This paper is involved with the formulation of rudimentary quick human movements. Examples can be cited in softball [1] [2] , kicking a soccer ball [3] - [5] , jumping off the ground [6] - [8] and performing a back somersault [9] - [11] . These movements are also referred to as ballistic movements [12] [13] . The ballistic movement, as formulated here, may be triggered by and synchronized with external events, may have to be reversed in time [14] [15] , and requires three consecutive phases in time. The three time intervals and the required central processing in each are vital to the execution of the movement. The timing and time event processing [16] - [19] may be carried out in the hippocampus [20] [21] , the cerebellum [22] , basal ganglia, and the reticular formation [23] .

The dynamics can be derived using different principles [24] [25] . Substantial research that merits a review of its own has been carried out by Soechting and colleagues [26] - [29] on the human arm. Iqbal and Roy [30] present a sagittal neuromusculoskeletal system. Simpler systems also have been proposed [31] . Three degrees of freedom are assumed at every joint here. More physiologically realistic constraints at the elbow and the wrist [32] [33] could be considered and related to experimental observations [34] [35] .

The three-link arm is equipped with 15 pairs of muscles and has already been used in some upper extremity research [36] - [38] . A virtual neuromusculoskeletal system is considered in [39] . A model, suitable for simu- lation has been formulated [40] . A model, emphasizing joint coordination, is presented in [41] . Studies of fingers, the palm of the hand, and forearm are carried out in [42] . Mathematical investigations of the arm and the kinematic issues are respectively addressed in [43] [44] . More physiologically accurate moment arms of the muscles, muscle masses, and muscle lengths and moment arms are respectively investigated in [45] - [47] .

The underlying philosophy of control presented here is to centrally command the arm to execute a specific maneuver. Further, it is intended that the three-link model be a building block in larger skeletal systems that would be composed of many such three-link modules. This philosophy of modeling and control merits further study of its own and is not undertaken here.

The ballistic movement from one equilibrium point to a second is brought about by an initial phase of acceleration, a second phase of coasting and a third phase of deceleration and stop. Co-activation [48] guarantees stability. Smoothness of motion and fast stop are implemented by spinal reflexes. Similar strategies of control are cited in the robotic field [38] and for anthropomorphic robots [49] . A number of hypotheses [30] [50] , experimentally-based observations [51] , and animal observations [52] - [54] support the model presented here.

In the present paper, the trajectory of the desired movement of the arm is articulated in the Central Nervous System (CNS) by specifying the nine Bryant angles [55] - [57] as functions of time. Further, the shape of the trajectories here is consistent with Kornhuber’s [51] [58] hypotheses that simple point-to-point movements are ramp type and that the basal ganglia is responsible for the synthesis of the linear trajectories [55] [59] [60] . As stated before, three phases of central and peripheral activity are required. A simple cerebellum model [61] takes care of programming the turning on and off the agonist and antagonist activations [62] and the necessary spinal reflexes [63] . Simple models of the needed basal ganglia and the cerebellum are presented. Specifically, the presentation here is consistent with Evarts’ formulation [64] [65] (namely, the attention set and attention mechanisms [66] ).

Timing is a major component of the implementation of fast movements, motivated by internal events or conditioned by external events, in activities such as sports. More complex related events in the CNS are storage and recall of information, delaying or advancing events in time, synchronizing, time scaling and reversing events in time. Three-phase timed events in tapping, stepping and rocking [67] , are part of a longer sequence in dance. Other examples are compression or expansion of signals in time, i.e, speeding up a movement or slowing it down. A timed event may be part of a pattern generator or an oscillator.

In a dynamic system, described by an Artificial Neural Network (ANN) or a differential equation [68] , a set of programmable gains control the behavior of the system as in cardiac myocytes [68] [69] and ( [70] , page 250). A majority of such motor functions in living systems are described qualitatively, e.g., ( [54] , Figs 41-14 and 42-5). Examples of conditioning and measurement of duration of events that need to be timed and the neural processing needed for timed events are presented in [71] [72] . A description of the conditioned reflex is provided in ( [20] , chapter 10) and in ( [73] , chapter 13).

Chemical agents and neuro-transmitters induce the needed gain changes in many instances. Serotonin is one agent known to increase the duration of action potentials in abdominal and pleural sensory neurons of Aplysia ( [70] , page 250). It inhibits and delays repolarization due to potassium channels. Serotonin may further decrease a voltage-dependent delayed rectifying potassium current and thus prolong the action potential ( [70] , page 254).

