CONTENTS

14.1 Introduction

14.2 Problem Formulation

14.3 Materials and Methods

14.4 Results

14.4 Discussion

14.5 Conclusions

References

14.1 Introduction

In order to understand and describe how the brain organizes limb movement control, we aim to design a mathematical model of this control The study is based on a comprehensive neurophysiological characterization of the cerebellar system for voluntary arm-hand control as described in Apps and Garwicz (2005) This system may be viewed as a vast network of interconnected neurons that involves many parts of the brain, which all are interconnected through a specific area of the cerebellum (Bengtsson and Jörntell 2009; Jörntell and Ekerot 1999). A foundation for the model system is a previous, detailed characterization of all constituent neuron types and a systematic description of connectivity patterns both within the cerebellum and in those brain regions outside the cerebellum, which are part of the network devoted to this specific control as shown in previous publications from our group (Bengtsson and Jörntell 2009; Ekerot and Jörntell 2001, 2003; Jörntell and Ekerot 2002, 2006; Jörntell and Hansel 2006)

As for detailed electrophysiological neuron modeling, the Goldman-Hodgkin-Katz voltage equation (or the Goldman equation) is the standard model used in cell membrane physiology to determine the equilibrium potential across a cell membrane, taking into account all of the relevant ion species active through that membrane (Junge 1981)

Brain function may be viewed as a result of the transformation functions of individual neurons and their precise interconnections However, the network of neurons that constitute the brain is very well organized into discrete subcomponents Each subcomponent is connected to a limited set of other subcomponents in specific, well-conserved connectivity patterns. Viewed in this way, it is possible to make a control system-inspired interpretation of the function of the brain in movement control (Fujita 1982; Ito 1972; Kawato and Gomi 1992; Miall and Wolpert 1996; Schweighofer, Arbib, and Kawato 1998; Schweighofer et al. 1998; Wolpert, Miall, and Kawato 1998) and we can interpret the neuronal system for arm-hand movement control as being organized into a number of distinct functional units, in a similar fashion as a control system Each functional unit, or subcomponent of brain circuitry, hence has a specific function, which can be expressed in mathematical terms. In the case of the brain, this function is carried out by a limited set of neuron types, which typically can have relatively simple internal circuitry connectivity

Conversely, a mathematical problem formulation of optimal control problem and related adaptive control with tentative solutions was published in Johansson (1990a, 1990b)

In the present study, we aim to provide a mathematical description of the function of one of these functional subunits, namely, the cuneate nucleus, which carries out the first-order processing of movement-generated sensory feedback. We illustrate the firing patterns of the primary afferents, main cuneate neurons, and local inhibitory interneurons and how system identification methods can be used to obtain a mathematical expression of the function carried out by the cuneate nucleus In ongoing work, we also simulate in detail how the underlying neuronal information processing is carried out, with the aim of providing neuroscientific correlates for specific features in these mathematical expressions.

14.2 Problem Formulation

Because our approach rests on the understanding of the biological system, that is, the brain, much work needed to be devoted the collection of the biological data The data collection should be from primary afferents; that is, nerve fibers that mediate sensory input from peripheral receptors to the central nervous system, from the main neurons of the cuneate, which project the processed sensory information to the cerebellum (Bengtsson and Jörntell 2009) Ideally, we should record from the primary afferent and its target cuneate neuron simultaneously, but this is technically very difficult. An approximation is to record from primary afferents and cuneate neurons driven by the same inputs To compare the primary afferent data with the data from the cuneate neurons, comparisons should only be made between primary afferents and neurons that are activated by the same modality of sensory information and from sensory input activated from the same topo-logical area (on the skin or in the joints/muscles). If we can fulfill these criteria, we can take advantage of previous findings suggesting that single primary afferents can have a dominant influence on the cuneate neuron (Ferrington, Rowe, and Tarvin 1987) The transformation taking place between the primary afferent input and the cuneate neuron output for a given stimulus can then be characterized through system identification.

