Chapter 2: Neurobiological Underpinnings:

In this chapter, we will sketch in a simple way the basic circuitry of the Purkinje cell of the cerebellum. We encourage you to consult a variety of sources for additional information. A wonderful place to start is the aptly named Human Brain Coloring Book (Diamond, 16) which although not completely current has many detailed drawings of the circuitry we wish to discuss at a level that is appropriate to our needs --- that is, not overly technical, yet also not too simplistic. Our discussions are of course derivative of the early Albus papers on the cerebellum (Albus, 1, 4). These papers led Albus to propose a simple model of the Purkinje cell whose implementation in software is known as the Cerebellar Model Articulated Controller or CMAC architecture which was developed by Albus in a series of fundamental application papers [Albus, 2, 3]. In the discussions that follow, you will see, the details of the CMAC structure follow directly from the basic wiring diagram of the Purkinje cell. After you are finished with this material, we invite you to explore further in this area by looking at selected research papers that try to model various aspects of the cerebellum (and other pertinent things) with a variety of tools. Clearly, the CMAC architecture is a rather simple model of how a real Purkinje cell processes information. The following papers suggest alternative modeling approaches to a number of areas of interest and therefore suggest different varieties of CMAC structures.

Chapeau--Blondeau et. al., 14:: ``A Neural Network Model of the Cerebellar Cortex Performing Dynamic Associations''; this paper discusses another type of model of the Purkinje cell circuitry.
Shadlen et. al., 51:: ``Noise, Neural Codes and Cortical Organization''; this paper discusses the neural codes that may subserve information transfer in cortical circuitry. There is much food for thought here on how we should represent information in our CMAC model.
Konishi, 24:: ``Similar Algorithms in Different Sensory Systems and Animals''; this paper discusses certain commonalities in how different systems solve sensory problems. What can we abstract from this?
Berthier et. al., 5:: ``A Cortico--Cerebellar Model That Learns to Generate Distributed Motor Commands to Control a Kinematic Arm''; this paper presents motor control models based on feedback loops and clusters of Purkinje Cells.
Burgess et. al., 11:: ``Hippocampus--Spatial Models''; this paper discusses neuronal models of spatial processing.
Burgess et. al., 12:: ``A Model of Hippocampal Function''; this paper discusses some aspects of hippocampal navigation modeling using radial basis functions.
O'Reilly et. al., 42:: ``Hippocampal Conjunctive Encoding, Storage and Recall: Avoiding a Tradeoff''; this paper propses that there are unique anatomical and physiological properties of the hippocampus that serve to minimize the tradeoff between separating patterns and reconstrucing patterns from partial and/ or noisy pattern pieces.
McClelland et. al., 28:: ``Why are there Complementary Learning Systems in the Hippocampus and Neocortex: Insights from the Successes and Failures of Connectionists Models of Learning and Memory''; this paper discusses how we might model the fact that memory traces over time become less dependent on the hippocampal system.
Bullock et. al., 10:: ``A Neural Model of Timed Response Learning In the Cerebellum''; this paper discusses how to build a cerebellum model that can learn timed responses.
Thach et. al., 52:: ``The Cerebellum and the Adaptive Coordination of Movement''; this is a review paper on the neurophysiology of the cerebellum.
De Schutter et. al., 15:: ``An Active Membrane Model of the Cerebellar Purkinje Cell I. Simulation of Current Clamps in Slice''; A careful Genesis model (Genesis is a very good tool for modeling real physiology) of a certain aspect of the Purkinje cell. Can we incorporate any of this insight into the CMAC models?
Bower et. al., 7:: ``Variability in Tactile Projection Patterns to Cerebellar Folia Crus IIA of the Norway Rat''; this paper suggests that the physiology that the CMAC architecture is based on is flawed.
Jager et. al., 23:: ``Prolonged Responses in Cerebellar Purkinje Cells Following Activation of the Granule Cell Layer: An Intracellular in vitro and in vivo Investigation''; this paper also suggests that the physiology on which the CMAC architecture is based is flawed.

2.1 The Purkinje Cell Wiring Diagram:

Let's begin by looking at a drawing that presents a simplified view of the wiring diagram of a typical Purkinje cell. The cerebellum for our purposes consists of three carefully defined layers with incredible geometric regularity. In Figure 2.1, this is represented by sharply defined volume elements drawn as boxes. The bottom of the figure represents matter deepest in the brain and the top of the figure is material closest to the inner portion of the skull. The deepest layer is called the Granule Layer and it is densely packed with vast numbers of Granule Cells and Golgi Cells. The middle layer holds the cell bodies of the specialized neurons called Purkinje Cells which are orders of magnitude fewer in number. Finally, the outermost layer is populated with the dendritic arbors of the Purkinje and the Golgi Cells.