Internal timed events are involved in motor learning, natural reflexes, and tight input-output coupling [34] [52] [74] [75] . Psycho-physical experiments [76] point to processing in the CNS that is governed by laws of physics [77] [78] . Arbib ( [79] , page 53) discusses leaky integrators as more realistic physical models of neurons. In Arbib’s view, a neuron circuit with a long time constant becomes an approximate integrator and, hence, a basic memory element. Doeringer and Hogan [50] have estimated the neural transmission delay in the spinal reflex loop of the upper extremity to be in the order of 30 ms, with a transfer function that has a frequency range of zero to less than 2 Hz. An inverse compensator is proposed for the delay, a form of predictor [80] . Predictors are also discussed in ( [81] , page 62) and [82] .

A single trajectory of force or motion [54] [77] can be stored in a tapped delay line [83] [84] , compatible with the structure of short-term memory ( [20] [77] , page 86 and chapter 9). The fading away of the information can be modeled as attenuation in a lossy delay line. Larger units of the CNS function can be envisioned that are involved in signal generation and processing. Time scaling and piece-wise linear signals are examples. Time scaling involves producing a slower or faster movement.

Timed events are also involved when the CNS has to relate internal decisions to external events. Timing [85] - [87] relative to an external cue or alarm could involve associative learning ( [54] , Chapter 65). Timing is also involved in mechanisms of attention [64] , preparation for a movement or in a motor set [73] . The role of the hippocampus in time reversal of movement has been noted [20] [21] , According to Foster and Wilson [14] , while rats rest after running a maze, their brains replay the movements in reverse. The idea of reversing events in time is a case of spatial synchronization of internal action with external events that is expounded later in Section 4 of the paper.

There are cases in the CNS where different sensory neurons fire in synchrony ( [54] , chapter 30, and [88] ). A simple example of this is the eye-blink experiment [89] . In this experiment [20] [90] , blowing a puff of air in a rabbit’s eye will cause the rabbit to blink [90] . Grethe and Thompson [90] review and compare models of this synchronization. Tapped delay lines of different lengths [78] [91] [92] were proposed. Artificial Neural Net- works (ANNs) can mimic the coincidence model of Buhusi and Meck [93] , granule cells begin to oscillate at the start of the conditioned stimulus, and then the outputs of these granule cells are recorded at the time of the stimulus. Nicholls et al. ( [94] , page 552) discuss, for medullary neurons that generate the respiratory rhythm in neonatal opossum, the location of synapses and the incremental time delays that are important in temporal processing and signal shaping. Simultaneous patterns may have to be recalled from storage or from pattern generators ( [70] , chapter 17). Two alternative schemes to generate patterns are by linear filters and neural oscillators [60] [68] [95] - [97] . The transient outward current in the bag cell neuron is shown to contain one exponential function of time with a time constant of 76 ms, and, under the influence of cyclic AMP, to contain a second exponential of a much faster decay (and, hence, a rationale for a second order system). Neuro-trans- mitters change the shape of an action potential ( [70] , page 249), suggesting the possibility of filtering and pro- cessing mechanisms that are implicit. In this case, that mechanism was the narrowing of the action potential pulse in the dorsal root ganglion of a chick. Other patterns are involved in voluntary movements or in elicited maneuvers [98] [99] or slipping and sliding [100] [101] .

For ease of presentation, afferent and efferent neural transmission delays are ignored in this paper [102] [103] . Spinal reflex delays are included because they are crucial to the stability of the ballistic movement. Spinal reflexes and spinal cord injury [104] - [106] , as a whole, merit more serious consideration that is not warranted here. A rudimentary spinal reflex for controlling forces of contact in pushing or pulling is considered in [107] . A simple spinal reflex is modeled here and more physiologically accurate models [80] [108] [109] remain to be included and investigated in the future. Similarly, it is assumed that gravity is fully compensated and does not affect the motion directly [110] . All discussions, models and formulas derived and developed here are limited to the simple ballistic movement and should not be construed to represent universal models of the cerebellum, the basal ganglia, the reticular formation, or the hippocampus.

The structure of the paper is as follows. The dynamics and control of the three-segment arm are briefly discussed in Section 2. The modeling of fast movements is presented in Section 3. The timing issues and central neural structures for the ballistic movement, its reversal in time and conditioning are presented in Section 4. Simulations of a movement (similar to one in throwing a softball) with and without delays in the spinal reflex path are presented in Section 5. Discussions and conclusions follow in Section 6.

