A computational model of Parkinsonian handwriting that highlights the role of the indirect pathway in the basal ganglia

Human Movement Science 28 (2009) 602–618

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A computational model of Parkinsonian handwriting that highlights the role of the indirect pathway in the basal ganglia G. Gangadhar a, D. Joseph b, A.V. Srinivasan c, D. Subramanian b, R.G. Shivakeshavan b, N. Shobana c, V.S. Chakravarthy b,* a

Machine Learning Group, IDIAP Research Institute, CH-1920 Martigny, Switzerland Department of Biotechnology, Indian Institute of Technology – Madras, Chennai, India c Institute of Neurology, Madras Medical College, Chennai, India b

a r t i c l e

i n f o

Article history: Available online 31 August 2009 PsycINFO classification: 2330 4160 Keywords: Handwriting Parkinson’s disease Micrographia Basal ganglia Oscillatory neural networks

a b s t r a c t Parkinsonian handwriting is typically characterized by micrographia, jagged line contour, and unusual fluctuations in pen velocity. In this paper we present a computational model of handwriting generation that highlights the role of the basal ganglia, particularly the indirect pathway. Whereas reduced dopamine levels resulted in reduced letter size, transition of STN–GPe dynamics from desynchronized (normal) to synchronized (PD) condition resulted in increased fluctuations in velocity in the model. We also present handwriting data from PD patients (n = 34) who are at various stages of disease and had taken medication various lengths of time before the handwriting sessions. The patient data are compared with those of age-matched controls. PD handwriting statistically exhibited smaller size and larger velocity fluctuation compared to normal handwriting. Ó 2009 Elsevier B.V. All rights reserved.

Abbreviations: BG, basal ganglia; BP, back propagation; df, degrees of freedom; DP, direct pathway; F, F ratio; GPe, Globus Pallidus externa; GPi, Globus Pallidus interna; IGP, input gate pulse; IP, indirect pathway; OGP, Output Gate Pulse; P, probability of null hypothesis; PD, Parkinson’s disease; SNc, Substantia Nigra pars compacta; SNr, Substantia Nigra pars reticulata; STN, subthalamic nucleus; STR, striatum. * Corresponding author. Tel.: +91 44 2257 4115; fax: +91 44 2257 4102. E-mail addresses: [email protected] (G. Gangadhar), [email protected] (D. Joseph), [email protected] (V.S. Chakravarthy). 0167-9457/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.humov.2009.07.008

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1. Introduction Handwriting has been gaining attention as a source of diagnostic information (Cobbah & Fairhurst, 2000; Kuenstler, Juhnhold, Knapp, & Gertz, 1999; van Gemmert, Teulings, Contreras-Vidal, & Stelmach, 1999), for a variety of neurological disorders including Parkinson’s disease (PD) (van Gemmert et al., 1999; Teulings, Contreras-Vidal, Stelmach, & Adler, 2002), schizophrenia (Gallucci, Phillips, Bradshaw, Vaddadi, & Pantelis, 1997), obsessive–compulsive disorder (OCD) (Mavrogiorgou et al., 2001), Huntington’s disease (Phillips, Bradshaw, Chiu, & Bradshaw, 1994), and more. Being a high-level motor activity, handwriting engages several cortical and subcortical regions including supplementary motor area (SMA), premotor area (PM), primary motor area (M1), basal ganglia (BG), cerebellum and spinal cord. Pathology of any of these regions is manifest as a characteristic distortion of handwriting. For example, PD, a neurodegenerative disorder involving pathology of BG, is associated with motor symptoms like tremor, rigidity, bradykinesia, and postural abnormalities (Marjama-Lyons & Koller, 2001). Principal etiology of PD is the loss of dopaminergic neurons in the Substantia Nigra pars compacta (SNc), a nucleus in BG. Parkinsonian handwriting is characterized by reduced letter size or micrographia (see Fig. 1), jagged handwriting contour, and abnormal fluctuations in velocity/acceleration profile (Teulings & Stelmach, 1991; van Gemmert et al., 1999). Although various handwriting features have been used in the past for PD diagnosis, most such approaches are based on drawing empirical correlations between disease condition and handwriting parameters. Handwriting features like stroke size, peak acceleration, stroke duration, and ratio between mean and standard deviation of stroke length or duration, have been used for PD diagnostics (Phillips, Stelmach, & Teasdale, 1991; Teulings & Stelmach, 1991). Tucha et al. (2006) found increased movement time, reduced maximum and minimum velocity, and increased number of velocity inversions in PD handwriting compared to that of controls. In a study in which PD patients were asked to copy ‘‘lll” patterns of increasing size, it was observed that PD patients undershot target size if the target size was greater than 1.5 cm (van Gemmert, Adler, & Stelmach, 2003). Lange et al. (2006) found disturbed kinematics of handwriting movements in PD patients, as the patient moved from ON state of medication to OFF state medication. A deeper insight into PD pathology could probably be obtained if the handwriting features can be mapped onto neurobiological parameters of BG circuitry, since PD is a disorder involving BG. Let us now present a brief description of the functional anatomy of BG. 1.1. Functional architecture of BG – relation to PD Fig. 2 shows the major nuclei and pathways that constitute BG both during normal and PD conditions. The striatum (STR) serves as a major target for the inputs from cortex to BG (Harner, 1997). Striatal output projections form two distinct parallel channels within the cortico-striato-pallidal pathways: the direct pathway (DP), formed by the inhibitory projections from the striatal output to the GPi, and the indirect pathway (IP) consisting of three components viz., inhibitory projections from striatum to GPe, inhibitory projections from GPe to STN, and excitatory projections from STN to GPi. Activation of the indirect activity is thought to have an opposite effect to that of activation of the direct

