Chapter 4 Sensitivity analysis of discharge patterns of subthalamic nucleus in the model of basal ganglia in Parkinson disease

Parkinson disease is one of the many diseases that are collectively known as movement disorders. We know that deficiency of dopamine is one of the major cause of Parkinson disease. Dopamine is produced in the area of brain called substantia nigra pars compacta (SNc). Parkinson disease leads to the death of SNc neurons that produce dopamine [7, 31]. which further leads to the deficiency of dopamine. As Parkinson disease progresses, the amount of dopamine produced in the brain decreases, leaving a person normally unable to control movement. The cause to the depletion of dopamine is not well known, and


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hence there is presently neither diagnosis no cure is available at early stage.There are treatment options at the later stages of the disease, such as medication and surgery to manage its symptoms [33,44,83,84].
It has been acknowledged by researchers that Parkinson disease neurological disorder can be characterized on the basis of non-motor and motor features and it has a central origin in the brain [85].Researchers also focus on the dopamenergic neurons and their localization but still the localization to the origin is not clear [82].Models of the basal ganglia and recent physiological studies have different aspects about and role of neuronal activities in the pathophysiology of Parkinson disease and oscillations and synchrony has been considered as one of the sign for Parkinson disease [86,87].There are few hypothesis about oscillation, bursting and tremor generation as proposed in [88,89].Some researchers suggest that the circuits in basal ganglia are itself oscillating and bursting generating circuits [90,13,61,91,92].While other researchers [13] suggested that a quantitative measure of motor signs would enable physicians to strengthen the medication regime more easily for a specific patient.suggested that subthalamic nucleus plays an important role in motor function and also suggested the role of calcium current in the regulation of subthalamic neuron.Dorian [91] suggested that non-linear propagation of density discharge contains meaningful information that is only revealed during spiking activity.

Model under consideration
The basis of this study is a single-compartment conductance based model of basal ganglia [31,7].It focuses on the importance of interaction between subthalamic nucleus and globus pallidus external neurons in indirect and hyper-direct pathway in the basal ganglia network [7,93]

Analysis of electrophysiological properties
This section presents the analysis of the electrophysiological properties of the STN-GPe neurons within the subthalamic neuron, which is is an oval-shaped small nucleus, that receives inhibitory and excitatory inputs from other neurons within basal ganglia as shown in Figure 3.1 [7].In this study, subthalamic nucleus and globus pallidus external has been considered as single neuron and also that subthalamic nucleus received input directly from cortex in hyper-direct pathway.Different studies [78,45] reveals that electrophysiological experiments results may differ from case to case.
Computation of correlation coefficient is described below: Here subthalamic nucleus in healthy primate is termed as STNH and in Parkinson disease condition is termed as STNP.The CCF k between ST N H t and ST N P t+k is called the k th order cross correlation of ST N H and ST N P .The sample estimate of this correlation coefficient, called CCF k , is calculated using the formula, where k (Lag) is considered to be zero.Most of our results are for k = 0 and are denoted by CCF unless mentioned otherwise in (4.1).This equation is derived by taking the reference from http://paulbourke.net/miscellaneous/correlate.

Sensitivity analysis for feedback membrane potential
In this sub-section, I have studied the effects on STN-GPe model by varying feedback membrane potential V f s .The strength of feedback mechanism between STN-GPe loop become stronger with increase in V f s .It has been found experimentally that STN-GPe feedback mechanism is affected by dopamine deficiency.The question now is the dopamine deficiency which leads to disruption of feedback mechanism has on the activity patterns of subthalamic nucleus.In our model the strength of the feedback is modeled through feedback potential V f s .As V f s decreases (increases) the feedback decreases (increases).
Our study shows that variation of V f s has negligible effect on the activity patterns.This has been shown in with the value of V f s .Figure 4.5 shows that there is lag between STNH and STNP but is much smaller as compared to I app and V presyn .This can also be seen for Figure 4.6, which shows that CCF ≡ 0.6 with increase in V f s .Figure 4.6 also shows that increase in CCF is negligible.Hence our earlier statement that variation of V f s in the activity patterns leads to negligible change is proved.Results are also simulated for sodium and potassium ionic currents, which is described in the section given below.

