the difference between the activations of *Tx* and *Ty*, as showed in Table 4. Therefore the mesothalamic dopamine hypoactivity forces the attention to focus on the stimulus *Y*, implying in attentional shifting difficulty and mental rigidity. On the other hand, the almost identical neural activity of *Tx* and *Ty*, enhanced by the mesothalamic dopamine hyperactivity, lead to a no-winner competition between stimuli *X* and *Y,* which may represent distraction or lack of attentional focus. These results are compatible with the ones provided by our previous model.


Table 3. SN spiking frequency and the thalamic tonic state.