The derivation of the dynamics of the three-link arm system are briefly sketched here. The formulation starts with the free body equations. The 18-dimensional vector Z represents the nine translational position states and the nine rotational Bryant angles. The arm is assumed to be connected to an stationary torso (the inertial coordinate system) and, as a consequence, all translational degrees of freedom are eliminated to derive the systems equations in terms of the rotational degrees of freedom. The 18-dimensional state space equations of the system are symbolically written here:

There are four nine-vector terms of the second equation. All muscular inputs are included in vector. The are two nonlinear terms: and. The centrifugal term is and the gravity term is. The steps in the definition and the derivation of these vectors are given in [8] [66] .

Stability is achieved by the simultaneous co-activation of agonist-antagonist pairs of muscles by at least three pairs at every joint [111] - [113] . The co-activation produces sufficient position and velocity feedback to bring about stability in the vicinity of the vertical equilibrium position. It can be shown that with no limitation on the amplitudes of the feedback, the system can be globally stabilized [114] . Similarly, for finite feedback amplitudes, the size of the domain of stability can be estimated. For this purpose, Lyapunov functions as the sum of the translational and rotational kinetic energies of both the passive and active tissues and the gravitational energy are considered. The system is stable when the loss of potential energy due to the motion of the system is more than compensated by the energy stored in the elastic structure of the system and sufficient loss occurs in the system.

Point-to-point movements can be slow or fast. Slow movements, where the trajectory of motion is defined by linear segments between the initial and final positions, have been considered before [115] . Relatively fast move- ments are discussed in [1] [116] . Hannford and Stark [117] have discussed three actuation phases for such movement. In the beginning phase, initial velocities launch the system. In the middle phase, the system coasts. In the final phase, the movement is quickly brought to a halt. The three phases [118] . are considered below.

Three timing issues involved in the movement are presented here and the possible neural involvement of the hippocampus, the reticular formation, the basal ganglia and the cerebellum are delineated. The three timing issues are:

・ reversal of motion timing,

・ parallel event timing, and

・ serial event timing.

For ease of presentation and communication, the functions are illustrated with simple daily events.

The simulation has four components here: the basal ganglia for time and trajectory construction, the cerebellum for setting the gains, the musculoskeletal system and the spinal reflexes. The ramp-type trajectories for time measurement and pattern generation are constructed in the basal ganglia [60] and ( [55] , Table 23.1, page 448). The first phase of movement starts as output of a block. When the first phase is complete, a trigger starts the second phase and, subsequently, a third phase. Asynchronous “one-shot” electronic circuits without a clock could be used to measure the three time intervals.

The cerebellum is involved in the selection of the activation levels of the muscles in phase one, in inhibition of position-dependent forward activation and disinhibition of spinal reflexes in phase two, and in both co- activation and maintenance of precise spinal reflexes in phase three. These inhibitions and disinhibitions occur in the motor cortex for preprogrammed movements according to Evarts ( [65] , page 178), and also conform to the summary of functions of the cerebellar outputs as given by Brooks ( [73] , page 287). Other activities may be involved in this path to the cortex, as discussed by Ghez ( [124] , page 637).

From input position angles and known gains K, the cerebellum produces products. In a better model than this, the cerebellum would produce many output functions for a corresponding input signal. The following assumptions are made:

・ The basket, Golgi and stellate cells regulate internal gains and internal stability of the cerebellum.

・ The cerebellum is assumed as a memoryless algebraic function generator, consistent with Pellionisz and Llina’s hypotheses [125] - [127] , where there are outputs from Purkinje cells (or sums of Purkinje cell outputs) and two classes of inputs.

・ Climbing fibers as (possibly binary) discrete events and conditions are one input class.

・ Mossy fiber inputs are trajectories and other functions of time.

・ The possible addition (or summing) of Purkinje cell outputs to construct larger signals (in frequency), which may require delicate timing mechanisms, is also ignored. Simple amplitude addition is employed here.

The requirement that all the signal processing in the cerebellum be formulated in frequency domain and with pulse signals merits a challenging and independent study. It also requires that the Purkinje cell outputs involved for the same large function have differently timed windows such that their pulses add together in adjacent intervals of time. This mechanism allows construction of signals with large intensity. It could also explain the “inhibitory” nature of the Purkinje cells.

A sketch of the control activity of the three phases is shown in Figure 5.

We also note that reversing the control here does not reverse the movement. The movement is from a vertically extended position and takes place in the sagittal plane. Initially, all 18 positions and velocities are zero.

In the terminal state, the three sagittal (pitch) angles are, respectively, radians and the rest of the position states (roll and yaw angles) and all the velocities are zero. The stiffness matrix is constructed from nine pairs of single-joint and six pairs of biarticular muscles, and is assumed to be constant throughout the movement. It is given in the Appendix.

