The Cerebellum as a Time-delayed Adaptive Filter

There has been a lot of work on trying to figure out what the cerebellum actually does. Much research has focused on the well-conserved neural circuitry of the cerebellum. There have been many proposed functions that this circuit can perform, including the Marr–Albus–Fujita model, simply known as an adaptive filter model. An adaptive filter is a filter that can modify its internal parameters using error signals. Likewise, modern studies have slowly validated the adaptive filter model, showing that climbing fibers induce long term depression LTD and long term potentiation LTP at the parallel fiber synapse in the absence of an error signal from the climbing fiber. This provides a succinct way to modify the adaptive filter’s weights and parameters, and allows the cerebellum to perform its job. 

Theoretically, this allows the cerebellum to control and compensate for external perturbations the body experiences during movement. Specifically, the cerebellum is believed to be critical for refining motor output to produce movements that are both precise in trajectory and smooth in execution. However, the cerebellum receives its inputs about 100 ms to 150 ms delayed. Moreover, its control signal takes another 100 ms to propagate to the target muscles. To deal with this time delay, the cerebellum has to modify its synaptic weights to compensate. A recent study by Dr. Ignacio Abadía, Dr. Francisco Naveros, Dr. Eduardo Ros, Dr. Richard R. Carrillo, and Dr. Niceto R. Luque built a cerebellar-based spiking neural network that can control a robotic arm while a variable Wi-Fi delay is introduced between the robot arm and the SNN controller. What’s more, the cerebellar architecture outperformed a PID controller and an ANN controller. Perhaps most strikingly, the cerebellar CNN could compensate for variations in the Wifi time delay lag, really demonstrating how powerful the cerebellum is as a time delayed controller.

 However, I think what is missing in the discussion of time delays and adaptive filters is how a simple PID (Proportional–integral–derivative) controller can compensate for time delays. Understanding how difficult the problem of designing a PID controller with delay is, and how well it can perform at controlling time-delayed systems, can help us understand how an adaptive filter like the cerebellum can perform a control task more robustly. 

