A model-based theory on the origin of downbeat nystagmus

Abstract

The pathomechanism of downbeat nystagmus (DBN), an ocular motor sign typical for vestibulo-cerebellar lesions, remains unclear. Previous hypotheses conjectured various deficits such as an imbalance of central vertical vestibular or smooth pursuit pathways to be causative for the generation of spontaneous upward drift. However, none of the previous theories explains the full range of ocular motor deficits associated with DBN, i.e., impaired vertical smooth pursuit (SP), gaze evoked nystagmus, and gravity dependence of the upward drift. We propose a new hypothesis, which explains the ocular motor signs of DBN by damage of the inhibitory vertical gaze-velocity sensitive Purkinje cells (PCs) in the cerebellar flocculus (FL). These PCs show spontaneous activity and a physiological asymmetry in that most of them exhibit downward on-directions. Accordingly, a loss of vertical floccular PCs will lead to disinhibition of their brainstem target neurons and, consequently, to spontaneous upward drift, i.e., DBN. Since the FL is involved in generation and control of SP and gaze holding, a single lesion, e.g., damage to vertical floccular PCs, may also explain the associated ocular motor deficits. To test our hypothesis, we developed a computational model of vertical eye movements based on known ocular motor anatomy and physiology, which illustrates how cortical, cerebellar, and brainstem regions interact to generate the range of vertical eye movements seen in healthy subjects. Model simulation of the effect of extensive loss of floccular PCs resulted in ocular motor features typically associated with cerebellar DBN: (1) spontaneous upward drift due to decreased spontaneous PC activity, (2) gaze evoked nystagmus corresponding to failure of the cerebellar loop supporting neural integrator function, (3) asymmetric vertical SP deficit due to low gain and asymmetric attenuation of PC firing, and (4) gravity-dependence of DBN caused by an interaction of otolith-ocular pathways with impaired neural integrator function.

Appendix

Model equations (healthy system)

The model equations are given in Laplace notation, with s denoting the complex frequency (see Fig. 9 for graphical illustration of the model structure and the equations). Thus, the temporal derivative of a variable is given by multiplying it with s, and the temporal integration is given by dividing by s. As an example, the first equation is given both in the Laplace notation and as differential equation.

Fig. 9

Model outline as in Fig. 1, but with visible subsystems (except for the saccadic burst generator). See legend to Fig. 1 and Appendix for details

The model output, vertical eye position e, is determined by the motor command m according to the eye plant equation

$$ e = m \cdot \frac{1}{{1 + \tau _{\text{e}} s}} $$

(1)

with τ _e = 0.2 s being the dominant time constant of the eye plant, or, as differential equation, by

$$ \dot e = \left( {m - e} \right)/\tau _e $$

Thus, the eye plant is modeled as first-order low-pass filter (see Glasauer 2007 for a review on eye plant models).

Inputs to the system are head angular position α in the pitch plane and vertical target position t. The semicircular canal transfer function relates head angular position α to afferent canal output ω _c by

$$ \omega _{\text{c}} = \alpha \cdot \frac{{\tau _{\text{c}} s}}{{1 + \tau _{\text{c}} s}} \cdot \frac{s}{{1 + \tau _{\text{d}} s}} $$

(2)

with a dominant time constant τ _c  = 5 s (e.g., Zupan et al. 2002) and a secondary time constant τ _d  = 0.01 s.

Utricular afferent output u is proportional to the sine of the head pitch angle with a gain factor g _u (see below):

$$ u = g_u \cdot \sin (\alpha ) $$

(3)

Thus, for the upright position (u = 0) there is no otolith influence on eye movements.

