James W. Gnadt, PhD
A schematic representation of the Breznen/Gnadt model is shown in Fig 1. Quantitative details can be found in Breznen and Gnadt (1997) and Jackson and Gnadt (1999). Basically, the model is a modification of the model of Jürgens et al. (1981) with the addition of a leaky feedback integrator and dynamic synaptic adaptation in the feedback circuit. As illustrated in Fig 1, the superior colliculus (SC) is a feedforward input of desired change in eye position (ÄEP) to the brainstem CPG. The output of the circuit to the motor plant comes from a class of pre-motor burst neurons (BN) in the pons and midbrain known as medium-lead burst neurons (MLB) or excitatory burst neurons (EBN). The efference copy is implemented as a recurrent feedback loop with a time constant of approximately 60 ms at rest. Connections between circuit elements are made with multiplicative gain coefficients. As a function of the circuit’s own activity, the circuit gains change from their resting values by up to 80% over a time constant of 50 ms. More recent analyses have adjusted downward our estimate of this parameter to about 20 ms.Limitations of the saccadic feedback model: The original implementation of our computational model was derived from generic analytical considerations so that we could test published models with the step response. We were quite pleased with the utility of our numerical simulations to predict the monkeys’ behavior and to provide testable hypotheses about the biological mechanisms. We are especially pleased to provide our new simulation tool for use by the greater research community from our lab’s web site. We are well aware that any numerical model is limited by the assumptions and simplifications it engenders. While our model is a simulation tool built from systems of differential equations, we have judiciously added biologically relevant nonlinearities at critical points in the system and we have been cautious to apply the model only to experimental issues for which it is appropriate. At present, the model does not attempt to solve the spatial-to-temporal transformation from the superior colliculus to the temporally-encoded neurons of the brainstem CPG. Nor is the model designed for two-dimensional output. We plan to upgrade the model to account for these two properties.
Because of assumptions about the OPN during the step response, we have also pointed out that the original model was constructed with an “unsophisticated” implementation of the pause cells in the circuit (Breznen & Gnadt, 1997). At this point, we find that this simplified version of the OPNs fails to completely account for the monkey’s frequency response. Thus, we are in the process of implementing a more realistic representation the OPN in the model.
I also point out that the model makes no attempt to suggest the actual neural mechanism for the feedback integration circuit. It was designed primarily as an analytic tool and simulation instrument, not a truly morphometric network of the circuit. It has proven useful in testing assumptions about the levels at which the feedback closes in the circuit and about the dynamics of feedback behavior. At present, there is no generally accepted model or experimental consensus for the actual neural circuit that implements the local feedback for the saccadic circuit.
Systems engineering study of the saccadic feedback circuit
One of our experimental strategies has been to use a systems engineering approach to reveal fundamental insights into the mechanisms of saccadic motor behavior in primates. The neural circuit that generates saccades is a central pattern generator (CPG) that is distributed within the brainstem where it is adapted to produce high velocity movements with high precision (Sparks & Mays, 1990; Wurtz & Optican, 1994; Moschovakis et al, 1996). Direct command inputs to the circuit are thought to come from two high-order movement centers; the frontal eye fields and the superior colliculus (Schiller et al, 1987; Keating & Gooley, 1988).
In terms of systems engineering, the saccadic circuit can be thought of as a biologic machine that creates movement behavior by transforming inputs into outputs. Like any machine, one can attempt to understand how it works by “reverse engineering.” That is, given the machine’s output, how do the internal circuits function to produce that output? Because we have a fairly detailed structural and functional circuit diagram of the biologic machine that generates oculomotor outputs, we can use systems control theory to make quantitative predictions about the system’s output in response to injecting characteristic input signals at critical points in the circuit. The compliance of the actual behavior with the predicted behavior, as well as experimental deviations from the idealized output, reveals fundamental information about the circuit.
“Local” feedback within the brainstem CPG: The relatively long latency of visual responses in the brain excludes the possibility that vision can supply dynamic feedback during the high velocity saccadic eye movements. Thus, either saccades are feedforward, ballistic movements or there must be a short latency “local” feedback mechanism (Robinson, 1975). According to the local feedback concept, the eye movements are driven by a signal that compares the current eye position with a desired eye position. The saccadic circuit generates a pulse of activity proportional to the size of this difference, which is called motor error. Through both direct and indirect connections, this motor error signal is used to drive the pattern of activity for the ocular motor neurons. This driving signal is then progressively reduced by dynamic negative feedback of a copy of the neural command to the eyes. This feedback copy of the output signal is known as “efference copy,” the existence of which was first inferred from behavioral observations nearly a century ago by Helmholtz (1909). During the movement, as the efference copy of the current eye position approaches the desired eye position, the output of the circuit is reduced to zero as the eyes land on target. Definitive evidence for this dynamic feedback control was provided when experiments showed that experimental perturbations of volitional saccades were quickly and precisely compensated (Sparks & Mays, 1983; Keller, 1977). A ballistic controller would have failed to compensate for short latency and mid-flight perturbations.