 Let’s get our feet wet with a very simple example of a time-delayed control problem. Let’s imagine a system $$ \frac{dx}{dt} = -a\,x(t) + b\,\sin(\omega t) + u(t), $$ where \(x\) is the variable we want to control, \(b\sin(\omega t)\) is a perturbation we wish to reject, and \(u\) is the control signal. In this example we want \(x\) to be zero, but it is being forced away from zero by the \(b\sin(\omega t)\) term. If we examine the uncontrolled system $$ \frac{dx}{dt} = -a\,x(t) + b\,\sin(\omega t), $$ we can calculate its gain via its transfer function: $$ s\,X(s) + a\,X(s) = b, $$ $$ H(s) = \frac{1}{s + a}, $$ $$ G(\omega) = \bigl|H(j\omega)\bigr| = \frac{1}{\sqrt{\omega^2 + a^2}}. $$ Now we construct a controller to ensure that the closed-loop gain \(G(\omega)\) is below some desired threshold. We can use a proportional–integral–derivative (PID) controller: $$ \frac{dx}{dt} = -a\,x(t) + b\,\sin(\omega t) + K_{p}\bigl(0 - x(t)\bigr) + K_{i}\int_{0}^{t}\bigl(0 - x(\sigma)\bigr)\,d\sigma + K_{d}\,\frac{d}{dt}\bigl(0 - x(t)\bigr). $$ Again, we compute the transfer function. Collecting like terms gives $$ (1+K_{d})\,\dot x + (a+K_{p})\,x + K_{i}\int_{0}^{t}x(\sigma)\,d\sigma = b\,\sin(\omega t). $$ Taking Laplace transforms, with \(X(s)=\mathcal{L}\{x(t)\}\) and \(\mathcal{L}\{b\sin(\omega t)\}=\tfrac{b\,\omega}{s^2+\omega^2}\): $$ (1+K_{d})\,s\,X(s) + (a+K_{p})\,X(s) + \frac{K_{i}}{s}\,X(s) = \frac{b\,\omega}{s^2+\omega^2}. $$ Factor out \(X(s)\): $$ X(s)\Bigl[(1+K_{d})s + (a+K_{p}) + \tfrac{K_{i}}{s}\Bigr] = \frac{b\,\omega}{s^2+\omega^2}. $$ Multiply numerator and denominator of the bracket by \(s\): $$ X(s)\,\frac{(1+K_{d})s^2 + (a+K_{p})s + K_{i}}{s} = \frac{b\,\omega}{s^2+\omega^2}. $$ Hence the disturbance-to-output transfer function is $$ H_{d}(s) = \frac{X(s)}{\tfrac{b\,\omega}{s^2+\omega^2}} = \frac{s}{(1+K_{d})s^2 + (a+K_{p})s + K_{i}}. $$ Evaluating at \(s = j\omega\): $$ H_{d}(j\omega) = \frac{j\omega} {-(1+K_{d})\,\omega^2 + j\,(a+K_{p})\,\omega + K_{i}}, $$ and the magnitude (gain) is $$ G(\omega) = \bigl|H_{d}(j\omega)\bigr| = \frac{\omega} {\sqrt{\bigl(K_{i} - (1+K_{d})\,\omega^2\bigr)^2 + \bigl((a+K_{p})\,\omega\bigr)^2}}. $$ Since this is a single-input single-output system, choosing \(K_{p}\), \(K_{i}\), and \(K_{d}\) sufficiently large will keep the gain low. However, a small gain alone does not guarantee stability. For example, consider $$ \frac{dx}{dt} = +a\,x(t), $$ whose gain is $$ G(\omega) = \bigl|\tfrac{1}{j\omega - a}\bigr| = \frac{1}{\sqrt{\omega^2 + a^2}}, $$ yet if \(a>0\), there is a pole at \(s=+a\) in the right half-plane and the system is unstable. To analyze stability of the non-delayed closed-loop system, we examine its characteristic equation—the denominator of \(H_{d}(s)\): $$ (1+K_{d})\,s^2 + (a+K_{p})\,s + K_{i} = 0. $$ To analyze the stability of such systems we can make use of the Routh–Hurwitz criterion. For a second‐order polynomial $$ a_2\,s^2 + a_1\,s + a_0 = 0, $$ stability requires $$ a_2 > 0,\quad a_1 > 0,\quad a_0 > 0,\quad a_1\,a_2 > a_0. $$ Identifying $$ a_2 = 1 + K_d,\quad a_1 = a + K_p,\quad a_0 = K_i, $$ the closed‐loop stability conditions become $$ \begin{aligned} 1 + K_{d} &> 0,\\ a + K_{p} &> 0,\\ K_{i} &> 0,\\ (a + K_{p})(1 + K_{d}) &> K_{i}. \end{aligned} $$ Hence, selecting \(K_{p}\), \(K_{i}\), and \(K_{d}\) to satisfy these inequalities ensures both a low disturbance gain and a stable closed loop. However, what happens when our PID controller has a time delay? We can consider the system described above: $$ u(t) = -K_{p}\,x(t-\tau) -K_{i}\int_{0}^{\,t-\tau}x(\sigma)\,d\sigma -K_{d}\,\frac{d}{dt}x(t-\tau), $$ $$ \frac{dx}{dt} + a\,x(t) = b\sin(\omega t) -K_{p}\,x(t-\tau) -K_{i}\int_{0}^{\,t-\tau}x(\sigma)\,d\sigma -K_{d}\,\frac{d}{dt}x(t-\tau). $$ When computing the transfer function, we must include an $e^{-s\tau}$ term. This yields: $$ G(s) =\frac{X(s)}{D(s)} =\frac{\displaystyle \frac{1}{s+a}} {\,1 + \frac{1}{s+a}\,\bigl(K_{p} + \tfrac{K_{i}}{s} + K_{d}\,s\bigr)\,e^{-s\tau}} =\frac{1}{(s+a)+\bigl(K_{p} + \tfrac{K_{i}}{s} + K_{d}\,s\bigr)\,e^{-s\tau}}. $$ Evaluated on the imaginary axis: $$ G(j\omega) =\frac{1}{\,a + j\omega +\bigl(K_{p} + \tfrac{K_{i}}{j\omega} + K_{d}\,j\omega\bigr)\,e^{-j\omega\tau}}. $$ Its magnitude (gain) is $$ \bigl|G(j\omega)\bigr| =\frac{1} {\sqrt{ \bigl[a + K_{p}\cos(\omega\tau) +\bigl(\omega K_{d}-\tfrac{K_{i}}{\omega}\bigr)\sin(\omega\tau)\bigr]^{2} + \bigl[\omega +\bigl(\omega K_{d}-\tfrac{K_{i}}{\omega}\bigr)\cos(\omega\tau) -K_{p}\sin(\omega\tau)\bigr]^{2} }}. $$ Although the gain is straightforward to compute, assessing stability is much more challenging. We must solve the transcendental equation $$ s + \bigl(K_{p} + \tfrac{K_{i}}{s} + K_{d}\,s\bigr)\,e^{-s\tau} = 0. $$ To simplify, set $K_i=0$ and $k_d =0$: $$ s + (K_{p})\,e^{-s\tau} = 0, $$ has solutions given by $$ s = \frac{1}{\tau}\,W\!\bigl(K_{p}\,\tau\bigr), $$ where $W(z)$ is the multivalued Lambert~$W$ function defined by $z\,e^{z} = s$, implying infinitely many branches. This is analogous to $\ln(z)$ having infinitely many roots as $\ln(z) = s \pm 2\pi n i$ for any integer $n$. You can see why this is such a hard problem as we massively simplified the problem but still need to consider infinite poles from a obscure function. Critically, if $K_i$ or $K_d$ are not zero, the poles have no closed‐form solutions. One must rely on advanced techniques (e.g. Nyquist, D-subdivision, ect.) to determine whether any roots lie in the right half of the complex plane. 