Retinal slip r _s, which is the input to the smooth pursuit system, depends on head angular position, eye position, and target position by

$$ r_{\text{s}} = \left( {t - \alpha - e} \right) \cdot s \cdot e^{ - \Delta t \cdot s} $$

(4)

where $e^{ - \Delta t \cdot s}$models a time delay of Δt = 100 ms due to visual processing. Retinal error, used to drive saccades, is simply given as $r_{\text{e}} = \left( {t - \alpha - e} \right) \cdot e^{ - \Delta t \cdot s} .$

The brainstem integrator is modeled as leaky integrator (e.g., Zee et al. 1981) with a time constant τ _b = 5 s. It receives the burst input b from the saccadic burst generator (see below), the canal afferent signal ω _c, the utricular afferent signal u, and the Purkinje cell (PC) afferent discharge p. Its output e _i can thus be written as

$$ e_{\text{i}} = \left( {b - \omega _{\text{c}} - p - u} \right) \cdot \frac{{\tau _{\text{b}} - \tau _{\text{e}} }}{{1 + \tau _{\text{b}} s}} $$

(5)

Note that the stead state gain is not unity, but slightly lower, to achieve an inverse model of the eye plant together with the direct pathway. The motor command m is composed of the weighted direct pathway carrying b, ω _c, and p, and the integrator output:

$$ m = \left( {b - \omega _{\text{c}} - p + c_{{\text{ft}}} } \right) \cdot \tau _{\text{e}} + e_{\text{i}} $$

(6)

with c _ft being a constant bias term at the level of the FTN compensating for the resting discharge of the PC discharge p (see Eqns. 8 and 9 below). Note that the utricular signal is supposed to pass exclusively through the integrator (Eq. 5), as suggested by studies on static ocular counterroll (Glasauer et al. 2001, Crawford et al. 2003).

The floccular PC population is supposed to receive an efference copy of the motor command m processed by an internal model of the eye plant (Eq. 1) to yield an internal estimate of eye velocity v _e (Glasauer 2003):

$$ v_{\text{e}} = \frac{s}{{1 + \tau _{\text{e}} s}} \cdot m $$

(7)

Additionally the PCs receive a copy of the canal signal, a copy of the burst command b, and visual input v from MST via the pontine nuclei. The output of the PCs is thus given by

$$ p = F\left( {g \cdot (v_{\text{e}} + \omega _{\text{c}} - v - b) \cdot \frac{1}{{1 + \tau _{{\text{pc}}} s}}} \right) $$

(8)

with g being a gain parameter (see below), τ _pc = 0.01 s the time constant of the PCs, and F(x) being the non-linear activation function which determines the resting discharge rate of the PCs and limits their discharge to positive values. F(x) was modeled as log-sigmoid function (e.g., Dayan and Abbott 2001):

$$ F(x) = \frac{{g_{{\text{pc}}} }}{{1 + e^{ - c \cdot x} }} $$

(9)

with the saturation parameter g _pc = 1 and the factor c = 4 yielding a resting discharge c _ft = 0.5 and unity slope at zero input (Fig. 3a). In the range of normal gaze holding and smooth pursuit velocity, this non-linearity can safely be ignored, since the PC input remains in the linear range. For very high target velocity, pursuit would break down due to the non-linearity.

The visual input driving the smooth pursuit response is thought to originate in middle superior temporal area (MST) as internal estimate of target velocity in space. Therefore, it consists of retinal slip, the internal estimate of eye velocity, and the canal output as internal estimate of head angular velocity multiplied by a gain factor g _v (see below):

$$ \begin{gathered} v = 0\quad {\text{if}}\,{\text{target}}\,{\text{is}}\,{\text{invisible }}({\text{light}}\,{\text{off}}) \hfill \\ v = g_{\text{v}} \cdot \left( {r_{\text{s}} + \left( {v_{\text{e}} + \omega _{\text{c}} } \right) \cdot e^{ - \Delta t \cdot s} } \right)\quad {\text{if}}\,{\text{target}}\,{\text{is}}\,{\text{visible }}({\text{light}}\,{\text{on}}) \hfill \\ \end{gathered} $$

(10)

Note that estimated gaze velocity in space (eye-in-head plus head-in-space) has to be delayed by the same amount of time Δt as the retinal slip (Eq. 4) to allow for reconstruction of target velocity. If light is off, e.g., during VOR in darkness, this signal is set to zero, i.e., desired gaze velocity in space then becomes zero.

