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A NEW LINEAR MUSCLE FIBER MODEL FOR NEURAL CONTROL OF SACCADES

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A NEW LINEAR MUSCLE FIBER MODEL FOR NEURAL CONTROL OF SACCADES
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  International Journal of Neural Systems, Vol. 23, No. 2 (2013) 1350002 (17 pages)c  World Scientific Publishing CompanyDOI: 10.1142/S0129065713500020 A NEW LINEAR MUSCLE FIBER MODELFOR NEURAL CONTROL OF SACCADES JOHN D. ENDERLE Biomedical Engineering, University of Connecticut 260 Glenbrook Road, Storrs, Connecticut 06269, USA jenderle@engr.uconn.edu  DANIEL A. SIERRA Electrical and Electronic Engineering School Universidad Industrial de Santander Cra 27 con Calle 9, Bucaramanga, 68002 Colombia dasierra@uis.edu.co Accepted 20 December 2012Published Online 28 February 2012 A comprehensive model for the control of horizontal saccades is presented using a new muscle fiber modelfor the lateral and medial rectus muscles. The importance of this model is that each muscle fiber has aseparate neural input. This model is robust and accounts for the neural activity for both large and smallsaccades. The muscle fiber model consists of serial sequences of muscle fibers in parallel with other serialsequences of muscle fibers. Each muscle fiber is described by a parallel combination of a linear lengthtension element, viscous element and active state tension generator. Keywords  : Saccade; time-optimal control; muscle fiber; oculomotor plant; system identification. 1. Introduction A fast eye movement is usually referred to as a sac-cade, and involves quickly moving the eye from oneimage to another image. This type of eye movementis very common, and it is observed most easily whilereading — that is, when the end of a line is reached,the eyes are moved quickly to the beginning of thenext line.This paper updates a neural network thatcontrols the eyes during horizontal saccades andintroduces a new muscle fiber model. The previouslypublished model uses an anatomically and physio-logically correct model of the oculomotor plant andneural network. 1 , 2 A key element of the neural net-work involves the autonomous burst firing of theexcitatory burst neuron (EBN) a and post inhibitoryrebound burst (PIRB) firing in the paramedian pon-tine reticular formation. In that study, the neural fir-ing rate for the agonist and antagonist motoneuronswere separately estimated using the system identifi-cation technique. Here, each muscle fiber has a sepa-rate neural input that allows a more precise controlof the saccade, which allows the investigation of someshortcomings of the previous model.The time-optimal controller described by Enderleand Zhou has a firing rate in individual neurons thatis maximal during the agonist pulse and indepen-dent of eye orientation, while the antagonist muscleis inhibited. 1 In this model, the activity of all neu-rons is summarized into the firing of a single neuron.Thus, as the magnitude of the saccades increases,the firing rate of the single neuron increases up to a Also known as a medium lead burst neuron. 1350002-1    I  n   t .   J .   N  e  u  r .   S  y  s   t .   D  o  w  n   l  o  a   d  e   d   f  r  o  m   w  w  w .  w  o  r   l   d  s  c   i  e  n   t   i   f   i  c .  c  o  m   b  y   1   8   6 .   8   7 .   1   7   9 .   2   3   4  o  n   0   3   /   1   9   /   1   3 .   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y .  