Journal of Integrative Neuroscience 16 (2017) 441–452 DOI 10.3233/JIN-170027 IOS Press 441 Sensitivity analysis of discharge patterns of subthalamic nucleus in the model of basal ganglia in Parkinson disease Jyotsna Singh a,∗ , Phool Singh b and Vikas Malik c a Department of Computer Science & Engineering and Information Technology, The NorthCap University, Gurgaon, Haryana, India E-mail: singhjyotsna1@gmail.com b Department of Applied Sciences, The NorthCap University, Gurgaon, Haryana, India E-mail: phool.singh24@gmail.com c Department of Physics, JIIT Noida, Uttar Pradesh, India E-mail: vikasm76@gmail.com Received 21 January 2017 Accepted 23 May 2017 Abstract. Parkinson disease alters the information patterns in movement related pathways in brain. Experimental results performed on rats show that the activity patterns changes from single spike activity to mixed burst mode in Parkinson disease. However the cause of this change in activity pattern is not yet completely understood. Subthalamic nucleus is one of the main nuclei involved in the origin of motor dysfunction in Parkinson disease. In this paper, a single compartment conductance based model is considered which focuses on subthalamic nucleus and synaptic input from globus pallidus (external). This model shows highly nonlinear behavior with respect to various intrinsic parameters. Behavior of model has been presented with the help of activity patterns generated in healthy and Parkinson condition. These patterns have been compared by calculating their correlation coefficient for different values of intrinsic parameters. Results display that the activity patterns are very sensitive to various intrinsic parameters and calcium shows some promising results which provide insights into the motor dysfunction. Keywords: Spiking patterns, Parkinson disease, dopaminergic neuron, subthalamic nucleus, globus pallidus, hyper direct pathways 1. Introduction Brain disorders constitute a major health problem and pose a challenge to the society (Arnulfo et al. [1]; Dorsey et al. [10]; Dhar et al. [9]). These disorders are among the most puzzling of all the diseases, and our ignorance of the underlying disease mechanism is a major obstacle to the development of better diagnostic systems. One of the disease caused by brain disorder is Parkinson disease (PD). PD is the one of many diseases that are collectively known as movement disorders. PD primarily affects neurons in the area of the brain called substantia nigra pars compacta (SNc) (Kang and Lowery [17]; Santaniello et al. * Corresponding author. E-mail: singhjyotsna1@gmail.com. 0219-6352/17/$35.00 © 2017 – IOS Press and the authors. All rights reserved 442 J. Singh et al. / Sensitivity analysis of STN in PD [28]). These dying neurons produce dopamine, a chemical that sends messages to the part of the brain that controls movement and coordination along with other activities. As PD 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 hence there is presently neither diagnosis no cure at early stage. There are treatment options at the later stages of the disease, such as medication and surgery to manage its symptoms (Bergman et al. [3]). Basal ganglia has four main nuclei, which are (1) substantia nigra, (2) striatum, (3) Subthalamic nucleus (STN) and (4) Globus pallidus (GP). Substantia nigra itself is further divided into two sub regions pars compacta and pars reticulata and GP is further divided into internal Globus pallidus (GPi) and external Globus pallidus (GPe). It has strong connections with the cerebral cortex, thalamus and other brain areas. The striatum receives input directly from cortex and delivers neurotransmitter which is inhibitory in nature to the GP through different pathways (1) direct and (2) indirect. STN directly receives inputs from cortex through (3) hyper direct pathway (Santaniello et al. [28]; Bergman et al. [3]; Hammond et al. [14]). The two pathways direct and indirect affect the basal ganglia network in opposite ways and are simultaneously involved in the control of voluntary movements (Kang and Lowery [17]). It has been acknowledged by researchers (Jankovic [15]; Gelb et al. [12]; Jellinger [16]; Rivlin-Etzion et al. [26]; Moran et al. [22]; Wichmann et al. [35]; Weinberger et al. [33]) that there is a central origin to PD but the localization to the origin is still not clear. There are few hypothesis about oscillation, bursting and tremor generation as proposed in (Jellinger [16]; Deuschl et al. [8]). Some researchers suggest that the circuits in basal ganglia are itself oscillating