Finally, the saccadic burst b is computed from the retinal error in a simplistic fashion via an internal feedback loop so that the saccade simply has constant velocity and reaches the target. Note that the exact implementation of the burst generator is not of importance for the overall performance of the model. Therefore, it is described separately below.

The ten equations above comprise the complete model for slow eye movements or saccades, if the burst command is regarded as additional model input. The equations can be reduced to a single overall transfer function for each input. Ignoring the time delay for smooth pursuit and the small time constant of the PCs, this transfer function is a leaky integrator with a time constant $\tau = \tau _{\text{b}} \cdot \left( {1 + g} \right)$ (see also Eq. 12 below).

Free parameters of the model are the utricular gain factor g _u , the PC gain g, the visual pathway gain g _v, the PC saturation term g _pc, and the bias term c _ft. All of these parameters can be determined on the basis of the desired ocular motor responses. As mentioned above, g _pc and c _ft are closely related, since c _ft compensated for the PC resting discharge, which is determined by g _pc (see below). PC saturation has to be large enough to keep the PC response from reaching zero within the range of smooth pursuit eye velocity, and was set to g _pc = 1, which results in c _ft = 0.5. The gain factor g _u has been chosen as g _u  = 1/τ _b, which results in a slow ocular drift towards an eye position $e = \sin \left( \alpha \right).$ That is, around the upright position, the static vertical eye position would perfectly compensate for head tilt. Accordingly, the model predicts positional nystagmus as found in normal subjects (see Fig. 7). However, whether the assumed gain is valid would necessitate a more detailed analysis of positional drifts. The PC input gain has been chosen to g = 10 to achieve an overall time constant τ for gaze holding of about 50 s. The visual pathway gain g _v is set to g _v = (1 + g)/g to result in an appropriate smooth pursuit response, i.e., eye velocity equals target velocity except for the delay and the effect of the leaky integration.

Damage to the PC population

In the patient simulations (Figs. 6, 7), the parameter g _pc of the activation function of the PC population (Eq. 9) was changed to g _pc = 0.6 to mimic, e.g., a loss of Purkinje cells such as in cerebellar atrophy. No other parameters were changed. The scaling of the non-linearity resulted in a lower resting discharge of the PC population and a lower sensitivity (see Fig. 3a). Simulations in Fig. 8a were done by applying the same change in g _pc to second PC population. For the simulation in Fig. 8b, we set g _pc = 0 to abolish floccular output completely, and set c _ft = 0.25 to avoid having nystagmus SPV of more than 25° per s (cf. Fig. 3b). Note that changes in c _ft do not always imply that FTN resting rate changed: due to the feedback loop, the same effect can be achieved by a horizontal shift of the activation function (see Eq. 11 below).

Due to the feedback loop, the fixed point of the PC population for gaze straight ahead is not equal to its resting discharge. The resting discharge, i.e., the discharge for zero input is 0.5·g _pc. The discharge of the PC population for gaze straight ahead is determined by:

$$ \frac{{g_{pc} }}{{1 + e^{ - c \cdot x} }} = c_{{\text{ft}}} - \frac{{\tau _{\text{b}} }}{{g \cdot \left( {\tau _{\text{b}} - \tau _{\text{e}} } \right)}} \cdot x $$

(11)

with x being the PC input. x/g is, in this case, equivalent to the slow phase eye velocity of DBN (see Fig. 3b). Equation 11 has no closed-form solution for x, but for small g _pc its solution x converges to $x = \left( {c_{{\text{ft}}} - g_{{\text{pc}}} } \right) \cdot g \cdot \left( {\tau _{\text{b}} - \tau _{\text{e}} } \right)/\tau _{\text{b}} .$ Thus, the fixed point is shifted away from the point of symmetry (x = 0) of the non-linearity towards saturation. Consequently, additional pursuit input will cause an asymmetric output due to the saturation.