The initial phase is assumed to take 10 ms and the three sagittal angles, i.e., the three pitch angles of the upper arm, lower arm and hand are expected to rise to 5 rad./sec. The triangular pulse scheme is used for 10 ms to initiate the movement. The three angles, angular velocities and the negative feedback moments of forces at the joints are shown in Figure 6. The other six angles remain zero and are not plotted. If the rest of the body, i.e., the torso, etc. was modeled (under co-activation) it would provide a stable platform as an inertial coordinate system from which the arm’s movement is launched.

At t = 0.5 second, i.e, the end of the free coasting cycle, the three nonzero sagittal positions are

The three non-zero sagittal velocities are

As can be seen, only the sagittal pitch position and angles are non-zero. Suppose it is desired to stop the system at that state. For the last phase, co-activation is implemented and, further, spinal reflex feedback is used to slow the system to an over-damped halt with no terminal oscillations. The gain of the spinal reflex is 0.2 in the coasting phase and unity in the stopping phase. This means about one fifth of co-activation gains in coasting and unity in the stopping phase. The co-activation gains are kept constant in the simulation of the ballistic movement. The three angles, angular velocities, and the negative feedback moments of forces at the joints are shown in Figure 7.

In the next three simulations, neural transmission delays are added to the spinal reflex. In Figure 8, a delay of 5 ms is added in the neural path of the spinal reflex.

In Figure 9, the delay is increased to 10 ms. In Figure 10, the delay is kept at 10 ms, but the gain of the spinal reflex feedback loop is reduced by half for all three muscles involved.

Neural modules were presented for the implementation of ballistic movements that may be reversed in time or conditioned and synchronized with external events. The role in the execution of movement of four central neural structures, hippocampus, the reticular formation, the basal ganglia and the cerebellum, was outlined.

The role of procedural memory was illustrated. Time scaling by basal ganglia circuits was demonstrated. Neural structures involved in time reversal of trajectories were discussed. Reversal of control sequences was not considered and remained a challenge for the future. Use of LIFO stacks for synchronization of large numbers of signals was formulated.

The point-to-point fast movement is implemented by assuming that the CNS has information about the nine Bryant angles at the initial and terminal position of the arm. Such information could be inferred by the CNS from the stretch responses of all the muscles that are relayed to the CNS at any position of the arm, and then kept, in an associative memory, with the visually encoded position of the arm. More research is necessary to establish the above hypotheses.

The central feed forward signals were the desired limb trajectories. The co-activation of 15 pairs of agonist- antagonist muscles produced the necessary muscular feedback. The tendons played a role here in controlling of their length allow control of stiffness.

It is instructive to compare the co-activation strategies of this paper with the ones attributed to the human system ( [73] , Fig. 4.11). The model here has to be contrasted with “length follow-up servo” as in ( [73] , Fig 4.11 A and B).

The gains of the feed forward path are set centrally, and not mediated by Golgi and afferent signals involved in feedback from the spinal cord ( [73] , Fig 4.11 C). Finally, precisely controlled spinal reflexes of velocity are needed in the third phase of the ballistic movement. The ballistic movement was specified and controlled in three phases of initial acceleration, coasting, and stopping. All three phases require construction of reference input trajectories in the basal ganglia and inhibition and disinhibition of muscular activities, prompted by the CNS and spinal reflexes by the cerebellum. Computer simulation showed how the system works, and the effect of delays in the spinal reflex loop was demonstrated.

Existing measurements could be used to tune the model parameters and improve the model and the simulation results. With recent posture estimation techniques [128] [129] , one could train or teach robots to play softball [130] . The performance of the above model can be improved [131] for human applications [132] - [134] and by anatomical accuracy [45] - [47] . More quantitative studies of the sensory and processing deficits, injuries and deficiencies could be undertaken [104] [106] [132] .

The authors are very grateful and indebted to Professor H. C. Ko, former chair and professor emeritus of the Electrical and Computer Engineering Department at The Ohio State University, for his generosity, vision, sustained support and encouragement of this work. We would like to acknowledge support of R. J. Garbacz, Lee Potter, Patrick Roblin and Vadim Utkin. We are indebted to Darla Peelle and David B. Ball for editorial assistance and inspiration.

Hooshang Hemami,Behzad Dariush, (2016) Neural and Spinal Modules in Implementation of a Simple Ballistic Movement. Journal of Software Engineering and Applications,09,326-345. doi: 10.4236/jsea.2016.97023

The physical parameters used in the computer simulations are given in the table below.

The gain matrix for the simulations is given in table below. The linear viscosity, in the first phase, is assumed to be 1/8 of the stiffness for all actuators.