The Purkinje cell has a very unusual physical shape. It consists of a single body called the soma from which a single axonal output moves down the innermost layer into the interior of the brain. The dendritic arbor that collects sensory stimulation for the Purkinje cell is two dimensionsal in appearance as is suggested in the illustration. For all practical purposes, it looks like a common fan that is oriented in the molecular layer as shown. This two--dimensional dendritic arbor receives input in two ways:

The Climbing Fiber Pathway:: The axons of these fibers arise from the inferior olive (a massive nerve fiber bundle) and interdigitate (that is, intertwine) directly with the Purkinje cell dendritic arbor. These inputs are excitatory. The climbing fibers also send excitatory input into a information pathway known as the intracerebellar and vestibular nuclei (again, a very massive nerve fiber bundle). This is where the axon of the Purkinje cell also lands. The Purkinje cell output is inhibitory. So, the ultimate ouput from the intracerebellar and vestibular nuclei is due to the combining of these separate outputs (as well as others).
The Mossy Fiber Pathway:: The axons of the mossy fibers emanate from the spinal cord and brain stem as well as other pathways. These outputs are also excitatory. The manner in which the mossy fibers interact with the Purkinje cells is very interesting and it is this physiology that is coarsely modeled with the CMAC architecture.

First, the mossy fibers also contribute excitatory output to the intracerebellar and vestibular nuclei. The mossy fiber, climbing fiber and Purkinje outputs all are combined in this nerve bundle to generate an output pulse.

Each mossy fiber will interact with about one hundred or so Granule cell dendritic arbors in a special bundle of connections called a Glomerus. The axonal output of the granule cell is sent straight up the layers as shown in the figure until it reaches the outermost layer, the molecular layer. There the axonal fiber branches and runs in both directions perpendicular to the Purkinje cell dendritic arbor as shown. This axonal output is also excitatory. The portion of the axonal output from the granule cell that runs in the molecular layer is called a parallel fiber.

Now look at the illustration again. In the molecular layer, there are also the large three--dimensional dendritic arbors of the Golgi Cells. The parallel fiber of a given granule cell will connect with some dendrite. The inputs into the Golgi cell are processed in the Golgi soma that lies in the granule layer, and the Golgi output is fed back into the already mentioned glomerus that connects the mossy fiber axons to the granule cell dendrites. Hence, the parallel fiber to Golgi cell circuit provides feedback of the transformed mossy fiber input signal to the granule cell.

On the other hand, the parallel fiber also connects with a dendrite of the Purkinje cell arbor. This provides excitatory input into the Purkinje cell.

Two additional cells also connect to the Purkinje cell arbor: the stellate and the basket cells. These cells accept parallel fiber input (excitatory) and output an inhibitory pulse.

Figure 2.1:

We can redraw the Purkinje cell wiring diagram in abstract form by concentrating on the lines of information transfer. This is shown in Figure 2.2. In Figure 2.3, we have shown a similar abstraction of the biology for more than one Purkinje cells.

Figure 2.2:

Figure 2.3:

2.2 The Purkinje Cell Abstraction:

We can abstract out of this wealth of neurobiological detail a number of interesting observations:

One PF excites a longitudinal array of about 300 Purkinje neurons. Roughly, the PF is 3000 microns (µ) long and the PF intersects a Purkinje fan every 10 µ or so. This implies each PF crosses about 300 Purkinje cells. Moreover, each PF connects to 1-2 dendrites per Purkinje neuron.
Each Purkinje cell receives excitatory input from 200,000 PF and one CF. However, only about 1% of the dnendrites in the Purkinje fan are activated at any one time.
Each Purkinje cell receives inhibitory input from basket, stellate, and Golgi cells. The Golgi inhibitory input is from the feedback loop previously discussed which cuts off the axonal output from the granule cell after a brief delay. Since mossy fiber input is sent up through these granule cell connections, this feedback loop essentially cuts off the mossy fiber input channeled through this granule cell. From the details of the neurobiology of the Purkinje cell, we can thus abstract an important feedback loop, as evidenced by the illustration Figure 2.4.