Fig. 1. Handwriting samples (the word written is ‘SriRamajayam’ in Tamil language) (a) Normal handwriting (word written once), (b) Handwriting (word written thrice) of a PD patient (52 years old).

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Cortex

GPe

Striatum

SNc STN

GPi

SNr

Thalamus

Cortex Fig. 2. Functional architecture of basal ganglia. Light gray arrows indicate excitatory connections, dark arrows indicate inhibitory connections (GPe – Globus Pallidus externa; GPi – Globus Pallidus interna; STN – Subthalamic nucleus; SNc – Substantia Nigra pars compacta; SNr – Substantia Nigra pars reticulata).

pathway on the GPi neurons. Activation of the indirect pathway tends to increase the activity of GPi cells, and therefore closes the ‘‘gate,” via inhibition of the thalamus, while activation of the direct pathway opens the gate. These pathways may be involved in modulating movement parameters (Contreras-Vidal & Stelmach, 1996). Dopaminergic transmission from SNc has a differential effect on striatal neurons according to its receptor types, D1 and D2. The direct pathway is selected when D1 receptors are activated and the indirect pathway is selected when D2 receptors are activated. Further, increase in striatal dopamine shifts the balance towards the direct pathway, thereby increasing overall motor activity. Thus the indirect pathway is the normally active pathway. The balance is switched just before movement onset, when dopamine release to striatum activates the direct pathway (Clark, Boutros, & Mendez, 2005). An important conceptual breakthrough in the functional understanding of BG came with the idea that dopamine (DA) release from the SNc, a BG nucleus, is thought to represent a reward signal, with the help of which rewarding responses to stimuli can be reinforced (Schultz, 1998). But the machinery for implementing reinforcement learning must consist, in addition to a reward signal, of a stochastic signal (the ‘‘explorer” component of reinforcement learning) that explores the space of possible responses. Although there have been models of exploratory behavior driven by dopaminergic neurons, neuroanatomical substrates of such exploration have not been addressed (Montague, Dayan, Person, & Sejnowski, 1995). Under normal conditions there is very little synchrony among the neurons of STN and GPe nuclei – both within and across the two nuclei (Heimer, Bar-Gad, Goldberg, & Bergman, 2002; Loucif, Wilson, Baig, Lacey, & Stanford, 2005; Raz et al., 2001). We had suggested earlier that this complex activity of the STN–GPe system qualifies this system to play the role of an ‘‘explorer” an important component in reinforcement learning machinery (Sridharan, Prashanth, & Chakravarthy, 2006). It is known that, under PD conditions, this natural complex activity degenerates into more regular forms of activity like synchronous bursting and traveling waves (Bergman, Whichman, Karmon, & DeLong, 1994; Brown et al., 2001; Heimer et al., 2002; Raz et al., 2001; Terman, Rubin, Yew, & Wilson, 2002; Wichmann & Soares, 2006). Such highly regular, pathological activity of STN–GPe system has been linked to PD

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tremor (Hurtado, Gray, Tamas, & Sigvardt, 1999; Terman et al., 2002). PD has been dubbed as a ‘‘dynamical disease” where neurons in the brain region associated with the disease operate in an abnormal dynamic regime (Beuter & Vasilakos, 1995). There has been significant prior effort towards developing neuromotor models of PD handwriting. Contreras-Vidal and Stelmach (1995) present a model of BG that explains normal and Parkinsonian movements. The model consists of a model of BG, in which each nucleus is represented by a single unit, combined with the Vector-Integration-to-Endpoint (VITE) model of (Bullock & Grossberg, 1988) capable of learning and generating simple handwriting movements. The model predictions were compared later with PD handwriting samples taken from van Gemmert et al. (1999). This model has been shown to reproduce many aspects of the normal and PD movement control including hypometria, bradykinesia, akinesia, impairments in the coordination of multiple joints, micrographia, effects of levodopa on movement size and speed, and pallidotomy. However, in the above model of PD handwriting just described, BG nuclei are modeled as lumped units, with activity levels represented by rate codes. BG dynamics is described in terms of fixed-point behavior only. Recently we presented a model of Parkinsonian handwriting in which altered dynamics of STN–GPe is shown to result in jagged handwriting contour and higher fluctuation in pen velocity (Gangadhar, Joseph, & Chakravarthy, 2008). A key idea underlying the present work is to show that rhythm-related changes in PD that occur at neuronal level, can be used in a natural fashion to explain rhythm-related changes in PD like tremor, etc. that are manifest, for example, in handwriting. We also present handwriting data from PD patients, which reveal smaller letter size and higher velocity fluctuations than that of normals. The article is organized as follows: Section 2 presents a neuromotor model of handwriting generation in which stroke velocities are expressed as a Fourier-style decomposition of oscillatory neural activities. Aspects of network function like preparation of oscillator layer, timing and training are briefly discussed. Section 3 presents a model of BG particularly highlighting the function of the indirect pathway consisting of the STN–GPe loop of BG. An integrated model consisting of the handwriting model of Section 2 combined with the BG model of Section 3 is presented in Section 4. Various handwriting distortions that arise due to the introduction of PD pathology into the model are described in Section 5. A coarse mapping of the integrated handwriting model and distortions in handwriting due to the PD pathological conditions are discussed in Section 6. 2. A neuromotor model of handwriting generation The essence of the proposed model of handwriting generation is to produce a stable rhythm in a network of oscillators and resolve the stroke output in Fourier-style, in terms of the oscillatory activities of network oscillators (Gangadhar, Joseph, & Chakravarthy, 2007). The architecture of the net-