The negative velocity feedback loop

The negative velocity feedback loop constituting the central element responsible for gaze holding and integrator function (ignoring the PC time constant and linearizing the non-linearity) basically consists of the following three equations (here for a velocity command ω):

$$ m = \left( {\omega - p} \right) \cdot \tau _{\text{e}} + e_{\text{i}} ;\quad e_{\text{i}} = \left( {\omega - p} \right) \cdot \frac{{\tau _{\text{b}} - \tau _{\text{e}} }}{{1 + \tau _{\text{b}} s}};\quad p = g \cdot \left( {m \cdot \frac{s}{{1 + \tau _{\text{e}} s}} - \omega } \right) $$

which can be reduced to a standard negative feedback system of the form

$$ m = F_{\text{f}} \cdot \left( {\omega - F_{\text{r}} \cdot m} \right) $$

where F _r is the feedback transfer function (basically the internal model estimating eye velocity) and F _f the feedforward controller (the brainstem integrator and direct pathway generating the motor command). Its solution is

$$ \frac{m}{\omega } = \frac{{F_{\text{f}} }}{{1 + F_{\text{f}} \cdot F_{\text{r}} }} $$

For stability, the real part of every pole of the right-hand-side transfer function must be negative. For our special case

$$ \frac{m}{\omega } = \frac{{\left( {1 + g} \right)\tau _{\text{b}} \left( {1 + \tau _{\text{e}} s} \right)}}{{1 + \left( {1 + g} \right)\tau _{\text{b}} s}} $$

(12)

and, therefore, for g > −1 the system is stable.

So far, we have assumed that the internal eye plant model and the feed-forward inverse model are both perfectly tuned. It is therefore worthwhile to consider a more general case. Let the forward transfer function be F _f = A/B and the feedback function F _r = gs/C with A, B, and C being first-order polynomials of s, i.e., A, B, and C are of the form $1 + \tau _{\text{i}} s$ with positive τ _i and g (additional gain factors can be ignored without loss of generality). The feedback function is thus a high-pass filter. Then the overall transfer function becomes $F = \frac{{BC}}{{BC + Ags}}.$ By solving the denominator for its roots, one can show that the two resulting roots always have negative real parts. Thus, the negative velocity feedback loop is stable independently of the time constants of the inverse or forward models or the brainstem integrator. Note that overall instable integrator behavior (centrifugal drift) as seen in some DBN patients can be generated by the model, if the inverse model is no longer a highpass filter, but contains a negative position component (effectively leading to a positive position feedback).

Saccadic burst generator

The motor error necessary to trigger a saccade is derived by first reconstructing target position in head from retinal error r _e (subject to a delay Δt = 100 ms, see above) and delayed eye position, and then subtracting actual eye position from this estimate of target position in head:

$$ m_{\text{e}} = r_{\text{e}} \left( t \right) + e\left( {t - \Delta t} \right) - e\left( t \right) $$

If the motor error was lower than a threshold, the burst generator was silent:

$$ b = 0\quad {\text{for }}\,\left| {m_{\text{e}} } \right| < 2^\circ $$

If the motor error exceeded this threshold, a constant saccadic burst in the appropriate direction was issued which continued until the motor error reached zero:

$$ b = 0.1 \cdot \text{sgn} \left( {m_{\text{e}} } \right)\quad {\text{for }}\left| {m_{\text{e}} } \right| > 0 $$

where sgn() is the sign function (sgn(x) = 0 for x = 0; sgn(x) = 1 for x > 0; sgn(x) = −1 for x < 0). Accordingly, the burst terminates if eye position equals target position in head.