**2.2 Ionic currents**

510 Etiology and Pathophysiology of Parkinson's Disease

that dopamine acts on the calcium dependent potassium channels, possibly by increasing its conductance. By this way, the level of potassium that leaves the cell increases, which consequently inhibits neuron spiking. Thus, the GABA release becomes inhibited – for more

Given such a structure, external stimuli *X* and *Y* are projected through excitatory pathways to neighboring thalamic regions, *Tx* and *Ty*, respectively. Once stimulated, *Tx* activates *TRNx* beyond collaterals of an ascending glutamatergic projection, whose final destiny is the PFC. Since our work does not model the PFC explicitly, such excitatory projection ends up in *TRN* in the model. Also, through an excitatory glutamatergic descending pathway, the cortical region enhances the activation of *Tx*, and also sends collateral axons to *TRNx*. Thus, once activated, the *TRNx* inhibits the thalamic region *Ty* through a GABAergic inhibitory projection. Summarizing, the thalamocortical circuit activation by an external stimulus *X* excites a central thalamic region *Tx* and inhibits its neighborhood, represented by the neuron *Ty*. As the mesothalamic dopamine inhibits neurons in *TRN,* a rise in the dopaminergic level contributes to deactivating such cells. This leads to a more active thalamic region *Ty*. Conversely, a reduction in the dopaminergic level activates *TRN*, and increases the inhibition of *Ty*. A symmetrical case involves the *Ty* and *TRNy* neurons, as illustrated in

**Prefrontal Cortex** 

**+ +** 

**SN**

**-** 

**-** 

**-** 

Fig. 1. The modeled thalamocortical network architecture: excitatory glutamatergic synapses (arrows), inhibitory GABAergic synapse (black sphere), inhibitory dopaminergic synapse

**X Y** 

**+ +** 

*Tx Ty* 

**+** 

*TRNy*

details see (Madureira et al., 2010).

*TRNx*

**+** 

Figure 1.

(white sphere).

To better investigate alternate thalamic states, this model incorporates physiological features related to the tonic and the bursting modes of thalamic spikes. Thus, we address the neuron spike by considering the sodium, *INa*, and the calcium, *ICa*, currents, which depolarize the neuron, and the potassium current, *IK*, which restores the cellular membrane potential (Kandel, 2000). With relation to the patterns of intervals between spikes, associated to different potassium currents (McCormick et al., 1995), our model incorporates the calcium dependent-potassium current, *Ic*, particularly, in the *TRN* neurons: it is a transient current, whose amplitude increases with intracellular calcium concentration, and it suffers dopaminergic influence (Florán et al., 2004; Madureira et al., 2010). Essentially associated to the bursting mode, the *Iahp*, a current that underlies neural hyperpolarization, is modeled as described below.

The model deals with a network of ionic and synaptic events that leads to a specific mode of spiking. Particularly, under this approach we examine if dopamine is able to influence the spikes of thalamic neurons. In this model we assume that a high inhibitory dopamine action on D4 receptors in TRN neurons hyperpolarizes such cells, thus facilitating the activation of calcium currents. As a consequence, and according to thalamic properties (Carvalho, 1994), even a small membrane depolarization is capable of triggering an action potential, due to the low threshold calcium-currents. Whenever the inflow of Ca2+ in the cell, due to spikes, increases the Ca2+ concentration above a threshold, then the hyperpolarizing current *Iahp*, becomes activated, hyperpolarizing the neuron. Therefore, the calcium currents become activate – or remain activated, depending on their previous state – and a cyclic or oscillatory behavior takes place in the TRN neuron. This is the burst thalamic mode of spiking, associated to sleep states (Steriade et al., 1993; Llinás & Steriade, 2006). It may be interrupted due to the calcium currents inactivation, which occurs whenever the neuron does not suffer hyperpolarization for around 100ms.

Based on this model, we speculate another possibility that is directly related to the PD origins: the generation of bursting in thalamic neurons, due to a strong inhibition imposed by TRN neurons. Such situation is plausible to occur in case of mesothalamic hypodopaminergy, which allows the inhibitory TRN neurons to become atypically over stimulated (Madureira et al., 2010, Florán et al., 2004).