Inhibitory “latch” of the CPG: As is the nature of any high-gain controller, the high velocity saccadic controller tends to be unstable and is susceptible to oscillations, as has been demonstrated in pathologic states in humans (Zee & Robinson, 1979; Ashe et al, 1991). We now know that the saccadic circuit is “latched” into quiescence between movements by an inhibitory input from the so-called omnipause neurons (OPN) (Curthoys et al, 1984). In the awake animal, these pause neurons are tonically active except just prior to and during saccades. Thus, saccadic movements are thought to be triggered by the sudden pause of this inhibitory latch, which releases the high-gain feedback circuit to react to the motor error imposed by the inputs from higher centers.
The CPG as a displacement controller: An important modification of the original local feedback model was to implement the circuit as a displacement controller, rather than an end-position controller (Jürgens et al, 1981; Scudder, 1988; Moschovakis, 1994). This incorporated the now recognized fact that the outputs from the superior colliculus and frontal eye fields are encoded as desired changes in position. Thus, the saccadic CPG generates a desired change in position, independent of the initial starting position. This innovation required that the feedback efference be a second, separate loop than the output of the motor neurons themselves, which always do reflect the current eye position. This concept also demands that the local feedback loop, which dynamically accumulates eye trajectory during the movement, must be reset to zero at the end of each movement.
Systems control analysis: The actual mechanisms of the CPG that generate saccades can be investigated using a systems control analysis of injecting characteristic input functions at critical points within the circuit. Three useful inputs are 1) the impulse function, an infinitely large and infinitesimally short input pulse; 2) the step function, an instantaneous step from one input value to a new steady-state value; and 3) the frequency function, a steady-state sinusoidal input. Technical considerations and system non-linearities preclude the use of the impulse function in the primate oculomotor system, but we have tested the saccadic circuit in monkeys for the step response (Breznen et al, 1996, 1997). Assuming a prolonged train of microstimulation pulses in the superior colliculus could simulate a frequency-encoded input to the saccadic CPG, we challenged the saccadic circuit with a train of pulses at various steady-state values. We then compared the step response of published circuit models to experimental data from monkeys.
Formally, the unit step response (Phillips & Harbor, 1991) for a linear second-order feedback system as a function of time, t, is given as
The term ùN is the natural frequency of the system, æ is the damping ratio and è is a phase shift equal to cos-1 æ. The damping ratio is a dimensionless factor that conveniently describes the behavior of the system. Critically damped systems have a damping ratio of 1.0 and approach their new steady-state value with an optimal time course. Underdamped systems (æ<1.0) tend to be unstable; exhibiting oscillatory, self-correcting overshoots according to the sinusoidal term on the right. Overdamped systems (æ>1.0) are well behaved because the exponential term predominates, but can be sluggish when æ >>1.0. Breznen et al. (1996, 1997) showed that the high-gain biological circuit behaved like an underdamped feedback system for which the output could be well described as the oscillatory behavior of a second-order system similar to that in equation 1. Each oscillation of the circuit produced a saccade-like movement, thus moving the eyes along in a ratchet-like fashion - a behavior that has been described as “staircase saccades” due to the appearance of plotting eye position as a function of time.
However, we also showed that the biological circuit behaves in ways not captured by past models of the saccadic mechanisms. We suggested a modified model that recapitulates the experimental behavior from the monkey (Breznen & Gnadt, 1997). One important modification from past models was to implement the recurrent feedback loop of the displacement controller as a “leaky integrator”. Thus, the reset of the local feedback is accomplished by the biologically plausible mechanism of a slowly-adapting relaxation of the activity that accumulated during the movement. A second modification, demanded by analytical considerations, was that the gains and time constants of the system were required to change dynamically during the movements. We hypothesized that this phenomenon was mediated by activity-related synaptic adaptation within the circuit (Breznen & Gnadt, 1997).
It was actually quite amazing that such a simple systems control analysis could account for so much of the empirical behavior from the monkey. Thus, we found it important to test the validity of the model. In keeping with the approach to reverse engineer this behavior, we tested both our model and monkeys with another characteristic input signal - the frequency response. This provided a rigorous test of the assumptions and parameters of the model. Since our model and the monkey’s neurons operate using a frequency of pulses or action potentials, we applied the input using a frequency modulated (FM) stream of stimulation pulses - both in the model and in the monkey’s colliculus. If our model is correct, it should accurately predict the behavior of the monkey to a prolonged train of frequency modulated pulses.
