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Linking brain dynamics, neural mechanisms and deep brain

stimulation

Anne Beuter and Julien Modolo Laboratoire d’Intégration du Matériau au Système

UMR CNRS 5218University of Bordeaux

May 16th, 2008

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Outline

1. Parkinson’s disease (PD), Basal ganglia (BG), and deep brain stimulation (DBS)

2. Mathematical model: population based approach3. Can we explain DBS paradoxes?4. Conclusion

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1. Parkinson’s disease (PD), Basal ganglia (BG), and deep brain

stimulation (DBS)

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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

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Static model of the network

(Fig from McIntyre, 2005) 5

21/04/23 (Fig modified from McIntyre, 2005)

Direct pathway

Static model of the network (2)

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Indirect pathway

(Fig modified from McIntyre, 2005)

Static model of the network (3)

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Hyperdirect pathway

(Fig modified from McIntyre, 2005)

Static model of the network (4)

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(Rubin, 2008)

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Zoom on basal ganglia

Modolo J., Mosekilde E., Beuter A., J Physiol Paris, 2007 10

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Deep brain stimulation (DBS)

video.mp4

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McIntyre et al (2005)

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Deep brain stimulation (DBS) – stimulator off

(from Johns Hopkins Parkinson's Disease and Movement Disorders Center )

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Deep brain stimulation (DBS) – stimulator on

(from Johns Hopkins Parkinson's Disease and Movement Disorders Center )

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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

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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.

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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 » ?

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Levesque and Parent (2005)

The subthalamic nucleus: the prefered target

Parent et al. (1995)

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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:

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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

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Neuron 1

Neuron 2

Neuron N

…..

Single neurons

CouplingEmerging activity(physio/pathological)

Neuronal network

Paradox 1 (cont’d)

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Neuron 1

Neuron 2

Neuron N

…..

Single neurons

CouplingEmerging activity(physio/pathological)

Neuronal network

DBS

Paradox 1 (cont’d)

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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)

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2. Mathematical model: population based approach

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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.

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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)

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A simplification: the Izhikevich model

Izhikevich, 2003 27

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From single neuron to neuronal populationWhat 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

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Population equationsGeneral form of a conservation equation

Detailed form of the main population equation

Population density Neural flux

Individual dynamics Neural interactions

Mean-field variable

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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

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Rest

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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

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Where is biology in the equations?

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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

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t

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Multiscale properties of the approach

Modolo J., Mosekilde E., Beuter A., J Physiol Paris, 2007 33

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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

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Mathematical model of the subthalamo-pallidal complexSystem of PDE describing the dynamics of STN and GPe depending on connectivity, delays and individual firing patterns:

Individual population dynamics Populations coupling

Modolo, Henry, Beuter, J Biol Phys (submitted) 35

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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

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STN and GPe dynamics

« Physiological » state « Pathological » state

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3. Can we explain DBS paradoxes?

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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).

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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)

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Paradox 1 (cont’d)

Response of STN model cells to DBS with/without excitatory coupling.

Supression of internal excitatory connections.

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Paradox 1 (cont’d)

To GPiFrom cortex

Illustration of Stimulation-Induced Functional Decoupling (SIFD).

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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 DBS DBS No DBS 43

Paradox 1 (cont’d)

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

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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).

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Paradox 1 (cont’d)

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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.

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Paradox 2 (cont’d)

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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.

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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)

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4. Conclusion

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Conclusion: a cascade of SIFDs?

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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

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Acknowledgements (model)

Dr J. Henry

University of Bordeaux 1

Dr A. Garenne

University of Bordeaux 2

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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

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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.

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Appendix

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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

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Multiscale properties of the approach

• Infinite number of neurons

• Identical dynamical behaviour

Modolo J., Mosekilde E., Beuter A., J Physiol Paris, 2007 57

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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

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Boundary conditions

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