Because of the dopaminergic modulation in the TRN, two types of neurons are modeled: the thalamic ones, *Tx* and *Ty*, and the *TRN* neurons. Both are single point spiking and are presented next.

#### **2.2.1 Thalamic neurons**

We define a simplified neuron model with a single compartment where dendrites, soma and axons are concentrated, and whose electric potential is *V*. The neural membrane is modeled according to the equation:

Mesothalamic Dopaminergic Activity: Implications in Sleep Alterations in Parkinson's Disease 513

d [ ]

The following equation describes how its conductance, *gahp*, behaves with respect to the *Iahp*

1, if *[Ca]* ≥ ΘCa

0, if *[Ca]* < ΘCa.

*ahp ahp ahp ahp*

τ

*gf g*

β

Once in the activated state, the *ICa* current facilitates the occurrence of LTS, and the consequent increase of the calcium concentration, *[Ca]*. As a result, the conductance *gahp*

Since TRN and thalamic neurons main properties are similar, the equations for the *TRN* neuron are quite similar to the ones considered in 2.2.1, except for the inclusion of the

We assume the final target of dopaminergic action is the calcium-dependent potassium channel, whose ionic current is *Ic* (Madureira et al., 2010; Florán et al., 2004). Thus, the

> d *TRN k ahp c syn l <sup>V</sup> C II II I*

Indeed, *Ic* = *gc* (*Ek* - *V*), where *Ek* represents the potassium reversal potential, and *gc* the

The conductance *gc* suffers dopaminergic influence, via D4 receptor, and depends on the

stands for the dopaminergic action on *gc* , and *S*( *[Ca]* ) stands for a sigmoid function of the intracellular calcium concentration, which increases by virtue of a neural spiking. We set

> ( ) ( ) <sup>1</sup> [ ] 1 exp [ ]

where the constant *a* controls the slope of *S*, and rhe calcium concentration behaves according to the equation ( 2 ). Indeed, in ( 2 ) the term *s* raises the calcium concentration

*a Ca* <sup>=</sup> + −

=+ ++ +

 *D4*\* *S*( *[Ca]* ), where *ĝc* is a constant, *D4*\*

<sup>−</sup> <sup>=</sup>

d d

membrane equation for the *TRN* neuron incorporates the ionic current *Ic* as:

*t*

d

*S Ca*

where *βahp* represents a variation rate of *gahp*, and *τahp* a time constant.

grows, and the neuron suffers a hyperpolarization.

*f([Ca]) =* 

intracellular calcium concentration. Thus, *gc* = *ĝ<sup>c</sup>*

calcium-dependent potassium current *Ic*.

conductance of ionic current *Ic*.

*t*

where *βCa* represents the rate of calcium concentration variation, and *τCa* a time constant. If the intracellular calcium concentration reaches a given threshold ΘCa, the potassium channels related to the hyperpolarizing current are opened. The step function f([Ca])

*Ca s Ca*

β

*Ca Ca*

τ

<sup>−</sup> <sup>=</sup> (2)

d

describes such event as:

current:

**2.2.2** *TRN* **neurons** 

*t*

$$C\_T \frac{\mathbf{d} \, V}{\mathbf{d}t} = I\_k + I\_{alpy} + I\_{syn} + I\_{l,s}$$

where we recall that *Ik* represents the potassium current, *Iahp* the hyperpolarizing current, *Isyn* the dendritic current induced by synaptic action, and *Il* the leak current, i.e., currents that are not modeled.

Considering the relation between membrane voltage and ionic currents, as

$$\begin{aligned} I\_k &= \mathbf{g}\_k \left( E\_k - V \right), \\ I\_{alp} &= \mathbf{g}\_{alp} \left( E\_k - V \right), \\ I\_l &= \mathbf{g}\_l \left( E\_l - V \right), \\ I\_{syn} &= \mathbf{g}\_{syn} \left( E\_{syn} - V \right) \end{aligned} \tag{1}$$

where the constants *Ek*, *Eah*p, *El* and *Esyn* are reversal potentials for the currents *Ik*, *Iahp*, *Il* and *Isyn*, respectively, and *gk*, *gahp*, *gl* and *gsyn* represent the conductances corresponding to these currents.

