A possible mechanism for the efficacy ofsubthalamic nucleus deep brain stimulation

Jonathan Rubin

Department of Mathematics

University of Pittsburghrubin@math.pitt.edu, http://www.math.pitt.edu/∼rubin

Universiteit Twente, BrainGain lecture

January 30, 2008

funding: U.S. National Science Foundation, Pittsburgh Institute forNeurodegenerative Disorders

1

Deep brain stimulation (DBS):

2

Focus on DBS of the subthalamic nucleus:

3

How can STN-DBS help alleviate parkinsonian motorsymptoms?

Boxes and arrows...

−−2 = inhibition

−−� = excitation

thalamocentric view:

• VL thalamus: job is torelay inputs betweencortical areas

•GPi/SNr: inhibitory“interference”

• idea: STN-DBS canhelp if it allowsappropriate thalamicfunction

• questions: underwhat conditions canthe thalamus do agood job? does STN-DBS set up theseconditions?

4

An interesting paradox:

= inhibition = excitation

normalCortex

Striatum

SNcGPe

STN

Thalamus

GPi/SNr

Cortex

Striatum

GPe

STN

Thalamus

GPi/SNr

SNc

classical view of PD

• PD: ↑ inhibition from GPi to thalamus compromises relay

• STN-DBS: data show GPi activity ↑ further

5

Hashimoto et al., J. Neurosci., 2003: data from GP

tightly correlated to the time of the stimulation pulse at higherstimulation frequencies (Fig. 3).

The consistent pattern and precise latencies of the responsesthat occurred during stimulation changed the spontaneous irreg-ular firing of GP neurons into a high-frequency regular pattern ofdischarge (Fig. 4). The interstimulus interval (ISI) histograms ofthe prestimulation and poststimulation periods show widelyvarying ISIs with a small peak at 4 –5 msec. The ISI histogram ofthe 136 Hz stimulation period, on the other hand, revealed largepeaks at 4 and 8 msec, indicating a regular firing pattern withmost ISIs occurring at regular intervals after the stimulationpulse. The increased mean firing rate reflected the dominant ef-fect of the short-latency excitatory responses after each stimula-tion pulse (Fig. 5A,B).

During HFS at 136 and 157 Hz, the discharge rate quicklyattained its maximum and then gradually decreased in approxi-mately half of GPe and GPi neurons in which a change was noted.Although the discharge rate gradually decreased during stimula-tion, it remained higher than the prestimulation period (Fig.5A,B). During periods of prolonged stimulation lasting �5 min,a sustained increase in mean discharge rate occurred in seven ofseven GPi neurons and two of two GPe neurons (Fig. 5C).

Changes in the shape of action potentials were observed dur-ing HFS in more than half of recorded neurons in which the firingrate increased. During HFS there was a decrement in the ampli-tude of both the negative and positive phases (Figs. 3, 5A,B). Thedegree of amplitude decrement was �20% in most neurons, buta few neurons showed changes of nearly 50%. Action potentialamplitude began to change with a time lag of several seconds afterthe start of stimulation and returned to normal within severalseconds after termination of stimulation.

Figure 3. Overlay of 50 sweeps of neuronal activity of a GPi cell during 2 Hz (top), 136 Hz(middle), and 157 Hz (bottom) stimulation at 3.0 V (R370). Depolarization (negative potential)is shown as an upward deflection. Each stimulation frequency is associated with excitationpeaks at 2.5– 4.0 msec and 5.5–7.0 msec after the onset of stimulation. Short-latency excita-tion was greater and more tightly coupled to each stimulation pulse during higher-frequencystimulation. Arrows indicate residual stimulation artifacts after artifact template subtraction.

Table 1. Percentage of neurons showing short-latency responses after 2 Hz STNstimulation

Latency (msec) Response

R7160 (2.4 V) R370 (3.0 V)

GPe(n � 16)

GPi(n � 22)

GPe(n � 18)

GPi(n � 27)

1.0 –2.5 Inhibition 31 77 56 672.5– 4.5 Excitation 63 41 83 524.5–5.5 Inhibition 25 41 11 705.5–7.0 Excitation 19 73 33 637.0 –9.0 Inhibition 6 45 28 44

Figure 4. Rasters (A) and ISI histograms (B) of GPi neuronal activity (R370). The firingchanged from an irregular pattern with varying ISIs into a high-frequency regular pattern withmost ISIs occurring at 4 or 8 msec during 136 Hz, 3.0 V stimulation.