 Thus, to ensure stability of a PID filter in a time-delay system—like a human with nerve conduction delay—ensuring stability and optimal control is a very difficult problem. However, that is where the beauty of adaptive filters comes in. Because they have an algorithmic way to reduce error between the input and the output, by adjusting the weights we can find stable solutions to the control problem without needing to analytically find stable solutions. Moreover, a lot of work has gone into mapping cerebellar LTP and LTD to the adaptive filter model.

 This raises several extremely interesting points, some philosophical, some practical. First, just as the PID controller has no model of the system that it is controlling, nor an explicit mention of what the exact time delay of the system is. It is just three constants, controlling the strength of the error, derivative of error, and integral of error. This, coupled with a learning algorithm to adjust the values of the three constants, is sufficient to act as a controller for a linear time-delayed system. A few things to stress here: having three simple constants precludes having a complex model of the system that you are controlling. It can "predict" the timing of the crests and troughs of a wave. Admittedly, this is really just a phase delay of almost a full cycle (or several cycles) that makes it appear to be a prediction of the wave the controller is trying to cancel. 

 What makes the cerebellum more powerful than a simple PID controller is its access to billions of neurons (or even the hundreds in the robotic WiFi controller). Instead of 3 parameters, we have billions or perhaps trillions of parameters that can be tuned by LTD and LTP. However, to return to the robotic example—the hundreds of neurons in the cerebellar SNN—there is no need to explicitly specify the model or the time delays. The adaptive filter properties of the cerebellum determine the best way to control the body.