Fig. 3. Architecture of the neuromotor model of handwriting generation. The stroke selection vector, n, is presented at the input layer. Output of the network are pen velocities, Ux(t) and Uy(t). The timing signals from the BG model coordinate events in the network. The BG model gates the input and output layers using input gating pulse (IGP) and output gating pulse (OGP) signals, respectively. It prepares the oscillatory layer using a preparatory pulse (PP) signal.

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work that learns to produce strokes has three layers – (1) input layer, (2) oscillatory layer, and (3) output layer (see Fig. 3). The nodes in the input layer are ‘‘stroke selection” nodes (one node for each stroke). All the components of the stroke selection vector (n) are set to zero in resting condition. To produce the kth stroke, the k-th neuron in the input layer is taken to a ‘‘high” (=1) state. In that state, velocity profiles (Ux(t) and Uy(t)) of the corresponding stroke are generated at the output layer. All the strokes are assumed to be of the same duration. The oscillatory layer has several sublayers. All the neurons in a sublayer have the same oscillation frequency. In each sublayer, neurons are connected in a ring topology. Each neuron in the input layer is connected to all neurons in the oscillatory layer. The output layer has two outputs representing horizontal and vertical velocities (Ux and Uy) of the pen tip. Every neuron in the oscillatory layer is connected to both the output neurons. Events in the above 3-layered network are controlled by the timing signals coming from the BG model. Each sublayer consists of a network of nonlinear oscillators connected in a ring topology. All the oscillators in a sublayer are set to the same frequency. The sublayers themselves are set at harmonic frequencies, f, 2f, 3f, etc., where f0 is the frequency of the first sublayer. (This condition may be relaxed and the frequencies of the sublayers may be uniformly distributed over the frequency range of interest (Gangadhar et al., 2007).) For details of the model the interested reader is referred to Gangadhar et al. (2007). Preparing the network state. Note that in the model described above accurate reconstruction of a stroke depends not only on the Fourier coefficients but also on the phases of the oscillators. Since each sublayer has a stable limit cycle attractor, once the initial phase is fixed, each sublayer settles in a specific phase pattern. Therefore, setting the initial phases of the neural oscillators is known as ‘‘preparing the network state”. We compare this step with the process of ‘‘motor preparation”, which is described variously in the literature as system configuration, motor programming, coordinative structure gearing, preparation, planning, schema build-up, etc. (Schomaker, 1991). (A detailed description of preparatory dynamics, available in Gangadhar et al. (2007), is omitted here for reasons of space.). Thus a single movement consists of two stages: preparation + execution. A PP signal is given at the beginning of preparation. Preparation ends when the network approaches a special state known as the standard state, Vs, at which time the execution begins. Execution ends after a fixed duration, Ti. Training. To train the network on lth stroke, nl, the lth input component in input vector n = {n1, n2, n3  nl  nn, 1}, is set to 1, and all other input components are set to 0. The corresponding target output is a set of stroke velocities, Vx(t) and Vy(t). Since this is a supervised problem, the network is trained using Backpropagation with (BP Momentum) and without momentum (Plain BP) (Haykin, 1998). An illustration of a letter reconstructed by a trained network is shown in Fig. 4. In the next section we describe the BG model.

Fig. 4. A sample output of the oscillatory network that is trained to produce the letter ‘‘a”. The original stroke is seen on the left extreme. The original x and y velocity profiles can be seen on the right of the original stroke. The central image shows the evolution of the state of oscillatory layer (Vik(t)) through time. The remaining images on the right are network-reconstructed x and y velocities and reconstructed ‘‘a” respectively.