J. D. Enderle & D. A. Sierra  8 ◦ , after which, the neuron fires maximally. However,the firing rate of a real neuron is maximal and doesnot change as a function of saccade magnitude. Toexplain this, the time-optimal controller was hypoth-esized to operate in two modes, one for small sac-cades and one for large saccades based on the numberof neurons activated by the Superior Colliculus. Thesubject of this paper is to introduce a muscle fibermodel of muscle, which, when incorporated into theoculomotor plant, allows the use of multiple neuronsto drive the eyes to their destination.Zhou and coworkers presented a linear third-order model of the oculomotor plant for horizontalsaccadic eye movements using a lumped parametermuscle model. 2 The muscle model used in the ocu-lomotor plant shown in Fig. 1, first published byEnderle and coworkers, 3 consists of a Voigt ele-ment (viscosity and elasticity elements in parallel)in series with another Voigt element in parallel withan active state tension generator. We refer to thismodel as the whole muscle model in this paper.The tension created by the muscle is  T  , and thevariable  x i  is the change from equilibrium for eachnode. This linear muscle model has been shown toexhibit accurate nonlinear force–velocity and length–tension relationships for the medial and lateral rectusmuscles. B  M1  K   Mlt  F B  M2  K   Mse T x 2 x 1 Fig. 1. Diagram illustrates the linear muscle model con-sisting of an active state tension generator  F   in parallelwith a length–tension elastic element  K  M  lt  and viscouselement  B M  1 , connected to a series elastic element  K  M  se in parallel with a viscous element  B M  2 . Upon stimula-tion of the active state tension generator  F  , a tension  T  is exerted by the muscle. We refer to this model as thewhole muscle model. (Adapted from Enderle, Engelken& Stiles, 1991.) To accurately describe the neural input to themuscles for small saccades, it is necessary to modelthe muscle at the basic building block level of themuscle fiber. Models of muscle typically used inthe oculomotor system during saccades are lumpedparameter models, that is, information about mus-cle fibers and other features is reduced to a smallset of parameters. Further, the single neural inputto the lumped parameter muscle model captures theentire population of neurons that fire and innervatethe muscle. With a muscle fiber model, each musclefiber has its own neural input allowing the impact of the number of actively firing neurons to be investi-gated. As demonstrated, the number of neurons fir-ing significantly affects the control of saccades ratherthan variations in the firing rate among neurons.Muscles are actuators that perform differenttasks controlled by the central nervous system. Illus-trated in Fig. 2 is the anatomy of a muscle that con-sists of two tendons, and a serial and parallel networkof muscle fibers. The muscle fiber (an individual cell)is the smallest independent muscle unit that displaysthe same mechanical properties as the whole muscle.Models explicitly defined to study coordination andforce generation during motor tasks include math-ematical descriptions of the muscle behavior thatrange from the microscopic properties of the muscleto the analysis of their input–output characteristics. 4 Goldberg and coworkers report that cat lateralrectus muscle contains approximately 15,000 musclefibers and that it is innervated by 1100 motoneurons,which means that the average motor unit consists of 15 muscle fibers and one motoneuron. 5 Rather thanmuscle fibers arranged in columns of series linkedmuscle fibers as described here, the cat has muscle Fig. 2. Anatomy of a muscle that consists of two ten-dons, and a serial and parallel network of muscle fibers. 