In order to verify that the mathematical model obtained accurately represents the transformation between the primary afferent and the cuneate neu-ron in the more general case, data from both the primary afferent and cuneate neuron should also be obtained from a very different type of stimulus If the mathematical model is correct, it should also be able to reproduce the input-output transformation in this case

14.3 Materials and Methods

The experiments were carried out with single-unit metal microelectrodes and patch clamp recordings in the acute animal preparation of the decer-ebrated cat as described in Bengtsson and Jörntell (2009) and Jörntell and Ekerot (2003) Primary afferent axons were recorded on their pathway into the cuneate nucleus and cuneate neurons were recorded inside the cuneate nucleus. Stimuli were delivered in two different ways. The first, which was used to produce the model through the system identification, was a standardized manual skin stimulation A miniature strain-gauge device was mounted on the tip of the investigator in order to control that the same amount of force and the same stimulation time were used A second mode of stimulus was electrical skin stimulation applied through a pair of needle electrodes inserted into the skin with a spacing of 3 mm Stimulation intensity was 1 0 mA with a duration 0 1 ms

As for empirical model estimation, standard methods and validation methods of system identification were used (Johansson and Magnusson 1991).

Then, the recorded data were processed using an action potential pattern recognition software, in order to reduce noise For every action potential, only the time after stimulation was determined, transforming data to spike-time data. This in turn could be added over several stimulations, yielding histogram data This was then exported to MATLAB (The Mathworks, Inc, Natick, MA, USA) For the mathematical modeling of the neuron transmission, the MATLAB System Identification Toolbox was used (Ljung 2002). Various different model structures were tested—e.g., prediction error estimate (PEM), ARX, ARMAX, OE, BJ, N4SID (Ljung 2002)—to determine the model that provided the most accurate representation The starting point was to find the model that most accurately represented the transformation for a standardized manual skin stimulation Subsequently, we simulated the primary afferent input evoked by electrical skin stimulation and compared the simulated cuneate neuron response with actual responses recorded from the cuneate neuron with the same electrical skin stimulation

14.4 Results

Figure 14.1 illustrates the spike responses of a primary afferent and a cuneate neuron to manual skin stimulation The data are displayed as peristimulus histograms; that is, the same stimulation was repeated many times and the spike counts for each bin represent the sum of spike responses for thirty to fifty consecutive, nearly identical stimuli. All stimuli were aligned so that they started at 0 ms and ended at 50 ms. The primary afferent, which conveys the input to the cuneate neuron via a synapse (Figure 14.1a), displays spike only on stimulation. Therefore, the time before the stimulation is devoid of spikes, even though the spike response to the stimulation is quite intense (Figure 14.1b, top). In contrast, the cuneate neuron has a spontaneous activity, and hence the spike activity before the stimulation, and in addition it seems to have a more intense spike response to stimulation (note the steep increase in spike activity immediately after the onset of the stimulation; Figure 14.1b, bottom)

In the next step, we simulated that the primary afferent input was driven by electrical skin stimulation This type of stimulation evokes a precisely timed primary afferent spike, about 4.5 ms after the stimulation, with a near 100% fidelity and a coefficient of variation of 0% (at 0.1 ms resolution; data not shown). Figure 14.2 illustrates the actual cuneate neuron response when this primary afferent spike input was evoked experimentally

Figure 14.2 illustrates the actual cuneate neuron response when this input was evoked experimentally (upper diagram) and the experimental response (lower diagram) evoked by a single pulse input comparable to the spike response of the real primary afferents (Figure 14.1b).

Figure 14.3 exhibits the spike response recorded from a cuneate neuron subject to electrical skin stimulation.

The traces illustrated in Figure 14.4 show the simulated response of the mathematical model with the best fit. Interestingly, this model captured the fast variations in the initial part of the response quite well but failed to capture the subsequent inhibitory response that followed the initial excitation. It also failed to capture the release from the inhibitory response (at 200 ms+). These two late phases of the response are likely generated by the local inhibitory interneurons of the cuneate nucleus

With the necessary data at hand, we next attempted to use system identification to find an appropriate mathematical model representing the transfor-mation of the information conveyed by the primary afferent to the cuneate neuron. We found that a PEM, discrete-time state-space model of fifth order gave the best fit, in the form of a state-space model:

xk+1=Axk+Buk+Kwk

yk=Cxk+wk

The fit for the PEM, ARX, ARMAX, OE, and BJ models were 66, 61, 66, 64, and 64%, respectively.