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3. A model of basal ganglia We have noted earlier in Section 1 that at elevated striatal dopamine levels, striato-pallidal transmission switches from the indirect pathway under resting conditions to the direct pathway, thereby initiating movement. In dopamine-deficient conditions of PD, it is probable that such transfer may not occur effectively, in which case activity of the indirect pathway may contribute significantly even during movement, introducing distortions like tremor and velocity fluctuations. Therefore, the indirect pathway will necessarily dominate a model of BG under dopamine-deficient PD conditions. In the following paragraphs, we describe a model of the STN–GPe, a subsystem of the BG, constituting the indirect pathway, presented as a complete BG model in PD conditions. Below we present a concise description of the BG model; a more expanded version with equations was presented in Gangadhar et al. (2008). Fig. 5a shows a single STN–GPe neuron pair consisting of glutamatergic (+) (STN ? GPe) and gabaergic () (GPe ? STN) connections. Pairs of such neurons are replicated and connected in a 2D grid fashion for realizing the STN–GPe system. STN and GPe are modeled as identical-sized 2D layers of neurons. Each STN neuron is connected to a GPe neuron at the corresponding location in the GPe layer; the GPe neuron in turn sends a recurrent connection to the STN neuron. There are lateral connections in the GPe layer but not in the STN layer. Thus each unit has a negative center and a positive surround; the relative sizes of center and surround are determined by e. Smaller e implies more negative lateral GPe connections. In the absence of input from the striatum, as e is varied from 0 to 1, the activity of GPe neurons varies from uncorrelated behavior to highly correlated behavior. GPe neural dynamics are designed such that the fraction of active neurons in GPe is proportional to striatal dopamine (DA). For example, note that in Fig. 6 (top row), where DA = 50 corresponding to a ‘‘normal” value, at any instant approximately 50% of the GPe neurons are in an active state. Under conditions of PD pathology, this value is reduced to model dopamine-deficient conditions. Fig. 6 shows the temporal evolution of the STN–GPe system in three dynamic regimes. For a fixed DA (=50), as e is increased from 0, the STN–GPe system transitions systematically from (1) complex activity (e < 0.1) to (2) traveling waves (0.1 6 e < 0.4), to (3) synchronized cluster (e P 0.4). In the synchronized cluster regime (bottom row of Fig. 6), a group of neurons in the center fire in synchrony but out of synch with all the other neurons. Thus at any given instant a large number of background neurons are in a synchronized state. Similar dynamical regimes are observed in electrophysiological model of STN–GPe studied by Terman et al. (2002). Terman et al. (2002) relate the regular dynamic regimes (traveling waves and clusters) to the tremor-like symptoms of Parkinsonism. Therefore, we associate the first regime (complex activity (Fig. 6, top row)) with normal function, and the other two (traveling waves (Fig. 6, middle row) and synchronized clusters (Fig. 6, bottom row)) with PD pathology. Above we presented a PD model of BG in which only the indirect pathway is highlighted, since the direct pathway normally remains unselected under dopamine-deficient conditions. Two control parameters, (1) striatal dopamine (DA) and (2) GPe connectivity parameter (e), determine the nature of STN–GPe dynamics in the model. Normal conditions are associated with DA = 50 and e = 0. PD

Fig. 5. (a) STN–GPE neuron pair showing exitatory – inhibitory connections; (b) grid of STN–GPe networks; (c) the lateral connections strengths in the GPe network, with ‘‘a”, as the height of the inverted gaussian, e, as the positive bias to it.

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Fig. 6. Snapshots of GPe network activity over time for the values of e = 0.0, 0.3, and 0.6. The black square corresponds to ‘‘negative” and the white square corresponds to ‘‘positive” values. We can observe (a) the complex activity regime for e = 0.0, (b) traveling waves for e = 0.3 (observe the third black stripe moving from left to right as wave) and (c) synchronized cluster activity for e = 0.6. GPe layer size is 20  20 (DA = 50).

pathology is induced by reducing DA and/or increasing e from these normal values. Reducing DA (<50) reduces the activity levels of GPe layer; increasing e (>0) pushes the STN–GPe dynamics to more regular regimes (traveling waves and synchronized clusters). We now present the integrated model by combining neuromotor system and BG for studying handwriting in both normal and PD conditions. 4. Integrating models of basal ganglia and neuromotor system We now integrate the BG model, in which the role of the indirect pathway in PD pathology is emphasized (described in Section 3), with the neuromotor model of handwriting generation (described in Section 2) to produce a model of Parkinsonian handwriting. Output of the BG model modulates the stroke velocities arising from the handwriting model. Generation of timing signals (PP, IGP, and OGPs) of BG’s indirect pathway is not modeled explicitly with a network in the current paper; they are simply specified (for details of timing signals see Gangadhar et al. (2007)). Only the activity of the indirect pathway is modeled in a network structure as described earlier in Section 3. A conceptual schematic of the integration is depicted in Fig. 7. The event sequence of the integrated Parkinsonian handwriting model during the execution of a single stroke are (i) the initial settings: The IGP is set to ‘‘zero”, so that input (n) cannot effect the oscillator layer of handwriting model before it is prepared. Also the OGPx and OGPy are set to ‘‘zero”, so that

Stroke Velocities

OGP PP IP

DP IGP

Model of HW Generation

Stroke Selection Fig. 7. The conceptual schematic of integrated BG and neuromotor model for generating handwriting. The acronyms: BG-basal ganglia, IP-indirect pathway, DP-direct pathway, PP-preparatory pulse, IGP-input gating pulse, OGPx and OGPy for gating x and y velocities, respectively.