 However, this raises a question about the internal model of the cerebellum. In some sense, it is not needed to perform its function—at least not in a way that is conceptually easy to digest. One could argue that an internal model of the muscle-environmental system is encoded in some inaccessible way into the synaptic weights of the cerebellar network. However, given that the PID system only has three weights, it seems unlikely that an internal model has been learned—only that optimal control weights have been found.

There is one major counterpoint to this. In the PID model, the delay cannot predict the arrival of the signal. When the sinusoidal input turns on, there must be no response until the controller becomes aware of the sinusoidal input (that is, $\tau$ milliseconds have passed). Then the integral term must wind up, and there will be a transient response to the sinusoid until the controller has responded to the situation.

 By contrast, if there is a predictable preceding signal that has been associated with an imminent incoming perturbation to the body, the cerebellum can prepare and actively control the muscles to compensate for that incoming perturbation. There have been many studies that have shown this behavior. While the robotic cerebellum and a PID controller do not show this behavior, it is a major advantage the cerebellum has over a PID controller, adaptive or not. Still, while the cerebellum can predict the future in a sense, it does not require an explicit internal model of the external environment or the body's coupling to that environment. Perhaps one can argue that the model is distributed over the synaptic weights in some abstract sense, but it seems this is by no means a requirement. 

 Admittedly my pessimism on this point of the lack of an internal model is more of my interpretation rather than a hard and fast rule. One could potentially argue, that statistical associations and optimal control weights are an internal model, just an abstract one that humans cannot imagine in our heads. 

 One last thing to touch on is the universal cerebellar transform. This is perhaps the "alternative" hypothesis to what the cerebellum is doing. This hypothesis was developed in response to the discovery that the cerebellum is involved in cognition and mental processes, even has a role in autism. How a time-delayed controller can help in cognition is a non-trivial task, especially in social situations. The universal cerebellar transform essentially states that the repeated, conserved nature of the cerebellar cortex implies that the cerebellum is performing a single universal function. Given its role in cognition and social cues, it is hypothesized that there is a representation of the external world stored in the cerebellum, and that the cerebellum uses this model to predict and then correct its actions based on this model. Therefore, cognitive and social models can be stored in the cerebellum.

 This is speculative, but as we saw above, even with a few parameters, a controller can be highly effective. Therefore, perhaps the cerebellum does not need an internal representation to enhance task performance—only to be tuned to the right values. Take, for example, the mental object rotation task, where one rotates a 3D object in one's mind. To summarize: humans with cerebellar damage cannot rotate the object quickly and take longer to rotate the object in their mind's eye. The cerebellum may have a role to play, not because it has any concept of the 3D object, but because the control signal is broken, and the cortex overshoots and takes a while to correct the error "manually" with cortical processing alone. This is, of course, just speculation on my part. If ChatGPT has shown us anything, it's that when neurons and parameters are scaled up to millions or billions, understanding seems to just fall out of the system. Whether or not this is truly a mental model may just be a meaningless semantic distinction at that point. 

 Still, it remains an open question how an adaptive filter can play a role in social cognition and autism. If the universal cerebellar transform hypothesis is true, and if the cerebellum is an adaptive filter, some creative way to apply adaptive filters to complex and abstract social and linguistic processing is going to need to be done. How that problem is solved is by no means obvious today in 2025. We have only been able to chip away at it in terms of mental models of more physical manipulations of 3D objects in the minds eye, which of course is a far cry from a true universal cerebellar transform. 

 Philosophy aside, the cerebellum offers a powerful neural architecture for control. Whether or not there is an explicit model is almost beside the point. Detailed control from the cerebellum can compensate for external perturbations and can take into account uncertainty in time delays. My hope is that by understanding the PID controller, one can get a sense of the difficulty of time-delayed controllers and the absolute magic that the cerebellum is able to compensate for uncertainty in time delay—and the extreme uncertainty of the real world. Mapping this to non-motor control tasks is the next step in understanding the cerebellum.

Author: Alexander White