The occurrence of a spike is associated to a step function *s(V)*, whose unitary value indicates the action potential depolarizing phase, depending on the *INa* or *ICa* currents:

$$s(V) = \begin{cases} 1, \text{ if } V \ge \theta\_{\text{spike}} \\\\ 0, \text{ if } V < \theta\_{\text{spike}} \end{cases}$$

where θspike is a voltage threshold for the channel opening. It is defined as:

$$\theta\_{\rm spike} = \left\{ \begin{array}{c} \theta\_{\rm Na\nu} \text{ if } I\_{\rm Ca} \text{ is activated} \\\\ \theta\_{\rm Ca\nu} \text{ if } I\_{\rm Ca} \text{ is activated} \\\\ \end{array} \right\} $$

where, θNa is the voltage threshold for the sodium channels opening, and θCa for the calcium channels. We have θNa > θCa, and the spikes triggered by the calcium ionic channels are the low threshold spikes (LTS).

During the network activity, the membrane potential, *V*, is monitored. When strong inhibitory events lead to periods of hyperpolarization, around 100ms, the *ICa* currents become activated (Carvalho, 1994). Once in activity, *ICa* currents cause the LTS.

Following a spike, the conductance *gk* of the restoring current *Ik* increases rapidly, bringing the neuron back to a resting potential. Such a process is described by

$$\frac{\mathbf{dg}\_k}{\mathbf{dt}} = \frac{s\boldsymbol{\beta}\_k - \mathbf{g}\_k}{\sigma\_k}$$

where the constant *βk* represents a variation rate of *gk*, and *τ<sup>k</sup>* a time constant associated to the potassium channel.

According to the frequency of spikes, the calcium concentration increases – and decreases due to calcium buffers and pumps (Carvalho & Roitman, 1995). Then, we have:

$$\frac{\text{d}\left[\text{Ca}\right]}{\text{d}t} = \frac{s\beta\_{\text{Ca}} - \text{[Ca]}}{\tau\_{\text{Ca}}} \tag{2}$$

where *βCa* represents the rate of calcium concentration variation, and *τCa* a time constant. If the intracellular calcium concentration reaches a given threshold ΘCa, the potassium channels related to the hyperpolarizing current are opened. The step function f([Ca]) describes such event as:

$$f(\operatorname{[Ca]}) = \left\{ \begin{array}{c} \text{1, if } \langle \operatorname{Ca} \rangle \cong \Theta\_{\operatorname{Ca}} \\\\ \text{0, if } \langle \operatorname{Ca} \rangle \le \Theta\_{\operatorname{Ca}}. \end{array} \right\}$$

The following equation describes how its conductance, *gahp*, behaves with respect to the *Iahp* current:

$$\frac{\mathbf{d} \,\mathbf{g}\_{alip}}{\mathbf{d}t} = \frac{f \,\mathcal{J}\_{alip} - \mathbf{g}\_{alip}}{\sigma\_{alip}}$$

where *βahp* represents a variation rate of *gahp*, and *τahp* a time constant.

Once in the activated state, the *ICa* current facilitates the occurrence of LTS, and the consequent increase of the calcium concentration, *[Ca]*. As a result, the conductance *gahp* grows, and the neuron suffers a hyperpolarization.

#### **2.2.2** *TRN* **neurons**

512 Etiology and Pathophysiology of Parkinson's Disease

d *T k ahp syn l <sup>V</sup> C II I I*

where we recall that *Ik* represents the potassium current, *Iahp* the hyperpolarizing current, *Isyn* the dendritic current induced by synaptic action, and *Il* the leak current, i.e., currents that

( )

*k kk ahp ahp k l ll*

*I gE V I g EV I gE V*

= −

= − = −

( )

*syn syn syn*

*I gE V*

= −

where the constants *Ek*, *Eah*p, *El* and *Esyn* are reversal potentials for the currents *Ik*, *Iahp*, *Il* and *Isyn*, respectively, and *gk*, *gahp*, *gl* and *gsyn* represent the conductances corresponding to these

The occurrence of a spike is associated to a step function *s(V)*, whose unitary value indicates

1, if V ≥ θspike

0, if V < θspike,

where, θNa is the voltage threshold for the sodium channels opening, and θCa for the calcium channels. We have θNa > θCa, and the spikes triggered by the calcium ionic channels are the