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the velocity of the pen tip is maintained at zero (this represents the ‘‘action gating” function of BG). (ii) Motor preparation: On the arrival of stroke selection input at the handwriting model (in Fig. 3, handwriting model), a copy of the same is sent to the BG model. The BG model then sends a preparatory signal (PP) to the oscillator layer (see Fig. 3). (iii) Stroke execution: Once the preparation is complete, the BG allows the OGP signals (OGPx and OGPy for horizontal and vertical velocities respectively) to vary according to the activity of the STN–GPE system. The OGPx and OGPy signals modulate the output of the handwriting model. The stroke velocities produced by the handwriting model are integrated to get current pen position. Once the stroke generation is complete the stroke selection input (n) to the model is reset. This resets the value of IGP (‘‘low” or ‘‘zero”). Thus the signals arising from the indirect pathway of BG modulate the velocity signals of handwriting model as follows,

U x ðtÞ ¼ OGPx ðtÞ

Nk Ns X X

W xik V ik ðtÞ

ð4:1Þ

W yik V ik ðtÞ

ð4:2Þ

k¼1 i¼1

U y ðtÞ ¼ OGPy ðtÞ

Nk Ns X X k¼1 i¼1

where

OGPx ðtÞ ¼

n X n X i¼1

OGPy ðtÞ ¼

ð4:3Þ

j¼1

n X n X i¼1

x W STN U STN ij ij ðtÞ

STNy

W ij

U STN ij ðtÞ

ð4:4Þ

j¼1

x x where W STN and W STN are the weights connecting STN neurons to the GPi neurons, OGPx(t) and OGij ij Py(t) are the output gating signals arising from GPi, V ik ðtÞ is the state of the ith oscillator in the kth sublayer, W xik and W yik are the weights from the ith oscillator in the kth sublayer to the x and y outputs respectively of the handwriting network. In the next section we describe the clinical methods used to collect data from Parkinsonian patients.

5. Methods 5.1. Participants The study was done on 34 patients and 25 age-matched controls. The patients were between 45–83 yr old (mean age = 63.5 yr, 28 males and 6 females) (Table 1). The controls were between the 50–74 yr old (mean age = 63.1, 19 males and 6 females). Informed Consent from the patients and the controls was obtained. All patients were on regular medication, but at the time of testing the last time they had medication varied between 0.5 h to more than 24 h. All participants were right handed. 5.2. Experimental setup The study included copying the cursive ‘‘elel” pattern (the participants drew some other patterns, but these results are not included in the present study). The participants were instructed to copy the patterns approximately 1 inch below the sample. The participants wrote with an electronic pen with a ballpoint tip on a white paper placed on the digitizer tablet (Genius, G-note, sampling rate 160 Hz, spatial resolution 0.01 cm). They were made to sit comfortably and the digitizer tablet was placed on a table (height of the table = 84 cm, height of the chair = 60.5 cm). Once the participant was comfortable with the setup, clear instructions were given regarding the test. They were instructed to copy the original pattern as accurately as possible. They were also asked not to lift the pen during the task, so that each pattern was copied as a single stroke.

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Table 1 Details of PD patient. S. No.

Age

Gender

Duration of disease (yr)

Time since medication (h)

UPDRS

Hoehn & Yahr

1 2 3 4 5 6 7 8 9 10 11 12 13

55 60 55 67 66 69 73 61 72 78 65 60 45

Male Female Male Male Male Male Male Male Male Female Male Male Female

21 3 7 5 3 8 2 3 1 1 4 2 4

4 3 13 13 15 3 1 24 24 22 3 13 3.5

27 44 48 49 31 36 52 45 32 37 42 27 49

1 1 2 2 2 3 3 3 2 2 2 1 3

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

49 63 74 49 52 66 83 71 72 83 65 67 48 54 62 65 55 62 78 55 60

Male Male Male Male Male Male Male Male Male Male Male Male Female Female Male Male Female Male Male Male Male

4 1 2 3 1 1 6 5 3 12 2 2 4 3 4 1 3 3 3 1 1

1 1 3 3 24 2 3 2 4 4 3 3 12 24 1 1 24 12 24 24 24

33 24 31 42 33 33 14 38 13 18 21 38 24 58 41 10 29 32 63 43 22

2 1 2 2 1 1.5 1 2.5 1 1 1 2 2 3 3 1 1 1 2 2.5 1

5.3. Simulation procedures The handwriting generation network is trained to produce the ‘‘el” pattern. Target velocity profile to train the network is generated as follows. The ‘‘el” pattern consists of repeated occurrence of two loops: a smaller loop for ‘‘e” and a larger one for ‘‘l”. Loop patterns can be generated by an elliptical velocity profile given as (Hollerbach, 1981):