Analysis of firing patterns for sodium membrane potential
In order to further study the behavior of model, I have fixed other parameters as in [1] and checked the sensitivity with respect to sodium membrane potential V N a .This is another important parameter after the calcium potential which needs to be observed in Parkinson disease.Initial value of V N a was 55mV for both the cases i.e. healthy primate and Parkinson disease condition.In order to check the sensitivity of Parkinson disease patterns with respect to V N a , I have increased it in the range from 50mV to 95mV and seen the trend of spiking.varies with variation in all the reported parameters.These results can further be used to study the effect of calcium membrane potential on the model with respect to other intrinsic parameters like synaptic conductance and incorporating sodium, potassium and also the time to observe the results of the study.
. To observe the behavior of activity patterns in subthalamic nucleus and globus pallidus external (STN-GPe) network in healthy primate and Parkinson disease condition, improved mathematical model discussed in chapter 3 by equation (3.10) has been considered [1].Analysis and simulation work of STN-GPe network is performed using using ODE45 in MATLAB 7.14 (i7 Intel processor, 4GB RAM machine) for time period from 0 to 250 ms for calcium and from 0 to 500 ms for the other parameters under observation.The objective of this analysis is to provide insights into major causes for sensitivity of the model in Parkinson disease.Therefore, in this work the activity patterns generated in subthalamic nucleus model, in healthy primate and Parkinson disease condition, has been analyzed, to explore the dynamics of the STN-GPe loop subjected to various ions.This analysis of electrophysiological properties compares the spiking patterns generated in healthy primate and in Parkinson disease condition, and a distinction is made between the two types.Comparison between discharge patterns in healthy primate and Parkinson disease condition is performed by computing correlation coefficient (CCF) between these two.Correlation coefficient (CCF) is measure of similarity of two series as a function of the lag of one relative to the other.
ST N P ) j − (ST N P )) 2 (4.1)To analyze and quantify the sensitivity of subthalamic nucleus model, the comparison between healthy primate (STNH) and Parkinson disease condition (STNP) has been performed for various ionic and synaptic currents.To investigate the model assumptions about physiological properties and connectivity patterns that lead to the best explanation of (i.e.closest fit to) our electrophysiological data, I studied different activity patterns generated by the mathematical descriptions presented above.All the discharge patterns were generated using (3.10).The simulations are run for 250ms and 500ms respectively.I am displaying the results for parameters which are very sensitive for the model.These parameters are applied current I app , feedback membrane potential V f s , presynaptic potential V presyn and V Ca .The discharge patterns showing sensitivity tradeoffs as compared to the discharge patterns obtained from [1], are presented in the following sections.All the figures have been plotted as a function of time.Detailed sensitivity analysis and results are given in the following sections.While simulating the above model I have used parameters as given in [1].4.4 Results and discussions 4.4.1 Sensitivity analysis for applied current Activity patterns displayed within STN-GP network in basal ganglia are typically irregular and are correlated with the activity inside these cells.Firstly, I have analyzed the activity patterns with respect to applied current I app .Initial reference value of I app =32 pA/mm 2 is taken from [1].To analyze the sensitivity of these patterns against applied current I app , I varied the value of I app in the range 30 to 37 pA/mm 2 .These patterns have shown the sensitivity for I app for both STNH and STNP.Comparison of the two time series STNH and STNP in Figure 4.1 and Figure 4.2 show that STNP lags the STNH time series.For all values of I app there is lag and lag is also increasing with time.Values of parameters is given in Table 3.2 Minor change in the value of applied current shows sensitivity tradeoff in the correlation coefficient of STNH and STNP computed as shown in Figure 4.3.It is observed from Figure 4.3 that maximum correlation coefficient attained is .0666at I app = 34pA/mm 2 between I app = 30 to 37 pA/mm 2 .Increasing the value of I app gradually from 30 to 37 pA/mm 2 does not improve the correlation coefficient between the two discharge patterns of STN and STNP.Thus the Parkinson disease patterns (STNP) are not very sensitive to variability in I app .4.4.2 Sensitivity analysis for pre-synaptic membrane potentialV presyn is the membrane potential of a pre-synaptic neuron.The effect of V presyn on synaptic variables has been analyzed, which ultimately affect subthalamic nucleus model in healthy primate and Parkinson disease condition.The reference value of V presyn was considered as 31mV by[1].Pre-synaptic potential may increase or decrease depending upon the connections between neighboring neurons.If the connections are disrupted, it

Figure 4 . 1 :
Figure 4.1: Discharge patterns generated in STN-GPe network of basal ganglia in healthy primate for I app = 32pA/mm 2 and Parkinson disease condition at I app = 30pA/mm 2 and keeping other parameters fixed [1].