The estimated state-space model is as follows:

xk+1=Axk+Buk+Kwk

yk=Cxk+wk

with the system matrices

A = [−0..0744−0.0207−00002000781..0281 −0.0056−0.0528−0.19590.91670.1103

B={0.0027−0.0163−0.0004−0.00950.0018},

0.8474 0.3213

−0⋅0113−0.03290.1257 −0.3575−0.0376−0.0179 −−0000005700289401507723360] −0.4983 0.7843

Kr={0.00260.00220.00330.0070−0.0039}w

A = [−0..0744−0.0207−00002000781..0281 −0.0056−0.0528−0.19590.91670.1103

B={0.0027−0.0163−0.0004−0.00950.0018},

0.8474 0.3213

−0⋅0113−0.03290.1257 −0.3575−0.0376−0.0179 −−0000005700289401507723360] −0.4983 0.7843

Kr={0.00260.00220.00330.0070−0.0039}w

C = [20.5466 −9.6495 4.4225 1.8089 5.8488 ]

Figure 14.5 shows the impulse response for the PEM in the form of a state-space model of the cuneate spike response. In this case, the fit between the experimental data and the prediction accuracy of the simulation appears to be very high, although the temporal relationships seem to need some further refinements of the model. In the recorded data (see Figure 14.3) the spontaneous firing activity of the cuneate neuron provides a background against which inhibitory effects in the neuron can also be recorded (as a temporary depression of the spontaneous firing activity). This occurs immediately after the initial excitatory response. In the model, inhibition is represented by negative output, something that is not possible in the brain for neurons that make excitatory synapses (which is the case for the cuneate neurons that make excitatory synapses with neurons of the cerebellum and thalamus). However, spontaneous background activity also participates in determining the spike output of the downstream neurons (i e, in the cerebellum and thalamus), so the reduction of this background activity can actually be viewed as being equivalent to negative output. Therefore, we can conclude that the mathematical model reflects the response properties of the primary afferent-to-cuneate neuron junction well. Because it is able to do so for two widely different inputs (manual skin stimulation that evokes a primary afferent spike response lasting 50 ms and electrical skin stimulation that evokes a spike response lasting for 0.1 ms), it promises to be able to account for the response properties over a wide range of inputs and perhaps covering the majority of the physiological range of inputs

As an illustration of model variability, Figure 14.6 shows the impulse response corresponding to electrical skin stimulation for the ARX model. It is evident that these models do not capture the cuneate neuron firing mechanics in response to electrical stimulation as the PEM model did. These empirical mathematical models are of the form

• ARX:

A(z)yk=B(z)uk+wk

• OE:

yk=B(z)F(z)uk+wk

Figure 14.7 shows the Bode diagram of the PEM model. Interestingly, the diagram shows deriving properties for a large part of the spectrum with phase lead These are favorable properties from a control systems point of view This indicates that these properties could be important features of the cuneate neuron in its role of transforming sensory information from primary afferent neurons to the cerebellum, which in turn can be viewed as a controller of body movements

Finally, a Ho-Kalman impulse response model reproducing cuneate histogram spike-response data was made with high modeling accuracy (Figure 14.8). Note that a long-term response was reproduced by the model.

14.4 Discussion

Based on the findings of Figures 14.5, 14.7, and 14.9, the empirical cuneate model exhibits differentiating characteristics, suggesting a phase lead action of the cuneate response. When a longer impulse response is considered (see Figure 14.9), the differentiating aspects are less pronounced.

14.5 Conclusions

Our preliminary findings suggest that system identification can be used to identify the mathematical properties of a local neural structure with an uncomplicated network structure. The failure to reproduce the later phases of the response to manual skin stimulation (see Figure 14.2) can be explained by the fact that this response occurred well after the cessation of the primary afferent spike train. In the brain, this type of response is likely created by the local inhibitory interneurons, which is an input that was not known to the model Future modeling experiments will show whether the addition of this factor can make the model reproduce this aspect of the response

Based on these findings and the previous experimental results of Ekerot and Jörntell (2001, 2003), Bengtsson and Jörntell (2009), Jörntell and Ekerot (1999, 2002, 2003, 2006), and Jörntell and Hansel (2006), further modeling of neural structures—inside and outside the cerebellum, in order to understand control system aspects—could potentially give new insights into cerebellar movement control. This aspect—that is, how the cerebellum achieves movement control—could be of great interest not only in the field of neuroscience but in robotics and other control applications as well

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