A Neural Model of Cerebellar Learning for Arm Movement Control: Cortico-Spino-Cerebellar Dynamics Jose L

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A Neural Model of Cerebellar Learning for Arm Movement Control: Cortico-Spino-Cerebellar Dynamics Jose L. Contreras-Vidal, 1 Stephen Grossberg, 2 and Daniel Bullock Department of Cognitive and Neural Systems and Center for Adaptive Systems Boston University Boston, Massachusetts 02215

Abstract discharges maintains stable learning. Long-term potentiation (LTP) in response to A neural network model of opponent uncorrelated parallel fiber signals enables cerebellar learning for arm movement previously weakened synapses to recover. control is proposed. The model illustrates Loss of climbing fibers, in the presence of how a central pattern generator in cortex LTP, can erode normal opponent signal and basal ganglia, a neuromuscular force processing. Simulated lesions of the controller in spinal cord, and an adaptive cerebellar network reproduce symptoms of cerebellum cooperate to reduce motor cerebellar disease, including sluggish variability during multijoint arm movements movement onsets, poor execution of using mono- and bi-articular muscles. multijoint plans, and abnormally prolonged Cerebellar learning modifies velocity endpoint oscillations. commands to produce phasic antagonist bursts at interpositus nucleus cells whose feed-forward action overcomes inherent Introduction limitations of spinal feedback control of tracking. Excitation of ct motoneuron pools, Clinical and experimental data on both oculo- combined with inhibition of their Renshaw motor and skeletomotor systems suggest that an cells by the cerebellum, facilitate movement intact cerebellum is neccssary for the learning and initiation and optimal execution. performance of accurate, rapid movements, espe- Transcerebellar pathways are opened by cially movements involving multiple degrees of learning through long-term depression freedom. Systems with multiple, mechanically (LTD) of parallel fiber-Purkinje cell linked degrees of freedom are difficult to control synapses in response to conjunctive because the inertial, gravitational, and interaction stimulation of parallel fibers and climbing forces, which must be compensated whencver fiber discharges that signal muscle stretch they do not assist desired motion or stasis, are con- errors. The cerebellar circuitry also learns to figuration- and rate-dependent. In very slow move- control opponent muscles pairs, allowing ments, there can be sufficient time to use scnsory cocontraction and reciprocal inhibition of feedback to adjust motor commands before errors muscles. Learning is stable, exhibits load caused by uncompensated forces can grow very compensation properties, and generalizes large. But as movement speeds increase, feedback better across movement speeds if begins to arrive too late to be useful for on-line motoneuron pools obey the size principle. control of the current movement. In such cases, The intermittency of climbing fiber accuracy cannot be guaranteed unless motor commands to entire groups of musclcs can be timed accurately and scaled prior to the arrival of feedback. 1present address: Motor Control Laboratory, Arizona State University, Tempe, Arizona 85287-0404. Such considcrations, in combination with ob- 2Corresponding author. servations regarding the cerebellum's typical "side-

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Contreras-Vidal et al. loop" embedding (Fig. 1, below; Ito 1984) within gists (degrees of freedom) be activated at different the sensory-motor system, support a view of the times after detection of a context stimulus. In con- cerebellum as an adaptive controller capable of trast, several features of cerebellar architecture and calibrating parallel feed-forward motor commands dynamics qualify it as the best known candidate for that can substitute for slow feedback-based com- such a sensory-motor memory: the cerebellum's mands and thereby enable high speed and accu- side4oop embedding, the large-scale projection of racy without iterations (Grossberg and Kuperstein state-specifying afferents to the granule cell layer of 1986). Relatedly, the cerebellum has been viewed the cerebellar cortex, the further fanout from gran- by some as an adaptive calibrator of internal for- ule cells of the parallel fibers (PFs), and the long- ward models that allow computation of the ex- term plasticity of PF-Purkinje synapses--now dem- pected effects of motor commands (Kawato and onstrated both in vitro and in vivo (e.g., Crepel et Gomi 1991; Miall et al. 1996). This application of a al. 1996). Moreover, recent modeling studies indi- cerebellar side-loop would allow substitution of cate how the specialized biochemistry of Purkinje fast, "internal feedback" for slow, sensor-based cells, particularly the slowly developing calcium feedback. These two views are not incompatible responses of the metabotropic glutamate receptor (Stein and Glickstein 1992). In both perspectives, cascade, can allow the cerebellum to learn timed the cerebellum learns to substitute a faster, antici- sequences of motor control signals (Fiala and Bul- patory process for a slower, reactive one. Also lock 1996; Fiala et al. 1996; Kano 1996). compatible with both approaches is the hypothesis Analyzing the distinct functions of elements of that external feedback, although rendered progres- the cerebellar system requires systematic modeling sively less important for on-line control as cerebel- and simulations. By this we mean simulations in lar learning progresses, retains a critical teaching which the model cerebellar system includes all of role; such feedback is ultimately the only reliable the major cell types; the model cerebellum's affer- source of information regarding the adequacy of ent signals are generated by simulated receptors the timing and scaling of feed-forward movement and circuits that are part of the simulation; and its commands, or of internal model calibration. At the efferent signals affect the simulated motor plant via level of the cerebellar system, these approaches a simulated spinal circuit. In such a model, all ma- thus postulate that feedback error signals in motor jor elements of the cerebellar system can be exam- coordinates are converted by the inferior olive into ined for their contribution, if any, to improved op- discrete teaching signals, which guide incremental, eration of a sensory-motor system whose dynamics trial-by-trial adjustments in the cerebellar adaptive are realistic enough to appropriately challenge cer- weights that control anticipatory actions. ebellar provisions for learning, memory, and per- Although supported by a wealth of data, the formance. This report describes such a model, view of the cerebellum as an adaptive controller with a focus on how the cerebellar-olivary side- capable of using feedback to learn how to substi- loop can exert learned control of opponently or- tute a faster for a slower "solution" in repeatable ganized limb spinal circuits and thereby improve task contexts has not gone unchallenged. Brain the performance of rapid two-joint movements, stem and spinal cord circuitry apparently provide whose desired form is determined by output from some basis for task-specific recruitment and con- a central pattern generator. trol of muscle synergies, and experimental studies Because the cerebellar model interacts with a of decerebrate/decerebellate animals have pro- two-joint limb moved by sets of opponent muscles, duced evidence suggesting that such circuitry has the simulations are pertinent to issues of oppo- some of the same associative competence widely nency both within the cerebellar system and in the attributed to the cerebellum (Bloedel and Bracha descending pathways from the cerebellum to the 1995). Although suggestive of partial redundancy, spinal cord. Regarding the latter, there has been such studies have not shown that the brain stem/ great interest in recent years in the role of the red spinal circuitry affords a large, content-addressable nucleus (RN) of the reticular formation and the memory for learning and storage of the relatively associated cerebello-rubro-spinal pathway. In many arbitrary task contexts and action groupings re- vertebrates, this pathway is a key element in the quired for a large repertoire of learned skills. Nor system for voluntary movement, and its analysis has it been shown that such circuitry is capable of promises to illuminate cerebellar function because learning to generate novel movements whose ac- the magnocellular division of the RN is dominated curate performance requires that different syner- by inputs from the nucleus interpositus (NIP) of

L E A R N ! N G & M E M O R Y 476 Downloaded from learnmem.cshlp.org on September 27, 2021 - Published by Cold Spring Harbor Laboratory Press CEREBELLAR LEARNING FOR ARM MOVEMENT CONTROL the cerebellum (e.g., Robinson et al. 1987). Much form of the model, these learning results also re- prior modeling work has focused on the role of the veal that muscle opponency imposes no con- NIP/RN in eye blink conditioning (e.g., Bartha et straints on prewiring at the level of the cerebellar al. 1991; Fiala et al. 1996). The comprehensive cortex other than the constraint that CFs carrying skeletomotor simulation strategy followed here re- signals of errors from opponent muscle channels fines and elaborates prior treatments of the limb- should project to distinct parasaggital microzones control role of the cerebello-rubro-spinal pathway. (/to 1984). Some aspects of this work have been Our simulations show a strong advantage, within reported briefly in Bullock et al. (1993a, and tm- the opponent muscle system, of the reported dual publ.) and in a dissertation (Contreras-Vidal 1994). action--a-motoneuron (oL-MN) excitation with Renshaw cell inhibition (Henatsch et al. 1986)--of descending signals from the NIP/RN. Materials and Methods Because the cerebellar cortex model includes all major cell types, as well as the interactions im- Figure 1 schematizes the components needed plied by their connectivity and sign of action, the to model the cerebellar system as it is embedded results illustrate how interactions in cerebellar cor- within the reaching control system that we stud- tex may assist rapid switching between the NIP ied; but it also exemplifies the typical embedding, zones that alternately activate opponent muscles in which there is some primary generator of move- during rapid joint rotations. Such rapid switching ment, such as a reflex circuit, a locomotor pattern appears to be necessary for appropriate shaping of generator, and so on, that sends projections to mo- the motor command pulses needed to achieve de- toneurons as well as to the cerebellum. The cer- sired acceleration and deceleration of limbs. Taken ebellum in turn sends a projection to the same together with prior results regarding adaptive tim- motoneurons that are addressed by the pattern ing, these results clarify how the cerebellum can generator. In addition, there is a feedback signal contribute to generation of critically timed and pathway to one of the olivary nuclei, typically an shaped, multiphasic burst patterns, which are ex- error feedback signal such as a retinal slip, muscle hibited by muscles during rapid movement in nor- stretch, unexpected contact, or a signal reporting mal animals but are disordered during cerebellar compensatory action by an error feedback-driven deactivation (/-Iore et al. 1991). motor command generator. The cerebellum is thus The third aspect of our results concerns learn- part of a side-loop with respect to the direct pro- ing. The model includes a local circuit for signal jection from primary generator to motoneurons, processing by the olivary nucleus. Its inclusion al- and the side-loop circuit receives teaching signals lowed a demonstration that there is no incompat- generated by residual errors of movement. ibility between data indicating that the olive is in- The Figure 1 system is shown in greater detail herently oscillatory (Llinfis 1989), and the hypoth- in Figure 2. As the kinematic central pattern gen- esis that the olive's output via climbing fibers erator (CPG), we chose the VITE (Vector Integra- constitutes a discrete error feedback-driven teach- tion To End point) model (Bullock and Grossberg ing signal. The results also show that the climbing 1988), which embodies key properties of the vol- fiber (CF) signals produced by the olivary model untary, and largely cortical, arm trajectory genera- are suitable for regulating incremental weight ad- tion network that outputs to lower brain centers justments in the cerebellar cortex. In the case stud- through area 4 (Bullock et al. 1997). The VITE CPG ied here, PF-Purkinje cell synapses exhibit both generates desired position and desired velocity long-term depression (LTD) and long-term poten- commands. As the spinal recipient of commands tiation (LTP), depending on whether PF activations from both the VITE CPG and the cerebellum, we do or do not correlate with climbing fiber activa- chose the FLETE model (Bullock and Grossberg tions of Purkinje cells (Sakurai 1987). Elsewhere, 1988, 1991, 1992; Bullock and Contreras-Vidal we have extended the present learning concept to 1993), an opponent muscle control model of how model how the metabotropic glutamate receptor spinal circuits afford independent voluntary con- system can enable LTD at the PF-Purkinje cell syn- trol of joint stiffness and joint position. This model apses to adaptively time depression of Purkinje cell incorporates second-order dynamics, which play a firing. This event enables subcortical pathways to large role in realistic limb movements. read out appropriately timed learned movement FLETE is an acronym for Factorization of gains (Fiala et al. 1996). Because of the opponent LEngth and TEnsion, which summarizes the most

