1. 26/12/2015 Linking brain dynamics, neural mechanisms and deep brain stimulation Anne Beuter and Julien Modolo Laboratoire d’Intégration du Matériau au Système UMR CNRS 5218 University of Bordeaux May 16th, 2008 1

2. 26/12/2015 Outline 1. Parkinson’s disease (PD), Basal ganglia (BG), and deep brain stimulation (DBS) 2. Mathematical model: population based approach 3. Can we explain DBS paradoxes? 4. Conclusion 2

3. 26/12/2015 1.Parkinson’s disease (PD), Basal ganglia (BG), and deep brain stimulation (DBS) 3

4. 26/12/2015 Parkinson’s disease (PD) and Deep brain stimulation (DBS) PD: 800 000 persons in Europe (65 000 new cases each year), 6 millions in the world DBS: Standard and efficient symptomatic procedure to improve motor symptoms Main targets: Vim, GPi, STN (favorite) Mechanisms of DBS: many hypotheses proposed, but mechanisms still unclear today 4

5. 26/12/2015 Static model of the network (Fig from McIntyre, 2005) 5

6. 26/12/2015 (Fig modified from McIntyre, 2005) Direct pathway Static model of the network (2) 6

7. 26/12/2015 Indirect pathway (Fig modified from McIntyre, 2005) Static model of the network (3) 7

8. 26/12/2015 Hyperdirect pathway (Fig modified from McIntyre, 2005) Static model of the network (4) 8

9. 26/12/2015 (Rubin, 2008) 9

10. 26/12/2015 Zoom on basal ganglia Modolo J., Mosekilde E., Beuter A., J Physiol Paris, 2007 10

11. 26/12/2015 Deep brain stimulation (DBS) 11

12. 26/12/2015 McIntyre et al (2005) 12

13. 26/12/2015 Deep brain stimulation (DBS) – stimulator off (from Johns Hopkins Parkinson's Disease and Movement Disorders Center ) 13

14. 26/12/2015 Deep brain stimulation (DBS) – stimulator on (from Johns Hopkins Parkinson's Disease and Movement Disorders Center ) 14

15. 26/12/2015 PD, DBS and paradoxes Reversibility of symptoms (sleep, somnanbulism or emergencies, pharmacology, DBS)  PD = dynamical disease (Beuter et al, 1995), defined by Mackey and Glass (1977) Lesion versus stimulation: excitation and/or inhibition of the stimulated area? Frequency dependent stimulation effect 15

16. 26/12/2015 Paradox 1: Reversibility of symptoms PD: reversible under DBS or L-DOPA, symptoms re-appear is DBS or L-DOPA is stopped.  DBS acts on a control parameter of the motor loop network to re-etablish physiological dynamics. 16

17. 26/12/2015 Role of the STN-GPe complex in basal ganglia STN and GPe: tightly interconnected nuclei STN: main excitatory structure in basal ganglia STN  weak activity in healthy state, strong and synchronous activity between 3 and 8 Hz in PD STN-GPe: can oscillate spontaneously, « bad pacemaker » ? 17

18. 26/12/2015 Levesque and Parent (2005) The subthalamic nucleus: the prefered target Parent et al. (1995) 18

19. 26/12/2015 Goal: understand these paradoxes to elucidate DBS mechanisms Develop a mathematical model Formulate candidate physiological mechanisms interpreted at several scales of description Confront with numerical simulations, experimental and clinical observations Our methodology is: 19

20. 26/12/2015 Philosophy of the modelling approach A multi-scale description: DBS current impacts the cellular, population and « network of populations » levels (Beuter and Modolo, 2007) A dynamical description with a fine temporal resolution: functional models are useful, but not sufficient (static) A model not too cumbersome easily re-used by other researchers in the field 20

21. 26/12/2015 21 Neuron 1 Neuron 2 Neuron N ….. Single neurons Coupling Emerging activity (physio/pathological) Neuronal network Paradox 1 (cont’d)

22. 26/12/2015 22 Neuron 1 Neuron 2 Neuron N ….. Single neurons Coupling Emerging activity (physio/pathological) Neuronal network DBS Paradox 1 (cont’d)

23. 26/12/2015 23 Neuron 1 Neuron 2 Neuron N ….. Single neurons Disruption of coupling Neuronal network  Stimulation-Induced Functional Decoupling. Emerging activity (physio/pathological) Paradox 1 (cont’d)

24. 26/12/2015 2. Mathematical model: population based approach 24

25. 26/12/2015 A reminder on the Hodgkin-Huxley model Izhikevich, 2007 Hodgkin & Huxley (1952): Study of the giant squid axon, measurement of the membrane potential under different stimulation currents + ionic channels hypothesis. 25