Hashimoto et al. • Effects of STN Stimulation on Pallidal Neurons J. Neurosci., March 1, 2003 • 23(5):1916 –1923 • 1919

DiscussionOur experimental setting closely reproduces the HFS system usedin humans, and the results demonstrated that STN stimulationproduces short-latency excitatory responses that tonically in-crease the average firing rate in both GPi and GPe. The results arein stark contrast to previous studies in anesthetized rats thatshowed a reduction of EP and SNr activity consistent with sup-pression of STN neuronal output during STN HFS (Benazzouz etal., 1995; Beurrier et al., 2001). We also observed a decrease in theratio of action potential amplitude to duration in many neurons,similar to the change preceding depolarization block as observedin spinal or nigral dopamine neurons (Curtis et al., 1960; Holler-man et al., 1992). Although the increased firing rate in GP neu-rons was sustained in the present study, it is possible that STNneurons could enter a state of depolarization block with inhibi-tion of STN neuronal activity, as observed in previous studies(Benazzouz et al., 1995; Beurrier et al., 2001). Although activationof GPi neurons coincident with inhibition of STN during HFS in

the STN may appear paradoxical, it is consistent with previousmodeling studies suggesting that axons exiting the stimulatedstructure may be activated at stimulation parameters that inhibitneuronal activity (McIntyre and Grill, 1999). Our studies werealso performed in awake primates using a stimulating lead that issimilar to that used in humans. Thus, the difference in the exper-imental conditions could also contribute to the different obser-vations and may explain the discrepancy between our results andthat of the previous study.

Although we did not directly explore the mechanism(s) un-derlying the short-latency excitation and inhibition observed inthis study, it is likely that the earliest excitatory response (peak,3– 4 msec after the stimulation pulse) occurred as the result oforthodromic activation of STN3 GPe and STN3 GPi axons.The latency of monosynaptic EPSPs of STN3 EP nucleus pro-jections of 1.7 � 0.5 msec (mean, SD) reported in the rat (Na-kanishi et al., 1991) and that of STN3 GPe, GPi projections of2–10 msec in the monkey (Nambu et al., 2000) would be consis-tent with this assumption. The second facilitation (peak, 5.5–7.0msec after the stimulation pulse) could occur as a result of split-ting the early excitation peak into two excitation periods by in-terposition of the second inhibition. The second inhibition at4.5–5.5 msec may have been elicited through disynaptic STN3GPe and GPe3 GPi pathways or by the refractory period of theearliest excitation. The inhibition preceding the earliest excita-tion evoked by STN stimulation in GPi could occur as the resultof antidromic activation of GPe3 STN collaterals to GPi, con-sistent with single neuron staining studies in the rat (Kita andKitai, 1994) and the monkey (Shink et al., 1996; Sato et al., 2000).This would also be consistent with previous observations in Jap-anese macaques of antidromic activation of GPe at 1.0 � 0.3 msec(mean, SD) (Hasegawa and Hamada, 2000) after stimulation inthe STN and may provide physiological confirmation of the ex-istence of axon collaterals from GPe to GPi (Smith et al., 1994;Nambu and Llinas, 1997). In addition to the above mechanisms,resetting of pallidal neuronal activity after IPSPs has been re-ported (Kita, 2001), and this mechanism could also contribute tothe induction of a more rhythmic discharge pattern, time-synchronized to inhibitory inputs.

A leading hypothesis regarding the development of move-ment disorders of basal ganglia origin suggests that hyperkineticand hypokinetic disorders occur as a result of changes in themean discharge rate in the GPi and SNr, which in turn suppressthalamocortical output (Albin et al., 1989; DeLong, 1990). Met-abolic studies using positron emission tomography (Playford etal., 1992; Eidelberg et al., 1994) and physiological recordings ofneuronal activity in the basal ganglia (Miller and DeLong, 1987;Filion and Tremblay, 1991) are consistent with this hypothesis. Inaddition to changes in mean discharge rate, however, altered fir-ing patterns in the pallidum have also been reported in MPTP-induced parkinsonian animals. Increases in bursting activity(Filion and Tremblay, 1991; Bergman et al., 1994), periodic os-cillatory activity (Bergman et al., 1994; Nini et al., 1995; Raz et al.,2000), and synchronization of GPi (Bergman et al., 1994; Nini etal., 1995) or GPi and striatal neurons (Raz et al., 2001) have allbeen reported to occur in this model. Increases in bursting andoscillation patterns of STN neurons in parkinsonian animalshave also been observed (Bergman et al., 1994; Hassani et al.,1996; Plenz and Kitai, 1999), suggesting that the altered STNoutput may contribute to the observed changes in pattern in GPineurons. Synchronized oscillatory bursts of the GPe, GPi, or STNhave been proposed to have a causative relationship for the de-velopment of parkinsonian motor signs (Bergman et al., 1994;