V x ðtÞ ¼ A sinðwx tÞ þ c V y ðtÞ ¼ B sinðwy t þ uy Þ

ð5:1Þ

where Vx(t) and Vy(t) are velocity profiles, A and B are sinusoidal amplitudes, xx and xy are frequencies of horizontal and vertical dimensions respectively; ‘‘c” represents the horizontal drift velocity. Eq. (5.1) describes cycloidal trajectories. We use A = B = 1.26 and c = 0.5 to produce the smaller loop (‘‘e”) and A = B = 3.145 and c = 1.26 to produce the larger loop (‘‘l”). Other parameters are: xx = xy = 3.141 rad/s and / = p/3 rads. Switching between the two loops is done at the points of minima. Vx(t) and Vy(t) corresponding to a single cycle of ‘‘el” is used to train the handwriting network. The handwriting network has four sublayers with 25 oscillators in each sublayer. Training is performed using BP with momentum. Learning rates for the first and second stage weights are 0.000005, and 0.0001, and the momentum factor is 0.7.

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6. Clinical and simulation results Two parameters represent PD pathology in the model: DA level and e. ‘‘Normal” values of DA and e are DA = 50 and e = 0. A reduction of DA from this baseline level results in smaller values of OGP (OGPx and OGPy) and therefore reduction in stroke velocity. An increase of e from the normal value of 0, alters the dynamics of STN–GPe, pushing the system towards more regular forms of dynamics. We now study effects of deviations from on the handwriting generated by the model. Three aspects of PD handwriting distortion are studied: (1) abnormal fluctuations in velocity, (2) reduced letter size, and (3) progressive micrographia. 6.1. Velocity fluctuations The trained network was instructed to write the word ‘‘el” with the value of DA kept constant at 50 throughout the experiment and the value of e varied from 0.0 to 0.6 in steps of 0.1 across trials. Fig. 8 depicts the stroke output of the network (left column), along with velocity magnitude profile (right column). Increasing standard deviation of velocity magnitude, indicating higher fluctuation in velocity, is shown in Fig. 9. Higher e value produces synchronized oscillations in the STN–GPe system. These synchronized oscillations induce fluctuations in the output of GPi, which is simply the Output Gate

Fig. 8. The output stroke sequence generated by the network in the left column and the magnitude of the velocity (1 unit is equal to 3 cm/s) of the pen tip in the right column with a DA level of 50 with values of e is varied. The pen velocity magnitude for various values of epsilon are as indicated (a) e = 0.0, (b) e = 0.4, (c) e = 0.8.

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Pulse or OGP. Since OGP modulates both GPx and GPy, the resulting handwriting has a jagged contour and abnormal fluctuations in velocity. We now compare the corresponding data from normal controls and PD patients. Fig. 10 below compares the respective values of standard deviation of the vertical velocity of the ‘‘elel” pattern drawn by age-matched normals (Group-I), with PD patients who have taken the drug less than 3 h before (Group-II), and longer than 10 h (Group-III) before the handwriting sample was taken. Note that Group-II and Group-III PD patients roughly correspond to patients in ON and OFF states of medication. From Fig. 10 it is evident that velocity fluctuation in normals is less than in PD patients; Group-II PD

Fig. 9. Variation in the standard deviation of the magnitude of the velocity of the pen tip as a function of e (DA = 50). On the y axis one unit is equal to 3 cm/s.

Fig. 10. Standard deviation of vertical velocity of the ‘‘elel” pattern drawn by (1) normals, (2) PD patients who received drug <3 h ago, (3) PD patients who received drug >10 h ago. The standard deviation was calculated over all participants in the group.

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Fig. 11. The output of the model for various values of e and DA. Horizontally from left to right, e is varied from 0.0 to 0.8 (steps of 0.4); vertically from top to bottom, DA is varied from 10 to 50 (steps of 20). As the value of DA decreases, the stroke size decreases. As e increases, the writing contour becomes more distorted.

Fig. 12. Average height of the ‘‘elel” pattern drawn by (1) normals, (2) PD patients who received drug <3 h ago, (3) PD patients who received drug >10 h ago.