1350002-2    I  n   t .   J .   N  e  u  r .   S  y  s   t .   D  o  w  n   l  o  a   d  e   d   f  r  o  m   w  w  w .  w  o  r   l   d  s  c   i  e  n   t   i   f   i  c .  c  o  m   b  y   1   8   6 .   8   7 .   1   7   9 .   2   3   4  o  n   0   3   /   1   9   /   1   3 .   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y .  A New Linear Muscle Fiber Model for Neural Control of Saccades fibers whose architecture involves serially arrangedand branching networks.There are twenty to thirty thousand muscle fibersin the human lateral and medial rectus muscle. 6 Leigh and Zee report that the medial and lateralrectus muscle differ anatomically, physiologically andimmunologically from skeletal muscle. 7 Some differ-ences include muscle fibers that are smaller and morerichly innervated. These muscles, consisting of sixdifferent muscle fiber types, are the fastest contract-ing muscles in the body and are fatigue resistant.There are two different layers in the medial and lat-eral rectus muscle consisting of a central global layerand a peripheral orbital layer, with each having dif-ferent ratios of muscle fibers that can either sus-tain contraction or provide brief rapid contraction.80% of the orbital layer has singly innervated andfatigue resistant muscle fibers that provide the brief rapid contraction responsible for driving the eyes totheir destination during saccades — these fibers havenumerous mitochondria in dense clusters that are notpresentin skeletal muscle. The global layerhas a mix-ture of fibers, with singly and multiply innervated,fatigue resistant and fatigable, and provide sustainedor rapid brief contraction — these muscle fibers arethought to keep the eyes at their destination after asaccade.A scalable muscle fiber-based muscle model isintroduced here that exhibits accurate nonlinearforce–velocity and length–tension relationships, andis incorporated into an oculomotor plant. At themuscle fiber level as shown, one can investigate theimpact of the number of actively firing neurons dur-ing saccades. This type of system was suggested bySparks as an ideal vehicle for investigating the ocu-lomotor system. 8 In this paper, the focus is on the neural input tothe orbital layer of muscle fibers, rather than focus-ing on the different types of muscle fibers. We alsoinvestigate the synchrony of neuron firing and varia-tions of firing frequency in the neuron population. 2. Muscle Fiber Model The muscle fiber model of muscle is shown in Fig. 3.The tendon is described with the viscous and elas-tic elements,  B 2  and  K  se , at the top and bottom of the figure, and the muscle fiber is described with theactive state generator  F  ij , viscous element,  B 1 , and B 1  K  lt B 2  K  se T B 1  K  lt B 2  K  se 1 x 12 x B 1  K  lt 1 m + 1 x  n m + 1 x 1 m + 2 x  n m + 2 x 2 n x B 1  K  lt B 2  K  se  K  se B 2 12 F   2 n F  1 m + 1 F   n m + 1 F  13 x 3 n x Fig. 3. Muscle fiber model of muscle. elastic element,  K  lt , where  i  refers to the  i th mus-cle fiber column and  j  refers to the  j th series musclefiber in column  i . In Fig. 3, there are  m  muscle fibersin series with two tendon elements, in parallel with n  columns of other tendons and muscle fibers. Theoverall tension created by the muscle is  T  , variable x 1 is the change from equilibrium length for the musclein the lengthening direction, and variable  x ij  is thechange from equilibrium at node  j  in the muscle fibercolumn  i .We have