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Contreras-Vidal et aL

"('-MN Figure 1: Cortico-spino-cerebellar model /-N desired structure: Excitatory pathways from a central ~/-'~'trajectory pattern generator (VlTE) send velocity commands to MNs in the spinal cord as well as 1 mossy fiber collaterals, which may bifurcate 4 spindle and send branches to both the cerebellar cor- Olive Renshaw tex and nuclei. The deep cerebellar nuclear cells ultimately project through the RN to kinje MNs-and spinal interneurons such as Ren- shaw cells. Spindle error signals resulting from a comparison of the desired with the ) actual trajectory project to MNs (not shown) and to the inferior olive, where they act as teaching signals via CFs to the Purkinje cells. I corticalVITE !] trajectory These teaching signals regulate LTD at syn- -O Red N. apses where active PFs contact Purkinje NIP - ~~-MN cells. It is hypothesized that for purposes of stability and economy of effort the path for velocity commands through the cerebellum is normally closed by Purkinje inhibition of nuclear cells while the gain of excitatory signals through the MN stage (both from descending commands and from proprioceptive reflexes) is low because of Renshaw inhibition of ot-MNs. However, LTD opens the gate in the transcerebellar side path and allows transient commands to simultaneously excite MNs and inhibit Renshaw cells. (CPG) Central pattern generator; (NIP) nucleus interpositus cells; (pf) parallel fibers; (cf) climbing fibers; ('y-MN) y-motoneurons; (oL-MN) ot-motoneurons. Arrow-terminated pathways are excitatory; solid-circle-terminated pathways are inhibitory.

important design principle used to explain why shows that one proposed teaching signal derives the spinal circuitry evolved to its modern form. For from feedback from muscle spindle organs. Experi- all aspects of motor control, it is critical that the ments on the stretch reflex (e.g., Matthews 1981) higher nervous system be able to exercise indepen- have demonstrated that muscle spindles normally dent control over muscle length and the force or signal the discrepancy (or error) between an intra- tension developed at that muscle length. The fol- fusal muscle length setting and the actual extra- lowing system of equations describes a subset of fusal length. Such discrepancies would also arise the spinal circuitry that can provide this indepen- naturally during voluntary movement, under con- dent control property in a robust way. ditions of ~-y coactivation by a descending desired The scenario for the simulation studies can be kinematics signal, if the rate of limb movement/ described with reference to Figure 1. Entering on joint rotation either undershoots or overshoots the the left is a velocity command (output by the VITE desired rate. In summary, to model cerebellar con- CPG; detail in Fig. 2), which is projected to both tributions to rapid arm movements, we treat the cerebellar cortex and to a deep cerebellar nuclear deep nuclear stage as a normally closed gate site, here interpreted as a zone within the NIP which, after learning in particular contexts, will be (Martin and Ghez 1991; Gibson et al. 1985). The transiently opened by timed Purkinje cell pausing NIP projects to and dominates the RN, which in whenever those contexts recur. turn excites cx-MNs and inhibits associated Ren- We now present a mathematical specification shaw cells (Henatsch et al. 1986). However, exci- of the model used to simulate this scenario. The tation of motoneurons by this pathway is pre- equations used to specify the model mathemati- vented by Purkinje inhibition of NIP unless PF-sig- cally can be divided into three sets: the CPG (VITE) naled contexts from the CPG and spinomuscular equations, the spinomuscular circuit (two-joint system have reliably led to teaching signals from FLETE model) equations, and the cerebellar system the olive that have induced PF-Purkinje LTD and equations. Because the first two sets of equations thereby transiently reduced Purkinje inhibition of are not a focus of this report, they are introduced the NIP, as has been demonstrated for the analo- with just enough detail to achieve a self-contained gous case of eye-blink conditioning (for review, see exposition. They define the system that is to be Bullock et al. 1994). The right side of the diagram assisted by the model cerebellum, which is of pri-

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CEREBELLAR LEARNING FOR ARM MOVEMENT CONTROL

CEREBELLUM VITE

Figure 2: Neural network representation of the neuromuscular control system, including feed-forward cerebellar control. (Top left) The VlTE model for variable-speed trajectory generation; (bottom) the FLETE model of the opponently organized spinomuscular system. Dotted lines show feedback pathways from sensors embedded in muscles. The two lateral feedback pathways arise in spindle organs sensitive to muscle stretch and its first derivative. The two medial feedback pathways arise in Golgi tendon organs sensitive to muscle force. Signals A 1 and A e specify the desired position vector, and the signals GVI and GV2 specify the desired velocity vector; signals 7-1 and Te specify the target position vector; signal P r scales the level of coactivation, and signal GO scales speed of movement. (Top right) Feed- forward cerebellar model computes transient inverse-dynamic signals that excite MNs and modulate the gain in spinal circuits. (b) Basket cell; (P) Purkinje cells; (n) nucleus interpositus cells; (O)inferior olive; (z)long-term memory weights. Paths terminated by solid circles are inhibitory; all others are excitatory.

mary concern. On the other hand, Tables 1 and 2 vector (PPV) command with components a I and summarize neurobiological evidence for each one A 2 from a target position vector (TPV) command of the processing stages of the VITE and FLETE with components T 1 and T 2. These relationships models. lead to the following system. GO signal dynamics are defined by the sigmoi- dal functions

CENTRAL PATTERN GENERATOR; VITE MODEL (t- ~-)2 The VITE circuit (Bullock and Grossberg 1988) G(t) = G O u[t- ~i] (1) 0.5 + (t- ~-i) 2 is a multichannel central pattern generator capable of generating desired arm movement trajectories where parameter G O scales the GO signal, ~i is the by smoothly interpolating between initial and final onset time of the ith volitional command, and u[t] length commands for the synergetic muscles that is a step function that jumps from 0 to 1 to initiate contribute to a prescribed multijoint movement. movement. Difference vector dynamics are de- The rate of the interpolation, and thus the velocity fined by of movement, is controlled by the product of two signals: a difference vector (DV), which continu- ously measures the residual vector of muscle d dt Vi = 30(-Vi + T~- At) (2) length changes needed to reach the final position, and a volitional gating signal, called the GO signal. As shown in the upper part of Figure 2, the DV where Ti is the target position command for with components V1 and V 2 is computed in the muscle channel i, i = 1, 2, and A i is the present CPG by subtracting an outflow present position position command. The difference vector activities

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Table 1: Proposed correspondence between VITE model elements and brain elements

Cell type by Model element physiology References

Desired velocity vector (DVV) area 4 phasic movement Georgopoulos et al. (1982); Fromm et al. time (MT) (1984); Kalaska et al. (1989) Present position vector (PPV) area 4 tonic Fromm et al. (1984); Kettner et al. (1988); Kalaska et al. (1989) Difference vector (DV) posterior area 5 phasic Chapman et al (1984); Crammond and Kalaska (1989); Kalaska et al. (1990); Lacquaniti et al. (1995) Target position vector (TPV) area 5 or area 7b Robinson and Burton (1980); Anderson (1987); Dum and Strick (1990); Lacquaniti et al. (1995) GO signal globus pallidus Horak and Anderson (1984a,b); Kato and Kimura (1992)

(Vi, Vj) are half-wave rectified to generate output where {i,j} = {1,2} designate opponent muscle signals [V~]§ and [Vj]§ to the next processing stage, commands. By equation 3, the PPV integrates the where Figure 2 shows that they are multiplied, or product of opponent GO times DV signals. These gated, by the GO signal to form the vector (G[Vt] § equations treat a case where there are are no cer- G[Vj]§ This vector is called the desired velocity ebellar inputs to the CPG, but this need not always vector (DVV) because opponent subtraction of its be the case. output signals determine the rate of change of the Prior studies (Bullock and Grossberg 1988, PPV at the next processing stage: 1991) have demonstrated that the VITE equations can be used to explain a large number of robust d kinematic features of voluntary point-to-point ~-~Ai= G[Vi]* - G[-Vj]+ (3) movements including smooth, bell-shaped velocity

Table 2: Evidence for connectivity and physiology incorporated in FLETE

Connection type References oL-MNi ~ R i Renshaw (1941, 1946); Eccles et al. (1954) Ri _7__>oL_MN i Renshaw (1941 ); Eccles et al. (1954) R ~-~ lalNi Hultborn et al. (1971) R i 7--> ~/-MN i Ellaway (1968); Ellaway and Murphy (1980) R, :--> Rj Ryall (1970); Ryall and Piercey (1971) lain i -7-->o~-MNj Eccles and Lundberg (1958); Araki, et al. (1960) laiN i :--> laiN] Eccles and Lundberg (1958); Hultborn et al. (1971, 1976); Baldiserra et al. (1987) la i fiber ~ laiN i Hultborn et al. (1971); Baldiserra et al. (1987) la i fiber ~ ot-MN i Lloyd (1943) IblN i -7-> oL-MN i Laporte and Lloyd (1952); Eccles et al. (1957); Kirsch and Rymer (1987) IblN i -~ oL-MNj Laporte and Lloyd (1952); Eccles et al. (1957) IblN i ~ iblNj Laporte and Lloyd (1952); Eccles et al. (1957); Brink et al. (1983) Nonspecific Humphrey and Reed (1983); P -~ spinal Deluca (198) motor pools