26. 26/12/2015 Modelling the effects of DBS with a population based model Why? : Complex systems imply numerous interactions between the elements of the system: analytical solving is difficult or impossible. Key concept : Average interaction for each element. Previous models: mainly based on the LIF model (Nykamp and Tranchina, 2000; Omurtag et al., 2000). Advantages: multi-scale, dynamic model.  Seems appropriate to model the effects of DBS in PD. (Fig from Paul De Koninck Laboratory) 26

27. 26/12/2015 A simplification: the Izhikevich model Izhikevich, 2003 27

28. 26/12/2015 From single neuron to neuronal population What do we need to describe a neuronal population of N neurons? 2) Quantify neuronal individual dynamics (using the Izhikevich model) 1) A population density function (number of neurons per state) such as 3) Quantify neuronal interactions (using a mean-field variable) If: N neurons, W afferences per neuron on average Then: if M spikes at time t, each neuron receives (W/N)*M spikes 28

29. 26/12/2015 Population equations General form of a conservation equation Detailed form of the main population equation Population densityNeural flux Individual dynamicsNeural interactions Mean-field variable 29

30. 26/12/2015 Synaptic events modelled by instantaneous « jumps » of amplitude ε in the membrane potential t v(t) Excitatory spike Inhibitory spike ε ε Where is biology in the equations? Membrane potential 30 Rest

31. 26/12/2015 Reception rate of neurotransmitters for each neuron: included in the spike reception rate This holds too for the neurotransmitters production rate. The mean-field variable expresses as: Mean connectivity degree Number of neurons Axonal conduction delays Past activity of the network 31 Where is biology in the equations?

32. 26/12/2015 Train of biphasic, charge-balanced pulses such as those used in Medtronic® stimulators Izhikevich model for STN cells (Modolo et al., 2008) Simplification: DBS current modelled as a current directly injected through the membrane IDBS(t) Modelling the DBS stimulation current 32 t

33. 26/12/2015 Multiscale properties of the approach Modolo J., Mosekilde E., Beuter A., J Physiol Paris, 2007 33

34. 26/12/2015 Modelling the subthalamo-pallidal network Terman and Rubin (2002, sophisticated and realistic cell models), Gillies and Willshaw (2004, firing rate model) The STN-GPe complex activity pattern can change under the following conditions:  Inhibition from Striatum to GPe increases  Intra-GPe inhibitory synapses weaken Relevance of modelling the STN-GPe network during DBS: currently not measured experimentally 34

35. 26/12/2015 Mathematical model of the subthalamo- pallidal complex System of PDE describing the dynamics of STN and GPe depending on connectivity, delays and individual firing patterns: Individual population dynamicsPopulations coupling Modolo, Henry, Beuter, J Biol Phys (submitted) 35

36. 26/12/2015 STN and GPe neurons modelling STN neurons with new parameters for the Izhikevich model 3) Post-inhibitory bursting 2) Increased spiking frequency under excitatory input 1) Spontaneous spiking activity Modolo, Henry, Beuter, J Biol Phys (submitted) 36

37. 26/12/2015 37 STN and GPe dynamics « Physiological » state« Pathological » state

38. 26/12/2015 3. Can we explain DBS paradoxes? 38

39. 26/12/2015 Paradox 1: Why do STN stimulation and lesion produce similar benefits? DBS: intuitively increases the firing rate of STN neurons. Lesion: destruction of the STN (subthalamotomy), thus completely suppresses STN activity, dramatically improves tremor (BUT is not reversible!) How can we explain this paradox? We propose the following mechanism:  Stimulation-Induced Functional Decoupling (SIFD): DBS current neutralizes the impact of glutamatergic synapses within the STN (cortical afferences or axon collaterals within the STN). 39

40. 26/12/2015 We propose the following mechanism: Stimulation Induced Functional Decoupling (SIFD) is the situation where neuronal interactions become negligible with regards to individual neuronal dynamics. Thus, the network becomes «unwired» and neurons seem independent from one another. Mathematically speaking: Individual neuronal dynamics (+DBS) Neuronal interactions Paradox 1 (cont’d) 40

41. 26/12/2015 41 Paradox 1 (cont’d) Response of STN model cells to DBS with/without excitatory coupling. Supression of internal excitatory connections.

42. 26/12/2015 42 Paradox 1 (cont’d) To GPiFrom cortex Illustration of Stimulation-Induced Functional Decoupling (SIFD).