Figure 6. Change in the mean firing rate of GPe and GPi neurons during 136 Hz STN stimu-lation in the two monkeys. A–C, Monkey R7160. During 1.4 V stimulation, the firing rates didnot change significantly in either GPe or GPi, but during 1.8 V stimulation, the mean dischargerate increased significantly, and the percentage of neurons displaying a change in the firing rateshowed a strong shift to excitation in both GPe and GPi. D–F, Monkey R370. Stimulation of 2.0V did not induce a significant change in the firing rate, but 3.0 V stimulation increased the firingrate significantly and shifted the percentage of neurons to excitation in both GPe and GPi.Asterisks indicate a significant difference in on versus off stimulation conditions. Significantdifferences: *p � 0.01, **p � 0.001, ***p � 0.001; t test.

Hashimoto et al. • Effects of STN Stimulation on Pallidal Neurons J. Neurosci., March 1, 2003 • 23(5):1916 –1923 • 1921

6

McIntyre et al., J. Neurophysiol., 2004: simulations

McIntyre 2001; McIntyre and Grill 2002). This hypothesis, ifsupported, resolves the apparently conflicting experimentalresults indicating that DBS both inhibits and excites the stim-ulated nucleus.

The second goal of this study was to determine the spatialextent of activation and/or inhibition of neurons surroundingthe electrode generated with therapeutic stimulation parame-ters. Presently, the stimulus parameters used in clinical DBS(monopolar or bipolar stimulation; 120 to 180 Hz stimulusfrequency; 0.06 to 0.2 ms pulse duration; 1 to 5 V stimulusamplitude) are derived by trial and error (O’Suilleabhain et al.2003; Volkmann et al. 2002). Understanding the impact ofparameter variation on the effects DBS represents an importantstep in developing rational methods for system design andtuning (Benabid et al. 2000).

M E T H O D S

This study combined a finite-element model of the electric fieldgenerated by a DBS electrode and a multi-compartment cable modelof a thalamocortical (TC) relay neuron to examine the cellular effectsof high-frequency extracellular stimulation. The electric field modelincluded a representation of the DBS lead surrounded by homoge-neous brain tissue and was used to determine the extracellular poten-tial distribution generated by DBS. The neuron model included ex-plicit geometrical representations of the dendritic arbor, cell body, andmyelinated axon and nonlinear membrane dynamics representing theion channel distributions in each section of the neuron. The resultingintegrated model allowed the study of the neuronal output as afunction of extracellular stimulation with DBS electrodes.

DBS electrode model

We developed an axisymmetric finite-element model of theMedtronic 3387 DBS lead (Medtronic, Minneapolis, MN) positionedin a homogeneous isotropic volume conductor to solve for the poten-tial distribution generated in the tissue medium (Fig. 1). The modelwas implemented using 4-node quadrilateral elements in a commer-cially available software package (ANSYS 5.7, ANSYS, Houston,PA). The voltages (V) at the nodes of the finite-element mesh werecalculated using a frontal solution method of the Laplace equation

� � ��V � 0

where � � 350 � cm�1 (Sances and Larson 1975) was the electricalconductivity of the tissue medium (Fig. 1). The model consisted of

15,782 nodes. Along the surface of the electrode contact the separa-tion between the nodes of the finite-element mesh was 25 �m, and inthe tissue within 5 mm surrounding the electrode the spacing betweennodes was �50 �m. The tissue box surrounding the electrode was50 � 50 mm with the outer boundary set to 0 V, and the electrodecontact set to the stimulus voltage. Doubling the density of the meshor doubling the distance of the boundary from the electrode (i.e., thesize of the tissue box) generated a potential distribution that differedby �2% when compared with the default model.

The extracellular potential (Ve) generated by the DBS electrodemodel was applied to the TC relay neuron model (see descriptionbelow) during time periods when the stimulus pulse was on to deter-mine the influence of the electric field on neural excitation. Eachcompartment of the neuron model (n) was assigned a Ve[n] thatcorresponded to the compartment position (X[n], Y[n], Z[n]) relativeto the electrode. These extracellular potentials were determined fromthe DBS electrode model by defining the radial distance from elec-trode to the neuron compartment as R[n] � (X[n]2 � Z[n]2)1/2 andY[n] as the vertical position of the neural compartment relative to thecenter of the electrode contact. The position of each of the neuralcompartments (R[n], Y[n]) did not necessarily correspond to theposition of a node in the finite-element mesh. Therefore a 4-point2-dimensional linear interpolation algorithm was used to calculateVe[n], and the mesh density in the tissue medium was such that cubicand linear interpolation of the potential differed by �1% (McIntyreand Grill 2001). During each time step of the simulation the Ve foreach compartment was set either to zero when no stimulus pulse wasbeing applied or to the potentials generated by the DBS electrodewhen a stimulus pulse was being applied (McNeal 1976).