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CEREBELLAR LEARNING FOR ARM MOVEMENT CONTROL profiles, voluntary speed control that leaves direc- where the force threshold F F = 1, M s is ot-MN pool tion and amplitude nearly invariant, peak velocity activity in muscle control channel i, [3i is contrac- and acceleration as a function of movement dura- tile rate, and B; measures the number of contractile tion, and a Fitts-type speed-accuracy tradeoff. The fibers. The origin-to-insertion muscle lengths for model's prediction of duration-dependent velocity opponent monoarticular muscles depend on joint profiles has been verified experimentally (Nagasaki angle O as follows: 1989). Recently, the cell types and connectivity of a self-consistently extended version of the VITE model have been used to simulate neuro-physi- L 1 = ~/(cosO) 2 + (20 - sinO) 2 (6) ological data on six identified cell populations in motor and parietal cortex during a variety of move- and ment tasks (Cisek et al. 1996; Bullock et al. 1997). This model extension uses proprioceptive feedback to adjust perceived position estimates when L 2 = ~//(cosO) 2 + (20 + sinO) z (7) obstacles are encountered and to compensate for static forces. The processing stages used in the pre- Equations 6 and 7 indicate that a change of joint sent simulations are shown in Table 1. Also, it has angle always implies a length increment in one been shown recently how basal ganglia structures muscle and a length decrement in its opponent. can generate a time-varying gating signal that has Limb dynamics for a single joint are specified by the properties of the GO signal in the VITE model (Contreras-Vidal and Stelmach 1995).

at e 0 = V 1 -F 2 + F e - n-~tO (8)

OPPONENT SPINO-MUSCULAR CIRCUIT; FLETE MODEL WITH CEREBELLAR INPUTS where F e represents an external force, F~ is the This part of the model, shown in the lower force associated with muscle i, dO/dt is angular part of Figure 2, incorporates an opponent force- velocity in radians, I m is the moment of inertia (set generating circuit that (1) attempts to implement to 1 unless specified otherwise), and n is the joint the CPG movement commands, (2) measures viscosity coefficient (which is used here at a value movement errors and returns error signals on a that allows it to stand in lieu of muscle-related muscle-by-muscle basis when execution is imper- damping; for example, because of the force-veloc- fect, and (3) is modulated by descending cerebellar ity characteristic of muscle). The model shows outputs. As shown in Table 2, the FLETE model qualitatively similar behavior over a range of vis- equations are based on the known neuroanatomy cosities n spanning at least 0.1 ~< n ~< 0.3. and physiology of the spinomuscular circuits, Both contraction rate and the number of con- which include neural, muscular, and sensory parts. tractile fibers are known to increase with excit- The quadratic force-length relationship of atory input to the oL-MN population, among other muscle is approximated in the model by factors. This is the size principle of motor unit organization (Hennemann 1957, 1985). Accord- F i = k ([L i - F i + Ci]+) 2 (4) ingly, contraction rate in equation 5 depends on the level of excitatory input to the oL-MN according where indices i = {1,2} designate antagonist muscle to pairs, F t is muscle force, L i is muscle length, F i - 20.9 is resting muscle length, and C~ is muscle con- ~i = 0.05 + 0.01(A i + n i + P + Eg) (9) tractile state, thus an activation level, not a length variable. The scaling parameter k was fixed at a where A; is the descending present position com- value of 1 in the current simulations. The contrac- mand, P is a coactivation signal (set equal to 0.3 tile state dynamics are defined by unless specified otherwise), E; is stretch feedback from spindles, and n~ is the activity of the NIP/RN d zone associated with muscle i. Likewise, the num- at C = 13g[(Bi- Ci)Mg - C~] - [Fg - F~] + (5) ber of contractile fibers recruited into force pro-

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Contreras-Vidal et al. duction also depends on the net excitatory drive to These equations indicate that model IalNs are re- the ot-MN: ciprocally inhibitory (Hultborn et al. 1971) and are not directly affected by descending signals from B i = 0.3 + 3(A i + n i + P + El) (10) the NIP/RN. They are, however, indirectly affected by NIP/RN inhibition of Renshaw cells. The net Figure 2 shows that axons emerging from moto- effect is disinhibition of the IaIN associated with an neuron pools Mi send excitatory collaterals to Ren- NIP/RN-activated muscle channel. This effect al- shaw cells. The Renshaw population activity is lows signals from NIP to reinforce reciprocal inhi- modeled by the membrane equation bition. The IbIN population activity is excited by d -~tRi = (5B i -- Ri)ziM i - Ri(0.8 + mj + 25n;), pathways originating in force-sensitive Golgi tendon organs: (11) where the Renshaw cell recruitment rate z; de- d --dtxi = 0.2(5 - x;)F~ - xi(0.8 + 0.2x) (18) pended on the level M i of c~-MN activation:

zi = 0.05(1 +M;) (12) Two other Golgi tendon organ feedback-related activities are a force-derivative related activity: The Renshaw population output signal is d R + = Max[0, Ri] (13) ~Y,. = 0.2(5 - YI)Fg- Y~(1 +Xi) (19) which equals R~, as R~ i> 0 by equation 11. Max[0,X] means the maximum of 0 or X. The and an interneuron population activity: oL-MN population activity is also governed by a membrane equation, as are all subsequent model d neural populations. Here, ~tzi -- 0.2(5 - z,)v,. - zi (20)

d -~tMi = (kB i - Mi)(A i + n i + P + E i + Z;) This population's output signal is

- (M i + 1.6)(0.2 + R~ + X; + I~) (14) Z+=Max [O,Z;-O. 21 (21) where X; is the type Ib interneuron (IblN) force feedback (see equation 18) and Zf is a signal de- Equations 19-21 and 14 say that the spinal net- pendent on the rate of change of IbIN force feed- work computes the derivative of force in one chan- back in the opponent muscle channel (see equa- nel, delays it, thresholds it, and then uses the re- tion 20). The oL-MN population output signal is sultant signal to excite ot-MN's in the opponent channel. Such an operation helps brake rapid M + i = Max[O,Mi] (15) movements and allows the network to exhibit an inverse myotatic reflex (Bullock et al. 1993b). Equations 14 and 11 say that descending signals n; The static ~/-MN activity is from the NIP/RN have an excitatory effect on ~-MNs and an inhibitory effect on Renshaw cells, d as reported by Henatsch et al. (1986). The type I a ~S; = 5(2 - Si)(a i + P) - (S i + 1.2)[0.2 + 0.3h(Ri)l, interneuron (IaIN) population activity is defined by (22)

d w and its output signal is -~tli= (lO- Ii)(Ai + P + Ei) - (Ii + 1)(1 + Ri + I;) where h(w) - 0.3 + w' (16) S~i=Max[0,Si] (23) and its output signal is The intrafusal muscle contraction associated with Ii=+ Max[O,/i] (17) static ~/-MN activation obeys

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d tween joint flexion or extension, whereas others -~tUi=( 2-U~)S+-U, (24) show responses over roughly half of the joint's range. The dynamic ~-MN activity satisfies Consistent with experimental data (Millar 1975; Tracey 1979) on half-range receptors that d fire more strongly as extremes are approached, we 2;= (8 - Di)(100G[V,.] + + P) modeled the joint receptor's dependence on joint angle in equation 8 with -(Di+ 1.2)[1 + 100G[Vj] + +0.5h(Ri) ] (25) Jf= 0.75013 if O > 0 rad (30) and its output signal is for shoulder flexion relative to the "zero" angle in the middle of the joint's range and D + = Max[O,Di] (26) Je = 0.75101 ~3 if O < 0 rad (31) The intrafusal muscle contraction associated with dynamic y-MN activation is for joint receptors activated by shoulder extension relative to the midpoint. d This two-component neural feedback signal -fftNi=O.l(2 - Ni)D~- ION i (27) specifies joint angle under both static and dynamic (motion) conditions regardless of direction or The spindle receptor activation was modeled with speed of angular changes. The signal is bounded because of biomechanical limits on the range of d joint angle O. For the two-dimensional simulations, ~W~= (2 - W,)([U~ + Li - ri] +) the shoulder joint receptor feedback from the agonist and the antagonist channels was used as addi- G,,([Nr d 0.3 + + d-~L;]+ ) -10W,. (28) tional excitatory inputs to the agonist and antagonist ot-MN equations for the elbow. Thus, signal Jf excited the elbow flexor and signal Je excited the where G v = 2, and the resting length F; = 20.9 for elbow extensor channel. This was accomplished these simulations. The stretch feedback signal was by simply adding these inputs to the excitatory a linear function of spindle receptor activation parts of equations otherwise identical to equation 14. E i = GsW ~ (29) The FLETE model equations have been used in prior reports (Bullock and Contreras-Vidal 1993; Bullock and Grossberg 1989, 1991, 1992) to ex- where feedback gain G s was set equal to 1. plain a number of properties of neuromuscular The two-dimensional planar arm simulations control, including yielding-compensating proper- used joint receptor feedback from the shoulder ties of the size principle of motoneuron recruit- joint. For reasons described later, it was assumed ment, the ability to voltmtarily vary joint stiffness that joint receptors activated by shoulder flexion without inadvertently changing posture, and emer- would project only to ot-MN pools that control el- gence of multiphasic burst patterns in the electro- bow flexion, and that only those joint receptors myograms (EMGs) associated with rapid, self-termi- activated by shoulder extension would project to nated joint rotations. ot-MNs that control elbow extension. The shape of joint receptors' response functions seems to vary from joint to joint (see Burgess ADAPTIVE SIDE-LOOP PROCESSING; THE et al. 1982). For example, elbow and wrist joint CEREBELLO-OLIVARY MODEL receptors discharge primarily near the end of the joint range (Millar 1975; Tracey 1979), but those in The cerebello-olivary model network is sche- the hip signal over a large fraction of the working matized in Figure 2 (top right) and in Figure 4, range of this joint (Carli et al. 1979). In addition, below. Table 3 contains current anatomical and some articular receptors do not distinguish be- neurophysiological data on the cerebellum that

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Contreras-Vidal et al. lable 3: Evidence for connectivity and physiology incorporated in the cerebello-olivary model