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patients have less fluctuation than Group-III PD patients (ANOVA results: F = 3.96, p = .051, df = 58, STD ERROR = 0.025). 6.2. Variation of letter size To evaluate the variation of letter size in the model, we systematically vary the values of both parameters, viz., DA from 10 to 50 and e from 0.0 to 0.8, over different trials (see Fig. 11). The output of the network as a function of DA and e is shown below. The results of this experiment allow us to map the handwriting features onto the level of PD, which, we suggest, can be parameterized by the level of dopamine (DA) and the degree of regularity in STN– GPe activity, controlled by e. The output of the network clearly demonstrates that as dopamine is decreased, the overall size decreases (micrographia); as e is increased, stroke contour becomes more and more jagged and the velocity profile manifests large fluctuations. We now look for similar changes in PD handwriting compared to control participants. Fig. 12 compares the respective sizes of the ‘‘elel” pattern drawn by age controlled normals (GroupI), with the corresponding values from PD patients who have taken the drug less than 3 h before (Group-II), and longer than 10 h (Group-III) before the handwriting sample was taken. Note that normals write larger letters than PD patients and Group-II PD patients draw larger letters than Group-III PD patients (ANOVA results: F = 4.18, p = .045, df = 58, STD ERROR = 0.075) (see Fig. 12). 6.3. Progressive micrographia Micrographia in PD is known to occur in two forms: (a) constant micrographia, and (b) progressive micrographia. In the former the letter size is consistently small, while in the latter, letter size reduces

Fig. 13. The word ‘‘elelel” is produced by the PD handwriting network. DA is allowed to decrease exponentially over the 3 cycles to produce progressive micrographia. (A) e = 0.2, (B) e = 0.4, (C) e = 0.6. See text for details.

Fig. 14. Samples of progressive micrographia exhibited by three patients. (a) Patient #1: 60 yr female; disease duration: 3 yr; medication taken 3 h before the test, (b) patient #2: 75 yr male, disease duration: 3 yr; medication taken 1 h before the test, (c) patient #3: 61 yr male; disease duration: 9 yr; medication taken 30 min before the test.

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progressively during the course of a sentence or a word (see Fig. 14). In the simulation studies shown so far, striatal dopamine levels are assumed to be tonic. However, in reality, striatal dopamine is depleted gradually until replenished by a fresh dopamine burst from SNc. Therefore, one must consider the effect of a variable dopamine signal on handwriting. Kilpatrick, Rooney, Michael, and Wightman (2000) show how striatal dopamine activity habituates to bouts of rewarding stimulation given to median forebrain regions. (An excellent discussion of the computational significance of phasic changes in striatal dopamine is available in Chapter 3 of Daw (2003).) Depletion of striatal dopamine is understandable since extracellular levels of released neurotransmitter deplete over time and there are also regulatory mechanisms that control transmitter release. It may be expected that in PD conditions, with reduced numbers of SNc dopaminergic neurons, this depletion may occur over shorter durations. It will be interesting to see the effect of depletion of striatal dopamine over the time taken to write a sentence or a single word on handwriting output. To this end we make the model described above to produce the ‘‘elelelel” pattern (four cycles of ‘‘el”). Since the network described above is trained to produce a single ‘‘el” pattern, to produce a sequence of ‘‘el”s the network is fired four times consecutively. However, unlike the previous studies described in this section, in the present case dopamine is assumed to fall exponentially with each cycle as follows: DA in 1st cycle: 50; DA in 2nd cycle: 50 exp (0.5) = 30.326; DA in 3rd cycle: 50 exp (1) = 18.39; DA in 4th cycle: 50 exp (1.5) = 11.15. The network produces the ‘‘elelel” pattern under the above conditions of DA depletion for three values of e = 0.2, 0.4, and 0.6 (Fig. 13). Note that there is a gradual reduction in letter size for e = 0.2 and 0.4. However, for e = 0.6, the letter size does not vary significantly during the first two cycles but falls abruptly during the third cycle. Thus the simple, nearly monotonic relationship between DA value and letter size (observed in Fig. 10) is not true anymore for larger e values since the synchronized bursts in STN–GPe activity introduce sharp modulations of stroke velocity that are not strictly related to DA levels. Fig. 14 shows samples of progressive micrographia in handwriting samples of PD patients. In Fig. 14a, there is a more or less gradual reduction in letter size. In Fig. 14b, there is a nonmonotonic variation of letter size from cycle to cycle. In Fig. 14c, there is an abrupt drop in size and change in shape of letter ‘‘e” beyond the third cycle.

7. Discussion We present a model of Parkinsonian handwriting in which an oscillatory neural network model of handwriting model is integrated with a BG model that emphasizes the role of the indirect pathway under PD conditions (Gangadhar et al., 2007). PD pathologies are simulated by varying (1) striatal dopamine (DA) and (2) GPe connectivity parameter (e). Although the model of Contreras-Vidal and Stelmach (1995) also combines a model of BG with a model of handwriting generation, the two approaches have fundamental differences because in the BG network model of Contreras-Vidal and Stelmach (1995) each nucleus is represented in a lumped fashion by a single unit. Only magnitude related aspects of handwriting – faster/slower, larger/smaller, etc. – can be captured with such a model. Our model takes a fundamental departure from previous models in that it links altered dynamics of STN– GPe to handwriting distortions. Oscillatory activity of STN–GPe system has been linked to PD tremor in earlier studies. Synchronized high frequency oscillations are found in STN neurons of PD patients with tremor; dopaminergic medication reduced the synchrony of STN neurons (Levy et al., 2002). In the present model, we observe that a reduction of complexity of STN–GPe activity translates into increased velocity fluctuations in handwriting. Since our main objective is to represent PD pathologies as manifest in handwriting we present a model of BG in which the indirect pathway is emphasized. Two changes mark PD conditions in the BG model: reduced dopamine represented by the parameter DA, and altered dynamics of STN–GPe controlled by the parameter e. Although these two parameters are varied independently in handwriting results of Fig. 10, synchronized oscillations of STN–GPe seem to be the result of reduced dopamine conditions in PD. Therefore, strictly speaking, in order to simulate PD conditions in the model, the two parameters have to be varied (reducing DA and increasing e) together. To address the question of the proper relationship between DA and e, we consider some experimental data below.