43. 26/12/2015 Intuitively, electrical stimulation of neurons should increase spiking activity (assumed in Rubin and Terman, 2004) However: in vivo recordings in MPTP monkeys show a decrease in STN neurons activity! (Meissner et al., 2005) Furthermore: GPi cells (target of STN cells) are activated at high-frequency (Hashimoto et al., 2003)  how is this compatible? McIntyre et al. (2004): DBS inhibits STN somas, and excites STN axons soma axon No DBSDBS No DBS 43 Paradox 1 (cont’d)

44. 26/12/2015 44 Decrease of somatic activity during DBS (model, top; experimental data in humans, bottom) (McIntyre et al., 2004)

45. 26/12/2015 Let us list STN neurons dynamical properties:  Dampened oscillations of their membrane potential (Bergman et al., 1994)  Post-inhibitory bursts of action potentials (Bevan et al., 2002)  Two stable equilibria (bistability) (Kass and Mintz, 2006)  STN neurons have their equilibrium near an Andronov-Hopf bifurcation (Izhikevich, 2007) and can be classified as resonators.  Eigen-frequency of STN neurons: low-frequency, thus: high-frequency (≥100 Hz) can delay or decrease the response.  STN neurons dynamical properties underlie their activity decrease during DBS (Modolo and Beuter, in preparation). 45 Paradox 1 (cont’d)

46. 26/12/2015 Paradox 2: Why are DBS effects frequency- dependent? Low-frequency (<20 Hz) DBS: has no effect on motor function, sometimes worsens symptoms (Timmermann et al., 2007). High-frequency DBS (>100 Hz): provides dramatic relief of symptoms. Modolo et al. (2008): low-frequency DBS current may cause a resonance with STN neurons eigen-frequency.  Low-frequency DBS appears to exacerbate pathological activity, while high-frequency DBS suppresses it. 46

47. 26/12/2015 Model results. (Modolo, Henry, Beuter, J Biol Phys, submitted) Paradox 2 (cont’d) 47

48. 26/12/2015 How does DBS facilitate motor function? DBS appears to mimic lesions by decreasing STN activity, and lesions improve motor function. Synaptic decoupling between motor cortex and STN during DBS via SIFD. Lalo et al. 2008  decrease bêta coupling between M1 and STN during the execution of movement (experimental study). Our interpretation: the STN becomes functionnaly decoupled from M1 following SIFD, facilitating the execution of movement. 48

49. 26/12/2015 Confirmation of insights from simulations DBS mimics the decoupling of the STN from internal excitatory connections and maybe from cortex, that normally occurs in the presence of dopamine (Magill et al., 2001) The effects of DBS are frequency-dependent, i.e., the stimulation frequency is close or away from the resonance frequency of the stimulated area (Timmermann et al., 2007) 49

50. 26/12/2015 4. Conclusion 50

51. 26/12/2015 Conclusion: a cascade of SIFDs? 51 Axonal activation of GPi efferences Decreased somatic activity Cortical afferent spikes Antidromic activation (Li et al., 2007) Cancellation by collision SIFD Afferences from primary motor cortex (M1) Efferences to GPi

52. 26/12/2015 Acknowledgements (model) Dr J. Henry University of Bordeaux 1 Dr A. Garenne University of Bordeaux 2 52

53. 26/12/2015 Acknowledgements BioSim European Network Of Excellence, No AB LSHB-CT-2004- 005137, Professor Erik Mosekilde, coordinator Aquitaine Region (France), No 20051399003AB Financial support of the project 53

54. 26/12/2015 Recent publications Modolo, Henry, Beuter. Dynamics of the subthalamo-pallidal complex during deep brain stimulation in Parkinson’s disease, J Biol Phys, submitted. Modolo, Mosekilde, Beuter. New insights offered by a computational model of deep brain stimulation, J Physiol Paris, 101:58–65, 2007. Modolo, Garenne, Henry, Beuter. Development and validation of a population based model based on a discontinuous membrane potential neuron, J Integr Neurosci, 6(4):625–655, 2007. Pascual, Modolo, Beuter. Is a computational model useful to understand the effects of deep brain stimulation in Parkinson’s disease? J Integr Neurosci, 5(4) :551–559, 2006. 54

55. 26/12/2015 Appendix 55

56. 26/12/2015 The diffusion approximation Let us remind the main population equation In the limit(EPSP low amplitude), the interaction term expresses as which gives a Fokker-Planck equation 56

57. 26/12/2015 Multiscale properties of the approach Infinite number of neurons Identical dynamical behaviour Modolo J., Mosekilde E., Beuter A., J Physiol Paris, 2007 57

58. 26/12/2015 Population equations Modolo J., Garenne A., Henry J., Beuter A., J Integr Neurosci, 2007 In summary, each population is described by a population density function Where the mean-field variable expresses as 58

59. 26/12/2015 Boundary conditions 59