Connection type References mossy fiber #--> gi, k.> ni Shinoda et al. (1992) t i ~-~ n i Courville et al. (1977) mossy fiber ~/i Eccles et al. (1967) gi +-'> li li 7__>gi Llinas (1981 ) Pi ~> ni Ito and Yoshida (1966); Ito et al. (1970) Fractured somatotopy Eccles et al., (1971 a,b); Bloedel and Courville (1981 ) t i electrotonic coupling Llin~s et al. (1974); Mano et al. (1989) ti +----)pi Bloedel and Courville (1981) mossy ~ gi ~ Pi .7__>ni Eccles (1977) ti ~ Pi dendrites mossy ~ gi ~ bi z_> Pi LTD and LTP at gi ~ Pi synapses Sakurai (1987); Hirano (1990, 1991 ); Crepel et al. (1996) n i +---) c~-MN Henatsch et al. (1986); Robinson et al. (1987) n i -~ Renshaw cells Henatsch et al. (1986) supports the cerebeno-olivary network. Purkinje Thus, the simulated model made no distinction be- cell activation obeys tween NIP and RN processing and assumes that each NIP/RN zone receives only one desired veloc- d ity signal. An earlier report (Bullock et al. 1994) ~-tpt = 2[- 2pi + (1 -Pi) showed that such specificity could be learned in x ( 25~g,z~ + t~ +f(Pi) + 0.3 ) NIP because climbing fibers (and the teaching signals that they carry) project both to a limited cortical microzone and to the deep nuclear cells that - (0.8 +pi)(O.lpj + bi) ] (32) receive inhibition from that region of cerebellar cotrex. Such learning was not modeled in this where zk; is an adaptive weight that multiplies the study, although its outcome was assumed. sampling signal gk carried by the parallel fibers Elements of the NIP/RN stage were modeled from the kth granule cell to the ith Purkinje cell, tg with is a CF teaching signal generated by the ith olivary zone, j*Iw] = w3/(0.25 + w 3) is a serf-excitatory d sigmoid signal function, and bf is stellate/basket ~gi = 2[(-2g, + (1-gt)(0.2 + 25,000G [Vi]+) cell inhibitory input. Inhibitory input. Inhibitory - (0.8 + gi)li] (35) stellate/basket-type interneurons were modeled with where G[V,.]§ is a desired velocity input assumed to be projected to the granule by the mossy fiber ~-~b~ =-b, + 3(2 - b~) [gk - 0.41 + , (33) pathway, and Ii is Golgi cell input. This equation indicates that granule cells were excited by desired velocity signals and inhibited by Golgi cells. The where gk is the activity of the kth granule cell, Golgi cells were modeled as excited by both de- implies that these cells share parallel fiber recep- sired velocity inputs and granule cell outputs: tive fields with the Purkinje cells that they inhibit. Elements of the NIP/RN stage were modeled d with --~1 i -- --1 i + (2 --/;)(25,000G [Vy[gk] +) (36)

d This type of granule-Golgi interaction filters the -~tni= 2[(-2ni + (1 - ni) (0.2 + 2500G [V,.]+) input to produce transient outputs from granule - (0.8 + n~)pt]. (34) cells with increasing and greater-than-average ex-

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CEREBELLAR LEARNING FOR ARM MOVEMENT CONTROL citatory input levels. In these simulations, Golgi and modulation served to shorten the duration for which the desired velocity input produced granule d output and to differentiate the velocity signal to -~tUi = O.I(--U i + t i) (39) obtain a phase lead. The adaptive weights zki from parallel fibers where the spindle activation E i is defined by equa- onto Purkinje cells were adjusted up (LTP) and tion 29. By transforming a step increase into a down (LTD) within a range from 0 to 1 according pulse, this circuit improves stability by preventing to the learning law: the path through the olive to the NIP from oper- ating as a continuous negative feedback pathway. d It also delimits the time during which the teaching 100~ Zki= gk [30(1 -- Zki ) -- lOOtiPiZki ] signal acts in cerebellar cortex, which is needed (37) for precise adaptive timing. However, use of recurrent feedback from the slow inhibitory interneuron where gk is the parallel fiber signal from the kth also makes the olive inherently oscillatory. In the granule cell to the ith Purkinje cell. Term t i is the present context, this experimentally observed climbing fiber teaching signal. This equation em- property (Llinfis 1989) is treated as a side effect bodies the hypothesis that LTP occurs whenever rather than as a key functional property. PF synapses are active without coincident CF activity, whereas LTD requires that both PF and CF signals be nonzero (Ito 1991; Ito and Karachot 1992; Fiala et al.). In addition, if an antagonist Pur- kinje cell activity pj inhibits Pi, as in equation 32, DYNAMICS FORMULATION OF 2 DEGREES OF then learning may not occur in equation 37 even if FREEDOM PLANAR ARM t i and gk are positive. The learning law (equation Some simulations of the model incorporate 37) hereby realizes a type of opponent learning via equations (Asada and Slotine 1986) for a 2 degree the voltage-dependent pi. If it is empirically the of freedom planar arm (Fig. 3) that were based on case that the CF burst invariably depolarizes the parameters of the human arm. Here we replace Purkinje cell, then a learning law withoutPi would equation 8 as follows. Assuming moment arms con- be functionally equivalent. stant and equal to 1, the net torque produced by The CF teaching signal t~ was set equal to the the muscles [pectoralis (pec) and posterior deltoid output of the inferior olive network. Each muscle- (pd)] acting at the shoulder becomes related channel i of this network, indicated by a site O, in Figure 2, was composed of two model neurons: an excitatory projection neuron with ac- Fpe c - Fpd=X 1 -- Hl1( ~ + H12(~ - h(i) 2 _ tivity t i and an inhibitory interneuron with activity u i (Ellias and Grossberg 1975). The olivary projec- 2h(~ + G 1 + bl(~ (40) tion neuron is inhibited by the interneuron and excited by spindle error feedback signals from the The net torque produced by muscles [biceps (bic) associated muscle. The olivary interneuron is ex- and triceps (tri)] acting at the elbow becomes cited by, and slowly tracks the activation level of, the projection neuron. This system generates a phasic excitatory burst whenever the excitatory Phi c -- Ftr i = ,I"2 = H22(~ + H120"C + h0 2 + G 2 + b2(i) input from the spindle exhibits a significant in- (41) crease in amplitude. However, if the input then Hll = rail21 + I 1 + m2[l 2 + & remains the same, or decreases, the phasic burst + 211 lc2COS(~) ] + I 2 (42) will not be repeated, because of the inhibitory feedback to the projection neuron via the interneuron. Thus, H22 - m2122 + I 2 (43)

d H,2 = m21,1c2COS(~ ) + m2122 + I 2 (44) -~ttt, = -2t, + (1-ti)(33.3[t ~ - 0.4] + +E i

- 33.3ti[u i - 0.4] + (38) h = m2lllc2sin(~) (45)

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Contreras-Vidal et aL

velocity, and joint stiffness, in this study it is assumed to be constant. The product of viscosity and angular velocity is important in achieving stability of the limb. We used typical estimates of segment masses (mi) and segment lengths (li) and inertial characteristics from anthropometric data (see Table 4) of Zatsirosky and Seluyanov (1983) and Karst and Hasan (1991). In our simulations, the shoulder and the elbow are restricted to one rotational degree of freedom (flexion-extension). The muscles whose actions are modeled in these simulations are listed in Table 5 and depicted in Figure 3: (PDEL) posterior deltoid; (PEC) pectoralis major; (BIC) biceps; and (TRD triceps. To compute muscle lengths, we assumed rotary joints affected by two opponent muscles (i.e., agonist and antagonist), each of which is inserted in the moving segment distal to the axis of rotation. The distance from muscle origin to the axis of rotation was given by/j, j = c, s, (see Table 1), and the midpoint of the joint's 180 ~ excursion was stipulated to be at joint angle O = 0 ~ and (I)= 0 ~ for the shoulder and elbow, respectively. Origin-to-insertion muscle lengths are a function of angle O for the single-joint muscles but are functions of O and (I) for biarticular muscles. It is assumed that the distance, lc~ i = 1,2, to the center of mass of each segment link equals half of the Figure 3: (A) Geometry of origin and insertion points length of the line joining the two ends of the link. for mono- and biarticular muscles. Monoarticular The PEC and PDEL muscles are single-joint muscles (PDEL and PEC) cross one joint, biarticular muscles, and their length at a given rotation can be muscles (BIC, TRI) cross two joints. (B) Schematic of PEC computed as follows: muscle geometry (and that of its antagonist PDEL). (C) Schematic of BIC muscle geometry (and that of its an- tagonistic TRI). (O) Shoulder joint angle; (~) elbow joint Lpa =~/[/c + a sin()] 2 + a 2 cos20 (48) angle; (a) insertion point for shoulder monoarticular muscle; (aE) insertion point for elbow biarticular 2 muscle; (I c) distance from clavicle/scapula to shoulder joint; (I S) length of humerus; (b) muscle origin distance Table 4: Anthropometric parameter values for from shoulder joint; (D) normal to the line of pull of the upper limb muscle. Parameter Value

m 1 1.97 kg G 1 = mllclg cos(O) + m2g{lc2cos(O + ~) /1 1.3 x 10 -2 kg/m 2 + llCOS(O)} (46) 11 = I s 36.0 cm m~ 1.64 kg (47) G 2 = m21c2mh; lq cos(O + ~) 12 2.7 x 10 -2 kg/m 2 /2 47.0 cm where O represents the shoulder joint angle in a Ic1 = 11/2 18.0 cm planar horizontal arm, @ represents the elbow G = 12/2 23.5 cm joint angle, g represents the acceleration of gravity bl, b2 0.3 Nm.sec/rad along the negative y-axis, b I and b 2 are viscosity I c = 11/2 18.0 cm coefficients that account for friction at each joint. b 1 cm Though viscosity may be a function of position,