Thalamocortical relay neuron model

The DBS electrode model was coupled to a three-dimensional(3-D) multi-compartment cable model of a TC relay neuron. Themodel consisted of a dendritic tree, cell body, and myelinated axonwith a geometry obtained from a 3-D reconstruction of a filled cell(Destexhe et al. 1998) (Fig. 2, Table 1). The membrane of the TCmodel consisted of the membrane capacitance (1 �F/cm2: cell bodyand dendrites; 2 �F/cm2: myelinated axon), in parallel with a com-plement of linear leakage and nonlinear calcium, potassium, andsodium conductances distributed in the different sections of the neu-ron (Fig. 2). The compartments were connected together with linearresistors based on the geometry of the connecting compartments andthe intracellular resistivity (300 � cm�1: cell body and dendrites; 70� cm�1: myelinated axon).

The membrane models of the cell body and dendritic tree of the TCrelay neuron were based on the results of previous modeling and exper-imental studies (Destexhe et al. 1998; Huguenard and McCormick 1992;McCormick and Huguenard 1992; Williams and Stuart 2000). The cellbody and dendritic compartments included the parallel combination ofnonlinear fast Na�, delayed rectifier K�, slow K�, T-type Ca2�, andhyperpolarization-activated cation conductances, Na� and K� linearleakage conductances, and the membrane capacitance (Fig. 2; Table 2;see APPENDIX). The initial segment compartments included the parallelcombination of nonlinear fast Na�, delayed rectifier K�, and slow K�

conductances, a linear leakage conductance, and the membrane capaci-tance (Fig. 2; Table 2; APPENDIX).

The myelinated axon (diameter: 2 �m) used a concentric double-cable structure with explicit representations of the nodes of Ranvier,myelin attachment segment (MYSA), paranode main segment(FLUT), and internode segment (STIN) regions of the fiber (Fig. 2;Table 3; APPENDIX) (McIntyre et al. 2002). Between each node therewere 2 MYSA compartments, 2 FLUT compartments, and 3 STINcompartments. The nodes included the parallel combination of non-linear fast Na�, persistent Na�, and slow K� conductances, a linearleakage conductance, and the membrane capacitance. The paranodaland internodal compartments included 2 concentric layers, each in-

FIG. 1. Finite-element model of the electric potential distribution generatedby a deep brain stimulation (DBS) electrode. An axisymmetric model of theMedtronic 3387 DBS electrode was constructed (dark gray: electrode insula-tion; gray: electrode contact; white: tissue medium) using a nonuniform meshdensity to increase accuracy near the electrode contact (left). Right: potentialdistribution generated in the tissue medium by a 1-V stimulus with the DBSelectrode.

1458 C. C. MCINTYRE, W. M. GRILL, D. L. SHERMAN, AND N. V. THAKOR

J Neurophysiol • VOL 91 • APRIL 2004 • www.jn.org

axonal firing was achieved without stimulation-induced trans-synaptic inputs.

In general, presynaptic axon terminals have lower thresholdsfor activation than local cells with DBS stimulation parameters(Baldissera et al. 1972; Dostrovsky et al. 2000; Gustafsson andJankowska 1976; Jankowska et al. 1975; McIntyre and Grill2002). Therefore a large number of synaptic inputs impingingon local cells will be activated by DBS, and can thus alter thefiring of local cells during DBS. We determined the effect ofstimulation-induced trans-synaptic activity on the TC relayneuron with a distribution of AMPA, NMDA, GABAa, andGABAb synaptic conductances on the dendrites and cell body(see METHODS). Figure 6 shows the influence of stimulation-induced trans-synaptic inputs on the efferent output of the TCneuron. The threshold to drive the efferent output of the neuronat the stimulus frequency decreased with increasing stimulusfrequency, and this relationship was unaffected by the presence

or absence of trans-synaptic inputs (also see Fig. 8C, below). Inmodeling the stimulation-induced trans-synaptic inputs thegoal was not to reproduce a specific type of input to TC relayneurons but to simulate the effect of strong synaptic actiongenerated during DBS as a result of simultaneous activation oflarge numbers of presynaptic terminals. We conducted a sen-sitivity analysis by examining the threshold of the TC relayneuron as a function of stimulus frequency after doubling orhalving each of the synaptic conductance values (Fig. 6). Ingeneral, increasing or decreasing the individual synaptic con-ductances had limited effects on threshold. Most of the synap-tic input parameter sets (including default) were unable togenerate action potentials solely from the applied synapticinputs because of the balance between the excitatory andinhibitory postsynaptic currents. However, in the cases of2 gAMPA or 0.5 gGABAa the balance was shifted toward exci-tation and the stimulation-induced trans-synaptic inputs were