Equation 2 describes the frequency response (Phillips & Harbor, 1991) of a linear second-order system as a function of time,
where the additional term, ù, is the input frequency. A purely linear version of the model shown in Fig. 1 would have behavior described exactly by equations 1 and 2. However, our model simulations are derived from a numerical implementation which includes the dynamic gain changes that simulate synaptic adaptation and several other biologically relevant non-linearities (Breznen & Gnadt, 1997; Jackson and Gnadt, 1999). These complexities make analytic solution of the non-linear implementation intractable. However, numerical simulations of the non-linear model produces similar behavior to the purely linear case with some biologically important deviations.
Manuscripts presenting our studies of the frequency response are currently in preparation and should be available soon.
In summary, we have been quite pleased with the utility of using a systems engineering approach to investigate fundamental issues about the saccadic circuit. We have proposed a simple numerical model of the saccadic feedback controller that accounts very well for the behavior of the system in vivo. By testing both the monkey and our model for the step response and frequency response, we have addressed several contemporary issues in oculomotor physiology, including dynamic synaptic gains, “leaky integration” via a slowly-adapting relaxation process, circuit oscillations in a central pattern generator (CPG) and implications for whether the superior colliculus lies within the CPG or upstream as a feedforward input. It is important to point out that these artificially-induced trains of activity would never occur naturally during normal voluntary behavior. The purpose is to challenge the saccadic circuit with characteristic input patterns for which we have explicit quantitative predictions for the output based on assumptions about the circuit. Since there are physiologic correlates to the numerical considerations (e.g., synaptic adaptation for dynamic gain changes), comparison of the monkey’s behavior to the model’s predictions serves to test the biological implications of the model.
References
Ashe, J, TC Hain, DS Zee and NJ Schatz, Microsaccadic flutter, Brain, 114: 461-472, 1991
Breznen, B, S-M Lu and JW Gnadt. Analysis of the step response of the saccadic feedback: System behavior, Exp
Brain Res, 111:337-344, 1996
Breznen, B and JW Gnadt, Analysis of the step response of the saccadic feedback: Computational models, Exp
Brain Res, 117:181-92, 1997
Curthoys, IS, CH Markham and N Furuya, Direct projection of pause neurons to nystagmus-related excitatory burst
neurons in the cat pontine reticular formation, Exp Neurol, 83: 414-422, 1984
von Helmholtz, H, Handbuch der Physiologischen Optik, Verlag von Leopold Voss, Hamburg and Leipzig, 1909
Jackson, ME and JW Gnadt, Numerical simulation of nonlinear feedback model of saccade generation circuit
implemented in the LabView graphical programming language, J Neurosci Meth, 87:137-145, 1999
Jackson, ME and JW Gnadt, Testing assumptions of the interrupted saccade paradigm: Consistency with the concept
of a leaky integrator, submitted
Keating, EG and SG Gooley, Disconnection of parietal and occipital access to the saccadic oculomotor system, Exp
Brain Res, 70:385-98, 1988
Keller EL, Control of saccadic eye movements by midline brain stem neurons. In: Control of gaze by brain stem
neurons (R Baker and A Berthoz, eds), Amsterdam: Elsevier, 327-336, 1977
Moschovakis, A.K, Neural network simulations of the primate oculomotor system I. The vertical saccadic burst
generator, Biol Cybern, 70: 291-302, 1994
Moschovakis, A.K, CA Scudder and SM Highstein, The microscopic anatomy and physiology of the mammalian
saccadic system, Progress in Neurobiology, 50:133-254, 1996
Phillips, CL and RD Harbor, Basic feedback control systems, Prentice Hall, Engelwood Cliffs, 1991
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Scudder, CA, A new local feedback model of the burst generator, J Neurophysiol, 59: 1455-1474, 1988
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Relevent Publications:
Breznen, B, S-M Lu and JW Gnadt. Analysis of the step response of the saccadic feedback: System behavior, Exp. Brain Res., 111:337-344,1996.
Breznen, B, and JW Gnadt. Analysis of the step response of the saccadic feedback: Computational models, Exp. Brain Res., 117:181-92, 1997.
Gnadt J.W, O. Litvak and B. Breznen Oscillations of the Saccadic Feedback Circuit in the Monkey, Soc. Neurosci. Abstr., 23:842,1997.
Jackson ME and JW Gnadt, Frequency Response of the Saccadic Generation Circuit in Primates: Resonant Frequency, Soc. NCM Abstr., 1998.
Gnadt JW and ME Jackson, Colliding saccades for the step and frequency responses in the monkey: interference patterns, Soc. Neurosci. Abstr., 24:419, 1998.
Jackson ME and JW Gnadt, Testing assumptions of the interrupted saccade paradigm: reset of the neural integrator, Soc. Neurosci. Abstr., 24:419, 1998.
Jackson ME and JW Gnadt, Numerical simulation of nonlinear feedback model of saccade generation circuit implemented in the LabView graphical programming language, J. Neurosci Meth, 87:137-145, 1999.
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