assumed that the muscle fibers are iden-tical for simplicity, and that each muscle fiber hasan active state tension generator that can be indi-vidually stimulated using different neural inputs. Inreality, extraocular muscles contain at least six differ-ent muscle fiber types that can be described as slowand fast. 8 More importantly, the neural input to themuscle fiber can be appropriately selected depend-ing on the experiment, which allows us to examinethe interaction of motoneurons on the muscle (e.g.synchrony in firing, some neurons firing at differentrates, and some neurons not firing), which has a pro-found impact on the neural control model.The tension developed from the muscle fibermodel is given by T   = n  i =1 T  i = n  i =1 ( K  se ( x 1 − x i 2 ) + B 2 (˙ x 1 −  ˙ x i 2 )) ,  (1) 1350002-3    I  n   t .   J .   N  e  u  r .   S  y  s   t .   D  o  w  n   l  o  a   d  e   d   f  r  o  m   w  w  w .  w  o  r   l   d  s  c   i  e  n   t   i   f   i  c .  c  o  m   b  y   1   8   6 .   8   7 .   1   7   9 .   2   3   4  o  n   0   3   /   1   9   /   1   3 .   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y .  J. D. Enderle & D. A. Sierra  where  T  i  is the tension developed by each mus-cle fiber column. The notation ˙ x  is shorthand for dxdt . Using D’Alembert’s principle, the equations thatdefine each muscle fiber column are given by T  i  =  K  se ( x 1 − x i 2 ) + B 2 (˙ x 1 −  ˙ x i 2 ) ,K  se ( x 1 − x i 2 ) + B 2 (˙ x 1 −  ˙ x i 2 )=  K  lt ( x i 2 − x i 3 ) + B 1 (˙ x i 2 −  ˙ x i 3 ) + F  i 2 ,K  lt ( x i 2 − x i 3 ) + B 1 (˙ x i 2 −  ˙ x i 3 ) + F  i 2 =  K  lt ( x i 3 − x i 4 ) + B 1 (˙ x i 3 −  ˙ x i 4 ) + F  i 3 , ... (2) K  lt ( x im − x im +1 ) + B 1 (˙ x im −  ˙ x im +1 ) + F  im =  K  lt ( x im +1 − x im +2 ) + B 1 (˙ x im +1 −  ˙ x im +2 )+ F  im +1 ,K  lt ( x im +1 − x im +2 )+ B 1 (˙ x im +1 −  ˙ x im +2 )+ F  im +1 =  K  se x im +2  + B 2  ˙ x im +2 . To make the muscle fiber model more compact andeasier to simulate, a state variable approach is usedwith  y i 1  =  x 1  − x i 2 , and  y im +2  =  x im +2 , and for  j  =2 ,...,m + 1,  y ij  =  x ij  − x ij +1 . It then follows that x 1  = m +2  j =1 y 1 j  = ··· = m +2  j =1 y nj  .  (3)The model is now given by˙ y i 1  =  T  i − K  se y i 1 B 2 ,  (4)and for  j  = 2 ,...,m + 1,˙ y ij  = T  i − K  lt y ij − F  ij B 1 ,  (5)and˙ y im +2  =  T  i − K  se y im +2 B 2 .  (6) 2.1.  Scalability and steady state The structure of the muscle fiber muscle model allowsone to calculate the viscosities and elasticities as afunction of the whole muscle parameter model asfollows K  se  = 2 K  M  se n , K  lt  =  mK  M  lt n ,B 1  =  mB M  1 n , B 2  = 2 B M  2 n . (7)Further, assuming the same active state tension ineach muscle fiber, gives the following relationship F  ij  =  F n .To evaluate steady-state conditions for the mus-cle fiber muscle model, we start with Eq. (3) andsubstitute steady-state conditions from Eqs. (4)–(6)(i.e. from ˙ y i 1  = 0 =  T  i − K  se y 1 ( ∞ ) B 2 , we get  y i 1 ( ∞ ) = T  i K  se and from ˙ y im +2  = 0 =  T  i − K  se y im +2 ( ∞ ) B 2 , we get y im +2 ( ∞ ) =  T  i K  se , and for  j  = 2 ,...,m  + 1, with˙ y ij  = 0 =  T  i − K  lt y ij ( ∞ ) − F  ij B 1 , we get  y ij ( ∞ ) =  T  i − F  ij K  lt ).The notation  y ( ∞ ) refers to  y  at steady state or  y  at t  = ∞ . Once substituted for any muscle fiber column i , gives x 1 ( ∞ ) = m +2  j =1 y ij =  T  i ( ∞ ) K  se +  T  i ( ∞ ) − F  i 2 K  lt + ··· +  T  i ( ∞ ) − F  im +1 K  lt +  T  i ( ∞ ) K  se =  2 K  lt  +  mK  se K  se K  lt  T  i ( ∞ ) − 1 K  lt m +1  j =2 F  ij , (8)or in terms of tension T  i ( ∞ ) =   K  se K  lt 2 K  lt  + mK  se  ×  1 K  lt m +1  j =2 F  ij  +  x 1 ( ∞ )  .  (9)Using Eqs. (4)–(6) and (9), the steady state for thestate variables are given by y i 1 ( ∞ ) =  T  i K  se =   K  lt 2 K  lt  + mK  se  1 K  lt m +1  j =2 F  ij  + x 1 ( ∞ )  , (10) 1350002-4    I  n   t .   J .   N  e  u  r .   S  y  s   t .   D  o  w  n   l  o  a   d  e   d   f  r  o  m   w  w  w .  w  o  r   l   d  s  c   i  e  n   t   i   f   i  c .  c  o  m   b  y   1   8   6 .   8   7 .   1   7   9 .   2   3   4  o  n   0   3   /   1   9   /   1   3 .   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y .  A New Linear Muscle Fiber Model for Neural Control of Saccades and for  j  = 2 ,...,m + 1 y ij ( ∞ ) = T  i − F  ij K  lt =   K  se 2 K  lt  + mK  se  ×  1 K  lt m +1  j =2 F  ij  + x 1 ( ∞ )  − F  ij K  lt ,  (11)and y im +2 ( ∞ ) =  T  i K  se =   K  lt 2 K  lt  + mK  se  1 K  lt m +1  j =2 F  ij  + x 1 ( ∞ )  . (12) 2.2.  Static and dynamic propertiesof the muscle fiber model of muscle As shown here, the static and dynamic characteris-tics of the muscle fiber muscle model are identicalto those of the whole muscle model demonstrated byEnderle and coworkers. 3 It should be noted that thewhole muscle model is the only linear model thathas the nonlinear force–velocity and length–tensionrelationships observed in data.2.2.1.  Static properties  To compare the length–tension characteristicsbetween the two models, assume that all of themuscle fiber active state tensions are identical, andthen substitute the parameter values given in Eq. (7)into Eq. (2), which gives T  ( ∞ ) = n  i =1 T  i ( ∞ ) = n  i =1   K  se K  lt 2 K  lt  + mK  se  x 1 ( ∞ ) + 1 K  lt m +1  j =2 F  ij  K  se = 2 KM  se n K  lt = mKM  lt n = n  i =1  2 K  M  se n mK  M  lt n 2 mK  M  lt n  + m 2 K  M  se n  x 1 ( ∞ ) + m   2 K  M  se n 2 mK  M  lt n  + m 2 K  M  se n  F  ij  = n  i =1  1 n   K  M  se K  M  lt K  M  lt  + K  M  se  x 1 ( ∞ ) +   K  M  se K  M  lt  + K  M  se  F  ij  =   K  M  se K  M  lt K  M  lt  + K  M  se  x 1 ( ∞ ) +  n   K  M  se K  M  lt  + K  M  se  F  ij  F  ij = F n =   K  M  se K  M  lt K  M  lt  + K  M  se  x 1 ( ∞ ) +   K  M  se K  M  lt  + K  M  se  F.  (13)Equation (13) is the same as Eq. (2) in Enderleand coworkers, 3 except that it is written in termsof lengthening instead of shortening. Thus, themuscle fiber muscle model has the same length–tension characteristics as shown in Fig. 3 of Enderleand coworkers, 3 which matches the data extremelywell.2.2.2.  Dynamic properties  To investigate the force–velocity characteristicsbetween the two models, we once again assume thatthe muscle fiber active state tensions are identicaland attach a mass to the end of the muscle, whichrests on a platform, as shown in Fig. 4. The equationsthat define this system are given as Mg  + M  ¨ x 1  + K  se ( x 1 − x i 2 ) + B 2 (˙ x 1 −  ˙ x i 2 ) = 0 ,K  se ( x 1 − x i 2 ) + B 2 (˙ x 1 −  ˙ x i 2 )=  K  lt ( x i 2 − x i 3 ) + B 1 (˙ x i 2 −  ˙ x i 3 ) − F  i 2 ,K  lt ( x i 2 − x i 3 ) + B 1 (˙ x i 2 −  ˙ x i 3 ) − F  i 2 =  K  lt ( x i 3 − x i 4 ) + B 1 (˙ x i 3 −  ˙ x i 4 ) − F  i 3 , ... 1350002-5    I  n   t .   J .   N  e  u  r .   S  y  s   t .   D  o  w  n   l  o  a   d  e   d   f  r  o  m   w  w  w .  w  o  r   l   d  s  c   i  e  n   t   i   f   i  c .  c  o  m   b  y   1   8   6 .   8   7 .   1   7   9 .   2   3   4  o  n   0   3   /   1   9   /   1   3 .   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y .
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