FIG. 3. Excitability properties of tha-lamic neurons. Model exhibited excitationproperties that matched well with experi-mental records including membrane poten-tial dependent firing properties [A: experi-mental comparison from Jahnsen and Llinas(1984)], rebound excitation [B: experimentalcomparison from Pape and McCormick(1995)], and firing frequency as a function ofstimulus amplitude [C: experimental com-parison from Pape and McCormick(1995)].D: intrinsic bursting was obtained by reduc-ing the somatic and dendritic Na� leakageconductance (gNaL � 0 S/cm2), increasingthe T-type Ca2� permeability (PCaT �0.0002 cm/s), and shifting the voltage de-pendency of Ih (�5 mV shift) [experimentalcomparison from Emri et al. (2000)].

FIG. 4. Action potential initiation occurs in the axon during extracellular stimulation of model thalamocortical (TC) relayneurons. Left: neuron model drawn to scale and superimposed on the potential distribution generated by the electrode model. Activeelectrode contact is black and the dashed line represents the line of axial symmetry. Application of the extracellular potentials toeach compartment of the neuron model resulted in the membrane polarization displayed on the right. False color plot of membranepotential displays the neuron polarization induced by a �1 V stimulus, 0.1 ms after onset. Recordings from the soma and nodesof Ranvier show the membrane potential as a function of time for a subthreshold (�1 V) and suprathreshold (�2 V) 0.1 msstimulus.

1461THALAMIC DEEP BRAIN STIMULATION

J Neurophysiol • VOL 91 • APRIL 2004 • www.jn.org

sufficient to generate efferent output at the stimulus frequency,independent of the applied field. These 2 cases were not plottedin Fig. 6 because the threshold stimulus amplitude was notdetermined by application of the extracellular electric field tothe TC relay neuron, but instead by the activation of axonterminals projecting to the TC relay neuron. Thus with weakGABAa and/or strong AMPA conductances the threshold todrive the efferent output of the TC relay neuron would bedetermined by the threshold for activation of afferent inputs.

The threshold to activate the TC relay neuron was dependenton the stimulus pulse duration (Fig. 7). We determined thestrength–duration (SD) relationship for the TC relay neuronpositioned 1.5 mm from the electrode center (as in Fig. 4). TheSD relationship can be characterized by rheobase (thresholdstimulus amplitude to generate an action potential with infi-

nitely long stimulus pulse) and chronaxie (pulse duration atwhich threshold stimulus amplitude is 2 � rheobase). Wecalculated chronaxie using the methods described by Holshei-mer et al. (2000a). Driving the TC relay neuron at the stimulusfrequency resulted in a chronaxie of 0.21 ms with stimulation-induced trans-synaptic inputs and 0.19 ms without them. Thesechronaxies were substantially shorter than the single pulseintracellular chronaxie of 6.3 ms, but slightly longer thanchronaxies determined for tremor reduction during thalamicDBS (0.05–0.1 ms) (Holsheimer et al. 2000b).

The response of the TC relay neuron was dependent on itsposition relative to the electrode. Figure 8 shows the pattern ofneural output as a function of the position of the soma withrespect to the electrode. In Fig. 8A each neuron was orientedwith its axon parallel to the electrode shaft and threshold wascalculated to drive the efferent output at 150 Hz using 0.1 ms

FIG. 5. Intracellular and extracellular high-frequency stimulation generateddifferent firing patterns in the model TC relay neuron. When stimulatingintracellularly, suprathreshold (30 nA) 0.1-ms stimuli were applied in thesoma. When stimulating extracellularly, the neuron was positioned as in Fig.4 and activated with �3 V, 0.1 ms stimuli.