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CEREBELLAR LEARNING FOR ARM MOVEMENT CONTROL

Table 5: Upper arm muscles Muscle Origin Insertion Type

Biceps-long head (BIC) glenoid fossa radius biarticular Triceps-lateral head (TRI) fossa of scapula ulna biarticular Pectoralis major (PEC) clavicule humerus monoarticular Deltoid-posterior (PDEL) scapula h u meru s monoarticu lar

where Lpa is the length of the PDEL muscle, l I is Results the distance from the clavicle/spatula to the shoul- In this section we present three sets of results. der joint, a = 10 cm is the insertion distance from First, we show how adaptive weights evolve in the proximal (shoulder) joint, and O is the angle of cerebellar cortex when the simulated limb is exer- rotation centered at ~) = 0 ~ (see Fig. 3). For the PEC cised by a long series of attempted movements. We muscle, show that weight evolution converged and was such as to establish a reciprocal pattern of function Lpe c = ~/[l c - a sin(O)] 2 + a 2 COS2(O) (49) at this level of the system. Thus, the learning rule is both stable and capable of pruning an initial con- Moment arms were assumed to be constant over nectivity to increase efficiency. Second, we show the whole range of limb movements, and their val- that model NIP/RN activity after learning exhibits ues were fixed at 1.0. Alexander (1981) has noted phasic bursting, with burst amplitude a function of that the angle of pennation--namely, the angle at desired movement rate. This result corresponds to which the muscle fibers pull with maximum observations of cells in the NIP and the magnocel- force--is approximately a constant between 0.87 lular portion of the RN, which gives rise to the and 1.0, which is why the moment arm can be rubro-spinal projection (e.g., Gibson et al. 1985; considered constant. In the case of the BIC and Martin and Ghez 1991). We also show that both TRI, the muscles cross two joints; therefore, the aspects of the dual projection from RN to the spi- shortening of these muscles is affected by both the nal cord are important. In particular, the inhibition elbow joint angle and the shoulder joint angle. We of Renshaw cells transiently enhances the gain of can approximate the length of these muscles as the stretch reflex (Hultborn et al. 1979; Henatsch follows (see Fig. 3). In the case of TRI, et al. 1986) and prevents premature truncation of o~-MN bursts needed to launch and brake rapid movements. Third, we show that the model learns /[I s + a e sin(q~) - b sin(O)] 2 (50) to substantially improve the execution of a two- Lt + (ae - b) 2 COS2((I)) joint movement by compensating for mechanical interactions between the two moving segments. where a e = 15 cm is the muscle insertion distance Because this improvement is measured relative to a from the elbow joint, b = 1 cm is the muscle origin decerebellate version of the model, which lacked distance from the shoulder joint, 12 is the humerus both the model cerebellum and its output to spinal length, O is the angle of rotation of the humerus circuits via the rubro-spinal pathway, the improve- segment with respect to the vertical, and 9 is the ment is directly attributable to the adaptive cer- rotation angle of the forearm with respect to the ebello-rubro-spinal system. humerus. In the case of the BIC muscle,

/[l s -- a e sin(q~) - b sin(O)] 2 lb (51) LEARNING AND EMERGENCE OF RECIPROCAL + (a e - b) 2 cos2((I)) PATTERNING IN CEREBELLAR CORTEX

Note that the lengths of BIC and TRI cannot be A major feature of the cerebellar cortex is the related simply to angle q) of the elbow joint. The asymmetry between the impressive fanout of sig- muscle lengths for biarticulated muscles depend nals carried by mossy and parallel fibers to widely on both the elbow angle and the shoulder angle. distributed Purkinje cells and the restricted distri-

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Contreras-Vidal et aL bution of signals carried by CFs to bounded sets of desired. Figure 4 schematizes the network before Purkinje cells within parasagittal microzones (Ito (Fig. 4A) and after (Fig. 4B) the learning that oc- 1984). This is mimicked in the model by the dis- curred during a series of single joint flexions and tribution of desired velocity signals, via the mossy- extensions performed by the three component granule-PF pathway, to all Purkinje cells in the model. Figure 4C shows that the weight change is model, combined with distribution of each CF to initially fast but becomes asymptotic over this se- only one Purkinje cell associated with a single NIP/ ries of learning trials. After learning, the PF projec- RN zone and its target muscle; namely, that muscle tion of the velocity signal G[VI] + command in whose stretch receptors project to the olivary zone equation 35 to Pl in equation 32 has been func- giving rise to that CF. tionally deleted from the network by LTD but the A first question to ask is whether the restricted projection to P2 remains. As a result, G[V1] + onset CF distribution leads to selective disinhibition of acts at the NIP stage to excite n 1 and in the cer- only that NIP/RN site that also receives the desired ebellar cortex to inhibit Pl. The latter effect is velocity input and whose projection to spinal cord achieved because G[V1] + excitation of p2 has been can help activate the muscle whose contraction is removed, whereas the PF projection of G[V1] + con-

Figure 4: Opponent cerebellar connectivity before (A) and after (B) learning. All of the synaptic weights between PFs and Purkinje cells were initially set to 1. During learning, weights in signal pathways Z22 and Zll shown in A are diminished through PF to CF LTD, resulting in the connectivity shown in B. (C-) Model simulations with cerebellar feed-forward learning. (Top) Random gain schedule for the GO signal multiplier during arm movement involving cyclic 25 degree flexion and extension; the GO schedule was random- ized over 500 trials (uniform distribution between 1 and 20) to vary the movement speed. (Middle) Learn- ing at PF-Purkinje cell synapses is stable and approaches a steady state after -400 movement trials; (bottom) joint position (O) and velocity (0). traces during the last five movement trials. The joint velocity trace shows that the system is able to dynamically track the desired joint velocity 0 d.

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CEREBELLAR LEARNING FOR ARM MOVEMENT CONTROL tinues to excite two sources of inhibition to Pl, channel burst at the NIP/RN was shortened signifi- namely bl in equation 33 and P2 in equation 32. cantly, and a following burst emerged in the an- Such unopposed inhibition of Pi leads it to pause, tagonist channel of the NIP/RN. This bursting pat- which opens the gate for passage of part of the tern was more reminiscent of the multiphasic burst G[V1] § command through the NIP stage. A recent pattern that is observed at the level of the EMG report on adaptive timing in cerebellum (Fiala et al. during rapid joint rotations, in which a large brak- 1996) proposed an additional source of Purkinje ing burst in the antagonist becomes necessary to cell pausing that may complement those simulated prevent limb inertia from causing a large overshoot here. relative to the intended end point of the movement (Lestienne 1979). The emergence of a braking burst in the model RN is not surprising, as the role of the cerebellum BIPHASIC BURST ACTIVITY OF DEEP NUCLEAR in the current model is to help generate the CELLS AND EFFECTS OF THE DUAL PROJECTION torques needed to achieve the desired velocity and FROM RN TO SPINAL CIRCUITS position. That is, its role is to compute solutions to inverse dynamics problems. The model thus pre- An important benchmark for a cerebellar dicts that if RN activity is examined appropriately model is data collected in recent decades on phasic during rapid practiced movements, the uniphasic activities of cells in the NIP and in its primary pro- pattern reported in many prior experiments should jection target, the magnocellular zone of the RN. In be replaced with a biphasic pattern distributed many studies, phasic activity increments in RN across two subpopulations. have appeared only in cells associated with the The model also clarifies the functional signifi- agonist muscle (Gibson et al. 1985; Martin and cance of the dual projection from RN to the spinal Ghez 1991) and the extended durations of these circuits. In earlier work, it was proposed (Bullock bursts have led some to interpret the RN signal as and Grossberg 1989; Akazawa and Kato 1990) that a velocity command. In our simulations as well, one role of the Renshaw cells is to make the re- shown in Figure 5, medium- and low-speed move- sponse of the a-MN pool to excitatory inputs more ments produced agonist channel bursts whose du- linear. In the case of an excitatory cocontractive ration was similar to that of the entire movement, signal sent to both opponent ~-MN pools, such a and therefore might be interpreted as velocity linearizing effect helps ensure independent control commands (if one were to ignore the actual struc- of joint angle and joint stiffness. Such indepen- ture of the model). However, when the speed of dence is quite important in postural control, when desired movement was increased, the agonist an animal may want to stiffen a joint without changing the difference between the forces acting across the joint. Movement, however, often requires rapid se- AGONIST ANTAGONIST quencing through large differences in the forces 0.5 / ~--A I , A I acting across the joint. Rapid rises of force toward 0.1 -.qt--- B 0.4 the maximal voluntary force level are impossible 0.0 0.3-- B C - unless Renshaw inhibition is curtailed during the D rise time. Because one leg of the dual-descending 0.2 -0.1 projection from the RN inhibits Renshaw cells, the 0.1 -0.2 cerebellum is capable of transiently shifting the

0.0 e~-MN pools out of the near-linear regime created -0.3 I 0.0 5.0 0.0 5.0 by Renshaw feedback and into the nonlinear regime made possible by the size principle of oL-MN Figure 5: Parametric analysis of the NIP response to recruitment. Figure 6 depicts the tracking capabili- movement of different speeds. (Left) Response of the ties of four networks corresponding to the entire agonist NIP cell; (right) antagonist NIP pool. Condition A corresponds to the faster movement, and condition D circuit of Figure 2 or various subsets thereof. In all corresponds to the slower movement. Two intermediate cases, the same desired trajectory was generated, conditions (B) and (C) are also presented. The input was for a flexion movement that started from an ini- the desired velocity vector (G[V] § signal from the VlTE tially extended joint position. The desired velocity circuit. profile appears in each panel as dotted trace c,

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Contreras-Vidal et al.

A DECEREBELLATE B INTACT Joint variables Joint variables

10.00 l- ~ ' ' a a 1 5.~176i)';7\, b .... lO.OO! b 0.00 ~ "~ "----::::: ...... ~ .... -5.00 ~174i J '~\:7 _10.00 ~- -15.00 -10.00I -20.00 -25.00 -20.001 -30.00 -30.00 ? -35.00 S -40-~176I / -40.00 -45.001 // ~50~ ~ J -50.00 p l ___. i 0.00 I0.00 20.00 0.00 10.00 20.00 C RENSHAW LESION D o~-MN LESION

Joint variables Joint variables

10.00 k- ' ' ' a i; J . t a 10.00 ,4~s~. "b...... :t,\ 5.00 /_,/;,:~,,,: i~r" r~ r,,. .~, ;. .... b 0.00 ' "~-§189 "-~- 5.oo f :si! ',! -~.... \l ~t 'd \/ .i ",./ -F ~,-. o.oo ':'..c-l~\:,~...... -5.00 Figure 6: VITE-FLETE-CEREBELLUM -5.00 -10.00 model simulations for decerebellate net- -10.00 -15.00~ work (A), intact system (B), missing inhibi- -15.00 t -20.00 ]- tory projections from NIP/RN to Renshaw -25.00 -20.00 I -3o.oo~- cells (C), and missing excitatory projections -3s.oo~ /,,/ from NIP/RN to o~-MN (D). Trace a repre- -30.00 -4o.oo) sents the joint trajectory, b represents the -35.00 t -45.00~- -40.00 ~- / ...... -5o.~176 , joint velocity, and c represents the desired t velocity command. 0.00 10.00 20.00 0.00 10.00 20.00

whereas the realized velocity appears as trace a seen in cerebellar patients. Without RN inhibition and the realized position as trace b. Figure 6A of Renshaw cells, the o~-MN burst and consequent shows the decerebellate case, where the dynamics force development are truncated prior to reaching of the movement are poor compared to the ideal the levels needed for the desired movement rate. velocity profile. The movement develops slowly The effects of specific simulated lesions in the and the reaction time is prolonged as seen from the NIP/RN projections to the spinal circuit, corre- difference between the onsets of desired and ac- sponding to the simulations shown in Figure 6, C tual joint velocities. The joint velocity has a long and D, were analyzed in terms of o~-MN activity and fight tail, and the rise time is slow. spindle feedback for the network without a NIP/ When only the RN projections to Renshaw RN-to-Renshaw projection (Fig. 7A,B) and the net- cells are eliminated (Fig. 6C), the system shows work without a NIP/RN-to-oL-MN pool projection end-point oscillations that are abnormally pro- (Fig. 7C,D). Both oL-MN activity and spindle feed- longed. Cutting the NIP/RN projection to the Ren- back reflect the oscillatory behavior caused by the shaw cells removed input related to the phasic and cerebellar lesions. Figure 7A shows that the agonist the tonic baseline components of NIP/RN dis- burst is stronger than the antagonist, but there is charge. A side effect is that the equilibrium values coactivation of both muscles. A comparison of Fig- of the system are modified, because the Renshaw ures 7B and 8D indicates that this coactivation re- cells do not receive the inhibitory tonic baseline sults in part from an abnormally large collapse of activity from the NIP/RN cells. When only the RN agonist spindle feedback, which releases the an- projections to the oL-MN pools are eliminated from tagonist oL-MN pool from IaIN-mediated inhibition. the system (Fig. 6D), the joint initially undershoots The collapse of the agonist spindle feedback is in the equilibrium position and oscillates around the turn attributable to strong agonist Renshaw inhibi- end point. This persistent tremor is similar to that tion of the agonist ~/-MN pool. Thus, the NIP/RN-