and bursting generating circuits (Llinás [21]; Guehl et al. [13]; Paré et al. [24]; Brewer et al. [6]; Song et al. [29]; Aur et al. [2]; Wichmann and DeLong [34]). While other researchers (Brewer et al. [6]) suggested that a quantitative measure of motor signs would enable physicians to strengthen the medication regime more easily for a specific patient. Wen-Jie et al. (Song et al. [29]) suggested that STN plays an important role in motor function and also suggested the role of calcium current in the regulation of STN Neuron. Dorian (Aur et al. [2]) suggested that non-linear propagation of density discharge contains meaningful information that is only revealed during spiking activity. To identify the cause to the origin of tremor and bursting generation and further to narrow down, we are considering a model of basal Ganglia as shown in Fig. 1. In this model, we focussed on the network of subthalmic nucleus (STN) and globus pallidus external (GPe). Connections are disrupted between different nuclei in PD condition (Santaniello et al. [28]). D1, the dopamine receptor, is excited by dopamine in healthy primate. It produces inhibitory neurotransmitter to GPi, which in turn suppresses the activity of GPi and increases the activation in thalamus. D2, the other dopamine receptor, when activated by dopamine, produces GABA to GPe. This suppresses GPe activities and excites the activities in STN and GPi. This result in the inhibitory activities in thalamus neuron. A possible viewpoint is that the bursting and oscillation generator circuits are basal ganglia thalamuscortical circuits (Deuschl et al. [8]). The basal ganglia circuits do not produce tremor oscillations in healthy basal ganglia circuits (Zirh et al. [36]; Raethjen et al. [25]; Lenz et al. [19]; Volkmann et al. [32]; Surmeier et al. [30]; Bevan et al. [5]). On the contrary, the tremor oscillations related activity is observed in STN in PD condition (Levy et al. [20]). However, the reason underlying tremor oscillations generation is still to be understood. Above studies suggest that STN could be considered as the target region to analyze the motor symptoms and the origin of PD. Therefore, in this paper we have analyzed the activity patterns generated in STN Model and compared the behavior of STN Model in healthy primate and in PD condition to explore J. Singh et al. / Sensitivity analysis of STN in PD 443 Fig. 1. Modified Basal ganglia-thalamo-cortical circuit (Dovzhenok and Rubchinsky [11]). Bold lines show the connection in healthy primate and dotted lines show the connection in Parkinson disease. the dynamics of the STN-GPe loop subjected to various currents. Work has been distributed in various sections. Section 2 explains the mathematical model and underlying methods. Section 3 is dedicated to the sensitivity based study of the model for various currents and their results. Section 4 is the conclusion of the work. 2. Methods and models The base of this study is a single-compartment conductance-based model of basal ganglia (Santaniello et al. [28]) in which STN receives excitatory input from cortex through hyper-direct pathway (Kang and Lowery [17]) and GPe receives inhibitory input from the striatum and excitatory input from STN. In addition, there are inhibitory synaptic connections within GPe cells. The STN receives inhibitory input from the GPe and excitatory neurotransmitters through cortex. It increases the frequency of the discharge patterns in GPi neurons (Kitai and Deniau [18]; Nambu et al. [23]). This result leads to the importance of interaction between STN and GP neurons in direct and hyper-direct pathway in the basal ganglia network (Kang and Lowery [17]; Lenz et al. [19]). To observe the behavior of activity patterns STN-GPe network in PD and healthy primate, mathematical model, conductance and time scales are taken from (Dovzhenok and Rubchinsky [11]). Analysis and simulation work of STN-GPe network is performed using MATLAB 7.14 (i7 Intel processor, 4GB RAM machine) using ODE45 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 was to provide insights into major causes for sensitivity of the model in PD. This model (Eq. (2.1)) (Dovzhenok and Rubchinsky [11]), includes a leak current (Il ), fast spikeproducing potassium (Ik ) and sodium currents (INa ), low threshold T-type (IT ) and high-threshold Ca2+ currents (ICa ), Ca2+ activated voltage independent after-hyperpolarization K + current (IAHP ), synaptic current (Isyn ) and applied current (Iapp ), so that the equation governing the membrane potential V takes the form: C dV = −Il − Ik − INa − IT − ICa − IAHP − Isyn + Iapp , dt (2.1) 444 J. Singh et al. / Sensitivity analysis of STN in PD where different membrane currents are given by (Dovzhenok and Rubchinsky [11]) Il = gl · [V − Vl ], (2.2) Ik = gk · n4 · [V − Vk ], (2.3) INa = gNa · m3∞ (V ) · h · [V − VNa ], (2.4) 3 · (V ) · r · [V − VCa ], IT = gT · a∞ (2.5) ICa = gCa · · (V ) · [V − VCa ], (2.6) [Ca] · [V − Vk ], [Ca] + k1 (2.7) 2 s∞ IAHP = gAHP · where k1 is the dissociation constant of Ca2+ dependent AHP current. Iapp current is used to adjust the membrane resting potential with the experimental data (Terman et al. [31]; Rubin and Terman [27]). Calcium current is described by the equation: d [Ca] = −ICa − IT − kCa[Ca] , dt (2.8) where characterizes the calcium influx and kCa is calcium pump rate. The equation for gating variable m, n, h, s and a is given as under dx φx [x∞ (V ) − x] = , dt τx (V ) (2.9) where φ is the constant in the equation for gating variables. The time constant function τx is given by τx (V ) = τx0 + τx1 , [1 + exp[−[V − θxτ ]/σxτ ]] (2.10) where θ is half activation/inactivation variable and τ is time constant functions. Synaptic current Isyn in the STN neuron is computed as a sum of synaptic currents from the GPe and other neighboring neurons (Dovzhenok and Rubchinsky [11]). Synaptic current Isyn is defined by considering the synaptic inputs from GP to STN and from feedback neurons to STN. Isyn = ggs · sg [V − Vgs ] + gf s · sf [V − Vf s ], (2.11) where ggs and gf s is the synaptic conductance and sg and sf are synaptic variables. As suggested in (Dovzhenok and Rubchinsky [11]), dopamine degeneration is one of the main cause to the Parkinson Disease. Synaptic inputs play a major role to show the degeneration of dopamine. To present this, we have considered two synaptic variables, sg and sf , to modulate the strength of synaptic input currents. sg and sf vary in (Gelb et al. [12]) range, so that the lower values of sg and sf correspond to lower dopamine levels and stronger conductance and higher values represent the opposite. All the synaptic J. Singh et al. / Sensitivity analysis of STN in PD 445 variables in the model circuit are modeled by the following 1st order kinetic equation, which describes the fraction of activation channels ds = α · H∞ (Vpresyn − g ) · [1 − s] − β · s, dt (2.12) where 1 H∞ (V ) = 1 + exp −[V −θgH ] σgH (2.13) is a sigmoid function, Vpresyn is the synaptic potential from neighboring neurons. The values of synaptic variables α, β, θ and σ are taken from (Terman et al. [31]). The values of synaptic strengths in the “normal” state (high dopamine level) are ggs = 0.695 and gf s = 0.215, and the maximal conductance of the AHP current in STN neuron was set to gAHP = 4.23 nS/mm2 . The values of synaptic strengths corresponding to the Parkinsonian (low dopamine level) state are ggs = 1.39, gf s = 0.43, with STN cell’s AHP conductance set to gAHP = 8.46 nS/mm2 . 3. Analysis of electrophysiological properties This section presents the analysis of the electrophysiological properties of the STN-GPe neurons within the STN neuron. STN is an oval-shaped small nucleus which receives inhibitory and excitatory inputs from other neurons within Basal ganglia as shown in Fig. 1 (Kang and Lowery [17]). In our study, we have considered STN and GPe as single neuron and also that STN received input directly from cortex in hyperdirect pathway. The analysis of electrophysiological properties compares the spiking patterns generated in healthy primate and PD, and a distinction is made between the two types. Different studies (Beurrier et al. [4]; Terman et al. [31]) reveals that electrophysiological experiments results may differ from case to case. The spiking patterns generated in healthy primate are termed as STN and those generated for PD are termed as STNP. Comparison between STN and STNP is performed by computing correlation coefficient (CCF) between these two. CCF is measure of similarity of two series as a function of the lag of one relative to the other. Correlation coefficient can be computed as described below: The CCF between STNt and STNPt+i is called the ith order cross correlation of STN and STNP. The sample estimate of this correlation coefficient, called CCFk , is calculated using the formula: n−k − (STN))((STNP)j +k − (STNP)) 2 n 2 1 ((STN)j − (STN)) 1 ((STNP)j − (STNP)) j =1 ((STN)j CCFk = n (3.1) To analyze and quantify the sensitivity of STN Model, the comparison between STN and 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, we studied different activity patterns generated by the