FIG. 6. Stimulation-induced trans-synaptic inputs had asmall impact on the excitability of model TC relay neurons.A: distribution of somatic and dendritic compartments thatreceived either excitatory or inhibitory synaptic inputs.Plots in B and C quantify the threshold to drive the efferentoutput of the neuron, positioned as in Fig. 4, at the stimulusfrequency with 0.1 ms extracellular stimulus pulses. B:default synaptic inputs had little impact on the stimulationthreshold. C: sensitivity of the influence of the differenttypes of synapses (AMPA, NMDA, GABAa, GABAb) onexcitation threshold were examined by doubling or halvingtheir individual conductance values. In general, altering thesynaptic conductances had a slight impact on the excitationthresholds; however, doubling the AMPA conductance orhalving the GABAa conductance generated firing at thestimulus frequency resulting solely from the stimulation-induced trans-synaptic inputs and are not plotted.

FIG. 7. Strength–duration relationship for extracellular stimulation of themodel TC relay neuron positioned as in Fig. 4. Threshold to drive the efferentoutput of the neuron at the stimulus frequency decreased with increasingstimulus pulse duration.

1462 C. C. MCINTYRE, W. M. GRILL, D. L. SHERMAN, AND N. V. THAKOR

J Neurophysiol • VOL 91 • APRIL 2004 • www.jn.org

7

Back to the paradox:

= inhibition = excitation

normalCortex

Striatum

SNcGPe

STN

Thalamus

GPi/SNr

Cortex

Striatum

GPe

STN

Thalamus

GPi/SNr

SNc

classical view of PD

• PD: ↑ inhibition from GPi to thalamus compromises relay

• STN-DBS: data show GPi activity ↑ further (e.g. Hashimoto etal., 2003); consistent with simulations (McIntyre et al.)

•Why should this ↑ in inhibition relieve PD symptoms?

8

Main idea:

versus

• In PD, GPi outputs become rhythmic (woodpecker), not juststronger.

• STN-DBS cuts the rhythmicity: stronger inhibition is less of anuisance if it’s more regular (window fan)

9

This talk - simulation and analysis results in support ofthis idea:

1. GPi data into biophysical thalamicortical relay (TC) cell model

2. biophysical models for basal ganglia/thalamic neurons →mechanism

papers:

• Terman, Rubin, Yew, and Wilson., J. Neurosci., 22(7):2963-2976,2002

• Rubin and Terman, J. Comp. Neurosci., 16:211-235, 2004

• Rubin and Josic, Neural Comp., 19:1251-1294, 2007

•Guo, Rubin, McIntyre, Vitek, and Terman, J. Neurophysiol.,accepted

10

Data-driven computational model:

• use GPi spike trains recorded from primates innormal/PD/sub-therapeutic DBS/therapeutic DBS togenerate inhibitory inputs

• use elevated spike time (EST) as measure of input bursti-ness

• feed inhibitory inputs to conductance-based TC cellmodel

• consider TC cell relay of simulated excitatory inputs

• quantify relay performance with

error index = (misses + bursts)/(excitatory inputs)

11

KEY RESULTS:

12

WHY? Biophysical basal-ganglia-TC network model:Individual TC cell equations:

STN

GPi+ −

+

−

TC (out)

GPe (in)Cmv′ = −IL− INa− IK − IT − IGPi→TC − Isignal

h′T = (hT∞(v)− hT )/τhT(v)

h′ = (h∞(v)− h)/τh(v)

s′ = α(1− s)exc(t)− βs, exc(t) = ΣH(t− ton)(1−H(t− toff))

IL = gL(v− vL) IT = gTm2T∞(v)hT (v− vCa)

INa = gNam3∞(v)h(v− vNa) IGPi→TC = gGPi(v− vinh)Σj(sGPi)j

IK = gKn4(v− vK) Isignal = gsignals(v− vexc)

X∞(v) = (1 + exp(v− θX)/σX)−1; X ∈ {m, h, mT , hT}

STN voltage equation:

Cmv′STN = −IL− INa− IK − IT − ICa− IAHP − IGPe→STN + DBS

GPe voltage and synaptic equations (GPi is similar):

Cmv′GPe = −IL− INa− IK − IT − ICa− IAHP − ISTN→GPe− IGPe→GPe

s′GPe = αGPe(1− sGPe)inh(vGPe, t)− βGPesGPe

13

Simulation results with model - Normal case:

← excitatoryinputs

14

Parkinsonian case:

15

DBS case (extreme example):

16

More examples and summary:

periodic input

normal

park

DBS

Poisson input

error index

1

error index = (misses + bursts)/(excitatory inputs)

17

Similar qualitative results in a reduced thalamic model:

v′Th = −(IL + IT )− Isignal− Iinh

h′Th = (h∞(vTh)− hTh)/τh(vTh)

periodic input (1) periodic input (2)