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CEREBELLAR LEARNING FOR ARM MOVEMENT CONTROL

A B Alpha-MN Spindle feedback x 10-3 , I t F 6.00 E 640.00.i 5.50 620.00 ! 5.00 600.00 4.50 580.00 I f'~,, 4.00~ [ ...."-- ...... 560.00 .j / ~' ,,j \J 'v--' ".. '- ..... / i 540.00 kf 350L / 520.00 i i t i ...... 1 i t 0.00 10.00 20.00 0.00 10.00 20.00 D Alpha-MN Spindle feedback x 10-3 66::i\ F 6"506oo ;q.I i~\i\ "v"-,/~, f.,,/--..v..-~v-.-v--.' l F 5 50 i Figure 7: VITE-FLETE-CEREBELLUM model simulations showing o~-MN activity 620.00 5.00~~ and spindle feedback when the projection 4.50 600.00 from NIP/RN to Renshaw cells is eliminated 58o.oo~ /i fi :i ,~ ,, (A,B), and when the NIP/RN-to-o:-MN pro- 560.00 -j i/ i/i/~ /i /~ /i/~/~ jection is damaged (C,D), in the simulations 3.50 of Fig. 6. Both simulated lesions produce ] ' . 540.00 ~" il ~i (i ~-' 3.00[ , '/ i i end-point oscillations, but the latter lesion is 0.00 10.00 20.00 0.00 10.00 20.00 more damaging.

to-Renshaw projection normally prevents Renshaw EQUILIBRIA OF THE TWO-DIMENSIONAL PLANAR cells from inhibiting y-MNs and thereby disrupting ARM the IalN-mediated pattern of reciprocal activation TO illustrate the cerebellar opponent learning of agonists and antagonists. Another source of co- capabilities using a two-dimensional planar arm activation is the agonist Renshaw inhibition of the with mono- and biarticular muscles, we used two antagonist Renshaw pool, which disinhibits the an- independent single-joint VITE-FLETE systems (Bul- tagonist MN pool. lock et al. 1993a, and unpubl.). Each FLETE system Figure 7C shows that the simulated lesion generates the muscle forces needed to move its of the NIP/RN-to-oL-MN projection led to an corresponding arm segment (e.g., shoulder or el- abnormally small ot-MN launching burst (cf. Figure bow joint) by contracting the appropriate muscles 8C) and to much less coactivation than in the according to the planned movement. The biome- case of the NIP/RN-to-Renshaw lesion. The de- chanics of the two-dimensional planar arm in- layed antagonist braking burst is nearly nor- cluded the full dynamic equations described earlier mal. This is because the surviving NIP/RN-to- (equations 40 and 41), except that the gravitational Renshaw projection disinhibits the antagonist effects were neglected by assuming a pure horizon- MN pool at the time when the antagonist spindle tal movement; that is, g = 0 in equations 46 and 47. feedback signal is growing and the Ialn-mediated Figure 9 depicts the steady-state response of the inhibition is at a minimum because of collapse planar limb controlled by a decerebellate multi- of agonist spindle activity. Thus, there is a large joint VITE-FLETE system. Joint angles for the transient enhancement of the gain of the antago- shoulder and elbow as a function of the difference nist stretch reflex. These effects combine to between the descending PPV commands, that is, produce the initial undershoot observed in Fig- A1-A 2 (see Fig. 2 and equations 9 and 10), are ure 6D. shown for each joint.

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A B PPC+DCN Muscle force x 10-3 ' ' F-- 1.12/' ' -~---V 600.00 E

500.00 1-~ I/I ii

400.00 1.04 1.02 300.00 l.OO~i il I" 200.00 ~,~ o96l!i/ 0.94; i!1 100.00 0.90 ~ 0.00 ~ t i 0.00 10.00 20.00 0.00 10.00 20.00 Figure 8: Key variables are depicted C D from flexor and extensor channels for the Alpha-MN Spindle feedback intact network simulation of Fig. 6. (A) The x 10-3 net input to the o~-MN pools. This input is 680.00 -, -F-- 7.00 composed of phasic, burst-like NIP/RN 660.00 output signals superimposed on ramp-like 6.50 640.00 opponent outputs from the PPV stage of 6.00 the CPG. The simulated EMG activity in C 620.00 5.50 shows a biphasic patterning of muscle ac- 600.00 5.00 tivity (AG followed by an ANT burst), 580.00 whereas the simulated muscle forces in B 4.50 560.00 represent a triphasic pattern (large AG, L 4.00 540.00 J iSi, ...... ANT, small second AG). The reciprocal 3.50 ~S spindle feedback activities are shown 520.00 ~ t ~ i r t in D. 0.00 10.00 20.00 0.00 10.00 20.00

In the absence of external loads, the difference articular and depend on the shoulder angle. Fur- A 1 -A 2 would specify a unique equilibrium joint thermore, during shoulder flexion, the BIC muscle angle in a one-joint FLETE system. Because we have is unloaded, and an imbalance is produced in favor a coupled system here, each set of plots represents of the antagonist TRI muscle resulting in the elbow a joint angle for a given joint (e.g., shoulder or extension. Thus, there was a partial breakdown of elbow) versus A 1 - A2, whereas the other joint re- the basic FLETE property that the joint angle asso- ceives constant input. Positive values of A 1 -a 2 ciated with any choice ofA 1 - A2 remains invariant mean joint flexion, and negative values mean joint across changes of joint stiffness because of varying extension. Simulations are shown for a system with the coactivation signal P in equations 9, 10, 14, 16, and without joint afferent feedback from the shoul- 22, and 25. To compensate, feedback signals de- der joint to the elbow motoneurons. Figure 9A denoted by equations 30 and 31 from the shoulder picts the steady-state behavior of the shoulder joint joint receptors act on the elbow motoneuron angle to the difference A 1 -A 2 when the elbow pools. Joint receptors respond to increased flexion joint received constant input (A 1 - A 2 = 0.08). The of the shoulder joint by projecting to the flexor simulation shows that the shoulder angle is ap- ~-MN pool for the elbow, thereby increasing el- proximately a linear function of the difference bow flexion. Because this feedback is position-de- A 1 -A2, but the elbow angle rotates (in the exten- pendent and monotonic, the outcome is elbow sta- sion direction) as the shoulder moves across the bilization. This is shown in Figure 9A. In particular, work space. The rotation of the elbow joint is in- the addition of joint receptor feedback allows the creased for higher coactivation values (e.g., the ef- elbow angle to remain constant for a given choice fect is greater for a cocontraction value in equa- of A m-A 2 over the full range of shoulder angles. tions 9 and 10 of P - 0.4 than for P - 0.3). This is Figure 9B illustrates that the compensation suffices understandable because the elbow muscles are bi- across the work space.

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CEREBELLAR LEARNING FOR ARM MOVEMENT CONTROL

reaching produced by the interplay of two pairs of A "~ ip=0. 4 witht joint feedback antagonist muscles admits four cases, as tabulated in Figure 10A. Let us assume that the stretch signals .~ 40 .... ~.,,~--'---~'~"~'~'A~_~--'---~'~"~'~"~ ~ - generated during two-joint arm movements in the O1 C P=0.3 " /" "" "'" horizontal plane follow a nonoverlapping (alternating, albeit with some degree of coactivation), dis- ~ o 0 / shoulder crete agonist-antagonist pattern (Fig. IOA). Figure lOB shows a representation of the network system -40 before learning, C, during learning, and D, after learning. The rows of panel B correspond to dis- I I i tinct PFs (denoted by GrC1-GrC4), carrying error -0.05 0 0.05 A1-A21shoulde r information for each muscle channel. Each PF projects nonspecifically to all of the Purkinje cells B AILA2=0.08 I [/ (P1-P4) and cerebellar interneurons (not shown). -- elbow / - Initially, the strength of the PF to Purkinje cell con- // nections is set to 1 (Fig. lOB). Activity in PFs and _o CFs is represented by a box enclosing the activated A1-A2=O.O r-01 cells. Each column representing a given Purkinje m 0 cell may receive specific activation from its CF