mathematical descriptions presented above. All the discharge patterns were generated using Eq. (2.1). These discharge 446 J. Singh et al. / Sensitivity analysis of STN in PD patterns are then simulated for selected ionic current. The simulations are run for 500 ms and 250 ms respectively. We are displaying the results for parameters which are very sensitive for the model. These parameters are applied current Iapp , feedback membrane potential Vf s , presynaptic potential Vpresyn and VCa . The discharge patterns showing sensitivity tradeoffs as compared to the discharge patterns obtained from (Dovzhenok and Rubchinsky [11]), 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 we have used parameters as given in (Dovzhenok and Rubchinsky [11]). 3.1. 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 we have analyzed the activity patterns with respect to applied current Iapp . Initial reference value of Iapp = 32 pA/mm2 is taken from (Dovzhenok and Rubchinsky [11]). To analyze the sensitivity of these patterns against applied current Iapp , we varied the value of Iapp in the range 30 to 37 pA/mm2 . These patterns have shown the sensitivity for Iapp for both STN and STNP. We have displayed the discharge pattern for Iapp = 30 in Fig. 2. Figure 2 shows that the patterns of variation between two time series of STN and STNP is linear in nature, except that the time series of STNP is slightly delayed. But the correlation coefficient computed for Iapp between Iapp = 30 to 37 pA/mm2 shows non-linearity. Minor change in the value of applied current shows sensitivity tradeoff in the correlation coefficient of STN and STNP computed as shown in Fig. 3. It can be observed from Fig. 3 that maximum correlation attained is 0.0666 at Iapp = 34 pA/mm2 between Iapp = 30 to 37 pA/mm2 . Increasing the value of Iapp gradually from 30 to 37 pA/mm2 does not improve the correlation coefficient between the two time series of STN and STNP. 3.2. Sensitivity analysis for pre-synaptic membrane potential Vpresyn is the membrane potential of a pre-synaptic neuron. We have analyzed the effect of Vpresyn on synaptic variables which ultimately affect STN model in normal and PD condition. The reference value Fig. 2. Spiking patterns in STN-GPe Model in healthy and PD condition at Iapp = 30 pA/mm2 . J. Singh et al. / Sensitivity analysis of STN in PD 447 Fig. 3. Correlation coefficient for healthy and PD spiking patterns for Iapp = 30 to 37 pA/mm2 . Fig. 4. Discharge patterns generated in STN and STNP at Vpresyn = 30.5 mV. of Vpresyn was considered as 31 mV by (Dovzhenok and Rubchinsky [11]). Pre-synaptic potential may increase or decrease depending upon the connections between neighboring neurons. If the connections are disrupted, it may decrease and if the connections are strong, it may increase (de Lau and Breteler [7]). Hence, we have analyzed it for disrupted and strong connection. Change in the pre-synaptic value does not affect the correlation coefficient for STN model in healthy and PD condition. Activity patterns have been shown for Vpresyn = 30.5 mV in Fig. 4. Correlation coefficient of STN and STNP for reference range has been tabulated in Table 1. Table 1 shows non-linearity in different values of correlation coefficient when we compare them with activity patterns showing linear behavior. At lag = 0, correlation coefficient is maximum for Vpresyn = 31.5 mV. Correlation coefficient is improving at Vpresyn = 31 mV at lag = 3. It is observed that in STN model there is time lag in activity patterns in normal and PD condition. This information reveals a linear shift between STN and STNP and the cause is not yet identified. 448 J. Singh et al. / Sensitivity analysis of STN in PD Table 1 Cross correlation coefficient for Vpresyn Lag 0 −1 −2 −3 −4 1 2 3 4 CCF at Vpresyn = 30.5 mV 0.06109 −0.06389 −0.1434 −0.1613 −0.1693 0.1968 0.2592 0.2544 0.2844 CCF at Vpresyn = 31 mV 0.05992 −0.06501 −0.1456 −0.1631 −0.171 0.1932 0.2539 0.3037 0.2895 CCF at Vpresyn = 31.5 mV 0.063652 −0.062725 −0.14553 −0.16363 −0.17221 0.19683 0.25803 0.28 0.28479 Fig. 5. Discharge patterns generated in STN and STNP at Vf s = 1–3. 