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The STN has often been described as the ‘‘clock” or the ‘‘pacemaker” of the BG (Beurrier, Garcia, Bioulac, & Hammond, 2002). STN neurons exhibit two forms of activity: single-spike mode and mixed bursting mode. The mixed busting mode of STN is often found in animal models of PD and in human PD patients. The mixed bursting mode becomes a particularly active mode of STN after degeneration of dopaminergic neurons of SNc. What exactly is the mechanism by which reduction of dopaminergic transmission results in change of dynamic mode of STN neurons? It was observed that local destruction of dopamine fibers into STN altered STN activity much less than destruction of SNc neurons themselves (Ni, Bouali-Benazzouz, Gao, Benabid, & Benazzouz, 2001). Application of dopamine agonist like apomorphine did not completely abolish the bursts. Baufreton et al. (2005) suggest a possible explanation that links loss of dopaminergic transmission with change of dynamics of STN. Under normal conditions, activation of D1 receptors in STN increases firing rates of STN neurons in tonic firing mode, and activation of D2 receptors switches the firing pattern from bursting to tonic firing. Activation of D5 receptors amplifies bursting mode further, but this does not happen in normal conditions. However, under prolonged dopamine-deficient conditions as in PD, D1 and D2 receptors in STN are no longer activated; only D5 receptors are activated resulting in bursting mode firing (Baufreton et al., 2005). In our model, we assume that dopamine controls the activity of STN–GPe neurons. This feature of the model is justified in light of the above-mentioned experimental data, though detailed mechanisms underlying the interaction are not clear. Although we tried to link synchronized oscillations of STN–GPe system with PD handwriting distortions, we have not explained the purpose of complex activity of STN–GPe in normal conditions. An important class of BG modeling examines BG from the perspective of reinforcement learning (Doya, 2000). An exploration of output space modulated by reward information is a key component of reinforcement learning. Sridharan et al. (2006) hypothesized that complex STN–GPe dynamics is necessary for the exploration of output space. Collapse of complexity in STN–GPe dynamics, then, results in inadequate exploration of the space of actions. We suggest that motor aspects of PD symptomology are a manifestation of this inadequate exploration. This broader function of STN–GPe is developed elsewhere (Joseph, Garipelli, & Chakravarthy, submitted for publication). Several instances have been discovered in physiology when chaotic activity of a system is essential for its normal function (Goldberger, Rigney, & West, 1990). A need for chaotic and desynchronized neural activity seems to arise in the problem of motor unit recruitment in skeletal muscle also (Prashanth & Chakravarthy, 2007). It appears that we have a similar situation in the STN–GPe system wherein complex activity corresponds to normal function, and loss of complexity to disease. Such a perspective of PD is in tune with the notion that PD is a ‘‘dynamic disease” (Beuter & Vasilakos, 1995). Handwriting produced under PD ‘‘pathological” regimes of these parameters exhibits classic PD handwriting distortions as follows: (a) Micrographia. Reduced dopamine levels in the model shows reduced handwriting size analogous to micrographia. This was also reflected in experimental data (Fig. 11). (b) Progressive micrographia. When striatal dopamine levels are reduced exponentially, a progressive reduction in stroke size is observed in the model. However, a monotonic reduction in size is observed only for smaller values of e. For larger values of e = 0.6, height variation is more labile. At e = 0.6, STN–GPe system operates in a ‘‘synchronized cluster” regime. Bursting activity of STN–GPe in this regime modulates the stroke velocity in such a way that there is a sharp local increase in velocity destroying the tendency of monotonic fall in size. The model suggests that whether micrographia is of progressive or constant type depends probably not only on rapid dopamine depletion but also on the dynamics of STN–GPe. Confirmation of this prediction can come from detailed experimental studies that link the type of micrographia with the state of BG pathology. (c) Bradykinesia. Since reduced dopamine in BG model reduces BG output levels, which in turn attenuate stroke velocity, bradykinesia comes about as a straightforward consequence of dopamine reduction. Further, experimental studies reveal that average stroke duration of the patients with PD is no longer than the average stroke duration of normal controls indicating that the principle of isochrony is spared in PD patients (van Gemmert et al., 2003). The property of isochrony emerges naturally in the handwriting model, since the stroke output essentially rides over a cycle of the rhythm generated by the oscillatory layer. The model generates all the strokes in more or less the same time, which does not vary in dopamine-depleted conditions.

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