.-j'5 (C1-C4). A co-occurrence of PF and CF activation is indicated in Figure I OC by shading the matrix -50 cell corresponding to the Purkinje axodendritic "~ I ] e,aow I = zone where these two signals could converge. In -0.05 0 0.05 A1-A2lshoulde r summary, each Purkinje cell can receive PF inputs from all granule cells and CF inputs only from the 9: (A) Steady-state response of the planar two- Figure unique inferior olive afferent innervating it. joint limb with and without feedback joint compensa- Figure 1OC shows the pattern of activations tion. Without joint feedback, at coactivation settings of either P = 0.4 or P = 0.3, elbow angle changes as shoul- elicited by conjunctive stimulation of PF and CF der angle changes despite constant settings of desired signals for each case shown in Figure I OA. In case elbow angle commands. Joint receptor feedback from 1, PF signals from GrCIj and GrC3j arrive at all Pur- shoulder joint to elbow joint allows the elbow joint kinje cells as shown by the boxes enclosing them. angle to remain constant over the full range of shoulder CF discharges signaling motor errors in channels angles if the descending position command for the el- C1 and C3 also arrive in close temporal and spatial bow is held constant. This simulation with hypothesized relationship with those PF signals. This would pro- joint receptor feedback used a coactivation value in duce LTD of the synapses GrCIj to P1, GrC3jto P1, equation 9 of P= 0.4. (B) The linearity is preserved GrCij to P3, and GrC3j to P3. After enough trials, across the work space. In these simulations, the input to encompassing all combinations in Figure IOA, the the FLETE elbow system was constant during three dif- resulting pattern of connectivity will be like that ferent input settings corresponding to three regions cov- ering the work space of the planar arm (A 1 - A 2 = 0.08, shown in panel D. In particular, only the weight 0, and -0.08). The input to the FLETE system controlling from the GrC projecting to the opponent Purkinje the shoulder joint was varied from -0.08 to 0.08. cell of the same joint will survive LTD. The resulting connectivity matrix shown in Figure IOD can produce the following two outcomes: (1) A single PERFORMANCE OF CEREBELLATE AND joint stretch (e.g., because of an external load) elicits reciprocal counteraction in the segment of ori- DECEREBELLATE VERSIONS OF THE MODEL IN THE gin as well as stiffening of the adjacent segment; CONTROL OF A TWO-JOINT ARM MOVEMENT and (2) a two-joint stretch elicits two-joint coun- Cerebellar learning in an opponent motor con- teraction. troller can adaptively discover and form muscle Using the connectivity matrix in Figure IOD synergies in the case of multijoint arm movements. we were able to study the performance of the tri- For a two-joint arm moved by both mono- and bi- partite model of Figure 2 with and without the articular muscles, the feedback learning process cerebellar component during a rapid two-joint can be summarized, as shown in Figure 10. Planar movement. Figure 11 shows the state space trajec-

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Figure 10: Map representation of the cerebellar connectivity before and after learning. (A) Four possible cases depicting alternating pattern of EMG activation during planar, horizontal, pointing movements. (B) Before learning, all of the synaptic PF --) Purkinje connections are strong (e.g., set equal to 1). (C) LTD-based cerebellar learning. (/9) Synaptic matrix after learning. (C1-C4) Climbing fibers; (P1-P4) Purkinje cells; (GrCI-GrC4) granule cell projections to Purkinje cells (e.g., PFs); (PEC) pectoralis major; (PDEL) posterior deltoid; (BIC) biceps; (TRI) triceps. tories of elbow and shoulder joints during two without cerebellar compensation corresponding to movements made with identical CPG settings. One the state space trajectories shown in Figure 11. movement was made with the cerebellar compo- The decerebellate system was simulated by elimi- nent (after learning trials with the two-joint sys- nating the DV*G commands to the cerebellar cir- tem) and one without. The coordinates for the cuitry but maintaining the tonic excitation to both plotted trajectories are error coordinates: the x-axis Purkinje and NIP cells, so that the tonic influence is joint position error, and the y-axis is joint veloc- of the NIP/RN cells upon spinal cord circuitry was ity error, with error measured relative to the de- intact, and only the phasic component was dis- sired trajectory generated by the CPG. A perfect rupted. The movement comprised a 65 ~ shoulder performance would be represented by a point at extension and a 34 ~ elbow extension. For this the origin, indicating zero velocity and zero posi- simulation, all of the circuitry from Figure 2 was tion error throughout the trajectory. That neither replicated to control two sets of opponent system approached perfect performance of the de- muscles, one monoparticular (P. DEL, PEC), the sired trajectory indicates that the model is not an other biarticular (TRI, BIC). Biomechanical equa- optimal controller in its present form. This is not tions were also modified appropriately, as noted surprising because of the restricted state informa- above. Heteronymous joint receptor feedback, tion available to the cerebellum in these simula- from the shoulder to elbow motoneurons, was also tions and because of our omission of the spectral added to help compensate for static shoulder joint timing component of cerebellar learning (Fiala et effects at the elbow. The decerebellate simulation al. 1996). However, what is more important in the shows smaller amplitude muscle activations and present context is that for each joint the excursion more sluggish limb response than in the intact sys- away from the origin was much larger for the de- tem. cerebellate model than for the model with intact In Figures 6-8, it was shown through simula- cerebellum. This is a graphic illustration of the er- tions of the one-joint case that interpositus lesions ror-reductive competence of the simulated cerebel- degraded the ability of the neuromuscular control lar embedding. system to track the desired velocity of the move- Figure 12 shows the simulated kinematics and ment, resulting in increased onset latency, rise EMG activities of the two-joint system with and time, and settling time. Here, through simulations,

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dE/dt ion. This simulation was performed by removing

40.00 Purldnje cell inhibition from the NIP model cells

~176...... ,.~.. =~ shown in Figures 1 and 2. The effect of this simu- 30.00 lated lesion was to take out the cerebellar process-

20.00 ing, except for the direct projections (collaterals) from mossy fibers to NIP cells. This implies that the 10.00 NIP cells would be disinhibited because of the removal of tonic and phasic Purldnje cell inhibition. 0.00 In these simulations, it was assumed that the ma- -10.00 trix of cerebellar connectivity was already learned (see Fig. IOD); in addition, climbing fibers were -20.00 r __L L not sent to the interpositus neurons. Therefore, 0.00 50.00 these simulations address the role of the mossy Figure 11: State-space representation of the shoulder fiber projection to the NIP cells alone. and elbow position and velocity tracking for both intact The plots of joint position and velocity of Fig- and decerebellate two-joint model simulations. Three ure 13B show that the movement evolves slower variables are shown: The plots show curves of joint po- compared to the intact system simulation shown in sition error (x-axis) vs. error derivative (y-axis) for both panel A. The oL-MN activities show a higher base- shoulder and elbow joints over time. E = O - O d (shoul- line of activities, which is correlated with the der) or E = ~ --~d (elbow). Plots A (shoulder) and C (el- higher level of tonic NIP output (Fig. 14) resulting bow) are for the intact system; plots B (shoulder) and D from removal of Purkinje cell inhibition. Note also (elbow) correspond to the decerebellate system. that the antagonist motoneuron pool is not inhibited during the early part of the movement and that we illustrate the effect of damage to the cerebellar the antagonist braking bursts are smaller in ampli- cortex in the two-dimensional system. tude. Figure 14 shows that only an agonist burst is Figure 13 shows the effect of a cerebellar le- generated, albeit of different duration, in the NIP sion on the performance of a multijoint limb flex- because of the corollary discharge from mossy fi-

DECEREBELLATE INTACT

5.6 PEC

5.4 5.2

5.6 DEL

5.4 i

-40 i i Figure 12" Simulations of the two-dimensional planar arm using two VITE- -20 ~ -20 FLETE-CEREBELLUM model systems ~ -40 coupled through joint receptors and the t i bio-mechanics of the arm. Rows 1, 2, 5, 5.9 and 6 show dynamics of MN activations 6.2 BIC BIC 5.8 associated with four muscles. Rows 3 and 4 show shoulder and elbow kine- 57i5.5 5.4 t i matics. (PEC) Pectoralis muscle; (P.DEL) posterior deltoid muscle; (BIC) biceps TRI muscle; (TRI) triceps muscle; 0, shoulder joint position; 0, shoulder joint velocity; 5.6 i i (I), elbow joint position; 4), elbow joint 20 t 0 2O velocity.

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A B

INTACT PURKINJE CELL LESION

Figure 13: Simulation of the dynamics of the two-dimensional planar VITE-

FLETE-CEREBELLUM system for the in- , IZ?.OEL .... ~lJ P. DEL 5.0 i , , 6.5 E H tact system (A) and when the cerebellar I I cortex is damaged (B) (e.g., Purkinje 1~ . 20 cells no longer inhibit NIP cells). Plots -20 0 show simulated joint position and veloc- -40 -=L l'Y," . 1 TM 01 ! ....!01 -40 I 5 ity for shoulder (O, 0) and elbow (@, ~) -60 -20 joints, as well as simulated mononeuron 0 t-L ' '' t~ ~ ' ' ' activities. The Purkinje cell output lesion o p ' t~ S------"~ 2o removes phasic cerebellar disinhibition .... ~'- " 0 of NIP cells while also tonically disinhib- -40 iting them. This results in higher peak ,-20 0 values, reduced ranges, and higher base- I I line levels of oL-MN activity, as well as '' BIC ' '' coactivation of antagonist muscles. The 6.5 trajectories of the Purkinje-lesioned sys- @0 --E-- 7.0 . TRI tem show decreased rise times and in- 5.5 I I I I 6.5 ~ - ' ' i] creased settling times. 40 50 60 t 40 50 60 t bets from the agonist channel. However, the am- duced by repetitive stimulation of PFs alone, can plitude of the agonist NIP response is reduced in lead to the discovery of opponent muscle synergies amplitude after the cerebellar cortex lesion. by cerebellar circuitry, which enables efficient op- These simulations suggest that the cerebellar ponent multijoint muscle control. It is shown that cortex plays an important role in modulating inter- the only requirement for opponent cerebellar con- positus nuclear cell discharge, so as to facilitate the trol is the specificity of CF projections from the initiation and performance of limb movements. In inferior olive to both the cerebellar cortex and the summary, a simulated lesion in the model's cer- deep nuclear zones that project back through the ebellar cortex, without damaging the deep nuclei, rubrospinal tract to control the same muscles that shows that although motor performance is degraded, arm movements are still performed, albeit with a slower time course and impaired regularity x 10 -3 or smoothness.

Conclusions 100.001 ...... An adaptive, context-dependent, preemptive 50.00t ~- ...... error reduction mechanism has been proposed to -0.00 i model cerebellar control of descending motor -50.00 i commands and tuning of proprioceptive reflexes 1 during multijoint movements. Specifically, the cer- _1++ i 1 , ) t t t ebellar model uses muscle-specific discrete error 40.00 50.00 60.00 70.00 signals from the inferior olive and nonspecific PF Figure 14: NIP cell discharge for the agonist and an- signals from granule cells to learn weight distribu- tagonist channels before [(a) agonist; (b) antagonist] and tions that improve inverse dynamic computations after [(c) agonist; (d) antagonist] Purkinje cell lesion. Af- on a trial-by-trial basis. ter the Purkinje cell lesion, the baseline of activity of NIP It is hypothesized that LTD of PF-Purkinje cell cells increases because of Purkinje cell disinhibition, the synapses, induced by conjunctive stimulation of PF duration of the agonist NIP burst is shorter, and no an- and CF, and LTP of PF-Purkinje cell synapses, in- tagonist burst is generated in the other NIP pool.