Figure 2.4:

Now, the neurobiological circuitry of the Purkinje cell and its interactions with stellate and basket cells as well as parallel fibers has a remarkable three--dimensional character. We have drawn this abstractly in Figure 2.5. We can further simplify this sketch and arrvie at a very simple three dimensional representation of the circuit as shown in Figure 2.6.

Figure 2.5:

Figure 2.6:

Further, the dendritic arbors of each golgi body are three--dimensional in nature; they roughly form a truncated hyperbolic paraboloid with the golgi body in the center throat of the surface. The radius at the top and bottom of the paraboloid is very large compared to the thickness and width of each Purkinje cell dendritic tree. This influences greatly the manner in which the Purkinje cell network functions. In Figures 2.7 and 2.8, we try to indicate the relevant size and geometry of this cells. Note each Purkinje cell dendritic fan is about 250 µ wide and 6 µ thick. Each golgi cell can receive stimulation from about 100,000 granule cells. The output of the golgi cell then inhibits perhaps two to three individual granule cells and each granule cell is inhibited by at least on golgi body.

Figure 2.7:

Figure 2.8:

Finally, the parallel fibers activated by a given mossy fiber input via the granule cell pathway form a tube or beam of closely aligned fibers that intersect the two--dimensional dendritic arbor of many Purkinje cells. We can illustrate this process with a very high level diagram as seen in Figure 2.9.

Figure 2.9:

We are now ready to abstract out of this neurobiological detail some principles of software design for an artificial Purkinje neuron. Each granule cell accepts information from mossy fiber input. This information is then output via the parallel fiber pathways. Each MF passes information to about 100 to 600 PFs; hence, there is at least a 1 to 100 ``expansion'' of information from the MF to the PF network. The granule layer thus functions as a N ® 100N recoder, where N denotes the number of MF input lines. We also know that there are about nine Purkinje cells. The signal carried on a given PF will intersect about 300 Purkinje cells. The value of this PF/ Purkinje dendritic fiber interaction can be thought of as a variable weight. Now there are about 10⁷ Purkinje cells in the Purkinje layer and about 10¹¹ granule cells in the granule layer. This implies that there are about 10,000 granule cells per Purkinje body.

The number of dendrites in a Purkinje cell dendritic arbor is on the order of 200,000. A single MF input will be split into say 100 PF lines and each of these PF fibers will interact with about two Purkinje dendrites. Hence, the ratio of active PF - Purkinje intersections to total dendritic fibers is

200

200,000

= .001

or about one percent.

There is also a massive amount of inhibitory feedback through the golgi body feedback loop. Excitatory input via a MF is transformed via feedback into inhibitory input to granule cells with time lag. So golgi cells tend to kill excitatory inputs off and hence shut down granule cells and their associated PF output lines. From this discussion, we are led to the concept of a 1 ® 100 input expansion recoder. In general, such a device would have the following properties: (see Figure 2.10).

N inputs (mossy fibers)
100N associated cells (granule cell/ mossy fiber interactions resulting in a PF output)
100N weights (PF - Purkinje dendrite intersections)
Approximately 1% of association cells fire per MF input. (Only 1% of the dendrites in the Purkinje arbor have an interaction with a given MF input via the PF pathways)

Figure 2.10:

We can modify the general abstract expansion recoder as presented in Figure 2.10 by showing more explicitly some of the neurobiological substrate information. In this figure, we explicitly show that all parallel fiber pathways interact with the Purkinje cell dendritic arbor either directly or indirectly via the stellate and basket cells. Since the output of the stellate and basket cells is inhibitory and the output of the parallel fibers is excitatory, the net effect at the Purkinje cell dendritic arbor can be excitatory or inhibitory. If we loosely think of inhibitory as a negative scalar and excitatory as a positive scalar, we see the value associated with a given dendrite of the Purkinje cell can be any real number.

Figure 2.11:

To summarize, mossy fibers play the role of the input lines in the recoder, the granule cells are the association cells, the lines connecting the association cells to the weights are the parallel fibers and the weights themselves are the values of the PF/ Purkinje dendrite interactions. The response cell is the Purkinje soma with the response cell output given by the value on the Purkinje cell axon. In the next chapter, we will begin to show how this notion of the N ® 100N input recoder is implemented mathematically. The basic idea is to find a way for a single input to be routed to multiple output lines having a unique address.