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Figure 4.2: Discharge patterns generated in STN-GPe network of basal ganglia in healthy primate for I app = 32pA/mm 2 and Parkinson disease condition at I app = 34pA/mm 2 and keeping other parameters fixed [1].

Figure 4
Figure 4.4: Discharge patterns generated in healthy primate at V presyn = 31mV and Parkinson disease condition at V presyn = 30.5mV Figure 4.5 and Figure 4.6.Figure 4.6 shows that correlation coefficient is constant for different values of V f s i.e.CCF ≡ 0.6 for V f s = 1mV-3mV, CCF≡ 0.61 for V f s = 4-7 mV and CCF ≡ 0.62 for V f s = 8-15mV.Correlation coefficient increases along After simulating the behavior of STN-GPe model for various intrinsic parameters, it has been observed that there was a lag in the Parkinson discharge patterns as compared to healthy primate patterns and this lag was increasing with time.To investigate the model assumptions about physiological properties and connectivity patterns that lead

Figure 4 . 5 :Figure 4 Figure 4 Figure 4
Figure 4.5: Discharge patterns generated in healthy primate at V f s = 0mV and Parkinson disease condition at V f s = 1mV to 3mV

Figure 4 . 9 :
Figure 4.9: Spiking patterns generated for healthy primate V Ca = 140mV for Parkinson disease condition at V Ca = 235mV Figure 4.11 shows the spiking trend of Parkinson disease and Parkinson disease patterns for this range.From Figure 4.11, I can see easily compare the spiking behavior of healthy primate and Parkinson disease patterns.The spiking range is from -80mV to 80mV for Parkinson disease patterns (STNP) and from 28mV to 42mV for healthy primate (STNH) patterns.Healthy primate discharge patterns are irregular whereas Parkinson disease patterns are regular comparatively, but spiking occurs in awider range.The correlation is also varying between a wide range of sodium value.The maximum value attained in the range V N a for 55mV to 95mV is 0.07 (approx.).Further the behavior of patterns has been analyzed in V N a range from 55mV to 275mV.Figure4.13 show the spiking trend, which has become slow for Parkinson disease and peak are also occurring at the same value i.e spikes are occurring at a fixed peak value i.e. 32.

Figure 4 .Figure 4
Figure 4.14 clearly shows that the maximum correlation coefficient CCF = 0.13 has been attained at V N a = 120mV (approx) and then it fluctuate between the range 0 to 0.08.

Figure 4 .Figure 4
Figure 4.12: Correlation coefficient between healthy primate at V N a = 55mV and Parkinson disease condition at V N a = 50mV to 95mV

Figure 4 .
Figure 4.14: Correlation coefficient between healthy primate at V N a = 55mV and Parkinson disease condition at V N a = 50mV to 300mV

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Figure 4.15: Spiking trend for healthy primate at V k = −80mV and for Parkinson disease condition at V k = −77mV

Figure 4 .Figure 4
Figure 4.16: Correlation coefficient between healthy primate at V k = −80mV and Parkinson disease condition at V k = −80mV to −40mV

Figure 4 .
Figure 4.18: Correlation coefficient between healthy primate at V k = −80mV and Parkinson disease condition at V k = −100mV to 150mV

Table 4
This information reveals a linear shift between STNH and STNP and Table4.1:Correlation coefficient for subthalamic nucleus at V presyn = 31mV and for Parkinson disease condition at modulated value of V presyn k(Lag) CCF at V presyn =30.5 mV CCF at V presyn =31 mV CCF at V presyn =31.5 mV