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CEREBELLAR LEARNING FOR ARM MOVEMENT CONTROL provide afferent inputs to specific inferior olivary tensions to the model proposed in Figure 2, zones. Thus, the model incorporates findings indi- through a more complete analysis of how NIP/RN cating that topographical order is maintained outputs to spinal cord affect the processing of sen- through circuits that traverse the olive and associ- sory feedback during arm movement control. ated deep nuclear zones, and shows that this design produces adaptive results when extended to include appropriate somatosensory feedback, in COMPARISON WITH OTHER CEREBELLAR MODELS this case, from stretch receptors. Large-scale simulations of the cerebellar model Grossberg and Kuperstein (1986) developed a in interaction with an arm movement cortical cen- model of error-based opponent learning of adap- tral pattern generator and a spinal force controller tive gain control by the cerebellum. This work show that (1) mossy fiber conduction of move- built on the cerebellar model of Grossberg (1969), ment velocity commands from the CPG to both the which suggested that both LTD and LTP occurred cerebellar cortex and the associated nucleus NIP/ at the PF-Purkinje cell synapse to learn to control RN leads to generation of phasic nuclear cell re- linearly ordered motor components. The early cer- sponses whose feed-forward action can greatly im- ebellar model of Marr (1969) assumed that only prove upon spinal feedback control of tracking, LTP occurred, whereas that of Mbus (1971) in- including reduction of oscillations; and (2) phasic voked LTD at PF synapses with both Purkinje and NIP/RN excitation of motoneurons, coupled with stellate cells to allow bidirectional weight changes. phasic inhibition of the Renshaw cells associated Smith (1981) proposed how the cerebellar connec- with those motoneurons, yields better tracking tivity could allow for agonist-antagonist opponent than either operation alone. These results help ex- muscle control. He hypothesized that during co- plain the need for the dual spinal level action of contraction of antagonist muscles, afferents from rubral stimulation that was discovered by Henatsch each opponent muscle converge on cerebellar in- et al. (1986). hibitory interneurons to remove the Purkinje cell inhibition of the deep cerebellar nuclei. Further- more, Smith suggested that by inhibiting groups of deep nuclear cells, Purkinje cells could mediate OTHER INFLUENCES OF THE CEREBELLAR RUBRAL disfacilitation of motoneuron pools resulting in an- PATHWAY IN SPINAL CIRCUITRY tagonist muscle relaxation. Recently, Smith (1996) Besides reports of direct monosynaptic rubro- proposed that pairs of Purkinje cells, as in our motoneuronal connections to the forelimb moto- model and that of Grossberg and Kuperstein neurons of the cat, particularly in the C8-T1 seg- (1986), could use reciprocal inhibition or antago- ments (Fujito et al. 1991), and differential effects of nist cocontraction to control agonist-antagonist stimulation of the cat's RN on lumbar a-motoneu- muscle groups. These models are compatible with tons and their Renshaw cells Otenatsch et al. the idea that the cerebellum is involved in the 1986), Hongo et al. (1972b) have shown that learning and coordination of muscle synergies stimulation of the RN produces monosynaptic ex- (Thach et al. 1992). Moreover, these models work citatory effects on cells monosynaptically acti- under the assumption that the command for move- vated, or disynaptically inhibited, from group I ment originates outside the cerebellum, that this muscle afferents and in cells di- or polysynaptically command provides a reference for computing er- activated from the flexor reflex afferents or exclu- ror feedback signals, and that the cerebellar output sively from cutaneous afferents. Hongo et al. projections to the spinal centers (e.g., the rubro- (1972a) demonstrated that stimulation of the RN spinal tract) form a side path with respect to the evokes a large dorsal root potential followed by descending corticospinal motor pathways (Ito pronounced primary afferent depolarizations in Ib 1984). This is in contrast to models that postulate and low threshold cutaneous afferents, and a dual the cerebellum as an adjustable pattern generator effect on Ia afferent terminals. These monosynap- (e.g., Houk et al. 1990, 1993) responsible for pri- tic effects on Ia afferents or IaINs could contribute mary trajectory generation. to the cerebellar tuning of proprioceptive reflexes Our opponent cerebellar model for arm move- and other control loops through changes in the ment control extends Smith's (1981, 1996) pro- organization, gain, and timing of these reflexes. posal considerably by demonstrating how an affer- These effects may be explained using modest ex- ent pattern suitable for regulating cocontraction or

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Contreras-Vidal et al. reciprocal inhibition of antagonist muscles can be opposite la interneurones in the human upper limb 9Exp. established by learning because of the spatiotem- Brain Res. 66:106-114. poral correlation at Purkinje of mossy and CF dis- Bartha, G.T., R.F. Thompson, and M.A. Gluck. 1991. charges in the same, but not in opposing or unre- Sensorimotor learning and the cerebellum. In Visual lated, channels. This cerebellar learning hypothesis structures and integrated functions (ed. M. Arbib and J. places few demands on a priori wiring for control Ewert), pp. 381-396 9Springer-Verlag, Berlin, Germany 9 of opponency in the cerebellum and extends to the Bloedel, J.R. and V. Bracha. 1995. On the cerebellum, control of a large number of muscles acting at sev- cutaneomuscular reflexes, movement and the elusive eral joints. As such, this competence appears to be engrams of memory. Behav. Brain Res. 68" 1-44. beyond what can be accomplished by the segmen- tal organizational scheme utilized by spinal stretch Bloedel, J.R. and J. Courville. 1981. Cerebellar afferent reflexes unassisted by learned descending modula- systems 9In: Handbook of Physiology 9Section 1. The nervous system. Vol. II, part 2, Motor control (ed. F. Plum, V.B. tion. The proposed cerebellar model allows a gen- Montcastle, and S.R. Geiger), pp. 735-829 9American eralization of rapid, and in that sense reflex-like, Physiology Society, Bethesda, MD. control beyond the one-joint, two-muscle level of organization (Thach et al. 1992; Bullock et al. Brink, E., E. Jankowska, D.A. McCrea, and B. Skoog. 1983. 1993b). Inhibitory interactions between interneurones in reflex pathways from group la and group Ib afferents in the cat. J. Physiol. (Lond.) 343: 361-373 9

Bullock, D. and J.L. Contreras-Vidal. 1993. How spinal Acknowledgments neural networks reduce discrepancies between motor We wish to thank Diana Meyers for her valuable intention and motor realization 9In Variability and motor assistance in the preparation of the manuscript 9 Thiswork control, (ed. K. Newell and D. Corcos), pp. 183-221. Human was supported in part by the Monterrey Institute of Kinetics Press, Champaign, IL. Technology (Mexico) (J.L.C.) and by the Office of Naval Research (ONR N00014-92-J-1309 and ONR Bullock, D. and S. Grossberg. 1988. Neural dynamics of N00014-95-1-0409) (D.B.). planned arm movements: Emergent invariants and speed-accuracy properties during trajectory formation 9 References Psychol. Rev. 95: 49-90. Akazawa, K. and K. Kato. 1990 9Neural network model for control of muscle force based on the size principle of motor 91989. VlTE and FLETE: Neural modules for trajectory unit 9Proc. IEEE 78" 1531-1535. formation and tension control. In Volitional action (ed. W. Hershberger), pp. 253-297. North-Holland, Amsterdam, The Albus, J.S. 1971. A theory of cerebellar function 9Math. Netherlands. Biosci. 10: 25-61 9 91991. Adaptive neural networks for control of Alexander, R.M. 1981. Mechanics of skeleton and tendons 9 movement trajectories invariant under speed and force In Handbook of physiology. Section 1: The nervous system 9 rescaling. Hum. Movement Sci. 10: 3-53. Vol. II, part 1, Motor control, (ed. F. Plum, V.B. Montcastle, and S.R. Geiger), pp. 17-42. American Physiology Society, 91992. Emergence of tri-phasic muscle activation from Bethesda, MD. the nonlinear interactions of central and spinal neural network circuits. Hum. Movement Sci. 11 9157-167. Anderson, R.A. 1987. Inferior parietal Iobule function in spatial perception and visuomotor integration 9In Handbook Bullock, D., J.L. Contreras-Vidal, and S. Grossberg. 1993a. of physiology. Section 1: The nervous system, Vol. V, Part 2. Cerebellar learning in an opponent motor controller for Higher functions of the brain (ed. F. Plum, V.B. Mountcastle, adaptive load compensation synergy formation 9In and S.R. Geiger), pp. 483-518. American Physiological Proceedings of the World Congress on Neural Networks, IV, Society, Bethesda, MD. pp. 481-486. Lawrence Erlbaum, Hillsdale, NJ.

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Received February 14, 1997; accepted in revised form April 14, 1997.

APPENDIX A: SIMUIATION METHODS All simulations were performed using the clas- sic Runge-Kutta fourth-order method or the Runge-Kutta-Nystrom method (Collatz 1966). Si- multations were performed by integrating the dynamic system that defines the model described in Materials and Methods. The simulations were started with all of the system's variables initially set to 0.0, except for the PF-Purkinje cell connectivity matrix, which was set to the identity matrix. For a typical single joint VITE-FLETE-CER- EBELLUM simulation, the following equations were integrated: Equations 1-3 for VITE, equations 4-29 for FLETE, and equations 32-39 for the cerebellar model. The input variables were Go, the Go signal multiplier, the target position vector (T1, T2) , and P, the coactivation signal; and the output variables were joint position (O) and velocities (0), muscle forces (/71, F2), oL-MN activities (3/1, M2), and spindle feedback (El, E2). For the two-dimensional simulations, we used equations 1-3 for each of the two VITE systems (one for each limb segment), equations 4 and 5, 9-31, and 40-51 for each FLETE system controlling the shoulder and the elbow joints, and equations 32-36, 38 and 39 for the cerebellar model extended to the two-dimensional case using the connectivity matrix depicted in Figure 10. As in the single-joint simulation, the input variables were Go, the GO signal multiplier; the target position vector (T1, T2) for each VITE; GO signal multiplier com- mon to both VITE systems; and the output variables were shoulder and elbow joint position (O and ~, respectively) and shoulder and elbow joint velocities (() and ~), respectively), muscle forces (F1, F2), for the shoulder and elbow; oL-MN activities (3/1, M2) for the shoulder and elbow MN pools. Note that as the moment arm was set equal to 1.0, the shoulder and elbow torque variables in equations 40 and 41 corresponded actually to the difference between agonist and antagonist muscle

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A neural model of cerebellar learning for arm movement control: cortico-spino-cerebellar dynamics.

J L Contreras-Vidal, S Grossberg and D Bullock

Learn. Mem. 1997, 3: Access the most recent version at doi:10.1101/lm.3.6.475

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