0 200 400 600 800 1000

−80−60−40−20

0

vTh

0 200 400 600 800 1000−100

−50

0

vTh

0 200 400 600 800 1000−100

0

100

time (msec)

vTh

0 200 400 600 800 1000−100

0

100

time (msec)

v

0 200 400 600 800 1000−100

−50

0

v

B

0 200 400 600 800 1000−100

−50

0

vnormal

DBS

park

118

Analysis

Quick reminder - relaxation oscillators and coupling:

v′ = f(v, h)− Iexc︸ ︷︷ ︸

<0

− Iinh︸ ︷︷ ︸>0

h′ = εg(v, h), 0 < ε� 1

h

v

h’=0 v’=0f>0

f<0

E

Iv’=0

h

v

19

Phase planes for reduced thalamic model:

v′Th = −(IL + IT )− Isignal︸ ︷︷ ︸IS or I below

−gGPi sGPi︸ ︷︷ ︸S below

(vTh − vinh)

h′Th = (h∞(vTh)− hTh)/τh(vTh)

key idea: for ANY constant inhibition, IT equilibrates and strongenough inputs at reasonable rate evoke reliable responses

20

Cartoon view:

inh

exc0

normal: DBS:

excinh+excinh

Numerics:

normal: DBS:

21

Parkinsonian:

0 200 400 600 800 10000

0.05

0.1

0.15

0.2

0.25

time (msec)

hTH

0 200 400 600 800 1000−100

−50

0

50

100

vTH

1

2 34

• responses 2 and 3 (partially) are blocked by left branch

• the response after 4 is inappropriately huge

22

Full thalamic model:

• constant inhibition (normal or DBS):

◦ INa, rather than IT , equilibrates

◦ hT is so slow that it is ∼ constant

• parkinsonian:

◦ phasic inhibition means INa and IT both participate

◦ reminiscent of bursts within sleep spindle oscillations

Bazhenov and Timofeev, Scholarpedia (from Timofeev and Bazhenov, 2005)

23

Normal case in the full model:

0 20 40−100

−80

−60

−40

−20

0

excitatory input

v

A

−2 0 2 4−100

−50

0

hT

v

input

no input

−100 −50 0 400

0.5

1

1.5

2

2.5

v

h

C

s

−100 −50 0 400

0.5

1

1.5

2

v

h

−100 −50 0 400

0.5

1

1.5

2

v

h

E

24

DBS in the full model:

−20 −10 0 10 20 30−100

−80

−60

−40

−20

0

excitatory input

v

−2 0 2 4−100

−50

0

40

hT

v

no inputinput

−100 −50 0 400

0.5

1

1.5

2

v

h

C

s

−100 −50 0 400

0.5

1

1.5

2

v

h

D

−100 −50 0 400

0.5

1

1.5

2

v

h

F

25

Parkinsonian case in the full model:

−100 −50 00

0.5

1

1.5

2With inhibition, no T−current

v

h

input on

input off

−100 −50 0 400

0.5

1

1.5

2With inhibition and T−current

v

h

input on

input off

−100 −50 0 400

0.5

1

1.5

2No inhibition, no T−current

v

h input off

input on

−100 −50 0 400

0.5

1

1.5

2No inhibition, with T−current

v

h

input on

input off

1 4

2 3

26

BACK TO DATA:Signals preceding each type of TC response

• average over 25 msec of GPi activity, consisting of 20 msec of ac-tivity preceding each TC response and 5 msec of activity followingresponse

• cluster by 3 TC response types (miss, bad, successful)

• result is consistent with predictions from theory!

0 5 10 15 20 250.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

time (ms)

missbadsuccessful

27

Apply GPi data to heterogeneous TC population:

TC miss at the same time in PD/subDBS, not in tDBS!

28

How general is model performance?

Consider stochastic inputs:

• let many GPi cells inhibit a biophysical model TC cell

• use Poisson processes to generate GPi firing

• vary correlation between GPi cells and GPi burst rate(EST)

TC

GPi1 GPi2

excitation

inhibition

randomsignals

overlap

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•PD ≈ moderate correlation and EST

•DBS ≈ high correlation and EST

• error index = (misses + bursts)/(excitatory inputs)

0 100 200 300 400 5000

0.5

1

GPi1

0 100 200 300 400 5000

0.5

1

time (msec)

GPi2

0 100 200 300 400 5000

0.5

1

GPi1

0 100 200 300 400 5000

0.5

1

time (msec)

GPi2

30

Simulation results:

error index varies with inhibitory input correlation and EST

31

Results from jittered data also separate correlation andburstiness:

unjittered jittered

32

Quantitative Analysis:Markov chains for arbitrary stochastic input trains

idea - use a Markov chain, based on transitions between bins definedin silent phase, to compute firing statistics:

step 1: define bins

wRK

wFP

w

w

V

excitation off

excitation onx

E

x

0

ELK

5 bins

• use time S, the min time between excitatory inputs

• bins (non-Markovian) are

Ik = [wERK · kS, wE

RK · (k + 1)S), if k + 1 < N ,

IN−1 = [wERK · (N − 1)S, wE

LK)

IN = [wELK, w0

FP ].