3.3. Sensitivity analysis for feedback membrane potential Vf s In this sub-section we have studied the effects on STN model by varying feedback membrane potential Vf s . However, the study of this model presents the occurrence of oscillations when STN-GPe loop becomes stronger. This phenomenon remains vigorous with respect to variations of various parameter which are dopamine dependent. It also remains vigorous for delays in the feedback loop. Whereas the delays remain unknown, which does not change in Parkinson disease (Dovzhenok and Rubchinsky [11]). The model is somewhat stable for various values of Vf s . This has been shown in Fig. 5 and Fig. 6. Figure 6 shows that correlation coefficient is constant for different values of Vf s i.e. CCF = 0.60 for Vf s = 1–3 mV, CCF = 0.61 for Vf s = 4–7 mV and CCF = 0.62 for Vf s = 8–15 mV. Correlation coefficient increases along with the value of Vf s . 3.4. Effect of calcium current on STN model After simulating the behavior of STN model for various intrinsic parameters, it has been observed that there was a linear shift in the Parkinson discharge patterns. To identify the cause of linear shift in the discharge patterns in PD condition, the model was simulated for VCa . It has been found out that calcium current greatly affected STN Model in PD condition. This can be clearly seen in the results displayed in J. Singh et al. / Sensitivity analysis of STN in PD 449 Fig. 6. Correlation coefficient for Vf s = 1–3, 4–7, 8–15. Fig. 7 for calcium current that shows the sensitivity of Parkinson discharge pattern to calcium current. Results displayed in Fig. 7(b) targeted towards the possibility of calcium to be the cause of linear shift in the Parkinson discharge patterns in comparison with patterns generated in healthy primate. In Fig. 7(a) we can observe the difference between activity patterns in STN and STNP at VCa = 140 mV. Initially the patterns are somewhat overlapping but gradually with time the process of generation of Parkinson discharge patterns is slowed down. Slowing down of discharge pattern might be due to the deficiency of various chemicals such as calcium, sodium and potassium. When we further analyzed the effect of VCa in the range from 100 mV to 350 mV for STNP. Results are displayed in Fig. 7(b). In this figure, we can see that the discharge patterns in Parkinson condition are overlapping with the discharge patterns in healthy primate. Correlation coefficient between these two patterns is then plotted for STN at VCa = 140 mV versus STNP at VCa = 235 mV in Fig. 7(c), which gives maximum correlation between the two. The correlation coefficient is approx. equal to 0.92. Therefore the higher the correlation coefficient, the better the results are for the parameter under observation. From this result we can conclude that increasing the concentration of calcium current improves the discharge patterns in PD. So calcium could be the major parameter under consideration which highly affects the STN model in PD. 4. Conclusion The inter-connection between STN-GPe neuron is strong and it is found that they are not engaged in rhythmic activity all the time. The activity patterns generated inside STN neuron and input from GPe neuron in healthy primate and Parkinson condition have been investigated in search of finding the parameters which alter the dynamics in Parkinson disease. We compared the discharge patterns by analyzing the sensitivity in STN model due to applied current, pre-synaptic membrane potential, feedback synaptic potential and calcium currents. The results in the above study show that STN model in Parkinson condition is sensitive to various parameters, but the model seems highly sensitive to calcium current. Results reveal the importance of observing the calcium current closely in order to identify the cause to this disease. These results can further be used to study the effect of calcium membrane potential 450 J. Singh et al. / Sensitivity analysis of STN in PD Fig. 7. Spiking patterns generated for calcium membrane potential, (a) at VCa = 140 mV for STN and STNP, (b) at VCa = 140 mV for STN and VCa = 235 mV for STNP. (c) Plot of correlation coefficient for for STN at VCa = 140 mV versus for STNP at VCa = 235 mV. on the model with respect to other intrinsic parameters like sodium and potassium to check the stability of the model. Acknowledgements We are highly thankful to the reviewers for their constructive and insightful comments and suggestions which helped us to improve the manuscript in a better way. We thank Department of Science and Technology, Government of India for financial support vide Reference No. SR/CSRI/166/2014(G) under Cognitive Science Research Initiative(CSRI) to carry out this work. We are also thankful to Prof. Karmeshu for providing valuable inputs in this work. J. Singh et al. / Sensitivity analysis of STN in PD 451 References [1] G. Arnulfo, A. Canessa, F. Steigerwald, N.G. Pozzi, J. Volkmann, P. Massobrio, S. 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