θCa, if *ICa* is activated,

θNa, if *ICa* is inactivated

During the network activity, the membrane potential, *V*, is monitored. When strong inhibitory events lead to periods of hyperpolarization, around 100ms, the *ICa* currents

Following a spike, the conductance *gk* of the restoring current *Ik* increases rapidly, bringing

*k kk k*

τ

β

<sup>−</sup> <sup>=</sup>

where the constant *βk* represents a variation rate of *gk*, and *τ<sup>k</sup>* a time constant associated to

According to the frequency of spikes, the calcium concentration increases – and decreases

*gs g*

become activated (Carvalho, 1994). Once in activity, *ICa* currents cause the LTS.

d d

due to calcium buffers and pumps (Carvalho & Roitman, 1995). Then, we have:

*t*

the neuron back to a resting potential. Such a process is described by

( )

,

,

,

(1)

( )

=+ + + ,

d

are not modeled.

currents.

low threshold spikes (LTS).

the potassium channel.

*t*

Considering the relation between membrane voltage and ionic currents, as

the action potential depolarizing phase, depending on the *INa* or *ICa* currents:

where θspike is a voltage threshold for the channel opening. It is defined as:

*s(V) =* 

θspike *=* 

Since TRN and thalamic neurons main properties are similar, the equations for the *TRN* neuron are quite similar to the ones considered in 2.2.1, except for the inclusion of the calcium-dependent potassium current *Ic*.

We assume the final target of dopaminergic action is the calcium-dependent potassium channel, whose ionic current is *Ic* (Madureira et al., 2010; Florán et al., 2004). Thus, the membrane equation for the *TRN* neuron incorporates the ionic current *Ic* as:

$$\mathbf{C}\_{TRN} \stackrel{\mathbf{d}}{\mathbf{d}} \frac{\mathbf{d}V}{\mathbf{d}t} = I\_k + I\_{alup} + I\_c + I\_{syn} + I\_R$$

Indeed, *Ic* = *gc* (*Ek* - *V*), where *Ek* represents the potassium reversal potential, and *gc* the conductance of ionic current *Ic*.

The conductance *gc* suffers dopaminergic influence, via D4 receptor, and depends on the intracellular calcium concentration. Thus, *gc* = *ĝ<sup>c</sup> D4*\* *S*( *[Ca]* ), where *ĝc* is a constant, *D4* \* stands for the dopaminergic action on *gc* , and *S*( *[Ca]* ) stands for a sigmoid function of the intracellular calcium concentration, which increases by virtue of a neural spiking. We set

$$S\left(\left[Ca\right]\right) = \frac{1}{1 + \exp\left(-a\left[Ca\right]\right)}$$

where the constant *a* controls the slope of *S*, and rhe calcium concentration behaves according to the equation ( 2 ). Indeed, in ( 2 ) the term *s* raises the calcium concentration

Mesothalamic Dopaminergic Activity: Implications in Sleep Alterations in Parkinson's Disease 515

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

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

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

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

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

**Prefrontal Cortex** 

**+ +**

*Tx Ty* 

**-**

**+**

**X Y** 

**+ +**

*TRN* 

modulates the activation of TRN neurons and, consequently, that of the thalamic cells.

factors as drugs that alter the dopamine level throughout synaptic clefts.

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)).

**3. Computational simulations** 

variations in external or cortical stimuli.

**SN**

**3.1 Asymmetrical architecture** 

our major concern was the attentional focus formation.

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

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

$$D\_4^{\ast \ast} = \hat{\mathcal{g}}\_{d4} \sum\_{i=1}^{N} \left( t - t\_i \right) \exp[-\left( t - t\_i \right) \text{ / } t\_{pd}] \,\mathrm{s}$$

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 et al., 2010).

#### **2.3 Synaptic projections**

Finally, we present the synaptic modeling (Carvalho, 1994). For the synaptic conductance *gsyn*, appearing in Equation (1), it follows that

$$\mathcal{g}\_{syn} = \hat{\mathcal{g}}\_{syn} \sum\_{i=1}^{N} \left( t - t\_i \right) \exp\left[ - \left( t - t\_i \right) \mid t\_p \right],$$

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 thalamus.

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 and inhibitory synapses respectively.

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 present glossaries with all necessary parameter values.


Table 1. Glossary of parameters.