• states of Markov chain are (Ik, m), m ≥ 1, where m−1 = numberof inputs since last fired

33

step 2: compute transition probabilities p(j,m)→(k,m+1):

example:

p(1,1)→(2,2) = 1, p(2,2)→(3,3) = λ,

p(2,2)→(4,3) = 1− λ, p(3,3)→(4,4) = 1,

p(4,3)→(1,1) = 1, p(4,4)→(1,1) = 1

in general, transition probabilities can be computed recursively

34

Theorem: This (excitation-only) Markov chain is aperiodic andirreducible, with limiting distribution Q, if and only if cells canmove across multiple bins in a single iteration (as in the aboveexample).

interpretation:

• probability of firing to next input = ΣNm=1Q[(IN , m)]

• expected number of failures = Ef = ΣN−1j=1 jFj, where

Fj = probability of j failures =ΣN−1

k=1 Q(Ik, 1)

ΣNk=1Q(Ik, 1)

ΣN−1

k=2 Q(Ik, 2)

ΣNk=2Q(Ik, 2)

···ΣN−1

k=j Q(Ik, j)

ΣNk=jQ(Ik, j)

Q(IN , j + 1)

ΣNk=j+1Q(Ik, j + 1)

• also extends to inhibition + excitation (see below), homogeneouspopulation with differing inputs, heterogeneous population withidentical inputs

35

example: 2-d TC cell with uniform distribution of intervals betweenexcitatory inputs

• 0 inhibition ⇒ bins (1,1), (2,1), (2,2), (3,2), (3,3) with

v0 = [.3404 .1135 .0922 .3617 .0922]

v0num = [.3276 .1289 .0878 .3629 .0878],

firing to ∼ 45% of inputs, ∼ 1.2 failures per spike

•max inhibition ⇒ 5 bins, 12 states with

vI = [.2290 .0763 .0859 .1622 .0215 .0569 .0624 .0005 .0004 .2211 .0833 .0005]

vInum = [.2272 .0784 .0726 .1927 .0194 .0388 .0669 .0022 .0007 .2171 .0820 .0002],

firing to ∼ 30% of inputs, ∼ 2.28 failures per spike

• similar results with other distributions

• E inputs + jumps in I ⇒ similar agreement, with firing to < 10%of inputs and large Q(I1,1) after onset of inhibition: PD case!

36

SUMMARY:

•Hypothesis 1: PD yields bursting oscillations (GPe, STN, GPi),not just changes in firing rates

– massive experimental support

– STN-GPe loop may generate (Plenz and Kitai, 1999; Terman,Rubin, et al., 2002)

•Hypothesis 2: phasic or bursty inhibition compromises TC relay

– seen in sleep spindles

– seen robustly in model due to IT , INa interaction

•Hypothesis 3: STN DBS restores TC relay reliability

– slow process can equilibrate during sustained inhibition

•Hypothesis 4: compromised relay is related to PD symptoms,while restored relay contributes to alleviation of symptoms

– only therapeutic DBS restores relay in model

– in a heterogeneous population, the effect is enhanced: rhyth-mic inhibition causes cells to fail together, while failures withregular inhibition are spread out

37

Tests:

• VL thalamic burstiness ↑ in PD, ↓ in therapeutic DBS?

• change in thalamus/cortex correlations in DBS, relative to PD?

•motor symptoms associated with thalamic burstiness?

• speed-up of T -current (de)inactivation in TC should help in PD

• do perturbations in VL inputs affect cortical processing when com-bined with other (e.g. Vim) thalamic inputs in cortex?

Future directions:

•more detailed analysis of local impact of DBS (McIntyre et al.)

•multi-cell GPi recordings and larger scale simulations

• optimization of DBS (Feng et al., 2007)

•multi-site stimulation (Tass et al.): mechanisms?

• role of hyperdirect pathway (Leblois et al., 2006)?

• thalamocortical relay/thalamic reticular/cortical interactions?

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